Transportation and Traffic Theory: Flow, Dynamics and Human Interaction (Proceedings of the 16th International Symposium on Transportation and Traffic Theory)

TRANSPORTATION AND TRAFFIC THEORY

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TRANSPORTATION AND TRAFFIC THEORY FLOW, DYNAMICS AND HUMAN INTERACTION Proceedings of the 16th International Symposium on Transportation and Traffic Theory University of Maryland, College Park, Maryland, 19-21 July 2005

edited by

HANI S. MAHMASSANI University of Maryland, USA

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© 2005 Elsevier Ltd. All rights reserved. This work is protected under copyright by Elsevier Ltd., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London W1P 0LP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

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Contents Preface

ix

International Advisory Committee

xi

Contributors

xii

Bilevel optimisation of prices in a variety of transportation models

1

MJ Smith A sequential experimental approach for analyzing second-best road pricing with unknown demand functions H Yang, WXu and Q Meng

23

Highway space inventory control system D Teodorovic and P Edara

43

Path size and overlap in multi-modal transport networks SHoogendoorn-Lanser, R van Nes andP HL Bovy

63

Pedestrian dynamics and evacuation: empirical results and design solutions D Helbing, A Johanssen and L Buzna

85

An empircal assessment of traffic operations C Chen, P Varaiya and J Kwon

105

Reliability of freeway traffic flow: a stochastic concept of capacity WBrilon , J Geistefeldt and M Regler

125

A critical comparison of the kinematic-wave model with observational data K Nagel and P Nelson

145

Average velocity of waves propagating through congested freeway traffic B Coifman and Y Wang

165

Microscopic three-phase traffic theory and its applications for freeway traffic control B Kerner

181

A behavioural approach to instability, stop and go waves, wide jams and capacity drop C Tampere, S Hoogendoorn and B van Arem

205

Controlling traffic breakdowns R Ktihne and R Mahnke

229

vi Contents Parameter estimation and analysis of car-following models S Hoogendoorn and S Ossen

245

Modeling Impatience of drivers in passing maneuvers M Pollatschek and A Polus

267

A simulation model for motorway merging behaviour J Wang, R Liu and F Montgomery

281

Freeway ramp merging process observed in congested traffic: lag vehicle acceleration model M Sarvi, A Ceder and M Kuwahara

303

A first-order macroscopic traffic flow model for mixed traffic including moving bottleneck effects S Chanut

323

A variational formulation of kinematic waves: bottleneck properties and examples C Daganzo and M Menendez

345

First-order macroscopic traffic flow models: intersection modeling, network modeling J P Lebacque and M Khoshyaran

365

Real-time estimation of travel times on signalized arterials A Skabardonis and N Geroliminis

387

Calibration and validation of dynamic traffic assignment systems R Balakrishna, H Koutsopoulos and M Ben-Akiva

407

Non-equilibrium dynamic traffic assignment W Y Szeto and H K Lo

427

Precision of predicted travel time, the responses of travellers, and satisfaction in the travel experience SKikuchi, S Mangalpally and A Gupta

447

Behavioral dynamics in activity participation, travel, and information and communications technology K Goulias and T G Kim

467

Modeling the joint labor-commute engagement decisions of San Francisco Bay Area residents

487

DTOry and P L Mokhtarian

Contents vii A model of daily time use allocation using fractional logit methodology X Ye and R Pendyala

507

Efficient estimation of nested logit models using choice-based samples L A Garrow, F S Koppelman and B L Nelson

525

Functional approximations to alternative-specific constants in time-period choice-modelling S Hess, J W Polak and M Bierlaire

545

Project-based activity scheduling for person agents

565

EJ Miller Modelling commercial vehicle empty trips: theory and application JHolguin-Veras, JCZorrilla andE Thorson

585

Rationality and heterogeneity in taxi driver decisions: application of a stochastic-process model of taxi behavior R Kitamura and T Yoshii

609

A rolling-horizon approach to the optimal dispatching of taxis

629

M GHBell, KI Wong and A J Nicholson Capacitated arc routing problem with extensions H Qiao and A Haghani

649

User-equilibrium route set analysis of a large road network H Bar-Gera and D Boyce

673

Comparison of static maximum likelihood origin-destination formulations H Rakha, H Paramahamsan and M van Aerde

693

A time-dependent activity and travel choice model with multiple parking options 717

HJHuang, Z CLi, WHKLam andS C Wong Doubly dynamic equilibrium distribution approximation model for dynamic traffic assignment N C Balijepalli and D P Watling

741

A combined model of housing location and traffic equilibrium problems in a continuous transportation system H W Ho and S C Wong

761

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Preface For many researchers and scholars in the inter-disciplinary field of traffic science and transportation system modelling, the book publication that contains the Proceedings of the International Symposium on Transportation and Traffic Theory (ISTTT) series provides a much-awaited field-defining milestone, featuring some of the best and most promising thinking and theoretical developments in the field. The ISTTT series is the main gathering for the world's transportation and traffic theorists, and those who are interested in contributing to or gaining a deep understanding of traffic and transportation phenomena in order to develop effective approaches to plan, design and manage these systems. The Symposium deals primarily with the scientific and fundamental aspects of transportation and traffic phenomena, including identification and characterization of previously undocumented or underappreciated phenomena, formulation, observation and testing of new models of and new perspectives on known phenomena, as well as extension, refinement, testing or critique of existing theories and models of known phenomena. This volume contains contributions selected for the 16th International Symposium on Transportation and Traffic Theory (ISTTT), held at the University of Maryland in College Park, Maryland in the USA on July 19 - 21, 2005. The 39 contributions selected for this volume do not represent all the papers accepted for inclusion in the ISTTT program. In view of the high level of interest and very high quality of submissions from all over the world, and the limited number that could be included in this volume, several additional papers were accepted for following a very demanding and rigorous refereeing process. These will be published through regular Journal channels. All papers in this volume passed a three-stage review process. In a first stage, nearly 200 submitted extended abstracts were reviewed by up to four members of the scientific and organizing committees, resulting in invitations to 88 authors to submit full papers. The second stage entailed the review of each of the 84 submitted full papers by two to five internationally recognized referees. Based on that process, 45 revised papers were submitted for final evaluation and review; 39 of these are included in the present volume, each reflecting the high quality, excellence, rigour and innovation that are the defining qualities of this series. The ISTTT 16 comes at a time when the field is experiencing a major renewal, reflected by the entry of a large number of new scientists from a variety of disciplines, and the mutual coexistence of a growing number of theoretical perspectives and modelling cultures. It is also a time when theories and models developed over the past 40 years are finding their way into professional practice through computer software packages to a greater extent than at any time in the past. The present volume reflects some of the most vibrant work taking place in the world's leading institutes on the science of traffic and transportation, and presents a view of the breadth of topics and richness of influences that are permeating the field.

x Preface The contributions included in this volume reflect a range of topics that pertain to modelling traffic and transportation processes as complex systems. Several themes are reflected in this collection, though perhaps none as strongly and visibly as the overarching desire to understand and characterise the fundamental processes that govern the behaviour and performance of our transportation systems. At the root of these characterizations is the recognition that traffic and transportation systems are a collective expression of individual human decisions. Modelling traffic and transportation systems is about modelling human interaction, amongst individuals as well as with the surrounding transportation infrastructure, controls and environment. The number of contributions in this volume that deal with some aspect of human behaviour, whether as travellers, drivers, passengers, operators, or regulators, is relatively greater than in past Symposia. This is reflective of great strides being made in our community of scholars in developing theories and mathematical representations of these phenomena, and in the combination of theoretical development with observation through novel measurement techniques. The underlying themes and approaches traditionally used in different functional areas, such as planning or operations, or transport modes, such as freight or passenger modes, appear to be converging towards greater focus on individual decisionmaking and human interaction in the context of flow processes, network dynamics and spatial systems. Special appreciation is extended to the members of the International Advisory Committee and the local organizing committee in the review and selection process of the papers included in this volume. Particular gratitude is owed to the referees who contributed considerable time and effort to the review process. Only through their dedicated and thoughtful effort can the ISTTT maintain its high standards of intellectual excellence. I am grateful to Ms. Elisabeth Fernandez-Kimmel for her most able assistance and tremendous effort in the final preparation of the camera-ready document. Hani S. Mahmassani March 2005

International Advisory Committee C F Daganzo

University of California, Berkeley, USA (Convenor)

R E Allsop

University College, London, UK

M G H Bell

Imperial College, London, UK

P H L Bovy

Delft University of Technology, The Netherlands

W Brilon

Ruhr-University, Bochum, Germany

A Ceder

Technion-Israel Institute of Technology, Haifa, Israel

N H Gartner

University of Massachusetts, Lowell, USA

R Kitamura

Kyoto University, Japan

R Kiihne

German Aerospace Center, Berlin, Germany

M Kuwahara

University of Tokyo, Japan

WLam

Hong Kong Polytechnic University, China

J-B Lesort

Institut National de Recherche sur les Transports et leur Securite, Lyon, France

H.S. Mahmassani

University of Maryland, College Park, USA

V V Silyanov

Moscow State Automobile and Road Technical University, Russia

M A P Taylor

University of South Australia, Adelaide, Australia

M Tracz

Cracow University of Technology, Poland

S C Wirasinghe

University of Calgary, Canada

Honorary members E Hauer

Canada

H Keller

Germany

M Koshi

Japan

W Leutzbach

Germany

H-G Retzko

Germany

D I Robertson

United Kingdom

S Yagar

Canada

Contributors R Balakrishna N C Balijepalli H Bar-Gera M G H Bell M Ben-Akiva M Bierlalre P H L Bovy D Boyce W Brilon L Buzna A Ceder S Chanut CChen B Coifman

C Daganzo P Edara

L A Garrow J Geistefeldt N Geroliminis K Goulias A Gupta A Haghani D Helbing SHess HWHo

Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Institute for Transport Studies, University of Leeds, UK Ben-Gurion University of the Negev, Be'er Sheva, Israel Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Institute of Mathematics, Ecole Polytechnique Federate de Lausanne, Switzerland Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Northwestern University Evanston, Illinois, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Department of Transportation Networks, University of Zilina, Slovakia Transportation Research Institute, Technion-Israel Institute of Technology, Haifa, Israel Laboratoire Ingenierie Circulation Transport LICIT (INRETS - ENTPE), Cedex, France University of California, Berkeley, USA Department of Civil and Environmental Engineering and Geodetic Science, Ohio State University, Columbus, Ohio, USA Institute of Transportation Studies, University of California, Berkeley, USA Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Falls Church, Virginia, USA Georgia Institute of Technology, Atlanta, Georgia, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Institute of Transportation Studies, University of California, Berkeley, USA University of California, Santa Barbara, USA University of Delaware, Newark, Delaware, USA Department of Civil and Environmental Engineering, University of Maryland, College Park, Maryland, USA Institute for Economics and Traffic, Dresden University of Technology, Dresden, Germany Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, The University of Hong Kong, China

Contributors

HWHo J Holguin-Veras S Hoogendoorn S Hoogendoorn-Lancer H J Huang A Johanssen B Kerner M Khoshyaran S Kikuchi TGKim R Kitamura

F S Koppelman H Koutsopoulos R Kiihne M Kuwahara JKwon W H K Lam J P Lebacque ZCLi RLiu HKLo R Mahnke S Mangalpally M Menendez Q Meng E J Miller P L Mokhtarian

xiii

Department of Civil Engineering, The University of Hong Kong, China Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands School of Management, Beijing University of Aeronautics and Astronautics, China Department of Physical Resource Theory, Chalmers University of Technology, Sweden Daimler Chrysler AG, Stuttgart, Germany ETC Economics Traffic Clinic, France University of Delaware, Newark, Delaware, USA The Pennsylvania State University, University Park, Pennsylvania, USA Department of Urban Management, Kyoto University, Kyoto, Japan, and Department of Civil and Environmental Engineering, University of California, Davis, USA Northwestern University Evanston, Illinois, USA Department of Civil Engineering, Northeastern University, Boston, Massachusetts, USA Institute of Transportation Research, German Aerospace Center, Berlin, Germany Institute of Industrial Science, University of Tokyo, Japan Statistics Department, California State University, Hayward, California, USA Department of Civil and Structural Engineering, Hong Kong Polytechnic University, China INRETS-GRETIA2, France School of Management, Beijing University of Aeronautics and Astronautics, China Institute for Transport Studies, University of Leeds, UK Department of Civil Engineering, Hong Kong University of Science and Technology, Hong Kong, China Uiversity of Rostock, Institute of Physics, Rostock, Germany University of Delaware, Newark, Delaware, USA Institute of Transportation Studies, University of California, Berkeley, USA Department of Civil Engineering, The National University of Singapore, Singapore Department of Civil Engineering, University of Toronto, Toronto, Canada University of California at Davis, Davis, USA

xiv Contributors F Montgomery K Nagel B L Nelson P Nelson A J Nicholson DTOry S Ossen H Paramahamsan R Pendyala J W Polak M Pollatschek A Polus HQiao HRakha M Regler M Sarvi A Skabardonis M J Smith W Y Szeto C Tampere D Teodorovic

E Thorson M van Aerde B van Arem R van Nes P Varaiya

Institute for Transport Studies, University of Leeds, UK Institute for Land and Sea Transport Systems, Technical University of Berlin, Berlin, Germany Northwestern University Evanston, Illinois, USA Dept of Computer Science, Texas A&M University, College Station, Texas, USA Deparment of Civil Engineering, University of Canterbury, New Zealand University of California at Davis, Davis, USA Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands Charles Via Jr. Dept. of Civil and Environmental Engineering, Virginia Tech, Blacksburg, Virginia, USA Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA Centre for Transport Studies, Imperial College London, UK Department of Industrial Engineering and Management, Technion-Israel Institute of Technology, Haifa, Israel Department of Civil and environmental Engineering, Technion-Israel Institute of Technology, Haifa, Israel The Robert H. Smith School of Business, University of Maryland, College Park, Maryland, USA Charles Via Jr. Dept. of Civil and Environmental Engineering, Virginia Tech, Blacksburg, Virginia, USA Institute for Transportation and Traffic Engineering, RuhrUniversity, Bochum, Germany Department of Civil Engineering, Monash University, Victoria, Australia Institute of Transportation Studies, University of California, Berkeley, USA Department of Mathematics, University of York, UK Department of Civil, Structural, and Environmental Engineering, Trinity College, University of Dublin, Ireland Department of Traffic and Infrastructure, Katholieke Universiteit, Leuven, Belgium Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Falls Church, Virginia, USA Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA Posthumously TNO Inro, Department of Traffic and Transport and University of Twente Faculty of Civil Engineering and Geosciences, Transport & Planning, Delft University of Technology, The Netherlands University of California, Berkeley, USA

Contributors xv J Wang Y Wang D P Watling K I Wong S C Wong W Xu H Yang X Ye T Yoshii J C Zorrilla

Institute for Transport Studies, University of Leeds, UK Department of Electrical and Computer Engineering, Ohio State University, Columbus, Ohio, USA Institute for Transport Studies, University of Leeds, UK Centre for Transport Studies, Imperial College London, UK Department of Civil Engineering, The University of Hong Kong, China Department of Civil Engineering, The Hong Kong University of Science & Technology, Hong Kong, China Department of Civil Engineering, The Hong Kong University of Science & Technology, Hong Kong, China Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA Department of Urban Management, Kyoto University, Kyoto, Japan Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York, USA

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1

BILEVEL OPTIMISATION OF PRICES IN A VARIETY OF TRANSPORTATION MODELS MJ Smith, Department of Mathematics, University of York, United Kingdom

INTRODUCTION The primary equilibrium model in this paper is similar to that suggested by Aashtiani and Magnanti (1983). This model combines the standard "user equilibrium" route-choice principle stated by Wardrop (1952) and a demand for travel between each OD pair which may vary with the costs of travel between the various OD pairs. There are many other equilibrium models. See for example: Beckmann (1956), Evans (1976), Gartner (1980), Bar Gera (2002) and Bar Gera and Boyce (2003). Optimising prices and signal controls within equilibrium models The need to optimise controls within equilibrium models is clear; as Beckmann often pointed out, control changes have a knock-on effect as users play their own games while the controller is playing his. The basic optimisation idea in this paper is intended to be a really quite simple approach to the problem; building directly upon the "half-space projection" direction developed in a series of papers: Smith et al (1997, 1998a, b), Clegg et al (2001) and Clegg and Smith (1998, 2001). We extend this previous work and also show how these methods apply to a range of different models. Fletcher and Leyffer (2000) and Rodrigues and Monteiro (2004) both consider descent methods which have some similarity with the simultaneous descent method outlined here. Using the simultaneous descent algorithm here the approach to a stationary point is typically via points which are not themselves approximate equilibria, so that the optimisation and the equilibration move in parallel and the need to compute a sequence of approximate equilibria is avoided.

2 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Abdulaal and Leblanc (1979), Tan, Gershwin and Athans (1979), Marcotte (1983, 1986), Fisk (1984), Tobin and Friesz (1988), Davis (1994), Yang and Yagar (1994), Yang (1996a, b, c), Chiou (1997), Clune et al (1999), Patriksson and Rockafellar (2002) and Clark and Watling (2002) have all sought to solve bilevel problems. Migdalas (1995) provides an excellent survey of the area. Bilevel problems in a wider context have been studied by for example Gauvin and Savard (1994), Luo et al (1996) and Outrata and Zowe (1998). The need to optimise signal controls subject to equilibrium within an equilibrium transportation model was pointed out in an early paper by Allsop (1974). In this paper we begin by considering optimisation within our primary model and then show that the same ideas work in several other models. Finally we give a general model embracing all of them. THE BASIC EQUILIBRIUM MODEL, MODEL 0 The model, model 0, presented here is the base or initial model considered in this paper. The main variables In model 0 we suppose given a network with K OD pairs, that OD pair ij is joined by Ny routes and so the total number of routes is N = SijNij. The main variables are as follows: Xjjr = the flow along the r* route joining OD pair ij (in vehicles per minute, say); X = the route flow vector comprising all the Xyr; Yy = the cost of travel between OD pair ij (in minutes per vehicle, say); and Y = the cost vector comprising all the Yy. Flows are in (flow per unit time) units and costs are in (time per unit of flow) units; so that a flow times a cost is dimensionless. Let 0N denote the zero N-vector, +coN denote the N-vector with infinite co-ordinates. Then we put [0N, +ooN) = [0, +oo)N and [0K, +ooK) = [0, +oo)K. Cost functions and demand functions We suppose given two functions; the cost function C(.) and the demand function D(.). Here Cjjr(X) is the cost of traversing the rth route joining OD pair ij when the flow vector is X > 0 and Dy(Y) is the total flow between OD pair ij when the OD cost vector is Y where Y > 0. We suppose that the cost function C(.) is defined throughout [0N, +ooN) and that the demand function D(.) is defined throughout [0K, +ooK). Thus including domains and co-domains, our two given functions are: C: [0N, +ooN) -^ [0, +ooN) and D: [0K, +ooK) [0K, +ooK). The functions T and S We define the two functions T:[0N, +<x>N) * [0K, +ooK) and S:[0K, +°oK) -> [0N, +°oN) as follows. For each ij and each ijr: Tij(X) = ZrXijr for all X e [0N, +ooN) and S1]r(Y) = Yy for all Y e [0K, +ooK). Tjj(X) is the total flow from i to j and Syr(Y) spreads each cost Yy over all routes joining i to j .

Bilevel optimisation of prices in a variety of transportation models 3 Assumptions: positivity, non-negativity, boundedness, monotonicity, smoothness We suppose that: C(X) > 0 for all X e [0N, +ooN) and D(Y) > 0 for all Ye [0K, +ooK). D is bounded and C is bounded on bounded sets. C and -D are both monotone. C and D are differentiable and their derivatives are continuous. C is monotone means that: [C(X') - C(X2)]-(X'- X2) > 0 for all X1 e [0N, +ooN) and X2 e [0N, +ooN). It is easy to show that if C and - D are monotone then (C(X) - S(Y), T(X) - D(Y)) is also a monotone function of (X, Y) e [0N, +ooN) x [0K, +ooK).

EQUILIBRIUM CONDITIONS AND SEARCH DIRECTION Suppose given a (flow-vector, cost-vector) pair (X, Y) in which all Xyr > 0 and all Yjj > 0. Then (X, Y) will be called an equilibrium if (X, Y) s [0N, +ooN) x [0K, +ooK), for each ijr the cost Cijr(X) of traversing the rth joining OD pair ij equals Yjj, and for each ij the demand Djj(Y) generated by the cost vector Y equals the total flow Tjj(X) = ZrXjjr actually occurring from i to j . So in this case (X, Y) is a variable demand equilibrium pair if and only if (X, Y) e [0N, +coN) x[0K,+ooK), Yjj - Cijr(X) = 0 for all ijr; and Dij(Y) - Tjj(X) = 0 for all ij. (1) The equilibrium equations (1) may be written in vector form: (S(Y) - C(X), D(Y) - T(X)) = 0. Here we are assuming initially that at equilibrium all Xjjr > 0 and all Yjj > 0. To relax this initial assumption and so to allow the possibility that a listed route may have a high cost and zero flow at equilibrium we revise (1) to introduce a partial complementarity condition. The revised equilibrium condition is as follows. For each ijr: Yy - Cijr(X) < 0 and Yy - Cijr(X) < 0 implies Xijr = 0; and Dij(Y)-Tij(X) = 0. (2) The set of equilibria, or solutions to (2), will be denoted by E. Throughout this paper we assume that this set is non-empty. The feasible set F and the objective function V Suppose given upper bounds ViUXy,- and !4UYy for Xyr and Yjj at equilibria; so that all equilibria lie inside ViF and so well inside F where F = [0N, UX] x [0K, UY] = n ijr [0, UXij,] x riijtO, UYy] c [0N, +roN) x [0, +ooK). This ensures that the set of equilibria E is well inside F. We do not here consider the problem of proving existence of a non-empty set of equilibria, and nor do we consider the specification of the set F. Some results concerning the existence of UXjjr, UY;J and F are given in Smith (2005). Given the set F, consider the objective function V: F -> R+ where, for all (X, Y) e F:

4 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Ljr{Xijr2[Cijr(X) - Y y ] + 2 + (UXijr- XijO^Yij - Cijr(X)]+2} +Lj{Yij2[Tij(X) - Dij(Y)]+2+ (UYrYij)2[Dij(Y) - T,j(X)]+2}. Here y+ = max{y, 0} for all real numbers y. It is easy to show that, provided the set of all equilibria is a subset of !4F then the set of zeros of V in F is exactly the set of equilibria.

V(X, Y) =

To seek a zero of V and hence an equilibrium state, consider the following search direction A = (A1, A2) where the ijrth component of the flow search direction A'(X, Y) is for each ijr: A'i]r(X, Y) = (UXijr- Xijr)2[Y;j - Cijr(X)]+ - Xijr2[Cijr(X) - Y S ] + ; and the ijth component of the cost search direction A2(X, Y) is for each ij: A2ij(X,Y) = (UYi]-Yij)2[Dij(Y)-Tij(X)]+-Yij2[Tij(X)-Dij(Y)]+. The whole search direction A(X, Y) = (A'(X, Y), A2(X, Y)) for all (X, Y) e F. This is precisely the algorithm (D) direction, introduced in Smith (1984a, b), in this setting; applied with the function - (C(X) - S(Y), T(X) - D(Y)) on the set F = [0N, UX] x [0K, UY]. We are given that C and -D are both monotone and it follows that (C(X) - S(Y), T(X) - D(Y)) is a monotone function of (X, Y) on F. This is the fact which is critical to what follows. Since (C(X) - S(Y), T(X) - D(Y)) is a monotone continuously differentiable function of (X, Y) on F, it follows from Smith (1984a, b) that A(X, Y) is a descent direction for objective V at (X, Y) (unless V(X, Y) = 0 in which case (X, Y) is an equilibrium) and also that there is for each (X, Y) a positive number T\ such that (X, Y) + tA(X, Y) e F if (X, Y) e F and 0 < t < n. The set E of equilibria is under the present conditions also convex, since (C - S, T - D) is monotone. Since V is continuous the set E = {x; V(x) = 0} is also closed. Thus since we are assuming that E is non-empty, E is a closed convex set such that E c [0N, V2UX]x[0K, ViUY].

AN EQUILIBRATION METHOD Motivated by the previous section, we suppose that the following basic condition holds. BASIC CONDITION: There is a fixed set of N routes joining K OD pairs as above, C and -D are monotone and continuously differentiable, F = [0N, UX]x[0K, UY], and the set of equilibria is a non-empty subset of >/2F = [0N, y2UX]x[0K, '/2UY]. Any pair (X, Y) in F will be called feasible. In order to estimate an equilibrium (X, Y), we start anywhere in F and update (X, Y) as follows: (X1, Y1) is any feasible starting value for (X, Y) and following some rule or algorithm: (X1, Y1) ^ (X2, Y2) * (X3, Y3) ^ . For any feasible start point all succeeding pairs are also to be feasible.

Bilevel optimisation of prices in a variety of transportation models 5 Definition of convergence We make the following definition. The above infinite sequence {(Xn, Yn)} of feasible (X, Y) pairs in F converges if and only if (1) the equilibrium set E is nonempty and (2) the Euclidean distance between (X", Y") and the set E of equilibria tends to zero as n tends to infinity. We will agree that an algorithm converges if and only if for any start point (X1, Y1) which generates an infinite sequence then this infinite sequence converges. Steepness of descent Suppose now that our basic condition above holds. Then Smith (1984a, b) shows that (given the basic condition above) there is a continuous real-valued function G such that G(X,Y)>Oif(X,Y)eF\E; G(X, Y) = 0 if (X, Y) e E; and A(X, Y)-gradV(X, Y) < - G(X, Y) for all (X, Y) e F. This means that V is a Lyapunov function for the dynamical system: d((X(t), Y(t))/dt = A((X(t), Y(t)) for all t > 0, (X(0), Y(0)) = (X°, Y°) e F. It also means that the steepness of the descent of the Lyapunov function V at (X, Y), as (X, Y) follows A(X, Y), maybe estimated by G(X, Y). That is: dV(X(t), Y(t))/dt = gradV(X(t), Y(t)) A(X(t), Y(t)) < - G(X(t), Y(t)) for all t > 0. The projection Proj Given F, for each (X, Y) e RN+1C the projection Proj(X, Y) of (X, Y) onto the feasible set F is defined to be the point of F closest to (X, Y).

DYNAMIC ARMIJO-LIKE STEP LENGTHS Given that the basic condition holds we reduce V to zero by moving (X, Y) e F continually in direction A. For step length choice we follow a dynamic Armijo-like scheme very close to that described in Smith (1984b). For now we let z stand for (X, Y) and Zn stand for (Xn, Yn). A (zn, tn_i)-updating dynamic Armijo-like equilibration algorithm Here we suppose that if we are at iteration n; at a non-equilibrium (Xn, Y") = z,, where the search direction is A(Zn) and the step length actually used at Zn is un then our next z will be Zn+i = PrOJ(Zn + UnA(Zn))

where Proj denotes the projection onto F. The real number un > 0 will be called a used step length and tn > 0 will be called a step length. For the purposes of most of this section we will suppose that if u^ is a used step length then Proj(zn + UnA(z^)) = Zn + UnA(Zn).

6 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction That is we suppose that the projection operator does not actually do anything; so the boundary of the feasible set F is here having no effect. To determine un we specify a dynamic Armijo-like scheme based on a continuous function G such as that given above. To motivate the scheme, it is clear that if the step length t at z were very small then the change in V = V(z + tA(z)) - V(z) would be more negative than - 3/4tG(z). So the slope of V(z + tA(z)) against t would be at least as steep as - 3AG(z) for small t. We do not wish to have such small steps t as the reduction in V might be very small, by virtue of the small step size t. On the other hand if t is large the slope of V(z + tA(z)) against t can be no steeper than -%G(z) on average; for if the average gradient of V(z + tA(z)) against t were < - %G(z) for large t then V(z + tA(z)) would sometimes be < V(z) - VitG(z) and this would be negative for large t, which is impossible. (Of course, by its definition, V > 0 always.) We do not wish to have steps this large as the reduction in V might be very small, due to a shallow negative slope (or perhaps V may even increase, due to a positive slope). So we seek step lengths which give rise to slopes between - 3/4G(z) and - lAG{z). Such step lengths have an Armijo property and allow convergence to equilibrium to be shown. The dynamic Armijo (zn, tn_i)-updating equilibration algorithm To be specific, we start at an arbitrary z\ e F and to = 1. This 0th or initial step length to is fairly arbitrary, but it must be positive. If we are at a current non-equilibrium point Z j e F , and the previous possible step length was tn_i, then we are to update Zn and tn-i (for n > 1) according to some fairly simple rules as follows. Firstly, zn is kept fixed and tn.i is halved to obtain: tn_i, (i4)tn.i, (/4)2tn-i, . . . . , (V2)ptn-i, where the halving ceases as soon as: V[z^ + (MOViAfe)] - V(zn) < - (l/8)(I/2)ptn.,G(zn) for the first time, p = 0 is allowed here; it may be that Vfc, + t^Afe)] - V(Zn) < - (l/8)tn.1G(Zn) already. The halving surely ceases by definition of G as Zn is a non-equilibrium. Then: let un = (Vy'tn-i for this p (this is the used step length at zn) and let Zn+i = Zn + UnA(Zn).

Finally update tn_i as follows: if V f e + unA(zn)] - V(zn) < -3/4UnG(Zn) put tn = 2u n ; if -3/4UnG(zn) < V[zn + unA(zn)] - V(Zn) < -1/4UnG(zn), put tn = u n ; and i f - V ^ G f e ) < V[z» + unA(zn)] - V(zn) < - (l/8)u n G(z n ) put tn ='/2un. These are three mutually exclusive possibilities and they together exhaust all eventualities since, by choice of un = ('/iytn-ijVtzn + unA(Zn)] - V(z,,) > - (1/8)^0(2,,) is not possible.

Clearly not all tn are in fact used to move z. The used step lengths are the un. It is clear that V[zn + unA(zn)] - V(zn) < - (l/SKGfe) for all used step lengths un. (Stop when V(zT1) is less than some preassigned positive number.)

Bilevel optimisation of prices in a variety of transportation models 7

CONVERGENCE TO EQUILIBRIUM Suppose that the basic condition holds. For any starting point zi e F and with an initial "previous" step length to = 1 we suppose that the algorithm specified above generates an infinite trajectory: z\, Zi, Z3,.. ., z,,, . . . . and an infinite sequence to, ti, Xi, \-$,. .., tn, . . . . of possible step lengths. We assume that these sequences are infinite: so we never actually hit the equilibrium set E. (If the sequence hits an equilibrium we simply stop.) To prove convergence we use proof by contradiction. Suppose that {z,^ tn-i} is an infinite sequence generated by our algorithm and that {zj does not converge to the equilibrium set E = {z e F; V(z) = 0}. Then, since F is closed and bounded, {Zn} must have a non-equilibrium limit point. Let w e F be such a non-equilibrium limit point of the sequence {zn}. Then we use the following lemma proved in Smith (2005). Lemma. Let the sequence {(z,,, tn_i)} be derived using the algorithm above so Zn+i = z,, + unAn(zn), let w e intF, let w be a non-equilibrium so that V(w) > 0 and suppose that w is a limit point of {z,,}. Then, under our assumptions, there is a suffix i(w) such that V(ZJ(W)) < V(w). This lemma shows (since V(Zn) decreases) that w cannot in fact be a limit point of the sequence {z,,}. It follows that the sequence {zj has no non-equilibrium limit points. Hence, since F is closed and bounded, {zn} converges to the set of equilibria.

OPTIMISING PRICES Now we suppose that there is a specified smooth function Z which we here regard as a measure of total disbenefit. We seek to minimise Z at an equilibrium by charging prices for traversing certain arcs or routes in the network, thus influencing the equilibrium traffic distribution. If Z itself depends on the prices charged then charges will also influence Z directly. We suppose throughout that Z is a continuously differentiable and non-negative function of the route-flow vector X, the OD cost vector Y and the new route-price vector P. Adding a price vector P Suppose that route ijr is (possibly) to be subject to a charge Pjjr and that the vector P of all possible route prices P;jr will be confined to some polyhedral closed bounded set Fcontroi of feasible control vectors. In this case the vector (X, Y, P) will now be called a user-equilibrium if (X, Y) e R+N+K, P e Fcontroi, and for all ijr, Yjj - Cijr(X) - Pijr< 0 and Yy - C1]r(X) - Pijr < 0 implies Xijr = 0, and Dij(Y) - Ty(X) = 0. (5) To optimise, as before with equilibration, we need a bounded set of (X, Y) pairs. So our basic condition will now be enhanced to allow for the set of controls. We now assume that the following control-enhanced condition holds. CONTROL-ENHANCED BASIC CONDITION: There is a fixed set of N routes joining K OD pairs as above,

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C and -D are monotone and continuously differentiable, F = [0 N , UX]x[0 K , UY], and for all fixed P e F control , the set of equilibria is a non-empty subset of [0 N , !/ 2 UX]x[0 K , '/ 2 UY]. The previous V and A will now involve P. Now for all (X, Y, P) e FxF contro i: V(X, Y, P) = L ]r {Xi jr 2 [P, ]r + C ijr (X) - Y S] ] + 2 + (UX1]r- X ^ f Y s - (P1]r + C 1]r (X))] + 2 } + LjJY/tTijCX) - Dij(Y)]+2 + (UY1J-Y1J)2[Dij(Y) - Tij(X)] + 2 }, AV(X, Y , P) = (UX ljr - Xi,-r)2[Yij - CPijr + C i]r (X))] + - X ijr 2 [(P ijr + C ijr (X)) - Yjj]+, A2,j(X, Y, P) = (UYtJ -Ys)2[DgCY) - T i] (X)] + - Yg2[Tij(X) - D y (Y)] + , and

A(X, Y, P) = (A'(X, Y, P), A2(X, Y, P)).

The previous equilibrium constraint (X, Y) e F and V(X, Y) = 0 will now be written (allowing for the possible prices): (X, Y, P) e FxFCOTtrol and V(X, Y, P) = 0. We will also write: H = FxFcontroi- Finally we suppose that all the Nc functions h; are linear and let H = {(X, Y, P); hi(X, Y, P) < 0 for i = 1,2, 3 , . . . , Nc}.

A constraint qualification and new notation Suppose that (X, Y, P) e H is not an equilibrium so that V(X, Y, P) > 0. It follows that for this (X, Y, P), A(X, Y, P) is a feasible descent direction for V with P fixed and hence that: ProjF[-grad(x, Y) V(X, Y, P)] * 0 if V(X, Y, P) > 0. Here only (X, Y) varies in F and P is fixed in Fcontroi. It follows at once that: ProjH[-grad(x, Y, P) V(X, Y, P)] * 0 if V(X, Y, P) > 0. Here (X, Y, P) varies in H. Replacing (X, Y, P) by just x, the equilibrium condition (X, Y, P) e H = FxFcontrol and V(X, Y, P) = 0 becomes: x s H and V(x) = 0. Also (6) above may now be written: ProjH[-grad V(x)] * 0 if V(x) > 0.

(6)

(7)

This may be thought of as a constraint qualification applying (for any positive e) to the set EE={xeH;V(x)<s}. Condition (7) implies that if x is any point in the set Es such that V(x) = s and hj(x) = 0 for i = 1, 2, 3, . . . , k and no others then there is a direction 8 = ProJH[-grad V(x)] such that

Bilevel optimisation of prices in a variety of transportation models 9 S-descV(x) > 0 and 5-deschj(x) > 0 for i = 1, 2, 3, . . . , k. That is: there is a feasible descent direction for V on the edge of any Ee provided s > 0. Now the proposed optimisation algorithm, where (X1, Y1, P1) is any starting value for (X, Y, P)inHand(X 1 , Y1, P1) ^ (X2, Y2, P2) ^ (X3, Y3, P3) ^ , becomes: x1 is any starting 1 2 3 . value for x in H and x -^ x -> x A simple simultaneous descent direction (for x in the interior of H) We suppose that we wish to minimise Z(X, Y, P) = Z(x) subject to V(X, Y, P) = V(x) < s so that we need to optimise V and Z simultaneously in some way. In this paper we let x lie in the interior of H. We also need our control-enhanced basic condition above. We make the further initial basic assumption that gradZ(x) ^ 0 for all x e H. Now, under our control-enhanced basic conditions, V(x) > 0 implies gradV(x) ^ 0. So we may now let (for x e H and V(x) > 0): descV(x) = - grad V(x)/||grad V(x)||, descZ(x) = - gradZ(x)/||gradZ(x)|| and desc(Z, V)(x) = /2descZ(x) +'/2descV(x). (8) This direction (8), if non-zero, reduces V and Z simultaneously. In any case this direction is never an ascent direction, for either V or Z. However descV is not defined where V = 0 and changes sharply in the vicinity of a point at which V = 0. So it is natural to change this direction (8) slightly so that it is defined everywhere and is smoothly varying. This may be done by enlarging the equilibrium set E. We are led to put (where s > 0): Es = {x e H: 0 < V(x) < s} and for all x e H: A£(x) = [ 1 -V(x)/e]+descZ(x) + [V(x)/s]desc(Z, V)(x) + [V(x)/e-1 ]+descV(x). (9) Our general approach here is also similar that adopted in Clegg and Smith (2001). The present direction (9) with the present definition of the simultaneous descent direction desc(Z, V)(x) has been introduced so as to allow an effective step length selection procedure to be more easily designed. [Estimation of Ae(x) is made more simple by supposing that the cost and demand functions have symmetrical Jacobians.] For any x in the interior of H the zeros of As(x) coincide exactly with points x in EE at which Z is stationary; or those points in E£ for which there is no descent direction for Z which remains inside Ee. We will now show this. Optimality conditions for x in the interior of H

Definitions ofs-feasible descent and s-linear optimality (for x in the interior ofH) The vector u is an s-feasible Z-descent direction at x (in the interior of H) if and only if

10 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction u-[descZ(x)] > 0 and either V(x) < s or {V(x) = s and u-[descV(x)] > 0}. x* (in the interior of H) is s-linearly-optimal if and only if V(x*) < s and there is no s-feasible Z-descent direction at x*. s-linear-optimality conditions (in the interior ofH) In this section we give optimality conditions for any x in the interior of H. These results are particularly useful if H has been chosen correctly; so that at any e-linearly-optimal x = (X, Y, P) the upper and lower bounds on x = (X, Y, P) are not binding. For such x, in the interior of H, we let A6(x) be given by (9) above and show how this vector may be used to classify points x according to whether they are s-linearly-optimal or not. Observe that desc(Z, V)(x) = !/->descV(x) + V2descZ(x) (if non-zero) is a direction in which both V and Z decline; and is never a direction of increase for either V or Z. So if xeintH and !4descV(x) + V2descZ(x) is non-zero then x is not E-linearly-optimal and also Ae(x) reduces both V and Z. This is the crux. It follows from these conditions that, at least for x s intH, AE(x) = 0 if and only if x is slinearly-optimal. In detail, there are four mutually exclusive cases (still supposing that x is in the interior of H): 1. V(x)>s, 2. 0 < V(x) < s, 3. V(x) = s and desc(Z, V)(x) * 0, and 4. V(x) = s and desc(Z, V)(x) = 0. We show that x is not E-linearly-optimal in cases 1 - 3 and A s (x) ^ 0 then; and that, in case 4, x is E-linearly-optimal and also that in this case A£(x) = 0.

1. V(x)>s In this case x ? E E and also Ae(x) = [V(x)/s]desc(Z, V)(x) + [V(x)/s - l]descV(x) is non-zero as descV(x) is non-zero and desc(Z, V)(x) is never a direction in which V ascends. Of course x is not s-feasible and so is not E-linearly-optimal. Here following As(x) reduces V. 2. 0 < V(x) < s In this case x e Ee and Ae(x) = [V(x)/s]desc(Z, V)(x) + [1 - V(x)/s]descZ(x) is again non-zero as descZ(x) is non-zero and desc(Z, V)(x) is never a direction in which Z ascends. Here following As(x) reduces Z maintaining V < s. 3. V(x) = s and desc(Z, V)(x) * 0 In this case x e Ee and Ae(x) = desc(Z, V)(x) is non-zero and so is a simultaneous descent direction for both V and Z. Thus As(x) is an Efeasible Z-descent direction at x so x is not s-linearly-optimal. Here again following As(x) reduces Z maintaining V < s.

Bilevel optimisation of prices in a variety of transportation models 11 4, V(x) = s and desc(Z, V)(x) = 0 In this case also x e Es. Now desc(Z, V)(x) = 0 and so descV(x) = - descZ(x). Consider any Z-descent direction u. Then u-[descZ(x)] > 0 and so u-[descV(x)] = u-[- descZ(x)] < 0, and u is not an s-feasible direction at x. Thus there is no s-feasible Z-descent direction from x and also V(x) < s, so x is s-linearly-optimal. In this case As(x) = 0. Thus we have shown that (at least for x e intH) zeros of A8(x) coincide with points x e Es at which there is no s-feasible descent direction for Z at x. Such points are e-linearly-optimal. We have also shown that A6(x) is an H-feasible descent direction for V if x is not in Es and that Ae(x) is an Es-feasible descent direction for Z if x is in EE and is not s-linearly-optimal. Thus, at least for x in the interior of H, As(x) is an excellent arbiter of s-linear-optimality at x; and for those x which are not s-linearly-optimal indicates a sound direction for moving x. An (xn, tn.i)-updating dynamic Armijo-like optimisation algorithm Here we change the previous equilibration algorithm to create an optimisation version. Now we suppose that if we are at iteration n; at a non-s-linearly-optimal xn e H where the search direction is Ae(xn) and the step length actually used at xn is un then our next x will be xn+i = Proj(xn + unA(xn)) where Proj now denotes projection onto H. Initially we suppose that if un is any used step length then Proj(xn + unAe(xn)) = xn + unAe(xn). That is, we suppose that the projection operator does not actually do anything; and so the boundary of the feasible set H is here having no effect. Let Gi(x) = - gradV(x)-Ae(x) and G2(x) = - gradZ(x)-Ae(x). These are continuous functions of x defined for all x e H. Both must be estimated using a two-point estimation where the second point lies in the direction A6 from the first. For our purposes here we specify a simple dynamic Armijo-like optimisation scheme based on the previous dynamic Armijo-like equilibration scheme but now taking account of the two objectives V and Z and the continuous functions Gi and G2 above, instead of just the previous V and the previous G. As before we update (xn, tn.i). To be specific, we start at an arbitrary xi e H and to = 1. Then if we are at a current non-s-linearly-optimal point xn and the previous possible step length was tn-i then we are to update xn and tn_i (for n > 1) as follows. To be most general we will, in this algorithm statement, let V[y] = V(ProjH(y)) for all y e RN + K. Let V(Xn) > e. In this case we follow the equilibration algorithm above with G replaced by Gi.

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Let V(x^ < e. Firstly, x n is kept fixed and V i is halved to obtain the sequence tn_i, (54)tn.i, ('/2) 2 t n .i,...., (Vytn.i where the halving ceases as soon as: Z[x n + (MOV.AeCx,,)] - Z(x n ) < - (l/8)C/ 2 ) p t n .iG 2 (x n ) and V[x n + (>/2)ViAE(xn)] < s for the first time, p = 0 is allowed here as before. The halving surely ceases by definition of G2 and the Z-descent property of Ae maintaining V < s, because xn is not s-linearly-optimal. Then let un = (Viftn-i for this p (this is to be the used step length at x n ) and xn+i = x n + u n A s (x n ). Finally update tn_i as follows: if Z[x n + unAE(xn)] - Z(x n ) < -3/4UnG2(xn) put tn = 2u n ; if-3/4UnG2(xn) < Z[x n + unAs(xn)] - Z(x n ) < -1AunG2(xn), put tn = un; and if-'/4UnG2(xn) < Z[x n + unAE(xn)] - Z(x n ) < - (l/8)u n G 2 (x n ) put tn = Vox*. These are three mutually exclusive possibilities and they together exhaust all eventualities since, by choice of u n = (Vi)ptr,--\, Z[x n + unAE(xn)] - Z(x n ) > - (l/8)u n G 2 (x n ) is not possible. The algorithm is terminated as soon as V(x n ) - s and G 2 (x n ) are both less than preassigned tolerances.

CONVERGENCE TO A STATIONARY POINT We assume that our control-enhanced basic condition holds.

Convergence preliminaries For any starting point xi e H and with an initial step length to = 1 we suppose that the algorithm specified above generates an infinite trajectory: xi, x 2 , X 3 , . . . , xn> . . . . and an infinite sequence to, ti, t2, t 3 , . . . , tn, . . . . of possible step lengths. To prove convergence in this case we will use proof by contradiction. So we suppose that {xn} is a given infinite sequence generated by our algorithm which does not converge to the set O e of slinearly-optimal points. Then not all limit points of the sequence {xn} belong to OE and since H is closed and bounded there is a limit point w which is not in OE. Thus we now make the basic assumption that w is not s-linearly optimal and is also a limit point of {x n }. We will show that this leads to a contradiction. This will show that, in fact, if the sequence {xn} is generated by the algorithm then all limit points of the sequence {xn} are in OE and so (as H is closed and bounded) dist(xn, O6) -> 0 as n -> 00.

Theorem Let our control-enhanced basic condition hold. Let {xn} be the sequence above. Suppose that {xn} lies in the interior of H, let x* be the limit of any subsequence of the above sequence {xn}, and let x* e intH. Then x* is s-linearly-optimal. [Note: these interior conditions may well happen naturally, especially within stochastic assignment models.]

Bilevel optimisation of prices in a variety of transportation models 13 Proof Suppose that x* is the limit of a subsequence of the above sequence and that x* e intH. We will show that none of the following three alternatives in 7.4.2 can occur: (l)V(x*)>s, (2) 0 < V(x*) < s, (3) V(x*) = s and desc(Z, V)(x*) * 0. It will then follow that (4) above holds, and so x* is s-linearly optimal. Ruling out (I) This is essentially as before with equilibration. Ruling out (2) Suppose that (2) does hold or that V(x*) < s. Then Ae(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]+descZ(x*) + [V(x*)/e-l]+descV(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]descZ(x*) * 0. [l-V(x*)/s]descZ(x*) is a descent direction for Z at x* (and desc(Z, V)(x*) is never ascent) so [gradZ(x*)]-AE(x*) < 0. A small enhancement of the equilibration argument now works in this case too, but with Z instead of V, ruling out (2). Ruling out (3) Suppose that (3) does hold or that V(x*) = s and desc(Z, V)(x*) is non-zero. Then A6(x*) = [V(x*)/s]desc(Z, V)(x*) + [l-V(x*)/s]+descZ(x*) + [V(x*)/e-l]+descV(x*) = desc(Z, V)(x*) * 0, and this (being non-zero) is a descent direction for both V and Z at x*. An enhancement of the equilibration argument embodied in the lemma in section 6 above now works in this case too, but this argument must in this case be applied to both Z and V. See Smith (2005) for the details. This means that x* cannot be a limit point and rules out (3). It follows from the arguments above that if x* is any limit point in the interior of H, (4) above must hold or: V(x*) = s and desc(Z, V)(x*) = 0. So x* is e-linearly-optimal, as we have seen from the s-linearly-optimality conditions above in section 7.4.2. and the proof is complete. Allowing for the boundary of H The boundary of H may be taken account of by re-designing the search direction Ae. We change descV(x), descZ(x) and desc(Z, V)(x) for those x close to the boundary of H. The new versions will be called descHV(x), descHZ(x) and descH(Z, V)(x) and Ae will become AHe. We do not give these details here. They are given in Smith (2005).

14 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

APPLICATIONS TO DIFFERENT MODELS The basic structure exploited above is as follows: there is a function O, defined on the product set FxFcontroi. such that - 0 for all ijr. Xjjo is essentially a slack variable added in to model 0. The rigid demand problem represents the given variable demand problem and is driven by the original costs Cyr(X) and also the new opportunity cost Cij0(X) = D^EeiX^) = D"!(T - Xij0). It is easy to check that the whole new cost function C is monotone if the original C and the original -D are both monotone. Thus we may add a price vector as previously and utilise the same simultaneous descent algorithm to optimise this equilibrium model. Equilibrium model 2: extra route when the inverse demand function is unknown The set of "flows" in this model is the set of all (Xo, X, Y) such that (Xo, X) e {(Xo, X); Er Xijr = Ty for all ij and Xijr > 0 for all ijr} where in the sum here r runs over 0, 1, 2, 3 , . . . Ny, and Y s {Y; 0 < Yy < Uy}. A typical "route-flow" vector is now (Xo, X, Y) where: (Xo, X, Y) e {(Xo, X); Zr Xyr = Ty and Xijr > 0 for all ijr}x {Y; 0 < Yy < Uy}. With this set of feasible "flows" this "rigid demand" model is driven by "costs" as follows: Yy: this cost is felt by each element of the flow Xyo, Cyr(X): this cost is felt by each element of the flow Xyr for r = 1, 2, 3 , . . . , Ny, and Ty - Xyo - Dy(Y): this "cost" is felt by Yy. Assuming for simplicity that all the listed routes are used at equilibrium: Yy and all Cyr(X) are equal so that the flows Xyi,. . ., Xyuy have no incentive to change and also Y represents these OD costs; all Ty - Xyo - Dy(Y) = 0 and so Yy e [0, Uy] also has no incentive to change.

Bilevel optimisation of prices in a variety of transportation models

15

Thus at an interior equilibrium as specified above the Wardrop condition holds, and, since Y does then represent the OD costs experienced on all routes, Ty - Xij0 - Djj(Y) = 0 implies that the total ij flow Tij - Xjjo on the real network does equal Djj(Y) as it should. Monotonicity of C and -D now implies that: (Y, C(X), T - Xo - D(Y)) is a monotone function of (Xo, X, Y). This means that the optimisation method designed for model 0 also works here. This formulation may of course be used to restate, in a more complicated way, a given rigid demand problem; just "reformulate" the given rigid demand problem as a variable demand problem with a flat (zero slope) demand function by making Dy(Y) = a constant Djj for all Y. Equilibrium model 3: A responsive control policy in the Payne-Thompson model In order to study ramp control on freeways, Payne and Thompson (1975) introduced an equilibrium model with queueing delays at bottlenecks. In their model as long as a bottleneck is saturated the queueing delay at the bottleneck is independent of flow, and is represented by an independent non-negative variable determined by the equilibrium conditions; but if the bottleneck is unsaturated then the bottleneck delay must be zero. Bringing in the delays at bottlenecks in this way, as separate variables, allows ramp metering to be modelled sensibly. The "new" independent bottleneck delays are here added to costs arising from more standard cost-flow functions ca which may be thought of as applying to the rest of arc a. We here consider the Payne-Thompson model with capacity constraints and explicit queueing delays. We insert signal green-times as in Smith (1987). There can be great advantages in real traffic situations if the signal controls respond appropriately and automatically to the flows and delays on the network and traffic responsive signal control is so often utilised in practice. This extends the type of equilibrium model which can be considered. Certain subsets of arcs a comprise stages; any stage contains only arcs a terminating at a single node and all arcs in a given stage are given green if the stage is green. The kth stage at node i will be called stage ik. Let va sa ba Qik ca(-)

be the traffic flow along arc a, be the saturation flow at the exit of arc a (both in vehicles per minute), be the bottleneck delay at the exit of arc a (in minutes per vehicle), be the proportion of time stage ik is green and be the cost function for arc a (this excludes the bottleneck delay at the arc exit).

Then we make the following agreements: the flow along arc a — V a ( X ) — 2-those ijr passing through arc a X j j r ;

the non-bottleneck cost of travel along route ijr ~~ ^ i j r ( X ) — 2^arcs a which belong to route r joining OD pair ijC a (V a );

the bottleneck delay on route ijr

16 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

)

Z_iarcs a which belong to route numbered r joining OD pair ijD a ,

the "pressure" felt on stage ik by the responsive signal setting algorithm =

Jik(b) — Z^arcs a which belong to stage ik Sa^ai a n a

the green-time awarded to arc a Qa

^--stages ik which include arc a Wik-

The special responsive policy PO seeks to equalise the pressures felt on the various stages by suitable allocations of green-time. Thus if one stage has a higher value of J than another then PO will reallocate some green-time from the less pressurised stage to the more pressurised stage. At equilibrium all stages with a positive green time have equal values of J and stages with smaller values of J have no green-time. This equilibrium allocation of green-time is similar to the Wardrop allocation of traffic flow to routes and is given by the fourth and fifth condition in the equilibrium condition below. The grand equilibrium flow/cost/queue/green-time equilibrium condition comprises the following conditions. These must hold (for a fixed price vector P): for all nodes i, j , routes ijr, arcs a, and signal stages ik. Yy - Cijr(X) - Bljr(b) - Pijr Dy(Y) - Ty(X) va(X) - saqa Jik(b) - m;

< = < <

0 and Yy - Cijr(X) - Bljr(b) - Pijr < 0 implies Xljr = 0; 0; 0 and va(X) - saqa < 0 implies ba = 0; 0 and J;k(b) - m; < 0 implies that Qjk = 0; and

(All variables must also satisfy non-negativity constraints.) The top three conditions is the equilibrium model introduced by Payne and Thompson. The bottom two conditions represent the PO control-equilibrium condition. It is easy to check that, for each P, - (the left hand side above) is a monotone function of (X, Y, b, Q, m). It follows that the optimisation method described previously in model 0 may be utilised also in this rather different equilibrium setting. A practical observation We assume here that in practice the signal green-times respond automatically according to PO, rapidly. Then we imagine optimising the price vector P in a leisurely long-run fashion. This is to ensure that we feed travellers predictable price information for their travelling decisions. Model 4: a general model embracing the previous five models This model here is a rigid demand model which includes the previous four models as special cases. This may be optimised just as we illustrated with our basic model 0. Let F be the polyhedron of those route-flow vectors X which meet the given rigid demands Ty. Let C(X) be a monotone function of X. Then we may follow Smith (1984a, b, c): firstly let F = niJF1J where the vertices of Fy are the Ny all-or-nothing flows for OD pair ij and a typical one is, say, Wjjr. This Ny vector has T;J in the rth place and zero elsewhere.

Bilevel optimisation of prices in a variety of transportation models 17 We will agree here that for each X, Wljr(X) denotes the vector which has N co-ordinates and is defined as follows: Wijri.jv(X)

= = =

Ty if i1 = i, j ' = j and r' = r; 0 if i' = i,j'=j andr'^r; and Xj]r otherwise.

Thus Wljr(X) leaves all components of X with a different ij unchanged. But the ijr'-component of the vector Wljr(X) becomes equal to Ty (if r' = r) or zero (if r' *'r). Hence if are at X and move to X + Wljr(X) then the flow on all non-ij routes remains the same but the flow on the ij routes is all re-allocated to the single route ijr from i to j . The algorithm (D) search direction is now defined for each route-flow vector X in F by: A(X)

=

Si]r [-C(X)-( Wljr(X) - X)]+ (Wljr(X) - X).

The algorithm (D) objective function is V where, for each X in F: V(X) =

Sljr{[-C(X>(Wijr(X)-X)]+}2.

Smith (1984c) showed that provided the cost function C(-) is monotone and continuously differentiable then, away from equilibrium, this A(X) is a descent direction at X for this V. Using the Armijo-like step length procedure above we surely converge to an equilibrium. In this model we may now add control parameters and the previous optimisation theory works in this general case just as it did with model 0. This model is a generalisation which embraces the previous equilibrium models 0, 1, 2 and 3. The only difference is the precise form of the cost function and the polyhedron F. Model 5: Stochastic equilibrium models To create a model with logit route choice it is now natural to define a new cost function as follows: Cst0Chijr(X) = Cijr(X) + G^lnXr for all ijr. Then a solution to model 4 above will under natural conditions be a stochastic user-equilibrium on a given set of routes with given rigid demands Ty for each OD pair ij. This may be considered as a variable demand stochastic user equilibrium model, following the "extra-route" model 2 above, as suggested by Connors et. al. (2004, 2005). Maher et al. (1999) give a different stochastic model with variable demand. In this model the "interior to H" assumptions in this paper may be expected to hold naturally and so the model seems very suited to the "simultaneous descent" optimisation method here. CONCLUSION This paper has given a method for approximately solving a bilevel optimisation problem within a specific but fairly general equilibrium model (model 0). In this bilevel problem a stationary point of a given objective Z is sought subject to the flow pattern being an

18 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction approximate variable demand equilibrium. If the cost function and - (the demand function) are monotone and smooth and certain other natural conditions hold then the suggested algorithm produces a sequence which converges to the set of points which are approximate equilibrium (at which V < s) and also stationary for the given smooth objective function Z. The method utilises the very simple simultaneous descent direction obtained under certain (interior) conditions simply by bisecting the angle between -gradV and -gradZ. In this direction V and Z simultaneously decline. All directions in this paper, including the simultaneous descent direction, may be modified close to the boundary of the polyhedral feasible region F and this modification allows the approach to work also when the feasibility constraints on the polyhedron F are active during the optimisation process or at a limit point. See Smith (2005) for the details. Finally the paper has shown that the same method will be applicable to a variety of formulations of the variable demand equilibrium model, including one where there is an responsive control system. Of course the optimisation problem here is non-convex so a variety of start points should be taken and the optimisation procedure followed from each.

ACKNOWLEDGEMENT Much of the material central to this paper was presented in the keynote presentation of the "Behaviour hi Networks Symposium", University of Seoul, South Korea, June, 2004.

REFERENCES Aashtiani H and Magnanti T (1983) Equilibrium on a congested transport network. SIAM Journal of Algebraic and Discrete Methods, 2, 213 - 216. Abdulaal M and Leblanc L (1979) Continuous network design problems. Transportation Research, 13B, 19-32. Allsop R E (1974) Some possibilities for using traffic control to influence trip distribution and route choice, Proceedings of the 7th International Symposium on Transportation and Traffic Theory, 345 - 374. Bar-Gera, H (2002) Origin-based algorithms for the traffic assignment problem. Transportation Science, 36, (4), 398-417. Bar-Gera, H and Boyce D (2003) Origin-based algorithms for combined travel forecasting models. Transportation Science, 37B, (5), 405 - 422. Beckmann M, McGuire C B and Winsten C B (1956), Studies in the Economics of Transportation. Yale University Press, New Haven, CT. Chiou S-W (1997) Optimisation of area traffic control subject to user equilibrium traffic assignment, Proceedings of the 25th European Transport Forum, Seminar F, Volume II, 53 - 64.

Bilevel optimisation of prices in a variety of transportation models 19 Clark S D and Watling D P (2002) Sensitivity analysis of the probit-based stochastic user equilibrium assignment model. Transportation Research, 36B, 617 - 635. Clegg J and Smith M J (1998) Bilevel optimisation of transportation networks, in Mathematics in Transport Planning and Control, the Proceedings of the Third International IMA Conference on Mathematics in Transport Planning and Control, Pergamon, 29 - 36. Clegg J, M J Smith, Y Xiang and R Yarrow (2001) Bilevel programming applied to optimising urban transportation. Transportation Research, B, 35, 41-70. Clegg J and Smith M J (2001) Cone projection versus half-space projection for the bilevel optimisation of transportation networks. Transportation Research, B, 35, 71 - 82. Clune, A, Smith M J and Xiang Y (1999), A Theoretical Basis for Implementation of a Quantitative Decision Support System - Using Bilevel Optimisation. Proceedings of the Fourteenth International Symposium on Transportation and Traffic Theory, Jerusalem (Editor: Avishai Ceder), Pergamon, 489 - 514. Connors R, Sumalee A and Watling D (2004) Understanding the variable demand probitbased network design problem. Paper presented at the Universities' Transport Study Group Meeting, University of Newcastle upon Tyne. Connors R, Sumalee A and Watling D (2005) Equitable Network Design. Presented at the Universities' Transport Study Group Meeting, University of the West of England, Bristol. Davis G A (1994) Exact Local Solution of the Continuous Network Design Problem via Stochastic User Equilibrium Assignment. Transportation Research, 28B, 61-75. Evans S P (1976) Derivation and Analysis of some Models for Combining Trip Distribution and Assignment, Transportation Research, 10(1), 37-57. Fisk C S (1984) Optimal signal controls on congested networks. Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, Delft (Editors: J Volmuller and R Hammerslag), VNU Science Press, Utrecht, 197-216. Fletcher R and S Leyffer (2000) Nonlinear programming without a penalty function. University of Dundee Numerical Analysis report NA 171. Gartner N H (1980) Optimal traffic assignment with elastic demands: A review. Part II: Algorithmic approaches. Transportation Science, 14, 192 - 208. Gauvin J and Savard G (1994) The steepest descent direction for the nonlinear bilevel programming problem. Operations Research Letters, 15, 265 - 272. Luo Z Q, Pang J S and Ralph D (1996) Mathematical programs with equilibrium constraints, Cambridge University Press. Maher M J, Hughes P C and Kim K S (1999) New algorithms for the solution of the stochastic user equilibrium problem with elastic demand. Proceedings of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem, Israel (Editor A Ceder). Marcotte P (1983) Network Optimisation with Continuous Control Parameters, Transportation Science, 17, 181-197. Marcotte P (1986) Network Design Problem with Congestion Effects: A Case of Bilevel Programming, Mathematical Programming, 34, 142-162.

20 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Migdalas A (1995) Bilevel Programming in Traffic Planning: Models, Methods and Challenge. Journal of Global Optimization, 7, 381 - 405. Murchland J D (1970) Road network traffic distribution in equilibrium, In Mathematical Models in the Social Sciences, II Oberwolfach, 1969, Anton Hain Verlag, Meisenheim am Glan., volume 8, 145-183. Outrata J and Zowe J (1995) A numerical approach to optimization problems with variational inequality constraints. Mathematical Programming 68, 105 - 130. Patriksson M and Rockafellar R T (2002), A Mathematical model and Descent Algorithm for Bilevel Traffic Management. Transportation Science 36, 271 - 291. Payne H J and W A Thompson (1975), Traffic assignment on transportation networks with capacity constraints and queueing. Paper presented at the 47th National ORSA/TLVIS North American Meeting. Available from the University of York. Rodrigues H S and M T Monteiro (2004) Solving mathematical programs with complementarity constraints (MPCC) with Nonlinear Solvers. Poster at the 12th FrenchGerman-Spanish Conference on Optimization, Avignon. Smith M J (1984a) A descent algorithm for solving a variety of monotone equilibrium problems, Proceedings of the Ninth International Symposium on Transportation and Traffic Theory, The Netherlands, VNU Science Press, Utrecht, 273-297. Smith M J (1984b) A Descent Method for Solving Monotone Variational Inequalities and Monotone Complementarity Problems. Journal of Optimization Theory and Applications, 44, 485 - 496. Smith M J (1987) Traffic control and traffic assignment in a signal-controlled network with queueing. Proceedings of the Tenth International Symposium on Transportation and Traffic Theory, MIT (Editors: N Gartner and N H M Wilson) 319 - 338. Smith M J, Xiang Y, and Yarrow R (1997) Bilevel optimisation of signal timings and road prices on urban road networks. Preprints of the IFAC/IFIP/IFORS Symposium, Crete, 628 - 633. Available from the University of York. Smith M J, Xiang Y, Yarrow R (1998a) Descent Methods of Calculating Locally Optimal Signal Controls and Prices in Multi-Modal and Dynamic Transportation Networks. Selected Proceedings of the 4th EURO Transportation Meeting, University of Newcastle, Editor: M G H Bell, 9 - 34. Smith M J, Xiang Y, Yarrow R and Ghali M O (1998b) Bilevel and Other modelling Approaches to Urban Traffic Management and Control. Paper presented at the 25th Birthday of the Centre de Reserche sur les Transports, University of Montreal (1996), and a chapter in the Proceedings, Equilibrium and Advanced Transportation Modelling, (Editors: Patrice Marcotte and Sang Nguyen) Kluwer Academic Publishers, Massachussetts, 283 - 325. Smith M J (2005) Simultaneous descent: some details. Available from the University of York. Tan H N, Gershwin S B and Athans M (1979) Hybrid optimization in urban transport networks. Laboratory for information and Decision Systems, Technical Report DOTTSC-RSPA-79-7; published by Massachusetts Institute of Technology, Cambridge Massachusetts, USA. Tobin R L and Friesz T L (1988) Sensitivity analysis for equilibrium network flow, Transportation Science, 22, 242 - 250.

Bilevel optimisation of prices in a variety of transportation models 21 Wardrop J G (1952) Some Theoretical Aspects of Road Traffic Research, Proceedings, Institution of Civil Engineers II, 1, 235-278. Yang H, and Yagar S (1994), Traffic assignment and traffic control in general freeway-arterial corridor systems. Transportation Research, 28B, 463 - 486. Yang H (1996a), Sensitivity analysis for queueing equilibrium network flow and application to traffic control. Mathematical and Computer Modelling, 22, 247 - 258. Yang H (1996b), Sensitivity analysis for the elastic-demand network equilibrium problem with applications, Transportation Research, 3 IB, 55 - 70. Yang H (1996c), Equilibrium network traffic signal setting under conditions of queueing and congestion. Applications of Advanced Technologies in transportation Engineering. Proceedings of the 4th International Conference (editors: Y J Stephanedes and F Filippi), American Society of Civil Engineers, 578 - 582.

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A SEQUENTIAL EXPERIMENTAL APPROACH FOR ANALYZING SECONDBEST ROAD PRICING WITH UNKNOWN DEMAND FUNCTIONS Hai Yang and Wei Xu, Department of Civil Engineering, The Hong Kong University of Science & Technology, Hong Kong, P. R. China; and QiangMeng, Department of Civil Engineering, The National University of Singapore, Singapore

INTRODUCTION Road pricing as a way to reduce traffic congestion and raise revenues to fund transportation improvements has received substantial added interest in recent years. Practical implementation has been progressing rapidly; for instance, electronic road pricing schemes have been proposed and implemented worldwide such as in Hong Kong, Singapore, California, and recently in London. With availability of new pricing technology and increasing concern over external cost, it is conceivable that there will be greater use of road pricing as an effective traffic management tool over the next few years. There has been a vast accumulation of literature that deals with the theoretical and empirical aspects, policy experiences, and environmental issues of road congestion and road pricing. In particular, the first-best and second-best congestion pricing have been widely discussed ranging from a single highway or bottleneck to a general transportation network. Moreover, there has been a great need for the development of practical and efficient methods for the actual design and implementation of road pricing. A fundamental question is how to choose the optimal congestion tolls in a simple yet practical manner. To this end the travel demand function tailored for road pricing is of special importance. Unfortunately, the demand curve is usually unknown and difficult to estimate in practice even with advanced transport modeling

24 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction techniques. To resolve this dilemma, Li (2002) proposed a practical iterative experimental procedure to derive the optimal congestion toll in the absence of a demand function and applied it to estimate the congestion toll for the area-licensing scheme and major expressways in Singapore. This enlightening implementation idea dates back to Vickrey (1993) and Downs (1993) who argued that, without necessarily using demand functions, congestion pricing could be implemented on a trial-and-error basis, based on observed behavior change. The basic logic behind the iterative implementation procedure can be summarized as follows (Li, 2002): With information on the underlying speed-flow relationship on the highway and the generalized cost, the optimal congestion toll occurs if and only if the intended speed level or traffic volume that is used to calculate the theoretical congestion toll is exactly equal to the observed speed level or traffic volume after the imposition of the toll. As a result, the exact knowledge of the demand function is not really necessary. The theoretical justification for finding the socially optimal level of congestion toll by Li (2002) is based on a given speedflow relationship and the true, but unknown downward-sloping demand curve. Thus, his method applies the simplified standard case of a homogenous traffic stream moving along a given uniform stretch of road such as an expressway connecting with a given entry and exit point. Li (2002) proposed to use a bisection method for iteratively adjusting the actual tolls, which is convergent and is in fact optimal within the standard demand-supply equilibrium framework for a single road. The above powerful idea of trial-and-error pricing implementation was further explored and extended to a general road network by Yang et al. (2004). hi the same spirit of Vickery (1993), Downs (1993) and Li (2002), Yang et al. (2004) proposed an iterative toll adjustment procedure based on the method of successive averages, and presented a rigorous theoretical proof of its convergence. However, the basic theory and method proposed by Li (2002) and Yang et al. (2004) are strictly valid only for the first-best pricing problem, which requires that a road user using each congested road should pay a toll equal to the difference between the marginal social cost and the marginal private cost. This perfect toll pattern could internalize the user externalities and thus achieve a socially optimal level of traffic flow by charging users of each link of the network equal to the user externality (Yang and Huang, 1998). Nonetheless, it is a rather theoretical construct and of little practical interest in view of the operating cost and public acceptance. Due to the imperfection of the first-best pricing from a political and technical implementation perspective, the second-best charging schemes have received ample attention in recent literature. This paper builds on research by Li (2002) and Yang et al. (2004) and makes a significant step towards efficient implementation of the sequential experimental mechanism for the practical second-best pricing problem. In contrast to the previous straightforward iterative procedure for first-best pricing, we deal with the price experimentation when only a subset of links is tolled and only a subset of link traffic counts is available in each trial. We aim at developing an effective and efficient procedure capable of at least working out a good

Sequential experimental approach for analyzing second-best road pricing 25 estimation or approximation of the relevant congestion tolls with incomplete set of data derived from the limited number of trial experiments. Our objective is achieved by virtue of advanced bi-level programming techniques, which are applied alternately and sequentially for estimating the origin-destination (O-D) demands from link traffic counts and determining the optimal link tolls with user equilibrium constraints on general networks. In what follows, we define some notation and outline the major idea used in the first-best pricing problem. Then, we introduce the trial-and-error implementation method for the second-best pricing problem with sequential price experimentation, and describe the necessary bi-level programming models and algorithms for the sub-problems of O-D demand estimation and link toll selection. After demonstrating the proposed sequential experiment approach with two simple numerical examples, we draw some conclusions and offer some perspectives on future research.

TRAFFIC EQUILIBRIUM AND FIRST-BEST PRICING Consider a road network with a set of directed links denoted by A, the flow on link a is denoted by va and v = (va,a e A) is a vector of all link flows. The travel time associated with traveling on link a is denoted by ta, which is assumed to be a function of flow on that link only (for simplicity, we consider separable link cost functions), ta = ta (v a ). We further use \a to denote the toll charge (in equivalent time unit) on link a e A . Let W denote the set of O-D pairs and Rw the set of all routes between O-D pair w e W. The route costs are given by the link travel time costs and toll charge (if any) as follows: cnv=Yj5l(ta(v)

+ xa), reRw,

weW

(1)

where 8™, = 1 if path r between O-D pair weW uses link a, and 8",. = 0 otherwise. Let cw = minjc^,, reRw]

be the minimum generalized travel cost between O-D pair w eW.

The true demand between O-D pair w e W is denoted by dw and is assumed to be a function of the minimum generalized travel cost between that O-D pair, dw = gw (cj) (for simplicity again we consider separable travel demand functions). Let d = (rfw, w e W)

The following standard assumptions are required throughout this study:

26

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Al. For each link a s A , the corresponding link travel cost ta (v 0 ) is a strictly increasing, convex and is a twice continuously differentiable function in link flow va. A2. For each O-D pair wsW , the corresponding O-D demand function gw (c^) is nonnegative, strictly decreasing, bounded and continuously differentiable in O-D travel time cw, 0 < gw (cw) < dw, where dw is the upper bound of the demand function g

(c).

A3. The route choice behavior of network users follows the user equilibrium (UE) principle for any given toll charges. Given the above assumptions, the traffic equilibrium problem with elastic demand can be formulated as the following mathematical program (Bell and Iida, 1997):

(2)

where gj (dw) is the inverse of the demand function gw(cw) , Q. is the feasible set of link flows and O-D demands defined by

n = \(v,d)\va = YjJ]frw8war,asA;frw>0,reRw,YJfm=dw
where fm is the flow on route rsRw

(3)

between O-D pair weW .

For any given set of link toll charges and with the assumptions made above, a unique set of link flows and O-D demands can be determined from the UE program (2). The resulting net economic benefit (EB) or social welfare is well defined for the separable demand functions as follows:

^EB (T) = I '"j g~: («)d«, - 2> a (va (*)) v, (T)

(4)

Sequential experimental approach for analyzing second-best road pricing 27 Therefore, the optimal congestion-pricing problem is to determine a set of link tolls to maximize the net economic benefit function (4) subject to the UE constraint (2). The problem is formulated as the following Bi-Level Programming Problem (BLPP) (Yang and Bell, 1997): max FEB (T) , subject to: (v,d) e Q solves UE problem (2)

(5)

where T is the feasible set of link toll charges. If every link is subjected to a nonnegative toll charge, then one immediate solution to the above BLPP is to use link tolls that are based on marginal-cost pricing and are given as (Yang and Huang, 1998): dt (v ) 1a=Va^TJLL

>

a e A

(6)

dv0

where v* = (v*,a e A\ and d" =(d"w,we W\ are the system-optimal solution that solves

max

V fg~'(co)dco-Y;„(¥„)v

(7)

In the iterative procedure developed by Yang et al. (2004) for marginal-cost pricing in the absence of demand functions, link tolls are adjusted with the change in observed link flows or according to the explicit formula (6) using the convergent method of successive averages. The observed link flows are the revealed aggregate result from the reaction to different trial tolls by individual road users, and are readily available in actual implementation. In other words, the socially optimal link tolls can be identified with repeated price experimentation and link flow observations in a straightforward iterative adjustment procedure.

A SEQUENTIAL SECOND-BEST PRICING POLICY Here we are interested in the selection of optimal link tolls in the same spirit of trial and error experimentation in a practical second-best pricing environment where only a subset of links are subjected to toll charge. We also show how a set of second-best link tolls can be obtained

28 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction by applying bi-level programming techniques sequentially for both the 0-D matrix estimation and the toll-pricing optimization sub-problems. Let Ac A denote a subset of links subjected to a toll charge and Ac. A a. subset of links whose flows are observed after a trial for each link toll pattern, A and A are not necessarily identical. The basic iterative estimation and optimization procedure works as follows. Suppose that, after k trials of toll pricing on link a e A, we obtain a set of calibrated, approximate demand functions g£' (c,6), weW where 6 e 0 is a vector of parameters in the demand function and 0 is the feasible set of these parameters. Then based on the updated, approximate demand function, we can determine a new subset of link tolls i*

, ae A for

subsequent price experimentation by solving the following BLPP:

where (V(T),C?(T)) solves the following UE problem for given x E f and 6 e 0 : v

v

d

<0 +x dM+ I } ' » d c o - £ f^1(«,e)dm ( T Q l}('«( ) «)

(v

'

>£

aeA o

aeA.atA o

(9)

weJf o

where g"1 («?w,9),w e W is the inverse function of the currently calibrated, approximate demand function gw (cw,&) and T is the feasible set of the subset of link tolls ia, a e A. After imposition of the newly determined link tolls tj,**1', as A, a subset of revealed link flows vj*+1), a s. A are observed. The observed link flows are the result of distribution of the realized O-D demands on the network, and thus provide useful information about the true but unknown demand functions. Thus we can use the BLPP technique to estimate or update the O-D matrix and the approximate demand functions from the newly available link traffic counts, thereby obtaining anew set ofupdated O-D demand functions gw(cw,6(*+1> j , we W . There are two ways to update the demand functions: one is updating the O-D matrix obtained in the last iteration from traffic counts, and then using the new O-D matrix to tune the

Sequential experimental approach for analyzing second-best road pricing 29 approximate O-D demand functions; the other is calibrating and updating the parameters of the assumed O-D demand functions directly from the observed new trial results. The O-D matrix-updating problem from traffic counts can be formulated as the following BLPP (Yang et al., 1992; Yang, 1995): mm F0D{d) = Fl(v(d),v) + F2(d,d)

(10)

where v(
V

r co +x dco+

™% E |( «( ) «) v

~

aeA o

fc(°>)dffi

Z

(n)

aeA.aeA Q

where

(12)

&d={(dw,weW)\0
Q

, (d) = \(Va>a

e A

) Va = E Z frX,'0

S

^ / ™ ^ 0, r E i?w, ^ / „ = dw, W E W\ (13)

In this model, v = (ya,a E ^4) represents the vector of link flows observed in the last trial; d = tdw, w e Wj is the O-D matrix (rearranged as a vector) obtained in the last trial; Ft (v,v) and F2(d,d\

are functions of the generalized distance or error measurement between v and

v , and d and d , respectively. The following Euclidean distance function (Yang et al., 1992) is frequently used:

(14)

where y, and y2 are two weighting factors reflecting the relative confidence in v and d respectively. Because traffic counts are revealed reliable information collected with a good accuracy in the pricing experiment, a larger y1 (> y 2 ) may be preferred.

30 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In the case that the approximate O-D demand functions are tuned by updating a set of parameters 0 e 0 directly, the problem is formulated as the following similar BLPP: mm F0D(Q) = Fi(v(Q),v) + F2(d{Q),d)

(15)

where (v(Q),d(6)) solves the following elastic-demand UE problem for given 0 6 0 and

xef: va

r™PQZj('»

+T

va

dw

« ) d ( 0 + £ J<0(a>)dcn-£J|;;1(a,e)d(»

(16)

The same quadratic function in (14) can be used for the objective function (15). The above iterative bilevel O-D demand function estimation and bilevel link toll optimization process is repeated until an appropriate convergence criterion is reached. To sum up, the general iterative trial-and-error procedure through sequential price experimentation for estimating the second-best link tolls is given below. Trial-and-Error Second-Best Toll Adjustment Scheme: Step 0.

(Initialization) Let \gw [cw,Q I, w e W\ be an initial set of approximate demand functions. Set k = 0 .

Step 1.

{Estimate link tolls) Solve the BLPP (8)-(9) to obtain a new set of link tolls

{
(Observe link flows) Observe the revealed link flows denoted by (v ( t ) , a e A\ after the imposition of the new link tolls |x ( t ) , a e A\.

Step 3. (Check convergence) If ( £(x[M) - i[k) f [ x ( £ ( T * f ] ' < s, then stop. \anA

Otherwise, go to Step 4.

)

\an'A

)

Sequential experimental approach for analyzing second-best road pricing 31 Step 4.

{Update O-D demand function) Solve the BLPP (10)-(l 1) or (15)-(16) to obtain a new set ofupdatedO-D demand functions (gw(cw,Q{-k+]) Y wew).

Let k:=k + \

and go to Step 1. In the algorithm, s is a positive number of convergence tolerance. The procedure will be terminated after the difference between two successive trial results is within a given tolerance. Remark 1. To execute the procedure, one needs to assume a specific functional form as an approximation of the true but unknown demand functions, an existing reference O-D matrix, if available, could be helpful for estimating/updating the O-D matrix and the approximate OD demand functions initially. Given a set of limited trial experiments around the optimum solution, the procedure is expected to derive the demand curve for a limited range of traffic levels that is relevant to road pricing together with the estimation of optimal link tolls.

Remark 2. The bilevel O-D matrix estimation and bilevel link toll optimization problems involved above have been studied extensively and significant progress has been made both computationally and theoretically in the transportation literature (Yang et al., 1992; Yang, 1995; Yang and Bell, 1997). Readers may refer to Yang and Bell (2001) and Yang and Bell (1998) for reviews of the various bilevel transportation network optimization problems with UE constraints. Among a handful of rigorous and heuristic algorithms, it is particularly worthy to mention the marginal function approach developed recently by Meng et al. (2001) for solving the transportation BLPPs. By exploring the differentiability property of the marginal function (Gauvin and Dubeau, 1982) associated with the DUE problem, Meng et al. (2001) transformed the transportation BLPPs into a single-level continuously differentiable optimization problem, which are amenable to the analysis and application of standard nonlinear programming problems (Bazaraa et al., 1993). Remark 3. As a special case of the general second-best pricing problem envisioned here, the proposed sequential linear demand function approximation can of course be used to determine socially optimal tolls in a first-best pricing environment without demand functions considered by Li. (2002) and Yang et al. (2004). We may reasonably expect that the Newton-Raphsonlike algorithm based on successive approximation would zone in on the optimal price much faster than the painfully slow process of the method of successive averages (Yang et al., 2004). This is important in practice, since the price experiments are generally limited to a few trials. Thus the proposed approach has an added bonus for the first-best pricing problem studied by Li (2002) and Yang et al., (2004): it involves fewer price changes and thus reduces the cost of implementing the pricing scheme, which is itself a social benefit.

32 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Remark 4. The sequential experimental procedure in general involves an update interval of multiple months, during which multiple or even daily observational values of link flows might be available in practice. In this case, a statistical OD flow or parameter estimation approach such as the generalized least square OD matrix estimation method is preferable by treating link flow observations as random variables, for which variance and covariance matrices can be easily derived from the multiple observation data (Yang et al., 1992, Yang, 1995).

SEQUENTIAL LINEAR APPROXIMATION As assumed throughout this study, we have to specify an approximate (but easily invertible and computable) function gw(cw,Q), weW,

9 G © , to use in place of the true, but unknown,

demand function in the neighborhood of the optimum point of the second-best pricing problem. One favorable choice is an interpolation and extrapolation scheme by a polynomial, including the simple linear and quadratic approximations. Here we adopt the following simplest sequential linear approximation scheme based on successively observed and estimated experimental data: L(cw,^))

= d(wt) + zit)(cw-clt)),

weW, * = 1,2,...

(17)

where 8'j0 = {zf\ ^ " , ^ " } , W E W are the function parameters, {z^\weW^ of the linear demand function approximations, \d^\we

are the slopes

W^ are the O-D demands estimated

from observed link flows at iteration k using the O-D matrix updating model (10)-(l 1), and fc'*', w e W\ are the corresponding equilibrium O-D travel cost inclusive of toll charges if any. By considering the straight line through the latest two experiment points, we have: z ^ S J * ' - ^ - 1 ' ) / ^ * ' - ^ * - 1 ' ) , weW, Jt = l,2,...

(18)

The line slope, jz^', weW}, are updated from iteration to iteration based on the observed results in each trial. This successive linear approximation, if convergent, is expected to become a tangential approximation of the true demand functions and the tangent point is the resulting equilibrium point associated with the second-best optimum toll charge.

Sequential experimental approach for analyzing second-best road pricing 33 Remark 5. When estimating the current O-D matrix d^, w e W via the model (10)-(l 1), the O-D matrix d{j~l), w e W obtained in the previous iteration is used as the reference matrix to ensure the sequence of the O-D matrices that are generated to converge to the final optimum. In this regard an initial seed or reference O-D matrix is used to estimate the initial O-D matrix for the linear approximation and an old matrix from a previous study could be used for this purpose. The quality of the initial seed O-D matrix is immaterial because its impact on the resulting O-D matrix estimates will diminish as the O-D matrix is successively converged. Remark 6. A rational linear approximation of the true demand curve requires that the slope {zf\

w E w) calculated by (18) from the latest two experiment points be negative (property

of a demand function as mentioned in Assumption 2). Nonetheless, this requirement is not always ensured as the iteration continues, particularly when the trial solutions approach near the real optimum. Because the O-D demands, d^

and d^~x), are merely estimates from

traffic counts, the slope z^} may become unstable as d(wk) and d^'

move closer to each

other due to estimation inaccuracy. Thus, a correction has to be made whenever a positive slope emerges by, for example, resetting it to a small negative value.

NUMERICAL EXAMPLES We now present two simple numerical examples to elucidate the applications of the proposed sequential price experimental approach to the first-best and the second-best road pricing problems with unknown demand functions, respectively. Example 1. In this example, we show that the algorithm based on successive linear approximations may involve fewer price changes than the method of successive averages does for the first-best pricing scheme. For simplicity, we consider the standard textbook case; a single O-D pair connected by a single road, with the following true demand and link travel time functions: g(c) = 600 exp (-0.05c)

(

f

Vs)

t(v) = 10 1 + 0.15 — v

'

{

(19)

UooJ j

(2°)

34 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Since a single road is concerned, the revealed or observed flow, v , is always equal to the OD demand, d , and hence the O-D demand estimation process is bypassed here. Now we briefly introduce the bisection method proposed by Li (2002) and the method of successive averages by Yang et al. (2004) for this simple example. Let rf'4'1' (or v'*'1') be the demand or flow used to estimate toll x'*"1' at iteration k -1 and d{k~l) be the observed flow after imposing t"" 1 '. The new intended traffic flow and new trial toll for the next iteration k are determined by:

rfw=^-1)+a(t-1)(rf
T

( 1 1

= V ^ d v

vhere OC

—1/2 for the bisection method and CC

= \jk

(21) v*«>

for the method of successive averages.

Figure 1. Comparison of the convergence rates by the bisection method, the method of successive averages and the sequential linear approximation method in Example 1

Sequential experimental approach for analyzing second-best road pricing 35

By equalizing the true demand function and the following marginal-cost function t (v)

F(v) = f(v) + v ^ £ i = l<)[l + O.75f—1 ] W W dv [ UOOJ J

(22)

the socially optimal level of flow, full trip cost and toll charge can be easily identified to be: v" =d" = 216.898; c =20.350; T* = 8.300 The equilibrium flow and full trip cost without tolling are equal to v<0) = dm = 276.581 and cm = 15.489. Starting with this observed flow with the corresponding initial toll charge t(1) = 21.933 calculated by (21), the results by the three methods are discussed below.

Figure 2. Convergence pattern of the sequential linear demand function approximation in Example 1

36 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction As already mentioned, the bisection method does converge for the simple single road case considered here, and is hence applied here for a comparison. For a given tight convergence criterion s = 0.001, the number of trials required by the bisection method, the method of successive averages and the linear approximation method are respectively 5, 18 and 7 (also refer to Figure 1). For this special example, the bisection method has the quickest convergence (but it does not work for a general network), the proposed sequential linear demand function approximation method outperforms the method of successive averages. Note that the relative convergence of the three methods depends on the degree of curvature of the real demand functions. Clearly, if the true but unknown demand function is linear, the linear approximation method will arrive at the optimum with two iterations only. Achieving a fast convergence of the sequential price experimental approach is important, because one may wish to employ as few trials as possible to obtain an acceptable estimation of the optimal congestion tolls in practical applications. The average private cost and the marginal social cost curves are plotted in Figure 2 together with the true but unknown demand curves (dashed). The linear extrapolation and interpolation lines are drawn through the two most recent experiment points (except the initial point which is the observed point without toll charge or in the do-nothing case). The points are numbered in the order that they are identified in the sequential price experimentation. For this simple single road example, the true O-D demand and O-D cost in each trial are observed directly without an estimation process and thus all trial points are located in the true demand curves. As observed from the figure, the sequence of experimental points converges to the social optimum (the intersection between the true demand curve and the marginal social cost curve). Example 2. In this example we use a small but general network to illustrate the application of the trial-and-error procedure for the second-best pricing problem. The network, as shown in Figure 2, consists of 4 links, 3 nodes and 2 O-D pairs (l->3, 2->3). The true demand functions are given below:

gM{cM)

= 500exP(-0.03cM)

ft-, (cM) = 400exp(-0.05c^3) The following link travel time functions are used for the 4 links respectively:

Sequential experimental approach for analyzing second-best road pricing 37

=12 1+0 15

10 1+0 15 4

^> ( - &J} ^>= ( - fe) ) The set of tolled links and observed links are assumed to be identical with two links: A = A = {3,4}-

Figure 3. The network used for Example 2

For given toll charges in each trial experiment, the observed link flows are simulated (generated) by using the elastic-demand traffic assignment method with the true nonlinear demand functions. In view of the small size of the example network, we used an extensive direct search method without derivatives (Hooke-Jeeves pattern search method, see, for example, Bazaraa et al. (1993)) for finding the global optimal solutions of both the bilevel OD estimation and toll optimization problems in each trial. For the purpose of validation, we identified the true optimum solution (but never known in reality in the absence of demand functions) by extensive direct search using the true O-D demand functions. With the known optimum toll solution, we can test whether or not the sequential alternate bilevel O-D estimation and toll optimization algorithm converges to the real optimum. To start the iterative procedure, the initial observed link flows are given as the equilibrium link flows without toll charge (do-nothing case), the initial O-D demand and O-D cost for the first linear approximation of demand function are estimated from the initial observed flows with an initial seed O-D matrix: d^3 = d2^i = 200. The starting point of the toll charge is (T3 0) ,T 4 0) ) = (4.0,4.0). The two weighting factors for the O-D matrix estimation problem

38 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

(14)are set to be (Y 1 ,T 2 ) = (1000.0,1.0), with the directly observed link flows carrying much higher weight. The convergence tolerance is set to be s = 0.01. Figure 4 plots the true optimum for the second-best pricing problem and the convergence pattern of the link tolls identified through the sequential price experimentation using the sequential BLPP methods. The true or exact optimum occurs at (i\,i\ J = (7.95,5.95). The algorithm converges after 6 iterations with a convergent point (T36),T(46)) = (7.84,5.80) which is sufficiently close to the exact optimum.

Figure 4. Convergence pattern of the sequential trial-and-error toll adjustment scheme for the second-best pricing problem in Example 2

Sequential experimental approach for analyzing second-best road pricing 39

Figure 5. Convergence pattern of the sequential linear demand function approximation in Example 2

Figure 5 displays the successive linear approximation to the true, but unknown, nonlinear demand functions. It is observed that, as the trial experiment continues, the linear approximation does become more tangential to the true demand curve near the true optimum points on the demand curve. In contrast to Figure 2 in Example 1, it is observed in this figure that the trial points are not necessarily located on the true demand curves, which are the direct results of the imperfection of the O-D demand estimation sub-problem mainly arising from the incomplete set of link traffic counts. As acknowledged in Remark 5, the instability problem including the positive slope estimates of the linear demand function approximation was indeed encountered when commencing the trial experiments with alternative initial toll charges. It is thus suggested that flow observations should be made on as many links as possible to enhance the accuracy and reliability of the O-D demand estimation, thereby leading to better linear approximations of the true demand curves.

40 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

CONCLUSIONS AND FUTURE EXTENSIONS We developed an encouraging procedure for the trial-and-error implementation of the secondbest congestion pricing on a general road network with unknown demand functions. The procedure involves sequential alternate solutions of the bi-level O-D demand estimation and the bi-level toll optimization problems. The technique presented here has the potential to allow a traffic planner to easily estimate the second-best link tolls from readily available traffic counts without resorting to the demand functions, and thus overcome the practical difficulty in designing a realistic second-best road pricing scheme. What is more, the estimation of the demand functions is in itself meaningful if one wishes to tackle the welfare and the re-distributional effects, both of which are critical to resolve the political issues of congestion pricing. A few interesting extensions are worthy of further exploration to address some behavioral and political issues in the actual implementation of the proposed sequential price experimentation approach. Here we provide a few meaningful examples. The trial-and-error experimental procedure obviates the explicit use of the demand functions. Nevertheless, it does require an accurate estimate of the generalized travel cost and hence the users' value of travel time saving, which is important to calculate the toll amount in actual dolls. Even though it might be easier to estimate users' value of time than to estimate the demand functions and there is indeed much published literature in this area, it is certainly meaningful to simultaneously estimate the optimal link tolls and the users' value of time through the proposed sequential price experimentation, particularly on networks with heterogeneous users. In this regard, the bi-criterion equilibrium models with either a discrete or a continuously distributed value of time over the user population should be useful (Leurent, 1996; Marcotte and Zhu, 2000; Yang and Huang, 2004). Throughout this study we have adopted the simplest functional approximation scheme: a successive linear demand function approximation, which may, as indeed observed in our study, encounter instability problem near the optimum point. It is expected to improve the stability and convergence of the successive function approximation by utilizing a more accurate high-order polynomial with more observed experiment points. For example, it is straightforward to use a quadratic approximation function, which is unique through the latest three experiment points. From an alternative perspective, one may also consider the methods of non-parametric curve estimation well established in econometrics (Efromovich, 1999) to derive the demands from the observed experiment results without the knowledge on the form of the demand function. In addition, as shown in our simple network example, the proposed sequential experimental procedure involves both upward and downward adjustment of the starting and intermediate toll levels, which might be politically unacceptable. It is thus desirable to design a procedure that optimally considers some plausible bounds for the trajectory of realistic toll evolution over time.

Sequential experimental approach for analyzing second-best road pricing 41 Our sequential experimentation procedure to uncover the demand curve is built upon the assumption that changes in demand are provoked by short run changes in price along the same demand curve. The demand revelation mechanism thus requires a series of price changes to derive the demand functions properly. Politically speaking, it is clear that a series of quick changes to a toll system are an unlikely social event, as it would make the newly introduced system unsustainable. A plausible toll adjustment by a willing authority would be in annual, semi-annual or at most quarterly intervals like the situation in Singapore. Nevertheless, such a socially acceptable long inter-trial period could create a behavioral dilemma for the analysis envisioned here. On the one hand, it would allow travelers to have sufficient time to learn and adjust their trip-making decisions in a new pricing environment, although this might not be a critical issue, as advanced information technology is nowadays available and travelers can be quickly informed of altered toll charges. On the other hand, the background variables such as the inflation and real income growth are changing over such length of time. The above issues suggest that the short run - long run differences in behavior as well as the time lag of user responses should be considered carefully in formulating a practical pricing mechanism. The political and behavioral limitations mentioned above could be mitigated in two different manners. First, albeit the price experiments are limited to a few trials and the inter-trial period could be considerably long, the traffic counts are relatively inexpensive to collect with automatic devices, even on a daily basis with good accuracy. These readily and frequently available traffic counts can be utilized effectively to update O-D demands and thus capture the long run demand shift prior to the next price trial. Second, apart from the actually observed behavioral changes, a set of stated preference experiments such as the travel behavior experiments initially suggested by Mahmassani and Chang (1986) can be conducted to analyze the repeated reactions of a set of respondents to price changes, thereby complementing the limited revealed preference experiments.

ACKNOWLEDGMENTS The authors would like to thank three anonymous referees for their helpful comments on an earlier version of the paper. This research described in this paper was substantially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKUST6033/02E).

REFERENCES Bazaraa, M.S., Sherali, H.D. and Shetty, CM. (1993) Nonlinear Programming: Theory and Algorithms. John Wiley and Sons, Inc., New York. Bell, M.G.H. and Iida, Y. (1997) Transportation Network Analysis. John Wiley & Sons, Inc., New York. Downs, A. (1993) Point of view: implementing peak-hour road pricing at fullscale: finding solution to practical problems. TR News 167, 7-9. Efromovich, S. (1999) Nonparametric Curve Estimation: Methods, Theory and Applications. Springer, New York.

42 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Gauvin, J. and Dubeau, F. (1982) Differentiable properties of the marginal function in mathematical programming. Mathematical Programming Study 9, 101-119. Leurent, F. (1996) The theory and practice of a dual criteria assignment model with a continuously distributed value-of-time. hi Transportation and Traffic Theory, ed J. B. Lesort, pp.455-477. Pergamon, Exeter. Li, M.Z.F. (2002) The role of speed-flow relationship in congestion pricing implementation with an application to Singapore. Transportation Research 36B, 731-754. Mahmassani, H.S. and Chang, G.L. (1986) Experiments with departure time choice dynamics of urban commuters. Transportation Research 20B, 297-320. Marcotte, P. and Zhu, D.L. (2000) Equilibria with infinitely many differentiated classes of customers. In Complementarity and variational problems—State of the Art, eds M.C. Ferris and J.S. Pang. SIAM, Philadelphia, PA. Meng, Q., Yang, H. and Bell, M.G.H. (2001) An equivalent continuously differentiable model and a locally convergent algorithm for the continuous network design problem. Transportation Research 35B, 83-105. Vickrey, W. (1993) Point of view: principles and applications of congestion pricing. TR News 167, 4-5. Yang, H. (1995) Heuristic algorithms for the bilevel origin-destination matrix estimation problem. Transportation Research 29B, 231-242. Yang, H. and Bell, M.G.H. (1998) Models and algorithms for road network design: a review and some new developments. Transport Review 18, 257-278. Yang, H. and Bell, M.G.H. (1997) Traffic restraint, road pricing and network equilibrium. Transportation Research 31B, 303-314. Yang, H, and Bell, M.G.H. (2001) Transport bilevel programming problems: recent methodological advances. Transportation Research 35B, 1-4. Yang, H., H. J. Huang. 2004. The multi-class, multi-criteria traffic network equilibrium and system optimum problem. Transportation Research 38B, 1-15. Yang, H. and Huang, H.J. (1998) Principle of marginal-cost pricing: How does it work in a general network? Transportation Research 32A, 45-54. Yang, H., Meng, Q., Lee, D.H. (2004) Trial-and-error implementation of marginal-cost pricing on networks in the absence of demand functions. Transportation Research 38B, 477-493. Yang, H., Sasaki, T., Iida, Y. and Asakura, Y. (1992) Estimation of origin-destination matrices from link traffic counts on congested networks. Transportation Research 26B, 1-18.

3

HIGHWAY SPACE INVENTORY CONTROL SYSTEM

Dusan Teodorovic and Praveen Edara, Charles E. Via Jr. Department of Civil and Environmental Engineering, Virginia Polytechnic Institute and State University, Falls Church, Virginia, USA

INTRODUCTION The number of trips by private cars has significantly increased in recent decades. At the same time, road network capacities have not kept up with this increase in travel demand. This imbalance of demand and supply has resulted in increased travel times, increased number of stops, unexpected delays, greater travel costs, inconvenience to drivers and passengers, increased air pollution and noise level, and increased number of traffic accidents. Expanding traffic network capacities is extremely costly, as well as environmentally damaging. Planners, engineers, and economists have introduced various demand management methods in an attempt to reduce the fast growing traffic congestion (park-and-ride, high occupancy vehicle lanes (HOV) (Giuliano et al., 1990; Yang and Huang, 1999; Dahlgren, 2002), high occupancy toll lanes (HOT) (Abdelghany et al., 2000; Murray et al., 2001), congestion pricing (Vickrey (1963, 1969, 1994), de Palma and Lindsey, 2002; Verhoef, 2002; Yang and Huang, 1999; Yang and Zhang, 2003; Zhang and Yang, 2004; Phang and Toh, 2004)). Successfully planned and implemented demand management strategies can result in significant toll revenues, decrease in total number of vehicle trips, decrease in total number of vehicle trips during peak periods, increase in number of vehicle trips during off-peak periods, increase in ridesharing, rise in public transit ridership, and in some cases increased cycling, walking, and telework. In this paper, a new demand management concept, Highway Space Inventory Control System (HSICS), is proposed and modelled. The basic idea of HSICS is that all road users have to make reservations in advance to enter the highway. The system allows highway operators to make real-time decisions whether to accept or reject travellers' requests to use the highway system in order to achieve certain system-wide objectives. The proposed HSICS model

44 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction consists of two modules - Highway Allocation System (HAS) and the Highway Reservation System (HRS). The HAS is an off-line module and determines the maximum number of trips from each user class (categorized based on time of departure, vehicle type, vehicle occupancy, and trip distance) to be accepted by the system given a pre-defined demand. It develops the optimal highway allocations for different traffic scenarios. The "traffic scenarios-optimal allocations" data obtained in this way enables the development of HRS. The HRS module operates in the on-line mode to determine whether a request to make a trip between certain origin-destination pair in certain time interval is accepted or rejected. The Highway Allocation System is modelled in the form of an integer program with the objective to maximize total travelled passenger miles. A neural network approach is used to model the online HRS. The paper is organized in the following way. Section 2 will present the literature review on highway booking, section 3 will explain in detail the concept and basic characteristics of the proposed HSICS, section 4 discusses the possible impacts of the proposed system, section 5 will present the modelling aspect of the HSICS, section 6 describes the numerical example and the obtained results, and sections 7 and 8 discuss the future research issues and conclusions.

LITERATURE REVIEW Highway booking concept is not a well explored topic in the area of travel demand management. Akahane and Kuwahara (1996) have studied the benefits of trip reservation systems that manage highway traffic demand on holidays based on a stated-preference survey. Wong (1997) proposed a qualitative approach for the highway booking problem. In the paper, he developed a conceptual framework of a highway booking system and discussed the advantages of such a system. Koolstra (1999) studied the potential benefits of slot reservation on highways by analyzing the difference between user equilibrium departure times and system optimal departure times, de Feijter et al. (2004) have proposed trip booking as a method for improving the travel time reliability and increasing the effective usage of road capacity. The proposed system aims at open dedicated infrastructure, such as bus lanes and dedicated freight lanes. Using simulation experiments they further prove the advantages of such a booking system. While all these papers have discussed the advantages and issues related to trip reservation systems, none of them have developed a comprehensive model of highway booking system that will allocate the highway space to potential users during different time intervals and accept/reject user travel requests. In this paper, we make an attempt to formulate the highway booking problem as a combination of offline optimization and online reservation problem.

HIGHWAY SPACE INVENTORY CONTROL SYSTEM: BASIC CHARACTERISTICS The basic idea of the highway-booking concept is that all road users have to book in advance. Airline industry, hotels, car rental, rail, and many other industries are utilizing reservation systems and revenue management concepts when selling their products (Littlewood, 1972; Belobaba, 1987; Brumelle and McGill, 1993; Cross, 1997; and Teodorovic et al., 2002).

Highway space inventory control system 45 Internet provides information on ticket prices to millions of users, and together with ecommerce, it has opened a new stage in the revenue management concepts and practice. The era of dynamic ticket pricing has already begun, opening a lot of complex problems to practitioners and transportation researchers. Revenue management could be described as a group of different scientific techniques of managing the company revenue when trying to deliver the right product to the right client at the right price at the right time. The basic characteristics of the industries to which different revenue management concepts were successfully applied are: (a) variable demand over time; (b) variable asset utilization; (c) perishable assets; (d) limited resources; (e) market segmentation; (f) adding new capacity is expensive, difficult, or impossible; (g) direct cost per client is negligible part of the total cost of making service available; (h) selling products in advance. Revenue management concepts that have been developed in various industries could be used in the development of highway flow management concepts. Let us elaborate on the abovementioned characteristics in more detail in the case of highways. It is well known that transportation flows change over time {variable demand). Changes are noticed by month, by week, by day in a week, and finally by hour in a day. Transportation flow changes over time can be determined by collecting appropriate statistical data, as well as by conducting passengers'/drivers' surveys. The highway space inventories are also perishable. Highway spaces not utilized during some time interval are lost. Highways also have limited resources. The available highway capacity is limited and cannot be expanded in a short time period. There is also market segmentation associated with the highway usage. Possible categories of potential highway users include private cars, transit vehicles, carpools, trucks, low emission vehicles (LEV), vehicles equipped with electronic toll collection, and others. The highway operator could offer to potential users during the same time interval a large number of different tariffs (or assign different priorities) based on a set of defined criteria. This is due to the highway operator's potential desire to achieve the greatest possible number of passenger-miles along the highway, to spread peak over time and space, to accept the greatest possible number of low emission vehicles (LEV) and/or vehicles equipped with electronic toll collection units, to achieve the highest possible total revenue, etc. Highway tariffs could be fixed or they could vary all the time (on an hourly basis). Highway operator would practically face the following dilemma all the time: Will I accept an early reservation from the user making low number of passengers-miles, and/or paying low tariff; or will I wait for a user who will make high number of passenger-miles and/or who is ready to pay higher highway tariff?

46 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In years to come, as the traffic congestion keeps growing, and with the progress made by Intelligent Transportation Systems (ITS), we believe that the highway space inventory control systems will be developed and implemented to manage the highway traffic flows effectively. Reservation requests can be made via telephones or electronically. Driver should specify entry time (within time interval whose width could be 5-10 minutes) on ramp, trip, and off ramp and inform the highway operator (by telephone or electronically). Requests would be usually made several days, hours, or minutes (last-minute request) before the planned trip. In this way, highway operator would know planned driver itineraries, as well as planned departure times. HSICS should explore in real- time, the possibility to accept driver's request. In such a system, drivers making reservations in advance would be protected from unexpected high traffic volumes that often result in increasing the travel time significantly. The assumption is that the highway operator would always accept limited number of driver requests, and in this way keep traffic flows within prescribed limits. When a driver's request is accepted, it means that a space on the highway, during a certain time interval is reserved for the driver.

IMPACTS OF THE PROPOSED SYSTEM Practical implementation of the proposed concepts depends on the reaction from various interest groups. Highway users are not a homogeneous group. They have different socioeconomic characteristics, as well as different reasons to enter the highway at a specific location at a specific time point. Income level, age, educational level, trip purpose, daily commuting, geographical location, and existence of transportation alternatives are the dominant factors leading to the heterogeneity of highway user groups. These groups, depending on their interests, will try to commend or condemn the proposed demand management strategy, debating on concepts of justice, fairness, ethics, traditional societal norms and tradition. Effectiveness of our proposed system can be measured exactly. On the other hand, there is always effectiveness perceived by public. Positive perceived effectiveness is one of the crucial factors for the successful implementation of the highway space inventory control system. Positive perceived effectiveness is among other things, influenced by successfully resolved issues of fairness and equity. One should expect that fairness issues could be the most preferred media's topic. One of the most important tasks in the implementation of the proposed strategy would be to convince various interest groups that highway reservation system is designed to partially mitigate traffic congestion, and not to collect additional taxes and additional revenue for the government and traffic authorities. The strongest opponents of the proposed concepts would be probably the advocates of the "tollfree highways". Members of this interest group claim that the government has the responsibility to supply free service of highways. It has been well known that "toll-free highways" also generate the controversy over equality between highway users and non-users. It is also probable that the public opinion would improve as the highway users experience the real system, as it was the case with the introduction of other well-known demand management strategies. We are not able, in this initial research, to fully answer all political, ethical, and economical issues regarding the implementation of the highway reservation system.

Highway space inventory control system 47

MODELLING OF HIGHWAY SPACE INVENTORY CONTROL SYSTEM The proposed HSICS model consists of two modules - Highway Allocation System (HAS) and the Highway Reservation System (HRS). HAS is an offline module that allocates the highway sections (different Origin-Destination pairs) to various vehicle types (single occupant cars, car pools, public transit, trucks) during different time periods with the goal of optimizing the objective function(s) value(s) subject to the existing supply and demand constraints. It develops the optimal highway allocations for different traffic scenarios. The "traffic scenarios - optimal allocations" data obtained in this way enables the development of HRS. HRS is the on-line system that makes on-line decisions regarding the possibility to accept driver requests. The developed HRS is able to generalize, adapt, and learn based on new knowledge and new information. Proposed HRS is based on neural network training. The neural network learns from the optimal solutions of the analyzed past traffic scenarios. For every vehicle class a separate neural network is trained for making decisions about accepting/rejecting driver requests. The algorithm to model the proposed HSICS consists of the following steps: Step 1: Generate various traffic scenarios. (Collect cumulatives that are based on a large number of drivers' requests on the highway in question). Step 2: Highway Allocation System: Formulate a corresponding integer programming problem and find the optimal allocation for each traffic scenario. Step 3: Highway Reservation System: Based on the statistical data resulting from Steps 1 and 2 train neural networks that will be capable to accept/reject driver's requests in real time. In the following section, the HAS module will be explained in detail. Highway Allocation System (HAS) The Highway Allocation System can be formulated as an integer program. One possible objective of the proposed HAS could be 'maximization of the total passenger miles over a defined time period, for a defined length of the highway'. It is possible to prioritize vehicle types based on some criteria. For every trip, it is possible to calculate the passenger miles travelled during that trip, by knowing the vehicle occupancy and distance covered in the trip. When summed over all the O-D pairs and all time intervals, the total passenger miles travelled on the highway can be calculated.

48 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 1: A typical highway section Different sections of a highway have different utilization percentages, some sections can be heavily congested while some others could have free flow conditions. It is vital for any allocation procedure to check for the time-varying capacity (vehicles/time) of different highway sections while allocating traffic. To carry out such an allocation procedure, the time period is divided into smaller time intervals, and the highway network is divided into a series of links (Figure 1), the road capacity remains same through out the link though it could vary with time. Also a 'trip' is defined as any combination of O-D pair, and Vehicle class. Each possible combination of these two attributes is a unique trip. In the HAS formulation, the objective function is to maximize the weighted (prioritized) total number of passenger miles over a defined time period and a defined length of the highway (Relation (1)). In this model, priorities can be assigned based on trip type (O-D pair and vehicle type). In a more general case, the users can be classified based on time of reservation requests and priorities assigned accordingly. However, in this model we do not consider that aspect. The allocation should be done in such a way that the total number of vehicles using a link during any time interval should not exceed the link capacity (Relation (2)). During any time interval, the traffic on a link consists of vehicles belonging to trip allocations in the current interval and vehicles belonging to trip allocations during earlier intervals that will be travelling on the link during the current interval. We define reserve capacity for every link in order to account for, a) providing access for emergency vehicles that cannot make reservations in advance, and b) uncertainty related to incidents or bad weather conditions. Also, the allocations for any trip should not exceed the corresponding demand value during that interval (Relation (5)). Based on these assumptions the HAS can be formulated as an integer program as shown below.
Maximize F = ^ ^ pru pmt- xu <=i i=i

subject to

(1)

Highway space inventory control system 49

^0^-x^C^R^aeAytET

(2)

;=i 1=1

6i

(1, if Posilt e a '"' \0,tfPosllt*a =

(3)

Posm = Posi0 + vme -St-{t-l)

(4)

xu
(5)

x,v>O,Vie/,V/er

(6)

xu are integer V? e / , V / e T

(7)

where: A 'trip' is defined as any combination of O-D pair and Vehicle class. Each possible combination of these two attributes is a unique trip. i - Trip index value / - S e t of all trips / - Time interval at which vehicles enter the highway t - Current time interval St - Size of time interval T - Set of all time intervals pru - Priority of/01 trip type during time interval' / ' pmi - Passenger miles traveled in the z'th trip xa - Number of vehicles of the f° trip type that can be accepted to enter the highway during time interval' / ' Da - Demand for ith trip to enter the highway during time interval' / ' A - Set of all defined links

50 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Ca - Maximum capacity of link ' a ' Ra - Reserve capacity of link ' a ' Posm - Current position (beginning of time interval 7') of vehicles belonging to i'h trip that entered the highway during time interval 7' Posia - Entry position of vehicles belonging to i'h trip vme- Average speed on the highway The assumption of average speed in the above formulation (Relation (4)) is very simple and does not take into account the traffic flow characteristics; speed does not vary with the current density on the link. This assumption limits the model accuracy in terms of determining the number of vehicles on each link in each time interval. A complex, but more accurate, update of the vehicle positions can be obtained as follows: Posilt=Posi0+

j^vUJ-St

(8)

H

where: j - index for an arbitrary time interval^' > / \/j,l e T (introduced mainly for usage in position calculations) vUj -j'h time interval speed of vehicles belonging to i'h trip that entered the highway during time interval '/' Speed vaj can be expressed in terms of the space mean speed of link 'a', uaj , as:

v w =i^-^

(9)

0=1

The space mean speed on link 'a', uaj, can be expressed as a function of the density on link 'a' using the Greenshields model as:

Kj=ufa\l-^-\

where:

(10)

Highway space inventory control system 51

uaj - Space mean speed of link 'a' during time interval 'j' ufa- Free flow speed of link ' a ' La - Length of link ' a' ^ - D e n s i t y of link ' a ' during time interval 'j' Ka{jam)- Jam density of link 'a' Density of link ' a ' during time interval ' j ' is: I'. ./ / ( / j imaj

im

(11)

Substituting for KaJ in relation (10) we get: I7! '

f

"l

YYfl -x 'maJ

L-^ L-i

(12)

!m

V

)

Using relation (12), relation (9) can be rewritten as:

v

=V

U

. 1 '=' "-1 L

«=1

.ft

(13)

«' A < 1 a«™) )

V

Substituting for vUJ in relation (8) we get: (

PoslU=Posl0+ X E ">

J

1 - ^

)

-O^St

(14)

In the HAS formulation, relation (4) could be replaced by relation (14). All other constraints remain the same.

52 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Speed estimation using density-speed relationship (similar to Greenshields model shown in relation (10)) has been used in the past by Mahmassani and Peeta (1993), and Jayakrishnan et al. (1995), while developing framework for dynamic traffic assignment. Mahmassani and Peeta (1993) have developed a discrete macroparticle traffic simulation model as a part of their dynamic assignment framework. In the simulation model, they assume that all vehicles on a link move at the same speed during a given time interval. Density and speed on each link are updated every time interval, as vehicles enter and leave a link. A similar approach is used by Jayakrishnan et al. (1995) in an optimization framework. They present additional insights into the usage of density-speed relationship rather than flow-speed relationship, including the fulfillment of first-in-first-out requirement and others. The formulated integer program is solved, and the allocations for all trips for all time intervals are obtained. Before we proceed to the next step of modeling the HRS, let us introduce the following notations (Figure 2) which will help for easier understanding of the HRS: Te - Time at which the user wants to enter the highway To- Time at which the advance booking opens (e.g.: 5 days before the scheduled trip departure) Tr - Time at which an user requests a reservation to enter the highway at time Te Tc - Time at which the advance booking closes and no more requests for that trip-departure time combination can be accepted by the reservation system (the time at which all the HAS allocations for that trip-departure time combination have been reserved). T - Advance booking interval is divided into smaller intervals of size 7 each AT = Tc - Tr, the amount of time after which the reservation system will be closed and no more travel requests from users of this trip type will be accepted. Positive value of AT means that the request can be accepted, and negative AT means that the request cannot be accepted. Dr - Cumulative number of requests by Tr Dc - Cumulative number of requests by Tc. It is the total number of HAS allocations for this trip-departure time combination (obtained from the integer program's optimal solution) D, - total number of requests from users willing to depart by Te The above defined variables can be better understood graphically. In Figure 2, the cumulative user requests over time are plotted for the demand related to one trip-departure time combination. The maximum number of allocations that can be done for this trip-departure time combination, Dc, is obtained from the HAS. From the plot, Tc can be obtained (the time corresponding to Dc), which is the time after which no more reservation requests for this trip-

Highway space inventory control system 53 departure time combination can be accepted. Users continue to make requests even after Tc until the trip's departure time Te, though they will be rejected.

Figure 2: Cumulative number of travel requests over time Brief Description of Artificial Neural Networks Artificial neural networks are inspired by biology i.e., they are composed of elements that function similarly to a biological neuron. Neural networks are able to learn from experience, to apply to new cases generalizations derived from previous instances, and to abstract essential characteristics of input data that often contain irrelevant information (Teodorovic and Vukadinovic, 1998). The model of an artificial neuron copied the properties of a biological neuron. The input signals y\, yi, ..., yn, representing the output signals of other neurons, are multiplied by associated connection strengths, w\, w^,..., wH. Each connection strength (weight) corresponds to the strength of a biological synaptic connection. The output signal (NET) is equal to the weighted sum of input signals. The range of the weighted sum of input signals, NET, is compressed by an "S" curve such that the value of the output signal, OUT, never exceeds a relatively low level regardless of the value of NET. The strength of a neural network is obtained as a result of the connectivity and collective behaviour of the neurons. Neural networks store the strength of synaptic connections (weights of the network's branches) by which the required data are reproduced. Learning refers to

54 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction estimating the appropriate weights in order to reproduce certain decisions under particular conditions. In other words, neural networks can be trained by adjusting the connection strengths in order to abstract the relations between the presented input/output data. Training algorithms are very simple mechanisms for adapting the weights of a network's branches. Highway Reservation System (HRS) The Highway Allocation System (HAS) allocates the highway space, off-line, to vehicles belonging to all trip-departure time combinations. This is the supply that is allocated. Users who wish to travel on the highway during any time interval should make reservations in advance. The reservation center could be opened several days before the trip's scheduled departure time, and the users can make requests for reservations either by calling or via internet. When the user makes a request, he/she has to furnish trip details including the preferred departure time, vehicle class used for the trip (single occupant car, carpool, transit, etc.), and the desired origin and destination for the trip (highway entry/exit locations). Using this trip information and the time at which the user is making the request, the operator has to make a decision whether to accept or reject the user request. This task of decision making is trivial in the event that the demand for travel during some time in the future is deterministic. However, it is difficult to predict the future demand patterns in an accurate manner. The operator has to make real-time decisions based on the cumulative requests at that instant, as he is unaware of the future requests. When a user makes a request at time Tr, the cumulative plot is known only until time Tr, so time Tc and time AT can not be calculated. To calculate these values we propose a Highway Reservation System (HRS) that uses data related to past demand patterns in training a group of artificial neural networks (ANNs) that will be used for real-time decision making. We propose one neural network for each vehicle class (see Figure 3). Let us illustrate using an example the procedure of defining the inputs/outputs of the neural network for any one vehicle class (say vehicle class 1). Using vehicle class 1, many trip-departure time combinations are possible. For each such trip-departure time combination there exists a cumulative requests plot as shown in Figure 2, that can be obtained from past data (there is no uncertainty in past data, the operator knows it). Using the HAS module, the maximum number of allocations, Dc, that can be done during any time interval can be obtained. From the cumulative requests plot, the closing time Tc corresponding to demand Dc is obtained. Also, the 'advance booking interval' is divided into 'n' smaller time intervals of size '? each. Assuming that the requests are made at the end of each small time interval (of size 'T'), for each of these requests, the AT value and the corresponding cumulative requests value Dr can be read from the cumulative plots (Figure 2). The inputs for the neural network are the cumulative requests Dr, and the corresponding outputs are the AT values (Figure 2). The number of inputs is equal to the number of trip-departure time combinations for this vehicle class (see Figure 3). Also, the number of outputs is equal to the number of inputs. The same procedure is repeated for several "traffic scenarios" and corresponding cumulatives are generated, this statistical data enables neural network training. Once the trained network is obtained, it can be used for real-time decision making. When the user makes a request at time Tr, then the Dr value is obtained from its cumulative plot (user specific trip-departure time

Highway space inventory control system 55 combination plot), and the Dr values for other trip-departure time combinations (but same vehicle class) are also obtained from their respective cumulative plots. Now this data is entered into the trained neural network and the output A rvalues are obtained. If the output AT value is positive then the operator will accept the user request, else rejects it. In this way, past data can be used for training the neural networks and solve the problem of uncertainty in future demand, and real-time decisions can be made. Same procedure is followed for developing neural networks for the remaining vehicle classes. Since the proposed HRS currently does not exist, the data related to reservation request rates has to be assumed for now, and later on once the system is in place and users start making requests for daily travel, the data can be collected and used in the future.

Figure 3: Neural network for one Vehicle Class

NUMERICAL EXAMPLE The proposed HSICS is illustrated using a numerical example. A three lane highway section consisting of three entry ramps and three exits as shown in Figure 4 is considered.

56 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 4: section of J lane highway considered in the numerical example Let us consider four vehicle classes - Single Occupant Vehicles, Car Pools (vehicle occupancy>l), Transit Vehicles, and Trucks. All the vehicle classes are assumed to have equal priority. The time period for which allocations are done is 7:00 am to 8:00 am. This one hour period is further divided into 15 minute intervals - 7:00 am to 7:15 am, 7:15 am to 7:30 am, 7:30 am to 7:45 am, and 7:45 am to 8:00 am. Four Origin - Destination pairs are considered - (entryl, exit2), (entry 1, exit3), (entry2, exit3), and (entry3, exit3). This means that there are vehicles belonging to 16 different trips that enter the highway during one time interval. Maximum capacity (Ca) and Reserve capacity (Ra) for all links of the highway section are assumed to be equal to 2200 vphpl and 500 vphpl respectively. Capacity available for allocation (Ca - Ra) for 15 min interval for 3 lanes is 1275 vehicles. In this numerical example, for the sake of simplicity, we use a constant speed value for position update calculations (Relation (4)). In the future, we will be developing a numerical example using position update calculation shown in Relation (14) that uses density-speed relationship. Relation (2) consists of variable 0tlal in the summation, which takes binary values based on whether Posm belongs to link 'a' or not. The position Posm as expressed in Relation (14) is a function of the allocations in earlier intervals, meaning that the position Posm cannot be calculated unless the previous intervals' allocations are known. Therefore, the problem turns out to be an implicit, constrained optimization model which is most likely very noisy and has many local optima. It is very difficult to solve the formulation using any of the available optimization algorithms, instead it would be more appropriate to abandon the search for optimal solution and seek a good solution using heuristics. We propose, in the future research, using meta-heuristics such as genetic algorithms or simulated annealing to solve the proposed formulation. In solving this numerical example, first the HAS is formulated as an integer program, and then it is solved to obtain the optimal allocations. This procedure is repeated for several traffic scenarios and the corresponding "traffic scenarios-optimal allocations" are obtained. This statistical data is used to develop the HRS. The HRS consists of four distinct neural networks - single occupant car network, car pool network, transit network, and truck network. As there are four time intervals, four O-D pairs, and four vehicle classes there will be 64 tripdeparture time combinations. For each vehicle class, there exist 16 trip-departure time combinations that will be used in building the neural networks. Several traffic scenarios (the total demand D, for all trips during all four time intervals) are randomly generated using

Highway space inventory control system 57 normal distribution (mean and variance are shown in Table 1). For each scenario, we also need to know the exact time points at which the reservation requests will be made in order to obtain the cumulative requests plots for each trip-departure time combinations. It is assumed that the two successive reservation requests have a time headway that is exponentially distributed (mean value 'X' is calculated using the demand D,, assuming that the advance booking system is opened 5 days before the trip's departure time, and is open for 12 hrs a day). For each scenario, we run the HAS and obtain the optimal allocations Dc and hence the closing time Tc for each trip-departure time combination. In this way we have the complete cumulative plots for all trip-departure time combinations for each scenario. As discussed in the methodology earlier, the 'advance booking interval' in the cumulative requests plot of all trip-departure time combinations is divided into smaller time intervals. In the numerical example used for illustrating HAS, 'advance booking interval' is divided into 25 smaller intervals (reservation system is assumed to be opened 5 days prior to the trip's departure time, and each day is divided into 5 intervals of 2.4 hrs each, starting at 7 am and ending at 7 pm). For each time point Tr, the corresponding cumulative demand Dr (neural network input) and time AT (neural network output) can be obtained from the cumulative plot. Table 1: Mean and standard deviation of demand Interval 7:00-7:15

7:15-7:30

7:30-7:45

7:45-8:00

Vehicle Class

(entl,ext2)

Single Occ Car Car Pool Transit Vehicle Truck Single Occ Car Car Pool Transit Vehicle Truck Single Occ Car Car Pool Transit Vehicle Truck Single Occ Car Car Pool Transit Vehicle Truck

V800 100 20 75 875 110 25 80 950 120 30 90 1000 130 35 100

a 75 20 5 15 75 20 5 15 75 20 5 15 75 20 5 15

(entl,ext3) V1000 75 25 100 1075 85 30 110 1150 100 35 120 1200 110 40 125

a 75 20 5 15 75 20 5 15 75 20 5 15 75 20 5 15

(ent2,ext3) a V750 75 100 20 30 5 75 15 825 75 110 20 35 5 85 15 900 75 120 20 40 5 95 15 950 75 130 20 45 5 110 15

(ent3,ext3)

a 900 75 25 80 975 85 30 90 1050 100 35 105 1100 110 40 115

75 20 5 15 75 20 5 15 75 20 5 15 75 20 5 15

For neural network training, 500 different traffic scenarios are randomly generated using the mean and standard deviation values shown in Table 1. Based on this data, a multilayer perceptron (MLP) network with low level of complexity (only one hidden layer) and full connectivity is developed using Neuro Solutions tool. MLP consists of 16 processing elements (PEs) in the input layer, one hidden layer with 10 PEs, and 16 PEs in the output layer. The number of epochs for which the training is done is 1000, and the transfer function used is tanhaxon. The calculated error backpropagates and the weights are adjusted using gradient descent rule. The learning rule used between the input layer and the hidden layer is based on momentum, with a normalized step size of 0.1, and momentum rate of 0.7. The

58 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction learning rule used between the hidden layer and the output layer is also based on momentum, with a normalized step size of 0.02, and momentum rate of 0.7. The trained network is tested using data that was not used for training (250 new scenarios are randomly generated from the data shown in Table 1); the results of testing are shown in Figures 5, and 6. For every vehicle class two types of plots are plotted (only plots of single occupant cars are shown in this paper). The first type of plot, 'Neural network model output vs Desired output', plots the neural network's predicted output on the Y axis against the desired output of the testing data (desired output is obtained from the optimal solution of the HAS) on the X axis. The data points (driver requests) in the first quadrant have positive value of'AT' for both the neural network model and the desired solution, which means that all these requests will be accepted by the HRS. Same is the case with data points in third quadrant, all these requests will be rejected by the HRS. So, if we view this plot as a way to check the HRS's accuracy in making decisions, then only the data points in second and fourth quadrant are the wrong decisions (see Table 2). The second type of plot shown is the 'Error frequency plot' which is a mere depiction of the frequency of the error in A T value (neural network predicted AT - desired AT). On X axis the error value is shown and on Y axis the frequency of this error is plotted. The results shown in Figures 5, 6, and Table 2 illustrate that the HRS is capable of achieving a very high percentage of correct predictions of the decisions (acceptance/rejection of the user reservation requests).

Figure 5: Neural network model output vs optimal desired output - Single occupant cars

Highway space inventory control system 59 Table 2: Percentage of correct predictions by the Highway Reservation System Vehicle Class Single Occ Car Car Pool Transit Vehicle Truck

Percentage of Correct Decisions (Acceptance/Rejection) 96% 94% 87% 93%

Figure 6: Error frequency plot - Single occupant cars

FUTURE RESEARCH There are still a lot of open questions that should be answered in the future research. The most important of them include, treatment of emergency trips, identification of all market segments, issues of privacy, consequences of rejections on the backgrounds of equity, responses of rejected travellers such as re-requests, and definition of fair pricing/priority mechanisms for various countries and various rural and urban geographical areas. In the current system, a person who does not reserve cannot use the highway. The person then may shift the time of departure, shift the mode, change the route, or completely quit the trip. Including such behaviour shifts in the model, could further assert the validity of the proposed HSICS.

60 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction It would be interesting to model the HSICS in combination with time-varying pricing strategy. For example, a driver needs to pay more if he or she wishes to use the highway facility in most preferable time. Daily commuters who know that they are going to work during every week day might want to reserve the highway space for the entire year in one day. Such trips may be given more priority as they are helping in decreasing the uncertainty in the travel demand for a year. Research related to the amount of priority that needs to be given to the commuters is also an important area for future research.

CONCLUSIONS In this paper, a new demand management concept - Highway Space Inventory Control System (HSICS) is proposed and modelled. The basic idea of the HSICS is that all road users have to make advance reservations to travel on the highway. Such a reservation system is flexible and could be applied during peak hours throughout the highway or only on highly congested sections of highways, or through out the day on highways depending on the requirements and considerations of road users, policy makers, and other relevant stakeholders. The proposed HSICS model consists of two modules - Highway Allocation System (HAS) and the Highway Reservation System (HRS). HAS is an offline module that allocates the highway sections to various vehicle types (single occupant cars, carpools, public transit, trucks) during different time periods with the goal of optimizing the objective function(s) value(s) subject to the existing supply and demand constraints. It develops the optimal highway allocations for different traffic "scenarios". The "traffic scenarios - optimal allocations" data obtained in this way enables the development of HRS. HRS is the on-line system that makes on-line decisions regarding the possibility to accept driver requests. The proposed model is illustrated using a numerical example of a highway section. The optimal allocation is obtained offline for a particular arrival pattern (the solution from the integer programming) and the performance of the proposed neural network can be checked against the optimal allocation. Many tests show that the outcome of the proposed neural network that is making real-time decisions to accept/reject the driver requests is nearly equal to the optimal solution. Further exploration of other objective functions will be necessary for specific application of the proposed procedure. On one hand, problem solving by optimization (or heuristic) technique and neural network training are time consuming procedures. On the other hand, the response time for the trained neural network is practically negligible. In other words, creation of the proposed Highway Space Inventory Control System might be time consuming, but once the system is created, it can be effectively used in real-time decision making. One of the goals of this research is also an attempt to develop new approaches for solving a class of complex real-time problems that are characterized by uncertainty. The proposed system could be treated as an "intelligent" system, because of its ability to recognize different

Highway space inventory control system 61 situations, as well as its ability to make the appropriate decision without knowing the functional relationships in effect between individual variables. The developed system is able to generalize, adapt, and learn based on new knowledge and new information. There are numerous transportation and logistic problems where this research could apply. The proposed concept is especially important for research activities whose unified themes are uncertainty (randomness, stochasticity, and fuzziness) and time-dependence (dynamic, real-time).

ACKNOWLEDGMENT Authors are grateful to the reviewers for their valuable comments and suggestions for the improvement of the paper.

REFERENCES Abdelghany, A. F., K. F. Abdelghany, H. S. Mahmassani and P. M. Murray (2000). Dynamic traffic assignment in design and evaluation of High-Occupancy Toll lanes. Transport. Res. Rec, 1733, 39-48. Akahane, H. and M. Kuwahara (1996). A basic study on trip reservation systems for recreational trips on motorways. In: Proceedings of the 3rd World Congress on Intelligent Transportation Systems, pp. 1-7, ITS America, Washington D.C. Belobaba, P. P. (1987). Airline Yield Management: An overview of seat inventory control. Transport. Sci., 21, 66-73. Brumelle, S. L. and J. I. McGill (1993). Airline seat allocation with multiple nested fare classes. Oper. Res., 41, 127-137. Cross, R. G. (1997). Revenue Management. Broadway Books, New York. Dahlgren, J. (2002). High-occupancy/toll lanes: where should they be implemented. Transport. Res. A-Pol, 36, 239-255. de Feijter, R., J. M. Evers and G. Lodewijks (2004). Improving travel-time reliability by the use of trip booking. IEEE T. Intell. Transport. 5, 288-292. de Palma, A. and R. Lindsey (2002). Private roads, competition, and incentives to adopt timebased congestion tolling. J. Urban. Econ., 52, 217-241. Giuliano, G., D. W. Levine and R.F. Teal (1990). Impact of High Occupancy Vehicle lanes on carpooling behavior. Transportation, 17, 159-177. Jayakrishnan R., W.K. Tsai and A. Chen (1995). A dynamic traffic assignment model with traffic-flow relation ships, Transport. Res. C-Emer., 3, 51-72. Koolstra, K. (1999). Slot allocation on congested motorways: An alternative to congestion pricing. In: Proceedings of Seminar D at the AET European Transport Conference, pp. 343-353, Cambridge, U.K. Littlewood, K. (1972). Forecasting and control of passengers bookings. In: AGIFORS Symposium Proceedings, Vol 12, pp. 95-117. Mahmassani, H. S. and S. Peeta (1993). Network performance under system optimal and user equilibrium dynamic assignments: Implications for ATIS. Transport. Res. Rec, 1408, 83-93.

62 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Murray P. M., H. S. Mahmassani and K. F. Abdelghany (2001). Methodology for assessing HOT lane usage and network performance. Transport. Res. Rec, 1765, 8-15. Phang S. Y. and R. S. Toh (2004). Road congestion pricing in Singapore: 1975 to 2003. Transportation, 43, 16-25. Teodorovic, D. and K. Vukadinovic (1998). Traffic Control and Transport Planning: A Fuzzy Sets and Neural Networks Approach. Kluwer Academic Publishers, Boston. Teodorovic, D., J. Popovic, G. Pavkovic and S. Kikuchi (2002). Intelligent airline seat inventory control system. Transport. Plan. Techn., 25, 155-173. Verhoef, E. T. (2002). Second-best congestion pricing in general networks: Algorithms for finding second-best optimal toll levels and toll points. Transport. Res. B-Meth., 36, 707-729. Vickrey, W. (1963). Pricing in urban and suburban transport. Am. Econ. Rev., 53, 452-465. Vickrey, W. (1969). Congestion theory and transport investment. Am. Econ. Rev., 59, 251261. Vickrey, W. (1994). Statement to the joint committee on Washington, DC, Metropolitan Problems. J. Urban. Econ., 36, 42-65. Wong, J. T. (1997). Basic concepts for a system for advance booking for highway use. Transport Policy, 4, 109-114. Yang, H. and H. J. Huang (1999). Carpooling and congestion pricing in a multilane highway with high-occupancy-vehicle lanes. Transport. Res. A-Pol., 33A, 139-155. Yang, H. and X. N. Zhang (2003). Optimal toll design in second-best link-based congestion pricing. Transport. Res. Rec, 1857, 85-92. Zhang, X. N. and H. Yang (2004). The optimal cordon-based network congestion pricing problem. Transport. Res. B-Meth., 38B, 517-537.

4

PATH SIZE AND OVERLAP IN MULTIMODAL TRANSPORT NETWORKS Sascha Hoogendoorn-Lanser, Rob van Nes, and Piet H.L. Bovy, Delft University of Technology, Faculty of Civil Engineering and Geosciences, Transport & Planning, The Netherlands

INTRODUCTION Overlap in route choice is a well-known problem in travel analysis. Due to overlap, alternatives are no longer independent, while it is unknown how overlap influences traveler's choice behavior. Having overlapping alternatives might be seen as an attractive phenomenon, since travelers have the opportunity to switch between routes during their trip, e.g. in case of incidents. In contrast, the fact that alternatives overlap leads to a statistical correlation between alternatives which should be accounted for modeling route choice behavior. Theoretically, the Probit model is able to model overlapping routes correctly (Bovy 1990; 1996). However, the fact that the Probit model cannot be solved analytically makes Logit models the most used tools for modeling travel behavior. The disadvantage of the basic Logit model, i.e. the Multinomial Logit model (MNL), is that all alternatives are assumed to be independent (IIA-property). Overlapping routes certainly do not meet that requirement. Therefore, in literature different extensions of the MNL model are proposed, ranging from methods focusing on the statistical correlation (e.g. Logit-Kernel) to methods that take overlap into account while maintaining the simplicity of the MNL (e.g. C-Logit and Path Size Logit). In this paper we focus on the Path Size Logit as proposed by Ben-Akiva & Bierlaire (1999) and Ramming (2002). Path Size Logit has a theoretical basis in discrete choice theory, while Ramming showed interesting results for route choice in uni-modal (road) networks. In our study we used Path Size Logit for analyzing pre-trip route choice in multi-modal transport networks under recurrent conditions. We found that in a multi-modal context the units, in which the Path Size is measured, require special attention. Furthermore, the interpretation of the results and a comparison with the results of Ramming, give reasons to review the way the

64 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Path Size is formulated and should be interpreted. A distinction should be made between a Path Size definition that is an approximation for the overlap in a choice set, and a definition that adds an additional behavioral element in the choice model. In the latter case, there appears to be strong asymmetry in the way the overlap between two routes is accounted for. This new component might be seen as an additional decision mechanism in the choice model. An alternative interpretation might be that specific Path Size definitions capture non-linear characteristics in choice behavior. The distinction in two types of Path Sizes also sheds new light on the findings presented by Ramming (2002) and Bekhor et al. (2002). The paper is structured as follows. First a brief classification of methods to account for overlap in route choice models is presented, with special attention to the way Path Size is defined. Next, an analysis of overlap characteristics in multi-modal networks is given, leading to additional measurements of the Path Size. This is followed by an analysis of Path Size Logit models for multi-modal route choice, using various definitions and measurements for the Path Size. The results will be compared with those found by Ramming (2002) for uni-modal networks. The new interpretation of the Path Size will be presented next, followed by a summary of the main conclusions.

MODELING OVERLAP MNL is a useful modeling approach for all kinds of choice processes. With respect to route choice, MNL has a serious drawback: it is assumed that error terms are independent and identically distributed (i.i.d.) Gumbel, leading to the Independence from Irrelevant Alternatives (IIA) property. In route choice, however, alternatives are often correlated. This limitation does not apply to the Probit model in which the error term covariance matrix allows to account for all kinds of correlations. The covariance matrix for a choice situation having n alternatives can be written as:

cov( e )= °» <7Bl

a

*

"

(1)

a]

with £=(Sl,...,Sn)

Gi = variance of route i ay = covariance of routes i undj Since this matrix is symmetrical, this description yields n(«+l)/2 additional parameters to be estimated. In order to simplify the estimation of Probit models the structure of the covariance matrix is often defined beforehand, based on characteristics of the choice situation. Bovy (1990, 1996) suggests that by assuming that

Path size and overlap in multi-modal transport networks 65 the correlation between alternatives only results from physical overlap, the overlap is independent of other route parts, the variance corresponds with the length of the overlap, Equation (1) can be replaced by the following expression, having only a single parameter &.

cov(s)«0-

7

7

7

Za

Zl

'

2

\n

'

(2)

Z

n

With Zt = length of route i Zy = length of overlapping route parts between routes i andy In this case the covariance matrix is indeed symmetrical, since the covariance is defined by the length of the overlap between two routes, which is independent of the order of the routes that are considered. It should be noted that although overlap is intuitively related to distance, the length of overlap might also be measured in different units, such as travel time. In the following section the measurement of overlap in case of multi-modal transport networks will be discussed in more detail. Looking at the extensions of the MNL we observe that all approaches borrow some properties from the way the Probit model deals with overlap. Three types of extensions can be distinguished: 1. models that apply a covariance matrix (e.g. Logit-Kernel); 2. models using a nested choice structure (e.g. Link-Cross-Nested, Paired Combinatorial Logit); 3. models including an overlap factor in the utility function (C-Logit, Path Size Logit). These basic types will be briefly discussed in the following subsections. Including covariance information The Logit-Kernel model was proposed by Walker (2001). Bekhor et al. (2002) and Ramming (2002) describe an application to real size road networks. The error term in the utility function is split into a part related to the correlations, based on i.i.d. random variables with zero mean and unit variance, and a Gumbel distributed error term, as follows: U = PTX + S = PTX + FT^ + U (3) where U = vector of utilities P = vector of unknown parameters X = matrix of explanatory variables F = factor loading matrix T = lower triangular matrix of unknown parameters 4" = vector of i.i.d. random variables with zero mean and unit variance v = vector of i.i.d. Gumbel variables with scale parameter ju

66 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This division in two parts yields the following formulation for the covariance matrix: cov(s) = FTTTFT+

-^ /

(4)

with g = variance of a standard Gumbel variable (n2/6) I = identity matrix In the applications of Bekhor et al. (2002) and Ramming (2002) matrix F is defined as the link-path incidence matrix, while matrix T describes the error variances of the link attributes, which are based on the length of the links. Given these assumptions the first part of the covariance matrix in Equation (4) becomes identical to the matrix defined in Equation (2). This approach has been applied to analyze uni-modal route choice in the Boston area, using choice sets for 188 respondents each having at most 51 alternatives. Results showed that the LogitKernel model based on this specification outperforms the MNL model, and even all other extensions of the MNL analyzed by Ramming: the log likelihood was improved by more than 5%. However, it should be noted that the values and significance of many parameters of the original MNL dropped substantially, hampering the interpretation of the resulting model. Furthermore, the estimation process required substantial computation time (up to 2.5 days), while the stability of the solutions is a point of concern. It can therefore be concluded that the Logit-Kernel model certainly is promising, but needs further research to be applicable for route choice problems.

Nested approaches The second approach uses the concept of choice hierarchy: travelers are assumed to choose first a set or nest of alternatives and subsequently choose an alternative within that set. In route choice problems, however, it is usually not possible to distinguish crisp sets of routes: routes overlap with many other routes (complex overlap patterns). Two approaches suited for route choice modeling can be found in literature: Cross Nested Logit (CNL) and Paired Combinatorial Logit (PCL). CNL allows an alternative to belong to multiple nests by using a weight factor describing the extent to which it belongs to a certain nest. Vovsha & Bekhor (1998) present CNL for route choice problems, where choice probabilities are given by:

XK-exp(^))i(EK.-exp(^))-f p

P(i\C.) = Z (i\C~YP(C~.\C,) = -

7

—

fy

—

t Z(v«p(»o)H »=i UeC™

)

with P(i\Cn) = probability of alternative i given choice set Cn Cn = set of alternatives Cmn = subset of Cn of alternatives that are included in nest m ami = allocation parameter expressing extent to which route i belongs to nest m jum = scale factor for nest m

(5)

Path size and overlap in multi-modal transport networks 67 M = number of nests Vovsha & Bekhor (1998) define a nest for each links in the choice set (upper level) with the routes being at the lower level. The allocation parameters for a nest are then defined as: a*=~** z

(6)

i

with za = length of link a Zj = length of route i Sai = link-route incidence indicator (1/0) It should be noted that this formulation for aai implies a kind of asymmetry: in case of two routes having one link in common the weight attached to each of these routes differs because of differences in route length. In case that the routes have similar lengths, the impact of asymmetry will be limited. It is obvious that the resulting nesting scheme is very large and complicated, making the estimation of CNL models computationally cumbersome. Application to the Boston area dataset showed that CNL model significantly improves the MNL model: an increase in log likelihood by 1.6% (Ramming 2002). Finally, it should be noted that from the traveler's perspective, the nesting does not look very realistic. It is not likely that travelers choose links prior to the decision which route to take. The PCL model defines all possible pairs of routes within the choice set (Koppelman & Wen (2000)) to be nests. The probability of route i thus depends on the attractiveness of route i and that off all pairs containing route /:

with rjij = similarity index between alternatives i andj The similarity index can be defined as: y]zi+zi

Please note, that the definition for rjy is symmetric for each pair of routes. Just as the CNL, the nesting structure is excessive, making it difficult to estimate. The PCL model is simple to implement, as can be seen by the application of PCL to multi-modal route choice by Benjamins (2001) and Benjamins et al. (2002). Finally, the behavioral perspective of PCL is unclear. In case of large numbers of alternatives, travelers are not assumed to be able to compare all pairs of alternatives systematically.

68 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Additional utility component The third category of extensions known to cope with overlap tries to maintain the simplicity and effectiveness of the MNL model. This is done by adding a component to the utility-function describing somehow the overlap with the other alternatives in the choice set. Cascetta et al. (1996) propose the C-Logit model in which a commonality factor CF is introduced: exp(F.+CF. ) J£Cn

Cascetta et al. (1996) present four definitions of the commonality factor, and used the second one to model intercity route choice for truck drivers. These definitions are:

f zY CFin = -Po In Y — ^ -

(10-1)

CFb=-0o-\1iY.TN»

(10 2)

"

aeT, A'

CFL—yVZ^rM^)

CF

"-PA1+

ihfrlf^]]

(10-3)

(104)

-

where Po, Y = coefficients to be estimated Pi = set of links in path i Nan = number of alternatives of set n using link a Only the formulation given in Equation (10-1) is symmetric, that is, the order routes i and/ are considered does not influence the value of the commonality factor. In the other cases differences in route length will lead to an asymmetry. The lack of theoretical guidance on the definition of the commonality factor, is reason for Ben-Akiva & Bierlaire (1999) to propose the Path Size Logit (PSL). The notion of Path Size is based on the application of discrete choice theory for aggregate alternatives, such as might be used for destination choice modeling (Ben-Akiva & Lerman (1985)). Ramming (2002) presents the basic Path Size model as:

exp(K. +p-\n(PS. ))

Xexp^.+ZMn^,)) where P

Sin=Y(— ] —

(12)

<«r, Vz i J 7V ™

The Path Size factor (PS) varies between 0 (full overlap) and 1 (no overlap or unique route), which makes it easy to interpret. Please note that compared to the commonality factor the natural logarithm and the weight /? are switched from the factor definition to the Logit speci-

Path size and overlap in multi-modal transport networks 69 fication. The Path Size factor also is asymmetric, because its value depends on the length of each route. Ben-Akiva & Bierlaire (1999) and Ramming (2002) propose an extension to the Path Size factor in order to prevent long routes having a low probability yielding a large amount of overlap for attractive short routes, which would make them less attractive. The Extended Path Size factor is defined as: EPS- = Y —

(13)

5z

' 'x(ff. with EPS = extended Path Size factor X = Size Assignment Parameter (SAP) The SAP works as follows. If route i is the shortest route, the ratio of the route lengths will be smaller than 1 for all routes except itself. If the SAP is high, the summation of the ratios will tend to go to 1, thereby assigning the size of the overlap related to link a fully to the shortest route. For all longer routes using link a, the ratio of the route lengths will be larger than 1. Combined with a high value of the SAP, this will result in assigning no size at all. Ramming (2002) found that for the Boston area data, high values of the SAP (his analysis suggested infinity) yielded best results and outperformed the C-Logit models: a log likelihood improvement of 4.3%. Discussion There appears to be two characteristics in which the methods differ: whether the overlap is expressed at route level (Zy) or at link level (za); whether there is symmetry in the overlap factors or not. Table 1: Summary of modeling approaches dealing with overlapping alternatives Category

Modeling approach

Covariance information

Probit Logit kernel Cross Nested Logit Paired Combinatorial Logit C-Logit

Nested approaches Additional utility component

Path Size Logit Extended Path Size Logit

Overlap expressed at route or link level Route level Route level Link level Route level Route (10-1,10-4)/link (10-2,10-3)level Link level Link level

Symmetry Yes Yes No Yes

Yes (10-1)/No No No

Table 1 summarizes these characteristics for the methods described. If overlap is expressed at route level, there is usually symmetry in the corresponding overlap factors. The only excep-

70 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction tion to this rule is fourth definition of the commonality factor of the C-Logit model (Equation 10-4). For the link-based methods, the overlap factors usually depend on the length of the route that is considered. In case of overlap, the same link may therefore lead to different overlap factors for the routes involved. However, when the differences in route lengths are small, the asymmetry will be of minor importance. Furthermore, link-based methods may include a weight for the number of routes using the link, which is not relevant for route-based methods. All methods described use the length of overlapping route parts as the basis in their analyses, be it as an estimate of the covariance matrix, as the weight of the nests, as the similarity parameter, as the commonality factor, or as the Path Size factor. The latter factors can be seen as a kind of summation of the rows of the covariance matrix given in Equation (2), while the weighing with the number of alternatives containing the same route parts is a proxy for the information in the column of the covariance matrix. Both methods can thus be seen as approximations of the information in the covariance matrix. The inclusion of the overlap reduces the attractiveness of routes having an overlap. For C-Logit and Path Size Logit this mechanism is obvious. In the CNL model the inclusion weight aai reduces the probability of route i in a similar way as the Path Size factor, as can be seen from the following equation: « o ,.-exp(^) = exp(^+In («„,.))

(14)

Similarly, in case of overlap the similarity index rjy in the PCL model also reduces the attractiveness of both alternatives i and_/. The introduction of the SAP in the Extended Path Size factor (Equation (13)), however, requires additional attention. First of all, the argument that long routes might excessively penalize shorter routes with respect to the Path Size factor, is also an assessment of the quality of the choice set. It might be argued that properly constructed choice sets should not contain routes that are substantially longer with respect to important variables in the utility function, since these are likely to be fully dominated routes. Second, and more important, the use of the SAP essentially changes the way the Extended Path Size factor works. Whereas the basic definition of the Path Size factor may be seen as an approximation of the covariance matrix, in the extended version this is no longer true. It is generally assumed that the structure of the covariance matrix is symmetric. The fact that in the Extended Path Size factor the size of the overlap is primarily, and if SAP equals infinity even fully, assigned to the shortest route, suggests an asymmetric covariance structure, or, more likely, a different mechanism related to overlap that explicitly favors the shortest route. In the following sections we will explore both the original Path Size factor as well as the Extended Path Size factor for route choice modeling in multi-modal networks, and show that the Extended Path Size indeed incorporates a different mechanism related to choice behavior and overlap.

OVERLAP IN MULTI-MODAL NETWORKS

Defining and measuring overlap In road networks the definition of overlap appears to be straightforward. Usually it is based on the 'length' of overlapping paths, either in time or in distance. In contrast to road networks,

Path size and overlap in multi-modal transport networks 71 however, route choice in multi-modal trip making has multiple dimensions, such as the time dimension due to the inclusion of scheduled transport services, and the distinction of the various modes and services that might be used. Furthermore, the nodes themselves, such as boarding, alighting and transfer nodes, may have important properties of their own, e.g. shops and travel facilities. Therefore, there are many possibilities for defining overlapping components relevant for route choice modeling: nodes (boarding and alighting stops and transfer locations); modes (private (walk, bicycle, car) or public transport modes (bus tram, metro, train)); services (local, express, intercity trains); paths (common public transport lines); links (roads, railways); runs (including the time dimension). From a behavioral perspective, routes having one or more of these components in common are not independent of one another. They share common unobserved aspects that are related to those specific components. Each option leads to different consequences for route choice modeling. Since travelers usually describe multi-modal routes as a set of legs, e.g. bike and train, we will apply a similar definition in our study. A leg then is defined as a part of a trip between two nodes in which a single mode or service type is used (without transferring). The only exception to this rule are walk access and egress to and from urban public transport, which are considered to be part of a urban public transport leg. Please note that this definition of a leg incorporates the first three options for overlapping components mentioned above: nodes, modes and services. More detailed descriptions, such as based on links and runs are left for further research. Compared to the formulae above, such as Equations (12) and (13), link a should be replaced by leg a. For the way the length of a leg is measured we will analyze three different measurements: occurrence of a leg in an alternative; travel time; i.e. expected (private modes and waiting) or scheduled travel time (invehicle times public transport). Using actual travel times would have the theoretical disadvantage that overlap might vary as travel times vary over time for instance due to congestion or to service irregularities; distance. Overlap characteristics of multi-modal route choice sets For the analysis of multi-modal travel behavior a survey has been conducted for train travelers in an urbanized corridor in The Netherlands (Hoogendoorn-Lanser (2005)). The survey focused on a (multi-modal) trip (modes used, transfer nodes, boarding and alighting nodes) and the alternatives known by the traveler. The survey data was extended with detailed data on all trip components, such as travel time and costs, as well as with similar data for all other reasonable non-reported alternatives for the same trip. In total, the dataset contains detailed trip data for 708 respondents. The number of legs of an alternative ranges from 3 to 6. The number of alternatives known by the traveler is limited, often only 1, while the maximum

72 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction number is 7. The number of reasonable alternatives, however, is very large. For one respondent even a total of 278 alternatives was generated. These reasonable routes were generated using a choice set generation algorithm that is combinatorial and generates an exhaustive set of alternatives, albeit within a set of temporal and behavioral constraints. Temporal constraints define the time band for alternatives relative to the timing of the chosen trip, whereas behavioral constraints refer to minimum and maximum values for distances, travel times, modal combinations, transfer characteristics and the like are derived from the Dutch National Travel Survey1. These constraints are further optimized using observed trips and reported subjective choice sets. As a result, generated choice sets include both chosen trips and reported non-chosen alternatives. A more detailed description of the data collection and the dataset can be found in Hoogendoorn-Lanser & Van Nes (2004) and in Hoogendoorn-Lanser (2005). For this data the overlap was determined according to the definitions given above: legs, travel time, and distance. For computational reasons, the maximum number of alternatives considered in this analysis is limited to 50 alternatives (that is 49 random draws from the full set of alternatives plus the chosen alternative). The average choice set size is 33 alternatives. The number of alternatives with which an alternative overlaps appears to be quite large. On average an alternative overlaps with 23 other alternatives with 16.5 alternatives having a single overlapping leg and 5.5 alternatives having two overlapping legs. These high numbers of overlapping alternatives suggest that the number of unique alternatives is relatively small. The number of overlapping legs thus is often restricted to one or two legs. Given the fact that most trips consist of three legs, the overlap is usually 1/3 or 2/3 of an alternative. Figure 1 shows the cumulative distributions of the number of overlapping alternatives for the main trip parts and for the main modes and services. The steeper the curve, the more unique that part or that mode or service is. With respect to overlap patterns it thus appears that: home-end legs are more frequently common in alternatives than activity-end legs and train legs: especially car legs are more often common than cycling and walking legs, while tram, bus and metro legs have equal degrees of commonality; Intercity train service legs are more common in alternatives than express train and local train service legs. The explanation for this pattern is that 'fast' legs, such as bike and car for access and egress parts and Intercity train services for the train part, offer more possibilities for combinations with alternative legs for the other parts of the trip, without yielding unreasonable alternatives, such as alternatives having excessive travel times. For instance, the total travel time of the combination consisting of urban public transport for the home-end and activity-end parts of the trip combined with local train services, is often too large to be considered as a realistic alternative. The dominant role of the home-end leg results from the fact that at home-ends of trips more 'fast' modes are available than at activity-ends. Please note, that these findings indicate that overlap in a multi-modal route choice set is likely to occur in case of attractive route parts. The differences in overlap among routes are clearest if the overlap figure is based on number of legs. Differences become smaller if other expressions are considered, especially The Dutch National Travel Survey is conducted by CBS, which collects travel data for more than 70.000 households annually, resulting in 600.000 trip records.

Path size and overlap in multi-modal transport networks 73 in case of distance. This suggests that the number of legs might provide the best insight into the influence of overlap on travel behavior. Furthermore, it should be noted that the alternatives vary in length when overlap is measured in terms of number of legs or travel time, while having equal lengths if based on distance. This can be seen in Figure 2, which shows the distribution of the ratio between route length and the median route length of the corresponding choice set. This finding implies that in the latter case only the basic Path Size factor should be used. For the first two definitions the Extended Path Size factor might be appropriate.

Figure 1: Cumulative distributions of number of overlapping routes for trip parts, train service types, private modes and public transport service types respectively (708 respondents and 24,300 alternatives).

74 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2: Distribution of ratios of route length and median route length of the corresponding choice set, where route length is expressed in number of legs, time, or distance respectively (708 respondents and 24,300 alternatives).

PATH SIZE IN MULTI-MODAL ROUTE CHOICE MODELING

Path Size factor characteristics In this section we will show some characteristics of the Path Size factor, and especially look at the impact of the SAP in the extended Path Size factor. Path Size factors analyzed in this section are computed expressing route length in terms of number of legs. Figure 3 shows the distribution of the mean Path Size value per choice set as a function of the choice set size. As expected the mean Path Size factor decreases with choice set size. A similar effect can be seen for the minimum. For the maximum Path Size value the same type of graph looks scattered, but the there is still a descending pattern. In this case, however, the pattern is less clear due some relatively unique alternatives, which include legs with local train services. In the Extended Path Size factor the SAP influences the Path Size values. The impact on the mean Path Size value, however, is small. An increase of the SAP leads to higher values for the Path Size factor, especially when the choice set size increases.

Path size and overlap in multi-modal transport networks 75

Figure 3: The relationship between mean Path Size (expressed in terms of number of legs) per choice set and size of choice set (708 respondents) (SAP=Q). Table 2: Path Size (expressed in terms of number of legs) statistics of routes for different Path Size definitions.

Mean value

Standard deviation

Chosen routes Subjective non-chosen routes Objective non-chosen routes Chosen routes Subjective non-chosen routes Objective non-chosen routes

Path Size SAP=0 0.202 0.185 0.183 0.145 0.114 0.103

Extended Path Size SAP=infimty SAP=\A 0.252 0.253 0.214 0.213 0.189 0.189 0.165 0.163 0.127 0.126 0.120 0.118

Since the (Extended) Path Size factor is included in the utility function using a logarithm, a higher value of the Extended Path Size factor would imply that its influence would be smaller. However, when we look at the impact for specific types of alternatives, we get a better view of what the influence of the SAP is. Table 2 shows the mean and the standard deviation of the (Extended) Path Size factors for chosen alternatives, non-chosen subjective alternatives (i.e. reported alternatives), and remaining non-chosen objective alternatives (i.e. generated alternatives). Introduction of the SAP leads to an increase of the (Extended) Path Size values: for chosen alternatives the increase is more than 25%, while for the remaining non-chosen objective alternatives the increase is only 3%. The Extended Path Size factor thus tends to favor the chosen alternatives. Furthermore, the variation of the (Extended) Path Size factor values increases 10% to 15%, thus increasing the variation in utilities as well. These findings indicate that although mean (Extended) Path Size values increase, the use of the Extended Path Size factor might lead to better estimation results.

76 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Modeling approach The impact of the (Extended) Path Size factor on route choice modeling is analyzed by including both Path Size definitions, measured in number of legs, time, and distance in the best MNL model that could be established for the dataset (Hoogendoorn-Lanser, 2005). This model makes a distinction between home-end parts, train parts, and activity-end parts of trips, reflecting the theoretical notion that tours are better suited for choice analysis in multi-modal networks. In this way the difference in availability of modes and in knowledge of the transport system can be explicitly accounted for. The best MNL model consists of 25 parameters, nearly all of them are significant at a 95%-confidence level, describing mode-specific constants for home-end and activity-end, and parameters for in-vehicle times per mode, train types, railway station types, costs for urban public transport (UPT) and parking, UPT headway, walking time to UPT-stops, and number of high- and low-frequency legs (the scaled parameter values are shown in the Appendix). The analysis focuses on model performance, i.e. final log likelihood, and the value of the parameter for the (Extended) Paths Size factor /?. Ben-Akiva & Bierlaire (1999) and Ramming (2002) state that following discrete choice theory p should have the value 1. However, as we have seen earlier the (Extended) Path Size factor is an approximation of the correlation between alternatives. This is certainly true for the Extended Path Size factor. Furthermore, travelers might have a positive or negative perception for overlapping route parts. Both arguments imply that ft should not be set to 1 (see also Hoogendoorn-Lanser et al, 2005). hi the analysis we will analyze both situations.

Modeling results: Path Size model In this section we will focus on the Path Size factor defined in Equation (12), which can be seen as an approximation of the covariance matrix. The results of the choice modeling can be seen in Table 3. Table 3: Modeling results for the Path Size Logit model. Model

Unit of overlap

/7 estimated or set to 1

Log likelihood

PS-6 PS-7 MNL PS-1 PS-2 PS-8 PS-3

Distance Time None Distance Time Number of legs Number of legs

1 1 02 0.211 1 0.55

-1656.72 -1634.95 -1616.39 -1616.39 -1615.05 -1613.92 -1608.10

Difference in log likelihood compare!i to MNL Relative Absolute -2.5% -40.33 -18.56 -1.1% Reference Reference 0.0% 0.00 1.34 0.1% 0.2% 2.47 8.29 0.5%

different from 0 at a 90%-confidence level P is estimated, but its value turned out to be not significantly different from zero, and thus PS-1 collapses into the MNL-model.

A first observation is that if the Path Size factor is expressed in time and distance, the impact on model performance is small, and in case ft is set to 1 the impact is even negative. Appar-

Path size and overlap in multi-modal transport networks 77 ently, overlap measured in time or distance does not match the way travelers perceive overlap in multi-modal route choice. However, if the Path Size factor is expressed in number of legs, there is an improvement in model performance, albeit small. It appears that J5 should be estimated and has a value 0.55, which is significantly different from 0 as well as 1. This low value means that the Path Size factor overestimates the correlation as it is perceived by travelers. The improvement of the log likelihood is limited to 0.5%, however, it is even higher than the improvement found by Ramming. Furthermore, nested hypothesis testing shows that the improvement in log likelihood value in the Path Size model is significant. In case that no SAP is applied and the Path Size factor is expressed in free flow travel time the improvement he found was 0.2%, while for two C-Logit definitions the increase was only 0.1% and 0.3% (using Equations (10-1) and (10-4) respectively). Modeling results: Extended Path Size model The Extended Path Size Logit model is applied for Extended Path Size factors expressed in number of legs and time. Since the differences in route length measured in distance are minimal (see Table 4), the SAP would have a minimal effect. Table 4 shows the results for both expressions for the optimal value for SAP, which is equal to 14 and 0 if the Extended Path Size factor is measured in number of legs and time, respectively. As references the results of the MNL and best Path Size model (PS-3) are included. It can clearly be seen that using a high value for the SAP, while expressing the Extended Path Size in time, leads either to an inferior model or to a /?that is not significantly different from 0. When Extended Path Size is expressed in time and j3 is set to one, the optimal log likelihood value is obtained when SAP is equal to 0, thus for a Path Size model instead of an Extended Path Size model. When Extended Path Size is expressed in time and /? is estimated, /? decreases from 0.21 (SAP=0) to 0 (&4P=oo). Table 4: Modeling results for the Extended Path Size Logit model. Model

Unit of overlap

P estimated

or set to 1

SAP

Log likelihood

Difference in log likelihood comparec1 to MNL Relative Absolute -

-00 00 Time 1 None Reference -1616.39 Reference oo Time 02 0.0% 0.00 -1616.39 Time 1.34 0.21' 0.1% 0 -1615.05 Number of legs -1608.10 0.55 0 8.29 0.5% 1 Number of legs 14 21.79 -1594.60 1.3% Number of legs 22.63 0.84 -1593.76 14 1.4% different from 0 at a 90%-confidence level 2 p is estimated, but its value turned out to be not significantly different from zero, and thus PS-4 collapses into the MNL-model.

PS-9 MNL PS-4 PS-2 PS-3 PS-1O PS-11

78 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In case of an Extended Path Size factor measured in number of legs, however, a substantial improvement (according to nested hypothesis testing) is achieved. Again, setting /? equal to 1 leads to a poorer performance. The best performance is achieved for SAP equaling 14 (determined after optimizing SAP), while jB is estimated. Apparently, higher values of the SAP overestimate the impact of overlapping alternatives. As can be seen in Figure 4, the variation in model performance is small for SAP-values greater than 10. In case of time, no optimal value for the SAP could be determined: the higher the value of SAP, the worse the performance. It should be noted that this finding is different from the findings by Ramming, who found that a value for the SAP of infinity yielded the best results. This difference is probably the result of the difference in the way in which the Extended Path Size factor is expressed. In case of free flow travel time, the variation in the ratio of route lengths for uni-modal networks will be small, compared to the variation in the ratio of route lengths expressed in number of legs for multi-modal networks. Including the Extended Path Size factor in the utility function has only a limited influence on the other parameters in the model (the complete modeling results for MNL, the Path Size factor, and the Extended Path Size factor for SAP=14 are included in the Appendix). Figure 5 shows the scaled parameters of the (Extended) Path Size logit models compared to those of the MNL model. Introducing the Path Size factor has the largest impact on the mode specific constants for the private modes and for bus and tram, which are reduced by about 10%. The Extended Path Size factor, however, also has a substantial impact on the parameters for the number of high and low frequency legs, which are reduced by more than 20%. A similar effect can be found in the results of Ramming. Introducing the Extended Path Size factor reduces the value of the parameters of variables associated with the units in which the Extended Path Size factor is measured. Finally, it should be noted that analyses with different choice sets yielded similar results.

Figure 4: Final log likelihood as a function of the Size Assignment Parameter of the Extended Path Size Logit model.

Path size and overlap in multi-modal transport networks 79

Figure 5: Relationship between parameter values of the MNL model and the (Extended) Path Size Logit models.

INTERPRETATION OF PATH SIZE MODELS Path Size The discussion on approaches dealing with overlap in Logit models suggested that the Path Size factor is an approximation for the covariance matrix. The application of the Path Size factor for multi-modal route choice showed that the modeling performance can be improved by introducing a Path Size factor, but the size of the improvement is limited to 0.5% at most. For uni-modal networks Ramming (2002) found similar results with improvements in the log likelihood of 0.3% at most, while the application of two other approaches showed clearly better results: 1.6% for CNL and 5.1% for Logit kernel. Both latter approaches take the overlap more fully into account than Path-Size logit or C-Logit. Hoogendoorn-Lanser (2005) also applied a Generalized Nested Logit model (Wen & Koppelman (2001)) for the analysis of route choice in multi-modal networks. In the GNL a distinction is made between public and private modes for both the home-end and the activity-end of the trip, which resulted in an improvement of the log-likelihood of 2.0%. Again the Path Size-Logit is outperformed. We conclude therefore that the Path Size factor accounts for the overlap in uni-modal and multimodal route choice in a limited way, while better modeling approaches are available. Furthermore, it should be noticed that the way in which the Path Size factor is expressed is clearly different for both applications. For multi-modal route choice, the Path Size factor should be measured in number of legs, while for uni-modal networks free flow travel time is more appropriate. However, habitual or perceived travel times might be more appropriate explanatory variables for route choice under congestion than free flow travel times.

80 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Extended Path Size The main reason for the introduction of the SAP in the Extended Path Size factor is the existence of long unattractive routes that might penalize attractive short routes too much. It stated before this reasoning also is related to the type of choice set. In our multi-modal analysis as well as in Ramming's analysis of uni-modal route choice, an extensive search for reasonable alternatives has been performed. Interestingly, both studies have choice sets of similar size of a maximum of 50 and 51 alternatives respectively. The results of both studies are therefore comparable in a certain way. Both the applications to multi-modal route choice and to unimodal networks show that the Extended Path Size factor yields a substantial improvement of model performance: 1.4% and 4.3% respectively. The difference in impact of the Extended Path Size factor might be explained by the fact that the models for the multi-modal case have more detailed utility functions containing twice as much parameters, which makes it more difficult to improve them. Furthermore, there is another difference between the multi-modal and the uni-modal case. For multi-modal networks an optimal value for the SAP could be established, while for uni-modal networks a value of infinity appeared to be optimal. The mtroduction of the Extended Path Size factor in the utility function leads to a drop in the value of the parameters related to the way the Extended Path Size is expressed. Apparently, the Extended Path Size factor captures part of the explanatory power of the related parameters. Since the Extended Path Size tends to penalize alternatives that are long in terms of the units in which the Extended Path Size is expressed, it is plausible that the Extended Path Size captures part of a non-linear relationship with respect to the number of legs for multi-modal route choice or free flow travel time for uni-modal route choice. A phenomenon found by Ramming is that the inclusion of the Extended Path Size factor in the CNL model and in the Logit-kemel model also improved the model performance: up to 4.9% and 7.0% respectively. In our discussion on approaches for accounting for route overlap, we stated that the Logit-Kernel takes overlap best into account. Furthermore, our analysis showed that that the Path Size factor does not properly account for the overlap in uni-modal and multi-modal route choice. The improvement of the performance of the Logit-Kernel (and for the CNL model as well) thus indicates that, for uni-modal route choice, the Extended Path Size factor captures a different phenomenon. For multi-modal route choice this finding could not yet be corroborated. Please note, that in the multi-modal case the specification of the nesting structure (private mode and public transport service types for the home-end and the activity-end of the trip) is strongly correlated with specification of the Extended Path Size factor, which complicates the combination of both approaches. If it is assumed that the Extended Path Size describes something else than the overlap of routes, the question then is what the Extended Path Size stands for. Please recall that the introduction of the SAP introduces a kind of asymmetry in the choice model. In case of overlapping routes, long routes are penalized by assigning a small Path Size, while short routes are favored by assigning a large Path Size. The analysis of the Extended Path Size factors above shows that this mechanism increases the differences between chosen and non-chosen routes. In case of multi-modal route choice, where the Extended Path Size is expressed in number of legs, this line of reasoning seems plausible. The chosen route in terms of number of legs is often 3 legs long. Routes having a longer length also have more transfers. Since it is well known that travelers dislike switching between modes or transport services, longer routes are thus

Path size and overlap in multi-modal transport networks 81 unattractive from the traveler's point of view. This also explains the high values for the parameters related to the number of high and low frequency legs in the basic MNL: -20.55 minutes and -24.70 minutes of train in-vehicle time, respectively. The difference between the MNL model and the Extended Path Size-logit model is that the MNL model penalizes all routes for having a number of legs, while the Extended Path Size factor only penalizes longer routes when it is sure that a shorter route exists. It might be stated that the Extended Path Size factor introduces a kind of if-then else decision rule for overlapping routes: shortest routes (i.e. in the units that Path Size is measured) gets full size, while all longer routes are penalized by a small size and thus a large disutility. Please note that such a decision rule is certainly defendable from a behavioral perspective, however, it is something else than accounting for the overlap. This mechanism also shows why an Extended Path Size in travel time did not lead to substantial improvements: transfers are considered to be more important from the traveler's point of view.

CONCLUSIONS This paper explores the way overlap is accounted for in route choice modeling, focusing on Path Size-Logit and multi-modal route choice. Different approaches accounting for overlap are discussed, showing similarities and differences. The analysis of Path Size Logit showed that its interpretation needs to be revised. A distinction should be made between the basic Path Size Logit, which can be seen as approximation of the error covariance matrix, and the Extended Path Size Logit, which introduces a new decision mechanism in choice models. In the Extended Path Size Logit a Size Assignment Parameter (SAP) is introduced that in case of overlapping routes penalizes longer routes and favors shorter ones. In case of multi-modal route choice (Extended) Path Size factors can be expressed in different ways, such as distance and time, but also in number of legs, in which a leg is determined by beginning and end node and the modes or transport services used. The latter definition proved to be most suitable for multi-modal route choice modeling. The inclusion of (Extended) Path Size factors in the choice model improves the performance of the choice models. However, in case of the basic Path Size factor the impact is limited to 0.5% improvement of the log likelihood at most. For the Extended Path Size factor the improvement is substantially larger: 2.0% up to 4.3%. In case of multi-modal route choice an optimal value for the Size Assignment Parameter could be established. The limited impact of the basic Path Size factor compared to the improvements obtained with more advanced methods, such as Cross-Nested Logit, Logit-Kernel, and Generalized Nested Logit, indicates that the Path Size factor only compensates for a small part of the correlation due to overlapping alternatives. The Extended Path Size factor does not so much correct for the correlation between alternatives, but introduces an additional decision mechanism in the choice model. The analysis showed that if the Extended Path Size factor is expressed in the right units, it penalizes long routes in a selective way, that is, only when shorter routes exist. As such, it can be regarded as a kind of if-then-else decision rule.

82 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction These conclusions are based on our analysis of (Extended) Path Size logit for multi-modal route choice compared with the results achieved by Ramming (2002). The arguments for our findings might be strengthened by application of other advanced choice models with our dataset of multi-modal route choice sets, for instance the Logit-Kernel model and the combination of Generalized Nested Logit and (Extended) Path Size factors. This kind of analysis will be performed in the near future.

ACKNOWLEDGEMENT This publication is a product of the research program Seamless Multi-modal Mobility, carried out within the Netherlands TRAIL Research School for Transport, Infrastructure and Logistics, and financed by the Delft University of Technology. The authors would like to thank Michel Bierlaire (EPFL) for providing Biogeme for the logit estimations.

REFERENCES Bekhor S., M.E. Ben-Akiva, M.S. Ramming (2002), Adaptation of Logit kernel to route choice situation, Paper # 02-3882 CD-ROM of the 81 st Annual Meeting of the Transportation Research Board, Washington Ben-Akiva M., M. Bierlaire (1999), Discrete choice methods and their applications to shortterm travel decisions, In: Hall R.W. (ed.) Handbook of Transportation Science, Kluwer Academic Publishers, Boston Ben-Akiva M.E., S.R. Lerman (1985), Discrete choice analysis, Theory and application to travel demand, The MIT Press, Cambridge Benjamins M., (2001) The assignment of trips to multi-modal networks using supernetworks, MSc Thesis Transport and Planning, Faculty Civil Engineering and Geosciences, TU Delft, Delft (in Dutch) Benjamins M., Ch.D.R. Lindveld, R. van Nes (2002) Multi-modal travel choice modeling using a supernetwork approach, Paper # 02-2948 CD-ROM of the 81 st Annual Meeting of the Transportation Research Board, Washington Bovy P.H.L. (1990), Traffic assignment in uncongested networks, Rotterdam (in Dutch) Bovy, P.H.L. (1996), Stochastic traffic assignment technique enhancements for congested networks. In: Bovy, P.H.L. (ed.) Transportation modeling for tomorrow, Rudi-mental contributions, Delft University Press, March 1996, pp. 129-145 Cascetta E., A. Nuzzolo, F. Russo, A. Vitetta (1996), A modified logit route choice model overcoming path overlapping problems: Specification and some calibration results for interurban networks, In: Lesort J.B. (ed.) Transportation and Traffic Theory, Proceedings of the 13th International Symposium on Transportation and Traffic Theory, Pergamon Hoogendoorn-Lanser S., R. van Nes (2004), Multi-modal choice set composition: Analysis of reported and generated choice sets, Paper # 04-3949 CD-ROM of the 83rd Annual Meeting of the Transportation research Board, TRB, Washington, Accepted for publication in Transportation Research Records

Path size and overlap in multi-modal transport networks 83 Hoogendoorn-Lanser S., R. van Nes (2005), P.H.L. Bovy, Path-size modeling in multi-modal route choice analysis, Paper # 04-1146 CD-ROM of the 84th Annual Meeting of the Transportation research Board, TRB, Washington Hoogendoorn-Lanser S. (2005) Modeling travel behavior for multi-modal transport networks, TRAIL Thesis Series T2005/, TRAIL, The Netherlands (to be published) Koppelman F.S., Wen C.H. (2000), The Paired Combinatorial Logit model: properties, estimation and application, Transportation Research Part B, Vol 34, pp. 75-89 Ramming, M.S. (2002), Network knowledge and route choice, PhD. Thesis, MIT, Cambridge Vovsha P., S. Bekhor (1998), Link-Nested Logit model of route choice, Transportation Research Record 1645, pp. 133-142 Walker J. (2001), Extended discrete choice models: Integrated framework, flexible error structures, and latent variables, PhD Thesis, MIT, Cambridge Wen C.H., F.S. Koppelman (2001), The Generalized Nested Logit model, Transportation Research PartB, Vol. 35, No. 7, Pages 627-641

APPENDIX Scaled parameter values of the home-end/activity-end MNL model and (Extended) Path Size Logit models

Home-end

Activity-end

Mode-specific constants Walk Bike Car Bus Tram Metro Train station constants Intercity train station Express train station Local train stations Mode-specific constants Walk Bike Car Bus Tram Metro Train station constants Intercity train station Express train station Local train station

Multinomial Logit

Path-Size Logit

Extended Path-Size Logit (SAP=14)

0 -16.99 -24.87 -33.53 -25.82 -17.17

0 -15.22 -22.33 -30.97 -23.63 -16.61

0 -14.58 -21.45 -31.03 -23.18 -16.95

0 0 -7.71

0 0 -8.08

0 0 -8.42

0 -16.99 -38.44 -31.19 -23.97

0 -15.22 -35.17 -28.53 -22.99

0 -14.58 -34.96 -28.06 -23.29

0 -7.26 -11.66

0 -7.11 -11.95

0 -6.99 -12.14

84

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Multinomial Logit

Path-Size Logit

Extended Path-Size Logit (SAP=U)

0 -5.15 -6.49

0 -5.23 -6.58

-2.25 -2.51 -4.11 -0.49 -0.89 0 -1 (-0.114)

-2.21 -2.52 -4.18 -0.51 -0.95 0 -1 (-0.114)

-23.25 -19.88

-18.69 -16.30

-2.34 -3.44

-2.31 -3.46

4.85 (0.55)

7.37 (0.84)

-0.16 -2.10 -1608.10 1548.30 0.32 26

-0.20 -2.05 -1593.76 1576.97 0.33 26

Train

Train constants 0 Intercity train -5.64 Express train -7.02 Local train Whole trip In vehicle times [min] -2.38 Walk Bike -2.61 -4.26 Car -0.40 Bus -0.84 Tram Metro 0 Train -1 (-0.104) Number of legs -24.70 Low-frequency (<8) -20.55 High-frequency (>8) Costs [€] -2.551 Parking costs UPT-costs -3.72 Path-size Based on number of legs Other [min] -0.131 Total headway Walking time to UPT-stops -2.21 Statistics Final log likelihood -1616.39 1531.71 Likelihood ratio test 0.32 Rho-square 25 Free parameters different from 0 at a 90%-confidence level

5

PEDESTRIAN DYNAMICS AND EVACUATION: EMPIRICAL RESULTS AND DESIGN SOLUTIONS Dirk Helbing, Institute for Economics and Traffic, Dresden University of Technology, Dresden, Germany; Anders Johansson, Department of Physical Resource Theory, Chalmers University of Technology, Sweden; and Lubos Buzna, Department of Transportation Networks, Faculty of Management Science and Informatics, University ofZilina, Slovakia

INTRODUCTION Mass events seriously challenge the safety concepts of stadia and other public buildings such as railway stations or airport terminals. Since 1945, thirty large crowd panics have occurred, in which 1,000 people were killed altogether and 3,400 were seriously injured. In 2001 there were two major crowd stampedes in soccer stadia of South Africa and Ghana with over 160 fatalities. However, even in the United States, many people died in 2003 due to crowd panics in night clubs on Rhode Island and in Chicago. While in many crowd disasters, people are killed by smoke (see Sime, 1990; Galea et al, 1996), in the above mentioned crowd stampedes people were often killed because of the behaviour of the crowd itself. The reasons for crowd stampedes may be various (for details see Helbing et al., 2005): Apart from real hazards such as a fire, one should mention fans forcing their way into a stadium or over barriers, a non-valued goal, an outage of power supply, pushy fans leaving a stadium, provocations and aggressions by drunken fans or hooligans, the return of leaving fans after a last-minute goal, fans fleeing a rain or hail storm, over-crowding, a gun shot, the use of tear gas, or competition of people for seats, jobs, or housing.

86 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 1: Panicking soccer fans trying to escape the football stadium in Sheffield. Hardly anybody manages to pass the open door, because of the clogging effect occurring at high pressures. Therefore, a huge crowd can turn into a deadly hazard not only in an actually life-threatening situation (Waldau et al, 2003), but also rather unexpectedly. In stampedes, the crowd starts pushing in order to get ahead faster. However, the people rather obstruct each other, and clogging effects may occur at exits and other bottlenecks (see Fig. 1). Shock waves emerge in the crowd (Virkler and Elayadath, 1994) and people may be crushed by the high pressure building up in the crowd (see Fig. 2). When people lose their balance and fall down, the mass tramples on them, as the pushing crowd is hardly manageable. (For a discussion of crowd management and control see Forsyth, 1999.) The injured people may turn into obstacles so that others fall as well, which can produce piles of people (see bottom photographs in Fig. 2). This urgently calls for particular architectural design solutions, which can reduce the potential hazards during mass events and increase the safety of heavily frequented buildings. The objectives of this paper are first to explain the dynamics of crowds and their characteristic behaviours (see section Self-organized patterns) and secondly to use this knowledge to propose improved design solutions for a number of examples: theaters, soccer stadia and pedestrian intersections (see section Some improved design solutions for pedestrian facilities). Finally, we describe some fundamentals of the implementation of our computer simulation model, which we use to assess pedestrian flow scenarios (see section Towards a universal pedestrian and evacuation simulator).

Pedestrian dynamics and evacuation 87

Figure 2: Illustration of typical problems during crowd stampedes: Top: The pressure in the crowd raises up to a painful and potentially lethal level. It can bend steel barriers. Bottom: Panicking people trample on others and are piled up one on top of each other.

OBSERVED SELF-ORGANIZATION PATTERNS IN PEDESTRIAN CROWDS Problems in pedestrian crowds typically arise due to counterflows, bottlenecks, or intersecting flows. These have been recently studied by video techniques (Daamen and Hoogendoorn, 2003a; Helbing et al., 2005). In the following, we will address some of these central issues of pedestrian dynamics in more detail. Counterflows Under everyday conditions, pedestrians with opposite directions of motion are not equally distributed over cross sections of the walkway. Instead, one observes the separation of pedestrian counterflows into lanes of uniform walking direction, see Fig. 3 (Oeding, 1963; Helbing, 1991, 1997, 1998; Yamori, 2001). This segregation phenomenon reduces the number

88 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction of encounters with oppositely moving pedestrians, i.e. the interaction frequency and number of necessary braking or avoidance maneuvers. In such a way, the efficiency of walking, i.e. the average velocity into the desired direction of motion is maximized. In cases of extreme densities, large disturbances, or nervous pedestrian crowds, ordered lanes can break down. Then, blocks of pedestrians with opposite desired directions of motion face each other, but cannot progress. This phenomenon is called "freezing by heating" and may emerge in counterflows of fleeing pedestrians (Helbing et al, 2000a). Note that we avoid the controversial term "panic", here (Keating 1982; Sime, 1990; Harbst and Madsen, 1996), as we are not interested in the psychological state of mind or social behavior, but in the dynamic behavior, i.e. the motion and (physical) interaction of pedestrians.

Figure 3: Representative photographs of lanes formed in bi-directional pedestrian streams (left: London, right: Budapest). An unexpected observed effect is that counterflows can be significantly more efficient than unidirectional flows (Helbing et al., 2005). Although this is rather surprising, it is consistent with observations by AlGadhi et al., 2002. The reason for this increase in capacity is probably the better coordination between people who meet each other in opposite directions, as they can react to each other (Goffman, 1971). In contrast, in unidirectional streams, pedestrians do not sufficiently react to what happens behind them (Helbing and Molnar, 1995). This causes conflicts (e.g. suppressed overtaking maneuvers) and coordination problems, which reduce the efficiency of motion. Finally, note that the increased efficiency in counterflows has not yet been found in today's pedestrian simulation models (Blue and Adler, 2000).

Pedestrian dynamics and evacuation 89 Bottlenecks In recent experiments, we have investigated mutual obstructions and perturbations at bottlenecks (Helbing et al, 2005). We found that the relative standard deviation of the time intervals between successive passings of a measurement cross section was more than doubled by the presence of a bottleneck, compared with a similar experiment with no narrowing. The related increase in the level of perturbations reduces the pedestrian speed and flow at the bottleneck. Long bottlenecks seem to be worse than local obstructions. If opposing flows at a narrow bottleneck disturb each other, we find oscillations in the passing direction. This is reflected by the large variation in the time gaps of successive pedestrians (see Fig. 4). The oscillation effect is more pronounced for longer bottlenecks, which have smaller oscillation frequencies. Typically, it is groups of people rather than single individuals who pass the bottleneck from one side to the other, before people from the other side have a chance to pass the bottleneck. As in the lane formation phenomenon, it is easier to follow someone than to move against an opposing stream. The resulting stream of people releases the pressure in the pedestrian crowd on one side, while the pressure on the other side increases, partially due to impatience. If the pressure difference becomes large enough, the unidirectional stream of people is stopped. Then, people from the other side start to occupy and pass the bottleneck.

Figure 4: Compared to the flow through a corridor without narrowings, the pedestrian flow directly after a bottleneck is less regular due to oscillations in the passing direction (middle). There is a high tendency that the bottleneck is passed by clusters of pedestrians with the same direction of motion rather than by single individuals in an alternating manner. This effect is caused by pedestrian interactions, not by gaps in the pedestrian inflow. Left: specification of the geometry. Right: Snapshot from the video used to evaluate the experiment.

In situations with bi-directional flows, bottlenecks should be avoided or, if unavoidable, bottlenecks should be short. Long narrow bottlenecks involve the risk that people try to turn the flow direction in an oscillatory way, but do not manage to do so. In these situations, the flow at the bottleneck drops significantly or stops completely. As time goes by, pedestrians reduce the distance to their predecessors, which eventually leads to a compression of the waiting crowd (see Fig. 5). This psychologically induced queuing behavior can be frequently observed and mathematically described (Helbing, 1991, 1997). The compression in the crowd is related with shock waves (Virkler and Elayadath, 1994), which produce the impression that the crowd is moving forward, even if nobody has passed the bottleneck. Therefore, the people in the back continue trying to get ahead. As the impatience in the crowd increases, people

90 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction start to push and to interact physically. Consequently, higher and higher pressures build up in the crowd. In extreme situations, this can cause clogging phenomena due to mutual obstructions, at least temporarily.

Figure 5:

Schematic illustration of observed queuing behavior of pedestrians (after Helbing, 1997). When the front of a queue (top) is stopped, one can often observe the following phenomenon: After some time, one of the waiting pedestrians begins to move forward a little bit, as the preferred (equilibrium) distance to the pedestrian in front decreases due to impatience (the perceived "pressure of time"). This causes the successor to follow up and so forth, producing a wave-like propagation of the gap to the end of the queue and a compactification of the queue (which decreases the comfort of waiting). Thereby, the tendencies of all individuals to move forward a little (in accordance with their steadily decreasing preferred equilibrium distance) add up towards the end of the queue, giving rise to larger and larger following-up distances. A more detailed explanation of this phenomenon considering the minimal length of pedestrian strides is presented elsewhere (Helbing, 1991).

In cases of crowd stampedes, clogging can even occur if all pedestrians are heading into the same direction (see Fig. 1). This leads to stopped or irregular outflows from a crowded place (as is reflected by the long steps in the curve representing the number of escaped persons as a function of the passing time of the door, see Fig. 9). The reason for this is friction effects, when pushy people interact physically. The "pressures" building up can reach orders of 1 to 4.5 tons per meter of railing of a wave breaker (Elliott and Smith, 1993; Smith and Dickie, 1993). This can bend steel barriers (see Fig. 2) or tear down brick walls. In such situations, people are frequently crushed or trampled, turning them into obstacles for others. Moreover, the flow of people managing to escape is significantly decreased (Helbing et al., 2000b).

Pedestrian dynamics and evacuation 91 Intersecting Flows Intersections of pedestrian streams are a big problem and practically unavoidable, as it is uncommon to use bridges for their separation. Let us first discuss what happens when two wide streams intersect. In this situation, the phenomenon of stripe formation has been observed (Ando et al., 1988). For an illustration see Fig. 6. Like lanes, stripes are a segregation phenomenon, but not a stationary one. Instead, the stripes are density waves moving into the direction of the sum of the directional vectors of both intersecting flows. Naturally, the stripes extend sidewards into the direction which is perpendicular to their direction of motion. Therefore, the pedestrians move forward with the stripes and sidewards within the stripes. This allows the pedestrian streams to penetrate each other in a way that pedestrians do not have to stop. Lane formation corresponds to the particular case of stripe formation where both directions are exactly opposite. In this case, there is no intersection of flows and motion takes place within more or less stationary stripes ("lanes").

Figure 6: Illustration of empirical findings for two intersecting pedestrian streams (after Ando et al., 1988). A characteristic observation is the formation of stripes perpendicular to the sum of the directional vectors of both streams. Previous investigations of intersecting flows (Helbing, 1997; Helbing and Vicsek, 1999) indicate that no stable pattern exists when three or more pedestrian streams intersect. Instead, one observes very short-lived patterns which destroy each other, producing a rather chaotic appearance. These temporary patterns include rotary traffic in clock-wise or counter-clockwise direction, dominant pedestrian flows in opposite directions with the perpendicular directions waiting, and short-lived stripes in one of the four possible diagonal directions. The steady competition and mutual destruction of these spatio-temporal patterns of motion leads to very inefficient pedestrian streams, including temporary blockages.

92 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

SOME IMPROVED DESIGN SOLUTIONS FOR PEDESTRIAN FACILITIES The complex interaction of several pedestrian streams can lead to completely unexpected results, which is a consequence of their non-linear dynamics. Therefore, the planning of pedestrian facilities with conventional methods does not guarantee the avoidance of jams, obstructions, or blockages, particularly not in emergency situations. In contrast, a skilful flow optimization utilizing the self-organization phenomena emerging in pedestrian streams can increase the efficiency and safety of pedestrian facilities at comparable costs and even with less available space. In the following, we will discuss some improved design solutions for exits and egress routes of stadia, theaters, and lecture halls, which may also be transferred to other highly frequented places such as hotels, convention centers, exhibition halls, business centers, shopping malls, railway stations, or airport terminals. These are a direct consequence of the various observed self-organization phenomena described above. For more examples of design solutions see Helbing et al. (2005). In this paper we have choosen to focus on three specific scenarios: a soccer stadium, a theater and a pedestrian crossing. The pattern of how the improved design solutions have emerged is the same in all cases: 1. Empirical observations have pointed our interest at some specific locations. 2. Ideas of architectural improvements have been developed. 3. Simulations of these new design solutions have been carried out to evaluate their (better) performance. Counterflows Counterflows are most efficient, when they are organized in a few wide lanes with stable interfaces. However, at mass events, opposite flow directions must be artificially separated, as the lanes become sensitive to perturbations (Helbing et al., 2000a) and tend to obstruct each other due to attempted overtaking maneuvers. If the lanes break down, dangerous blockages are likely to emerge, which may finally trigger crowd stampedes. Surprisingly, the desired separation of opposite flow directions can be supported by a series of obstacles such as railings, trees, or columns (see Fig. 7). Such kinds of solutions are partially permeable, but have psychologically and physically similar effects as a separating wall. They prevent pedestrians from using small gaps in the opposite flow for overtaking maneuvers, which could cause broader streams in one direction than the other, later on provoking an interactioninduced reduction in the walking speed and serious obstructions. However, when the other side is little frequented, a permeable design allows pedestrians to use the other side as well. For this reason, a series of obstacles is flexible and can reduce the frequency of deceleration and avoidance maneuvers. In this way, the average velocity is increased, although some space is used up by the series of obstacles.

Pedestrian dynamics and evacuation 93 Similar solutions are useful at doors: It is often better to have two separate doors for the two opposite directions of motion than one twice as big door. These will typically be used by one flow direction each (Molnar, 1996a, b; Helbing, 1997). In contrast, one door tends to produce oscillatory changes in the passing direction with intermediate periods of standstill during which both directions struggle with each other. This struggling can produce a dangerous pressure in the crowd, which does not build up when there is a reasonable and continuous flow in both directions through separate doors.

Figure 7:

The interfaces of opposite pedestrian flows can be efficiently stabilized by a series of columns (here in a tunnel of the metro system of Budapest at Deak Ter). Nevertheless, the permeability allows for a flexible usage in accordance with the respective pedestrian traffic volumes in both directions.

Bottlenecks and Clogging in Pushy Crowds One possibility to improve the situation at bottlenecks is a funnel-shaped design (which, by the way, requires less walkable space). However, a funnel-shaped design is not sufficient in big pedestrian crowds, when a large pressure builds up. This is well-known from mass events and can turn them into deadly stampedes under particular circumstances. Appropriately placed columns (pillars) can improve the situation. Their function is similar to conventional wave breakers. They can absorb pressure in the crowd and reduce it to a subcritical level. However, they do not seriously obstruct the movement towards the exits, as wave breakers (i.e. transversal railings) would do. The performance of various alternative design solutions for exit areas has recently been compared by means of computer simulations (Escobar and de la Rosa, 2003). This independent study is recommended to read.

94 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 8:

Snapshots of an experiment imitating conditions of crowd stampedes. The participants had comparable distances to the exit in the beginning. They were asked to rush towards the door after an acoustical signal was given and to behave in a pushy way. Without an obstacle, the experiment showed clogging effects and a tendency of people to fall (left). In another setup, a cupboard served as an obstacle (right). Despite of the strong forces in the crowd (the cupboard was shaking), the clogging effect could be significantly reduced. This increased the efficiency of escape and diminished the tendency of falling.

Figure 8 shows an experiment we have performed to determine the effect of an obstacle on a pushy crowd. Without an obstacle, we found a clogging effect, which considerably reduced the efficiency of escape and produced high pressures in the crowd. Placing an obstacle (here: a cupboard) in front of the exit could avoid the clogging effect. This becomes visible in the cumulative counts of the number of escaped persons over time (see Fig. 9): Without an obstacle, one observes long plateaus indicating a temporarily stopped flow. Placing an obstacle (a cupboard of width 45 cm) in front of the exit reduces the plateaus and increases the rate of escape. That is, an obstacle can reduce both, the mean value and the relative standard deviation of the time intervals between successive escapes from the room. The outflow under conditions of a pushy crowd was increased by about 30% (see Fig. 9). A similar thing should apply to crowds of very high density. According to recent simulation results, even larger improvements are expected for bigger crowds and other geometrical designs (Escobar and de la Rosa, 2003; Helbing et al, 2000b). Clogging effects and irregular flows typically occur in pushy crowds, when the pressure exceeds a certain level. They become more serious when more people are involved. Under conditions of normal leaving, however, the outflows for rooms with and without pillars will be about the same.

Pedestrian dynamics and evacuation 95

Figure 9:

Escape times and number of escaped persons in experiments imitating conditions of crowd stampedes with and without an obstacle in front of the exit (see Fig. 8). The left figure illustrates the slower and less regular escape without an obstacle. Long steps reflect the clogging effect, i.e. long periods during which nobody manages to leave the room. In the right figure, the lines are not shown in a step-wise manner in order to allow one to distinguish the different curves, which just connect the measurement points of leaving persons. The broken lines are a guide for the eyes. Their slopes allow to compare the average flows with and without an obstacle in front of the exit.

Intersections In intersecting flows, mutual obstructions are practically unavoidable. They are, therefore, the greatest challenge to architects, urban planners and organizers of mass events (e.g. Olympic games). However, computer simulations based on the social force model (Helbing et al 1991, 1995, 2002) confirm the result shown in Fig. 6, i.e. the emergence of stable, moving stripes in two wide and dense intersecting flows (Fig. 10). These allow pedestrian streams to penetrate each other without major obstructions. In contrast, two intersecting flows of small density are characterized by the formation of temporary channels (Daamen and Hoogendoorn, 2003b). When three or more flow directions intersect, coherent stripes have not been observed, as stripes in several directions would have to propagate through each other, which would destroy any existing stripes. Patterns in intersection areas of more than two flow directions are, therefore, often unstable, "chaotic" and inefficient.

Figure 10: Representative simulation results of two intersecting pedestrian streams in a rectangular area of 90m x 35m, where the pedestrian streams have sources and sinks in the corners. Our computational results, obtained with the social force model, reproduce the observed phenomenon of stripe formation quite well (cf. Fig. 6) (Helbing et al. 2005).

96 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Nevertheless, the different flows at an intersection (and turning manoeuvres) can be organized in a way that supports a stable pattern. It is remarkable that, in computer simulations, an obstacle in the middle of an intersection has been found to improve the efficiency of motion, although it reduces the available space for the pedestrians (Molnar, 1996a). This becomes understandable, when the different flow directions are separated in a way that there are four intersections of two flow directions each (see Fig. 11). Efficient stripes may, then, emerge in all four intersection areas, as described above. Good solutions will give rise to rotary traffic, when the crossing is heavily frequented, hi periods of light traffic, some pedestrians may try to move opposite to the main stream. During these periods, however, a deviation from rotary motion does not matter a lot. Our first empirical investigations confirm this assessment. Simulations of an intersection (see Fig. 11) have shown that the improved design results in higher pedestrian throughput for high crowdedness. Additional simulations have been made with either the pillar in the middle or the railings. It has been found that each single measure increases the throughput as well, if the pedestrian density is high enough and pedestrians have a preferred side of walking. The design also provides enough flexibility for efficient usage patterns at low densities. Finally, note that the optimal location of the obstacle and railings depends on the respective fraction of pedestrians in each direction and their turning fractions. It may be determined by evolutionary algorithms.

Figure 11: Left: At conventional intersections, one observes continuously changing patterns of motion, i.e. the pedestrian streams are unstable, "chaotic", and inefficient. Right: If four intersecting pedestrian flows are separated by barriers, one-way regulations, and an obstacle in the center of the intersection, stripes may form in all four intersection areas of two pedestrian streams and propagate perpendicularly to their sideward extension. As a consequence, efficient round-about traffic may emerge at high densities. The design uses the preference of the right-hand side in Central European pedestrian streams. In other countries such as Japan, it would be reasonable to invert the flow directions.

Pedestrian dynamics and evacuation 97 Egress Routes of Theaters In 1903, 602 people died in a theater fire, and in 1970, 46 people were killed by a fire in a ballroom in St. Laurent-du-Pont. (More examples are discussed by Waldau et al, 2003.) Even after these tragic events, theater fires have cost many lives and were considered to be so dangerous that fire fighters had to be on the spot during every show.

Figure 12:

Left: A uniform width of the corridors in a theater implies long waiting times for the more distant rows. Right: A funnel-shaped design can reach shorter waiting times for the same number of seats, which reduces the danger of pushing behavior in emergency situations. Columns in front of the exit could decrease the pressure on the leaving crowd at the exit, similar to wave breakers. The wider exits reduce the risk of jam formation and clogging.

Today, it should be possible to evacuate theaters within a few minutes, as specified by safety codes. Therefore, the minimum width of egress routes is regulated by law. However, it is not always reasonable that egress routes have a constant width. In reality, the top floor of a hotel, the lowest deck of a ship, or a theater's most distant row from the exit has to cope with the smallest pedestrian volume. The number of persons who wish to leave increases with each floor, each deck, or each row of seats that is closer to the exit. Therefore, the time to leave grows proportionally or even over-proportionally with the distance to the exit, which implies long waiting times. This causes impatience in the crowd, particularly in situations of danger, which can produce a pushy behavior with all its possible consequences: clogging phenomena, dead and injured people. As a consequence, it would be reasonable to have increasing widths of egress routes, similar to the stream of a river. In this way, it is possible to avoid long leaving times and to speed up the evacuation of people. The average width of egress routes does not have to be greater than today, so that the usable area and potential profit are not reduced. Figure 12 compares a typical design of theaters with an improved solution, in which the blocks of seats have a trapeziform rather than a rectangular shape. For large crowds it can also make sense to place columns at suitable places in front of the exits (see Figs. 8 and 9). These can serve to absorb

98 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the pressure in the crowd like a wave breaker, in order to relieve the strain on the escaping people at the exit, while allowing pedestrians to get around the obstacle. In many cases, columns are anyway needed for static reasons, so that no additional costs would incur. The visibility of exits and exit signs can be taken care of. For example, a suitable lighting concept could support orientation. The two scenarios shown in Fig. 12 have been simulated and the evacuation time is reduced for the improved design, as expected. Note that the changing of egress routes into funnel shapes does not change the total number of seats, since the average widths are the same and therefore the area occupied by the egress routes is constant. Staircases and Exits of Stadia Staircases are serious obstacles in any panic situation. Practically, it is only a matter of time until persons in the pushy crowd are falling, especially as it is not possible to see the exact location of the single stairs in a dense crowd. The danger is highest for staircases going down. The crowd stampede in a night club of Chicago on February 17, 2003, is characteristic for this. It was triggered by a relatively harmless event: Staff members tried to separate two fighting guests using pepper spray. However, other guests assumed a terror attack with poisonous gas, as the government had activated the second highest alert level (Code Orange). Therefore, some guests ran towards the exit and alarmed the other people. Most of them rushed towards the main exit as well, while the available side exits were used by a few people only. On the long staircase leading down to the exit, people lost their balance, and succeeding people fell over them. In the end, people were piled up in many layers (see, for example, the bottom photos in Fig. 2). Attempts to draw exhausted individuals out of this heap of people failed, with the result of 21 dead and 50 injured people. The danger of falling on staircases can be reduced by replacing them by ramps, where possible. However, the hazards of long downward staircases can also be reduced by dividing them into small enough segments pointing into different directions. Figure 12 illustrates the suggested zigzag design for a stadium exit. It limits the pushing-related build-up of pressure in a crowd and avoids that many fallen people are piled up one on top of each other. Although upward staircases may act as obstacles for escaping individuals as well, the problem of heaps of fallen people is typical for downward staircases. Upward staircases may even reduce the pressure of the crowd rushing towards the exit, as they separate the people in vertical direction. Consequently, it makes a big difference whether egress routes lead upwards or downwards. The two architectural designs in Fig. 13 have been simulated both for normal evacuation and a scenario in which one of the downward staircases was blocked due to fallen persons. In both cases the evacuation times were marginally lower for the improved infrastructure. Moreover, the pressure levels were reduced by the improved design by almost a factor of 2. Both designs have only one exit each, located in the center, but the three egress routes leading to this exit differ.

Pedestrian dynamics and evacuation

Figure 13:

Top: Conventional design of a stadium exit in an emergency scenario, where we assume that some pedestrians have fallen at the end of the downwards staircase to the left. The dark area indicates high pressures, since pedestrians are impatient and pushing from behind. The different queue lengths in the different rows are due to heterogeneous desired pedestrian velocities. Bottom: The zigzag design of the downward staircases changes the pushing direction of the crowd in the improved scenario. This avoids the build-up of an overcritical pressure, which could cause that many falling people are piled up (see bottom of Fig. 2). Another improvement is the increase in the diameter of the corridors, which can reduce waiting times, impatience, and evacuation times, even if the number of seats is the same.

100 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

TOWARDS A UNIVERSAL PEDESTRIAN AND EVACUATION SIMULATOR The implementation of our pedestrian microsimulator based on the social force model (Helbing et al, 1991, 1995, 2002) has been recently optimized for efficiency and scalability in terms of computational speed and memory. It is no problem to simulate 10,000 pedestrians in real-time on a normal PC. Parallel computing will even allow one to simulate more than 100,000 pedestrians in real-time. This is suitable for the simulation of large scenarios such as pedestrian flows in extended urban areas (see Fig. 14) or during mass events such as street parades or carnivals (Batty et al., 2003). In order to be suitable for scenarios of this complexity, we have implemented certain additional features complementing the social force model: 1. a powerful scenario generation module based on XML, 2. path finding and route choice modules to describe the selection of destinations, including spontaneous stops and impulse shopping, 3. calibration modules to adapt simulations to real measurements, 4. a statistics module evaluating various performance measures (such as densities, velocities, flow efficiency, pressure, overall evacuation time, etc.) to assess and compare different scenarios, 5. an optimization module based on evolutionary algorithms which allows to improve the geometrical boundary conditions of pedestrian facilities, 6. various visualization modes for the illustration of the simulation results, both on- and off-line, 7. a functionality reflecting density-dependent herding effects. The simulator should not only serve as a research tool, but can also be used as a design tool to assess the effects of new pedestrian sources and attractions (such as theatres and shopping malls) on the overall pedestrian streams and activity patterns.

SUMMARY AND DISCUSSION Pedestrian dynamics has recently received a high level of interest due to the evacuation of the World Trade Center on September 11, the crowd panics in night clubs on Rhode Island and in Chicago 2003, and the sad accidents during the holy pilgrimage in Makkah.

Pedestrian dynamics and evacuation

101

We have used video techniques to explore the dynamics of pedestrian streams at bottlenecks, around obstacles, or at intersections. Our empirical results on pedestrian streams in normal and pushy scenarios show that the geometric boundary conditions are highly relevant for the capacity of the elements of pedestrian facilities, due to self-organization phenomena. These findings can be used to improve design elements of pedestrian facilities and egress routes. The great challenge of simulation models is the reproduction of the observed collective phenomena in pedestrian crowds. This includes lane formation in corridors and oscillations at bottlenecks in normal situations, while different kinds of blocked states may emerge in pushy pedestrian crowds. By means of microsimulations based on a social force model of interactive pedestrian dynamics, the spatio-temporal patterns in pedestrian crowds can be successfully reproduced and interpreted as self-organized phenomena. In contrast to previous sociopsychological approaches, this allows an understanding of the observations from a physical point of view. In the social force model, the transition between normal and pushy behavior is controlled by a single parameter, the "impatience", which influences not only the desired speeds, but also the fluctuation strength (irregularity of motion) and the tendency of herding (Helbing et al, 2000b, 2002).

Figurel4: Simulation of pedestrian streams in a simulation of the city center of Dresden, Germany. The figure shows the average pedestrian density over about 10 minutes simulation of the lkm x lkm area, with about 30 000 simultaneously simulated pedestrians.

102 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The advantage of the social-force-based simulation approach is its simple form and its small number of parameters, which do not need to be calibrated anew for each situation. Therefore, the model is suitable for the predictions of pedestrian streams in novel architectures and new situations. Even counter-intuitive effects are well reproduced. This includes paradoxial effects like "freezing by heating", "faster is slower", and the ignorance of available exits (Helbing et al, 2000a, b, 2002). At the same time, the model suggests innovative measures to improve pedestrian flows, both in normal and panic situations. For example, we propose to use suitably shaped and placed "obstacles" to stabilize flow patterns and to make them more fluid. Moreover, we suggest increasing diameters of egress routes in stadia or theaters to avoid long waiting times for people in the back and shock waves due to impatience. In addition, zigzag-shaped geometries and columns can reduce the pressure in pushy crowds. The proposed design solutions are expected to increase the efficiency and safety of, for example, shopping malls, cinemas, theaters, public buildings, hotels, or passenger ships in the future. Depending on the numerical implementation, the social-force model is suitable for large-scale simulations on parallel PC clusters. In this way, more than 100,000 pedestrians could be simulated in real-time, which makes the approach suitable for the assessment of airport terminals, stadia, railway stations, hotels and mass events. Moreover, one can simulate pedestrian flows in extended urban areas. This allows one to assess the attractiveness of certain locations for new shops, but also the impact of new buildings like theaters or malls on the overall pedestrian flows.

ACKNOWLEDGMENTS D.H. is grateful to Stefan Lammer as well as Tilo Grigat for producing the schematic illustrations. Last but not least, we appreciate that Roger Hughes has drawn our attention to the interesting study by Ando et al. (1988).

REFERENCES Ando, K., H. Oto and T. Aoki (1988). Forecasting the flow of people [in Japanese]. Railway Research Review, 45, (8), 8-13. Batty, M., J. Desyllas, and E. Duxbury (2003). The discrete dynamics of small-scale spatial events: agent-based models of mobility in carnivals and street parades. Int. J. Geographical Information Science, 17(7), 673-697. Blue, V. J. and J. L. Adler (2000). Cellular automata microsimulation of bi-directional pedestrian flows, J. Trans. Board, 1678, 135-141. Bolay, K. (1998). Nichtlineare Phanomene in einem fiuid-dynamischen Verkehrsmodell. Master's thesis, University of Stuttgart.

Pedestrian dynamics and evacuation 103 Borgers, A. and H. Timmermans (1986a). A model of pedestrian route choice and demand for retail facilities within inner-city shopping areas. Geographical Analysis, 18, 115-128. Daamen, W. and S. P. Hoogendoorn (2003a). Experimental research of pedestrian walking behaviour. Transportation Research Record, 1828, 20-30. Daamen, W. and S. P. Hoogendoorn (2003b). Qualitative results from pedestrian laboratory experiments, Pedestrian and Evacuation Dynamics, E. R. Galea (ed.), 121-132. Elliott, D. and D. Smith (1993). Football stadia disasters in the United Kingdom: Learning from tragedy? Industrial & Environmental Crisis Quarterly, 7 (3), 205-229. Escobar, R. and A. de la Rosa (2003). Architectural design for the survival optimization of panicking fleeing victims. Presented at 7th European Conference on Artificial Life (ECAL 2003), Dortmund, Germany, September 14-17. Forsyth, D. R. (1999). Group Dynamics, Wadsworth Publishing, Ed. 3, Chap. 15. Galea, E. R., M. Owen and P.J. Lawrence (1996). Computer modelling of human behaviour in aircraft fire accidents, Toxicology, 115, 63-78. Goffman, E. (1971). Relations in Public: Microstudies in the Public Order, Basic, New York. Harbst, J. and F. Madsen (1996). The behaviour of passengers in a critical situation on board a passenger vessel or ferry. Technical Report, Danish Investment Foundation, Copenhagen. Helbing, D. (1991). A mathematical model for the behavior of pedestrians. Behavioral Science, 36, 298-310. Helbing, D. (1993). Stochastische Methoden, nichtlineare Dynamik und quantitative Modelle sozialer Prozesse. Ph.D. thesis, University of Stuttgart, 1992, published by Shaker, Aachen. Helbing, D. (1995). Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes. Kluwer Academic, Dordrecht. Helbing, D. (1996). Traffic modeling by means of physical concepts, in Traffic and Granular Flow, D. E. Wolf, M. Schreckenberg and A. Bachem (eds.), 87-104, World Scientific, Singapore. Helbing, D. (1997). Verkehrsdynamik [Traffic Dynamics], Springer, Berlin. Helbing, D. (1998). Pedestrian dynamics and trail formation, in Traffic and Granular Flow '97, M. Schreckenberg and D. E. Wolf (eds.), 21-36, Springer, Singapore. Helbing, D., L. Buzna, A. Johansson, and T. Werner (2005). Self-organized pedestrian crowd dynamics: Simulations, experiments, and design solutions, Transportation Science, 39(1), 1-24. Helbing, D., I. J. Farkas, P. Molnar and T. Vicsek (2002). Simulation of pedestrian crowds in normal and evacuation situations, in Pedestrian and Evacuation Dynamics, M. Schreckenberg and S. D. Sharma (eds.), 21-58, Springer, Berlin. Helbing, D., I. Farkas and T. Vicsek (2000a). Freezing by heating in a driven mesoscopic system. Physical Review Letters, 84, 1240-1243. Helbing, D., I. Farkas, and T. Vicsek (2000b). Simulating dynamical features of escape panic. Nature, 407, 487-490. Helbing, D. and P. Molnar (1995). Social force model for pedestrian dynamics. Physical Review, E 51, 4282-4286. Helbing, D., P. Molnar, I. Farkas and K. Bolay (2001). Self-organizing pedestrian movement. Environment and Planning, B 28, 361-383. Helbing, D. and T. Vicsek (1999). Optimal self-organization. New Journal of Physics, 1, 13.1-13.17.

104 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Hoogendoorn, S. P., P. H. L. Bovy and W. Daamen (2002). Microscopic pedestrian wayfinding and dynamics modelling, in Pedestrian and Evacuation Dynamics, M. Schreckenberg and S. D. Sharma (eds.), 123-154, Springer, Berlin. Keating, J. P. (1982). The myth of panic. FireJ., 147, 57-61. Molnar, P. (1996a). Modellierung und Simulation der Dynamik von FuBgangerstromen, Shaker, Aachen. Molnar, P. (1996b). Microsimulation of pedestrian dynamics, in Social Science Microsimulation, J. Doran, N. Gilbert, U. Mueller and K. Troitzsch (eds.), Springer, Berlin. Oeding, D. (1963). Verkehrsbelastung und Dimensionierung von Gehwegen und anderen Anlagen des FuBgangerverkehrs, in StraBenbau und StraBenverkehrstechnik, Heft 22, Bundesministerium fur Verkehr, Abt. StraBenbau, Bonn. Sime, J. (1990). The concept of panic, Chap. 5, 63-82, in Canter, 2nd edition. Smith. R. A. and J. F. Dickie (1993). Engineering for Crowd Safety, Elsevier, Amsterdam. Virkler, M. R. and S. Elayadath (1994). Pedestrian density characteristics and Shockwaves, in Proceedings of the Second International Symposium on Highway Capacity, Vol. 2, R. Akcelik (ed.), 671-684, Transportation Research Board, Washington, D.C. Waldau, N., M. Schreckenberg and P. Gattermann (2003). Design criteria related to orientation in buildings during high stress situation models and their applications, Pedestrian and Evacuation Dynamics, E. R. Galea (ed.), 121-132. Yamori, K. (2001). Going with the flow: Micro-macro dynamics in the macrobehavioural patterns of pedestrian crowds, Psychological Review, 105, 530-557.

6

AN EMPIRICAL ASSESSMENT OF TRAFFIC OPERATIONS Chao Chen and Pravin Varaiya, University of California, Berkeley; and Jaimyoung Kwon, Statistics Department, California State University, Hayward, California, USA

INTRODUCTION The California Freeway Performance Measurement System (PeMS) stores real-time data from 26,000 loop detectors. PeMS is accessed via an internet browser (http://pems.eecs.berkeley.edu/). It currently has 3 TB of data, growing at 2 GB/day. PeMS extracts useful information from these data and displays it in graphical or tabular form. These data provide an unparalleled opportunity to assess freeway performance and suggest ways to improve freeway management. Operational since 2001, PeMS receives real time data from 26,000 loops grouped into 8,040 Vehicle Detector Stations (VDS) covering 3,000 directional miles of freeways in major California urban areas. PeMS also collects incident data from the Traffic Accident Surveillance and Analysis System (TASAS) and the California Highway Patrol. The principal aim of this paper is to examine congestion as a performance measure and demonstrate that data can be processed to reliably estimate the causes of congestion, and the gains from better ramp metering, incident management, and traveler information. Each of the following six sections addresses a different aspect of congestion. Some sections report previous research by the PeMS Development Group. The section BPR CURVE suggests replacing the standard BPR curve by two curves: one for the free flow regime, the other for the congestion regime. For Los Angeles the two regimes separate at 50 mph. Drivers in Los Angeles spend 35% of their time in the congestion regime, so congestion delay can be reduced if this regime can be avoided. IDEAL METERING presents an empirical procedure to rapidly obtain a rough estimate of this reduction by preventing the onset of the congested regime at recurrent bottlenecks. For Los Angeles the procedure estimates an annual saving of 50 million vehicle-hours.

106 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Not all bottlenecks cause significant congestion. BOTTLENECKS summarizes an automated procedure to identify all bottlenecks and rank them by frequency of occurrence and severity of impact. For San Diego County the procedure locates 160 bottlenecks, the ten most severe of which account for 61 percent of the delay from all bottlenecks. To estimate the delay from a collision, its effect must be separated from congestion caused by bottlenecks. CONGESTION PIE reviews a technique that predicts what the congestion would have been had the collision not occurred. Collisions and bottlenecks cause congestion, and delay from bottlenecks can be reduced by ramp metering. Putting these considerations together yields three congestion pie slices corresponding to collisions; congestion that can be eliminated by ramp metering; and 'residual' congestion due to all other causes, the largest being 'excess' demand. Congestion delay measures system performance. Travelers experience congestion as large variations in travel time. Because the travel time stochastic process exhibits a large temporal autocorrelation, real time data can be processed to reliably predict travel time, as shown in PREDICTING TRAVEL TIME. Travel time prediction increases welfare: It can suggest a shorter alternative route if one is available; and it can reduce the uncertainty in travel time, even when that time itself cannot be reduced. The Bay Area provides a unique opportunity to study the impact of HOV lanes on non-HOV traffic because the HOV lanes are time-actuated. HOV LANE EFFECTIVENESS presents limited evidence suggesting that HOV actuation increases overall congestion, by imposing a congestion penalty on non-HOV traffic (which loses one lane) and a capacity penalty on the HOV lane (which acts as a one-lane highway with much lower speed).

BPR CURVE Figure l(a) is a scatter plot of speed vs. flow across all four lanes of I-10W in Los Angeles at vehicle detector station (VDS) 717162. Each point represents a one-hour average for the 30day period 13 June-13 July, 2004. Also displayed are two curves fitted to the BPR (Bureau of Public Roads) equation

v=

V

-l \ + a(q/Cy'

in which v is speed, v/ is free flow speed, q is flow, and C is capacity. Data with average hourly speed below 30 mph are discarded. The capacity C is estimated to be the maximum hourly flow observed during the 30-day period, and the free flow speed is the median speed when occupancy is below 10%. The parameters a and /? are either user-specified or obtained using a nonlinear least- squares, Marquardt-Levenberg algorithm (Martin, W., 1998).

Empirical assessment of traffic operations 107

Figure 1 (a) Scatter plot of 1-hour average speed vs. flow and (b) trajectory of 5-min average speed vs. flow. Two BPR curves are fitted to the scatter plot in (a). The shaded region, pcr, in (b) is the critical density separating free flow from congestion. Figure l(b) plots the temporal evolution of speed vs. flow at the same location as in figure 1 during 0400-1700 (4:00 AM-5:00 PM) on 13 July, 2004. Each point now represents a fiveminute average. The figure suggests a modified BPR procedure that divides traveler experience into two distinct 'metastable' regimes: the free flow regime implicit in the BPR curve, and a low-speed congestion regime, separated by a 'critical density' band, pcr. The likelihood of the two regimes can be empirically computed for any location, freeway, or an entire region. Figure 2(a) gives the frequency distribution of VMT (veh-miles traveled) and VHT (vehhours traveled) on I-10E during 0500-1900, 14 July, 2004. Drivers spent 35% of their time at an average speed of 30 mph and 65% at an average speed of 60 mph, suggesting the tworegime BPR model of Figure 2(b), separated at 50 mph. The free-flow BPR curve is as before. Two linear regressions are fitted to data in the congestion regime,

= 8 +e3.. vf

C

The solid line is obtained by least-squares; the dotted line is the least quantile regression, which is less sensitive to outliers. The likelihood of each regime, determined by frequency counting, is P(v > 50 mph) = 0.79, P(v < 50 mph) = 0.21.

108 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2 (a) Distributions of VMT and VHT vs. speed for I-10E and (b) A two-regime BPR model.

IDEAL METERING Figure l(b) suggests that holding back volume surges by metering on-ramps may prevent the occurrence of the congestion regime at some bottlenecks, and figure 2(a) implies a large reduction in delay if this can be done. Designing a ramp metering algorithm for a specific freeway section is arduous. Many set points and feedback gains must be selected (Papageorgiou, M., 1983; Papageorgiou, M. et al., 1991), based on a calibrated simulation model. But there is a simple procedure to roughly estimate the benefits from ramp metering without detailed simulations, based on the hypothesis that the congestion regime can be avoided by controlling flow according to the Ideal Metering Principle (IMP) (Jia, Z. et al., 2000): If volume surges at on-ramps are held back by a metering policy that always keeps flow below its capacity in every link, freeway speed will be maintained at 60 mph and congestion will not appear. As a consequence of metering vehicles may be stopped at the ramps for some time. The IMP hypothesis has two parts. One part is that if flow is always maintained below capacity, or equivalently, if density is always less than critical (pcr), traffic will be kept in the free flow regime. Data like in figure l(b) provide indirect support: If the traffic density is never allowed to enter the critical region, traffic will always stay in the free flow regime. The definition of 'capacity' is empirical: It is taken to be (say) 95% of the maximum sustained observed flow. The second part of the hypothesis is that maximum flow occurs at free flow speeds, nominally 60 mph, as in Los Angeles (Jia, Z. et al., 2001) and Orange County (Chen, C. and P. Varaiya, 2001). Of course not all congestion is due to volume surges at on-ramps and practical considerations, such as ramps of insufficient length, may prevent implementation of a proper metering policy. The planner should ask: "What will be the impact of implementing IMP-conforming ramp metering if the IMP hypothesis is true?"

Empirical assessment of traffic operations 109 A procedure to answer this question is illustrated in (Jia, Z. et al., 2000), using data for a 7mile section (postmiles 0-7) of I-405N in Orange County, during 0500-1000 for 10 weekdays in June 1998. The section is divided into 13 links, each corresponding to one VDS; eight links have one on- and off-ramp each. A virtual on-ramp is created at the beginning of the most upstream link in order to account for metering of on-ramps upstream of the study section.

Figure 3 The top graph is the time in VHT actually spent on the freeway section, every 5 minutes. Units are normalized to VHT per hour, so the total VHT on this section, between 0500 and 1000 is the area under the top graph. The middle graph is the VHT per hour under IMP metering, including time in ramp queues. The bottom graph excludes time spent on the ramps, so it is the VHT per hour that would be spent traveling at 60 mph. The area between the top and middle graphs is the time saved by metering. The area between the middle and bottom graphs is the time spent at the ramps. Source: (Jia, Z. et al., 2000). The capacity of each link is calculated as the maximum sustainable aggregate flow. Inflows at on- ramps and exit flows at off-ramps are assumed to remain unchanged despite the metering, whose impact is estimated as follows: 1. At each on-ramp, inflow is metered so that the link flow remains 5% below the link capacity; 2. Traffic on each link after metering is assumed to move at 60 mph; 3. The queue at each on-ramp is calculated by accumulating the net inflow. On average, two-thirds of the total delay (defined as additional vehicle-hours traveled (VHT) driving below 60 mph) is eliminated by ramp metering. More insight is gained from figure 3, which shows how metering holds back large surges in demand.

110

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

This procedure was repeated in (Chen, C. et al., 2001) for five freeways (1-5,1-10, US 101,I110 and 1-405) in Los Angeles during 0000-1200, 3-9 October, 2000. That exercise found that IMP-metering reduces delay by 70%. PeMS calculates the total congestion delay (from driving below 60 mph) for Los Angeles for 2003 to be 83 million vehicle-hours. The procedure suggests that ramp-metering may eliminate 57 million vehicle-hours of delay, which at $20/veh-hr is in excess of 1 billion dollars. Even if only one-half of the delay savings from IMP-metering can be practically realized, this indicates the large productivity gain that good management can achieve.

BOTTLENECKS Bottlenecks can cause congestion, which may be reduced by ramp metering. A bottleneck may be associated with physical features such as ramps, lane drops, grade changes, curvature, lane closures, and accidents; but traffic jams and congestion may 'spontaneously' arise in locations with none of these features. In the absence of a guide to locating bottlenecks and estimating their severity, we need an algorithm to automatically (1) identify all bottlenecks, and (2) calculate the delay each one causes. Such an algorithm is reported in (Chen, C. et al., 2004b), and applied using flow and speed data from 263 VDSs on 270 miles of seven freeways in San Diego. The algorithm uses a sustained speed gradient between a pair of upstream-downstream detectors to identify bottlenecks. We describe the algorithm. Consider a freeway with n detectors indexed i = 1,..., n, each giving speed and flow measurements, averaged over 5-minute intervals indexed t= \, 2,... Detector i is located at postmile x,-; v,-(f) = v(x,-, t) is its speed (miles per hour, mph) and qi{t) = q(xt, t) is its flow (vehicles per hour, vph) at time t. If*,- < Xj, it is understood that x, is upstream of x,. The algorithm has four steps. First, it declares an active bottleneck at certain locations and times if the data meet criteria (l)-(4) below. Second, it includes additional time periods as part of the same bottleneck activation, provided nearby time intervals are selected in the first step. The criterion for this is (5). Third, it calculates the delay caused by a bottleneck, using (9). Lastly, identified bottlenecks are ranked in terms of frequency of occurrence and severity to isolate recurrent from transitory bottlenecks and to help prioritize mitigation efforts. Step 1 Declare an active bottleneck between locations x,- < xj during t if all four inequalities hold: Xj -

2 miles,

v(xk, t) - v(xi, t)>0 ifxi <xk<xi<xh

(1)

(2)

V{XJ, t) - v(x.i, t) > 20 mph,

(3)

v(xh t) < 40 mph.

(4)

Thresholds in (l)-(4) are selected on the basis of experience. In Los Angeles free flow speed is 60 mph and, when a bottleneck is activated, speed drops rapidly to below 40 mph (e.g. figure l(b)). Hence the 20 mph minimum speed differential (3) and 40 mph congestion speed (4) thresholds. The maximum separation of 2 miles in (1) is designed to include locations where speed continues to drop as we go downstream, but the difference between each neighboring pair is small. Location xt is upstream of Xj, but there may be other detectors at xt,

Empirical assessment of traffic operations 111 x; between these locations. The constraint (2) that speed should drop continuously is the algorithm's characterization of an active bottleneck. Step 2 Sustained bottlenecks last longer than five minutes. Let A,{t) = 1 if there is an active bottleneck at location i and time period t; otherwise At(i) = 0. A bottleneck is sustained between times t\ and ?2 if ^4-(r)>9Ar,V?,
(5)

with N = 1 and q = 5/7. That is, a sustained bottleneck has at least five active bottleneck periods (25 min) within every seven consecutive periods (35 min). This ad hoc definition accounts for situations like in figure 4(a), in which at postmile 26 the bottleneck is continuously sustained between 0700 and 0800 except for several five-minute periods. The notion of sustained bottleneck allows treating this as a single bottleneck rather than two or three bottlenecks. The most downstream location of a sustained bottleneck is the location of an active bottleneck. Figure 4(a) shows the result of applying the algorithm to data from I-15S. The locations and times of detected bottlenecks are the squares superimposed on the speed contours. The contours visually suggest one sustained bottleneck between 0545 and 0945 at postmile 26, and another between 0645 and 0830 at postmile 15, and indeed both bottlenecks are identified by the algorithm. Step 3 To calculate the delay, the algorithm first delineates the space-time congested region of each bottleneck and then the delay in vehicle-hours associated with the region. As an example, the speed contour in figure 4(a) shows regions of congestion upstream of two bottleneck locations. The n detectors divide the freeway into n segments. A segment is declared congested at time t if its speed is below 40 mph. The congested region associated with a bottleneck is the contiguous group of congested segments immediately upstream of the bottleneck location. For an active bottleneck just downstream of segment 7 at time t, the congested region is the set of segments Bj(t), Bj{t) = {i: vk (t) < 40 mph, for all i < k<j) .

(6)

112 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

(a) Bottleneck location

(b) Delay distribution

Figure 4 (a) Bottleneck detection on 1-15 SB on 5/1/2003. Traffic flows in order of decreasing postmile. (b) Distribution of daily delay among 160 bottleneck locations; the 10 'outliers' account for 61% of the delay. Source: (Chen, C. et al., 2004b). The delay D/t) associated with the bottleneck during this period is the sum of the delays in

w> Dj{t)= £
(7)

in which d,{f) is the delay in segment i at time t. Segment delay is the additional vehicle-hours traveled driving below the free flow speed, 60 mph,

d,(t) = I, x q,(t)x 4 T " — I v/=

60 m h

P-

(8)

Here /,, q,{t), and v,{t) are the segment length, volume, and average speed on the segment at t. The total delay attributed to a bottleneck at segment j that is active between times t\ and ^ is

£>,(<„/,) = £/>,(/).

(9)

Step 4 The steps above found 1733 sustained bottlenecks distributed over 160 distinct locations on 64 workdays. These bottlenecks represent all locations and times that satisfy equations (l)-(5). Their causes are unknown, and may include incidents or recurring conditions. The delay associated with each detected bottleneck is computed using (9). The total delay associated with bottlenecks during the test period is 1.2 million vehicle-hours, which is 64% of the total delay measured on these freeways during this period. Of the delay caused by bottlenecks, 61% is attributed to the top ten locations alone. These are the outliers in figure 4(b).

Empirical assessment of traffic operations 113

THE PIE OF CONGESTION Bottlenecks cause congestion, some of which can potentially be removed by ramp metering. Collisions also cause congestion. These considerations lead to the 'congestion account' (10)(13), for a contiguous section of freeway with n detectors indexed i = 1, ... , n. Using the earlier notation, d(t) = lxq(t)x\

vehicle-hours,

Dm'YLd^t),

(10)

(11)

i=\ t=\

Drec=Dtol-Dcol, (12) Dtot = Dcoi+Dpot + Drem. (13) Here v/ is the reference speed, 60 mph. So d,{t) is the delay in segment i in interval /, and Dtot is the total delay in the section. Both d,{f) and Dtol(t) are directly obtained from PeMS. Dcoi is the delay caused by collisions, which has to be estimated. Drec, as defined in (12), is often called the 'recurrent' congestion, much of which occurs at bottlenecks. A significant amount of Drec can be potentially eliminated by ramp metering. We call this amount Dpot, which also needs to be estimated. Putting all these definitions together gives the summary (13), in which Drem is the 'residual' congestion. Drem is largely due to 'excess demand' whose impact cannot be eliminated by ramp metering, and shows up as delay at ramps. Drem also includes the contribution to congestion of all other causes, such as adverse weather and special events. The study in (Kwon, J. and P. Varaiya, 2005) proposes an automated procedure to estimate all three components in (13), using PeMS loop data and collision data from Traffic Accident Surveillance and Analysis System (TASAS) maintained by Caltrans. The procedure is applied to a 22.5 mile (postmile 4.5 to 27) section of I-15N in San Diego County. The time period is from 0500 to 2200, for 44 weekdays (2 September-31 October, 2002). Figure 5 summarizes the study's conclusions. The total average daily congestion pie is divided into three slices. If Dpot and Drem are reported together as Drec, recurrent congestion would amount to 70%. As Hallenbach et al. (Hallenbach, M.E. et al., 2003, p. 11) observe, this large 'recurrent' congestion may in part be caused by "unusual volume surges at ramps . . . that are not being effectively handled by the ramp metering program." Figure 5(a) indicates that 30% of the total congestion (or 60% of recurrent congestion) can be removed by IMP-metering that effectively handles these volume surges. Figure 5(b) summarizes traveler exposure to congestion in the study section. Travelers spend 89% of their time in the free-flow regime, and 11 % in the congestion regime. The pies in figure 5 are for the study section. We are in the process of constructing congestion portraits for all freeways in California for which PeMS and TASAS data are available. We now discuss the four-step procedure in (Kwon, J. and P. Varaiya, 2005). The first step delineates for each collision its time-space region of impact. The second step predicts how much delay this region would have experienced had that collision not occurred; this is the

114 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction recurrent congestion. The third step calculates how much of this recurrent congestion can be eliminated by IMP-metering. The fourth step puts the estimates together in the congestion pie. Step 1 Following a collision, congestion propagates upstream up to some maximum spatial extent. The congestion lasts a certain amount of time, called its duration. Empirically, freeway segment i is declared congested during a 5-minute interval t if the speed v,{t) < 50 mph. (This is slightly different from the 40 mph criterion in (3).) Formula (10) is then used to calculate the total delay in this duration-extent 'rectangle'. The precise algorithm is similar to that of (6)-(7). The step leads to the estimate Dtota(X) of the total delay at time t in the impact region of each collision a. (The same procedure can be used to delineate the impact of noncollision incidents.)

(a) Congestion pie

(b) Exposure pie

Figure 5 The congestion pie (a) and the exposure pie (b) automatically constructed for the I15N study section. Source: (Kwon, J. and P. Varaiya, 2005). Step 2 This step predicts Dreca{t), the recurrent congestion at time t that would have occurred in the absence of collision a. This is the AT-nearest neighbor prediction of the recurrent delay, based on historical data of the delay Da(t,d) during the same time t and over the same spatial extent, for several other days d = 1, ..., T. More precisely, the estimate is the median value Deceit) = median{Da{t, dk") \ k= 1, ... , K}, in which dt', k = 1, ... , K are K days with smallest values of | Da(t, d) - Dtota{ta) | for d = 1, ... , D. (hi the empirical study of figure 5, K = 3.) Here ta is the time just before collision a occurred. The recurrent congestion that would have occurred in the absence of collision a is predicted to be Drec.a = 2 j Drec,a if),

in which the sum is over the duration of impact. Finally, Dcoi,a = I , max(Ao<.a(0 - Drec,a(t),0),

(14)

is the contribution to congestion of collision a. Step 3 This step estimates the potential reduction in delay at recurrent bottlenecks by IMPmetering discussed in IDEAL METERING. The procedure first identifies all bottlenecks following the algorithm in BOTTLENECKS, and restricts attention to those that occur for more than 20% of the days. Next, if the bottleneck time-space region overlaps with the impact

Empirical assessment of traffic operations 115 region of a collision, that day is excluded. An estimate of the reduction in delay is then computed using the procedure in IDEAL METERING. This gives an estimate ofDpo, in (13). Details are in (Kwon, J. and P. Varaiya, 2005). Step 4 The three delays estimated above, together with overall VHT from PeMS are displayed in the pies of figure 5. An important side-effect of the procedure is an estimate of the delay caused by each collision, Dcoi,a (14). Of 74 collisions during the study period, two-thirds cause no additional delay. These occur either when recurrent congestion is very low or very high. Eight outliers (10% of collisions) account for 90% of total collision delay. Incident management could be made more effective if the high delay-causing accidents could be quickly diagnosed once they occur. TASAS provides crash information including type of collision, number of vehicles involved, and weather. From this information, we find the strongest predictor of high delay-causing accidents is the number of vehicles involved; adverse weather is a moderately strong predictor; all others, including injury and trucks, are weak predictors. Note, however, that the data set in the study contains only 74 collisions.

PREDICTING TRAVEL TIME The delay on a freeway on the same day of week varies much more than the total demand. Travelers experience this variation as large uncertainty in their travel time. Let T(t) be the travel time of a trip over a fixed route starting at time t. T(t) is a stochastic process with trends that can be calculated from historical data, and a large variance due to congestion. Let a (i) be the (unconditional) variance of T(t), and let a (t,s) be the variance of the predictor T(t,s) of this travel time conditioned on knowledge of traffic conditions up to time s < t. Because the travel time process has a large autocorrelation, a2(t, s) is much smaller than o2(t). We summarize a study (Chen, C. et al., 2004a) which estimates the benefits of prediction.

(a) Travel times

(b) Prediction vs. historical

(c) Prediction vs. optimum

Figure 6 The scatter plot of travel times along the two routes (a). Comparison of minimum predicted travel time vs. historical (b) and vs. true minimum travel time (c). Source: (Chen, C. et al., 2004a). The study compares travel time along two alternate routes between the I-5/I-805 interchange and the I-5/I-163 interchange in San Diego. Route 1 is entirely along I-5S, Route 2 has its first segment on 1-805 and the second segment on I-163S. Travel times T\{t) and T2(t) along the

116 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction two routes are computed for departure times t between 0500 and 2200 during the 22 weekdays between 1 and 31 August, 2002. There are 1320 departure times over the study period at every 17 minutes. Each point in the scatter plot of figure 6(a) represents (T\(t), T2(t)) with the same departure time /. There are 1320 points. Two features of the scatter plot are clear. First, the travel time distributions on the two routes are similar. Second, there is a large uncertainty: 90% of the distribution lies between 12 and 35 minutes, with a median below 20 min. A PeMS application predicts travel time T(t,t) for a trip starting at any time t, based on historical data and real time data available up to time t (van Zwet, E. and J. Rice, 2001). We now estimate the travel time savings using the PeMS prediction. Figure 6(b) compares the travel time that would be experienced by a traveler who selects the route with the shorter predicted travel time mini f;(t,t), with that of a traveler who selects the route with the shorter expected travel time min; ET,{t), which can be estimated from historical data alone. Most of the points lie on or below the 45 degree line, indicating that reliance on PeMS prediction is much better than historical experience. The travel time saving is the horizontal distance to the 45 degree line. Figure 6(c) compares the travel time based on PeMS prediction with that of a clairvoyant traveler who unerringly chooses the route with the shorter travel time, min,- Tfo). Naturally, all points lie below the 45 degree line, but the significant feature is how frequently the points lie on the 45 degree line, indicating that prediction correctly selects the ex post shorter route. When there are alternative routes as is the case here, accurate travel time prediction reduces both the average travel time and the uncertainty. Even when alternative routes are not available, the reduction in uncertainty increases traveler welfare. Estimates in (Chen, C. et al., 2004a) suggest that the benefits are significant for the example presented here. As a final remark we note that the travel time estimate in (van Zwet, E. and J. Rice, 2001) involves predicting the traffic conditions that the traveler will encounter along the route. Such a predictor performs much better than the commonly used predictor which simply adds up the most recently reported travel times on the segments along the route.

EVALUATION OF HOV LANE EFFECTIVENESS Several studies reach the obvious conclusion that HOV travelers benefit from lower travel times, see e.g. (DKS Associates, 2003; The PB Study Team, 2002). But these studies do not evaluate the impact of HOV lanes on overall congestion, including the congestion on mixedflow lanes. San Francisco Bay Area data are especially helpful in evaluating this impact, because its HOV lanes are time-actuated. To facilitate comparison, the evidence below is for freeways with heavier PM peak traffic. In all cases, lane 1 (the fast lane) is HOV actuated on weekdays between 0500-0900 (5:00-9:00 AM) in the morning and 1500-1900 (3:00-7:00 PM) in the evening; at all other times HOV is deactuated. We argue that in the Bay Area, HOV lanes increase overall congestion. Figure 7 shows speed and flow on all three lanes, l(HOV), 2 and 3, of SR-237E at a particular location on 18 August 2004. During the 0500-0900 HOV actuation period, the HOV lane is underutilized, but since overall traffic is low, all lanes are in the free flow regime. (Most HOV lanes in the off-peak direction are underused (DKS Associates, 2003, Table 3, p.7).)

Empirical assessment of traffic operations 117 Immediately after deactuation at 0900, speed and flow are (nearly) equalized on all lanes, and they remain in the free flow regime until HOV re-actuation at 1500. At 1500 HOV flow drops dramatically, compensated by increased flows in lanes 2 and 3. But until 1700, all three lanes remain in the free flow regime, and flows in lanes 2 and 3 reach a maximum. From 1700 until 1900, HOV flow increases and speed decreases, and the HOV lane remains in free flow. However, lanes 2 and 3 enter the congestion regime. They suffer a large reduction in both speed and flow. The decline in flow is severe enough to reach the level of the HOV lane at 1900. TnTj

Figure 7 Speed and flow in lanes l(HOV), 2 and 3 on 18 August 2004 at VDS 400104 on SR-237E. The impact of HOV actuation on overall congestion can be seen by comparing the behavior before and after HOV deactuation at 1900 in figure 7. HOV activation during 1700-1900 reduces capacity for non-HOV traffic (which loses one lane), pushes non-HOV lanes into the congestion regime, and reduces total non-HOV flow. Thus traffic suffers a non-HOV congestion penalty. Shortly after deactuation at 1900, all lanes enter the free flow regime, and total flow reaches a maximum over the entire day. More surprisingly, even HOV lane performance improves after deactuation: both speed and flow increase. Put inversely, both speed and flow in the HOV lane decline during HOV actuation, even though it is in free flow. We call this the HOV capacity penalty. In summary: HOV actuation imposes a congestion penalty on non-HOV lanes and a capacity penalty on the HOV lane. The HOV capacity penalty—increased HOV speed and flow after deactuation—is seen in the six freeway locations we examined. Figure 8 shows speeds in six different freeways during 1400-2000, starting one hour before the afternoon HOV actuation at 1500 and ending one hour after HOV deactuation at 1900. (Flows are not shown as they have the expected behavior, similar to that in figure 7.) In all cases, speeds in all lanes, including lane l(HOV), increase after deactuation; moreover, flow in lane l(HOV) increases, and flows in the other lanes decrease. The lane l(HOV) capacity penalty is explained as follows. The flow increases after deactuation because drivers in lane 2 move into the lower density lane 1. The speed decreases

118 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction during HOV actuation because the HOV lane becomes a one-lane highway whose speed is governed by the low speed vehicles—the 'snails'. As the non-HOV congested lanes are even slower, a faster HOV driver cannot pass the slower snail in front of it. However, as soon as HOV is deactuated, slower drivers move to the outer lanes and the fastest drivers move to (what was) the HOV lane. Speed in all lanes increase—usually dramatically as in figure 8. The hypothesis that during HOV actuation speed is controlled by snails is confirmed in the scatter plots of figure 9. Each point is a 5-minute average of flow and speed. Plot (a), during HOV actuation, shows a sharp decrease in speed as flow (and hence the number of snails) increases, even though the lane is in free flow. Plot (b) shows no decrease in speed, as only the fast drivers are in lane 1. The difference between plots (a) and (b) is typical of a one-lane vs. a multi-lane highway in free flow.

Figure 8 Speeds in all lanes at locations on six different Bay Area freeways, 1400-2000, beginning one hour before HOV activation (at 1500) and ending one hour after HOV activation (at 1900). In all cases, speed is highest in lane 1 (HOV), followed by lane 2, lane 3, etc. The notation 80E-400808-080804 means VDS 400808 on I-80E on August 8, 2004. Three different (non-exclusive) causes may account for snails. A proportion of HOV drivers may be intrinsically slow, so their number grows as HOV flow increases. Second, the slowdown may be caused by lane changes by HOV drivers (and SOV violators) from the slower lane 2 into the HOV lane. The lane changes increase in proportion with HOV lane flow, farther reducing HOV lane speed. Third, as the speed differential between the HOV and

Empirical assessment of traffic operations 119 the adjacent non-HOV lane increases, drivers in the HOV lane may slow down due to the increased perceived risk of a collision should someone from the non-HOV lane merge into the HOV lane. In the last two cases, an HOV lane that is physically separated from lane 2 would not exhibit the slowdown seen in figure 9 (a), hi either case, the slowdown would not be seen in freeways with two HOV lanes.

Figure 9 Speed vs. flow (5-min averages) in lane 1, (a) 1600-1900, HOV actuated, and (b) 1900-2100, HOV de-actuated, for five weekdays in August, 2004, at VDS 400352 on I-880S. We finally arrive at the interesting question: "Will the overall congestion in the six cases in figure 8 be reduced by eliminating the HOV lane?" The answer would be unreservedly "yes," but for two qualifications: one having to do with freeway management, the other with mode choice. It is obvious that a management strategy with no HOV lane and no metering will lead to greater congestion than a strategy with one HOV lane and no metering, because HOV actuation serves as a (one-lane) metering mechanism. So to fairly compare an HOV vs. a nonHOV regime, we must assume that proper ramp metering is in place to guarantee vehicle flow in non-HOV lanes that is close to maximum observed vehicle flow. The second qualification is more interesting. It is based on either of two claims: (1) HOV lanes move significantly more people overall (even if they don't move more vehicles), (2) HOV lanes induce enough drivers to switch from SOV to HOV to compensate for both the congestion penalty imposed on non-HOV lanes and the capacity penalty imposed on the HOV lane by HOV actuation. We cannot address the second claim because there are no empirical estimates of the SOV-HOV mode shift for the Bay Area. We evaluate the first claim that HOV actuation increases flow of persons/hour. We calculate flow of persons per hour (PPH) by multiplying vehicle flow (from PeMS) and AVO (average vehicle occupancy). Since the accuracy of vehicle counts exceeds 90-95%, the single most important empirical quantity in any study of HOV effectiveness is the AVO. Unfortunately, AVO estimates are very unreliable for many reasons (Levine, N. and M. Wachs, 1994), so we will use a range of estimates. According to (California Department of Transportation, District 4, Office of Highway Operations, 2002, p. 66) on the section of I-880S that includes VDS 400486 in figure 8,

120 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction during the afternoon peak hour, the HOV lane AVO is 2.1, and the AVO on the three nonHOV lanes is 1.1. We use these estimates for the HOV actuation period. (The HOV AVO rate should be reduced by a highly variable HOV violation rate measured at 5.8% on 5 July, 2002.) AVO estimates during HOV deactuation are not available, and we have several alternatives: the State Household Travel Survey gives an AVO of 1.5 for all trips and 1.1 for home to work trips; the Metropolitan Transportation Commission for the Bay Area gives an AVO of 1.4 for all trips and 1.1 for home to work trips; lastly, the California Life-Cycle Benefit/Cost Analysis Model uses a default of 1.38 for peak period AVO. We will use 1.25, 1.3 and 1.4 for AVO during HOV deactuation. Figure 10 (a) plots the flow in persons per 5-minutes, aggregated over all lanes, with HOV AVO = 2.1 and non-HOV AVO = 1.1 during HOV actuation, and AVO = 1.25, 1.3 or 1.4 during HOV deactuation.

Figure 10 (a) Flow in persons per 5-min using the indicated AVO values, and (b) cost index per person-mile and per vehicle-mile, 1400-2000, 18 August 2004, at VDS 400486,1-880S. With the two higher AVO estimates, HOV actuation causes a reduction in the flow of persons per hour compared with the period 1400-1500 before actuation. With the lowest AVO estimate, HOV actuation causes a small increase in PPH compared with the period 14001500. So the data do not support the claim that HOV actuation significantly increases (say by 10%) the flow in persons per hour. In comparing the HOV vs. non-HOV regime, we should not ignore the travel time cost imposed by HOV actuation. Knowing the speed and the flow in persons/5-min and vehicles/5min in each lane, we can calculate the amount of time that each person and vehicle takes to travel a fixed distance. This gives us a 'cost index', which will vary over time, as the flow and speed vary. Figure 10 (b) displays the two cost indices (AVO = 1.25 is used for these plots). Evidently, the average person (on all lanes) pays a travel time cost during HOV actuation (1700-1900) that is two-and-a-half times higher. Of course, a significant part of this higher cost is due to inadequate ramp metering. If we think of the freeway as a 'people-mover' and the cost of its operation to be travel time, we must conclude that the cost is increased during HOV actuation. This is a much better indicator of productivity loss than the productivity gain measured as the ratio between HOV AVO and non-HOV AVO in (DKS Associates, 2003, p.6,8). The latter productivity gain merely reflects the fact that HOV actuation causes carpools to move into the HOV lane.

Empirical assessment of traffic operations 121 We close this section with some remarks. First, the analysis above leads to conclusions that run counter to those reached by most studies of HOV effectiveness. Because the evidence presented here is fragmentary, the analysis must be repeated with a more complete data set before the conclusions can be trusted. Second, it is possible from Bay Area data to estimate the SOV-HOV mode shift, based on the hypothesis that the shift will be more pronounced the larger is the travel time differential between HOV and non-HOV lanes. Also, people may find the SOV-HOV shift to be less inconvenient on some routes than on others. Third, when a 2+ (i.e. two or more persons) HOV lane becomes congested, it is sometimes converted to a 3+ lane. The SOV-3+HOV shift will certainly be lower than the SOV-2+HOV shift. So the conversion from 2+ to 3+HOV lane may increase overall congestion. Lastly, because HOV lanes in the Bay Area are time-actuated, it is straightforward to estimate both the non-HOV congestion penalty and the HOV capacity penalty. This distinction is less obvious in a 24-hour HOV facility, although it, too, imposes both penalties. There is interest in increasing the utilization of underused HOV lanes by converting them into HOT (HOV/Toll) lanes. The snail phenomenon implies, however, that even modest increases in volume following conversion will bring down HOT speed to that of non-HOT lanes (which, moreover, will have higher speed because they carry less traffic). That is, the HOV capacity penalty does not leave much room for additional traffic, so that even the cautious estimates for revenue enhancement in the Bay Area may be overly optimistic (DKS Associates, 2003, p. 22). A recent proposal to permit hybrid vehicles into HOV lanes will certainly increase congestion. From a purely technical viewpoint, this discussion suggests that a better way to manage freeways is to eliminate HOV lanes, institute ramp metering, and permit HOV/HOT bypass at ramps. This will eliminate the HOV penalties, while encouraging mode shift from SOV to HOV. On the other hand, by having weak or no ramp metering, the HOV regime can always be made to look better. For policy considerations this technical viewpoint has to be weighed with many other factors.

CONCLUSIONS In its draft Transportation Management Systems (TMS) Master Plan (System Metrics Group, Inc., 2003), Caltrans proposes an action plan to improve incident management, traffic control, and traveler information. Central to the plan is its reliance on specific performance indicators to serve "as monitoring and evaluation tools, and establish an accountability framework for the implementation of planned TMS improvements." Caltrans has invested significant resources to develop a performance measurement system (PeMS). This paper illustrates why PeMS became a major source of 'performance indicators' and suggestions for performance targets. The paper uses PeMS data to study freeway congestion from six different perspectives, ranging from identification of bottlenecks to evaluating the benefits of ramp metering and the effectiveness of HOV lanes. In each study, the aim is to measure the severity of congestion and reveal the opportunity for improvement. The approach is to argue on the basis of

122 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction statistical models that the data are used to estimate. Qualitatively of course the models are inspired by prior theory, but the emphasis is always on quantitative conclusions. Partly motivated by the success of PeMS, some universities and DoTs are developing smallscale prototypes of PeMS-like systems. These efforts will have a small impact until state DoTs invest in data collection infrastructure. The availability of these data will shift DoT focus from construction to operations improvements. Academic research, too, will change as it exploits opportunities opened up by access to large-scale data sets and pays more attention to questions that address the opportunities for operations improvements and conducting experiments that demonstrate improvements.

ACKNOWLEDGEMENT We are grateful for comments and criticism from Professor Martin Wachs of U.C. Berkeley; Robert Copp, Fred Dial, David Seriani and John Wolf of Caltrans; and Tarek Hatata of the System Metrics Group. This study is partly based on research that was supported by grants from Caltrans to the California PATH Program. The contents of this paper reflect the views of the authors who are responsible for the facts and the accuracy of the data presented herein. The contents do not necessarily reflect the official views of or policy of the California Department of Transportation. This paper does not constitute a standard, specification or regulation.

REFERENCES California Department of Transportation, District 4, Office of Highway Operations. (2002) HOV lanes in the Bay Area. Chen, C , Z. Jia, and P. Varaiya. (2001). Causes and cures of highway congestion. IEEE Control Systems Magazine, 21(4):26-33. Chen, C. and P. Varaiya. (2001). Max flow in D12 occurs at 60 mph. http://pems.eecs.berkeley.edu. Chen, C , A. Skabardonis, and P. Varaiya. (2004a). A system for displaying travel times on changeable message signs. In Proceedings of 83rd Transportation Research Board Annual Meeting, Washington, D.C., January 2004. Chen, C , A. Skabardonis, and P. Varaiya. (2004b). Systematic identification of freeway bottlenecks. In Proceedings of 83rd Transportation Research Board Annual Meeting, Washington, D.C., January 2004. DKS Associates. (2002). 2002 High Occupancy Vehicle (HOV) Lane Master Plan Update. Prepared for Metropolitan Transportation Commission, Caltrans District 4 and the California Highway Patrol Golden Gate Division. Hallenbach, M.E., J.M. Ishimaru, and J. Nee. (2003). Measurement of recurring versus nonrecurring congestion. Washington State Transportation Center (TRAC). Jia, Z., P. Varaiya, C. Chen, K. Petty, and A. Skabardonis. (2000). Congestion, excess demand and effective capacity in California freeways, http://pems.eecs.berkeley.edu.

Empirical assessment of traffic operations 123 Jia, Z., P. Varaiya, C. Chen, K. Petty, and A. Skabardonis. (2001). Maximum throughput in LA occurs at 60 mph. http://pems.eecs.berkeley.edu. Kwon, J. and P. Varaiya. (2005). The congestion pie: delay from collisions, potential ramp metering gain, and excess demand. In Proceedings of 84th Transportation Research Board Annual Meeting, Washington, D.C., January 2005. Levine, N. and M. Wachs. (1994). Methodology for Vehicle Occupancy Measurement. Report submitted to the California Air Resources Board and the California Department of Transportation (Office of Traffic Improvement). Martin, W. (1998). Travel estimation techniques for urban planning. Technical report, NCHRP Report 365, Transportation Research Board, Washington, D.C. Papageorgiou, M. (1983). Applications of Automatic Control Concepts to Traffic Flow Modeling and Control. Lecture Notes in Control and Information Sciences, Vol. 50. Springer. Papageorgiou, M., H. Hadj-Salem, and J. Blosseville. (1991). ALINEA: a local feedback control law for on-ramp metering. Transportation Research Record, 1320, 58-64. System Metrics Group, Inc. (2003). Transportation Management Systems Plan. The PB Study Team. (2002). HOV Performance Program Evaluation Report. Los Angeles County Metropolitan Transportation Authority, van Zwet, E. and J. Rice. (2001). A simple and effective method for predicting travel times on freeways. In 2001 IEEE Intelligent Transportation Systems Proceedings, pp. 227-232, Oakland, CA, 2001.

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RELIABILITY OF FREEWAY TRAFFIC FLOW: A STOCHASTIC CONCEPT OF CAPACITY Werner Brilon, Justin Geistefeldt, and Matthias Regler, Institute for Transportation and Traffic Engineering, Ruhr-University, Bochum, Germany

INTRODUCTION Conventional measures of effectiveness for freeway facilities usually reflect travel time in the form of e.g. travel velocity or delay. Recently it has been becoming more and more obvious that these parameters are not sufficient for freeway traffic performance assessment since they place great emphasis on smaller differences in travel time, whereas the more significant difference between flowing traffic and congestion is not adequately represented. In addition, traditional quality assessment fails whenever demand exceeds capacity because it will simply attest a failure in this case. However, temporary freeway overloads are quite common. This is why quality assessments for different degrees of freeway congestion are required as well (cf. Shaw, 2003). The capacity of a freeway is traditionally treated as a constant value in traffic engineering guidelines around the world, such as the HCM (2000). Doubts about this nature of capacities as constant values were raised by Ponzlet (1996) who demonstrated that capacities vary according to external conditions like dry or wet road surfaces, daylight or darkness, and prevailing purpose of the freeway (long distance or metropolitan commuter traffic). Moreover, several authors affirmed that even under constant external conditions, different capacities can be observed on freeways in reality (Elefteriadou et al., 1995; Minderhoud et al, 1997; Persaud et al., 1998; Kuehne and Anstett, 1999; Lorenz and Elefteriadou, 2000; Okamura et al., 2000). Most of these authors only observed traffic breakdowns at different flow rates to demonstrate the variability of flows preceding a breakdown. For a more systematic analysis, however, a comprehensive theoretical concept is required.

126 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

BASIC CONCEPT OF STOCHASTIC CAPACITY Corresponding to the HCM (2000), the capacity of a freeway is defined as the maximum flow rate that can reasonably be expected to traverse a facility under prevailing roadway, traffic, and control conditions. With an identical meaning, the term "capacity", i.e. the maximum flow rate, could also be defined as the traffic volume below which the performance of the facility is acceptable and above which - in case of greater demand - proper operation fails. The transition between proper operation and non-acceptable flow conditions is called "breakdown". On a freeway, such a breakdown occurs when the average travel velocity is reduced from an acceptable speed level to a much lower value of congested conditions. These transitions usually involve a rather sudden speed reduction. The suddenness of this breakdown, however, may differ from one country to another depending on the general driving culture. From this definition, it is clear that capacity in this sense is by no means a constant value. A constant value would mean that, given a capacity of e.g. 3,600 veh/h, the traffic should be fluent at a demand of 3,599 veh/h and be congested at a demand of 3,601 veh/h. This clearly indicates that the demand volume that causes breakdown varies in real traffic flow and that the flow rate of a breakdown depends on the behavior of several drivers combined with the specific local constellation on the freeway. Thus, it is plausible that the breakdown volume should have all properties of a random variable. For using this concept of freeway capacity randomness it is necessary to know more about the capacity distribution function. Its determination is, however, not a trivial task. It is clear that any analytical approach must be supported by a broad empirical investigation. Observations of traffic flow on freeways deliver pairs of values of traffic flow rates and average speeds during predetermined observation intervals (index i). According to the definition of capacity, the observed volume will be below capacity if the average speed exceeds a certain threshold value (e.g. about 70 km/h for German freeway conditions). With an average speed lower than the threshold value, traffic flow is congested. Thus, the flow must have exceeded capacity during the time between two such observations. The capacity itself, however, can not be measured directly. In addition, higher demand volumes are less likely to be observed in the field since there is also a higher probability that, before they occur, the breakdown has already happened during preceding intervals at lower volumes. Both effects make it difficult to estimate the capacity distribution function, which is defined

Fc(q) = p(c
(1)

where Fc(q) c q

= capacity distribution function = capacity = traffic volume

[-] [veh/h] [veh/h]

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127

A practicable estimation method was first presented by van Toorenburg (1986) and discussed by Minderhoud et al. (1997). The investigations presented here are based on this idea. However, it seemed advisable to modify some of the basic assumptions of the method. The method proposed by van Toorenburg (1986) is based on an analogy to the statistics of lifetime data analysis. This statistics, in its basic formulation, serves to describe the statistical properties of the duration of human life. Moreover, it is usually applied to analyze the durability of technical components. In this context, the lifetime distribution function is: F(t) = l-S(t)

(2)

where F(t) T S(t)

= distribution function of lifetime = p(T t)

Lifetime distributions are often estimated on the basis of experiments of limited duration. Consequently, the lifetimes of several individuals in the sample exceed the duration of the experiment and therefore can not be measured. It is only possible to state that these lifetimes are longer than the duration of the experiment. However, even this information is valuable. These data are called "censored data" (cf. e.g. Lawless, 2003). If a traffic breakdown is regarded as a failure event, the methods for lifetime data analysis can be used to estimate the capacity c, which is the analogon of the lifetime T. The whole analogy between capacity analysis and lifetime data analysis is given in Table 1. Table 1: Analogy between lifetime data analysis and capacity analysis Analysis of Lifetime Data Timet Death at time t Lifetime T Lifetime T is longer than the Censoring duration of the experiment Survival function S(t)=l-F(t) Probability density function f(t) Probability distribution function F(t) Parameter Failure event Lifetime variable

Capacity Analysis Traffic volume q Breakdown at volume q Capacity c Capacity c is greater than traffic demand Sc(q) = l-F c (q) fc(q) Fc(q)

The statistics of lifetime data analysis can be used to estimate distribution functions based on samples that include censored data. A non-parametric method to estimate the survival function is the so-called "Product Limit Method" (PLM) by Kaplan and Meier (1958):

128 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

i—r n , — d ,

sw = n -—-

p)

where S(t) nj dj

= estimated survival function = number of individuals with a lifetime T>tj = number of deaths at time tj

[-] [-] [-]

Usually, each observed lifetime is used as one tj-value. In this case, dj in Eq. 3 is always equal tol. Transferred to capacity analysis, Eq. 3 together with Eq. 2 can be written as:

Fc(q) = l - n ^ L : i e { B }

(4)

where Fc(q) q qi k; d; {B}

= = = = = =

distribution function of capacity c traffic volume traffic volume in interval i number of intervals with a traffic volume of q >q; number of breakdowns at a volume of q; set of breakdown intervals (see below)

[-] [veh/h] [veh/h] [-] [-]

Using this equation, each observed traffic volume q is classified according to B: Traffic is fluent in time interval i, but the observed volume causes a breakdown, i.e. the average speed drops below the threshold speed in the next time interval i + 1. F: Traffic is fluent in interval i and in the following interval i + 1. This interval i contains a censored value. Its information is that the actual capacity in interval i is greater than the observed volume q;. Cl: Traffic is congested in interval i, i.e. the average speed is below the threshold value. As this interval i provides no information about the capacity, it is disregarded. C2: Traffic is fluent in interval i, but the observed volume causes a breakdown. However, in contrast to classification B, traffic is congested at a downstream cross section during interval i or i - 1. In this case, the breakdown at the observation point is supposed to be due to a tailback from downstream. As this interval i does not contain any information for the capacity assessment at the observation point, it is disregarded. The Product Limit Method does not require the assumption of a specific type of the distribution function. However, the maximum value of the capacity distribution function will only reach 1 if the maximum observed volume q was a B-value (i.e. followed by a

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129

breakdown). Only in this case, the product in Eq. 4 will be 0. Otherwise the distribution function will terminate at a value of Fc(q) < 1 at its upper end. Eq. 4 is a useful solution for estimating the capacity distribution function of a freeway from traffic observations. For practical application, two items remain to be defined: Duration At of observation intervals For analysis, only rather short observation intervals are useful. Otherwise the causal relationship between traffic volume and breakdown would be too weak. 1-hour counts, for example, are not adequate for this reason. Ideally the observation period should be 1 minute or even less. Considering both the availability of reliable data from loop detectors and the usefulness of the results, Brilon and Zurlinden (2003), after experiments with different At's, came to the conclusion that At = 5 minutes was the best compromise. Consequently, the analyses below are all based on 5-minute volume and speed values. Exact understanding of a breakdown The definition of a breakdown mentioned above (e.g. in Eq. 4) is a decisive aspect of the whole methodology. Van Toorenburg (1986; see also Minderhoud et al., 1997) defined breakdown capacity as the volume measured downstream of a queue at a bottleneck. In consequence, each congested flow volume is regarded as a B-value (see comments to Eq. 4). Within the meaning of the analogy (Table 1) this would be equivalent to including in a lifetime analysis an individual who died a while ago. This does not seem to be reasonable. Instead, only those intervals i that cause a breakdown are treated as B-intervals. As a breakdown of traffic flow usually involves a significant speed reduction, breakdown events can be detected using a time series containing both traffic volumes and average space mean speeds. This is done by using a constant threshold speed value. If the speed falls below the threshold value in the next interval i + 1, the traffic volume in interval i is regarded as a Bvalue. A threshold speed of 70 krn/h was found to be fairly representative for German freeways but may be different for other road types. In some cases, different or more detailed criteria may be required to reliably identify traffic breakdowns - e.g. a criterion that considers the minimum speed difference between the intervals i and i + 1. The capacity of a freeway section (one direction) can be analyzed most precisely if observations are made at a clearly distinguishable bottleneck, as Fig. la shows. At such a bottleneck, breakdowns should only be caused by oversaturation of the bottleneck itself. Tailback from downstream should not occur as greater capacities are always available in the succeeding section. Observations are therefore made at a point slightly upstream of the bottleneck. Such observations were performed by Brilon and Zurlinden (2003) to make sure that the external conditions were clearly in harmony with theoretical assumptions.

130 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

As an example, Fig. 2 shows speed-flow diagrams from two observation sections along the ring of freeways around the city of Cologne. Both sites are geometric bottlenecks with the road widening downstream of the observation point. The figure shows speed-flow data for 5minute counts across all lanes obtained from automatic loop detectors throughout the year 2000. Due to frequent oversaturation of both freeway sections, many congested intervals were observed. With the conventional approach (estimation of a regression model in the k-v diagram plus deriving the maximum flow rate from q = k v), a capacity of 4284 veh/h for the 2-lane example and 6720 veh/h for the 3-lane case was determined based on the speed-flow relationship proposed by van Aerde (1995).

Fig. 2: Speed-flow diagrams for two freeway sections (5-minute intervals) The Product Limit technique (Eq. 4) was used to estimate the capacity distribution function for both examples (see black lines in Fig. 3). 933 intervals with a breakdown (classification B) were identified on the Al freeway and 834 on the A3 freeway. It appears that, despite the large size of a one-year sample, no complete distribution function could be estimated since the highest q-values observed were not followed by a breakdown. This effect makes it difficult to find an appropriate estimate for the whole capacity distribution function.

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Fig. 3: Estimated capacity distribution functions for two freeway sections according to Fig. la (5-minute intervals, dry roadway conditions) To overcome this problem, it is necessary to know more about the mathematical type of the distribution function Fc(x), which did not have to be defined for Eq. 4. Various plausible function types like Weibull, Normal and Gamma distribution were tested (Brilon and Zurlinden, 2003). To estimate the parameters of the distribution functions, a maximum likelihood technique was used. The likelihood function is given by (cf. Lawless, 2003):

L = flf c (q i ) 5l -[l-F c (q 1 )] 1 ^

(5)

where fc(qi) F c (q0 n 8j 8;

= = = = =

statistical density function of capacity c cumulative distribution function of capacity c number of intervals 1, if uncensored (breakdown of classification B) 0, elsewhere

[-] [-] [-]

The likelihood function or its natural logarithm (log-likelihood) has to be maximized to calibrate the parameters of the distribution function (cf. e.g. Lawless, 2003). By comparing different types of functions based on the value of the likelihood function, the Weibull distribution turned out to be the function that best fitted the observations at all freeway sections under investigation. The Weibull distribution function is:

-f-T F(x) = l - e

w

for x > 0

(6)

where a

= shape parameter

[-]

132 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction (3

= scale parameter

[-]

The two examples in Fig. 3 show that the Weibull distribution fits very well into the PLM estimation. Of course, the Product Limit Method for capacity estimation could also be used with traffic densities k instead of volumes q. Eq. 4 and 5 remain unchanged, except that q is replaced by k. It was expected that k would be a better determinant for the occurrence of a breakdown than q and that, consequently, the analysis would reduce the variability of the resulting distribution function. This was tested by Regler (2004) using data from six 3-lane freeway sections. The median of the breakdown densities ranged from 70 to 90 veh/km with Weibullparameters a = 8.4 through 13.2 and p = 72 through 92 veh/km for the analysis of 5-minute intervals. The resulting distribution of breakdown density and breakdown capacity tended towards larger variances compared to the analysis over the q-axis. Therefore, it was not advantageous in this context to estimate capacity distributions from densities. Moreover, densities are artificial parameters that must be calculated from measured speeds v and volumes q (k = q / v). The k-based analysis also needs a more complicated definition of the speed threshold between fluent and congested traffic. Therefore, using the Product Limit estimation based on densities is not recommended.

APPLICATION TO FREEWAYS So far, only results from freeway bottlenecks according to Fig. la have been discussed. Under this assumption, the estimation technique would be restricted to rather specific geometric situations. Regler (2004) applied the methods described above to freeway sections without a distinct bottleneck (Fig. lb) whose geometric properties included no change in the number of lanes. To make sure that intervals with a tailback from downstream bottlenecks did not impair the results, classification C2 (see above) was considered as well. With this technique, the capacity distribution of quite a variety of freeway sections could be analyzed (Table 2). The analysis was based on 5-minute counts taken over several months, with an arrangement as shown in Fig. lb. All sites are on 3-lane freeway carriageways, mainly in level terrain. Periods of work zones were excluded from the data. It turns out that the shape parameter a in the Weibull distribution typically ranges from 9 to 15 with an average of 13. This magnitude seems to be characteristical for 3-lane freeways. The scale parameter p of the Weibull distribution varies over a wide range between the analyzed sections. This may be mostly due to different geometric and control conditions, different driver and vehicle populations, and diverse prevailing travel purposes (long distance travel versus metropolitan commuter traffic).

Reliability of freeway traffic flow 133 Table 2: Parameters p and a, expectation E(c) and standard deviation c of the estimated Weibull capacity distribution at 15 freeway sections (3-lanes, 5-minute intervals) Section A3-1 A5-1 A5-2 A5-4 A5-5 A5-6 A5-7 A5-8 A9-1 A9-2 A9-3 A9-4 A9-5 A9-6 A9-7

P [veh/h] 7441 6217 6074 6608 6392 6272 7194 6884 7937 7399 5988 6141 6648 7109 6648

a[-] 11,31 11,15 13,59 13,92 14,16 14,69 13,98 13,35 8,85 13,66 14,82 18,86 14,24 9,62 14,92

E(c) [veh/h] 7115 5941 5847 6365 6161 6053 6932 6622 7510 7124 5780 5969 6409 6752 6419

a [veh/h] 762 645 526 559 532 505 606 606 1013 637 478 392 551 842 528

Having found that the shape parameter a seems to be almost constant, we may transform the capacity distribution function to fit different interval durations A. According to Eq. 1, F5(q) is the probability of a breakdown during A = 5 minutes at flow rate q. Hence, (1 - Fs(q)) is the probability of no breakdown occurring in this interval. If we assume that breakdowns occurring in succeeding intervals are independent of each other, then the probability of fluent traffic flow during a whole hour is p60 (fluent traffic) = [l - F5 (q)] n

(7)

Using the Weibull distribution (Eq. 6), this is converted into i2

F « ( q ) = l - P 6 o (fluent traffic) = l - e

(-T

UiJ

(8)

which is again a Weibull distribution with an unchanged shape parameter a and a scale parameter p6o = r p 5 , where r = 12("1/a). With a = 13 (see above) we get r « 0.82 » 1/1.2. This means that for 5-minute observations the expected capacity should be in a range of 1.2 times the 1-hour capacity. This factor of 1.2 seems to be typical for the transformation of capacities from 5-minute intervals into 60-minute intervals, as was pointed out by Keller and Sachse (1992) or Ponzlet (1996) based on empirical capacity estimates obtained from the fundamental diagram. One might object to Eq. 8 that the traffic volume q usually is not constant during a whole hour. However, numerical calculations showed that volume variations during one hour did not

134 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction significantly change the results. It also seems to be realistic to assume that traffic breakdowns in succeeding intervals are independent of each other since there is no imaginable reason why the opposite should be true. This question may, however, be made a subject of further research. The new technique was used to investigate differences in performance between dry and wet road surfaces. At all sections under examination (see list in Table 2), it turned out very clearly that on a wet road surface the capacity was reduced by around 11 %. The effects of darkness were investigated as well. Contrary to Ponzlet's (1996) results, it was clearly found that darkness did not shift the capacity distributions. The results shown in Table 2 also demonstrate differences in the capacity distribution between an uncontrolled freeway (section A9-3) and a freeway with traffic adaptive variable speed limits (section A9-4): The mean capacity of the controlled section is slightly (by 3 %) higher compared to the uncontrolled section, but the standard deviation is significantly lower (cf. Fig. 4). The two analyzed sections are the two opposite carriageways of the freeway A9 near Munich and thus have similar geometric and traffic characteristics.

Fig. 4: Capacity distributions for a 3-lane freeway with and without variable speed control (13.5 % average truck percentage, 5-minute intervals) All these examples demonstrate that the statistical interpretation of freeway capacity together with the corresponding estimation technique provides a better understanding of freeway traffic operation. It improves the methodology of investigating differences between various external conditions.

FREEWAY TRAFFIC DYNAMICS So far, only capacities in the upper branch of the speed-flow diagram, i.e. under fluent traffic conditions, have been analyzed. However, the lower branch of the speed-flow diagram representing congested conditions must be considered as well.

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It is well known that dynamics in the speed-flow diagram (i.e. the sequence of v-q points over time) follow specific patterns. The first to report typical hysteresis phenomena within these dynamics were Treiterer and Myers (1974). More recently, Kim and Keller (2001) came to the conclusion that six different typical traffic states should be distinguished within the speedflow diagram (or the fundamental diagram). These dynamics were analyzed by Regler (2004) based on 5-minute data for the freeway sections A3-1 and A5-7 (cf. Table 2) extending over 4 months and 10 months, respectively. More than 120 breakdowns from fluent traffic to congested traffic were observed. The analysis came to the conclusion that there are three different states of traffic conditions to be distinguished: 1. Fluent traffic at high speeds (i.e. v > 70 km/h) and low densities. In this state, volumes q may range from 0 to the maximum flow rate. This is the ascending branch of the q-k diagram. 2. A transient state with an average velocity of around 60 km/h and rather high volumes. We like to call this state "synchronized flow", knowing that this term is used by other authors (Kerner and Rehborn, 1996) with a slightly different meaning. In this state, vehicles are forced to travel at fairly similar speeds on all lanes. 3. Congested traffic with low speeds and low traffic volumes. To illustrate typical dynamic patterns, Fig. 5a shows observations from the freeway A5 with only states 1 and 2 involved. In all examples, the transition from fluent to synchronized flow began at a rather high volume. Traffic flow stabilized at slightly lower volumes at an average speed of about 60 km/h. From here, the traffic flow recovered to fluent traffic conditions. All recoveries involved much lower traffic volumes than the preceding breakdown. This hysteresis phenomenon seems to be a characteristic of traffic dynamics. In addition to these 2state sequences, Fig. 5b shows those cases from the A5 where a breakdown from fluent to synchronized flow was followed by a subsequent transition into congested traffic with very low speeds. The recovery back to fluent traffic did never happened directly. Instead, each recovery process passed through the transient state of synchronized flow.

136 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 5: Two typical patterns of traffic dynamics during breakdown and recovery (freeway section A5-7, 5-minute flow rates) Empirical analysis of freeway traffic dynamics showed that: Transitions between traffic states usually happen suddenly, i.e. within rather short times and distances. Breakdowns from fluent traffic are first followed by the synchronized traffic state. From there, speed and flow rate may decline further to heavy congestion. All recoveries pass through the synchronized state. No recovery process jumps directly from congestion back to fluent traffic. Recovery from synchronized to fluent traffic always involves much lower volumes than the breakdown. The difference between breakdown volume and recovery volume (fluent traffic after a recovery) on the observed 3-lane freeway sections ranged from 500 to 1500 veh/h. Volumes in the synchronized state are always lower than the maximum flows in fluent traffic (= capacity). Observed differences on 3-lane freeways ranged from 200 to 600 veh/h, measured in 5-minute-intervals. This effect is called "capacity drop" (cf. following section). Because of the large amount of data that was analyzed and because of the remarkable analogy between all cases observed, we may say that these properties are typical for the dynamics of freeway traffic flow. It should, however, be admitted that these dynamic effects are predominantly due to driver behavior, concerning headways at high speeds in dense fluent traffic, together with braking and accelerating behavior. Therefore, different typical dynamics might be found for other driving cultures than in Germany. Even here, under variable speed control, a few traffic breakdowns were observed where the average flow in synchronized traffic was higher than the flow rate before breakdown.

CAPACITY DROP A number of investigations has proven the existence of different capacities under flowing and congested traffic conditions. Banks (1990) as well as Hall and Agyemang-Duah (1991) analyzed this "capacity drop" phenomenon for different North-American freeways. Capacity

Reliability of freeway traffic flow 137 drop values of between 3 and 6 % were measured. Ponzlet (1996) analyzed traffic flow on German freeways to see whether this phenomenon existed. He determined a 6 % drop for 5minute flow rates. Brilon and Zurlinden (2003) analyzed the capacity drop by comparing the stochastic capacity to flow rates in congested flow. They computed an average of 24 %, which is very high compared to other authors' results. There are different hypotheses about the reasons for the capacity drop phenomenon: Bottleneck downstream of the study site: The flow at the point under investigation will remain fluent until the section between this point and the bottleneck is filled with congested flow. After this time, the maximum flow will be the bottleneck's capacity. Different driver behavior: Drivers in fluent traffic accept shorter headways since they expect to be able to pass the vehicles in front. Once they have given up this idea, they switch to a more safety-conscious style of driving and keep longer headways. Restricted acceleration capabilities: At the front of the congested area, drivers need to accelerate. Some vehicles, however, have limited acceleration power, which opens a larger gap in front of them. These hypotheses and the value of the capacity drop were analyzed by Regler (2004) for the 15 freeway sections listed in Table 2, using different approaches. The main question was: What are the capacities under synchronized flow conditions as described in section 4? The conventional traffic flow model of van Aerde (1995) as enhanced by Ponzlet (1996) to account for the capacity drop phenomenon yielded an average drop of 270 veh/h in 5-minute flow rates. Using a distribution of breakdown flow rates (observed immediately prior to breakdown) and a distribution of queue discharge flow, an average drop of 250 veh/h in 5minute flow rates was determined, which is comparable to the result from the fundamental diagram. It is not easy to find a distribution of capacity after a breakdown that is adequate to the Product Limit estimate (Eq. 4). Brilon and Zurlinden (2003) applied a distribution of all flow rates in congested traffic and computed a very high capacity drop. Regler (2004) developed a method comparable to the Product Limit technique to obtain a capacity distribution in queue discharge flow. This method is based on the following assumptions (cf. Fig. 6): Flow rates during congestion do not represent the maximum possible flow of congested traffic. Capacity can directly be measured in queue discharge flow, i.e. in the last 5-minute interval before recovery of traffic flow (uncensored data). Flow rates during congestion are lower than capacity in queue discharge flow (censored data). Flow rates in free flow are not relevant for the capacity of queue discharge flow.

138 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 6: Flow rate and speed time series during congestion (freeway A5, 5-minute intervals) Based on these considerations, the observed flow rates in each time interval can be classified as follows: B*: Traffic recovers from congestion to free flow, i.e. the average speed exceeds the threshold value from time interval i to interval i + 1 F*: Traffic is congested in intervals i and i + 1, i.e. the average speed is lower than the threshold value in both intervals. This interval i contains a censored value. C*: Traffic is fluent in interval i, i.e. the average speed is above the threshold value. This interval is not relevant. After this classification, a capacity distribution for queue discharge flow according to Eq. 4 (where {B} is to be replaced by {B*}) can be computed from empirical data. This capacity level turned out to be always lower than the capacity before breakdown obtained from Eq. 4. The difference between both distributions (see Fig. 7), represented by the median value, for instance, may be regarded as the capacity drop.

Fig. 7: Capacity distributions for pre-queue (cf. section 2) and queue discharge flow (freeway section A5-7, 11.8 % average truck percentage, 5-minute intervals)

Reliability of freeway traffic flow

139

For the 15 freeway sections listed in Table 2, an average drop of 1,180 veh/h was estimated using this algorithm. It should, however, be mentioned that the results for the capacity drop varied widely between the sites investigated. All attempts to identify regularities within the variation failed. It might be that the capacity drop shows some chaotic properties, as was indicated by other authors (Kerner, 2000) using different methodologies.

TRAFFIC RELIABILITY Traffic reliability is an important factor for the assessment of the performance of highway segments and systems. In this context, the term "reliability" mainly refers to the variability of travel times. However, several definitions can be found in the literature. A comprehensive outline of these definitions is given by Shaw (2003). Here, traffic reliability is assessed by analyzing the probability that a freeway link is not congested, i.e. that the travel time does not exceed an acceptable level. This is becoming a question of increasing importance for "just-in-time" transportation in modern logistics chains. With the stochastic concept of capacity, it is possible to assess this kind of traffic reliability for a freeway link consisting of several sections. The overload probability for a single bottleneck (either distinct or virtual, cf. Fig. 1) is equal to the capacity distribution function Fc(q) given in Eq. 1. The probability of no congestion Pfree(q) represents the complementary event: Pfree (q)

= l-F c (q)

(9)

For the analysis of n subsequent (quasi-) bottleneck sections, the probability of no congestion in the whole system is the product of the single probabilities for each section: Pfree,1+2+...+n(q,,q2v..,qn) = fl[l-F c , i (q i )]

(10)

where qi = demand at section i Fc,i(qO = capacity distribution function of section i By applying the Weibull distribution function to each section, this can be written as:

P & ee,, + 2 + ... + n(q,,q2,-,qn)=e lP ' a *>'

* J = e ""'"

(11)

The overload probability for the whole system, which is equal to the capacity distribution function, is:

140 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

-t—

F c , + 2 + . . . + n ( q 1 , q 2 , . . . , q J = l - e "'P'"

(12)

In the special case of the traffic demand being identical in all sections (qi = q2 = ... = qn = q), the overload probability for the system is:

Fc,1+2+...+n(q) = l - e

«"'

(13)

This is again a Weibull distribution with the scale parameter pi+2+...+„:

Eq. 10 through 14 are based on the assumption that breakdown events due to an overload at the bottleneck sections and thus the capacity distribution functions are statistically independent. This assumption seems to be reasonable if the length of each section is sufficiently large. However, further empirical research may be required to establish the degree to which this assumption is justified. The fact that the volume of traffic arriving at bottleneck section i is influenced by congestion incidents at preceding sections (i - 1, i - 2, ...), e.g. due to the capacity drop, is not relevant because in case of an overload of one section, the whole system is regarded as overloaded according to the applied definition of traffic reliability. For the same reason, the impact of a queue spillback spreading over several sections does not affect the results.

TRAFFIC EFFICIENCY Brilon (2000) has proposed to use the parameter E = q-v-T

(15)

where E q v T

= = = =

traffic efficiency volume travel velocity over an extended section of the freeway duration of the time period for analysis of flow

[veh km/h] [veh/h] [km/h] [h]

as a measure to characterize the efficiency of traffic flow on a freeway. This parameter describes the "production per time unit" of a freeway. The more veh km a freeway produces per hour, the greater the efficiency with which the potential of the existing infrastructure is exploited.

Reliability of freeway traffic flow

141

By applying the concept of random capacities, each volume q has to be combined with the corresponding probability of a breakdown. Brilon and Zurlinden (2003) have derived: Ee Xp (q D ) = [ q D - v - ( l - x ) + q d c - v d c - x ] - T

(16)

where Eexp(qD) = qo = v = q dc =

expected efficiency at a demand volume qo demand traffic volume average velocity in fluent traffic for q = qo queue discharge volume

Vdc T

= queue discharge velocity = duration of the period under investigation

X

=

1

h

[veh km/h] [veh/h] [km/h] [veh/h]

[km/h] [h]

m

(

li-k)-( 1 -Pco I ,g,i-k-l)

H i-l k-0

n m ricong pe(q)

= expected proportion of congested intervals with q = qdc = number of 5-minute intervals during T

[-] [-]

= Minjncong-l;i-1} = average duration (number of intervals) of a congested period >1 = probability of a breakdown at volume q (here: q = qo) (e.g. after Eq. 4: p B = dF c (q)/dq)

[-] [-] [-]

m

Pcong,i =

XPefai-lJ-O-Pcong.i-k-l) k=0

= probability of congested flow in 5-minute interval

[-]

If we insert real data for ps(q) = fc(q) into this set of equations it becomes clear that the maximum expected efficiency is achieved for a demand volume qo that is lower than the average capacity. Sample calculations show that the highest efficiency of a freeway is to be expected at a demand of approximately 0.9 c, where c is the traditionally defined (constantvalue) capacity.

CONCLUSIONS As a result of a series of studies of German freeways, the concept of stochastic capacities seems to be more realistic and more useful than the traditional use of single value capacities. This probabilistic approach provides an improved understanding of both the variability of traffic flow observations and the typical dynamics in different traffic states on a freeway. The idea of random capacity is based on the work of authors like van Toorenburg (1986) and Minderhoud et al. (1997). Compared to their approach, the so-called Product Limit Method for capacity estimation has been modified and extended. It was rather distinctly shown that the capacity of a freeway section is Weibull-distributed. For German freeways, the shape parameter seems to be in a range of 13, whereas the scale parameter can be different for

142 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction specific freeway sections. One drawback of the methodology is that huge sample sizes are needed to estimate capacity distribution functions. With the application of ITS-methods, these data will become increasingly available. The concept of randomness permits to demonstrate the capacity reducing effect of wet road surfaces (-11 %) and the capacity increasing effect of traffic adaptive variable speed limits. The Product Limit Method was also used to estimate queue discharge capacity. It was confirmed that this capacity is usually lower than the capacity in fluent traffic. The studies did clearly show that three typical states in traffic flow exist: fluent traffic, congested traffic, and a transient state that occurs in each breakdown and recovery of traffic flow. The extent of the so-called capacity drop did not show any regularities, although many freeway sections, each with a large sample size, have been analyzed. It is concluded that the capacity drop has rather chaotic properties. The concept of random capacity reveals that the optimum degree of saturation for a freeway, based on data from Germany, ranges around 90 %. If the degree of saturation increases further, the risk of a breakdown becomes too high, so that the efficiency of freeway operation must be expected to be lower than at a saturation of 90 %. The stochastic concept can also be applied to freeways consisting of several succeeding sections. Overall, it is expected that the random interpretation of freeway capacity offers the potential for improved traffic engineering methodologies.

REFERENCES van Aerde, M. (1995). A Single Regime Speed-Flow-Density Relationship for Freeways and Arterials. Proceedings of the 74th TRB Annual Meeting. Washington D.C. Banks, J. H. (1990). Flow Processes at a Freeway Bottleneck. Transportation Research Record No. 1287, Transportation Research Board, National Research Council, Washington D.C. Brilon, W. (2000). Traffic Flow Analysis beyond Traditional Methods. Proceedings of the 4th International Symposium on Highway Capacity, pp. 26-41, TRB Circular E-C018, Transportation Research Board, Washington D.C. Brilon, W. and H. Zurlinden (2003). Ueberlastungswahrscheinlichkeiten und Verkehrsleistung als Bemessungskriterium fuer Strassenverkehrsanlagen (Breakdown Probability and Traffic Efficiency as Design Criteria for Freeways). Forschung Strassenbau und StrassenverkehrstechnikNo. 870. Bonn. Elefteriadou, L., R.P. Roess, and W.R. McShane (1995). Probabilistic Nature of Breakdown at Freeway Merge Junctions. Transportation Research Record 1484, Transportation Research Board, National Research Council, Washington D.C. Hall, F.L. and K. Agyemang-Duah (1991). Freeway Capacity Drop and the Definition of Capacity. Transportation Research Record 1320, Transportation Research Board, National Research Council, Washington D.C. HCM (2000). Highway Capacity Manual. Transportation Research Board, Washington D.C.

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143

Kaplan, E.L. and P. Meier (1958). Nonparametric estimation from incomplete observations. Journal of the American Statistical Association, 53, 457-481. Keller, H. and T. Sachse (1992). Einfluss des Bezugsintervalls in Fundamentaldiagramrnen auf die zutreffende Beschreibung der Leistungsfaehigkeit von StraBenabschnitten (Influence of Aggregation Intervals in the Fundamental Diagram on the Appropriate Description of Roadway Capacity). Forschung Strassenbau und Strassenverkehrstechnik No. 614. Bonn. Kerner, B. S. (2000). Theory of Breakdown Phenomenon at Highway Bottlenecks. Transportation Research Record No. 1710, pp. 136-144. Transportation Research Board, National Research Council, Washington D.C. Kerner, B. S. and H. Rehborn (1996). Experimental Properties of Complexity in Traffic Flow. Physical Review E, 53 (5), 4275-4278. Kim, Y. and H. Keller (2001): Zur Dynamik zwischen Verkehrszustanden im Fundamentaldiagramm (Dynamics between Traffic States in the Fundamental Diagram). Strassenverkehrstechnik, Issue 9/2001, pp. 433-442. Kuehne, R.D. and N. Anstett (1999). Stochastic Methods for Analysis of Traffic Pattern Formation. Proceedings of the 14' International Symposium on Transportation and Traffic Theory. Jerusalem. Lawless, J.F. (2003). Statistical Models and Methods for Lifetime Data. Wiley, New York. Lorenz, M. and L. Elefteriadou (2000). A Probabilistic Approach to Defining Freeway Capacity and Breakdown. Proceedings of the 4th International Symposium on Highway Capacity, pp. 84-95. TRB-Circular E-C018, Transportation Research Board, Washington D.C. Minderhoud, M.M., H. Botma and P.H.L. Bovy (1997). Roadway Capacity Estimation Methods Explained and Assessed. Transportation Research Record No. 1572. Transportation Research Board, National Research Council, Washington D.C. Okamura, H., S. Watanabe and T. Watanabe (2000). An Empirical Study on the Capacity of Bottlenecks on the Basic Suburban Expressway Sections in Japan. Proc. of the 4th International Symposium on Highway Capacity, pp. 120-129. TRB Circular E-C018, Transportation Research Board, Washington D.C. Persaud, B., S. Yagar and R. Brownlee (1998). Exploration of the Breakdown Phenomenon in Freeway Traffic. Transportation Research Record No. 1634, pp. 64-69. Transportation Research Board, National Research Council, Washington D.C. Ponzlet, M. (1996). Dynamik der Leistungsfahigkeiten von Autobahnen (Dynamics of Freeway Capacity). Schriftenreihe des Lehrstuhls fuer Verkehrswesen der RuhrUniversitaet Bochum, No. 16. Bochum. Regler, M. (2004). Verkehrsablauf und Kapazitat auf Autobahnen (Freeway Traffic Flow and Capacity). Schriftenreihe des Lehrstuhls fuer Verkehrswesen der Ruhr-Universitaet Bochum, No. 28. Bochum. Shaw, T. (2003). Performance Measures of Operational Effectiveness for Highway Segments and Systems. NCHRP Synthesis 311, Transportation Research Board, Washington, D.C. van Toorenburg, J. (1986). Praktijwaarden voor de capaciteit. Rijkswaterstaat dienst Verkeerskunde, Rotterdam.

144 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Treiterer, J. and J.A. Myers, (1974). The Hysteresis Phenomenon in Traffic Flow. Proceedings of the 6' International Symposium on Transportation and Traffic Theory, pp. 13-38. Reed Pty Ltd., Sydney.

8

A CRITICAL COMPARISON OF THE KINEMATIC-WAVE MODEL WITH OBSERVATIONAL DATA Kai Nagel, Inst. For Land and Sea Transport Systems, Technical University of Berlin, Berlin, Germany; and Paul Nelson, Dept of Computer Science, Texas A&M University, College Station, Texas, USA

INTRODUCTION The objectives of this work are to: critically summarize the predictive capabilities and predictions of the classical kinematic-wave model (KWM) of Lighthill and Whitham (1955) and Richards (1956), where by "classical" is intended the case of a strictly concave fundamental diagram; critically summarize the extant data that are and are not reproduced by those predictions, especially with a view toward identifying the disagreements between observations and the classical KWM that seem most firmly established. Whether an allegedly observed phenomenon is "firmly established" is subjective. An element is reproduction by multiple independent observers; however, that is not sufficient, because it is possible for multiple observers to build upon the same mistaken assumption or technique. It is therefore possible for two reasonable people to disagree as to whether existence of an alleged phenomenon has been firmly established; indeed the two authors do not completely agree in all instances. Our approach in cases where either of us has doubts is simply to summarize, as objectively as we can manage, both sides of the issue. There also arise issues of interpretation of the predictive variables internal to a theory. This is especially relevant to continuum models, of which the KWM is an instance, because these models have continuous functions (e.g., mean speed) as their predictive variables, whereas any particular instance of traffic flow consists of discrete vehicles in motion. Tying theoretical continuum results to predictions is particularly difficult because much of the available data

146

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

is obtained by aggregating over time at sparsely spaced detectors near ramps, yet there seems to be few studies of the effect of aggregation interval, detector spacing or detector location. The contents of this work are as follows. In the following section we summarize the predictions of the classical KWM, and associated issues, especially those involving computational employment of the KWM. The next section summarizes instances of agreement of the KWM with observations (qualitative features, cumulative flows), and provides a critique of such observations, including especially identification of areas where further observational studies seem warranted. The following section is devoted to a critical discussion of alleged observations that apparently disagree with the classical KWM. The specific alleged discrepancies discussed there include unstable flow and related phenomena (spontaneous breakdown, the two-capacity phenomenon), what we term empirical jammed flow (particularly "wide jams"), and internal structure in congested flow. Our final section contains conclusions and recommendations for further related work.

THE CLASSICAL KWM The classical KWM for traffic consists of the equation of continuity, d,p(x,t) + dxq{x,t) = 0, plus the assumption that flow (q=q(x,t)) can be written as a function (FD) of density (p=p(x,t)) and possibly explicitly of longitudinal position (x) and time (i), say q(x,t) = Q{p(x,t\x,t). Here Q is strictly concave in p, i.e. Q"<0, and vanishes at p=0 and at some jam density p=pjlm (see Fig. 1). If the FD is spatially homogeneous (independent of x), then density is a solution of the nonlinear first-order conservation law d,p(x,t) + dpQ(p,t)dxp(x,t) = O.

(1)

The FD commonly is assumed to be piecewise constant in x, so that Eq. (1) holds except at points of discontinuity. (See the final paragraph of this section for further brief discussion of the effect of such points.) The FD further is commonly assumed stationary (i.e., not to depend explicitly upon time), although nonstationary effects could be included (e.g., sudden changes in weather). In the homogeneous stationary case Eq. (1) implies the density is constant along the "characteristic curves" in the (AxJ-plane that are defined by the ordinary differential equations ^ = c(x,t)~dpQ(p(xM

(2)

These curves are lines in the (t,x)-p\ane, with slope equal to the (constant) wave velocity c=c(p) along that line. Momentarily consider an infinitely long roadway, with initial data given at t=0, say p(x,0)= Po(x). If through (t,x), with t>0, there passed exactly one characteristic intersecting the x-axis (i.e., t=0), then the density at that point would be uniquely determined as that initially specified at that point of intersection with the initial line. There exist situations in which this stipulation holds, but there are two ways it can fail.

Critical comparison of the kinematic-wave model with observational data 147

Figure 1 - A strictly concave FD, as typically conceptualized in regard to the KWM. First suppose density is initially increasing as one moves downstream, at the initial time (e.g., near the end of a rush period). Because of the strict concavity the wave speed is (algebraically) decreasing along that interval, and therefore any two characteristics through distinct points along this roadway will eventually intersect. At such a point the "constant density along characteristics" specification of the preceding paragraph therefore fails to specify a unique value of the density. This ambiguity is traditionally resolved via a shock wave, that is a curve x=xs(t) in the (t,x)plane that is the locus of the initial points of intersection of characteristics, and along which the density (and hence flow) are permitted to be discontinuous. The shock condition dx, = Aq =Q(pu(t))-Q(pd(t)) „. dt

Ap'

pu(t)~PA')

then follows from conservation of vehicles. Here pu(t)=p(xs(t)-,t) (pd(t)=p(xs(t)+,t)) is the density immediately upstream (downstream) of the shock wave at time t. If one applies the "constant density along characteristics" specification only so long as the characteristic does not cross a shock, then the solution again is uniquely specified everywhere. In particular, Eq. (3) is an ordinary differential equation that determines the trajectory of the shock wave. In case the initial density decreases as one moves downstream no corresponding difficulty arises; the characteristics diverge ("fan out") with increasing time, but one and only one characteristic passes through each point (t,x), t>0. However, in the limiting case of a jump discontinuity in the initial density at some xo, with the downstream density smaller (e.g., at a traffic signal that has just turned green), there are points (t,x), t>0, through which no characteristic passes. In this case one mathematical (weak) solution is a shock, with densities downstream (upstream), up to the downstream (upstream) characteristic passing through (t,x)=(O,xo), equal to p(xg+,0) {p(xo-,O)), and trajectory otherwise determined by Eq. (3), just as in the preceding paragraph. However a second solution that fills the gap between the up-

148 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction stream characteristic through (t,x)=(0, xo) and the downstream characteristic through the same point is given by the similarity solution

p( x , 0 = c -^£zfij,

(4)

where c1 is the functional inverse of the wave velocity, which exists because of the strict concavity of Q. In case of such initial discontinuities one can create infinitely many (weak) solutions by piecing together segments of this similarity solution with pieces of the preceding shock solution. It is generally deemed that the similarity solution is that best representing the true state of traffic. The simplest mathematical argument supporting that choice is "stability": if one considers a sequence of initial densities having linear segments about x=xo that converge to the discontinuous initial density, and otherwise are equal to that initial density, then the corresponding (now uniquely determined, as previously) solutions resemble and converge to the similarity solution. That is, the similarity solution is self-healing, in that small perturbations from it return to it, whereas (in the present case) small perturbations of the shock solutions diverge from that shock solution. The simplest argument supporting the choice of similarity and involving driver behavior is that any shock segment represents some drivers traveling more slowly than they safely could, according to the prevailing FD. The choice of the similarity solution also relates mathematically to the so-called "entropy condition," which in turn also has ties to driver behavior. In its application to the classical KWM the entropy condition essentially asserts (Ansorge, 1990) that drivers accelerate as soon as safely possible and wait as long as safely possible to decelerate, where safety is denominated by adherence to the given FD. But the entropy condition is relevant mostly because it can be given a quantitative formulation (e.g., Leveque, 1992) that can be applied to computational approximations. This is extremely important, because early within the computational development of traffic flow theory some investigators mistakenly concluded that the classical KWM was fundamentally flawed because they inappropriately employed simple and reasonable computational approximations that failed to satisfy the entropy condition. The corresponding computational approximations, for an initial discontinuity as above, then converged with mesh refinement to the often clearly incorrect unstable shock solution, rather than to the stable similarity solution; cf. Ross (1988), Newell (1989), and Ross (1989). This mistaken conclusion seems to have provided much of the impetus for development of so-called higher-order methods (Payne, 1971). Some of these have recently been shown (Aw and Rascle, 2000) to be seriously flawed, in that they can develop negative (i.e., upstream!) flows, even though all flows are initially downstream (and all initial densities nonnegative and less than jam density). The STRADA code (Buisson, 1997), which has been widely applied in France, is based on a computational solution of classical KWMs via the Godunov method (Lebacque, 1996), which has a discretization error that is first order in both space and time. Computational methods having higher-order computational approximations exist (Bale et al, (2002), but seem not to have been explored in traffic flow, perhaps because lower-order methods have some advantages for the real-time demands of traffic control. It is necessary, for any computational implementation, to consider a system of some finite length, and therefore to incorporate appropriate boundary conditions. See Lebacque (to appear) for a discussion of boundary conditions, and Nelson and Kumar (2004) for the extension of these to interfaces (jump disconti-

Critical comparison of the kinematic-wave model with observational data 149 nuities of the FD in x) and point constrictions (removable singularities, in x, of the FD). In these cases consideration of only the similarity solution, for downward initial jumps in density as one moves downstream, is again conventionally adopted to obtain uniqueness.

EMPIRICAL AGREEMENT WITH THE CLASSICAL KWM In this section we review empirical results that appear to reflect agreement between observations and predictions from the classical KWM. The next section is devoted to a discussion of empirical observations that are often considered to reflect disagreement between observational data and the KWM. In many instances there are widely held but contrary opinions regarding these matters, for reasons discussed in the Introduction. The classification of a particular allegedly observed phenomenon as reflecting agreement or disagreement with the KWM is therefore somewhat arbitrary. Qualitative consistency between observations and the predictions of the KWM, particularly the shock waves and acceleration waves delineated in the preceding section, comprise perhaps the most widely accepted type of agreement. This is not easily or widely documented, perhaps because it is so widely accepted. For example, much of the technology underlying the Highway Capacity Manual, which is a semi-official U.S. standard for the analysis and design of roadway facilities, is based on supply-demand analysis, which is the steady-state version of the KWM. The situation is also complicated because many empirical studies that might provide verification have been reported in the literature related to a particular type of facility (e.g., freeway or signalized intersection), as opposed to the traffic modeling literature, and therefore not necessarily considered in light of consistency between data and the KWM. Shock waves Shock waves have been primarily observed in the context of freeway bottlenecks, which is to say freeway sections at which the capacity (maximum) flow in the FD of Fig. 1 is lower than immediately upstream. This is presumably because (inductive-loop) data from such locations are widely available, and because an understanding of the manner in which congestion at such locations forms and behaves is considered important to mitigation of the undesirable increase in travel times associated with congestion. According to the classical KWM, whenever the demand (flow) just upstream of the bottleneck entrance exceeds the capacity of the bottleneck, a shock wave will form at the entrance to the bottleneck and propagate upstream, leaving behind (i.e., bounded upstream by the moving shock discontinuity and downstream by the fixed bottleneck entrance) a "queue" of vehicles in which flow is equal to bottleneck capacity. However, in that queue the density will be the higher of the two densities corresponding to that flow on the FD, and therefore mean vehicle speed (= flow divided by density) will be (typically much) lower than that either upstream of the shock wave or downstream of the bottleneck entrance. The speed of the shock should be predictable from the shock condition (3). Agreements between these KWM predictions and observations have been documented, perhaps most notably in a series of data-oriented papers (Cassidy and Mauch, 2001; Windover and Cassidy, 2001) and companion modeling-oriented work (Newell, 2002) emanating from the transportation group at the University of California at Berkeley; see also Banks (1999).

150 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This collective body of work even argues that a "zeroth order" (Newell, 2002) theory of highway traffic, which relies on an entirely triangular FD, is still able to predict accurately the most important effects of highway traffic, especially cumulative flows. Note however that cumulative flows are integral quantities, in the sense that at any location x they are the time integral of the differential flow q(x,t). Integral quantities are inherently easier to both measure and predict than the corresponding differential quantity; e.g., it appears that numerical differentiation, with the associated inherent loss of accuracy, will be required in order to obtain speeds and hence travel times from cumulative flows. The KWM characterization of a deceleration region as a "shock wave" of zero thickness is an idealization; nonetheless the actual structure of such transition zones upstream of queues has only recently been investigated empirically, by Mufioz and Daganzo (2003). These investigators describe detailed instances of shocks that are about 1 km wide, corresponding to a relatively gradual deceleration of about 1/3 m/sec2. This imposes some spatial (and presumably temporal) limitation on the scale of validity of the KWM, but these researchers further argue that even with such a wide shock the KWM can predict many relevant quantities quite accurately. As an example they cite vehicle positions to within 160 m, or about five vehicle spacings; however this bears further elaboration, because neither the KWM nor any other continuum model predicts vehicle spacing per se. Acceleration waves The focus in validation of the KWM has tended toward shock waves, perhaps because of the ready availability of data from "point" (e.g., inductive-loop) detectors on freeways, and the practical interest in congestion on freeways. There seems to be little observational data exemplifying the companion KWM prediction of acceleration waves. This is presumably because freeway detectors tend to be located upstream of potential bottlenecks, and therefore not in position to acquire data representative of the "queue discharge" that is the manifestation of acceleration waves downstream of a bottleneck. On the other hand, acceleration waves are extremely important to the behavior of traffic at signalized intersections; we believe the literature related to such intersections is an important source of potential data relative to validation of the KWM that has been too much neglected. A full realization of the potential use of such data would require spatially extensive measurements of flow during a green cycle, both upstream and downstream of the signal. With more widespread use of camera-controlled signals, such data seem destined to become more widely available. Even now some data are available, in the form of measurements of the "saturation flow" at a signal during the green phase; this parameter is extremely important to the performance of signalized intersections, and has been extensively discussed in the literature on signalized intersections. The prediction of the KWM is that (unopposed) queue discharges at such an intersection will be equal to the minimum of the capacity (maximum) flows immediately upstream of the intersection, at the intersection itself, or immediately downstream of the intersection, but in any event that it will remain constant during the green phase of the signal. In regard to this prediction, Lin, Tseng and Su (2004) state that "actual queue discharge patterns, however, often do not display an identifiable steady maximum rate." On the other hand, the variations in observed flow at the intersection that are reported by these investigators are only of the order of 10-15% (see also Bonneson, 1991). Thus here the KWM seems

Critical comparison of the kinematic-wave model with observational data 151 to be categorically neither a success nor a failure, but rather the appropriate view is dependent upon the level of accuracy required. Lebacque (2002) has considered replacing the entropy condition for the KWM by a "bounded acceleration" condition, because of doubts about accuracy of the entropy solutions for modeling ramp metering. It should be noted that a KWM with a strictly concave FD does not predict a constant flow rate out of the traffic light, except exactly at the position of the light. Boundary-induced breakdown The preceding section mentions the FD as a hypothetical representation of flow as a function of density. Let us call the FD so related to the KWM theory KWM-FD. One can also measure flows and densities in the field, leading to an empirical FD. Classically such observations lead to empirical FDs that resemble the KWM-FD well in the lower-density free-flow regime, but at higher densities display a tail extrapolating the free-flow data lying above a "cloud" of widely scattered data points (Drake, Schofer and May, 1967, Koshi, Iwasaki and Ohkura, 1981; Kerner, 2002, esp. Figs. 14 and 16); cf. Fig. 2 for a typical instance of such an "inverted-lambda" empirical FD. Often it is better to plot density p, flow q and velocity v as functions of time, as in Fig. 3. These time-series data tend to show that a typical transition is from the free-flow regime to a regime where flow is only somewhat diminished but densities are much higher, meaning much lower velocities (Mika, Keer and Yuan, 1969; Kerner and Rehborn, 1996a; Kerner, 1998, 1999a); such a transition often is termed "breakdown."

figure z - empirical iunaameniai diagram, as recoraea on me uerman ireeway A m. uensny (on the x axis) is given in veh/(km lane), flow (on the y axis) is given in veh/(hour lane). Source: Nagel, Wagner, and Woesler, 2003. Daganzo, Cassidy and Bertini (1999) note that a bottleneck downstream of a measurement location can easily generate such a plot, in the following way: The system starts with low flow at low densities. Both flow and density keep increasing, along the "free flow" branch of the fundamental diagram.

152 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This flow can be larger than can flow through the bottleneck. A queue thus begins to form upstream of the bottleneck, but that does not immediately influence the measurement. Eventually, the queue will reach the measurement location. At that point in time, data points will move to a much higher density, while the flow value will now drop to bottleneck capacity. This is exactly what the KWM predicts, in that there is a transition from free flow to congested flow that accompanies a shock wave propagating upstream. Thus this must be marked as a success of the classical KWM. Unfortunately, there are too many flow-density data in the literature where the associated geometrical constraints are not well enough documented. Such data are relatively useless for many of the points raised in the current scientific discussion. At the same time alternative descriptions, such as that involving three phases, free flow, congested or synchronized and jammed "synchronized flow," do not yet seem to have evolved to the point that they can provide quantitative predictions even capable of being tested. (But see Schonhof and Helbing, submitted, and also Kerner, 2004.) This KWM interpretation of empirical observations of breakdown has recently (Cassidy, 1998) given rise to questions regarding the data analysis techniques traditionally employed for empirical FDs. Specifically, the empirical FD is obtained from averaging, for example over fixed time intervals. If one assumes that the KWM describes traffic, then these averages can extend over several regimes. This means that linear combinations of points on the KWM-FD can become points of the empirical FD. For a concave KWM-FD this means that points within the KWM-FD can become an empirical data point.

Figure 3 - Time series of the three fundamental variables of traffic flow. Flow and velocity are averaged; density is computed as *, where > denotes the mean. The data have been recorded on the German freeway Al near an intersection with German freeway A59. Time is counted from 6/Jun/1996 midnight. Source: Nagel, Wagner, and Woesler, 2003.

Critical comparison of the kinematic-wave model with observational data 153 Cassidy and Bertini (1999) pursue this line of reasoning in some detail. They also describe how averages can be restricted to time intervals of (nearly) stationary traffic, ensuring that the empirical data points lie on a possibly existing KWM-FD as much as possible. Unfortunately, these results are not conclusive as regards the possibility of obtaining a KWM-FD via filtering empirical data for stationarity, because data so filtered are sparse both near capacity flow and in the congested flow regime. On the other hand, such sparsity is exactly what would be predicted by the KWM, in the region upstream of a bottleneck, which is where data customarily are taken (and indeed where the KWM predicts data must be taken, in order to observe any congested data). That is, the KWM predicts that once flow at an observation point upstream of a bottleneck exceeds the capacity of that bottleneck, it will do so for only a limited time, after which the queue formed at the bottleneck will spill back to the observation point and flow at that point subsequently will be equal to the bottleneck capacity (i.e., will correspond to a single density-flow point). Indeed, the chief inconsistency between the filtered nearstationary data and the KWM seems to be the existence of multiple data points in the congested regime (cf. Fig. 11 of Cassidy and Bertini, 1999). We believe much remains to be done in the area of data analysis related to empirical FDs.

EMPIRICAL DISAGREEMENT WITH THE CLASSICAL KWM In this section we review empirical results that seem to reflect disagreement between observations and predictions from the classical KWM. As before the difference between "agreement" and "disagreement" is considerably subjective. Further, the disposition toward perceiving consistency or inconsistency between the KWM and a particular set of observations often is reflective of familiarity with the KWM or some alternative theory. We make an effort to factor out such inherent bias, but perfection in that regard is elusive. Unstable flow, spontaneous breakdown and the two-capacity phenomenon In the preceding section we have already offered one hypothesis that might somewhat reconcile data such as Fig. 2 with the KWM-FD, but if one takes such data at face value - as has been done through much of the history of traffic flow theory - then it seems doubtful they can be represented adequately by any single continuous function. This observation is classical in traffic flow theory, and is behind (e.g.) the well-known "two-regime" theory (Ceder, 1967). The interpretation of this "scatter" regime is controversial, and even the terms one chooses to employ in referring to it can evoke bias toward one or another interpretation, and perhaps associated theory. In an effort to avoid such bias, we employ the following terminology: By empirically congested flow we shall intend any data clearly not falling within the free flow regime, as defined above. By empirically uncorrelatedflow we intend any data set within which there is considerable scatter in the density-flow plane, as illustrated in Figure 2. (Thus empirically uncorrelated flow is a subclass of empirical congested flow.)

154 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction By empirically jammed flow we shall intend yet another subclass of empirically congested flow that has been hypothesized by some investigators, as discussed in more detail in the following subsection. The word "empirical" and its derivatives will be omitted when the context permits. The traditional acceptance of these density-flow measurements in the (empirically) congested regime raises doubts regarding the validity of the KWM. This alone has been persuasive to many, and has fueled both substantial skepticism directed toward the KWM, and a search for alternative data interpretations and associated explanatory theories. On the other hand, the successes of the KWM discussed in the preceding section have made others considerably reluctant to accept any suggestion that this theory has no validity at higher densities. In the preceding section we have already indicated a recently suggested possibility for reconciling kinematic-wave theory with empirically uncorrelated flow. The fundamental hypothesis underlying that possibility is that breakdown is exclusively what nonlinear dynamicists would term a "boundary-driven phenomenon," which is to say that it derives from shocks propagating upstream from a bottleneck, as summarized from Daganzo, Cassidy and Bertini (1999) in the preceding section. In our opinion such bottleneck-driven breakdowns certainly occur, and empirical reports should include sufficient information to permit determination of the extent to which boundary effects contribute to data reflecting breakdown. This is a nontrivial challenge, because it requires spatially distributed data, or at least a very careful description of the roadway downstream of the point of observation. An extreme version of this suggestion is that empirical results should not be published at all, unless the data on which they are based are made publicly available, so that alternative hypotheses can also be tested against those data. Nonetheless, when one takes into account the possibility of bottleneck-driven breakdown, there remain observed phenomena that are difficult to explain absent spontaneous ("emergent") breakdown. The objective of this subsection is to summarize the observational data supporting such spontaneous breakdown. More generally, bottleneck-driven breakdown is an instance of boundary-driven phenomenon, which in turn is instance of an "external" phenomenon that stems from circumstances outside the system being modeled, while spontaneous breakdown is conceived as an emergent phenomenon that arises from phenomena internal to the system being modeled. However, to a very large extent this distinction is subjective, in that it depends upon what one is willing to assume is known a priori (i.e., an external variable), as opposed to being a quantity that the model is required to predict (i.e., an internal variable). As a very concrete instance of this dichotomy, the presence of a slow truck in the right lane certainly can be a significant factor in determining traffic flow. But should this presence be regarded as an external factor, and therefore known a priori, or as a stochastic fluctuation whose potential existence must be represented among the statistical distribution giving rise to the mean values that a continuum model should provide? There exist circumstances under which either of these views is reasonable, depending upon the purpose of the model, and therefore what one is willing to assume is known a priori. For example, if the objective of the model is to better understand the effect of a slow truck (e.g.,

Critical comparison of the kinematic-wave model with observational data 155 Daganzo and Laval, 2005), then it is perfectly appropriate to take the presence (and characteristics) of such a truck as a known external variable. However, if the purpose is to predict mean traffic behavior where it cannot be (continuously) measured then the presence or absence, characteristics, and effect of such a truck must be regarded as a stochastic internal variable to be appropriately reflected in the model predictions. The simplest way to determine internal effects for a model of traffic flow is to implement it on a homogeneous closed loop, which completely eliminates external (boundary) effects. Such "ring roads" are rare in practice, but Sugiyama et al. (to appear) recently reported a relevant experiment conducted on a test circuit. Briefly, they experimentally show that under appropriate circumstances, spatially homogeneous flow is unstable for sufficiently high densities, but rather tends "spontaneously" to transition, after several minutes, to "jammed flow." In this context the characteristic of jammed flow is a single jam, propagating opposite to the direction of traffic flow, within which vehicles come to a stop, or very nearly so. Quantitative details of the observed speed-headway distribution over the entire jam flow are not reported, but supporting simulations suggest they comprise a characteristic ("universal") limit cycle ("hysteresis loop"). This result suggests that spontaneous breakdown can occur in traffic flow. However, in itself it signifies neither failure of the KWM nor nonexistence of a KWM-FD, in the sense that flows and densities averaged over the characteristic limit cycles could still comprise a perfectly well defined function in the density-flow plane. This possibility is illustrated by the cellular automata model CA-184a of Nelson (submitted), which produces results similar to the empirical FD of Fig. 2, in simulations of traffic stream observations upstream of a bottleneck, but that on a ring road fall exactly on a well-defined KWM-FD in the density-flow plane, with multiple underlying limit cycles. However, at a minimum the results of Sugiyama et al. (to appear) suggest existence of a lower limit on the spatial and temporal scales for validity of the KWM; existence of some such scale limitations is to be expected for any continuum model; cf. Lesort et al. (to appear) for further discussion of scale issues in traffic flow. In addition, Jost and Nagel (2003) argue that if the spontaneous breakdown is related to a true (thermodynamical) phase transition, then the jam features will coarsen until they are visible on all spatial scales, even the largest. The mechanism for coarsening is that jams coagulate until there is eventually only one large jam left in the system. However, they cannot answer if their modeling results apply to real-world traffic or not. Observations of what we term as "two-capacity flow" have been persistently reported in the literature. This phenomenon is relevant here because it also is strongly suggestive of spontaneous breakdown, and is inconsistent with the classical KWM. Here by two-capacity flow we mean observations and related analyses that suggest bottlenecks can operate at two distinct maximum flows, typically a higher value prior to and early in the formation of a queue upstream of the bottleneck and a somewhat lower value subsequent to development of such a queue. This contradicts the classical KWM, because the (entropy solution of the) latter predicts flow at an enqueued bottleneck will be a constant, specifically the capacity in the bottleneck (Nelson and Kumar, 2004). Observations of two-capacity flow have been reported, among others, by Edie and Foote (1960), by Agyemang-Duah and Hall (1991) and by Cassidy and Bertini (1999), while other

156 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction studies (e.g., Persaud and Hurdle, 1991) have reported lack of conclusive evidence supporting a reduction in flow coincident with formation of a queue. From an empirical perspective, Banks (1991; cf. also Banks, 1990 and Banks, 1991a) indicates that "the hypothesis that flow decreases when it breaks down is confirmed, provided the hypothesis applies to individual lanes," but "when averaged across all lanes there was no significant change," and concludes that alleged two-capacity flow "is unlikely to provide a basis for metering." Note that if one fully subscribes to the KWM, then there is no rational basis for ramp metering, absent some societal decision to favor mainline traffic over that entering freeways. When differences between the levels of flow at a bottleneck, before and after queue formation, are reported, their magnitude tends to be only about 5-10%. Given that relatively small alleged difference, it is difficult to know whether the different conclusions reached in different studies are attributable to differences in methodologies for obtaining and analyzing data, or to actual differences in traffic behavior at distinct sites, or possibly some combination of these factors. This inherent difficulty in determining existence of the two-capacity phenomenon has been discussed by Persaud and Hurdle (1991). Nonetheless one can question whether we have yet advanced significantly beyond the status observed by Wattleworth (1963): "The question of whether or not the flow downstream of a freeway bottleneck decreases when congestion sets in is currently the subject of much discussion in engineering circles. Research findings support both the yes and no answers to this question. Several studies ... suggested that perhaps the question did not have a simple yes or no answer." Yet, on balance we tend to agree with Cassidy and Bertini (1999) that "the average rate vehicles discharge from a queue can be ... lower than the flow measured prior to the queue's formation." See Elefteriadou, Roess and McShane (1994), and Lorenz and Elefteriadou (2001) for a suggestion that the time of occurrence of this type of breakdown can usefully be viewed as stochastic in nature.

Empirically jammed flow Some researchers perceive an additional subclass of empirically congested flow, which is the empirically jammed flow already mentioned above. The characteristic signature of empirically jammed flow is so-called "wide jams," which are described by Knospe et al. (2002) as follows: "Wide (moving) jams are regions with a very high density and negligible average velocity and flow. The width of these structures is much larger than its fronts at the upstream and downstream ends where the speed of the vehicle changes sharply. Wide jams can propagate undisturbed through either free flow or synchronized traffic without impact on these states, which thus allows their co-existence." (See the following subsection for more on "synchronized flow.") The predominant evidence for wide jams is time-series data such as that shown in Fig. 23 of Knospe et al. (2002); i.e., data from a stationary (loop) detector that show at early times relatively large flows and speeds with small densities, followed by an intermediate period of small flows and speeds and high densities, finally followed by yet a third time period qualitatively resembling the first. The hypothesis commonly invoked is that such data represent a structure ("wide jam") traveling upstream, within which vehicles are closely spaced, and in which both upstream and downstream fronts remain sharply defined as the structure propagates. If this interpretation is correct, then it provides a challenge to the classical KWM, because (see the section entitled "The Classical KWM") that model predicts that an initially

Critical comparison of the kinematic-wave model with observational data 157 sharp front between a high-density upstream region and a low-density downstream region (such as the downstream front of the hypothesized wide jam) will dissipate (spread) into an acceleration fan (similarity solution). However, absent additional data from other spatial points there is an interpretation of these data that reflects a boundary-driven phenomenon, and is consistent with the classical KWM. What is required is the sequence of events suggested in the preceding section as providing an explanation of Fig. 3 consistent with the KWM, followed by a decrease of upstream demand to a value again below bottleneck capacity. The KWM then predicts a sequence of events as already described, until the upstream demand falls below the bottleneck capacity. Once that decreased demand reaches the tail of the queue the tail will remain a shock wave, and thus be a stable structure within the confines of the classical KWM, but now it will propagate downstream. It will eventually reach the detector, where it will be registered as a sharp transition from a high-density low-flow/speed regime to low-density high-flow/speed. As seen at the hypothesized single detector the traffic pattern will thus be exactly the observed signature of an alleged wide jam, as described in the preceding paragraph. Thus the question of whether the data supporting existence of wide jams is consistent with the classical KWM comes down to the whether the second observed sharp front is moving upstream or downstream. This seems to require data from more than one spatial location. At this time the totality of the data of the type described above that has been reported in the literature in support of the wide-jam hypothesis seems to be Kerner and Rehborn (1996a, 1996b), Knospe et al. (2002) and Schonhof and Helbing (submitted). Of these three works, only in Figures 3(a)-(c) of Kerner and Rehbom (1996a) and Figure 2 of Kerner and Rehborn (1996b) does one find explicit measurements from detectors at distinct positions along the roadway. These data definitely support existence of wide jams. It would be desirable if they were replicated by other investigators. Nonetheless, in the following sections we conditionally accept the hypothesis of existence of wide jams, and hence empirically jammed flow, and explore the potential for various models to reproduce that phenomenon. Some empirically properties of wide jams that have been reported (Kerner and Rehborn, 1996a, 1997) include the following: 1. Traffic jams, once created, are fairly stable and can move without major changes in their form for several hours against the flow of traffic. 2. The flow out of a jam is a stable, reproducible quantity, and can be up to 50% smaller than the maximal flow at the same location. Further internal structure in empirically uncorrelated flow The "wide jams" characteristic of the empirically jammed flow discussed in the preceding subsection constitute one instance of internal structure in empirically uncongested flow. However, other types of internal structure have been reported. In fact, "stop-and-go" flow under congested conditions is anecdotally familiar to virtually all drivers. Notwithstanding that this phenomenon was initially studied scientifically nearly forty years past (Edie and Baverez, 1967), there remains a paucity of related quantitatively reliable scientific data. (But

158 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction see Schonhof and Helbing, submitted, for study of start-stop waves via a novel filtering technique applied to data from multiple dual-loop detectors.) This may be because such data require short-term and short-distance temporally and spatially distributed data, which necessarily will be subject to a high degree of stochastic uncertainty (stemming from, e.g., variations between individual drivers). Nonetheless there are persistent reports of "something else" (e.g., Kerner, Klenov and Wolf, 2002), and there is considerable discussion (Daganzo, Cassidy and Bertini, 1999) as to whether these observations are or are not entirely explainable by KWM theory. This subsection is devoted to a brief review of these issues. Much of the discussion focuses on what Kerner and collaborators (Kerner and Rehborn, 1996b, 1997) term as "synchronized flow." One of the barriers to an objective discussion of synchronized flow is the lack of consensus as to its characteristic signature(s). An approximately common (i.e., "synchronized") speed across multiple lanes, and the existence of embedded very narrow (compared to the wide jams of the preceding subsection) jams are two signatures that have been mentioned by investigators who have considered the matter. Yet another barrier is lack of public availability of much of the data on which many of the alleged observations of synchronized flow are based, and the consequent inability of the community at large to reproduce the underlying data analysis, or suggest alternate analyses. Even among those who are disposed favorably toward existence of synchronized flow there is not agreement on the question of whether the associated strong scatter of the data has a dynamical origin, for example stemming from an on-ramp, or a statistical origin, such as being caused by a mixture of cars and trucks that display different driving characteristics. Sugiyama et al. (to appear) provide simulation results, based on the optimal-velocity model (Bando et al., 1995) that suggest a hypothesis regarding possible internal structure of congested flow. Briefly, their simulations reveal the following structure upstream of a bottleneck: immediately upstream a region of "laminar flow," within which vehicles are spaced rather regularly, followed by a region within which the flow consists of very narrow "jams," finally followed by a region even further downstream within which wide jams appear. These researchers hypothesize the laminar region is an inherently metastable region, the region of wide jams is inherently stable, and the region of narrow jams is a connecting unstable region (as dictated by Hopf bifurcation) that is somehow stabilized by presence of the bottleneck. Note that the latter interpretation also is consistent with considerable difficulty in observing the narrow jams, which in turn would be somewhat consistent with difficulties in observing synchronized flow, if one hypothesized an identification between synchronized flow and narrow jams, as do Sugiyama and Nakayama (2003). The above description, in particular that narrow jams coagulate to form wide jams, is also consistent with work by Jost and Nagel (2003). Jost and Nagel do not investigate bottlenecks, but rather use a closed homogeneous system, i.e. a homogeneous loop. Once of their initial conditions is a completely homogeneous distribution of cars, i.e. all vehicles with the same space headway, as given by the density. Also in that system, one obtains, for certain densities, initially "laminar flow,", then "narrow jams," and then wide jams. As pointed out earlier, the wide jams will further coagulate until there is only one jam left in the system, and that statement holds for arbitrarily large systems. Further, unpublished simulations of our own show that the same code, after the introduction of a bottleneck, displays the same behavior as the model of Sugiyama et al. (to appear). An interpretation is that, either by a bottleneck or by having a closed system, traffic can be "pinched" at a certain (average) density, at which the

Critical comparison of the kinematic-wave model with observational data 159 system displays the above-described behavior. However, even here it must be said that there are very few empirical observations that include the spatial picture, and at least one of them (Windover and Cassidy, 2001) displays little or no coarsening with increasing distance from the bottleneck. The principal point of interest here is the extent to which structure internal to empirically congested flow disagrees with the KWM. Continuum models of traffic flow make predictions regarding density, mean speed and flow, all of which are continuum quantities that can be interpreted as mean values over some underlying distribution of vehicular speeds. But continuum models themselves are silent on the form of the associated speed distributions, and indeed regarding the nature of these distributions (i.e., over time and space, in a particular instance, or over an "ensemble" of instances). From this perspective structure within empirically congested flow perhaps provides information (e.g., the appropriate nature of the associated speed distribution) beyond that available from the KWM itself, but that information is not necessarily in contradiction to the KWM. See Nelson (submitted) for an example of a simple model of traffic flow that has such an internal structure within its congested regime, but nonetheless can be modeled reasonably well via the KWM, with an appropriately constituted FD. Note also the recent work of Bagnatini and Rascle (2003) in which a macroscopic model is displayed that displays spatial structure in its solutions, but has the property that suitably homogenized solutions nonetheless are solutions of the KWM. However, once more note that in some models structures can coarsen beyond any arbitrarily large but fixed length scale.

NONCLASSICAL KINEMATIC-WAVE MODELS A classical (strictly convex) FD will not reproduce the observed stable wide jams from a KWM. This section is a brief summary of what is known or believed about the ability of nonclassical KWMs to reproduce wide jams. Once we allow strictly linear pieces in Q(p), then some sort of stability of both fronts can be achieved, even for entropy solutions of the KWM, as long as both the density within the jam and the surrounding densities are within the same linear segment (Lin and Lo, 2003). This is because the slope of Q(p) denotes the phase velocity of the wave features, and if all densities in the range of interest are within the same linear segment, then their wave features move with the same velocity. Thus at least some of the elements of wide jams seemingly can be reproduced within the context of the simple triangular FD suggested by Newell (2002). However, such solutions seem to be only "weakly stable," in the sense that if the initial densities are slightly "smeared," then the ensuing density profile retains this smeared form, rather than reorganizing into a sharp front (i.e., displaying "self healing"). The bounded-acceleration extension (modification?) of the KWM (Lebacque, 2002) does not require the FD to hold in circumstances under which this would require maximum accelerations that exceed some a priori specified bound, but rather replaces the FD by the stipulation that the maximum acceleration occurring is equal to that bound. This modification has been demonstrated to lead to stable wide jams; see the reference cited for details. It is, however, a matter of taste if a model with bounded acceleration still is a kinematic model in

160 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the strict sense, since bounded acceleration actually means that flow is no longer a function of local density only. hi unpublished work the authors have considered the KWM under an FD with a strictly convex but decreasing tail (i.e., positive second derivative and negative first derivative). The corresponding entropy solutions of the Riemann problem can be worked out, using results from the mathematical literature (Li, 2003, and citations therein). These solutions have some interesting features (e.g., deceleration fans and acceleration shocks). These can be explained in terms of hypothesized driver behavior. However, it does not appear that this approach is capable of leading to stable wide jams.

CONCLUSIONS This paper was written with the intent to first reach a firm basis of agreement between the two authors, and then to continue from there with the description and analysis of simulations that describe some of the more subtle effects of traffic flow, such as structure formation in queues. Somewhat surprisingly, just reaching the firm basis has exhausted the limits given for this paper. This seems to be caused by the following: (1) Both authors have their own intuitions about the dynamics of traffic flow. (2) Both authors agree that the other author's intuition is valid, although considerable explanation and thoughtfulness was necessary in order to have that agreement based on true mutual understanding. (3) Both authors' intuitions are consistent with the data sets that they were aware of; indeed, only very few data sets are able to answer at least some of the critical questions. The perhaps most significant example is the breakdown experiment of Sugiyama et al. (to appear), which, although it has been reported in 2001, has neither been published not been widely disseminated. Nevertheless, it is the only unequivocal example of spontaneous breakdown that we are aware of. Similar statements hold for the availability of data that are both temporal and spatial. The most important results of our effort are the following: The kinematic wave model (KWM) explains and predicts many if not most effects of real world traffic dynamics. As pointed out, e.g., by Daganzo, Cassidy, and Bertini (1999), much of the single detector data supposedly in support of spontaneous breakdown can also be explained by queue dynamics in conjunction with a geometrical constraint. On the other hand, there are some data sets that are, in both authors' views, very difficult to reconcile with the traditional KWM. This holds certainly for KWMs with strictly concave fundamental diagrams (FDs), which do not explain stable jams, spontaneous breakdown out of nothing, or structure formation in queues. Structure formation in queues could be argued as being beyond the spatial scale that KWMs claim to deal with, although a "true" phase transition interpretation of traffic flow would predict that these structures coagulate into larger and larger structures with increasing distance from the bottleneck. Finally, there seems to be only one unequivocal observation of spontaneous breakdown. As said before, more traffic data that include the spatial aspect is needed in order to make further progress. Further, the field of traffic-flow theory and analysis would benefit from an international effort to provide universal access to data supporting published analy-

Critical comparison of the kinematic-wave model with observational data 161 ses, and support from journal editors to make provision of such data to this project a prerequisite to publication would be helpful.

REFERENCES Agyemang-Duah, K. and F. L. Hall (1991). Some issues regarding the numerical value of capacity. In: Procs. Int. Symp. Highway Capacity and Level of Service (U. Brannolte, ed.), pp. 1-15, A. A. Balkema Press, Rotterdam. Ansorge, R. (1990). What does the entropy condition mean in traffic flow theory? Transp. Research B,24R, 133-144. Aw, A. and M. Rascle (2000). Resurrection of "Second Order" Models of Traffic Flow. SIAMJ. Applied Mathematics, 60, 916-938. Bagnerini, O. and M. Rascle (2003). A multiclass homogenized hyperbolic model of traffic flow. SIAMJ. Math. Anal, 35, 949-973. Bando, M. et al. (1995). Dynamical model of traffic congestion and numerical simulation. Physical Review E, 51, 1035-1042. Banks, J. H. (1990). Flow processes at a freeway bottleneck. Transp. Research Record, 1287, 20-28. Banks, J. H. (1991). Two-capacity phenomenon at freeway bottlenecks: A basis for ramp metering? Transp. Research Record, 1320, 83-90. Banks, J. H. (1991a). The two-capacity phenomenon: Some theoretical issues. Transp. Research Record, 1320, 234-241. Banks, J. H. (1999). Investigation of some characteristics of congested flow. Transp. Research Record, 1678, 128-134. Bale, D. S. et al. (2002). A wave propagation method for conservation laws and balance laws with spatially varying flux functions. SIAMJ. Scientific Computing, 24, 955-978. Bonneson, J. A. (1991). Modeling queued driver behavior at signalized junctions. Transp. Research Record, 1365, 99-107. Buisson, C. (1997). Decoupage objets de Strada. Internal report LICIT, ENTRE-INRETS, France. Cassidy, M. J. (1998). Bivariate relations in nearly stationary highway traffic. Transp. Research B, B32, 49-59. Cassidy, M. J. and R. Bertini (1999). Some traffic features at freeway bottlenecks. Transp. Research B, B33 25-42. Cassidy, M. J. and M. Mauch (2001). An observed traffic pattern in long freeway queues. Transp. Research A, A35 143-156. Ceder, A. (1967). A deterministic traffic flow model for the two-regime approach. Transp. Research Record, 567, 16-30. Daganzo, C. F., M. J. Cassidy, and R. Bertini (1999). Possible explanations of phase transitions in highway traffic. Transp. Research A, 33, 365-379. Daganzo, C. F. and J. A. Laval (2005). On the numerical treatment of moving bottlenecks. Transp. Research B, 39, 31-46.

162 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Drake, J. S., J. L. Schofer and A. D. May (1967). A statistical analysis of speed density hypotheses. Highway Research Record, 154, 53-87. Edie, L. C. and E. Baverez (1967). Generation and propagation of stop-start traffic waves. In: Vehicular Traffic science: Proc. 3r International Symp. Theo. Traff. Flow (L. C. Edie, R. Herman and R. Rothery, eds.) pp. 26-37, American Elsevier, New York. Edie, L. C. and R. S. Foote (1960). Effect of shock waves on tunnel traffic flow. Proc. Highway Research Board, 37, 492-505. Elefteriadou, L., R. P. Roess and W. R. McShane (1994). Probabilistic Nature of Breakdown at Freeway Merge Junctions. Transp. Research Record, 1484, 80-89. Hall, F. L. and K. Agyemang-Duah (1991). Freeway capacity drop and the definition of capacity. Transp. Research Record, 1320, 91-98. Jost, D. and K. Nagel (2003). Probabilistic traffic flow breakdown in stochastic car following models. Transp. Research Record, 1852, 152-158. Kerner, B. S. (1998). Experimental features of self-organization in traffic flow. Physical Review Letters, 81, 3797-3800. Kerner, B. S. (1999a). Phase transitions in traffic flow. In: Traffic and Granular Flow'99 (D. Helbing et al, eds.), pp. 253-284. Springer, Berlin. Kerner, B. S. (2002). Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks. Physical Review E, 65, Paper No. 046138. Kerner, B. S. (2004). The Physics of Traffic, Springer, Berlin. Kerner, B. S., S. L. Klenov and D. E. Wolf (2002). Cellular automata approach to threephase traffic theory. J. Physics A: Math. General, 35, 9971-10013. Kerner, B. S. and H. Rehborn (1996a). Experimental properties of complexity in traffic flow. Physical Review E, 53, R4275-R4278. Kerner, B. S. and H. Rehborn (1996b). Experimental features and characteristics of traffic jams. Physical Review E, 53, R1297-R1300. Kerner, B. S. and H. Rehborn (1997). Experimental properties of phase transitions in traffic flow. Physical Review Letters, 79, 4030-4033. Knospe, W. et al. (2002). Single-vehicle data of highway traffic: Microscopic description of traffic phases. Physical Review E, 65, 1-16. Koshi, M., M. Iwasaki and I. Ohkura (1981). Some findings and an overview on vehicular flow characteristics. In: Transp. and Traffic Theory: Proc. 8th ISTTT (V. F. Hurdle et al, eds.), pp. 403-426, University of Toronto Press, Toronto. Lebacque, J. P. (1996). The Godunov Scheme and what it means for first order traffic flow models. In: Transp. and Traffic Theory: Proc. 13th ISTTT (1. B. Lesort, ed.) pp. 647677, Pergamon, Oxford. Lebacque, J. P. (2002). A two-phase extension of the LWR model based on the boundedness of traffic acceleration. In: Transp. and Traffic Theory in the 21s' Century: Proc. 15th ISTTT (M. A. P. Taylor, ed.) pp. 697-718, Pergamon, Amsterdam. Lebacque, J. P. (to appear). Intersection modeling, application to macroscopic network traffic flow models and traffic management. In: Traffic and Granular Flow '03.. Lesort, J.-B., E. Bourrel and V. Henn (to appear). Various scales for traffic flow representation: Some reflections. In: Traffic and Granular Flow'03. Leveque, R. (1992). Numerical Methods for Conservation Laws. Birkhauser Verlag, Basel.

Critical comparison of the kinematic-wave model with observational data 163 Lighthill, M. J. and G. B. Whitham (1955). On kinematic waves II - a theory of traffic flow on long crowded roads. Proc. Royal Society, London, A229, 317-345. Li, T. (2003). Global solutions of nonconcave hyperbolic laws with relaxation arising from traffic flow. Journal of Differential Equations, 190, 131-149. Lin, F. B., P. Y. Tseng and C. W. Su (2004). Variations in queue discharge patterns and their implications in analysis of signalized intersections. In Proc. 83rd Annual Meeting Transp. Research Board, Wash. D.C., Jan. 11-15. Lin, W.-H. and H. K. Lo. A theoretical probe of a German experiment on stationary moving traffic jams. Transp. Res. B, 37, 251-261. Lorenz, M. R. and L. Elefteriadou (2001). Defining freeway capacity as function of breakdown probability. Transp. Research Record, 1776, 43-51. Mika, H. S., J. B. Keer and L. S. Yuan (1969). Dual mode behavior of freeway traffic. Highway Research Record, 279, 1-13. Mufioz, J. C. and C. F. Daganzo (2003). Structure of the transition zone behind freeway queues. Transp. Science, 37, 312-329. Nagel, K., P. Wagner, and R. Woesler (2003), Still flowing: Approaches to traffic flow and traffic jam modeling. Operations Research, 51, 681-710. Nelson, P. (submitted). On two-regime flow, fundamental diagrams and kinematic-wave theory. Submitted to Transp. Science, March 2004. Nelson, P. and N. Kumar (2004). Point-constriction, interface and boundary conditions for the kinematic-wave model: In Proc. 83rd Annual Meeting Transp. Research Board, Wash. D. C , Jan. 11-15. Newell, G. F. (1989). Comments on traffic dynamics. Transp. Research B, 23B, 386-389. Newell, G. F. (2002). A simplified car-following model: A lower order model. Transp. Research £, 36B 195-205. Payne, H. J. (1971). Models of freeway traffic and control. Simulation Council Procs., 1, Simulation Council, Inc., La Jolla, CA, 51-61. Persaud, B. N. and V. F. Hurdle (1991). Freeway capacity: definition and measurement issues. In: Proc. Int. Symp. Highway Capacity and Level of Service (U. Brannolte, ed.), pp. 289-307, A. A. Balkema Press, Rotterdam. Richards, P. I. (1956). Shockwaves on the highway. Operations Research, 4, 42-51. Ross, P. (1988). Traffic dynamics. Transp. Research B, 22B, 421-435. Ross, P. (1989). Response to Newell. Transp. Research 5, 23B, 390-391. Schonhof, M. and D. Helbing (submitted). Empirical features of congested traffic states and their implications for traffic modeling. See www.helbing.org. Sugiyama, Y. et al. (to appear). Observation, theory and experiment for freeway traffic as physics of many-body system, hi: Traffic and Granular Flow '03. Sugiyama, S. and A. Nakayama (2003). Understanding "synchronized flow" by optimal velocity model, hi: Modeling of Complex Systems: Seventh Granada Lectures (P. L. Garrido and J. Marro, eds.), pp. 111-115, American Institute of Physics. Wattleworth, J. A. (1963). Some aspects of macroscopic freeway traffic flow theory. Traffic Engineering, A35, 15-20. Windover, J. R. and M. J. Cassidy (2001). Some observed details for freeway traffic evolution. Transp. Research A, A35, 881-894.

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AVERAGE VELOCITY OF WAVES PROPAGATING THROUGH CONGESTED FREEWAY TRAFFIC Benjamin A. Coifman, Department of Civil and Environmental Engineering and Geodetic Science, and Yun Wang, Department of Electrical and Computer Engineering, Ohio State University, Columbus, Ohio, USA

INTRODUCTION Large portions of traffic flow theory are built upon the bivariate relationship between flow and density or the related relationship between headway and spacing. This scope begins with the first-order, macroscopic model of Lighthill, Whitham and Richards (LWR) to predict the propagation of signals or waves through the traffic stream (Lighthill and Whitham, 1955; Richards, 1956) and continues into higher-order macroscopic models and microscopic models that attempt to model the behavior of individual drivers. But because it is difficult to measure density and uncommon to find stationary conditions during congested periods, debate continues about the shape of the bivariate curve and thus, implicitly impacts the models predicated upon it. It is widely agreed among traffic flow theorists that during free flow conditions flow increases monotonically with density, with a slight drop in slope as conditions approach capacity. But little can be said for certain about the congested regime beyond the fact that in aggregate, flow drops with increasing density. Empirical studies by Hall et al. (1986, 1988) propose an inverted V shape for the flow-occupancy curve, while noting that the findings were consistent with earlier studies. A few years later, out of convenience, Newell (1993) also chose to simplify the relationships by using a triangular flow-density curve for illustrative purposes in the course of theoretical development. Given a triangular curve, LWR predicts that during congested periods, waves should propagate upstream with a constant velocity, independent of

166 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the given flow, density, or local traffic speed.1 Subsequent empirical analysis suggests that indeed, during congestion, waves do appear propagate upstream with roughly constant velocity. Windover and Cassidy (2001) showed that the upstream moving waves could be found by comparing the cumulative arrival curves across all lanes between two neighboring loop detector stations. They used data from a 2 km segment of northbound Interstate-880, just south of Oakland, California, and found that the wave velocities during congested periods were nearly constant (17-20 km/h) over the homogeneous freeway segment, independent of flow. Mauch and Cassidy (2002) observed traffic data over multiple days along a 10 km stretch of the Queen Elizabeth Way (Ontario, Canada). They measured wave velocities by plotting the cumulative arrival curve across all the lanes from many successive detector stations with vertical displacements in proportion to the physical distance separating the successive detector stations and found that the waves propagating upstream in congestion had nearly constant velocities in the range of 22-24 km/h, independent of the location and the flow. Smilowitz and Daganzo (2002) employed the LWR model to derive the wave velocity from the slope of the line passing through the two states in accumulation-flow relationship, where the accumulation is proportional to the density between two observers. They found wave velocities on a single lane, homogeneous highway segment were 17.2 km/h on one day and 18.8 km/h on another. Deviating from the earlier techniques of manually extracting wave velocities, Munoz and Daganzo (2001) applied cross correlation to cumulative arrival curves summed across selected lanes for a single sample period and found a wave velocity of 19.4 km/h. These earlier efforts suggest that the flow-density curve is indeed triangular, e.g., Windover and Cassidy (2001) note, "traffic on homogeneous freeway segments might better be modelled using linear (triangular-shaped) relations." But all of the earlier studies used small samples, often aggregated across several lanes, to find the velocity of waves during congestion. If the wave velocities are roughly constant, independent of traffic conditions, they can be used as inputs to estimate link travel times (Coifman, 2002), which is part of the motivation behind this paper. To provide a more comprehensive analysis of wave velocity during congested periods, this research builds off of the cross correlation idea presented by Munoz and Daganzo (2001) and seeks to extract the average wave velocity automatically for each sample. Our methodology can accommodate additional data with little human intervention, thereby allowing the study of much larger data sets to better model the phenomena. The following section of this paper uses cross correlation of cumulative arrivals measured at successive detector stations. Then the paper discusses the limitations arising from flow-based measures, such as cumulative arrivals, when attempting to measure waves propagating against the flow of traffic. The next section uses traffic speed measurements to address the limitations and then examines the influence of prevailing traffic speed, lane, and location on the results. This paper then closes with a brief discussion and conclusions.

' Throughout this paper, speed refers to motion of the traffic as it travels downstream and velocity refers to the motion of waves through the traffic stream. Furthermore, a positive velocity is used to denote when these waves travel upstream.

Average velocity of waves propagating through congestedfreeway traffic 167

AVERAGE WAVE VELOCITY FROM CROSS CORRELATION OF CUMULATIVE ARRIVALS BETWEEN TWO STATIONS Our first attempt to extract wave velocities automatically is to simply replicate Munoz and Daganzo's cross correlation method on one month (August, 2003) of cumulative arrival curves sampled every 2-seconds from seven successive detector stations in the westbound direction of the Berkeley Highway Laboratory (BHL) along Interstate-80, north of Oakland, CA (Coifman et. al., 2000). The method seeks to find the time lag at which the cross correlation of the two time series recorded at two successive detector stations reaches a maximum and that time lag corresponds to the dominant travel time of waves in the sample. The details for the cross correlation method are as follows. Assuming that waves propagate between the two stations, the upstream time series, y(t), could be regarded as a scaled and delayed version of downstream time series, x(t), with zero mean additive noise, n(t), i.e., y(t) = ax(t-ro)

+ n(t)

(1)

where cris the scale factor and z0 is time delay corresponding to the travel time of waves from downstream location to upstream location. Let R (T) represent the cross correlation of x(t) and y(t) , and R^ij)

be the auto-correlation of x(t) . From equation (1) and the

definitions of Rxy{t) and Rxx(t), we have: Rxy(r) = E[xQ)y(t + r)] = E[x(t){ax(t + T-T0) + n(t + T)}] = aE[x(t)x(t + T-T0)] = ORXX(T-T0)

Q)

where Ra (r) < R^ (0). Thus, Rxy (r) reaches its maximum when r = r 0 . For this study the search window for feasible time lags spanned -4096 sec to 4096 sec and in the rare cases that two maxima were found, the given sample is excluded. There are 5 lanes in each direction on the freeway segment used in this study (fig. 1). The innermost lane is reserved for high occupancy vehicles (HOVs), because the HOV lane generally exhibits little congestion compared to the general flow lanes it was excluded from the analysis of upstream moving waves. The seven dual-loop detector stations in the segment are denoted stl through st7, respectively. The stations are spaced about 1/2 km apart from each other and each detector station has a dual-loop detector in each lane, which allows for direct measurement of local traffic speeds.

168

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

The event data from each detector are recorded at 60 Hz and then aggregated to 2-second average traffic speed and flow measurements. This 2-second aggregation was chosen because it is roughly on the order of typical headways and is consistent with Munoz and Daganzo (2001), facilitating comparison. Unlike Munoz and Daganzo, who paired lanes together, we applied the cross correlation technique on individual lanes. Because this study is limited to the waves that arise in congested traffic, the following pre-processing step is used to exclude all non-congested periods. The lane is considered congested if the 5-minute average traffic speed remains below a pre-specified threshold for at least one-hour at both the upstream and downstream stations on a link. Each contiguous "congested period" in each lane for each station pair is divided into non-overlapping, one-hour long samples that are used in the cross correlation analysis. The exact threshold was chosen somewhat arbitrarily, with the specific intent to ensure that conditions were indeed congested throughout the entire link for an extended period. A traffic speed threshold of 48 km/h (30 mph) was used in most of the analysis presented in this paper, though a higher threshold at 72 km/h (45 mph) was used as well throughout the research to verify the results did not depend on the choice of the threshold. Other sample durations were studied, such as 45-min and 30-min long, with results similar to those presented herein, but they became noisier as the period decreased. Fig. 2 shows the cumulative distribution (CDF) of the average wave velocity from each sample, estimated via cross correlation on an individual lane basis applied to the 2-second cumulative arrival curves after subtracting a background flow equal to each lane's average flow during the given period. So each datum in this distribution represents the wave velocity corresponding to the maximum correlation from a one hour long sample for the given lane. Rather than reporting mean or median, the entire distribution is shown to allow detailed evaluation, e.g., the standard deviation could be evaluated from the CDF. Over all links and in all lanes, one sees mixed results. Many of the observations are consistent with the earlier research, with wave velocities between 15 km/h and 25 km/h, but many more of the measurements fall outside this range. For reference, table 1 shows the total number of samples in each of the curves from fig. 2. The research also considered several related variations. In the first case, before employing cross correlation, rather than subtracting a background flow the cumulative arrival curves were filtered to remove the least-squares fit of a straight line from the data. Next, the time series of 2-second "raw" flow and of "filtered" flow (using the same filtering processing as used in cumulative arrival curves) were investigated for

Figure 1.

Schematic of the Berkeley Highway Laboratory on Interstate-80, north of Oakland, California.

Average velocity of waves propagating through congested freeway traffic

169

estimating the average wave velocity, applied to the same congested periods used in fig. 2. Fig. 3 shows the resulting CDFs of each sample's average wave velocity from the three additional methods for a typical station pair (st4 and st5 westbound, the other station pairs exhibited similar trends), compare to fig. 2D. The filtered cumulative arrivals found almost half of the measurements with zero time lag because in many cases the background flow was the dominant component. As a result, fig. 3A shows the CDF with the zero time lag included as dashed lines and excluding all samples with zero time lag with solid lines for the same lanes. In any event, all four methods had more measurements between 15 km/h and 25 km/h than any other 10 km/h window, with the filtered flow having many more measurements in this range compared to the other three methods. Although encouraging, the results remain mixed for all four methods since several lanes had fewer than 40 percent of the observations falling in the window.

Figure 2.

Distribution of the average wave velocity from each sample using cross correlation applied to cumulative arrival curves with background flow subtraction, August 2003, WB, BHL.

170 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 3. Distribution of the average wave velocity from each sample using cross correlation applied to (A) filtered cumulative arrival curves (B) raw flow (C) filtered flow, st4 to st 5, August 2003, WB, BHL.

Table 1.

Number of Samples m each CDF for fig. 2 and fig. 5 Number of samples by lane st1 to st2

st2 to st3

st3 to st4

st4 to st5

st5 to st6

st6 to st7

Iane2

34

68

74

87

95

73

Iane3

62

70

84

75

76

63

Iane4

61

62

63

46

41

32

Iane5

44

44

56

37

13

10

PROBLEMS USING FLOW, OCCUPANCY AND CUMULATIVE ARRIVALS TO EXTRACT AVERAGE WAVE VELOCITY Given homogeneous vehicles and drivers, LWR predicts that for a convex flow-density relationship all waves should propagate upstream during congestion. Yet there are numerous examples in the literature demonstrating that some information actually travels downstream during congested periods, as observed in flow or occupancy measurements, e.g., Dailey (1993), Petty et al. (1997) and Cassidy and Windover (1998). Cassidy and Windover referred to this phenomenon as "driver memory" with drivers resuming similar headways after passing through disturbances while the other researchers simply used the phenomenon to measure travel time. None of these papers note the seemingly apparent conflict with LWR. To rectify this problem, one can derive an explanation based on the fact that real vehicles simply are not homogeneous.

Average velocity of waves propagating through congestedfreeway traffic 171 Consider fig. 4, showing a hypothetical example of several vehicles traveling in a freeway lane. The figure shows four vehicles moving from right to left. The i-th vehicle has a fixed effective length2, Lt, and a variable spatial gap between itself and the preceding vehicle, git that is likely to be a function of the driver, vehicles, and traffic conditions. The total spatial headway for this vehicle at any instant is simply, s

i = Li + Si

(3)

If two successive vehicles, i-1 and i, were traveling at the same constant speed, U, then the ith vehicle's headway passing a point in space is given by, h,=^-

(4)

or relaxing the assumption of constant speed to reflect the varying speeds experienced in congested traffic (as is done throughout the remainder of this paper), the relationship becomes, A,*^

(5)

where ut is the i-th vehicle's speed passing the specified point in space, e.g., a detector station. Similarly, as the vehicle passes a detector, it turns on for the following duration,

Figure 4.

2

A hypothetical example of the traffic stream showing the corresponding spacings, headways and detector status.

The effective length includes the physical length of the vehicle and the length of the detection zone.

172 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction L.

(6)

where an equal sign is used assuming the vehicle is traveling fast enough so that acceleration is negligible in the on-time measurement. Given these facts, if one measures flow, qJy and occupancy, ocCj, over a fixed number of vehicles, n, it should be clear that, n

1j=-

.

°CCj=—n

(7)

Most operating agencies use fixed sample periods, T, rather than fixed n, in which case the equalities in Equation 7 become "approximately equal" due to edge effects from partial headways observed at the start and end of a given sample.3 It should be clear from this derivation that both flow and occupancy are functions of ut and Lt. Obviously, Li travels downstream with the vehicles even during congested periods, e.g., Coifrnan and Cassidy (2002) explicitly used measured vehicle lengths to match the observation of a given platoon of vehicles at one station with the earlier observation of the same platoon from an upstream station on the same freeway during congested periods. This information traveling downstream will confound any attempt to extract information propagating upstream against the flow of traffic, e.g., although cut off by the horizontal axis of the plots, many velocities corresponding to downstream moving waves were found in the CDFs of fig. 2. The velocity of many of these downstream moving waves were on the order of the prevailing traffic speed, but they were not the dominant trend in the distribution, as evidenced by the majority of the velocities in fig. 2 indicating upstream moving waves. The relatively low frequency of downstream moving waves in the CDFs is due in part to the large sample period (1 hr), during which time the link travel speed experienced by drivers typically changes significantly and this range usually prevents the downstream moving waves from becoming the dominant feature in the crosscorrelation. The existence of downstream moving waves during congestion is consistent with the theoretical development of the mixed flow first-order macroscopic models of Zhang and Jin (2002) and Chanut and Buisson (2003), which extend LWR to heterogeneous flows of two or more vehicle types. Namely, in their models, trucks and cars had different lengths but travel with the same speed during congestion. These models predict that in congestion there are two kinds of waves, waves that move downstream with the vehicles as an artefact of the inhomogeneous vehicle (and driver) fleet and waves that move upstream according to LWR. If ui is independent of the inhomogeneities of the vehicle fleet, then one would expect that during congested periods LWR would better predict the evolution of traffic speed over time

3

In this case n changes from one sample to the next while T is the summation of the headways during a given sample. Provided the sample includes many vehicles, the contributions from the partial headways at the start and the end of the sample are usually negligible.

Average velocity of waves propagating through congestedfreeway traffic 173 and space than LWR does for flow during the same periods (or cumulative arrivals, which is simply a function of flow).

AVERAGE WAVE VELOCITY FROM CROSS CORRELATION OF LOCAL TRAFFIC SPEED BETWEEN TWO STATIONS In an attempt to improve the performance of the automated data extraction, and thus to analyze the characteristics of waves, the analysis now shifts to using local traffic speed from the detector stations to measure the average wave velocity in each hour long sample. As with the cumulative arrival data, the time series of local traffic speed measurements were filtered to remove the least-squares fit of a straight line from the data. Fig. 5 shows the CDFs of average wave velocity from each sample, estimated using the time series of 2-second traffic speed measurements corresponding to the same congestion samples used in fig. 2. The CDFs from local traffic speed exhibit a much tighter distribution compared to those from the cumulative arrival curves or flow measurements. Now all lanes in all links have over 90 percent of the measured wave velocities between 15 km/h and 25 km/h. Grouping all of the lanes and station pairs together, the median wave velocity is 18 km/h. Note that fig. 2 and fig. 5 only show the results for westbound traffic in August 2003 (WB03). The analysis was repeated for eastbound traffic (EB-03), as well as data from August 2002 (WB-02 and EB-02). In all four cases the results exhibited the same pattern moving from cumulative arrivals to local traffic speed when measuring the average wave velocity during congestion and all four cases exhibited the dominant wave velocity response in the range of 15-25 km/h, which is consistent with the previous research.

Characteristics of waves To investigate whether the upstream moving wave velocity is independent of local traffic conditions, this research examined the relationship between average wave velocity and the prevailing traffic speed in each sample. For each one-hour long congestion sample we measured the minimum, median, and maximum of the 2-second traffic speed measurements from the downstream detector station. The results were similar for all three parameters, so this paper only presents the results for the median. The points in fig. 6 show each sample's average wave velocity versus the median of the downstream station's traffic speed from the same data set presented in fig. 5. For brevity, all lane and station pairs are shown in a single plot. Of the 1370 data points in the figure only 8 percent are below 15 km/h. To extract the general trend, these data were binned by 5 km/h increments of the local traffic speed and the total number in each bin is noted in the first column of table 2. The median of each bin is shown in fig. 6 as a solid line, with the median being chosen over the mean to reduce the sensitivity to the small number of points that are far from the center of the distribution. Note that the slowest bin had

174 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 2. prevailing traffic speed range 0-5 km/h 5-10 km/h 10-15 km/h 15-20 km/h

Number of samples in fig. 6 - 9 Subsampling WB-02 WB-03 0

EB-03 0

5 71

3

WB-02 0

EB-02 0

0

0

0

0

2

34

0 3 21

3 12

20-25 km/h

184 271

15

71 104

25-30 km/h 30-35 km/h

310 286

48 61

132 111

55 82

35-40 km/h

197

40-45 km/h 45-50 km/h

45 1

92 26

66 13

4

0

Figure 5.

onramps 0

offramps lane 2 lane 3 lane 4 lane 5 0 0 0 0 0 0 1 2 2 2 18 31 30

8 49

27

113

63 85

33

107

96

18 24

91

3

55 8

101 52

20

43 28 2 0

0

0

0

0

84

18

5 0

21 42

15 30

49

24

58 64

49 30

51 17 1

39 15 0

Distribution of the average wave velocity from each sample using cross correlation applied to filtered traffic speed measurements, August 2003, WB, BHL.

only 5 samples and the fastest bin had only 1 sample. Ignoring the larger deviations arising from these two bins on the ends, the fitted line is relatively flat, ranging between 17 km/h and 19 km/h. Fig. 7 presents all four of the fitted lines resulting from each direction in each year.

Average velocity of waves propagating through congested freeway traffic

175

As with fig. 6, table 2 tallies the number in each bin, and again, both the first and last bin for each curve has few samples, they are included only for completeness. For each curve in fig. 7, one can see the wave velocities may exhibit a small correlation with the prevailing traffic speed, but given the fact that three of the curves increase slightly and one decreases, it would be premature to draw any conclusions about the details of any relationship. Investigating these finer points of the relationship will be the subject of future research. In any event, the range is less than 4 km/h of wave velocity over a span of 35 km/h of local traffic speed for all four of the curves. So if local traffic speed impacts the wave velocity during congested periods this impact is small and there are other unmeasured factors that impact the wave velocity with a similar magnitude. The congestion threshold was set to 48 km/h in fig. 7, but as noted previously, the analysis was repeated with a threshold of 72 km/h. The results from the higher threshold generally exhibit the same pattern but become slightly noisier at higher traffic speeds because many of the one-hour samples tend to include a mixture of free flow and congested conditions.

Figure 6.

Average wave velocity versus prevailing traffic speed for each sample, August 2003, WB. BHL.

176 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 9.

(A) Median wave velocity versus prevailing traffic speed by lane, August 2003, WB, BHL. (B) Detail of A

Finally, we explicitly investigate if there is a strong relationship between lane change maneuvers and wave velocity. Because it is difficult to measure lane change maneuvers directly, inflow is used as a proxy for the net number of lane change maneuvers during a given sample, where the inflow is defined as the cumulative downstream arrivals less the cumulative upstream arrivals in the lane over the hour. Inflow will underestimate the total number of lane change maneuvers since exiting vehicles cancel out entering vehicles during the sample. Noise also arises in the measurement due to the fact that the number of vehicles stored in the lane may change between the start and the end of the sample, though this noise should be unbiased. Fig. 10 shows inflow versus wave velocity. Lane 5 regularly exhibits an outflow (negative inflow) of 500 vph between st 1 and 2, and then an inflow of 400 vph between st 2 and 3 due to the ramps. A few of the other lanes have an inflow on the order of 200 vph for these links, with the magnitude of the inflow being much smaller elsewhere. Yet as shown in Fig. 5 for these same data, all four lanes exhibit similar trends in wave velocity over all of the links. Thereby suggesting that aggregate wave velocity is independent of inflow. Of course it is possible that a relationship to lane change maneuvers may have been obscured by the long sampling period, the use of a proxy measurement, or simply the noise in the data.

CONCLUSIONS This paper employed cross correlation to estimate wave velocities between several successive detector stations during congested periods and over a large data set. Given homogeneous vehicles and drivers, LWR predicts that for a convex flow-density relationship all waves should propagate upstream during congested periods. The analysis first employed cumulative arrivals — a function of flow — and then the local traffic speed at the detector stations. It was shown that the flow-based analysis yield mixed results, with many measurements being consistent with earlier research, but many more measurements falling outside the typical range of measured wave velocities from the literature. But vehicles are not homogeneous and it was also shown that flow and occupancy depend on effective vehicle length as well as the

Average velocity of waves propagating through congestedfreeway traffic 177

Figure 10. Inflow versus wave velocity from each sample using cross correlation applied to filtered traffic speed measurements, August 2003, WB, BHL. local traffic speed. Because the vehicle lengths travel downstream with the vehicles, this information will hinder attempts to extract wave velocities propagating against the flow of traffic when using flow-based measures. This fact is evident in fig. 2 and 3. Trucks are restricted from lane 2, thus the standard deviation of vehicle lengths is smaller and less information travels with the vehicles. As a result, comparing across lanes for all of the plots in these figures, lane 2 had the highest percentage of observations that fell within the 15 km/h to 25 km/h window. To reduce the influence of the confounding vehicle lengths, the analysis was repeated over the exact same samples using local traffic speed measurements. Because drivers are constrained by downstream vehicles in congestion, the traffic speed trends are much less dependent on specific vehicle or driver characteristics. It was confirmed that LWR does a better job predicting the evolution of average traffic speed over time and space, with over 90 percent of the samples having an average wave velocity propagating upstream in the range of 15 km/h to 25 km/h. Of course these results represent the aggregate performance over one-hour samples and individual waves could differ significantly from this range, e.g., it is likely that other sources of noise remain in the data, such as lane change maneuvers disrupting the propagation ofwaves. Unlike earlier studies, the method presented in this paper is not labor intensive, thereby allowing for the study of a large amount of data and extracting the general trends in the wave velocity during congestion. Our results over two months of data from both westbound and

178 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction eastbound lanes of the BHL show that wave velocities are nearly constant over a large range of local traffic conditions, which is consistent with earlier work and further supports the use of a triangular flow-density relationship at least for a first-order approximation. There was some evidence that the wave velocity may change with local traffic speed but the change is very small, less than 1 kmyh of wave velocity for every 10 km/h of local traffic speed. Given the fact that the direction of this trend changed across the four data sets, this small trend may simply be an artifact of the sample size or it is indicative of other factors that have a similar impact on the wave velocity but are yet to be identified, hi any event, these outstanding questions are the subject of ongoing research. Unlike most of the earlier efforts to measure wave velocity during congested periods, the analysis was applied on a lane-by-lane basis, allowing for comparisons between lanes and for the analysis to quantify the impact on the wave velocity from vehicles entering or leaving at ramps. To this end, wave velocity does not appear to have a dependence on ramps or the particular lane. However, future research will seek to verify this finding. For example, this analysis was limited to the BHL, which has five lanes in each direction, results may be different for a two or three lane freeway.

ACKNOWLEDGEMENTS This material is based upon work supported in part by the National Science Foundation under Grant No. 0133278. The authors also wish to thank the anonymous reviewers for their valuable feedback.

REFERENCES Cassidy, MJ. and J.R. Windover (1998). Driver memory: motorist selection and retention of individualized headways in highway traffic. Transportation Research: Part A, 32(2), 129-137. Chanut, S. and C. Buisson (2003). Macroscopic Model and Its Numerical Solution for TwoFlow Mixed Traffic with Different Speeds and Lengths. Transportation Research Record: Journal of the Transportation Research Board, No. 1852, TRB, National Research Council, Washington, D.C., pp.209-219. Coifman, B., D. Lyddy and A. Sabardonis (2000). The Berkeley Highway LaboratoryBuilding on the 1-880 Field Experiment. Proc. IEEE ITS Council Annual Meeting, IEEE, 2000, pp 5-10. Coifman, B. (2002). Estimating Travel Time and Vehicle Trajectories on Freeways Using Dual Loop Detectors. Transportation Research: Part A, 36(4), pp. 351-364. Coifman, B., and MJ. Cassidy, (2002). Vehicle Reidentification and Travel Time Measurement on Congested Freeways, Transportation Research: Part A, 36(10), pp. 899-917. Daganzo, C. F. (1997). Fundamentals of Transportation and Traffic Operations. Pergamon.

Average velocity of waves propagating through congestedfreeway traffic 179 Dailey, D. (1993). Travel Time Estimation Using Cross Correlation Techniques, Transportation Research: Part B, 27(2), pp 97-107. Hall, F. L., B. L. Allen and M. A. Gunter (1986). Empirical Analysis of Freeway FlowDensity Relationships. Transportation Research: Part A, 20(3), pp. 197-210. Hall, F. L and T. N. Lam (1988). The Characteristics of Congested Flow on a Freeway across Lanes, Space and Time. Transportation Research: Part A, 22(1), pp. 45-56. Lighthill, M. J. and G. B. Whitham (1955). On Kinematic Waves. I Flow Movement in Long Rives. II A Theory of Traffic Flow on long Crowded Roads. Proc. Roy. Soc, A. 229, 281-345 Munoz, J. C. and C. F. Daganzo (2001). The Bottleneck Mechanism of a Freeway Diverge. Paper presented at the 80th annual TRB meeting, Transportation Research Board, January 2001. Mauch, M. and M. J. Cassidy (2002). Freeway traffic oscillations: observations and predictions. International Symptom of Traffic and Transportation Theory, (M.A.P. Taylor, Ed.) Elsevier, Amsterdam, pp. 653-674. Newell, G. F. (1993). A Simplified Theory of Kinematic Waves in Highway Traffic, I General Theory, II Queuing at freeway bottlenecks, III Multi-destination Flows. Transportation Research, Part B, 27(4), pp. 281-313. Petty, K., P. Bickel, M. Ostland, J. Rice, F. Schoenberg, J. Jiang, and Y. Ritov, (1997). Accurate Estimation of Travel Times From Single Loop Detectors, Transportation Research: Part A, 32(1), pp 1-17. Richards, P. I. (1956). Shockwaves on the Highway. Operations Research, pp. 42-45. Smilowitz, K. R. and C. F. Daganzo (2002). Reproducible Features of Congested Highway Traffic. Mathematical and Computer Modeling, 35, 509-515. Windover, J. and M. J. Cassidy (2001). Some Observed Details of Freeway Traffic Evolution. Transportation Research Part A, 35(10), pp. 881-894. Zhang, H.M. and W.L. Jin (2002). Kinematic Wave Traffic Flow Model for Mixed Traffic. Transportation Research Record: Journal of the Transportation Research Board, No. 1802, TRB, National Research Council, Washington, D.C., pp. 197-204.

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10

MICROSCOPIC THREE-PHASE TRAFFIC THEORY AND ITS APPLICATIONS FOR FREEWAY TRAFFIC CONTROL Boris S. Kerner, DaimlerChrysler AG, Stuttgart, Germany

INTRODUCTION There are a huge number of traffic flow theories and models, which should explain diverse congested traffic patterns that are observed upstream of freeway bottlenecks (see e.g. references in the reviews by Gartner et al. (eds.) (1997); Helbing (2001)). However, it is only recently that a "puzzle" of spatiotemporal features of congested patterns has been solved and these pattern features adequately understood (Kerner, 1998, 2002). Consequently, earlier traffic flow theories and models are in a serious conflict with many of these empirical spatiotemporal traffic pattern features. Therefore, the author introduced "three-phase traffic theory" (see references in Kerner (2004)). In three-phase traffic theory, besides the "free flow" phase there are two other phases in congested traffic: "synchronized flow" and "wide moving jam." Thus, there are three traffic phases in this theory: 1. Free flow. 2. Synchronized flow. 3. Wide moving jam. Three-phase traffic theory explains the complexity of traffic based on phase transitions among the three traffic phases, and on their complex nonlinear spatiotemporal features. The empirical (objective) criteria that distinguish between the two traffic phases in congested traffic are related to spatiotemporal features of these phases (Kerner, 1998, 2002). A wide moving jam is a moving jam that maintains the mean velocity of the downstream jam front, even when the jam propagates through any other traffic states or freeway bottlenecks. In

182 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction contrast, the downstream front of the "synchronized flow" phase is often fixed at a freeway bottleneck. Within this front, vehicles accelerate from lower speeds in synchronized flow to higher speeds in free flow. Three-phase traffic theory is a qualitative theory (Kerner, 1998, 2002). The related microscopic theory (Kerner and Klenov, 2002, 2003) confirms hypotheses of this theory. This microscopic theory explains all known up to now reproducible empirical spatiotemporal congested pattern features at freeway bottlenecks. Note that in the book (Kerner, 2004) empirical features of reproducible freeway traffic patterns, three-phase traffic theory including a microscopic analysis of congested patterns, as well as their engineering applications (traffic pattern recognition, tracking, prediction, and control) have been considered. In this paper, simulation model results based on a numerical study of a microscopic traffic flow model in the context of three-phase traffic theory are presented. A deep connection between the diagram of congested patterns at bottlenecks with probabilistic theory of freeway capacity of free flow at bottlenecks, as well as application of the microscopic three-phase traffic theory for on-ramp inflow control and for a study of an ACC vehicle influence on spatiotemporal congested patterns at bottlenecks are discussed. STOCHASTIC MICROSCOPIC MODEL Here, we consider a stochastic microscopic two-lane model in the context of three-phase traffic theory (Kerner and Klenov, 2002, 2003). Model results presented below are related to numerical simulations of this model. We discuss only a hypothetical traffic flow with identical vehicles. A model of heterogeneous flow can be found in (Kerner, 2004). Basic rules of vehicle motion Basic rules of vehicle motion in the model of identical vehicles are as follows: vn+1 =max(0,min(v f r e e ,v s n ,v c n )),

Vc

jVn+An - "|vn+ant = n

f

Or for

x n+1 = x n + vn+1x,

«,n-Xn^Dn x,,n-xn>Dn'

(1)

X

An = max(-b n x,min(a n x,v M -v n )),

U

(3)

where index n corresponds to the discrete time t = nx, n = 0,1,2,...; x is the time step; vfrl,e is the maximum speed in free flow; the lower index £ marks functions (or values) related to the preceding vehicle; D n is a synchronized distance; an > 0 and b n > 0 (see below). In (1), vs n is a safe speed (KrauB et ah, 1997) that is a solution of the equation (Gipps, 1981)

Microscopic three-phase traffic theory and applications 183 v s ,^ + X d (v Sjn ) = g n +X d (v S j n ),

whereX d (u) = bT2(ap + a ( a - l ) / 2 ) ,

(4)

g n = x ^ n - x n - d is the space gap between vehicles, d is the vehicle length, b is a constant, a is the integer part of u / bx, (3 is the fractional part of u / bx. Speed adaptation within synchronized distance and steady model states Eqs. (l)-(3) describe the speed adaptation effect in synchronized flow. This speed adaptation effect takes place when the vehicle cannot pass the preceding vehicle, within the synchronization distance D n : at x<>n - x n n ),

(5)

D(u, w) = d + max(0, ktu + (^"'u^i - w)),

(6)

where k>l and (j> are constants; a is the maximum acceleration. Model fluctuations and random vehicle acceleration and deceleration As in other stochastic traffic flow models (see references in Helbing (2001)), at the first step of calculations of the model, the speed of each vehicle is vn+1 = v n+1 ,

(7)

where vn+1 is calculated based on Eqs. (l)-(6). At the second step, a noise component ^n is added to the calculated speed vn+1 (7) and then the final value of the speed vn+1 at time step n+1 is found from the condition vn+1 = max(0, min(vfree, vn+1 + ^ n , vn + a, v s n )).

(8)

Random deceleration and acceleration are applied depending on whether the vehicle decelerates or accelerates, or else maintains its speed: f-5b 4n =

if Sn+1 = -1

4a i f S n + 1 = l { 0 ifS n + 1 =O,

(9)

184 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where 4 a and t,h are random sources for deceleration and acceleration, respectively: (10) (11)

^=aT6(pa-r), *b=aT0(Pb-r),

p a and p b are probabilities of random acceleration and deceleration, respectively; r = rand(0,l), i.e., this is an independent random value uniformly distributed between 0 and 1; 9(z) = 0 at z < 0 and 9(z) = 1 at z > 0; S n+] denotes the state of vehicle motion (S n+1 = - 1 represents deceleration, Sn+1 = 1 acceleration, and Sn+1 = 0 motion at nearly constant speed):

Sn+1=

[-1 i f v n + 1 < v n - 8 1 ifvn+1>vn+8 0 otherwise,

(12)

where 8 is a constant (8 « at).

Fig. 1. Explanation of the model: (a) Hypothetical steady state model solutions, (b) Steady state solutions and the line / that represents a steady propagation of the downstream front of a wide moving jam. (c) Model of an on-ramp bottleneck. In (b), qout and p min are the flow rate and density in free flow formed in the wide moving jam outflow, p max is the density within a wide moving jam (see references in Kerner (2004)). To simulate a driver time delay either in vehicle acceleration or in vehicle deceleration, an and b n in (2), (3) are taken as the following stochastic functions an=a6(P0-ri), bn=a0(P1-r1),

(13) (14)

Microscopic three-phase traffic theory and applications 185

Po=!Po 0 [1

if

*-*1 Sn=l,

Pl

lf

S

lP 2

if

Pl=j 1

lf

"""1

(15)

(16)

Sn=-1,

where rx = rand(0,1); probabilities p o (v), p 2 (v) are given functions of speed, probability P! is a constant;

1 - Po and 1 - P, are the probabilities of a random time delay in vehicle

acceleration and deceleration, respectively. Models for two-lane traffic flow and on-ramp bottleneck Lane changing rules in a two-lane model are associated with conditions R^L:

v;>v,iI,+81andvI1Sv<>n,

(17)

L^R:

v;>v,>n+51orv>vn+51

(18)

for lane changing from the right lane to the left (passing) lane (R -» L) (17) and a return change (L —» R) (18). The safety conditions for lane changing are g>min(vnT,D;-d),

(19)

g;>min(v;T,D;-d),

(20)

D;=D(vnX),

(21)

D;=D(V;,vn).

Functions in (21) are associated with the function D(u,w) (6). In (17), (18), &l >0 is a constant, hi (17)-(21), superscripts + and - in variables and functions denote the preceding vehicle and the trailing vehicle in the "target" (neighboring) lane, respectively. In the model, there is an on-ramp on the main two-lane road (Fig. l(c)). The related on-ramp bottleneck consists of two parts: (i) The merging region of length L m , where vehicles can merge onto the main road from the on-ramp lane, (ii) A part of the on-ramp lane of length Lr upstream of the merging region, where vehicles move in accordance with the model (1)-(16). Open boundary conditions are applied. Main behavioral model assumptions and model parameters 1. In synchronized flow, a driver accepts a range of different hypothetical steady state speeds at the same space gap to the preceding vehicle (see (2) and Fig. l(a)). 2. A driver tends to adjust the speed to the preceding vehicle within the synchronization distance (5), (6).

186 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 3. Over-acceleration effect: In synchronized flow of a lower density, a driver searches for the opportunity to accelerate and to pass. Random acceleration ^a (9), (10) describes this effect: The over-acceleration is simulated as a "collective effect", which occurs on average in traffic flow, through the use of random vehicle acceleration. 4. Over-deceleration effect occurring if the preceding vehicle begins to decelerate unexpectedly. This well-known effect associated with a driver time delay in deceleration simulates moving jam emergence in synchronized flow. In the model, the over-deceleration is simulated as a "collective effect" through the use of random deceleration £b (9), (11). Thus, random deceleration and acceleration (9)-(ll) simulate driver time delays mentioned in item 3 and 4, which occur if the vehicle either accelerates or decelerates only, therefore, in (9) £,n = 0 if Sn+1 = 0. 5. At the downstream front of synchronized flow, which separates this synchronized flow upstream and free flow downstream, a driver in synchronized flow does not accelerate before the preceding vehicle has begun to accelerate. Formulae (13), (15) together with (l)-(3) simulate the associated driver time delay in acceleration that takes place if the vehicle does not accelerate (S n * 1) at time step n. 6. Moving in synchronized flow, a driver comes closer to the preceding vehicle over time that explains the pinch effect in synchronized flow (Kerner, 2004). Formulae (14), (16) together with (l)-(3) simulate a driver time delay in deceleration associated with this effect. We see that random model characteristics are used to simulate various driver time delays, which are assumed to be different in various local driving conditions. Quantitative validation of these driver time delays and of other model parameters have been made to have simulation parameters of congested patterns, in particular, the characteristic times of phase transitions as well as the velocities of the fronts that separate free flow, synchronized flow, and moving jams within the patterns in accordance with empirical results (Kerner, 2002). A detailed consideration of this model parameter validation can be found in (Kerner, 2004). DIAGRAM OF CONGESTED PATTERNS AT ON-RAMP BOTTLENECK IN THREE-PHASE TRAFFIC THEORY This section contains simulation results of congested patterns at an on-ramp bottleneck and of the associated congested pattern diagram determined from application of the model presented in the previous section. This congested pattern diagram contains regions of congested pattern occurrence and existence in the flow-flow plane (Fig. 2(a)) whose coordinates are the flow rate qin in free flow upstream of the bottleneck and the flow rate to the on-ramp qon (Fig. l(c)) (Kerner, 2002; Kerner and Klenov, 2002, 2003). There are two main types of congested patterns (Fig. 2): (i) a synchronized flow pattern (SP) and (ii) a general pattern (GP). The SP is a congested pattern that consists of synchronized flow only, i.e., there are no wide moving jams within the SP. The GP is a congested pattern, which consists of synchronized flow upstream of the bottleneck and wide moving jams that emerge spontaneously in that synchronized flow.

Microscopic three-phase traffic theory and applications 187 There are three types of SPs: (1) Widening SP (WSP) (Fig. 2(b)). The downstream front of an WSP is fixed at the bottleneck. The upstream front of the WSP propagates upstream continuously over time. (2) Localized SP (LSP) (Fig. 2(c)). As in the WSP, the downstream front of an LSP is fixed at the bottleneck. In contrast with the WSP, the upstream front of the LSP is localized at some distance upstream of the bottleneck. (3) Moving SP (MSP) (Fig. 2(d)). In contrast with the LSP and WSP, an MSP is an SP that propagates as a whole localized pattern on the freeway over time. However, in contrast with a wide moving jam, if an MSP reaches a bottleneck, the MSP is caught at the bottleneck: The MSP that propagates upstream can exist only for a finite time. There are two main boundaries in the pattern diagram, FS(B) and Sj B) (Fig. 2(a)). Below and left of the boundary FgB), free flow is realized at the bottleneck. Between the boundaries FS(B) and Sj B) different SPs emerge on the main road upstream of the bottleneck (Figs. 2(b-d)). Right of the boundary Sf } different GPs appear (Figs. 2(e-h)). If qin > q out , where qout is the flow rate in free flow formed in the wide moving jam outflow, then the width (in the longitudinal direction) of the farthest upstream wide moving jam of an GP gradually increases over time (Figs. 2(e, h)). If in contrast qin < q out , then the farthest upstream wide moving jam of an GP gradually dissolves over time (Fig. 2(f)). However, even if qin < q out , the region of wide moving jams in the GP can expand in the upstream direction. There is a region in the diagram right of the boundary Sj B) where a dissolving GP (DGP) occurs. In the DGP (Fig. 2(g)), continuous wide moving jam emergence is interrupted. As a result, a single wide moving jam emerges only and an LSP remains at the bottleneck, while the wide moving jam moves on the main road upstream of the bottleneck. It must be noted that several independent scientific groups have recently developed new microscopic traffic flow models in the context of the author's three-phase traffic theory (Davis, 2004; Lee et at, 2004; Jiang and Wu, 2004). Models of Lee et al. (2004) and Jiang and Wu (2004) exhibit features of the diagram of Kerner (2002) (Fig. 2(a)).

188 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 2. Congested patterns at on-ramp bottleneck (Kerner, 2002; Kemer and Klenov, 2002, 2003): (a) - Congested pattern diagram, (b-g) - Vehicle speed on the main road within different congested patterns: an WSP (b), an LSP (c), an MSP (d), an GP at qin > qout (e), an GP at qin < qout (f), and an DGP (g). (h) - Vehicle trajectories within the GP in (e).

Microscopic three-phase traffic theory and applications 189 CRITICAL DISCUSSION OF EARLIER DIAGRAM OF CONGESTED PATTERNS AT ON-RAMP BOTTLENECK In this section, we compare the congested pattern diagram of three-phase traffic theory discussed in the previous section with earlier model results associated with congested pattern emergence at an on-ramp bottleneck as well as with empirical features of the onset of congestion in free flow at the bottleneck. Results of three-phase traffic theory (Kerner, 1998, 2002, 2004) contradict qualitatively with conclusions of all earlier traffic flow theories and models that claim to explain the onset of congestion at the bottleneck (Helbing, 2001). Most of these microscopic, macroscopic, probabilistic, and other models of freeway traffic explain the onset of congestion in free flow at the bottleneck by moving jam emergence. Note that congested states occurring at the bottleneck, which are associated with different types of moving jams, are classified by Helbing et al. (1999) as MLC (moving localized cluster), TSG (triggered stop-and-go traffic), and OCT (oscillatory congested traffic) (Fig. 3(a)). (a) Diagram of Helbing

Fig. 3. Sketch of congested pattern diagrams at an on-ramp bottleneck at a high enough given flow rate qin upstream of the bottleneck as a function of the flow rate to the on-ramp qon : (a) - earlier traffic theories (Helbing et al, 1999). (b) - three-phase traffic theory (Kerner, 2002). However, the fundamental model result that the onset of congestion in free flow at freeway bottlenecks is associated with spontaneous moving jam emergence (Kerner et al, 1995; Lee et al, 1998; Helbing et al., 1999; Helbing, 2001) is in a serious conflict with empirical evidence (Kerner, 1998, 2002, 2004). In empirical observations, rather than moving jam emergence, a phase transition from free flow to synchronized flow (F -> S transition) governs the onset of congestion in free flow at the bottleneck. A first-order F —> S transition postulated in three-phase traffic theory discloses the nature of the breakdown phenomenon at freeway bottlenecks found in empirical observations (see Hall et al. (1992); Elefteriadou et al. (1995);

190 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Persaud et al. (1998); Kerner (2004) and references therein): The terms "F->S transition", "breakdown phenomenon", "traffic breakdown", and "speed breakdown" are synonyms. The crucial difference between the congested pattern diagram at an on-ramp bottleneck derived by Helbing et al. (1999) for earlier traffic flow models (e.g., Kerner et al, 1995; Lee et al., 1998; Helbing et al., 1999; Helbing, 2001) and the congested pattern diagram of threephase traffic theory (Fig. 2(a)) derived by Kerner (2002) can be seen in Fig. 3. If the flow rate upstream of the bottleneck qin is a given high enough value and the flow rate to the on-ramp qon increases gradually beginning from zero, then in the diagram of Helbing (Fig. 3(a)) the onset of congestion at the bottleneck is associated with moving jam emergence in free flow. This is inconsistent with empirical results mentioned above in which moving jams do not emerge spontaneously in free flow at the bottleneck. In contrast, in the diagram of Kerner (Fig. 3(b)) under the same conditions the onset of congestion is associated with synchronized flow emergence (different SPs appear). This is in accordance with empirical results. If qon further increases, then as follows from the diagram of Helbing (Fig. 3 (a)), homogeneous congested traffic (HCT) occurs in which no wide moving jams emerge. This model result is also inconsistent with empirical results in which at greater qon moving jams emerge in synchronized flow, i.e., GPs appear (Kerner, 1998, 2002). The latter is exact the result of the diagram of Kerner (Fig. 3(b)) in which GPs emerge at greater qon . Conclusions: Congested patterns at the bottleneck are qualitatively different in the diagram of Helbing (Fig. 3(a)) and in the diagram of Kerner (Fig. 3(b)). The diagram of Helbing is inconsistent with empirical congested pattern emergence, whereas the diagram of Kerner explains empirical congested pattern features. PROBABILISTIC THEORY OF FREEWAY CAPACITY AT BOTTLENECKS In this section, a deep connection between the congested pattern diagram based on threephase traffic theory (Figs. 2(a) and 4(a)) with a probabilistic theory of freeway capacity at bottlenecks is discussed (see references in Kerner (2004)). This diagram (Fig. 4(a)) found in model simulations and a probabilistic theory of traffic breakdown at an on-ramp bottleneck (Kerner and Klenov, 2005) can explain empirical features of probabilistic freeway capacity at the bottleneck (Elefteriadou et al., 1995; Persaud et al, 1998; Lorenz and Elefteriadou, 2000). Firstly note that in three-phase traffic theory (Kerner, 2004), traffic breakdown is associated with a deterministic, i.e., motionless and permanent perturbation (decrease in speed and increase in density) in free flow occurring at the bottleneck due to the on-ramp inflow. Even if there were no random perturbations in traffic flow, the deterministic traffic breakdown (deterministic F —> S transition) occurs within the deterministic perturbation at the bottleneck after a critical flow rate for deterministic traffic breakdown Isum

=

Ideterm, FS

(22)

Microscopic three-phase traffic theory and applications 191 is reached, where qsum = qon + q in . However, real random perturbations within the deterministic perturbation can cause random traffic breakdown (random F -> S transition) at the flow rate Isum

<

(23)

Ideterm, FS

In three-phase traffic theory, the boundary FS(B) in the congested pattern diagram (Figs. 2(a) and 4(a)) is associated with this random F - » S transition occurring during a given time interval Tob with probability PpB), which is equal to 1. hi empirical data, the role of the time Tob for observing traffic flow is played by the averaging time interval Tav for traffic flow.

Fig. 4. Probabilistic breakdown phenomenon at on-ramp bottleneck (Kerner, 2004; Kerner and Klenov, 2005): (a) - Simplified diagram from Fig. 2(a) with the threshold boundary Ft(hB). (b) - Boundaries for two constant values of the breakdown nucleation rate G(FB) (28) (curves 1 and 2) together with the boundary F ^ ^

s

for deterministic traffic breakdown, (c) -

B)

Probability P^ for traffic breakdown as a function of the flow rate qsum at a given qon . A random F - » S transition can also occur at the bottleneck during the time interval Tob if a point (q on , q in ) in the diagram is below and left of the boundary F^B) (dashed region in Fig. 4(a)). However, the probability P^B) for this F->S transition is less than 1. The more distant the point (q on , q in ) from the boundary FS(B), the less the probability P^B) for an transition at the bottleneck. There should be the threshold boundary FtbB) in the diagram: If a

192 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction point (q o n ,q i n ) in the diagram is below and left of this threshold boundary Ft(hB), then the probability PpB' for an F -» S transition at the bottleneck during the time interval Tob is equal to zero, i.e., no F —> S transition can occur at the bottleneck during this time interval. Thus, there are an infinite number of maximum freeway capacities in free flow at an on-ramp bottleneck, qJ^exeB). These maximum freeway capacities are equal to the critical (maximum) flow rates downstream of the bottleneck qsiim = qin + qon associated with the points (q on , q in ) in the diagram, which lie on the boundary FS(B): qSx eB) =qsu m (qon.q m )l F . B .The maximum freeway capacities related to the boundary FS(B)

(24) depend on the flow rates qon ,

q in , and the time interval T ob . The greatest of these maximum freeway capacities is the maximum freeway capacity at qon -> 0 (but qon * 0 ) denoted by q^raexeB^ (Figs. 2(a) and 4(a)). Each of the maximum freeway capacities cannot be observed at the bottleneck during a longer time interval thanTob . This is related to an F—»S transition that occurs spontaneously in this free flow at the bottleneck during the time interval Tob with the probability

p B>

- L,>r> =1 -

(25)

If Tob decreases, then the position of the boundary FS(B) changes: At a given q in , the maximum capacity q™aexeB) associated with a new boundary position increases. This increase has a limit given by the maximum capacity q ^ ^ p s (22) that lies on a limit boundary Fdeterm s

m tne

diagram associated with a deterministic F —> S transition (deterministic traffic

breakdown) (Fig. 4(b)). Thus, when q££ B) -> q FS , the boundary F Fd(B)erm, s . There are also an infinite number of minimum freeway capacities at the bottleneck, q(tB). These minimum freeway capacities are equal to the threshold flow rates downstream of the bottleneck qsum = qin + qon associated with the points (q on , q in ) in the diagram, which lie on the threshold boundary Ft(hB) in the diagram of congested patterns: q!hB)=qsum(qon>q,n)|Fr-

(26)

The minimum freeway capacities are the minimum flow rates downstream of the bottleneck at which an F—>S transition can still occur spontaneously in this free flow at the bottleneck during the time interval T ob . This mean that the flow rates downstream of the bottleneck qsum = qin +q<>n m at are related to points (q on , q in ) in the diagram, which lie below and left

Microscopic three-phase traffic theory and applications 193 of the threshold boundary Ft(hB) in the diagram of congested patterns, are less than any freeway capacity: At these flow rates qsum = qin + qon the probability PFB) for an F —> S transition at the bottleneck during the time interval Tob is equal to zero. Maximum freeway capacities are related to the boundary FS(B) in the diagram. Minimum freeway capacities are related to the threshold boundary FtbB) in the diagram (Fig. 4(a)). Different flow rates qsum = qin + qon downstream of the bottleneck associated with the points (q on , q in ) between the boundaries FgB) and Ft(hB> in the diagram are also equal to the infinity of freeway capacities in free flow at the bottleneck. This is because at each of the corresponding flow rates qsum =q i n +q on an F—>S transition occurs spontaneously at the bottleneck during the time interval Tob with the probability P F B) , which satisfies the condition O
(27) B

B)

When the flow rate qsum increases beginning from qsum = q ' \ the probability PF (27) for traffic breakdown exhibits an exponentional growth (Fig. 4(c)). Such growth, which is observed in empirical observations (Persaud et al., 1998; Lorenz and Elefteriadou, 2000), is usual for each first-order phase transition in systems of natural science. This confirms that traffic breakdown is a first-order F—>S transition. These conclusions follow also from a probabilistic model for random traffic breakdown at the bottleneck, hi accordance with this model, the nucleation rate GFB) of traffic breakdown is (Kerner and Klenov, 2005)

G (B) =

U . \l/2 a (B) qde,em1,Fs£dAJ

V2*Nd

f 3/2\ ISNgAg)

\

S^J

w h e r e A

(B) _ Qdete^FS q«um qdB>em>FS

(2g)

Idet'erm FS , N d , and ^d are functions of qon . The nucleation rate G F |' determines the mean frequency [ min"1 ] with which for given qon and qjn random traffic breakdown occurs at the bottleneck. As the probability PFf' (Fig. 4(c)), the associated generation rate GFB) for traffic breakdown exhibits an exponentional growth if Ac (28) decreases, i.e., when qsum increases (curves 1 and 2 in Fig. 4(b) are related to GFB) = 1/3.5 and 1/60 [miiT1], respectively). For traffic flow, this growth is very sensible to the critical flow rate qdB)erm FS (22). This emphasizes an important impact of deterministic traffic breakdown on freeway capacity.

194 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction METASTABILITY OF CONGESTED PATTERNS AT ON-RAMP BOTTLENECK In this section, we show that at the same traffic demand depending on initial conditions and fluctuations in traffic flow qualitatively different congested patterns can occur at the same bottleneck. This effect is called the metastability effect of pattern formation.

Fig. 5. Metastability of congested pattern (Kerner and Klenov, 2003): (a) Initial short-time perturbations to the flow rate qon used for pattern excitation, (b-d) Different traffic patterns at the same flow rates qon and qjn : Free flow (b), an LSP (c), and an GP (d). No perturbation to qon (line 1 in (a)) is applied for free flow (b); for the excitation of the LSP (c) (line 2 in (a)) and of the GP (d) (line 3 in (a)) short-time perturbations to the flow rate qon are applied. hi Fig. 5, depending on the amplitude of an initial short-time perturbation applied to qon (Fig. 5(a)) either free flow (Fig. 5(b)) or an LSP (Fig. 5(c)), or else an GP (Fig. 5(d)) is excited at the bottleneck. The stability of these three states is markedly different. Whereas free flow (Fig. 5(b)) and the GP (Fig. 5(d)) remain stable for 120 min in 10 different calculated realizations, the LSP has transformed spontaneously into an GP in 6 of 10 realizations. INTENSIFICATION OF DOWNSTREAM CONGESTION DUE TO UPSTREAM CONGESTION In this section, based on the microscopic model discussed above we consider some results of numerical simulations of complex congested patterns, which emerge on a freeway with two adjacent on-ramp bottlenecks (Kerner and Klenov, 2003). We call the bottleneck with the flow rate to the on-ramp q^ p) , which is at a freeway location x = x^ p) upstream, the on-ramp 'U' (upstream bottleneck) and the on-ramp bottleneck with the flow rate to the on-ramp

Microscopic three-phase traffic theory and applications 195 q^ own) ,

which is at a location x = x^ own) downstream, the on-ramp

'D' (downstream

bottleneck). Accordingly, we call congestion that emerges upstream of the downstream bottleneck as "downstream congestion" and congestion that emerges upstream of the upstream bottleneck as "upstream congestion". One of the non-linear effects of complex pattern transformation and interaction found in this case is the effect of intensification of downstream congestion due to upstream congestion, which is considered below.

Fig. 6. Intensification of upstream congestion due to downstream congestion. DGP (a) transforms into an GP (b). In an GP (c) the frequency of moving jam emergence increases (d). The initial flow rate in free flow on the main road upstream of the on-ramp 'D' q^own) = q i n + q ^ p )

in each of the left panels of Fig. 6 is equal to the flow rate

q(down) _ ^

nea£^ of^e

+

q^up) j

the flow rate q^ own)

flow rate q-^

p)

associated right panels of Fig. 6. However, in Figs. 6(a, c),

= 0, whereas in Figs. 6(b, d), the flow rate q^p) > 0. The equality of the

in Figs. 6(a, b)

is realized due to higher flow rate qin in free flow upstream

of the upstream bottleneck in Fig. 6 (a) in comparison with the flow rate qin in Fig. 6(b). This is also valid for Figs. 6(c, d). Depending on the flow rate q|^p) and on the congested pattern type at the downstream bottleneck, different intensification phenomena for the initial downstream congestion are found. We consider only two of them.

Transformation of DGP into GP An initial DGP at the on-ramp 'D', where only one wide moving jam emerges (Fig. 6(a)), transforms into an GP at this bottleneck where an uninterrupted sequence of wide moving jams emerges (Fig. 6(b)). After the wide moving jam in the DGP (Fig. 6(a)) emerges, the

196 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction current flow rate upstream of the on-ramp 'D' q,-nOwn:> cannot be greater than qout (Fig. l(b)): qtdown) <. q ^ ^ j]^is decrease in q^cwn) occurs because vehicles merging onto the main road from the on-ramp 'U' must slow down when they reach the wide moving jam. When the wide moving jam is between the on-ramps 'D' and 'U', the flow rate q^ p) does not lead to an increase in the flow rate q,^0™"': Due to the flow rate q["p) only the jam inflow increases. hi Fig. 6(b), an LSP is induced at the on-ramp 'U' after the wide moving jam from an DGP (downstream congestion) has reached the on-ramp 'U'. After the wide moving jam has passed the on-ramp 'U', there are no wide moving jams between the on-ramps 'D' and 'U'. For this reason, vehicles merging onto the main road from the on-ramp 'U' cause an increase in q(down) j^-g f ] o w r a t e j s m u c j j greater than q out . This leads to subsequent moving jam emergence at the on-ramp 'D', and so on: An GP is formed at the on-ramp 'D' (Fig. 6(b)). The mean period of moving jam emergence in the GP is determined by the time of wide moving jam propagation to the on-ramp 'U'. Before a wide moving jam has passed the onramp 'U' the condition q|nOwn) < qout is satisfied, and no moving jam can emerge at the given flow rate q^own) to the on-ramp 'D'.

Increase in frequency of moving jam emergence The onset of upstream congestion can lead to intensification of downstream congestion that results in an increase in the mean frequency of moving jam emergence in an initial GP at the on-ramp 'D' (Fig. 6(d)). This effect is similar to the transformation of an DGP into an GP discussed above. After the first wide moving jam has emerged in the GP (Fig. 6(c)), the current flow rate upstream of the on-ramp 'D', q^ own) , has dropped to q\^°'m) < q out . At the flow rate q^own) < q out , the frequency of moving jam emergence in the GP upstream of the on-ramp 'D' is relatively low at the chosen flow rate q^own) to the on-ramp 'D' (Fig. 6(c)). In contrast, in Fig. 6(d) after the wide moving jam has passed the on-ramp 'U', first there are no wide moving jams between the on-ramps 'D' and V. Vehicles that merge from the onramp TJ' cause an increase in the flow rate q^own) upstream of the GP. This leads to wide moving jam emergence within the GP during a considerably shorter time interval than the mean time between wide moving jams in the GP shown in Fig. 6(c). As a result, the mean frequency of wide moving jam emergence in the GP at the on-ramp 'D' increases Fig. 6(d). FEEDBACK ON-RAMP METERING hi this section, based on numerical simulations of the model considered above we compare the ALINEA method for feedback on-ramp inflow control of Papageorgiou et al. (1991) (see references in Papageorgiou and Kotsialos (2002)) whose basic idea is to maintain free flow at an on-ramp bottleneck with a "congested pattern control approach" (ANCONA for short) of

Microscopic three-phase traffic theory and applications 197 the author (see references in Kerner (2004)). In ANCONA, congestion at the on-ramp bottleneck is allowed to set in. However, in ANCONA a congested pattern that emerges at the bottleneck should not propagate upstream; rather than propagating upstream the congested pattern should be localized on the main road in a small neighborhood of the bottleneck. In ALINEA, the flow rate q^°nt) of vehicles that can merge onto the main road from the onramp is (Papageorgiou et ah, 1991) (cont)/f \_-(cont)/- t

\ + v ('f)

C29"l

One of the realizations of ANCONA is as follows (Kerner, 2004)

q

°"

Cl)

lq on2

1 fv<->(t s ) >v c o n g -

(30)

In (29) and (30), t s = Tavs, s = 0,1, 2,...; Tav = 1 min is the averaging time interval for data measured by feedback control detectors. In ALINEA, these detectors are on the main road downstream of the on-ramp bottleneck. In contrast, in ANCONA the detectors are upstream of the bottleneck. In (29), o is the occupancy, Ko is a positive given constant, o ^ ' is the optimal occupancy; oj£™'' is the occupancy measured at the detectors in ALINEA. In (30), v(det) is the speed measured by the detectors in ANCONA; vcong is a "congestion speed" that is constant; qonl and qon2 are constants (q on2 > qoni)- hi ANCONA, there is no on-ramp control as long as free flow is at the bottleneck. On-ramp inflow control is first realized only after the onset of congestion has occurred at the bottleneck. Feedback control is performed when v(det) < v cong , i.e., the average speed v(det) is equal to or drops below v cong . Then the flow rate qon°nt) is reduced via light signal operation. The decrease in this flow rate should lead to an increase in v(det) above v cong . Then greater flow rate qon°nt) is allowed via light signal operation. If under this greater flow rate q|,™nt) the onset of congestion occurs at the bottleneck once more, an incipient congested pattern begins to propagate upstream. As a result, the speed at the detector for feedback control decreases: The condition v(det) < vcong is satisfied once more. This leads to a new decrease in the flow rate qOn°nt>, and so on. To compare ALINEA with ANCONA, we chose the flow rates qin and qon related to an onramp bottleneck at which an GP appears at the bottleneck when there is no on-ramp inflow control. We find that an appropriate choice of the optimal occupancy o ^ ' in the ALINEA method leads to a suppression of GP emergence due to a decrease in the flow rate (29) in comparison with qon using light signal operation in the on-ramp lane (Fig. 7(a)). At the same

198 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction traffic demand in ANCONA, an LSP remains at the bottleneck (Fig. 7(b)), i.e., congestion occurs that is localized at the bottleneck. Nevertheless, because the speed within the LSP is not very low (about 80 km/h) the difference between the travel times on the main road in these methods cannot almost be distinguished (Fig. 7(c)). In contrast, in ALINEA the waiting time at the light signal (curve 2) is considerably longer than in ANCONA (curve 1 in Fig. 7(d)). Simulations show that the throughput in ANCONA is greater than in ALINEA. Moreover, it turns out that if short-time perturbations in the flow rate qon and/or qin occur, then ALINEA cannot prevent GP emergence at the bottleneck. This is associated with the metastability of free flow with respect to congested pattern formation, which is realized between the boundaries FS(B) and Ft(hB) in the diagram of congested patterns (Fig. 4(a)). Strictly speaking, ALINEA can maintain reliably free flow at the bottleneck if the flow rate downstream of the bottleneck is less than minimum freeway capacities of free flow at the bottleneck. This is because only in this case the probability for speed breakdown is zero, hi ANCONA, congestion is allowed to set in at the bottleneck. Therefore, greater throughputs are possible. In contrast with ALINEA, in ANCONA perturbations in the flow rate qon and/or qm cannot cause GP emergence at the bottleneck.

Fig. 7. Comparison of the ALINEA and ANCONA methods, (a, b) Vehicle speed on the main road in space and time for the ALINEA method (a) and for the ANCONA method (b). (c, d) Dependence of the travel time on the main road (c) and in the on-ramp lane (d) on time. In (c, d) curves 1 for the ANCONA method and curves 2 for the ALINEA method are shown. Thus, benefits of the ANCONA method in comparison with the ALINEA method are: 1. Achievement of greater throughput under very high traffic demand when congestion has to occur somewhere in a traffic network. In the latter case, a more homogeneous distribution of congestion among bottlenecks in the network may be possible via the application of ANCONA.

Microscopic three-phase traffic theory and applications 199 2. Considerably shorter vehicle waiting times at the light signal in the on-ramp lane(s). 3. Upstream propagation of congestion on the main road does not occur regardless of short-time perturbations in traffic flow and even if a congested pattern occurs at the bottleneck: In ANCONA, the congested pattern is spatially localized at the bottleneck. EFFECT OF AUTOMATIC CRUISE CONTROL ON CONGESTED PATTERNS AT ON-RAMP BOTTLENECK In this section, based on numerical simulations of the model considered above we discuss an influence of automatic cruise control (ACC) vehicles on traffic flow at an on-ramp bottleneck (see references in Kerner (2004)). ACC is one of the important ways of enhancing driver comfort and safety in freeway traffic (e.g., Becker et al, 1994; van Arem et al, 1996). An ACC vehicle measures the space gap gn = xln - x n - d and the relative speed vgn - v n . The ACC vehicle calculates the current time gap between the ACC vehicle and the preceding vehicle. In real ACC systems, two or more ranges of speed can be chosen where different dynamic rules are used. For simplicity we will discuss a hypothetical ACC system in which there is only one dynamic rule for the ACC vehicle in the whole possible range of vehicle speed. In most known ACC systems, at least in one of the speed ranges the dynamic behavior of the ACC vehicle can be approximately described by the well-known equation (e.g., Becker et al, 1994; van Arem et al, 1996; McDonald and Wu, 1997): a =K 1 (g n -vnT) + K 2 (v, >n - v n ) ,

(31)

where aJ,ACC' is ACC vehicle acceleration; tj A C C ) is a desired time gap of the ACC vehicle; K, and K2 are coefficients of ACC adaptation. In simulations of ACC vehicle influence on congested patterns at the bottleneck, there are vehicles that have no ACC system and ACC vehicles. Vehicles that have no ACC system move in accordance with the model described above. The ACC vehicles are randomly distributed on the road between other vehicles that have no ACC system. The ACC vehicles move in accordance with Eq. (31) where, in addition, some limitations of ACC vehicle acceleration and deceleration as well as the safe speed are used to avoid collisions. Automatic cruise control with quick dynamic adaptation To study the influence of ACC vehicles on congested patterns, we assume that at chosen qin and qon an GP occurs at the bottleneck, if there are no ACC vehicles (Fig. 8(a)). ACC vehicles can decrease the amplitude of moving jams in the initial GP if the coefficients of ACC adaptation Kj and K 2 are large enough. In this case, the ACC vehicle quickly reacts to changes in time gap and speed difference to the preceding vehicle. Moving jam suppression in the initial GP due to ACC vehicles can be seen in Figs. 8(b-d). Here, at the same qon and qin as those in Fig. 8(a) the percentage of ACC vehicles, y, in the flow rates qon and qin

200 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction increases (Figs. 8(b-d)). There is some critical y = y cr . When this critical percentage of ACC vehicles is reached, there are almost no moving jams in the congested pattern on the main road upstream of the bottleneck. If y further increases, no new moving jams occur in the congested pattern. Thus, ACC vehicles can prevent moving jam emergence.

50 100 ACC percentage Fig. 8. Influence of ACC vehicles with quick dynamic adaptation on an GP at an on-ramp bottleneck, (a-d) - Speed on the main road at different y. (e, f, g) - Discharge flow rate (e), the flow rate within the congested pattern (f) and travel time (g) as functions of y. TjACC) = 1 s, ycr = 39% , K, = K| 0 ) /max(v n ,v m m ), K<0) = 4.5 m/s 3 , K2 = 0.9 s"1, vmm = 5 m / s . At a chosen T^ ACC) , the discharge flow rate qou°ttle\ i.e., the flow rate downstream of the bottleneck is an increasing function of y (Fig. 8(e)), whereas the flow rate qcon within the congested pattern first decreases (Fig. 8(f)). This is because the speed in the pattern upstream decreases, when y increases. Thus, ACC vehicles with quick dynamic adaptation prevent moving jam emergence, i.e., a more comfortable and safer driving experience is possible. However, ACC vehicles cannot prevent or dissolve traffic congestion. They even lead to

Microscopic three-phase traffic theory and applications 201 some increase in the travel time Ta (Fig. 8(g)). T^y) has a maximum when y

further

increases. Automatic cruise control with slow dynamic adaptation Another case occurs when an ACC vehicle reacts slowly to changes in gap and speed differences to the preceding vehicle, i.e., K, and K2 in (31) are small. Then ACC vehicles can induce congestion at the bottleneck even if there is no congestion without ACC vehicles.

Fig. 9. Influence of ACC vehicles with slow dynamic adaptation on the onset of congestion at the bottleneck. Speed on the main road at different y. KJ 0 '=1 m / s 3 , K 2 = 0.18 s"1, Ycr = 9% . Other parameters are the same as those in Fig. 8. In Fig. 9(a), where there are no ACC vehicles, free flow occurs at the bottleneck at the chosen flow rates qon and q in . There is some critical y = y cr . At the same qon and q in , if ACC vehicles appear on the freeway, then at y < ycr there is no influence of ACC vehicles: Free flow remains at the bottleneck. If in contrast y > y cr , then ACC vehicles induce the onset of congestion at the bottleneck (Fig. 9(b)). As a result, an GP appears at the bottleneck. The greater y, the greater the frequency of moving jam emergence in the GP (Figs. 9(c, d)). CONCLUSIONS 1. Three-phase traffic theory can explain and predict all known reproducible empirical features of phase transitions and congested patterns at freeway bottlenecks. 2. An F—>S transition governs phase transition phenomena in free flow at bottlenecks. Wide moving jams can occur spontaneously only in synchronized flow.

202 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 3. There is a deep connection between the diagram of congested patterns at a bottleneck with freeway capacity for free flow at the bottleneck: The threshold boundary for an F—>S transition in this diagram determines an infinity of the minimum freeway capacities, whereas the boundary at which the F -» S transition occurs spontaneously during a given time interval with the probability, which is equal to 1, determines an infinity of the maximum freeway capacities. 4. There are metastable regions in the diagram of congested patterns at the bottleneck in which different types of patterns can coexist. In other words, at the same traffic demand depending on perturbations in traffic flow various patterns can occur in an initial free flow at the bottleneck. However, the lifetime of these patterns, which is associated with pattern stability, can be qualitatively different. 5. If two adjacent bottlenecks are close to one another, complex patterns and nonlinear pattern interaction effects are realized. In particular, congestion at an upstream bottleneck can intensify congestion at a downstream bottleneck. 6. Different kinds of freeway traffic control methods and vehicle assistance systems can be investigated in simulations based on three-phase traffic theory concerning their influence on the efficiency of control systems, as well as comfortable and safe driving. This is because a microscopic simulation environment based on three-phase traffic theory is in accordance with empirical spatiotemporal features of congested patterns. Thus, this is a high-value instrument in a study of different freeway traffic control strategies and traffic assistance systems before they are introduced to the market. ACKNOWLEDGEMENTS I thank Sergey Klenov for his help. REFERENCES Becker, S., M. Bork, H. T. Dorissen, G. Geduld, O. Hofmann, K. Naab, G. Nocker, P. Rieth, J. Sonntag (1994). Summary of Experience with Autonomous Intelligent Cruise Control (AICC). In: Proceedings of the 1st World Congress on Applications of Transport Telematics and Intelligent Vehicle-Highway Systems, pp. 1828-1843. Elefteriadou, L., R. P. Roess and W. R. McShane (1995). Probabilistic nature of breakdown at freeway merge junctions. Transportation Research Record 1484, 80-89. Davis L.S. (2004). Multilane simulations of traffic phases. Phys. Rev. E, 69, 016108. Gartner, N. H., C. J. Messer and A. Rathi (eds.) (1997). Special Report 165: Revised Monograph on Traffic Flow Theory (Transportation Research Board, Washington, D.C.). Gipps, P. G. (1981) A behavioral car-following model for computer simulation. Trans. Res. B. 15,105-111. Hall, F. L., V. F. Hurdle and J. H. Banks (1992). Synthesis of Recent Work on the Nature of Speed-Flow and Flow-Occupancy (or Density) Relationships on Freeways. Transportation Research Record 1365, 12-18.

Microscopic three-phase traffic theory and applications 203 Helbing, D. (2001). Traffic and Related Self-Driven Many-Particle Systems. Rev. Mod. Phys. 73,1067-1141. Helbing, D., A. Hennecke and M. Treiber (1999). Phase diagram of traffic states in the presence of inhomogeneities. Phys. Rev. Lett., 82, 4360-4363. Jiang R. and Wu, Q.-S. (2004). Spatial-temporal patterns at an isolated on-ramp in a new cellular automata model based on three-phase traffic theory. /. Phys. A, 37, 81978213. Kerner, B. S. (1998). Experimental Features of Self-Organization in Traffic Flow. Phys. Rev. Lett, 81, 3797-3800. Kerner, B. S. (2002). Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks. Phys. Rev. E, 65, 046138. Kerner, B. S. (2004). The Physics of Traffic (Springer, Berlin, New-York, Tokyo). Kerner, B. S. and S. L. Klenov (2002). A microscopic model for phase transitions in traffic flow. J. Phys. A: Math. Gen., 35, L31-L43. Kerner, B. S. and S. L. Klenov (2003). A microscopic theory of spatial-temporal congested traffic patterns at highway bottlenecks. Phys. Rev. E, 68, 036130. Kerner, B. S. and S. L. Klenov (2004). Probabilistic Breakdown Phenomenon at On-Ramp Bottlenecks in Three-Phase Traffic Theory. Cond-mat/0502281. E-print in http://arxiv.org/abs/cond-mat/0502281. Kerner, B. S., P. Konhauser and M. Schilke (1995). Deterministic spontaneous appearance of traffic jams in slightly inhomogeneous traffic flow. Phys. Rev. E, 51, 6243-6246. KrauB, S., P. Wagner and C. Gawron (1997). Metastable states in a microscopic model of traffic flow. Phys. Rev. E, 53, 5597-5602. Lee, H. Y., H.-W. Lee and D. Kim (1998). Origin of synchronized traffic flow on highways and its dynamic phase transitions. Phys. Rev. Lett., 81, 1130-1133. Lee, H. K., R. Barlovic, M. Schreckenberg and D. Kim (2004). Mechanical restriction versus human overreaction triggering congested traffic states. Phys. Rev. Lett., 92, 238702. Lorenz, M., L. Elefteriadou (2000). A probabilistic approach to defining freeway capacity and breakdown. Transportation Research Circular E-C018, 84-95. Persaud, B., S. Yagar and R. Brownlee (1998). Exploration of the Breakdown Phenomenon in Freeway Traffic, Transportation Research Record, 1634, 64-69. McDonald, M. and J. Wu (1997). The Integrated Impacts of Autonomous Intelligent Cruise Control on Motorway Traffic Flow. In: Proceedings of the LSC 97 Conference. (Boston, USA 1997). Papageorgiou, M., H. Hadj-Salem and J.-M. Blosseville (1991). ALINEA: a local feedback control law for on-ramp metering. Transportation Research Record, 1320, 58-64. Papageorgiou M., and A. Kotsialos (2002). Freeway ramp metering: An overview. IEEE Transactions on ITS 3, 271-281. van Arem, B., J. H. Hogema and S. A. Smulders (1996). The Impact of Autonomous Intelligent Cruise Control on Traffic flow. In: Proceedings of 3rd World Congress on Intelligent Transport Systems.

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A BEHAVIOURAL APPROACH TO INSTABILITY, STOP AND GO WAVES, WIDE JAMS AND CAPACITY DROP Chris Tampere, Department of Traffic and Infrastructure Katholieke Universiteit Leuven, Belgium; Serge Hoogendoorn, Delft University of Technology, Faculty of Civil Engineering and Geoscience; and Bart van Arem, TNO Inro, Department of Traffic and Transport and University ofTwente, Applications of Integrated Driver Assistance (AIDA)

INTRODUCTION Congested traffic flow dynamics have attracted much interest over the last decade. To explain congested traffic flow phenomena, some researchers have stressed the importance and validity of first order hydrodynamic models, while others point out the importance of second order phenomena (instability) for the formation of various congestion patterns, and of local perturbations initiating phase transitions. In this paper, a macroscopic traffic flow model is used to examine typical congestion related phenomena like traffic flow instability, stop-and-go waves, wide jams, hysteresis, and capacity drop, and the role of driver behaviour therein. The model relates individual carfollowing behaviour to macroscopic traffic flow dynamics, and accounts for the principal aspects of individual longitudinal behaviour like: finite reaction times, anticipation behaviour to conditions downstream (i.e. non-locality and anisotropy), and finite, speed dependent space requirements of drivers. Because of the strong relation with individual human driving behaviour and the gas-kinetic mathematics that underlie this model, it is called the humankinetic traffic flow model. Moreover, the parameters that govern the individual driver behaviour in the human-kinetic model are not necessary constant in time, but may vary with traffic flow conditions, location, weather, and other conditions. In this sense, the humankinetic traffic flow model is an experimental framework in which the role of variable individual longitudinal driver behaviour in macroscopic traffic flow dynamics can be examined. This model is used to examine why different congested traffic patterns that have been reported in the literature can emerge and under what conditions. The stability of the

206 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction deceleration process in the upstream jam front turns out to be of crucial importance as well as the queue discharge rate of the accelerating jam front downstream. To increase the insight into the nature of these macroscopic traffic flow properties, the human-kinetic model is used to identify the individual behavioural parameters having dominant influence. Furthermore, it is unlikely that these behavioural parameters remain constant in time. Therefore, the effect of some plausible assumptions regarding the variability of these parameters as a function of traffic conditions is analysed. Literature review Congestion formation was already accounted for in the earliest traffic flow models by Lighthill & Whitham (1955) and Richards (1956). Ever since, numerous authors have proposed more refined traffic flow models, motivated by the fact that some types of congested traffic, like for example stop and go waves, are not adequately described by LWR (or first order) theory. A well-known example is the second order model proposed by Payne (1971). In the second half of the 1990'.? the theme of congested traffic dynamics gained renewed interest when German physicists published empirical observations and theory of a previously unknown variety of congested patterns. Kerner and co-workers claimed the discovery of 'wide moving jams' (Kerner & Rehborn, 1996a) and of 'synchronised traffic' (Kerner & Rehborn, 1996b). The author synthesised and formalised a series of publications in his 'three phase theory' (Kerner, 1999), and provided simulation models corresponding to this theory (Kerner & Klenov, 2002). Helbing et al. (1999), almost simultaneously with Lee et al. (1999), published a theory based on simulation results and later empirical confirmation (Treiber et al., 2000; Lee et al., 2000), herewith broadening the spectrum of congested traffic patterns by discerning 'pinned' and 'localised clusters', 'triggered stop and go waves', 'oscillating congested traffic', and 'homogeneous congested traffic'. Empirical phenomena like traffic hysteresis, discovered as early as 1974 by Treiterer & Myers, were analysed and included in LWR and Payne-type models, for instance by Newell (1965) and Zhang (1998, 1999). Capacity drop, or the 'two-capacity phenomenon' (Banks, 1991) was introduced in LWR theory by Lebacque et al. (1998) by considering finite acceleration capability and by Daganzo (2002) via an adaptation of the fundamental diagram. The current article aims at providing insight into the key phenomena governing the aforementioned congested traffic flow patterns: traffic instability and the outflow or queue discharge from a jam. As such, the work presented in this article follows a tradition of theoretical in-depth analyses, like the work of Zhang (1999) on traffic hysteresis, that of Herrmann & Kerner (1998) on local cluster effects, and the critical review of macroscopic traffic flow models by Lebacque & Lesort (1999). Structure of the article The remainder of this article is structured as follows. The article starts with the definition of the traffic scenario that is considered in the simulations throughout the article and an overview of the assumptions on longitudinal individual driver behaviour, hi the following section, these behavioural assumptions are formalised, defining the macroscopic humankinetic traffic flow model. Then, we illustrate various congested traffic patterns that can be obtained with this model, which leads to the central questions of this article, inquiring into the

Instability, stop and go waves, wide jams and capacity drop 207 nature of these patterns, the conditions in which they occur, and the role of individual driver behaviour therein. The next three sections are then devoted to answering these questions, with the human-kinetic traffic flow model providing guidance in these analyses. First, we analyse the different congestion patterns at the macroscopic level, and highlight the role of traffic flow stability and the queue discharge rate. The next section dives deeper into these crucial aspects of traffic flow dynamics with the aim of gaining insight into the role of individual driver behaviour in these phenomena. We then elaborate on the effect of temporary variations of individual driver behaviour and their impact on traffic flow dynamics. Finally, the article concludes with a discussion of the main findings and an outlook to further research.

SPECIFICATION OF TRAFFIC SCENARIOS AND DRIVER BEHAVIOUR The scope of this paper is limited to the influence of longitudinal driver behaviour on traffic flow dynamics on motorways. A traffic scenario is analysed of a single motorway lane without overtaking. Congestion is triggered either by considering a jam in the initial conditions (without specifying any cause for the initial jam) or by considering merging traffic at an on-ramp. In all cases, the boundary condition upstream1 is a constant flow in equilibrium (no time-varying flows at neither on-ramp nor main lane) and open downstream (Figure 1).

Figure 1

Simulated scenario

In the model and analyses considered in this paper, the following assumptions hold for the longitudinal behaviour of drivers on the motorway: all drivers behave according to the same set of behavioural rules; every individual driver has the same parameters as all other members of this class, except for the desired speed that is drawn from a normal distribution (Hoogendoorn, 1999; Helbing 1997); a driver is always in one of two modes: free flowing (unconstrained) or car-following (constrained); in either case the control behaviour is characterised as a delayed stimulus-response process with: o apure time delay equal to the reaction time of the driver;

1 Note that this assumption does not hold when a queue spills back into this boundary. Solving this issue is not trivial in case of the non-local human-kinetic model. In the simulations in this article we have circumvented this issue by extending the spatial domain upstream if necessary and plotting only the relevant x-domain.

208 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction o

the response is a choice of acceleration; the vehicle however has limited acceleration and deceleration capability, so that the desired acceleration is bounded between a minimal and maximal value; o the stimulus of an unconstrained driver is the deviation of the individual speed with respect to the desired speed; o a constrained driver has two stimuli: the (perceived or anticipated) relative speed with the predecessor and the deviation of the gap to the predecessor with respect to a speed-dependent desired gap; the parameters of the stimulus response models may depend on the actual level of activation of the driver; this is the internal state of the driver that is normally equal to some comfortable or normal level, but can be raised or decreased due to more or less demanding traffic conditions experienced recently. More specifically, these behavioural assumptions are formalised for a single individual driver as follows. With the notation v for the speed and s for the gap, the actual acceleration v at time M-Twith 2" the reaction time of a driver j with a predecessor j - \ equals:

V{t+T>J^M

;

',(0-^M')) + y,.,(0-v,(0l

(1)

accmin < v < o,ccmax

with:

(2)

and parameters: : desired speed of driver7 Wj TW, TS, TV : relaxation factor for the desired speed, gap and relative speed : minimal and maximal acceleration capability of the accmin, accmax vehicle 5,rf, s* : linear and quadratic factors for the desired gap With the notation a for the activation level, the change of a in time is given by:

da _aL»al(v)-a dt ra

}

This equation expresses that the activation level a relaxes towards the speed-dependent normal level avmrnlal (v) with a relaxation factor za. To conclude the specification of individual longitudinal driver behaviour, a specification is needed for the dependency of the longitudinal behaviour of equations (1) and (2) on the activation level a. Little theoretical or experimental knowledge is available to date to validate any specification in this respect. In order to illustrate the potential influence of the activation level on driver behaviour, the plausible assumptions will be made further in this paper that less active driving can be identified either with more moderate levels of acceleration or with

Instability, stop and go waves, wide jams and capacity drop 209 larger values for the desired gaps (see section on variable driver behaviour). In the scenarios with on-ramps, no specific behavioural rules are specified with respect to individual merging behaviour, other than the following assumptions: the exact position along the on-ramp of merging into the main lane is drawn from a triangle-shaped probability distribution; prior to merging onto the main lane, drivers first adapt their speed on the on-ramp to the average speed of the main lane flow at the point of merging; drivers merging into the main lane have an activation level that is equal to the average activation level of drivers in the main lane flow at the point of merging.

FORMALISATION OF BEHAVIOURAL SPECIFICATIONS: THE HUMAN-KINETIC MODEL The human-kinetic traffic flow model The behavioural assumptions of the previous section have been formalised in a kinetic traffic flow model, according to the theory originally proposed by Prigogine & Herman (1971). The core of this model is the formulation of a generalised conservation law for vehicles with certain commonalities in their individual states S'. The conservation law expresses the conservation of the probability density p(t,S'), which is defined so that the expected number of vehicles having state »S"e[5",5"+dS") at time t equals p(t,S') dS" (Hoogendoorn ,1999; Leutzbach, 1988):

a

* r dt) { dt ) ,

M

In this equation, the dependency of p o n (t,S') has been omitted for notational simplicity; the product operator denotes the inner product and Vs, the Nabla operator for the state vector 5" with dimensions (x,si,S2, ...,sn). The method of moments is traditionally used2 to transform the kinetic continuity equation (4) into its equivalent macroscopic formulation (Helbing, 1997). With this method, dynamic equations for the ii order moment of the probability density function p for state variable s, are obtained by multiplication of both RHS and LHS of equation (4) by s" and consequent integration over all state variables in S'. It can be shown that with S'=(x,v) and for values of K equal to 0 and 1 respectively the following macroscopic 'momentum' equations are obtained:

It should be mentioned here that an alternative procedure exists, based on a Chapman-Enskog expansion of the kinetic traffic model (Nelson & Sopasakis, 1999). Although this approach has some elegant theoretical advantages, little is known about the practical applicability of the resulting sequence of traffic flow models, which was the motivation to use the more traditional, but more ad hoc approach of the method of moments and associated higher-order models.

210 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

»

+

8t dkV2

8kV +

8t

8x

dk® +

=

^x_

J^

convec ion

pressure

(5)

^ =W \dt)iiscrete , I dv\ k{—

Xdtj^ smootn

acceleration

f

+

(dp) v ——

J»

dv

(6)

\.dt)dhcrae

discrete

acceleration

In these equations, the following symbols and notations are used: k : macroscopic density : macroscopic or average speed V © : speed variance (the product k® is often indicated as the traffic pressure) tne {y(x))x expected value of a function y(x) with respect to all possible values of x Equation (5) is the macroscopic continuity equation for the vehicular density k. The term in the RHS stands for flow of vehicles into or out off the road section under

— ^

d t

) discrete

consideration. Equation (6) describes the dynamics of the flow rate Q=kV, also called the speed momentum equation or conservative speed equation. Note that there are two entries in equation (6) in which changes of the speed can be considered: smooth acceleration: change of the average speed V due to smooth individual accelerations; discrete acceleration: change of the average speed V due to any event causing a discrete change of p{tjc,v), the expected number of vehicles with speed v; The human-kinetic model of this paper uses this generic approach with two specific characteristics: acceleration and deceleration of vehicles in the flow are considered simultaneously and as smooth processes in the so-called acceleration integral, based on an individual longitudinal behavioural model; the state 5" of individual vehicle-driver combinations is extended with the internal state of the driver, the activation level a, in addition to the position and longitudinal speed: S'=(x,v,a). The characteristic specifications of the human-kinetic model are summarised in the next subsections. For a more elaborate description of the model, its specifications and properties the reader is referred to Tampere (2004). The acceleration integral The acceleration in the speed momentum equation (6) is generally determined by the specification of the smooth and discrete acceleration terms. Differently from existing kinetic traffic flow models, where deceleration interactions of faster with slower vehicles are

Instability, stop and go waves, wide jams and capacity drop 211 modelled as instantaneous events via the discrete acceleration term, no discrete changes of the speed (momentum) are considered in the human-kinetic traffic flow theory (except for contributions to the total speed momentum by in- or outflow of traffic near on- and offramps). The gross acceleration of the flow is accounted for in the smooth acceleration term, and calculated as a macroscopic resultant of individual acceleration or deceleration decisions, according to the individual longitudinal model presented in equations (1) and (2):

w

s

L^^J) ) =
with.

Ps (Sj)

= Ps (Sj \'>X>Vj)

1/^VJ)L^^'J)li_1p^v^)<'^^'sJ'v^)dv^dsJdvJdwJ(7) Pv(v,-) = /\,(v,.|r,*) Pv {Vj-l ) = Pv ( v j-l |*>X + S°j+ Sj,

Vj,Sj)

Equation (7) shows how the expected acceleration is found by integrating the individual acceleration function Vj over all potential combinations of a desired speed w,- and actual speed Vj of a potential vehicle j at x and a gap Sj with respect to a predecessory who is driving with speed y,-.] at location x+ s° +Sj, i.e. at a location downstream, equal to the position x of the own rear bumper + the (gross) vehicle length s° + the gap Sj . The probability that such combination occurs is given by the conditional probabilities (8), and the acceleration Vj that would be chosen by a driver j in that case is given by the individual behavioural model defined in (1) and (2). Note that, when the probability density functions in (8) are specified, the acceleration integral (7) constitutes a direct micro-macro relationship between the individual longitudinal behaviour of drivers and the gross or macroscopic acceleration of the flow. Moreover, any kinetic and macroscopic traffic flow model building on these definitions explicitly accounts for the anisotropy of driver behaviour (Daganzo, 1995) and the finite space requirements of drivers (Helbing, 1997), because the gap between vehicles is explicitly taken into account (by considering the probability ps and a non-local probability pv for the speed of the predecessor/1), in combination with finite space requirement of an individual driver defined by (2). Finally, let us mention that in the simulations shown in this paper, the conditional probability distributions in (8) have been approximated by normal distributions for the speeds and a lognormal distribution for the gaps. Moreover, we have assumed for simplicity of calculation that these distributions are mutually independent, herewith neglecting possible correlations between the speeds and gaps of successive vehicles (an assumption known as vehicular chaos, see for instance Shvetsov & Helbing (1999) for a theoretically more sound treatment of correlations). The activation level A feature of the human-kinetic model, also addressed by Treiber & Helbing (2003), is variable driver behaviour. We assume that drivers do not behave like machines with exactly the same control actions in all identical traffic conditions. Instead, the model uses the

212 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction activation level a of drivers as an additional variable that is associated with either an active or rather passive driving strategy. Mathematically, the activation level a is included in the state vector S' of the vehicle, so that S'=(x,v,a). The method of moments is again applied, so that after multiplication of equation (4) with a and consecutive integration over v and a, a macroscopic equation for the dynamics of the average activation level A is obtained: dkA

+

dkAV

,,.x = k(a)

+\

^Z^n

; ( (dp) \a——

'

= k(a)

"

7—^—-r

+A\ — I dt ul \

, , dvda ( 9 )

J discrete

The calculation of the continuous term in the RHS of equation (9) follows from a direct integration of the individual behaviour defined in equation (3):

\ pa"'{v)~a

^A\

dvda



If, for convenience of calculation, we assume a normal (or any symmetric) distribution for the activation level a and for the speeds v, equation (10) can be rewritten as: , .

a1

,(V)-A

Within the applications presented in this paper, the discrete term in equation (9) only accounts for an increase of the activation level momentum due to an inflow of traffic near an on-ramp. See Tampere (2004) for other applications that use the discrete term. Equilibrium solutions of the human-kinetic traffic flow model The acceleration integral replaces the equilibrium velocity of more traditional macroscopic models. It is a very important input to the model since it determines to a large extent the static and dynamic behaviour of the model. Normally, the equilibrium velocity is introduced as an empirical relation of macroscopic speed or flow against density. In kinetic traffic flow models however, the equilibrium velocity is theoretically obtained as a stationary solution of the model, resulting from the net effect of interactions between vehicles, in combination with the tendency to drive with the desired or free speed. Similarly, for the human-kinetic traffic flow model stationary or equilibrium solutions can be obtained. If equilibrium is defined as a combination of macroscopic density with an average speed for which the expected acceleration equals zero, then an equilibrium solution V (k) is implicitly defined by setting equation (7) equal to zero (Tampere et al. 2003). This equilibrium solution plays an important role in the solutions of the human-kinetic model. It is represented in the remainder of this article by a dotted line in diagrams of flow rate against density.

Instability, stop and go waves, wide jams and capacity drop 213

CONGESTED TRAFFIC PATTERNS Before proceeding to the theoretical analysis of congested traffic patterns, let us illustrate the potential different patterns that can be obtained when using the same initial and boundary conditions but slightly different model settings or assumptions. Note that at this point, the precise difference between the various simulations is not relevant, and is therefore not yet explained here; this will become clear in the remainder of the article. A single-lane motorway of 9 km length is considered with traffic entering upstream at a constant flow rate of 2240 veh/h. At x=6 km a constant additional inflow of 400 veh/h enters through an on-ramp.

Figure 2: Congestion patterns obtained using different model and/or parameter settings; shown is the average speed in space and time (K-axis pointing downward) Figure 2 shows three different congestion patterns that were obtained with different model and/or parameter settings of the human-kinetic traffic flow model (the difference in settings is discussed throughout the remainder of the article). Interestingly, these patterns contain some typical phenomena that have also been observed in empirical traffic data: Stop-and-go waves: present in Figure 2a and Figure 2b Wide jams: present in Figure 2b and Figure 2c Homogeneous congested traffic: present in Figure 2d

214 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Capacity drop: present in all four cases as illustrated in Figure 3.

Figure 3:

Queue discharge rates showing capacity drop in the simulations of Figure 2

These simulation results raise some intriguing questions about the essence of these congestion patterns and how they can emerge: How can slight variations to the same traffic model with identical initial and boundary conditions yield such different traffic patterns? Which mechanisms are responsible for the occurrence of these patterns? E.g. how and in which conditions can constant boundary conditions lead to an oscillating pattern? Why does the model produce various queue discharge rates, and how does this relate to the type of congestion that occurs? The remainder of this article gives answers to these fundamental questions.

MACROSCOPIC ANALYSIS: THE ROLE OF JAM FRONTS, STABILITY AND QUEUE DISCHARGE RATE In this section, the origin of the different jam types constituting the congestion patterns of Figure 2 is analysed at the macroscopic level, i.e. by considering flows, jam fronts, and other aggregate phenomena, and without considering the contribution of individual behaviour to these phenomena. The latter is discussed later in a separate section. Figure 4 summarises the rationale that is outlined in the following subsections. If a local perturbation causes a localised reduction of the speed, the first crucial aspect is whether or not the deceleration to this decreased speed is a stable process. If so, a homogeneous congested traffic pattern emerges if the source of perturbations in the flow remains active. If not, a second crucial aspect becomes important. If the highest potential queue discharge rate of the acceleration front after the speed drop (the downstream jam front) is larger than the flow rate upstream of the speed drop, a dynamic equilibrium is established: a jam cluster is formed with an inflow rate equal to the outflow rate; depending on the strength of the perturbation,

Instability, stop and go waves, wide jams and capacity drop 215 congestion patterns known in literature (Treiber et al, 2000) as 'pinned localised cluster', 'moving localised cluster', 'stop and go waves', or 'oscillating congested traffic' emerge. In the other case, the inflow exceeds the outflow of the jam, and the jam grows; this congestion pattern evolves into what is known in literature as 'wide jams' (Kerner, 1999). In the next subsections, the above mentioned rationale is motivated.

Figure 4 Dependency of the type of congestion pattern on upstream jam front stability and queue discharge rate Homogeneous traffic versus jam formation: the role of (in)stability For the sake of argument and without loss of generality, let us consider a bottleneck caused by merging traffic (Figure 5). Suppose that the stationary demand is A on the main lanes and B on the on-ramp. The maximal flow that can be carried by the motorway both before and after the merge is C, and the sum of A and B exceeds C. For simplicity, suppose that the merging flow B takes priority over the main lane traffic. Therefore, traffic breaks down near the merge and a queue builds up on the main lanes. Stable decelerating jam front First, consider the scenario of Figure 5a without instability in the decelerating jam front (upstream). Due to the capacity constraint in the bottleneck and the assumption of merging traffic taking priority over main lane traffic, the flow in the jammed area is limited to A'=C B. Therefore, the region A' has higher density and lower speed than the region with flow rate A. The deceleration of vehicles crossing the upstream jam front between region A and A' is stable, and hence also the flow rate inside the region A'. Now let us examine what happens at the downstream end of the congested region A' as a result of this stable flow. Consider the accelerating jam front (dotted line): at the left hand side of this front (upstream side), a total flow arrives equal to B+A\ whereas at the right hand side of this front (downstream side), traffic is evacuated at a flow rate C. Because these flows

216 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction are equal, traffic leaves the accelerating jam front at the same rate than the rate with which it is supplied, so that this front remains stationary at the location of the merge.

Figure 5 Flows determining the dynamic behaviour of the queue head at a bottleneck

Unstable decelerating jam front Now consider the scenario of Figure 5b with instability in the decelerating jam front (see the remainder of the article for more insight into the conditions for traffic instability). As in the previous case, traffic initially breaks down due to the merging towards a flow rate A' and the associated lower speed. However, instability implies that due to overshoot the speed inside the jam temporarily drops lower (and density increases), and hence also the flow rate drops below A' to A"
Instability, stop and go waves, wide jams and capacity drop 217 This discussion has shown that, dependent on the stability of traffic flow during breakdown at a bottleneck, two scenarios can occur. Either deceleration is stable, so that the downstream (or accelerating) jam front remains fixed at the bottleneck location. Or deceleration is unstable, so that a repetitive process is initiated: ajam is formed but the accelerating jam front of that jam travels upstream; flow at the breakdown location is restored to such an extent that a new breakdown occurs, with a new jam that travels upstream, and so on. This illustrates that instability is a necessary condition for the occurrence of oscillating (or periodic) congestion patterns like stop and go waves or oscillating congested traffic. Stop & go waves versus wide jams: the role of the queue discharge rate The previous section argued that either a stable homogeneous congestion pattern emerges, or a pattern based on traffic instability. The jam that occur in the latter case, can still have different properties, as was pointed out in the discussion of Figure 2. This section discusses the difference between jam clusters and wide jams, and the conditions in which they occur. Decisive for the occurrence of jam clusters or wide jams is the demand upstream, i.e. the flow rate
Figure 6 Evolution of an initial jam cluster dependent on in- and outflow rate

(i)

(ii)

lout = lout < am -1""" cluster develops into a wide jam. The inflow into the initial jam cluster exceeds the outflow, which is equal to the potential queue discharge rate. Due to instability of the deceleration into the jam, the density inside the jam grows and speed decreases until some minimum speed is reached, possibly standstill. From that moment on, the length of the jam grows until qin drops below the queue discharge rate. This type of jam is called wide jam and has two independent jam fronts and a region of traffic with speed V=Q in between (see also Kerner, 1999). q™"
218 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

(iii)

decelerating and accelerating jam fronts in which the outflow of the jam is limited by, and hence equal to, the inflow rate (see below). This type of jam is a jam cluster. qin < qmt = q™": cluster disappears. The inflow is too small to establish equilibrium, so that the outflow is higher than the inflow rate, and after some time the jam dissolves.

c)

Figure 7 (a) evolution of the speed in space and time (K-axis pointing downward) ofajam cluster and (b) of a wide jam; (c) representation of traffic state in the flow-density diagram during deceleration and acceleration for a. jam cluster and (d) for a wide jam.

Let us further examine jam clusters and wide jams. Figure 7a and c show the evolution of a jam cluster in space and time, and in the flow-density diagram respectively (the so-called phase trajectory in the q-k diagram). Instability during deceleration means that the phase trajectory evolves from a state on the free-flow branch of the equilibrium diagram in the direction of the congestion branch, after which it crosses the diagram instead of settling stably on the congestion branch. The flow rate (and hence the speed) associated with a given density is temporarily higher than the equilibrium flow (speed) for the same density. When traffic reaches its minimal speed in the jam (lowest point on the phase trajectory), acceleration starts immediately without vehicles standing still during some period. This is because due to the sharp acceleration immediately downstream, adaptation of the speed to that of their (accelerating) predecessor (term with rv in equation (1)) dominates over the concern to

Instability, stop and go waves, wide jams and capacity drop 219 maintain a desired distance (term with TS in equation (1)), which would otherwise direct traffic state towards the equilibrium solution. As a result of the immediate acceleration, no stock of jammed traffic is available inside the jammed region that would be able to feed the outgoing flow up to its highest attainable value qF"t , and a dynamic equilibrium is established: traffic is evacuated from the jam with the same rate as the rate with which it is supplied. A necessary condition for such equilibrium to exist is that the flow rate kjam) this means relaxation towards standstill. From that instant, a stock of vehicles is available inside the jam that supplies the outflow at the maximum attainable queue discharge rate q^t . The decelerating and accelerating jam fronts act independently from now on: the properties of the accelerating jam front are fixed (see also Kerner, 1999; Helbing et al., 2002), while those of the decelerating jam front depend on traffic demand upstream. The jam will therefore grow until the demand level q;n drops below q'™, after which it gradually dissolves. The discussion of this section has shown that jam clusters in pinned and localised clusters or in oscillating patterns like stop and go waves and oscillating congested traffic consist of two coupled jam fronts in dynamic equilibrium without traffic standing still in between. Wide jams have two independent jam fronts, with fixed outflow properties for the downstream or accelerating jam front. The conditions in which both types of jams occur have been outlined: the determinant factor is the proportion of traffic demand (inflow into the upstream jam front) with respect to the potential queue discharge rate of the downstream jam front.

DISCUSSION ON THE INFLUENCE OF DRIVER BEHAVIOUR In the previous section, the importance of traffic flow stability and of the potential queue discharge rate was explained. So far however, the question has not been answered why traffic flow is stable or unstable, and in which conditions. Neither was it explained why the queue discharge rate is limited to some potential maximum value. This section relates both issues to parameters of individual driver behaviour, hence providing a microscopic explanation for the crucial macroscopic traffic flow phenomena of the previous section. The influence of driver behaviour on traffic flow stability This section identifies and examines the behavioural parameters that have influence on traffic flow stability. For this purpose, we use the results of a linear stability analysis was performed

220 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction on the human-kinetic traffic flow model, the details of which are outlined in Tampere (2004). It is proven there that for any traffic flow model that can be written in the form: ^ + ^ =0 8t dx

(12)

8

JL+v8-L+^=f{k,v)

dt

dx

(is) y

kdx

'

linear stability is guaranteed if:

(3fy3P_k.2(3Tf_ a P 9 j r > 0 [dVJ 8k

[dk)

(14)

dVdVdk

Tampere (2004) shows that, under certain assumptions, the human-kinetic model can be linearised into the form of (12) and (13), with the following definitions:

) /(*,F)^l--^f]'l(i-/(F)]

(16)

Note that these solutions only hold when the vehicles are not in free flow according to the minimal operator in equation (1). Otherwise, stability is obtained for all equilibrium conditions (k*, V*) with:

V'=w s°j+sd(w) Above this density, combining (14) -(16) leads to the following simple stability criterion:

ifLL + ^ _ i > 0 2T

s

r

(18)

v

Equation (18) summarises the behavioural parameters that have influence on linear stability. Strikingly, stability does not depend on traffic conditions: neither the density k nor the speed Kare present in (18). Only parameters of individual driving behaviour determine whether or not traffic flow (that does not obey (17)) is stable. Of these behavioural parameters, the desired time gap S* has a stabilising effect: the greater the parameter, the more stable is traffic flow. This result is intuitive: maintaining larger distances in car-following is beneficial for the stability of the flow. On the other hand, large relaxation times rs or rv destabilise traffic flow. Also this result is intuitive: the more sluggish the reaction of the driver, the more traffic flow is sensitive to instability. A simulated illustration of the latter conclusion was already given in Figure 2, where the only difference between the stable simulation (d) and the oscillating solution (a) is a doubling of the relaxation time rv.

Instability, stop and go waves, wide jams and capacity drop 221 Note that in this respect, it is surprising to find that according to the linear stability condition (18) the pure reaction time Thas no influence on traffic flow stability (that is to say: as long as T is small, since otherwise the linear Taylor approximations that are necessary for linear stability analysis are invalidated, and hence also result (18)). This does not exclude the possibility that T has a non-linear influence on stability (which it actually has). However, the results do imply that the influence of the dynamic behavioural parameters that do occur in (18) is dominant over that of parameters that are not included, as for instance the reaction time. A practical implication is that if one intends to improve traffic stability by using traffic management measures or driver assistance systems, these systems should aim at influencing the parameters s* , TS or rv, rather than for instance reducing pure reaction time delay. A final remark with respect to parameters influencing stability concerns the potential saturation of the acceleration defined by equation (2). This aspect was omitted from the analysis in this section. However, the reader may simply verify from equation (1) that a temporary limitation of the acceleration within narrow bounds has the same influence as a temporary increase of the relaxation times TS and/or rv. Therefore, saturation of the acceleration renders traffic flow more sluggish, and hence destabilises traffic. Influence of driver behaviour on queue discharge rate In the macroscopic analysis of jam formation, an important influence was attributed to the potential queue discharge rate, i.e. the maximal flow rate that can be realised downstream of a jam when traffic can accelerate freely to its desired speed. Indeed, also in the international literature, the importance of queue discharge rate has been recognised, for instance in the discussion on traffic hysteresis and the capacity drop (see for instance Banks, 1991; Cassidy & Bertini, 1999; Daganzo, 2002). This section identifies the behavioural parameters contributing to capacity drop and hysteresis in the human-kinetic model. Figure 3 and Figure 7d already illustrated that the human-kinetic traffic flow model exhibits capacity drop and hysteresis. These simulations show that higher flow rates are stably maintained prior to congestion formation than the maximal flow out of a queue. Figure 7d reveals that the reason for this is hysteresis: i.e. the fact that deceleration and acceleration in the flow-density state space do not follow the same trajectory.

222 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 8 Simulation of a dissolving queue; shown are snapshots of the solution for (scaled) flow, speed and density at intervals of 25 seconds

Let us examine the acceleration process during recovery from congestion in more detail. For that purpose, the acceleration out off jammed traffic was simulated and analysed (Figure 8). The phase trajectory in the accelerating jam front (obtained as the evolution in time of the traffic state at any point in the originally jammed region x < 3000m) is represented in the flow-density diagram (Figure 9, solid line). As it appears, the traffic state follows the equilibrium solution quite accurately during the early phase of acceleration (high density end). When the flow rate reaches +/- 1800 veh/h, the actual flow rate becomes lower than the equilibrium flow corresponding to the same density. Therefore, equilibrium is attained on the free flowing branch, which is characterised by lower-than-capacity flow: the capacity drop. How does this relate to individual driver behaviour? From equation (1) it appears that acceleration can be caused by three stimuli: Freeflow stimulus: a deviation from the desired speed; Car-following stimulus 1: a deviation from the desired gap; Car-following stimulus 2: a relative speed difference with the predecessor. Since in the early stages of acceleration the density is high, the two stimuli for car-following determine the acceleration. Initially, the second car-following stimulus (term in zv for the relative speed) is dominant since the vehicle stands still and the predecessor starts to move. If the relaxation constant rv is small enough, the driver accelerates, and the gap remains small (compared to the desired gap). Apparently, this is the case in the solid line in Figure 9a and b, since the traffic state remains close to the equilibrium state, indicating that the first carfollowing stimulus remains almost zero (no deviation from equilibrium).

Instability, stop and go waves, wide jams and capacity drop 223

Figure 9 Representation of traffic state in the accelerating jam front in the flow-density diagram for different parameter settings (a) for the relaxation time of the relative speed and (b) for the maximum acceleration capability However, because the predecessor approaches his desired speed, he moderates acceleration, so that the relative speed with respect to the vehicle that we are considering decreases. Therefore, also the second car-following acceleration component - that dominated so far - is moderated. As a result, the traffic state gradually deviates from the equilibrium curve, with larger gaps than in equilibrium. This lasts until the first car-following stimulus becomes strong enough: the gap is noticeably too large, and the driver would correct this by accelerating more firmly. However, by that time, the own speed approaches the desired speed, so that the gap could only be closed by temporarily driving faster than desired. The driver does not do this, because he reaches a new equilibrium on the free flowing branch. From this analysis, the conclusion can be drawn that the following individual driving parameters have a dominant influence on the queue discharge rate: relaxation constants rv and TS: as an illustration, Figure 9a shows some results obtained with other values of TV (TS kept constant). With more sluggish behaviour, capacity drop becomes more pronounced (at least, for TV > 1/0.9, since smaller values eventually converge to approximately the same discharge rate, albeit via a slightly different trajectory). maximum acceleration capability accmax: following the same argument as we used earlier, a temporary saturation of the acceleration moderates the car-following response in a similar way as when the relaxation constants rv and/or rs would be increased. Therefore, lower acceleration capability also makes capacity drop more pronounced, as is confirmed by the simulation results of Figure 9b. Furthermore, the shape of the equilibrium curve, more precisely the slope of the congestion branch, also influences queue discharge rate (because car-following stimulus 1 does not allow too strong deviations from the congestion branch). An example of this was already given in Figure 7d, where we used a different (steeper) slope of the congestion branch upon deceleration than upon acceleration (see next section for an explanation why). The reader can easily verify the behavioural parameters determining the slope of the congestion branch:

224 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

ratio of the gross vehicle length s° over the desired time gap s* during acceleration: because these parameters determine the slope of the congestion branch, which represents desired gaps during acceleration, and an individual driver tends to approach his desired gap, these parameters strongly influence the acceleration process and hence capacity drop. The latter is especially true if the desired gap during acceleration is larger than during deceleration (see Figure 7d and its explanation in the next section). Let us finally remark that analyses of additional simulations have shown that the pure delay due to the reaction time Thas very little effect on the queue discharge rate: a delayed reaction of 0.5 to 1 seconds is easily compensated by slightly higher acceleration values. The conclusion is therefore similar as for the influence on stability: the effect of dynamic feedback parameters on hysteresis and capacity drop is by far dominant over that of the reaction time.

DISCUSSION ON VARIABLE DRIVER BEHAVIOUR Several studies in the literature (Cassidy and Bertini, 1999; Daganzo, 2002; Dijker, 1998; Munoz and Daganzo, 2001; van der Hulst et al., 1998; van Winsum, 1998; van Winsum and Heino, 1996; Zhang and Kim, 2001) suggest that drivers can change their motivation or activation level, which leads to changed dynamic driving parameters. In this section, concrete form is given to these assumptions in the context of the human-kinetic model discussed earlier. For now, assume that the only influence on the activation level of a driver is a gradual loss of motivation to adopt an active driving strategy (i.e. a gradual decrease of activation) as long as the driving speed remains below some threshold Vmotivatwn loss- For that purpose, equation (10) that defined the smooth or gradual dynamics of the activation level is combined with the definition of the speed-dependent normal activation level aymmal (v) as depicted in Figure 10.

Figure 10 Dependence of the normal activation level on the prevailing speed On the other hand, let us assume that the desired time gap is inversely related to the level of activation, as suggested by Dijker et al. (1998) and Munoz & Daganzo (2001). A simple specification for an activation-dependent desired time gap sf (a) is given in Figure lla.

Instability, stop and go waves, wide jams and capacity drop 225 With these specifications, the simulation result of Figure 2b was obtained (all other settings were unchanged with respect to the simulation shown in Figure 2a). Traffic arrives with free flowing speed at the bottleneck. Therefore, the activation level is normal, and according to Figure l l a also the desired time gap is normal. Once a traffic jam emerges and the speed drops below the motivation threshold, the activation level decreases according to Figure 10 and equation (10). Therefore, the equilibrium relation between flow and density is different for acceleration in the downstream jam front as compared to that for deceleration in the upstream jam front. This explains the different phase trajectories for deceleration and acceleration and the according capacity drop that were shown in Figure 7d: the jam evolves into a wide jam, as can be clearly verified from Figure 2b.

Figure 11 (a) Dependence of the desired time gap on the activation level; (b) Dependence of the maximal acceleration on the activation level Note that also alternative behavioural assumptions could be made. Suppose that the loss of motivation in low speed conditions is related to a reduction of the maximum acceleration that is applied by drivers (Figure 1 lb). This behaviour is plausible if drivers realise that they are caught in congestion and that it is no use to accelerate firmly out of a jam, since some distance downstream a new jam would oblige them to slow down anyway. In that case, queue discharge of the accelerating jam front is characterised by the curves in Figure 9b, which also leads to capacity drop, and hence the potential occurrence of wide jams. Indeed, this type of behaviour was implemented and simulated with all other settings unchanged (with respect to Figure 2a), yielding the simulation result of Figure 2c. The lesson learned from both behavioural assumptions is the same: although hysteresis and capacity drop may occur in models that neglect behavioural variations, the phenomena are much more pronounced if such variations are explicitly taken into account. Since, from a behavioural point of view, variations of control parameters are plausible and their impact on macroscopic dynamics is clearly observable, the conclusion must be that for a good representation of hysteresis and capacity drop, a traffic flow model should account for behavioural variations.

226 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

CONCLUSIONS This paper presented some theoretical analyses performed using a macroscopic human-kinetic traffic flow model. The model builds on a specification of individual longitudinal driver behaviour. A micro-macro link is established using an acceleration integral that implicitly defines the equilibrium solutions of the model. Moreover, the model allows considering drivers with variable driver behaviour, recognising the difference with machines that have fixed dynamic control parameters. For this purpose, the activation level of the drivers is considered as an additional variable describing the state of the driver in the flow. Based on simulated congested traffic patterns, questions were raised about the essence of different congestion patterns, their formation mechanism, and the role of (variable) driver behaviour therein. It appears that two properties of the traffic flow model are determinant for the type of congested traffic that can occur: the stability of the decelerating or upstream jam front: stable transitions lead to homogeneous congested traffic patterns, comparable to those from first order LWR theory; unstable transitions produce wide jams or jam clusters, possibly in oscillating patterns. the (potential) queue discharge rate in the accelerating or downstream jam front in proportion to the flow rate upstream of the decelerating jam front: lower flows upstream lead to jam clusters, whereas higher flows upstream lead to wide jams. Behavioural parameters with dominant importance on both traffic flow stability and queue discharge rate turned out to be: the dynamic feedback or relaxation parameters for the relative speed and the desired gap, the limited acceleration capability, and the desired time gap. Surprisingly, the pure reaction time delay proved less important in this respect. The validity of these results is limited, in that they are obtained with a specific car-following model. On the other hand, any car-following model contains parameters that are dynamically equivalent to those of the model used here: relaxation in feedback, finite space requirements etc. Furthermore, we argued that from a behavioural point of view it is very unlikely that these important parameters are constant during driving. For that reason, simulations were conducted examining the influence of plausible assumptions of variable driver behaviour. These assumptions had significant consequences on traffic dynamics and congestion patterns. The conclusion is that - although assumptions on variable driver behaviour are not necessary to build a traffic flow model that exhibits certain congestion patterns and phenomena like capacity drop - such variations are so plausible and their influence so significant, that consideration of variable driver behaviour is indispensable if one aims at modelling congested traffic dynamics correctly. The human-kinetic modelling approach illustrates that behavioural variations can be accounted for in macroscopic traffic flow models by an appropriate extension of kinetic traffic flow theory. However, much experimental work and research into individual driver behaviour is needed before a unique and validated behavioural framework can be established.

Instability, stop and go waves, wide jams and capacity drop 227

ACKNOWLEDGEMENT This research was commissioned by the T3 TNO-TRAIL co-operation program of the Dutch Organization of Applied Science TNO and the TRAIL Research School for Transportation Infrastructure and Logistics. Their support is herewith acknowledged.

REFERENCES Banks, J.H. (1991), Two-capacity phenomenon at freeway bottlenecks: some theoretical issues, Transportation Research Record 1320, pp. 234-241. Cassidy, M.J. & R.L. Bertini (1999), Observations at a Freeway Bottleneck, in: Transportation and Traffic Theory (ed: Ceder, A.), Proc. of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem 1999, pp.107-124. Daganzo, C.F. (1995), Requiem for second-order fluid approximations of traffic flow, Tr. Res. B, vol 29B, No.4, pp.277-286. Daganzo, C.F. (2002), A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections, Tr. Res. B vol 36B, issue 2, pp.131-158 Dijker, T., P.H.L Bovy & R.G.M.M. Vermijs (1998), Car-following under Congested Conditions: Empirical Findings, Transportation Research Record 1644, pp. 20-28 Helbing, D. (1997), Verkehrsdynamik, Neue physikalische Modellierungskonzepte, Springer Verlag, Berlin. Helbing, D., A. Hennecke & M. Treiber (1999), Phase Diagram of Traffic States in the Presence of Inhomogeneities, Phys. Rev. Let. 82 pp.4360-4363. Helbing, D., A. Hennecke V. Shvetsov & M. Treiber (2002), Micro- and Macrosimulation of Freeway Traffic, Mathematical and Computer Modelling 35, pp. 517-547. Hoogendoorn, S.P. (1999), Multiclass Continuum Modeling of Multilane Traffic Flow, Dissertation thesis, Delft University of Technology, Faculty of Civil Engineering and Geosciences, Delft University Press, Delft, The Netherlands. Kerner, B.S. & H. Rehborn (1996a), Experimental features and characteristics of traffic jams, Pysical Review E Vol.53 No. 2, pp. 1297-1300. Kerner, B.S. & H. Rehborn (1996b), Experimental properties of complexity in traffic flow, Physical Review E Vol. 53 No. 5-A, pp. 4275-4278. Kerner, B.S. & S.L. Klenov (2002), A microscopic model for phase transitions in traffic flow, Journal of Physics A: Mathematical and General, Vol. 35, pp. L31-L43. Kerner, B.S. (1999), Theory of Congested Traffic Flow: Self-Organization without Bottlenecks, Transportation and Traffic Theory, Proc. of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem 1999, pp.147-171. Lebacque, J.P. & J.B. Lesort (1999), Macroscopic traffic flow models: a question of order, Transportation and Traffic Theory, Proc. of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem 1999, pp. 3-25; Lee, H.Y., H.-W. Lee & D. Kim (1999), Dynamic States of a Continuum Traffic Equation with On-Ramp, Phys. Rev. E, 59, p.5101-5111.

228 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Lee, H.Y., R.-W. Lee & D. Kim (2000), Empirical Phase Diagram of Traffic Flow on Highways with On-Ramps, Traffic and Granular Flow '99, edited by: Helbing, D., R.J. Herrmann, M. Schreckenberg & D.E. Wolf. Leutzbach, W. (1988). An introduction to the theory of traffic flow. Springer-Verlag, Berlin. Lighthill, M.J. & G.B. Whitham (1955), On kinematic waves: II a theory of traffic flow on long crowded roads. In: Proc. of the Royal Society, vol. 229 (1178) of A, pp.317-345. Mufioz, J.C. & C.F. Daganzo (2001), Moving Bottlenecks: a Theory Grounded on Experimental Observation, In: Transportation and Traffic Theory in the 21st Century, Proc. 15th Int. Symp. on Transportation and Traffic Theory (Ed: Taylor, M.A.P.), Pergamon-Elsevier, Oxford,U.K, pp. 441-462. Nelson, P. & A. Sopasakis (1999), The Chapman-Enskog expansion: a novel approach to hierarchical extension of Lighthill-Whitham models, Proc. of the 14th International Symposium on Transportation and Traffic Theory, Jerusalem 1999, pp. 51-79. Payne, R.J. (1971), Models of freeway traffic and control, hi: Mathematical models of public systems, volume 1 of Simulation Councils Proc. Ser., ed. by: Bekey, G.A., pp, 51-60. Prigogine, I. & R. Herman (1971), Kinetic Theory of Vehicular Traffic, American Elsevier, New York. Shvetsov, V. & D. Helbing (1999), Macroscopic dynamics of multi-lane traffic, Physical Review E 59, pp. 6328-6339. Tampere, C.M.J., S.P. Hoogendoorn & B. Van Arem (2003), Gas kinetic traffic flow modelling including continuous driver behavior model, Transportation Research Record, No.1852, pp. 231-238. Tampere, C.M.J. (2004), Human-kinetic traffic flow modelling for the analysis of Advanced Driver Assistance systems in congestion, Dissertation thesis, Delft University of Technology. Available through: http://www.kuleuven.ac.be/traffic/dwn/P2004C.pdf Treiber, M., A. Hennecke & D. Helbing (2000), Congested Traffic States in Empirical Observations and Microscopic Simulation, Physical Review E 62, pp. 1805-1824. Treiber, M. & D. Helbing (2003), Memory effects in microscopic traffic models and wide scattering inflow-density data. Phys. Rev. E 68, No. 04611 Treiterer, J. & J.A. Myers (1974), The hysteresis phenomenon in traffic flow. Proceedings of the 6th International Symposium on Transportation and Traffic Theory. Ed.: D.J. Buckley, pp. 13-38. Van der Hulst, M., T. Rothengatter & T. Meijman (1998), Strategic adaptations to lack of preview in driving, Transportation Research Vol. IF, pp. 59-75. Winsum, W. van (1998), Preferred time headway in car-following and individual differences in perceptual-motor skills, Perceptual and Motor Skills, Vol. 87, pp.863-873 Winsum, W. van & A. Heino (1996), Choice of time-headway in car-following and the role of time-to-collision information in braking, Ergonomics, Vol. 39, pp.579-592 Zhang, H.M. & T. Kim (2001), A car-following theory for multiphase vehicular traffic flow, Proceedings of the 80th Annual Meeting of the Transportation Research Board. Zhang, H.M. (1998), A theory of nonequilibrium traffic flow, Tr. Res. B, Vol. 32B, No. 7, pp.485-498. Zhang, H.M. (1999), A mathematical theory of traffic hysteresis. Tr. Res. B, Vol. 33B, pp.l23

12

CONTROLLING TRAFFIC BREAKDOWNS

Reinhart Kiihne, German Aerospace Center, Institute of Transportation Research, Berlin, Germany; Reinhard Mahnke, University of Rostock, Institute of Physics, Rostock, Germany

INTRODUCTION Three main reasons can be observed for traffic breakdowns due to bottlenecks: 1. the on ramp traffic exceeds together with the main stream traffic capacity, 2. a bottleneck in the course of a road section, 3. the off ramp traffic exceeds the off ramp capacity and spills back. These reasons are presented in Fig. 1, and form the basis for the investigation presented in this paper. Breakdowns due to bottlenecks are regarded as stochastic events depending only on a few traffic parameters like traffic flow at the beginning of a breakdown. To complete the picture, it is necessary to mention that breakdowns may either be caused by an excess of upstream demand over downstream capacity or it may occur spontaneously. From a traffic engineering standpoint, a traffic breakdown is defined (usually based on 5 minutes measurement interval data) as a -speed drop

&v>l5km/h^ , „ , , v c , < 75km/ h - mean speed after speed drop '"ml , OOOveh h lane - traffic volume before speed drop 1 >l '' . The definition is not very critical with respect to the introduced thresholds as it was shown in Kiihne and Anstett (1977).

230 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction For the following probabilistic description, a traffic breakdown is defined as a car cluster formation process. For this we consider a model of traffic flow on a freeway section and study the spontaneous formation of a jam regarded as a large car cluster arising on the road. To get rid of some boundary conditions like entries and exits we can idealize the section by a circular road of length L with N cars moving on it. All the cars are assumed to be identical vehicles and can form two phases. One of them is the set of freely moving cars and the other is the congestion called car cluster. The cluster is specified by its size n, the number of aggregated cars. Its internal parameters, namely, the headway distance and, consequently, the speed of cars in the cluster are treated as fixed values independent of the cluster size n. We note that in the model under consideration there can be only one cluster on the road. The free flow phase is specified also by the corresponding headway distance that, however, depends strictly speaking on the car cluster size n. The larger the cluster is, the less is the number (N - n) of the freelv moving cars and therefore the laraer is the headwav distance.

On-ramp flow exceeds capacity Figure 1: Typical bottleneck situations described in the text, adapted from Kuhne (1996). When a vehicular cluster arises on the road its further growth is due to the attachment of the free cars to its upstream boundary, whereas the cars located near its downstream boundary accelerate to leave it, which decreases the cluster size. These processes are treated as random

Controlling traffic breakdowns 231 changes of the cluster size n by 1 (see Fig. 2) and the cluster evolution is described in terms of time variations of the probability function P(n,t) for the cluster to be of size n at time t. Then, following Mahnke (1997), we write the balance equation called master equation governing the cluster evolution

dtP(n,t) = w+(n-l)P(n-\,t)

+ w_(n + l)P(n + l,t)-[w+(n)P(n,t)

+ w_(n)P(n,t)].

(1)

Figure 2: Schematic illustration of cluster transformations The attachment rate w+ («) to the cluster takes into account strictly speaking the net time gap for a freely moving car to move up to the cluster. If net time gap and gross time gap are taken equal then this rate is just the traffic flow q w^=q

(2)

to which the clustered vehicles as standing vehicles do not contribute. The transition rate w+ is thus independent of the cluster size n. The rate of the cars escaping from the cluster at its downstream front is written as W-{n)~

(3)

where the value r(n) can be interpreted as the characteristic time needed for the first car in the cluster to leave it and to go out from its downstream boundary at a distance about the headway distance in the current free flow state. When the cluster is sufficiently large, it is reasonable to regard the characteristic time r(«) as a constant r m . For small clusters the dependence, however, requires special attention. We assume that the escape rate w_(n) exhibits substantial variations in the region n < na with n0 « 20.

232 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The parameter n0 actually divides the car clusters into the group of large clusters for n > n0, for which the escaping rate is constant 1 / „ , and the group of small clusters, n
n>n0 =-(l-Asign(n-n0))

(4)

n
where

Numerical values r = 1.5s and A = 0.2 can be choosen as reasonable first guesses. The influence of traffic control measures on these values will be discussed later. With this simple but effective assumption of the cluster size dependence of the escape rate we are able to describe a traffic breakdown. At least one of the kinetic coefficients w+ («) and w_ (n) has to be a direct function of the cluster size n for its relatively small values corresponding to the formation of the critical nucleus.

CONGESTION FORMATION: PROBABILISTIC DESCRIPTION hi order to apply well developed techniques of escape theory (Gardiner, 2002; Risken, 1996) to the analysis of the traffic breakdown probability, we approximate the discrete balance equation (1) by the corresponding Fokker-Planck equation with diffusion approximation because in the case under consideration the kinetic coefficients w+ («), w_ (n) , first, vary smoothly on scales about unity and, second, are approximately equal to each other. The latter conditions enable us to treat the argument n as a continuous variable and expand the balance equation illustrated in Fig. 2 into a Taylor series up to second order Td,P(n,t) = (dnO'(n) + 82n)p(n,t),

(6)

where the potential («) can be calculated from its derivative

. . w (n)-w, (n) ,
. (7)

Controlling traffic breakdowns 233 as (n) = -a(n-no)-A\n-no\,

a = qz-l.

(8)

Figure 3: Graph of the potential for different parameter configurations indicating different scenarios. Free flow at a < -A (left), bistability within the parameter range -A < a < A (middle) and congested traffic at a > A (right). The functional graph of the potential is shown in Fig. 3 for different scenarios where the system state tends either to n = 0 (free traffic flow) or shows instability or tends to completely congested traffic. The potential provides the driving force in the creeping movement of the system state under the influence of stochasticity represented by a correlated fluctuating force entering into Langevin equation ^

= -o'(n)+r(t),

(r(t)r(t')) = 2S(t-f)

(9)

FREQUENCY OF TRAFFIC BREAKDOWNS DURING A FIXED OBSERVATION TIME INTERVAL Up to now the probability of finding a congestion of exactly n vehicles in a car cluster has been calculated from a balance equation regarding transition rates for cluster formation and dissolution. As an observer one is interested in breakdowns and cluster formation independent of the cluster size. Empirically a traffic breakdown, as mentioned in the introduction, is typically analyzed by detecting a significant drop in the vehicle speed during a certain fixed time interval Tot,s. From time series as shown in Fig. 4 the relative frequency of breakdowns versus stable traffic flow situations of the same corresponding traffic volume as control parameters can be derived as shown in Fig. 5 together with the breakdown distribution for a corridor control section. There is clear indication that the relative frequency increases drastically with increasing traffic volume.

234 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In terms of probabilistic modeling, a traffic breakdown is an event that occurs when the system's state, which started at time t = 0 with n = 0 (free flow), reaches for the first time nex where the escape value nac is a given cluster size regarded as overcritical. To calculate this first passage time, the following boundary conditions must be considered: n = 0 : probability current vanishes (jj> + dn) P = 0 (reflecting wall); n = nac: probability vanishes P = 0 (absorbing wall). The corresponding stochastic movement of the state variable n described either by the Fokker-Planck equation (6) or the equivalent Langevin equation (9) when first reaching the critical cluster size nesc is sketched in Fig.6. Following Risken (1996), the distribution function of the first passage times Tis, given by the probability distribution flux through the absorbing boundary at n = «sec and of the domain of interest 0 < n < «,„.

Figure 5: Relative frequency of breakdowns for different traffic volumes. Data from 20000 intervals of 5 minutes observation on autobahn Munchen Holledau (after Schick, 2003) with and without automatic speed control.

Controlling traffic breakdowns 235

Figure 6: First passage times T of two realisations n(t) to leave the domain from free flow (n = 0) at escaping cluster size nesc.

Figure 7: First passage time distribution (left: density; right: cumulative distribution) as probabilistic interpretation of observing a traffic breakdown during a time interval (T,T+dT). Fig. 7 shows probability distributions calculated from numerical mapping of the corresponding Langevin equation (9). The density shows an offset since in any case it takes some time to observe a breakdown with formation of a critical cluster size. After reaching a maximum the distribution dies out pointing the fact that the possible breakdowns already happened. W(t)

= ~'jP(n,t)\^dn ot 0

(10)

236 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction From the probability density (10) the cumulative distribution of observing a breakdown within the observation time Tobs can be calculated:

W(Tobs)=

\w{T)dT.

(11)

Given a fixed observation time Tobs the control parameter a = qv — \ or the traffic volume q at beginning of the breakdown respectively can be varied which leads to the cumulative probability distribution for traffic breakdowns.

Figure 8: Cummulative probability distribution for traffic breakdowns for fixed observation times as a function of the control parameter q (traffic flow at beginning of breakdown) For the distribution shown in Fig. 8 a reasonable approximation can be given. Starting with the probability as the ratio of a short observation time At and the mean first passage time T

W =j

(12)

the formula can be extended to finite observation time Tobs or finite number of observation intervals v = Tobs I At. For this aim the probability that no breakdown occur within the v intervals is given by

* = (l-fj In case of infinitesimal observation intervals the limit gives

03)

Controlling traffic breakdowns 237

* = (l-f]"-«p(-^)

(14)

from which W = \-exp\—^

(15)

can be interpreted as transition probability from free flow to congestion during the observation time Tobs. This transition probability has to be multiplied with the possibilities f° P" (n) dn for starting from a free flow state leading to the total breakdown probability

W= £p"(n)dn + l-exp( - 5 ^ 1 )^P"(n)dn

= l-exp(-^jjV'(»)
(16)

(17)

The remaining challenge is to calculate the mean first passage time T . We follow the scheme given in Risken (1996) and Kiihne et al. (2002) by calculating the first nonzero eigenvalue of the investigated potential as rate for overcoming the barrier in the potential for the bistable parameter configuration (see appendix) leading to

in units of 7. The complete result is ( T W = \-exp\—2»L

\

£V=iVexp(-O)

(19)

as shown in Fig. 8. It can be transformed into a three parameter Weibull distribution (Muthly et al., 2004)

GfaH-expf-^yj for p = 2 when the ^-dependence of the front factor is neglected.

(20)

238 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The shape of W in any case is similar to the stochastic distributions used in reliability assessment, and supports the ideas of Regler (2004), Zurlinden (2003), Brilon and Zurlinden (2003), Lorenz and Elefteriadou (2000), and Ktthne (2001). These authors use Weibull distributions as fitting curves with enough parameters to match a broad variety of cummulative distribution functions.

FINDINGS FROM MEASUREMENTS For short observation times the simplification

W= l - e x p f - ^ V P" (n)dn*\-[° F'(n)dn Pexp(-O)d« yy-\_A

[i =

exp(-®)rfn

a>0

J

|exp(a« 0 ),

(21) (22)

a<0

can be used. Fig. 9 shows that the measurement points from a two lane freeway section which fits quite nicely the exponential increase leading to the critical cluster size no = 10 and the mean escape time r = 2.1s .

Figure 9: Approximation for the breakdown probability function for a two lane section. Data from Kiihne and Anstett (1997). Taking into account a more sophisticated approach regarding the observation time, a Weibull fit is shown from Regler (2004) confirming the stated overall behaviour, see Fig. 10.

Controlling traffic breakdowns 239

Figure 10: Fit of measurement for traffic breakdown probability distribution from field observation for German Autobahn A5, for 5 min intervals (Regler, 2004). As a rough first guess for the reported freeway sections can be stated as the following parameter values = \.Ss,

T^=2.0S,

r o = 1.2.5, A = 0.25

(23)

Finite size effects may play a role in some specific merging areas which are not considered here.

INFLUENCE OF CONTROL MEASURES ON BREAKDOWN PROBABILITY For assessment of the influence of control measures on the breakdown probability the operational definition of a traffic breakdown as a speed drop beyond a certain threshold is used. To quantify the influence of control measures such as traffic actuated speed limits, lane closures and keep in lane recommendations and eventually ramp metering field observations exploring more than 20 000 data points from measurements of freeway sections with and without corridor control systems are analysed. The investigations were carried out at the autobahn A9 from Miinchen to Holledau between July 27 and August 9, 2000 (Kiihne, 2001). Fig. 11 shows the speed-flow diagrams from oneminute intervals for a three lane section. There is no significant change in the maximum observed traffic flow due to traffic control measures.

240 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 11: Typical speed-flow diagrams from one minute intervals for a 3-lane-section with and without automatic traffic control system. The stabilizing effect of the control system reduces speed drops in the traffic volume range between 3000 and 5400 veh/h/3 lanes and stabilizes the final speed above 75 km/h as already shown in Fig. 5 the beginning of the cummulative distribution is altered by a factor of 3 while the tail of the cummulative distribution is expected to be unchanged as is the maximum observed traffic flow. Fig. 12 reflects these changes due to automatic traffic control system by altering the critical cluster size n0 and the difference A in the discharge rate for small and large clusters.

Controlling traffic breakdowns 241

Figure 12: Cummulative breakdown probability distribution as function of the traffic volume q as control parameters for different critical cluster sizes no and differences A in the discharge rate for small and big clusters modelling the influence of traffic control measures on the breakdown probability.

ACKNOWLEDGMENTS The authors gratefully acknowledge funding by German Academic Exchange Program (DAAD) to take part at the summer school on many-particle physics in Ekaterinburg (Russia), where part of this work was performed, as well as financial support by the German Science Foundation (DFG), grant MA 1508/6 (R. Mahnke). The authors are indebted to J. Hinkel (Rostock), J. Kaupuzs (Riga), I. Lubashevsky (Moscow), and P. Wagner (Berlin) for fruitful discussions and support.

REFERENCES Brilon, W.; Zurlinden, H. (2003). Tjberlastungswahrscheinlichkeiten und Verkehrsleistung als Bemessungskriterium for StraBenverkehrsanlagen. Forschung StraBenbau und StraBen-verkehrstechnik, Heft 870, Bundesministerium fur Verkehr, Bau- und Wohnungswesen, Abt. StraBenbau, StraBenverkehr Bonn (Hrsg.). Gardiner, C. W. (2002). Handbook of Stochastic Methods, Springer, Berlin, Heidelberg.

242 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Kiihne, R.(1996). Der Beitrag von Verkehxssystemmanagementprojekten zur Entscharfung der Verkehrsprobleme in Ballungszentren Dresden - Radebeuler Verkehrstag, Radebeul 26.09.1996, S. 126. Kiihne, R. (2001). Application of Probabilistic Traffic Pattern Analysis to Capacity, Siam Conference, San Diego, July 9 - 1 9 . Kiihne, R., and Anstett, N. (1997). New Methods for Determining Critical Sections on Freeways, Annual Meeting of the Transportation Research Board, Washington, DC. Ktihne, R., Mahnke, R., Lubashevsky, I., and Kaupuzs, J. (2002). Probabilistic Description of Traffic Breakdowns, Physical Review E, vol. 65, 066125. Lorenz, M., and Elefteriadou, L. (2000). A Probabilistic Approach to Defining Freeway Capacity and Breakdown. In: Brilon, W. (ed.) 4th Int. Symp. on Highway Capacity, Transp. Res. E Circular EDCO/8. Mahnke, R., and Pieret, N. (1997). Stochastic Master-Equation Approach to Aggregation in Freeway Traffic, Physical Review E, vol. 56, 2666. Mahnke, R., and Kaupuzs, J.:(1999). Stochastic Theory of Freeway Traffic, Physical Review £,vol. 59, 117. Muthly, D. N. P., Xie, M., and Jiang, R.(2004). Weibull Models, Wiley, New Jersey, 2004. Regler, M. (2004). Verkehrsablauf und Kapazitat auf Autobahnen (Traffic Flow and Capacity on Freeways), Dissertation, Universitat Bochum. Risken, H.(1996). The Fokker-Planck Equation, Springer, Berlin, Heidelberg, 1996. Schick, P. (2003). Einfiuss von Streckenbeeinflussungsanlagen auf die Kapazitat des Verkehrs von Autobahnabschnitten sowie die Stabilitat des Verkehrsflusses, Dissertation, Universitat Stuttgart. Zurlinden, H. (2003). Uberlastungswahrscheinlichkeiten und Verkehrsleistung als Bemessungskriterium fur StraBenverkehrsanlagen, FE-Nr.: 03.327/1999/ KGB der Bundesanstalt fur StraBenwesen.

APPENDIX: CALCULATION OF THE MEAN FIRST PASSAGE TIME To calculate the mean first passage time T for overcoming the barrier in the bistable potential shown in Fig. 13 starting at n = 0 (free flow conditions) the following boundary conditions reflecting wall at n = 0 ; absorbing wall at n = nesc are considered.

Controlling traffic breakdowns 243

Figure 13: Potential of the Fokker-Planck equation in the bistable regime. The reflecting wall condition leads to S = -(0 + dn)P(n,t)\no=O

(24)

shown in the above potential on the left boundary, see Fig. 13. In order to determine the mean passage time we follow Risken (1996) by calculating the first non zero eigenvalue which yields the mean first passage time

(25)

X=l. The corresponding Fokker-Planck equation in time units of r dP(n,t) = [dK0'(n) + dl)P(n,t)

(26)

is transformed into a Schrodinger equation by the separation 9P(n,t) = ^Pst (n)
with

P" =A r exp(-O),

K=-O'2--O" 4 2

(27) (28)

where the Schrodinger potential Vs reads 0 < n < nac

Vs^

+hSiS

f-n^-^n-n0)

(29)

244 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction while the first excited state as odd function ^sinA^ + iJcosA:^

H

Csmk2(n-nesc)

: 0
cpl(O) = 0 reads

k, +

: no
=X

rAv

k\ + ^ ~

(30)

=^

with continuity condition of n = n0 (p(n0 +e^ = (p{n0

» ^4sin^,n0 +5COS^,M 0 = CsinA;2 (n —nac)

(31)

and jump condition at n = n0 -(p"(n) + Vs(n)
["+° dn

-(p'(n0+s) + cp'(n0 -s)-cp{n0) £ > -^A-k2

s

(n)dn = 0

cotk2 («0 - « e s c ) - ^ tan(A:1w0 - a ) = 0

with tanh a =

. For £, = 0 it leads directly to the mean first passage time

(32) (33) (34)

13

PARAMETER ESTIMATION AND ANALYSIS OF CAR-FOLLOWING MODELS Serge P. Hoogendoorn, Transport & Planning Department, Delft University of Technology Saskia Ossen, Transport & Planning Department, Delft University of Technology

INTRODUCTION With increasing computation power, microscopic simulation models have become widely applied tools for numerous applications in traffic engineering. Parameter identification remains one of the difficult tasks when using these microscopic traffic flow models. The reasons for this are manifold, e.g. parameters are not directly observable from common traffic data, they are not transferable to other situations (locations, periods of the day, etc.), driving behaviour is variable in time and space, etc. Several approaches to structured model calibration using macroscopic and in some cases microscopic traffic data have been proposed. For the sake of illustration, let us consider two examples of automated approaches to model calibration. (Brockfeld, Kiihne et al. 2004) cross-compare different microscopic traffic flow models using data from car-following experiments. The error rates of the different models in comparison to the data varied between 9 and 24 %, for validation between 12 and 30%. No single model appeared to be significantly better than the others (even the more complex ones). The authors argued that these errors could probably not be suppressed, irrespective of the model that is used, due to the different behaviour of each driver, which is in line with the results that will be presented in this contribution. (Schultz and Rilett 2004) proposed a methodology to introduce and calibrate a parameter distribution using measures of central tendency and dispersion (i.e., mean and variance) to generate input parameters for carfollowing sensitivity factors in microscopic traffic simulation models. The approach is applied to IH-10 in Houston, Texas using the CORSIM model, and subsequently calibrated utilizing an automated genetic algorithm. An overview of calibration approaches as well as parameter values is found in (Brackstone and McDonald 1999). Automated approaches generally entail optimisation of some pre-specified error measure, after which they provide time-invariant and condition invariant estimates of the model

246 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction parameters. Here, we propose an approach allowing estimation of dynamically and spatially changing model parameters, which could be caused by for instance changing traffic or road conditions, or changes in driver behaviour. The method allows straightforward, even on-line calibration, as well as investigating variability in driving behaviour. The method which is used involves two main steps. In the first step, we propose the static model identification approach yielding constant estimates of the parameters of the considering car-following model (sensitivity, reaction time, etc.). This step is used to establish which of the considered car-following models is most suitable for the subsequent dynamic analysis, given its complexity and predictive performance. The second step entails using these estimates as initial conditions in the dynamic parameter estimation approach. This shows if and how the parameters of the car-following model change over time and space, and as such provide insight into the adaptive behaviour of drivers. The main contributions of this paper are a new identification framework and estimation algorithms to identify parameters of microscopic traffic flow models. In particular, the problem of static and dynamic estimation of the reaction times of the retarded car-following models has been solved. The static model estimation reveals that to accurately describe car-following behaviour, both the distance to the predecessor and the relative speed need to be included. Furthermore, empirical evidence is present showing that drivers also tend to react to the 'second leader', although improvements in predictive performance are relatively minor. We furthermore discuss application of the dynamic estimation approach to both synthetic data and real-life trajectory data, using the linear model of Helly (chosen due to satisfactory outcomes of the static estimation). The dynamic estimation results can be used to improve traffic modelling as well as for the purpose of increasing the insights into car-following behaviour. In this respect, the results of application are interpreted from the viewpoint of adaptive driver behaviour, and turn out to be in line with the theory on changing driver behaviour.

CAR FOLLOWING THEORY AND MODELLING A microscopic model provides a description of the movements of individual vehicles. These movements are the result of the characteristics of drivers and vehicles, the interactions between drivers, and between the driver and road characteristics, external conditions and the traffic regulations and control. In general, two types of driver tasks are distinguished: longitudinal tasks (acceleration, maintaining speed, distance-keeping relative to leading vehicle) and lateral tasks (lane changing, overtaking). This contribution considers the former, i.e. the car-following task. hi this contribution, the term car-following model is used for the general class of dynamic microscopic models describing the longitudinal behaviour of a driver in relation to the driver(s) in front. Driver /' driving, who is following driver i+\, may for instance react on (changes in) the spacing Ar/1' between the vehicles, or its his or her speed compared to vehicle z'+l. Many models have been proposed that aim at describing this longitudinal behaviour. It is beyond the scope of this contribution to provide a comprehensive overview of all models. Rather, a small selection of relatively simple models has been considered and analysed. The static and dynamic parameter identification approaches put forward in this contribution are nevertheless generally applicable, given that sufficient data is available.

Parameter estimation of car-following models 247 All considered models describe a delayed reaction (acceleration) to different stimuli offered to driver i. These stimuli entail amongst other things the distance and relative speed with respect to the vehicle in front (referred to as the first leader, vehicle z'+l), or the second leader (i.e. vehicle i+2). Single leader car-following models The well-known model of (Chandler, Herman et al. 1958) is based on the intuitive hypothesis that the acceleration of driveri is proportional to the relative speed: where Avf = vM - v ; (1) ai(t + Tr) = a^Avf\t) where Tr denotes the reaction time, a denotes the so-called sensitivity, and Ay/1' denotes the relative speed with respect to vehicle z'+l. This simple model has several undesirable and unrealistic properties (Leutzbach 1988). For instance, vehicles tend to get dragged along when the vehicle in front is moving at a higher speed. To remedy this deficiency - at least to a certain extent - the sensitivity ai can be defined as a decreasing function of the distance. In more general terms, the sensitivity can be defined as follows V

where Ar(1) -r —r (T\ (> [Ar (0T ' ~M ' Eqn. (2) implies that the following vehicle adjusts its speed v,{t) proportionally to both distances and speed differences with delay Tr. The extent to which this occurs depends on the values of ceo, I and m. a - a

(1)

The model described by Eq. (1) has been generalized in a number of ways. A well known generalisation is the model of (Helly 1959): 1) ) (3) ai(f + Tr) = «1Av< (0 + A ( ^ ° ( 0 - 5 1 ) which has been further extended by Tampere (2004): 0 c,
~ min \ a,Avm (t) + /?, (Ar(1) (t) - S{),V ~ V; ^ \

.

I

T

elsewhere

J

In both Eq. (3) and Eq. (4), Si denotes the desired or safe distance, for instance given by: Si=Sl(vi) = s0+s1vi (5) In Eq. (4), the parameters c\ and ci defined a region in changes in the distance headway do not yield a response of the driver. The model also includes acceleration towards the free speed v° with acceleration time r. Multi-anticipative car-following modelling Another generalisation is the inclusion of the relative speed compared to the second leader i+2 (as well as vehicles even further downstream), yielding the model of (Bexelius 1968): a,(t) = a,Avf'(t-T r ) + a2Av,<2)(t-Tr) where Avf' = v;+2 -v,. (6) The resulting model includes multi-anticipative car following behaviour.

248 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Other multi-anticipative generalisation are due to (Lenz, Wagner et al. 1999), which is based on the car-following model of (Bando, Hasebe et al. 1995). The parameters of various car-following models have been estimated using macroscopic and microscopic data by a large number of researchers. It should be noted that no generally applicable set of parameter estimates has been found so far, i.e. estimates are site-specific. An overview of parameter estimates can be found in (Brackstone and McDonald 1999). Adaptive driving theory The previous sections discussed briefly car-following theory and outlined some basic carfollowing models. In general, it is assumed that the parameters describing car-following behaviour (e.g. reaction time, sensitivity, etc.) are constant over time and space. On the contrary, in literature indications are presented that driving behaviour is subject to variations. In this section, we will recall a few of these contributions. Please note that this overview is not intended to be exhaustive, but serves mainly to illustrate why it is plausible to assume that parameters capturing driving behaviour may change over time and space (more information can be found in (Tampere,2004)). (Mufloz and Daganzo 2001) report increases in time headways of vehicles in a moving bottleneck. The authors suggest that the first drivers caught behind the moving bottleneck (a slow truck) were motivated to follow their predecessor closely, since they saw the cause of the queue and knew they would be able to overtake the truck soon. As the queue grew longer, drivers further upstream were less motivated and adopted longer headways. The capacity drop and hysteresis phenomena have also been the inspiration for behavioural hypotheses. (Daganzo 2002) suggests that the transition from high prior-congestion flows to lower queue discharge rates also corresponds to a loss of motivation. The author argues that before the transition to congestion multilane traffic flow essentially splits into a faster and slower flow {rabbits versus slugs or two-pipe regime), with the rabbits driving on the leftmost lanes. When traffic becomes denser, the speed difference disappears because the frequency at which some faster slugs change lanes to the fast 'rabbit' lanes increases, the speed in the fast lane decreases and therefore the rabbits lose their motivation to follow each other closely. Once the speed difference has disappeared the initially faster rabbits occupy both the faster and the slower lanes and congestion is maintained during a longer period. Essential in this theory — which is based on macroscopic empirical observations — is the loss of motivation and consequent expansion (longer headways) of the initially densely packed faster vehicles. Although using different terminology, (Zhang and Kim 2001) propose a similar behavioural theory in an attempt to reconstruct and explain the reversed-lambda shape and hysteresis of the fundamental diagram. Again these authors need assumptions on driver behaviour to obtain macroscopic traffic behaviour consistent with observations; the minimal set of assumptions that yields realistic macroscopic behaviour includes multiphase behaviour, i.e. driving strategy dependent on traffic conditions. Besides the indirect arguments above, direct evidence for variable driving behaviour has also been presented. For instance, (Dijker, Bovy et al. 1998) analysed differences in car-following behaviour between congested and non-congested flow with data from two sites on Dutch motorways. They found that - at the same speeds - passenger car drivers follow with smaller

Parameter estimation of car-following models 249 headways in non-congested than in congested conditions. Car-following of truck drivers does not show differences between regimes. The authors suggest that the driver's motivation to follow closely or take over slower cars is lower in congested conditions, causing a different driving strategy. Other studies, e.g. (Van der Hulst 1998), show that drivers can change driving strategies, for instance accept temporary reductions of the normal following distance while still being able to respond accurately to deceleration manoeuvres, at the cost of a higher alertness, attention level or vigilance. The author states that drivers apply a hierarchy of adaptive strategies which are aimed at the control of time pressure (stress) in driving. In normal conditions, drivers adopt an anticipatory driving strategy. When the possibilities for anticipation are reduced, drivers compensate by speed reductions and increased time headway. When this compensatory strategy is impossible or undesirable, drivers compensate by higher alertness to react accurately to unpredictable events. However, drivers try to avoid increased driving effort costs when they have the opportunity to do so.

STATIC AND DYNAMIC PARAMETER IDENTIFICATION In this contribution, we propose a parameter identification method that allows estimation of (slowly) changing parameters. This is achieved by applying an Dual Extended Kalman Filter (DEKF). A Kalman filter is a set of mathematical equations that provides efficient computational (recursive) means to estimate the state of a process which dynamics are described by linear equations. It is very powerful in several aspects: it supports estimations of past, present, and even future states, and it can do so even when the precise nature of the modeled system is unknown. For non-linear processes, so-called Extended Kalman filters (EKF) have been developed. The DEKF applied in this contribution is a special class of the EKF that allows identification of unknown or uncertain model parameters by solving the dual {or joint) estimation problem of estimating the states of the vehicles and the unknown model parameters from noisy data. Before presenting the DEKF, the static estimation procedure used to cross-compare the model and to find initial estimates for the parameter values is described. State of an platoon and its relation with observations We consider sets or platoons of n vehicles, where n > 2. Let r,{t) and v,{t) denote the position and the speed of vehicle i at time instant t. Let i = 1 be the last vehicle in the platoon, and let i = n denote the platoon leader. The so-called state of the platoon is defined by the positions and speeds of all vehicles in the platoon: x = (r1>v1,..,rB>vJ (7) It is assumed that all drivers in the platoon share the same parameters describing their carfollowing behaviour. In fact, the main reason for considering platoons is to enable parameter estimation, even if there is relatively little data available. If sufficient data is present, which is the case for the data sets considers in this contribution, a platoon will generally consist of two vehicles (or three if a multi-anticipative car-following model is considered). The observations y(tk) relate to the state x(/*) according to the measurement equation: y(tt) = Hx(tt) + r](tk)

(8)

250 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where H is a linear mapping between the state and the observations; T\(tt+l)~ N(0,R(tk)) is a zero-mean Gaussian noise term with covariance i?(4) describing the observation errors. Note that in our case, measurements y(4) of the vehicle positions and (derived from that) the speeds are available for each 0.1 s. As a result, the measurement matrix H is the identity matrix. Static estimation of car-following parameters Let us first assume that the parameters of the car-following model are stationary and homogeneous. To estimate these parameters, observed vehicle speeds and positions are compared to speeds and positions predicted by the considered car-following model. To this end, let us again consider an n vehicle platoon. For all vehicles in the platoon, the positions r,{ti) and (derived from these positions) the speeds v,(4) at sampling instants 4, for k = \,...,K have been observed empirically. For the predictions, the behaviour of the platoon leader (i=n) is assumed known (equal to the observed positions and speeds). Now consider vehicle ;' following vehicle i+l (starting with i = n-\). We can apply the considered car-following model to predict the dynamics of vehicle i dynamics based on the (observed or predicted) behaviour of the preceding vehicles in the platoon, yielding predictions r,{4|0) v,{4|0) of the position and speed of the vehicle at the sample instants 4. 0 denotes the vector of parameters describing the car-following behaviour of vehicles i = 1,...,»-1 in the platoon (including the reaction time Tr). To determine the predicted performance, we define the (generalised) Root Mean Squared Error (RMSE) by

RMSEl(Q) = ^fJ(rrd(tt

I B)-r?(tk)f +^ft(vTl(tt

] Q)-^(tt)f

(9)

where r| > 0 is a weighing parameter. The static estimation procedure entails finding the set of parameters minimising the average RMSE for all vehicles in the platoon: e = argmin.RMS£(e) where RMSE(Q) = —

VRMSE.(Q)

(10)

Besides the RMSE, the performance of the model is also assessed by considering the coefficient of determination p2. To account for the number of parameters in the model, the adjusted p 2 is also determined. Dynamic parameter identification of non-delayed car-following models When parameters are expected to change in time and space, a dynamic (on-line, adaptive) estimation procedure is required. The development of this approach is done here in two steps. We first assume that the reaction times of the drivers can be neglected. The dynamics of the state are governed by the car-following rule of the model under consideration, expressing acceleration as a function of the state x(f) of the platoon. Let h > 0 denote the time-step, and let tk = kh denote the (discrete) time-instants. We use the following formulation: >; C M ) = t) h ( ^ 1 ) + /2.( a; (x|e) + w,.(rt)) for i = \,...,n-\ \\(tk+l) + h-w,{tk)

fori = »i

(11)

Parameter estimation of car-following models 251 where x denotes the state of the platoon, defined by (7), and 9 denotes the vector of (unknown) parameters of the car-following model, described by the acceleration a, as a function of x and 8. To account for modelling errors, zero-mean model or process noise w = (w\,..., wn) has been included in the state dynamics. The noise is assumed to be additive, white, and Gaussian, with covariance defined by:

£[wft)w'(*,)] = fa'^)

/=

*

(12)

[ 0 elsewhere For the sake of simplicity, explicit first-order time-discretisation was used to determine the discrete time system (11) from the continuous-time car-following model. The approach applies however to general time-discretisation schemes. When noisy measurements of the (part of the) state (positions and speeds) are available, the Extended Kalman Filter (EKF) can be used to generate (approximate) Maximum Likelihood estimates and predictions of the state by combining the predicted state with the noisy measurements. (Singhal and Wu 1989) propose using the EKF to estimate parameters of nonlinear models from clear data. In most practical applications, a clean state observations is however not available and a dual or joint estimation approach is required, such as the approach presented in (Haykin 2001). The approach entails joint estimation of the state (in this case, the positions and speeds of the vehicles in the platoon), and parameters G of the nonlinear car-following model. To apply this joint estimation approach, we assume that the car-following parameters 6 (e.g. the sensitivity parameter oci and the reaction time Tr) are also driven by a zero-mean white process (just as the platoon state dynamics, i.e. Eq. (11)), reflecting again modelling errors: 9(tt+l)=Q(tM) + h-a(tt) (13) with E(<x>(tk)o)'(tt)) = QSh)- Next, we augment the state x by including the parameters of the car-folio wing model: x = (r1>v1,...,rn,vB,91,...,em) (14) where m denotes the number of parameters in the car-following model under consideration. Using this definition, we can formulate the discrete-time system describing the (augmented) systems dynamics:

= W w ) / = W 4 + 1 ) + Ma,(x|e) + w;(O)j =f(I(O,w(/ t )) , 8('*+i) J [ Q{tt+l) + h-m(tk)

(15)

where f denotes the right-hand-side, and where the model noise satisfies:

Emk)*%)] = \Qlk)

/ =*

where QJ&

°1

(16)

[ 0 elsewhere ^ 0 QJ The elements of the error covariance matrix Q express the reliability or accuracy of the model: smaller elements imply a higher model accuracy and vice versa. The performance of the filter turns out to be relative insensitive to the specification of the covariance matrix. The measurement equation is given by:

252 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction (I, y(t t ) = Hx(tt ) + Ti(/t) where H = \»

0°) I

(17)

where hn denotes the 2M by 2« identity matrix (showing that the locations and speeds, but not the parameters themselves are observed). To illustrate the filter workings, let us consider a two-vehicle platoon the dynamics of which are governed by the GHR model (Eq. (1)). The follower i = 1 is assumed to react to the leader i = n=2 according to Eq. (1). Furthermore, we assume that ai = ao is constant, that is / = m = 0. and only relevant for the follower i=\. The augmented state becomes: x = (r1)v1>r2,v2,a1) (18) For this particular example, the augmented state dynamics equal Eq. (18), with V

l

'1

a,(v 2 -v,)

v,

1

v, f(x,w) =

r2 +h--

v2

v2

0

,aJ

[{

+w

X

2

x5(xA-x2)

= r2 +h-v2

J J U J {{

x4

+w

(19)

0

J

Dual Extended Kalman Filter summary The DEKF provides approximate Maximum-Likelihood (ML) estimations of the augmented state (i.e. including the unknown parameters). More specifically, the EKF yields estimates for the augmented state that minimizes the cost function J: K-l

) = X[(v* + i -Hxt+JRk(yk+l

-flx i + 1 ) + (xt+1 -x t + l t )'e t (x t + 1 - x t + l t ) ]

(20)

*=o

The DEKF consists of two steps, prediction and correction. A double index is used to express whether a prediction or correction result is considered. We provide an overview of the most important steps of the DEKF. For the sake of notational convenience, we drop the tilde from the notation in the remainder (x = x). 1. Initialisation: Initialisation pertains to computing the initial state as well as estimating the initial covariance: x0|0 = E [x0 ] and Poo = var (x 0 ) (21) where the covariance matrix Po|o describes the uncertainty of the initial state xo = x(fo). 2. Prediction: The prediction step entails application of the state dynamics to predict the state as well as the covariance matrix, i.e.: x i+n*=f(x*i*>°) (22) Here, Ft is the linearised state dynamics:

Recall that the matrix g* denotes the covariance matrix of the process or model error w(4).

Parameter estimation of car-following models 253 3. Computation of Kalman filter gain The Kalman filter gain Kt describes the way observations are used to correcting the prediction. The filter gain yields an ML estimate of the linearised system: KM = Pt^tH't+] [HMPtmH't+1

+ Rt+1 ]"'

(24)

Recall that Rk = R(tk) denotes the covariance matrix of the measurement error. For the expression, first insights into the workings of the filter can be determined. For one, it can be seen that if the state covariance Pk+i\k is small compared to the measurement noise Rk, the filter gain will converge to a matrix which entries are small, and vice versa. 4. Correction The final step entails correction of the predicted state and covariance by using the measurements V A

—V

t+i;*+i ~

^+1,4+1

=

A

t+u

A- HT T Jv

P

fv

—W

* + r t+ijt |_y*+i

J

Y

1

A

*+i t+i|*J

P

(^~-^*+A+i)- *+i:,t

Eq. (25) nicely shows how the DEKF works. Given the differences between measurements and predictions, and the Kalman gain and state covariance, the prediction is corrected accordingly: if the gain Kk+\ has small entries (i.e. when the state covariance Pk+\\t is small compared to the measurement noise), little correction is performed, implying that Xi+i|4+i = Xt+\\k and Pk+\\k+\ = Pk+\\k- If the gain Kk+\ has large entries, the influence of the measurements becomes much more elaborate. Let us finally mention that besides the covariance matrix P provides information of the accuracy of the estimates, since by definition: ^«=cov(vtt,xt)t) (26)

Adaptive parameter identification of delayed car-following models So far, we have assumed that the reaction time of drivers Tr can be neglected. It is well known from literature that the reaction time plays an important role in the characteristics of traffic flow. Therefore, we will consider the following general delayed car-following model:

v(t

, J v i ( ' M ) + M i i W l ) , ^ - y i « ) + «' i fe))

for i = l

Ti-1

(27)

Here, a,- = a,- (x,^|9) denotes the acceleration function as a function of the current state x=x(t) and the retarded state %=\{t-Tr). As explained in beginning of this section, we can easily determine the reaction time Tr assuming that it is time-invariant. To include the reaction time in the dynamic model identification is however more involved. The most straightforward would be to apply a simple linearisation to the car-following model, i.e. x(/-rr)«x(0-rri(0 (28) yielding: a,(x(0,x(<-7,) I 9) *

fl,(x(0,x(0-r,i(0

19) . a,.(x(0,x(0 j 6)-Tr

(29)

254 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Clearly, this approximation will only be valid when the reaction time is relatively small. Application has shown that this is not the case for the problem considered, and that the approximation is only reasonable when Tr < 0.3 s. We therefore propose linearisation around a prior estimate of the reaction time Tr (e.g. determined from static estimation). It is shown easily that this yields:

a,(x(0,x(* -r r -sr r ) | e)« a(x(t)Mt-Tr) I e) - s r , i ( o 5 "

(30)

for sufficiently small perturbations hTr of the reaction time. If we for instance consider the GHR car-following model (Eq. (1)), we have: a.Xx(tXx(t-Tr-hTr)\al,Tr)x.v,,Avii\t-Tr)-uxhTrAaf (31) In applying the DEKF to this model, we define the augmented state by the positions, speeds, the sensitivities cxi and perturbation 8Tr for i= l,...,n-l. E.g., for a two-vehicle platoon, the augmented state equals x = (rl,vl,r2,v2,a1,8Tr). The retarded relative speeds Avfl\t-Tr) and relative accelerations Aa,(1)(f-rr) are determined directly from the observations, and do not enter the augmented state (i.e. the are consider autonomous). Considering another example, the linear Helly model (Eq. (3)), we find: <1) (r-^) + p 1 ( A ^ ) ( ; - r r ) - 5 1 ) a ,(x(0,x(/-7;-8r r )|a 1 ,p i ,5 0 ,5 1 ,r r )«a 1 Av,

-5J r (a 1 Aaf ) a-r r ) + P1(Av,<1)(?-rr)-^1a,.a-rr))) In this case, the augmented state equals x =

(32)

(r1,v1,r2,v2,ai,$1,s0,sl,8Tr).

Verification of dynamic parameter identification Let us consider the workings of the algorithm by its application to synthetic data, showing that the approach is able to estimate car-following parameters from (noisy) trajectory data. Furthermore, we show how the approach effectively adapts to changing car-following parameters. The latter is important to investigate whether it is likely that the behaviour changes during driving. The synthetic data used for verification was established as follows: from the empirical data, a vehicle pair was selected. The behaviour of the leader vehicle was determined directly from the collected trajectory data; the behaviour of the follower was determined by application of the simple car-folio wing rule Eq. (1) using different settings for the sensitivity parameter ao and the reaction time Tr. Various levels of noise were applied to the data, reflecting measurement errors. In the first step of the verification, we have considered the results in case both ao and Tr are assumed constant, while considering different noise levels. Without going into detail here, we observed that the approach can accurately reproduce the constant model parameters. It also turned out that the variability (and thus the accuracy) of the parameter estimates changes with changing noise levels. The second part of the verification entails showing the ability of the filter to track changes in driver behaviour expressed by changes in the parameter values. In the first case, we assume that the sensitivity parameter changes suddenly from 0.4 1/s to 0.6 1/s. Fig. 1 shows the results of this experiment. Without going into detail, from the figure it can be concluded that

Parameter estimation of car-following models 255 the filter is able to track the sudden change in ao with reasonable accuracy and within a reasonable time. This holds equally for the second test, where an sudden reduction from 1.4 s to 1.1 s in the reaction time was simulated (Fig. 2). hi both cases, the performance of the filter depends strongly on the filter design, i.e. the covariance matrices Q and R for the model error and the measurement error, as well as on the noise applied to the synthetic data, hi general, assuming high model noise for either parameter yields a quickly adapting parameter estimation, which is however noisy. Lower model noise leads to slower but more robust parameter adaptation. In the remainder, Q and R are chosen such that for the synthetic data, results are deemed of good quality. The same values are then used for the empirical analysis presented in the next section.

Fig. 1 Parameter identification for suddenly changing sensitivity parameter. The filter identifies changes in the sensitive sensitivity parameter ct0.

Fig. 2 Parameter identification results for suddenly changing reaction time Tr. The right figure shows how the filter adapts quickly to the changed reaction time.

The filter has been applied to more complicated situations (two parameters changing at the same time, slowly changing parameter, etc.). Without going into detail, the filter shows overall good performance. This holds equally when other models are used. Let us finally note that considering multiple vehicle platoons with common car-following parameters increases accuracy of the dynamic parameter estimates due to the increase in the

256

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

sample size. We will however not consider this approach further in the remainder of this contribution.

TRAJECTORY DATA COLLECTION The vehicle trajectory data used here was collected using a new data collection approach (Hoogendoom and Van Zuylen,2004) using an air-borne observation platform (a helicopter), mounted with a high-frequency digital camera and frame grabber. Using image processing software, the vehicles are detected and tracked. This yields trajectory data covering approximately 500 m of motorway roadway stretch; the spatial resolution is smaller than 40 cm, while the temporal resolution is 0.1 s. Besides the trajectories of all vehicles present, the system also determines the vehicles' lengths and widths. Vehicles driving in both roadway directions were detected and tracked. Only one driving direction is considered here. Two datasets have been used in the parameter identification, both pertaining to the afternoon rush-hour. One of these was collected during the afternoon peak hour at the three-lane A15 motorway to the South of the Dutch city of Rotterdam (referred to as the Waalhaven site). Fig. 3 shows a subset of the collected trajectories for a ninety second period. During the entire period in which data were collected, congestion was quite heavy (average speeds of 7 m/s). The other dataset (the Everdingen site) was collected at the A2 motorway near the Dutch city of Utrecht and is characterised by stop-and-go traffic conditions.

Fig. 3 Sample of vehicle trajectories for data collected at A15 site. The small dots represent time instants which are 2.5 second apart; data is collected at a temporal resolution of 0.1 s.

Parameter estimation of car-following models 257

STATIC ESTIMATION RESULTS AND MODEL CHOICE In this section, we apply the static estimation approach described earlier. The choice for the model used in the remainder will in part be based on the relative performance of the model steaming from the static estimation. The estimation procedure furthermore yields prior parameter estimates (including reaction time) that will be used in the dynamic approach. Fig. 4 depicts example model estimation results for three models (GHR, Bexelius and linear Helly). The figure shows observed and predicted relations between distances and speeds (Fig. 4a, c, and e), and between relative speed and retarded acceleration (Fig. 4b, d, and f). Example shows small differences in performance between the models. Fig. 4 shows that the Helly model outperforms the other models (in terms of the RMSE), although the (absolute) reduction in the RMSE is only minor (compared to the additional number of parameters involved in the linear Helly model). Note that on average, the performance of the Helly model is much better that the GHR model, as is discussed in the ensuing.

Fig. 4 Parameter estimation results; a,b) GHR, c,d) Bexelius, and e,f) linear Helly models.

Static model identification has been applied to a variety of models; Tab. 1 and Tab. 2 provide overview of estimation results, pertaining to data collected at the A15 motorway (Tab. 1) and the A2 (Tab. 2) respectively. Both tables provide insight into the distribution of the prediction

258 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction errors (in terms of the median, mean, and standard deviation of the RMSE's determined for each observed vehicle trajectory), and show coefficients of variation (normal and adjusted). Both tables show large differences between the different models. The model of (Lenz, Wagner et al. 1999) exhibits the worst performance for both the A15 and A2 datasets. The GHR-type models (including the multi-anticipative model of (Bexelius 1968)) all show reasonable performance; the model of (Bexelius 1968) outperforms the single-leader models, indicating that multi-anticipative driver behaviour can be identified empirically from the available microscopic data. The linear model of (Helly 1959) and the model of Tampere (2004) show comparable performance and both outperform the other models. Also note that the Tampere models shows smaller RMSE especially for the A2 dataset. This can be explained by the fact that given the prevailing traffic conditions, driving behaviour is likely to be determined (in part) by acceleration towards the free speed v° (included in the Tampere model, but not in the Helly model). Small performance differences are due to the inclusion of indifference boundaries. Note that for all models, the distribution of the RMSE is skewed rather than symmetric, and the error standard deviations are relatively high. In other words, large differences exist is the ability of the car-following model to correctly predict the behaviour of individual drivers. For most drivers, the predictions are accurate, whilst for some, the error is very large. Tab. 1 Overview of estimation results from A15 dataset. Helly GHK Model GHR Number of parameters RMSE median RMSE mean RMSE standard dev. P"

(moMJ!)

<m=0./=l)

1 4.8 30.5 95.8 0.88 0.88

1 4,7 30.1 95. t 0.86 0.86

Tab. 7 Overview of estimation results from A2 dalase GHR GHR Model Number of parameters RMSE median RMSE mean RMSE standard dev. P" PV

<»!=/=()) 1

(m=0,/=l) 1

3.3

3.4 7.5 13.4 0.64 0.61

N.I

15.0 0.67 0.62

Tampere

Bexelius

Lenz

4 2.4 12.1 42.5 0.90 0.90

7 2.4 11.9 42.2 0.90 0.89

2 2.8 26.6 133.1 0.87 0.87

4 38.0 70.2 89.4 0.19 0.19

Helly

Tampere

Bexelius

Lenz

4 0.7 1.5 1.9 0.82 0.82

7 0.6 1.4 1.9 0.83 0.82

2 1.8 5.5 11.2 0.71 0.70

4 3.7 16.2 30.9 0.64 0.61

Static estimation results for the linear Helly model In the remainder of this contribution, the linear Helly model will be used, due to its good performance and its (relative) simplicity. Let us briefly describe some of the results of the model estimation using this model.

Parameter estimation of car-following models 259 Fig. 5a, c and d shows the empirical distribution functions of the parameter estimates for the Waalhaven and the Everdingen case. Clearly, inter-driver differences are considerable. The most considerable differences between the measurement sites pertain to the reaction times Tr (Fig. 5a) and the sensitivity ai with respect to the speed differences (Fig. 5c). These can be explained by differences in prevailing traffic conditions for the two sites, but also in the different in traffic composition. Note that the estimation results for the two sites show small inter-site differences both for the sensitivity Pi (Fig. 5d), and for the estimated relations between the speed v and the desired distance S\(v) (Fig. 5b; notice the difference in the ranges on the observed speeds in both situations). The estimated parameter values are in line with those found in other studies (see (Brackstone and McDonald 1999) for an overview). Different parameter estimates are found for the two data collection sites, hi general, the reaction times at the Everdingen site turn out to be somewhat higher, as are the sensitivities. The differences between the estimation results from the respective sites are considerable, and the reasons behind them are manifold. An explanation may be the differences in traffic conditions: on the Waalhaven site, congestion was heavy and had been for a long time. The Everdingen site is characterised by modelatly congested traffic conditions. From this perspective, the changed parameter values may indicate changes in driver behaviour.

Fig. 5 Estimation results for linear Helly model showing Waalhaven and Everdingen results (latter are indicated by dotted lines), showing; a) distribution of the reaction times,; b) average relation between driving speed and safe distance headway; c) and d) distributions of the sensitivities <Xi and Pi.

260 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

APPLICATION OF ADAPTIVE PARAMETER IDENTIFICATION This section presents the results of application of the filter to the empirical data collected at the Waalhaven site and at the Everdingen site. The results show that in case we assume that drivers follow the leader according to the linear model of (Helly 1959), application of the DEKF show an overall improvement of the RMSE, accompanied by dynamically changing model parameters. For the Everdingen site, a plausible behavioural interpretation for the parameter changes can be given. For the Waalhaven site, an explanation of the estimation results is less straightforward. Initial filter settings To apply the DEKF, initial choices concerning the covariance matrices Q and R (for the process noise and the measurement noise respectively) are to be made. The measurement noise Rk = R was chosen such that it reflects the expected measurement error. These are in particular due to the spatial resolution of the measurements (Hoogendoorn and Van Zuylen,2004)). The process noise variance for the vehicle positions r,{tk) was chosen equal to zero; for the speeds v,(4) we have chosen non-zero error variance (value of 10 was chosen for the trials shown in the remainder), as for all other variables of the augmented state (all parameters having an error variance of 0.01). For the measurement noise on the location and speed, the variance was chosen equal to 100 and 500 respectively. General results of dynamic parameter identification For both the Everdingen and Waalhaven sites, it was observed that the DEKF decreased the RMSE of the estimates considerably (on average with a factor of 3.3 for the chosen settings of the error covariances). This is not surprising, since not only the parameters, but also the predicted speeds are corrected according to the observations. Fig. 6a,b shows two examples of application of the DEKF. The figure shows how filtered traffic state closely tracks the observed positions (in the figure expressed in terms of the gap) and the speeds, especially when compared to the static model predictions. Results from application at the Waalhaven site are similar.

Fig. 6a,b Estimation results from application of the DEKF to Everdingen data. In the remainder, we iscuss example results from dynamic parameter identification. The parameter estimates cause the prediction error Eq. (19) to be minimised, given the current

Parameter estimation of car-following models 261 filter settings. As such, care is to be taken in interpreting estimation results since strictly speaking, they represent only the result of statistical optimisation. If the model cannot explain car-following behaviour due to its inadequate structure and the used explanatory variables, the estimations will lack a behaviour interpretation. Results from data collected at the Everdingen site Fig. 7 shows some results of applying the dynamic estimation procedure to the trajectory measurements collected at the Everdingen site. At this location, traffic conditions were moderately congested. More specifically, a high-density wave propagated upstream across the entire roadway during the measurement period. Fig. 7 shows estimates of the sensitivities cti and pi, as well as the reaction time Tr as a function of time. The figure also shows the vehicle speed (dotted line) for reference purposes, indicating how the parameter estimates change in relation to prevailing traffic conditions. Note that the estimates for the parameters so and s\ describing the desired distance headway S\(v) turn out to be nearly constant. This observation is in line with our expectations given the small inter-site differences in static estimates, so and s\ are therefore not considered further. On average, sensitivity values ai and pii increase and reaction times Tr decrease when road users drive into the smooth shock. An increase in sensitivity thus generally occurs when speed reduces rapidly, while at the same time a decrease of the reaction time occurs. From Fig. 7 we also conclude that changes in Pi generally precede changes in at.

262

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 7a-i Example results from application of the adaptive parameter estimation approach for the Everdingen measurement site. To quantify the relation between the parameter changes and the changing traffic conditions, statistical relations between the rate of change in the parameter values (oci, Pi and Tr) and the rate at which the traffic conditions changed have been considered. In doing so, the dynamic estimation results for a random group of individual drivers were combined and linear regression was applied to determine a relation between the aforementioned quantities. The following statistically significant relation between the rate of change of the reaction time Tr and the speed v was established (at 95% confidence level): ^ * -0.015-^ dt dt

(33)

For the sensitivity cxi we find: ^ L » 0.0013-^ (34) dt dt Both relations are roughly in line with our expectations: when traffic conditions deteriorate, sensitivities increase and reaction times decrease. Note that changes in the reaction time are much more sensitive to the changing traffic conditions. For the sensitivity Pi, no statistical relation could be established between the rate of change and the change in traffic conditions.

Parameter estimation of car-following models

263

Results from data collected at the Waalhaven site For the Waalhaven measurement site, clear conclusions on the relation between the carfollowing parameters and prevailing traffic conditions cannot be given. For one, the carfollowing behaviour at this heavily congested site appears to be more stationary than at the Everdingen site (smaller changes in the car-following parameters). Increasing sensitivity estimates are found in both situations where the speed increases and decreases. Reaction times show a small overall increase over time, providing weak evidence for motivational loss due to the prolonged congested conditions. The same statistical analysis has been applied in an attempt to find a relation between the rate of change in traffic conditions and the considered car-following parameters. In line with our expectations, no such relation could be established.

Fig. 8a-i Example results of application of the estimation approach to data collected at the Waalhaven measurement site, showing dynamically changing parameters.

CONCLUSIONS This contribution presents a new approach to static and dynamic parameter estimation for carfollowing models. The approach enables identification of model parameters of general carfollowing models. Its dynamic nature makes it possible to investigate smoothly or rapidly

264 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction changing driver behaviour as expressed in terms of for instance changing driver sensitivity or reaction times. From verification using synthetic car following data, it is shown that the approach is able to track changes in the parameters describing the car-following behaviour. The static parameter estimation is relatively straightforward and entails minimisation of the difference between predicted and observed behaviour. Using vehicle trajectories collected at two motorway sites (with different traffic conditions), the static approach was applied to test and compare a number of car-following models (including multi-anticipative car-following models, such as the model of Bexelius). Estimation results showed that the linear Helly model was the most suitable model for the subsequent dynamic estimation, due to its performance as well as its relative simplicity. The data analysis revealed differences in car-following behaviour for the two data collect sites. These differences may be explained by hypothesizing that drivers have adapted to the prevailing traffic conditions present at the two data collection sites, but also the differences in traffic composition, roadway geometry, etc. Besides inter site differences, also large intra-site differences have been found. The static estimation results for linear Helly model show considerable but plausible differences in terms of driver sensitivity as well as reaction times. Application of the dynamic estimation approach presented in this contribution shows clearly that car-following parameters yielding the optimal fit of the model and the trajectory data change dynamically during the trip. Depending on the measurement site and the prevailing traffic conditions at the site, the variation in the parameters is considerable. At the site characterised by heavy congestion, changes in the car-following parameters cannot however be convincingly attributed to changes in the prevailing traffic conditions and the consequent changes in driver motivation, alertness or awareness. For the site characterised by moderate congestion, the driving parameters generally changed according to changes in the prevailing traffic conditions: when driving into a Shockwave, road users appeared to change their sensitivity and reaction time accordingly. The opposite is observed when drivers drive at the same speed for a while, or when the speed increases again.

ACKNOWLEDGEMENTS The research presented in this paper is part of the research programme "Tracing Congestion Dynamics - with Innovative Traffic Data to a better Theory", sponsored by the Dutch Foundation of Scientific Research MaGW-NWO. The data was used at the courtesy of the Traffic Research Centre ( A W ) of the Dutch Ministry of Transportation.

REFERENCES Bando, M., K. Hasebe, et al. (1995). "Dynamical Model of Traffic Congestion and Numerical Simulation." Physical Review E(51): 1035-1042. Bexelius, S. (1968). "An extended model for car-following." Transportation Research 2(1): 13-21. Brackstone, M. and M. McDonald (1999). "Car-Following: A Historical Review." Transportation Research Part F2: 181 -196.

Parameter estimation of car-following models 265 Brockfeld, E., R. D. Kiihne, et al. (2004). Calibration and Validation of Microscopic Traffic Flow Models. Transportation Research Board Annual Meeting Pre-Print, Washington D.C. Chandler, R. E., R. Herman, et al. (1958). "Traffic Dynamics: Studies in Car-Following." Operations Research 6(2): 165-184. Daganzo, C. F. (2002). "A Behavioral Theory of Multi-Lane Traffic Flow. Part I: Long Homogeneous Freeway Sections." Transportation Research B vol 36B, issue 2: 131158. Dijker, T., P. H. L. Bovy, et al. (1998). "Car-Following under Congested Conditions: Empirical Findings." Transportation Research Record 1644: 20-28. Haykin, S. (2001). Kalman filtering and Neural Networks. Ontario, John Wiley & Sons, Inc. Helly, W. (1959). Simulation of Bottlenecks in Single Lane Traffic Flow. International Symposium on the Theory of Traffic Flow, New York. Hoogendoorn, S. P. (2002). "Microscopic Traffic Data Collection by Remote Sensing." Transportation Research Records 1855: 121-128. Hoogendoorn, S. P. and H. J. van Zuylen (2004). Tracing Congestion Dynamics with Remote Sensing. Management and Information Systems 2004, Malaga, Spain, WIS. Lenz, H., C. K. Wagner, et al. (1999). "Multi-anticipative car-following model." The European PhysicalJournal B(7): 331-335. Leutzbach, W. (1988). "An Introduction Into the Theory of Traffic Flow." Mufioz, J. C. and C. F. Daganzo (2001). Moving Bottlenecks: a Theory Grounded on Experimental Observation. Transportation and Traffic Theory in the 21st Century, 15th International Symposium on Transportation and Traffic Theory, Pergamon-Elsevier, Oxford, U.K. Schultz, G. and L. Rilett (2004). An analysis of the distribution and calibration of carfollowing sensitivity parameters in microscopic traffic simulation models. Transportation Research Board Annual Meeting, Washington D.C, Transportation Research Board. Singhal, S. and L. Wu (1989). "Training multilayer perceptrons with the Extended Kalman Filter." Advances in Neural Information Processing Systems 1: 133-140. Tampere, C.M.J. (2004), Human-kinetic traffic flow modelling for the analysis of Advanced Driver Assistance systems in congestion, Dissertation thesis, Delft University of Technology. Available through: http://www.kuleuven.ac.be/traffic/dwn/P2004C.pdf Van der Hulst, M. (1998). Adaptive control of safety margins in driving. Groningen, University of Groningen. Zhang, h. m. and T. Kim (2001). A car-following theory for multiphase vehicular traffic flow. 80th Annual Meeting of the Transportation Research Board., Washington D.C.

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14

MODELING IMPATIENCE OF DRIVERS IN PASSING MANEUVERS Moshe A. Pollatschek, Department of Industrial Engineering and Management, and Abishai Polus, Department of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Haifa, Israel

INTRODUCTION Drivers in many traffic situations make decisions based on their experience during the previous few minutes of driving. The learning-from-driving experience is based on the prevailing traffic conditions. Such learning actually involves an estimation of the existing traffic parameters, and this results accordingly in modification of driver behavior. A prime set of examples is provided by drivers' gap acceptance whether in entering an unsignalized intersection or a roundabout, in merging into a freeway, or in passing on a two-lane highway. This learning however, often leads to impatience, which stems from a realization delays could be excessive without additional risk. This paper will concentrate on impatience during the passing process against the oncoming traffic on two-lane rural highways. The need to consider this impatience developed from the necessity to simulate two-lane traffic when estimating when to widen a two-lane highway to four lanes. A major criterion for widening is an economic consideration, involving a comparison of the cost of building the additional lanes with the cost of delay resulting from the inability to pass. In heavy flows, gaps between on-coming vehicles are small, and the probability of finding a suitable gap for passing is small. The delay then increases, and drivers develop tendency to accept more risk by reducing the gap they are willing to accept. A similar phenomenon was discovered and recently modeled by Polus, Craus, and Reshetnik (1996) and by Polus, Shmueli-Lazar, and Livneh (2003) and by Pollatschek and Polus (2003) for the behavior of drivers while accepting gaps at conventional intersections and roundabouts.

268 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The delay to traffic can be evaluated by both simulation and queuing models. Simple theoretical models assume a fixed critical gap, above which all gaps are accepted and a passing maneuver occurs, and below which no passing occurs. However, as will be shown, the traffic flow in general and platoon formation in particular are very sensitive to the critical gap. It is assumed that the critical gap is dependent on the traffic in both directions, and more specifically on waiting time in the platoon until passing can be performed. Therefore, it is necessary to estimate the function that describes the reduction in critical gap with the increase in traffic flow. This function cannot be estimated directly, because it is almost impossible, or at best prohibitively expensive, to measure the duration of time in which a specific driver remains in a platoon immediately behind a slow vehicle. Consequently an indirect approach must be utilized; this methodology will be described below. The purpose of the paper is to present these indirect observations of platoon lengths, and from them to derive the impatience function. Validation of this function is conducted by integrating it into a simulation software that simulates flow on two-lane highways. The simulation is used to generate the percent-time-spent-following by drivers, as a function of volumes and speeds, and to compare the percentage to HCM 2000 suggested values for this parameter. The findings are evaluated here, and a discussion is presented regarding the applicability of the approach.

LITERATURE REVIEW There are three aspects of the present paper that have already been discussed in the literature: first, passing maneuvers and related times, speeds and distances; second, application of queuing theory to traffic phenomena of queue lengths and time delays; third, impatience during driving and its impact on drivers' behavior, particularly on the willingness to accept more risk in return for reduced delay. Several studies have dealt with passing maneuvers on two-lane highways; two will be noted here. Polus, Livneh, and Frisher (2000) analyzed over 1,700 passing maneuvers and developed models to estimate the distances and times required to perform the passing. Rao, Sahoo, and Kumar (1996) conducted a study on the acceleration characteristics of vehicles during passing maneuvers and found several speed and acceleration relationships for each stage of the passing process. There are many examples of applying queuing theory to traffic phenomena. Tae Son (1999) developed a model of queue formation at the two ends of a lane closure on two-lane highways. The model was based on enhanced fluid approximation, which captures fluctuations in vehicle arrivals and departures. Harris (1976), who developed an M/G/l model for a single-lane entrance ramp, presented the concept that the delay of the lead vehicle was composed of the sum of successive gaps faced by the entering driver. Using a critical gap (minimum gap for entering the main freeway) concept for merging, he suggested lowering the critical gap for ramp-metering projects when the queue on the ramp became

Impatience of drivers in passing maneuvers 269 longer. This approach is also applied to the present study, except that our critical gap depends on the traffic volume (intensity), whereas Harris's development was based on the length of the queue. For that reason theoretical developments are easier to formulate and lead to closed forms of the expected queue length. The impatience of drivers during gap acceptance was treated by Polus, Craus and Reshetnik (1996). Pollatschek and Polus (2003) presented a decision model for gap acceptance and capacity at intersections with application to roundabouts. Both these models assumed that drivers were more willing to take a risk in order to reduce delays when traffic volume and delay increased.

QUEUING MODEL FORMULATION The assumption for the theoretical development regarding the queuing model development is that the road contains both slow and fast-moving vehicles and that each of the two groups has a constant speed. Platoons are formed when a few fast vehicles accumulate behind a slow vehicle, thus creating a moving queue. The first vehicle behind the impeding vehicle tries to perform a passing maneuver on the lead vehicle. The time interval between the arrival of a fast vehicle behind the slow vehicle and the time when it can start the passing process is denoted To- The latter does not occur until a sufficient gap in the oncoming traffic is available for the fast driver. The critical passing gap is defined in this study as the minimum gap that is sufficient for passing; it is denoted Tc. The gap is considered sufficient, thereby leading to the passing maneuver, when it is greater than the passing critical gap Tc. It is assumed that the behavior of all drivers is homogeneous and, therefore, they all have the same critical gap. If we assume that the arrival of cars in both directions is a homogeneous Poisson process, with volumes (intensities) of X\ and ^2 , in the main direction and the opposing flow, respectively, then the first two moments of TD may be given as

E(TD) =

c

(1)

Where Tc is the critical passing gap, and E(TD2) = ^E(TD)

+ 2[E(TD)f -T*

A2

The derivation of Eqs. 1 and 2 is shown in the appendix.

(2)

270 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction If X] is the rate of arrival of vehicles (intensity) in the main direction relative to a standing observer, then the rate of arrival of fast vehicles behind and relative to a slow-moving vehicle, Xo, is given as ^o=^)

(3)

where Xo and X\ are defined above and, s = speed of fast vehicles u = speed of slow vehicles It is assumed that s and u are two deterministic parameters, representative of the fast and slow vehicles, respectively. This passing situation can be formualted by a queuing model with a single server, in which the service time is To and customer arrivals are homogenous Poisson with volume (intensity) XQ. The Khinchin-Polacek formula for the expected value of the number of cars behind a slow car (L) (number of fast cars in the platoon) is given then by the following: L = A0E(TD)+

%E(-T^

(4)

where all the parameters have been defined above. Note that L is a function of Tc and that the expected platoon length is (L+l) because L does not include the slow-moving lead vehicle. In order to calibrate this function, the critical gap (Tc), which is included in E(TD) and E(TD2), should be chosen such that L+l best fits the observed average platoon lengths. The assumptions behind Equation 4 (homogeneous Poisson process, fixed speeds, and fixed critical gap) are not true in real driving situations, however, the question is, how closely they approximate the real conditions? Another question concerns the values of u and s that should be assumed for the analyses. Finally, the value of Tc may change in reality because of a loss of patience after long driving in a platoon. This last point will be discussed in the next section; the other questions will be addressed in section 6. From Little's theorem, the expected value (W) of the total delay of the fast vehicles in the queue (from their arrival at the back of the platoon until the start of the passing maneuver) is given by: W = -^ Xo where all parameters have been defined above.

(5)

Impatience of drivers in passing maneuvers 271 Formation of platoons and delay to traffic are sensitive to the critical gap of drivers, as illustrated by Figure 1 and Figure 2. Figure 1 shows the expected delay W, which is a function of the volume (intensity) of arrivals of the opposing traffic (12), and the critical gap for passing. Figure 2 shows the sensitivity of the expected number of cars behind a slow vehicle to the critical passing gap.

Figure 1: Interrelationship of Critical Gap, Volume (Intensity) in Each Direction, and Total Delay (W)

Figure 2: Impact of the Critical Passing Gap on the Expected Number of Cars Behind a Slow-Moving Vehicle

272 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

DATA AND PRELIMINARY ANALYSIS The data for the study was collected on 10 segments of two-lane rural highways by recording the speeds and time of passing of vehicles with a pair of detectors. The summary statistics of the data are presented in Table 1 and Figure 3; each line in the table represents the summary for one direction of travel. The length of the queue was determined by identifying platoons for which the headway between successive vehicles was 5 sec. or less, which is according to the definition of a platoon in the Highway Capacity Manual (2000). Table 1: Summary Statistics of Traffic Data on Five Two-Lane Rural Highways Road

Average Queue Length (cars)

77-1A 77-1B

Mean Gaps (sec)

One-way Volume (vph)

4.4

5.09

699

4.49 6.49

802

77-2A

5.0 3.9

77-2B

3.9

5.79

774

R77-43-n

3.2

6.94

512

R77-43-S R77-47-n

3.1

7.56

3.5 3.7 2.6

5.89 5.90 13.34

466 610

1.6

13.85

R77-47-S 79-3A 79-3B

535

600 265 245

Figure 3 shows the histogram of speeds of all vehicles on all the roads investigated. The data was aggregated because there was no significant difference among speed observations on the different roads. The median speed was 85.6 km/h; the 25th percentile was 77.9 km/h; and the 75th percentile was 93.3 km/h. For the model in Eqs. 3 and 4, the 75th percentile was selected as the fixed speed of the fast vehicles, and the 25th percentile as the speed of the slow vehicles. The "slow" vehicles were those with speeds below the median speed, and the "fast" vehicles had speeds higher than the median speed. The 25th and 75th percentiles are the mid-points of the slow and fast vehicles, respectively.

Impatience of drivers in passing maneuvers 273

Figure 3: Observed Distribution of Speeds of All Vehicles on All Roads

THE IMPATIENCE FUNCTION The value of the critical passing gap for all drivers varies between a minimum value, below which it is impossible to pass, and a maximum value, above which all drivers are willing to accept a gap and perform a passing maneuver. As discussed in the introduction, the longer a driver remains in a platoon, the more one's impatience increases, making the driver more ready to accept smaller gaps than he would otherwise have been accepted in lighter traffic, without substantial platoons. Therefore, it was hypothesized that the critical passing gap is a continuous, monotonously decreasing function of the prevailing traffic, represented by X1+X2 . In order to validate this hypothesis, there was a need to estimate the critical passing gap, Tc, such that the calculated queue length (Eq. 4) was equal to the observed queue length for each road, as given in Table 1. The resulting critical gaps, as a function of the two-way volume, are shown as data points in Figure 4. The two-way volume is the sum of the two traffic intensities, X\+Xi and is denoted by V. It can be seen that the critical gap indeed decreases with the increase two-way hourly volume. It was now necessary to choose the functional form that best describes the estimated critical gaps according to the following considerations: (a) a monotonously decreasing function; (b) a continuous function; and (c) a function that varies between a minimum and a maximum values.

274 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The simplest function that satisfies the above-mentioned criteria is given by Eq. 6; it is needed to be shown that this function, termed the impatience function, provides good fit to the empirical data of queue lengths.

Tc = Tcmin + AT/[1 + e P*P-1+u-a>]

(6)

Where: Tcmm = minimum critical passing gap, below which it is impossible to pass A.i= volume (intensity) of traffic in the main direction %2 = volume (intensity) of on-coming traffic AT, a, and (3 = parameters to be estimated Polus, Livneh and Frisher (2000), in a study of passing behavior on two-lane rural highways, found that the Tcmm was 9 sec. and that passing gaps of 31 sec. and above were always accepted by all fast drivers. Therefore, the maximum value of Tc is derived from Eq. 6, and given as Tcmax = Tcmm + A T = 3 1

(y)

AT can be derived from Eq. 7. The parameters a and b of the impatience function were not available from previous studies and had to be found by minimizing the sum of the squares of the differences between the estimated Tc values and the values calculated by the logistic function in Eq. 6. The values obtained for a and f> were 872 and 0.00538, respectively. The minimum and maximum critical gaps and the fitted logistic curve are shown in Figure 4.

Impatience of drivers in passing maneuvers 275

Figure 4: Critical Gaps Obtained by Equating Theoretical Queue Lengths with Average Observed Queue Lengths and the Fitted Logistic Curve

VALIDATION OF THE IMPATIENCE FUNCTION The impatience function given by Eq. 6, which is also presented in Figure 4, now had to be validated in order to check the assumptions that to led the development of the function. These assumptions were that the traffic flow behaved according to a Poisson process and the speeds of the slow and fast vehicles were fixed, constant values. Clearly, these assumptions are not true in reality: the Poisson process which assumes independent arrivals of vehicles, is not valid because of the platoons, particularly in heavy flows. Additionally, the speeds of vehicles are distributed as shown in Figure 3, are not fixed. In order to validate the impatience function (Eq. 6), it was decided to use a specially developed simulation, called TwoLaneSim, which has characteristics closely imitating real traffic behavior. The simulation included the impatience function for gap acceptance as given in Figure 4. The simulation assumed a two-lane highway segment of 100 km. with cars entering on at either end, depending on a given volume. Different speeds, drawn randomly from a triangular distribution are assumed, with minimum and a maximum speed as input to the simulation, depending on actual speed observations. Depending on the available gaps, and the gap acceptance according to the impatience function, platoons are formed when fast vehicles trail slow vehicles.

276 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Data was collected only in the middle section of 50 km., while 25 km. at each end served as a warm-up length and their data was ignored. The simulated flow in the middle section does not constitute, therefore, a Poisson process. Figure 5 presents a screen showing a plan-view of a middle 3 km. section of a 100 km. road with relatively heavy traffic of 1,500 vehicles/hour in the two directions. The long platoons can be clearly seen. The formation of the impatience function of drivers is due to the intolerance that develops while trailing slow, impeding vehicles.

Figure 5: Simulated Two-Lane Highway Showing Vehicles Moving in Platoons

In order to validate the simulation and, consequently, the impatience function, it was decided to compare the results of the simulation with a commonly used parameter of traffic flow on two-lane highways. The percent-time-spent-following (PTSF) is a measure of the average percent of time that vehicles travel in platoons. It was selected because it is a key parameter in the analysis of two-lane highways according to the Highway Capacity Manual (HCM, 2000) and because it is strongly related to the gap-acceptance behavior of drivers. The percent-time-spent-following is the closest proxy variable in HCM to time spent in platoons (W, Eq. 5), and hence to queue length (L, Eq. 4). The simulation outputs of the PTSF parameter were collected, and an exponential curve that best fit the results was obtained. Six curves for different average speeds of fast vehicles are shown in Figure 6, along with the HCM 2000 curve for this parameter. The close proximity between the HCM curve and the simulation curves can be observed. This result shows the validity of the simulation software and, therefore, the validity of the imbedded impatience function. Figure 6: Percent-Time-Spent-Following as a Function of Volume and Mean Speed of Fast Vehicles

Impatience of drivers in passing maneuvers 277

SUMMARY AND APPLICABILITY The paper has presented a function that describes the impatience of drivers while accepting gaps during passing maneuvers on two-lane rural highways. The function was found by fitting the theoretical expected value of queue length to the empirical average queue lengths that were observed in the field on 10 two-lane sections. This fitted function, which is presented in Eq. 6 and graphically in Figure 4, was validated by a specially developed simulation package, TwoLaneSim, that chooses gaps for passing according to it. The measure of performance that was selected for the validation of the simulation was the percent-time-spent-following (PTSF), which is a well-established parameter in the Highway Capacity Manual for evaluating level-of-service on two-lane highways. Additionally, PTSF is strongly related to drivers' gap-acceptance behavior. A close proximity was shown to exist between the HCM 2000 results and the PTSF values that were obtained by the simulation package with the imbedded impatience function. The findings of this study show that drivers' impatience reduces the accepted gaps while performing passing maneuvers. This behavior was also shown previously Polus, Craus, and Reshetnik (1996) and Polus, Shmueli-Lazar, and Livneh (2003) and Pollatschek and Polus (2003), in regard to gap acceptance at unsignalized intersections. An explicit function for the reduction of critical passing gaps was derived and it can be used in simulation and other traffic analyses. The applicability of the findings is two-fold: (1) they show that the throughput (capacity) of two-lane highways in reality is higher than may be obtained by assuming a fixed critical passing gap; (2) they indicate that when traffic volumes increase, drivers are willing to accept a higher risk by dangerously reducing their critical gaps. This last conclusion is has implications for safety aspects and road-widening criteria.

REFERENCES Harris, C. M. (1976). State Dependent Gap Acceptance, Naval-Research LogisticsQuarterly, 23(4), 653-662 Highway Capacity Manual (2000). Transportation Research Board, National Research Council, Washington. Pollatschek, M., A. Polus, And M. Livneh (2002). A Decision Model for Gap Acceptance and Capacity at Intersections, Transportation Research. 36B, 7, 649-663. Polus, A., J. Craus, And I. Reshetnik (1996). Non-Stationary Gap Acceptance Assuming Drivers' Learning and Impatience, Traffic Engineering and Control 37, 6, 395-402. Polus, A., M. Livneh, And B. Frisher (2000). Passing Behavior on Two-Lane Rural Highways, Transportation Research Record 1701, 53-60. Polus, A., S. Shmueli-Lazar, And M. Livneh (2003). Critical Gap as a Function of Waiting Time in Determining Roundabout Capacity, Transportation Engineering Journal of ASCE, 129, 5, 504-509.

278

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Rao S. K., P. K., Sahoo And V. M. Kumar (1996). A Study of Acceleration Characteristics of Vehicles During Overtaking Maneuvers. Indian Highways. Nov. Tae Son, Y (1999). Queueing Delay Models for Two-Lane Highway Work Zones. Transportation Research, 338,459-471.

APPENDIX: DERIVATION OF EQUATIONS 1 AND 2 The time interval, To, is defined as the time between the arrival of a fast vehicle behind a slow vehicle and the time when it can start the passing process. Assume that the passing is possible only if the gap in the opposing traffic is greater than a critical gap, Tc, and the gaps in the traffic stream, T, are exponentially distributed with mean I/X2, where X2 is the volume (intensity) in the opposing flow. Thus, the probability density function (pdf) of T is

fj(t) = X2e~X2t,

t > 0 , \2>0.

Consider that the first gap that is greater than Tc is the (N + l) th gap (N = 0, 1,...,2). The gaps are indexed by i, i = 1, 2,...,N. Then the pdf of Ti given Ti
^e-^/fl-e-^c)

0
0

elsewhere

f T (t|T ;
(la)

and the TD for a given N is: TD = I T ;

(2a)

where all parameters have been defined above. It is useful to apply the Laplace transform, 3 x ( z ) f° r Oa)> a s follows:

As the Laplace transform of convolutions is the product of Laplace transforms, the Laplace transform of TD, given N gaps till passing, is:

.3D(z/N)= - ^ —

[X2

+z

X

l-e"^2Tc

Now, N is a geometrically distributed random variable with probability function (Harris, 1976):

Impatience of drivers in passing maneuvers 279

p(N = n) = Q~X^C (i _ e-^Tc )°. Therefore, the Laplace transform of TD is:

3 D (z)= Z 3 D ( z | N = n)p(N = n) = e-^ T c f n=0

n=0^2+ z

e-(*+Wc

f

=

)

The last equation follows as - ^ — [ l - e " ^ z + X 2 ^ T c J<1 for any X2 > 0 and z > 0. Equations 1 and 2 in the text are the first two derivatives of 3 D (z) at z = 0.

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15

A SIMULATION MODEL FOR MOTORWAY MERGING BEHAVIOUR Jiao Wang, Ronghui Liu and Frank Montgomery, Institute for Transport Studies, University of Leeds, Leeds LS2 9JT, UK

INTRODUCTION AND BACKGROUND Motorway merging has long been regarded as a major source of conflicts and congestion on motorways (Hounsell and McDonald, 1992; Evans et al., 2001). Traditionally, merging behaviour has been modelled based on the analysis of the headway distributions of the motorway traffic and gap acceptance by the merging traffic (e.g. Drew, 1967; Darzentas, 1981). However, for motorway merging, there is usually an acceleration lane running in parallel with the motorway along which vehicles can merge into the motorway traffic, i.e. there is continuous gap-acceptance and acceleration behaviour over the entire length of the acceleration lane (Michaels and Fazio, 1989; Zheng, 2002). Obviously, a merging operation involves two interactive traffic streams: the ramp traffic and the motorway traffic. Most existing models simplify the resulting complex dynamic interactive behaviour by assuming that merging traffic has no influence on the motorway traffic (e.g. Michaels and Fazio, 1989; Yang and Koutsopoulos, 1996). However motorway traffic has in fact been seen to exhibit a kind of cooperative behaviour by changing to the inner lanes upon approaching the merge section or by yielding to create gaps for the merging traffic (Hounsell and McDonald, 1992; Sarvi et al., 2002; Elefteriadou et al. 1995; Evans et al, 2001; Troutbeck, 2002; Bunker and Troutbeck, 2003). Such a cooperative process, which is also called "limited priority system", has recently received some attention. It may seem to help the joining traffic to merge, therefore reducing conflicts and increasing capacities.

282 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This paper presents a model of motorway traffic merging behaviour which tries to capture both the acceleration and the gap-acceptance behaviour of the merging traffic, and also the cooperative behaviour of the motorway traffic. The paper presents the model formulation, sensitivity analysis of some of the key model parameters, initial simulation tests on a standard merging section, and their comparison with observations.

INTERACTIONS IN A MERGING PROCESS AND A SIMULATION MODEL FRAMEWORK The behaviour at motorway merges referred to in the previous section is summarised in Figure 1. The driver of a merging vehicle takes into consideration two surrounding factors- the nearside motorway traffic actions and the remaining distance to the end of the merging lane and decides to accelerate or decelerate to maximize the chances of available gaps.

Fig. 1 The interactive behaviours between the merging traffic and the nearside motorway traffic.

Aware of the arriving merging traffic, the drivers in the motorway nearside lane may choose to ignore the merging traffic by continuing with their desired movements, or they may slow down to create gaps for the merging vehicle, or they may even change lane on the motorway to allow the merging traffic to join. The latter two behaviours are termed courtesy yielding and cooperative lane-changing respectively. Both courtesy yielding and cooperative lane-changing are reactions to the ramp traffic; ignoring these reactions will lead to an inaccurate description of the traffic phenomena and poor evaluation of the merging operations (Kita et al., 2002; Zheng, 2002). A simulation-based model of motorway merging behaviour has therefore been developed, aimed at capturing all the merging behaviours involved. Before describing the new merge model, we illustrate in Figure 2 a typical merge area, where a merging vehicle (C) interacts with its putative leader (PL) and putative follower (PF) on the nearside motorway lane. The merging vehicle will examine the original gap between PL and PF (the first motorway gap to be faced by C when it arrives at the acceleration lane as shown in Figure 2), the previous gap in front of PL, and the following gap behind PF. A PL (or PF) exists if the lead (or the lag) gap is less than 5 seconds.

A simulation model for motorway merging behaviour 283 The new merge model tries to capture the above behaviour through a number of sub-models described as follows: (a) Cooperation Model. Cooperative yielding behaviour and cooperative lane-changing are both modelled as a random decision made by the PL as to whether or not to reduce its speed to create gaps for C or to move to the offside lane(s). (b) An acceleration model. This models the acceleration or deceleration of C towards its target gap while maintaining a safe distance away from the vehicle in front in the acceleration lane. (c) A gap selection model. Based on its speed and location relative to its PF and PL, the driver of vehicle C will select a target gap to merge. (d) A gap-acceptance model. Here the acceptable lead and lag gaps are calculated as a function of the speed, merging driver's reaction time and maximum deceleration of vehicle C, PF and PL by considering the forecast of their actions in the merging process. (e) A merge model. When an acceptable gap is found, vehicle C merges into the motorway traffic. However, if the vehicle has not found an acceptable gap before reaching the end of the acceleration lane, a merge failure will be registered.

Fig. 2 A schematic diagram of a merge area. Using the symbols in Figure 2, we introduce the following notation to be used in this paper: at (t) the acceleration of vehicle i at time t (m/s2); bi

the maximum deceleration of vehicle i, negative value (m/s );

b[j

the most severe braking of vehicle i as perceived by vehicle,/, a negative value (m/s2);

v; (t) the speed of vehicle i at time t (m/s); v'u(t) the speed of vehicle i following its current traffic condition as perceived by vehicle j at xt (t)

time t (m/s); the position of vehicle i at time t (m);

Lt

the length of vehicle i (m);

284 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction T; 6 gz Gz

the reaction time of the driver of vehicle i (s); a flag indicating whether courtesy yielding is provided by PF, where 6=1 indicates a positive yielding by PF and 6=0 otherwise; the lead (z=Lead) or lag (z=Lag) distance gap (m); the lead (z=Lead) or lag (z=Lag) time gap (s).

The framework of the simulated merging process is displayed in Figure 3, and the model formulation of each of the sub-models is described in the following sub-sections.

Fig. 3 Simulation framework of merging interaction process.

Model of cooperation by the motorway traffic This model attempts to capture the cooperative behaviour of the PF, which includes courtesy yielding and cooperative lane changing behaviour. These cooperation effects can facilitate the merging manoeuvres by creating gaps for the merging vehicles. The decision on whether or not to provide cooperative lane-changing is drawn randomly from a binomial distribution with a given probability of a/, the value of which can be obtained from direct video observation of the apparent cooperative lane-changing movements. In the current model, such a decision is made at every instant of time by the motorway drivers in the merging area where there is a merging attempt in front. A binomial distribution is applied here as it is able to represent these two possible decisions. In the simulation, the cooperative lane-changing can be perceived by C by means of a flag indicating whether lane-changing is occurring or not (in the

A simulation model for motorway merging behaviour 285 real world, the direction indicators of the lane-changing vehicle) and will re-set its new current gap to that between the new PL and PF. Similarly, the decision on courtesy yielding is a constrained binomial distribution of a rate 05, subject to the current Giag being within the range Gmin ^ GLag < Gmax. The two limits, Gmin and Gmax are applied to represent the circumstances where no courtesy yielding is necessary: these are when the current Giag is either too small (below 0.25 second) or too large (above 4.0 seconds) based on the observation of Junction 11 of the M27 motorway, UK (Zheng, 2002). Hence the courtesy yielding value indicator (8) can be generated as: [Bin (a,) [0

for

G - < G,


otherwise

In the simulation, once a driver decides to show courtesy yielding, he/she will begin to follow vehicle C instead of his/her current leader on the motorway, and the car-following rule will be applied accordingly. Courtesy yielding is perceived by driver C by noticing an apparent change in the relative speed to vehicle PF. A constant deceleration is assumed by vehicle C in its estimation of the speed of PF. Hence the speed of vehicle PF, as estimated by C will be: [vPF{t) v'c(t)

'

=\

PFW

W(0 + bPFTc

for

0=0

for

0=1

J

(2)

Model of acceleration of the merging vehicle The driver of the merging vehicle in the acceleration lane needs to conduct multiple tasks: to look for potential gaps on the motorway; to keep a safe distance away from the vehicle in front on the acceleration lane to avoid collision; and to keep an eye on the remaining length of the acceleration lane to avoid running into the end. Thus, three factors influence C's acceleration behaviour: (1) target gap on the motorway; (2) current leader on the acceleration lane and (3) the remaining distance to the end of the acceleration lane. The influence of the target gap Generally, the speed of the merging traffic is slower than that of the through motorway traffic (Michaels and Fazio, 1989; Kita, 1993). When it arrives at the acceleration lane, the merging vehicle will first accelerate to increase its speed and aim to reach the speed of the motorway traffic. If its speed is greater than that of the motorway traffic, the driver of vehicle C will decelerate to adjust its speed and position so as to fit in the gaps. Eq. (3a) represents the

286 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction acceleration of vehicle C needed to reach the speed of the PL; while eq (3b) gives the deceleration of vehicle C at a rate controlled by the relative speed and gap with its PL. mm(a-c(t),amMc a

= Kx " " ^ " " ^

c(>) =

for

^

mm(ac(0,CK^

^

vc(t)
(3 a )

2

'

^

for v rW >v,, W

(3b)

where a+max and a"max are the maximum acceleration and deceleration of a passenger car (listed in Table 1); the values for HGVs being three-quarters of those of a passenger car (Goodman, 2001). Kx is a driver aggression factor which is uniformly distributed in the range of (0,1). We broadly clarify the driver types according to Kx values as described in Table 2. This parameter is used to model the behaviour whereby more aggressive drivers may be able to adjust their speeds and positions more rapidly than more timid drivers. Table 1 The acceleration and deceleration of passenger cars for different speeds (ITE, 1999) 0-32 Speed (km/hr) 32-48 48-64 64-80 80-96 >96 Max acceleration a ^ (m/s2) 1.4 2.4 1.8 1.6 1.4 2.0 Max deceleration amax (m/s2) 4.9 Table 2 The applied values of driver factor Kx uniformly distributed related to driver type X Driver type Very timid Timid Average Aggressive Very X Aggressive Kx 0-0.2 0.2-0.4 0.4-0.6 0.6-0.8 0.8-1 When the speeds and longitudinal positions of the merging vehicle and the motorway nearside vehicle(s) become very close, a situation termed as closing situation, equation (3) will yield a very small acceleration of the merging vehicle which may prevent it reaching its desired gap. To overcome such a problem in the model, the acceleration of the merging vehicle will be such as to create a larger lag (eq. 4(a)) if it is in closing situation with PF, or larger lead gap (eq. 4(b)) if with PL. If vehicle C is in closing situation with respect to both PL and PF, eq. (4b) will be applied. if \VPF-Vc\
) =

mm(0;«),(C,)

0 ;(0 = - g ,

2[A? ( y

" ' ''t('):X'(')"Z'")]

if

\Vn-Vr\<\Vand

\XFL - Xc - Ln\ < AS

(4b)

The speed of the merging vehicle C as a result of interaction with PF and PL will be: vc°(t + Tc) = vc(t) + ac(t)zc

(5)

It should be noted that if PF and PL do not exist (according to the definition given previously), then vehicle C will merge into the nearside traffic as soon as it arrives at the merging area and

A simulation model for motorway merging behaviour 287 will move according to the normal car-following rule used for the motorway traffic. The logic of the acceleration control (when C is interacting with PF and PL) is summarised in Figure 4.

Fig. 4 Acceleration control for C interacting with PF and PL

The effect of other vehicles in the acceleration lane and its remaining length If there is a vehicle (L) in front of vehicle C in the acceleration lane, C will try not to collide with L. In addition, driver C will continuously monitor the distance to the end of the acceleration lane so as not to overrun it. In the model, a maximum acceptable deceleration (be) is applied as an urgency effect on C to prevent it from running into either the vehicle in front or the end of the acceleration lane. The value of be is related to the distance to the end of the acceleration lane, subject to the vehicle's maximum deceleration a^. The driver aggressiveness factor is also applied here, hence the proposed equation for be is as follows: bc=msx(-Kx

2

^ (*m

~

x

,-<*„„)

(6)

c)

where xmax is the location of the end of the acceleration lane. The maximum deceleration be is used as a pressure for the merging vehicle to accept a smaller gap to the preceding vehicle on the acceleration lane and to the PL on the motorway. Applying be to the Gipps collision avoidance model (1981), the speed of vehicle C following its leader in the acceleration lane is calculated as: 4(t + tc) = bcrc+^bc2tc2-bc{7[xL(t)-LL-xc(t)]-vc(t)rc-vL2(t)/bLC}

(7)

Then the actual speed of vehicle C on the acceleration lane which is constrained by both the traffic on the motorway and that in front in the acceleration lane is given by: vc(t + TC) = mm{v"c(t + Tc),vbc(t + TC)}

(8)

288

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

This speed is also used as the forecast speed of C in selecting acceptable gaps, as described in the next sub-section.

The model of gap selection and acceptance By default, the target gap is assumed to be the original gap according to Zheng's observation (2002). However, the merging driver may choose to select a different gap as the target gap in different situations: a fast moving merging vehicle may overtake PL on the acceleration lane and choose to take the previous gap as its target gap; conversely, a slow moving merging vehicle may then choose to target the following gap; alternatively, if neither lead nor lag gap is acceptable, the merging vehicle will have to wait for other gaps. The gap-acceptance model developed here has adopted the game theory idea proposed by Kita et al. (2002), where the merging vehicle C chooses its best action by considering its forecast of the other drivers' actions (in terms of their speed changes and cooperation) and its own actions (in terms of its speed and acceptable braking). By applying be as a pressure for vehicle C to accept smaller gaps, and with the application of a gap acceptance sensitivity parameter /3, the acceptable lead and lag gaps can be derived from an interpretation of the Gipps car-following model (1981) as follows:

^(O=^[^-^^- + 2, c v^,(O + v w ( ,, r c ]

W

The gap acceptance sensitivity parameter /3 ( < 1 ) which directly affects the lead and lag thresholds (eqs. (9)), is applied based on the finding (Wilson, 2001) that the gaps modelled by Gipps' car-following model are generally longer than observed. The effects of this parameter on the merging process will be discussed in detail in the later section on sensitivity tests. To account for human errors and variability in driving proficiency, a normal distribution is applied to the acceptable lead and lag (eq. (10)) with g'Lmd (t) and g~Io (t) being the mean values and a2 the variance, limited by a lower boundary gmm at 4.5m (Zheng, 2002). It should be emphasized that this variability refers not to that across driver population, but only to the inability of the driver to maintain a constant gap due to driving proficiency or human error.

glead it) = max {N{gUad (tW),g™}

(10a)

gLag(t) = maxMg i 0 S (0 V),g"""}

(1Qb)

A simulation model for motorway merging behaviour 289 Note from equations (9) and (10) that the higher the absolute value of be (when vehicle C is closer to the end of the acceleration lane), the smaller the lead threshold, and that the lower the perceived speed of PF (for example when courtesy yielding is provided by PF), the smaller the lag threshold. It can also be seen that the longer the reaction times, the higher the value of lead and lag thresholds and that the lower the gap acceptance sensitivity parameter, the smaller the lead and lag thresholds. The merging model When an acceptable gap is found, vehicle C will merge into the motorway traffic and be placed behind PF; otherwise, C will adjust its speed and look for another gap. If however, on reaching the end of the acceleration lane, the vehicle C has not found an acceptable gap, it will be removed from the acceleration lane and a merge failure will be registered. This treatment of the merging failures, though subjective, is consistent with the observation that merging drivers very rarely stop at the end of the acceleration lane (Zheng, 2002 and our own video observations of the merging traffic on the M8 motorway, UK).

MODEL SIMULATION AND SENSITIVITY TESTS

Experimental design and model performance Simulation tests were conducted on a 500-metre long merge section with a multi-lane motorway and one parallel acceleration lane (Figure 5). Both the motorway traffic and the ramp traffic enter the test section at point X=0, 100 metres before the start of the actual merge area and the acceleration lane (point Xmjn in Figure 5). In the tests, only the merging traffic on the acceleration lane and the traffic on the nearside lane of the motorway are simulated. It is assumed that the traffic on the other lanes of the motorway will not change to the nearside lane during the merge section, and hence will not have direct interaction with the merging traffic. Any nearside vehicle which chooses to carry out cooperative lane-changing will simply be removed from the simulation. Both the motorway and the ramp traffic arrive at the simulated section randomly; their headways follow a negative exponential distribution with mean equal to the inverse of the average arrival flows. The nearside motorway traffic arrives with an average flow qm° and an entrance speed vm°. The ramp traffic is generated from an average flow qr° and enters the section with an initial speed

290 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction vr°. The simulation is updated every 0.2 seconds, and the default simulation parameter values are listed in Table 3.

Fig. 5 Motorway merging configuration used in simulation. The simulated section starts from an arbitrary point 100m upstream of the merge area, denoted as X—0; all positions are measured relative to this point.

Table 3 The default values of the model parameters. Denotation

Parameters

Default value

Reference 2

The max deceleration of vehicle /

. . _ -2Ar(1.7,0.3 )-3.0.

Gipps(1981)

perceived by vehicle./ (m/s2) bPF

The maximum deceleration of vehicle

-4

Gipps(1986)

Zheng (2002); Brackstone et al

PF (m/s2) (AS, AV)

Closing thresholds (m, m/s)

(4.5, 2)

g™1

Lower boundary of gap acceptance

4.5

Zheng (2002)

(2002)

thresholds (m) ?c

Reaction time of driver C(s)

0.4

Toledo (2003)

Lace

Length of the acceleration lane (m)

147

Junction 27, M8, UK

(q r U ,v r u )

Arrival merging traffic

(600, 55)

flow and

speed (veh/hr, km/hr) (q m °,v m °)

Detector information of Junction 27, M8, UK

Arrival motorway traffic flow and

(1500,90)

speed respectively (veh/hr, km/hr) <*i

Probability

of

cooperative

lane-

0.07

Video observation of Junction 27, M8, UK

changing (^2

Probability of courtesy yielding

0.2

13

Gap acceptance sensitivity factor

0.5

Assumption

A simulation model for motorway merging behaviour 291 The simulation outputs a selection of the different gaps taken by the merging vehicle to illustrate the performance of the gap acceptance behaviour modelled, and how smooth the merge is. Figure 6 presents examples illustrating merges in which an original, a following and a previous gap were accepted. Figures 7-9 show the speed-distance-time profiles of three different merging operations. The above three examples demonstrate the ability of the model to represent different merging behaviours including forced merging (Figure 7), merging under courtesy yielding (Figure 8) and cooperative lane-changing (Figure 9). In the next section, sensitivity tests are carried out on some of the key model parameters to investigate the model's response to the changes in parameter values. Sensitivity analysis This section reports the results of a sensitivity analysis of some of the key model parameters as listed in Table 4. The parameter values have been systematically varied in the simulation tests. The indicators used in the sensitivity analysis are the numbers of successful merges which took the original gap, the following gap, and the number of failed merges. Figures 1 0 - 1 2 show the average results of the simulation tests, presented as percentages of the total number of merges.

Table 4 Parameters for sensitivity tests Parameter

Default Value

Test Range

Test Increment

0

0.5

0.2-1.0

varies

«i

0.07

0.0 - 0.4

varies

Oil

0.2

0.0-0.8

0.2

i«,(m)

147

50 - 250

50

t(s)

0.4

0.4 ~ 1.0

0.2

q" (veh/hr)

600

200 ~800

300

vm" (km/hr)

90

80-120

varies

?m °(veh/hr)

1500

900-1800

300

The results show that lower values of the gap-acceptance factor (/3) leads to more original gaps being taken, fewer failed merges and fewer merges taking the following gap. This is intuitively correct as the parameter jS directly affects the gaps acceptable for the merging vehicles (via equations (9)-(10)). However, the effect of |3 becomes less significant when its value is higher than 0.8. The analysis shows that the model is sensitive to the cooperative lane-changing rate a.\. When cooperative lane-changing is more common, merging vehicles are more likely to take the original gap and less likely to take the following gap or fail to merge.

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Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 6 The space-time trajectories of a merging vehicle (in broken lines) and of its PL and PF vehicles on the nearside motorway (solid lines) where the merging vehicle accepted (a) the original gap, (b) the following gap and (c) the previous gap.

Fig. 7 The profiles of (a) speed-time, (b) distance-time and (c) speed-distance of the merging vehicle and those of its PL and PFs during a forced merge. Points A and B are the corresponding points where a forced yielding was executed.

A simulation model for motorway merging behaviour 293

Fig. 8 The profiles of (a) speed-time, (b) distance-time and (c) speed-distance of the merging vehicle and those of its PL and PFs with courtesy yielding provided. Points A and B are the corresponding points where a courtesy yielding was executed.

Similarly, with a higher probability of courtesy yielding, more merges take place using the original gaps and fewer merges with the following gaps. Here, the merge failures are less sensitive to courtesy yielding when the yielding rates are higher than 0.2. A possible explanation for this is that excessive courtesy yielding can improve the chances for the merging drivers to accept the original gaps but does not help those unskilled drivers who have poor acceleration control. The results show that the longer the reaction time, the lower the percentage successfully merging. The analysis shows that the chance of taking an original gap does not vary significantly with the length of the acceleration lane; however, the chance of merging failures decreases and the chance of accepting a following gap increases with increasing length of acceleration lane. The model is also found to be sensitive to the length of the acceleration lane and to the average motorway flow. It appears that the longer the acceleration lane and the lower the motorway traffic speed, the fewer the merging failures and the greater chance that a following gap will be chosen. In order to achieve smooth merging (taking the original gaps), it appears that the optimal acceleration lane length (for all motorway traffic speeds), is approx 100m. With acceleration lanes longer than 100m, the percentage of vehicles taking original gaps drops, because more

294 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction vehicles tend to take the following gaps (as shown in Figure 11 (c)). These results are consistent with simulation tests of the standard motorway design (with typical design speeds and associated acceleration lane lengths as applied in the UK, Table 5).

Fig. 9 The profiles of (a) speed-time, (b) distance-time and (c) speed-distance of the merging vehicle and those of its PL and PFs with cooperative lane-changing. Points A are the corresponding points where the cooperative lane-changing was provided.

Simulation tests using the design speeds and merge lengths of Table 5, suggest that, with the design merge length, merging failures are generally below 15% of the total merging attempts and that with higher design speeds and longer merging lengths, more vehicles tend to take the following gaps and fewer take the original gaps. This result may have some practical implications for motorway merging design: taking smooth merging (i.e. taking original gaps) as the main criterion, the optimum length of acceleration lane is approx 100m; taking merging failure as the main criterion, a longer acceleration length such as 200m is required. That is, the design may be different depending on which criterion is chosen. Table 5 Design speed and the associate length of the acceleration lane (DMRB, 2001) Design Speed (km/hr) 85 100 120

Merging Length (m) 90 110 130

A simulation model for motorway merging behaviour

295

Fig. 10 Distribution of the percentages of merging failure, successful merging into the original gap and into the following gap in the sensitivity tests on (a) gap acceptance sensitivity value /3, (b) cooperative lane-changing fraction au (c) courtesy yielding fraction a2, (d) reaction time Tc and (e) the length of acceleration lane.

Figure 12 shows that the accepted lead and lag gaps decrease as the flows on the motorway increase. However, they do not seem to vary with increasing mergmg traffic flows. Both of these results are in agreement with real-world observations (Zheng, 2002). The effects of the ramp merging flow and the motorway traffic flow on successful merging are shown in Figure 13. In contrast with Figure 12, these results suggest that the percentage of

296 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction successful merges decreases with increasing nearside motorway traffic flow and also with increasing merging traffic flow. This is because the greater the motorway traffic flow, the smaller are the gaps for the merging vehicles, hence the lower the merging success rate. To summarize, although the ramp flow traffic has an insignificant effect on the accepted gap values (Figure 12), it does affect merging success because the higher the ramp flow, the lower the merging opportunities, hence the lower the percentage of merging success.

Fig. 11 The effects of acceleration lane length and motorway arrival speed on the distribution of (a)merging failure, (b) successful merging into the following gap, and (c) merging into the original gap-

Fig. 12 Variation of the accepted lead and lag gaps with motorway and ramp flows.

A simulation model for motorway merging behaviour 297

Fig. 13 Successful merging with motorway and ramp flows.

CALIBRATION OF THE MERGING MODEL Calibration is the process of quantifying the model parameters using empirical data, and making reasonable adjustments to match the model's output with real-life observations (May, 1990). Some parameters can be obtained from the literature or field observations. For those parameters whose values can not be obtained easily in this way, simulation runs with a range of feasible values are performed and the results are compared with observation (Kosonen, 1999). The parameter values producing the best fit are then chosen as the calibrated ones. The calibration of the proposed merging model was conducted on a real-world merge section using data collected from Junction 11 on the M27 motorway during a two-hour morning peak period (07:00-09:00) between 21st May and 17th July 2001 (Zheng, 2002). The observed data listed in Table 6 were used as input to the model calibration. The model parameters to be calibrated here are the gap acceptance factor (/3) and the drivers' reaction time (ic). Table 6 Observed data from Junction 11, M27 motorway (Zheng, 2002) Parameters Lace (m)

v,° (km/hr) vrau (km/hr) qru (veh/hr) qm° (veh/hr) «i

Percentage HGV flows (%)

Parameter Values 182 72 86 932 1000 6.63% 0 (courtesy yielding was not considered in Zheng's study) 5

Varying the values of the reaction time and the gap acceptance factor, the simulated percentages of those merging vehicles successfully taking the original gaps (denoted as Pfm) have been obtained for each test run i. Thus, the calibration step can be formulated as an optimization problem which seeks to minimize the difference between the observed percentage of successful

298 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction merges using the original gap and that simulated. A range of values of @ and TQ have been tested. The set of parameter values 0=0.4 and TQ =0.4S produced P,slm = 84% which was found to be the closest to that observed (at 87%). Based on the above calibrated parameter values and those fixed as the inputs, the modelled accepted lead and lag gaps were compared with those observed. The observed lead and lag gaps were within 0.3 and 0.4 seconds accuracy respectively (Zheng, 2002). Table 7 lists the observed and simulated distributions of the accepted gaps in cumulative percentiles; the comparison is shown in Figure 14. Table 7 shows that the averaged simulated lead gap is 1.75 seconds, a mere 0.2 seconds higher than that observed, whilst the simulated average lag gap is the same as the observed one. A regression of the individual percentiles gives R2 values above 0.9, indicating that the distributions of the simulated lead and lag gaps compare well with those observed.

Fig. 14 Cumulative distributions of simulated lead and lag gaps versus the real observation.

Figure 15 displays the simulated individual accepted lead and lag gaps as a function of the relative speeds between the merging vehicle and its PF and PL on the motorway compared to the thresholds derived by Zheng (2002) (eq. (11)). Most of the accepted gaps from the proposed merge model are above the thresholds calculated by Zheng's model with only a few shorter gaps. This is because this new merging model considers factors such as the merging conditions (i.e. forecasts of the actions of vehicle PF and PL and C, the urgency effects) as well as the relative speeds considered in Zheng's gap acceptance model (eq. (11)). L

=6.7553 x e - 0 - 2 6 8 4 ( " » - ^

glag = 5.3472 x e °- 5 I M " - - < >

for for

vc-vPL>0 vPF - v c > 0

A simulation model for motorway merging behaviour 29 Table 7 The statistics of the simulated gap structure versus the observation by Zheng (2002). Lead (s)

Lag (s)

Time Gap Simulation

Observation

Simulation

Observation

Observation No.

903

79

903

79

Percentiles

10

0.32

0.55

0.48

0.32

20

0.69

0.71

0.82

0.52

30

1.04

0.87

1.14

0.76

40

1.36

1.19

1.43

1.23

50

1.62

1.36

1.69

1.76

60

1.89

1.73

1.95

2.24

70

2.21

2.21

2.29

2.73

80

2.53

2.80

2.61

90

3.14

3.08

Multiple R

0.968

0.982

Regression

R2

0.936

0.965

Statistics

Adjusted R^

0.926

0.958

Std. Error

0.206

Average value

1.75

Std. Deviation

1.17

0.131 1.52

1.81

1.81

1.10

Fig. 15 The simulated acceptable gaps versus Zheng's gap acceptance model.

CONCLUSIONS A micro-simulation model of motorway merging has been developed based on the behaviour of not only the drivers of the merging vehicles but also those on the motorway. To mimic the dynamic interactions between the motorway and merging vehicles, the model includes the gap selection, acceleration and deceleration behaviour of the merging traffic on the acceleration lanes,

300 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction as well as the cooperative behaviour of the motorway traffic in terms of gap creation and lane changing. Sensitivity tests have been carried out on key model parameters; the results suggest that the model responds well to changes in these parameter values. Simulation tests on a standard UK merge section were carried out and the results compared with those observed. The results indicate that the model can correctly reproduce the observed gap distributions very well While the existing parameter values from the literature have been evaluated and the range of feasible values illustrated in this paper, we have not set out specifically to directly estimate the proposed model parameters. Considerable advances have been made recently both in the development of sophisticated estimation models and in testing alternative theories against field data (e.g. Brackstone et al., 2002; Brilon et al., 1999; Hagring, 1998). The next step in our research is to apply some of the advanced estimation methods in the calibration of the model using data collected at inter-urban and urban motorway merges. Research is also planned to investigate the effect of cooperative merging on the stability of motorway traffic flow.

ACKNOWLEDGEMENTS The authors would like to thank the staff of NADICS, Glasgow, UK for their continuous cooperation during the research on driving behaviour in motorway merging sections. Jiao Wang deeply appreciates the financial support to her PhD study by a scholarship from the Overseas Research Students Award Scheme, UK, and the University of Leeds.

REFERENCES Brackstone, M., Sultan, B. and McDonald, M. (2002). Motorway driver behaviour: studies on car following. Trans Res F, 5, 31-46. Brilon, W., Koenig, R., Troutback, RJ. (1999). Useful estimation procedures for critical gaps. Trans Res A, 33, 161-186. Bunker, J. and Troutbeck, R. (2003). Prediction of minor stream delays at a limited priority freeway merge. Trans Res B, 37, 719-735. Darzentas, J. (1981). Gap acceptance: myth and reality. Proc 8th Int Symp on Transportation and Traffic Theory (eds. V. F. Hurdle, E. Hauer and G. N. Stewart), Toronto, 175-192. DMRB (2001). Design Manual for Road and Bridges. Vol 6(2), Highways Agency, London. Drew, D. R. (1967). Gap acceptance characteristics for ramp-freeway surveillance and control. Highway Res Rec, 157, 108-134. Elefteriadou, L., Roess, R. and McShane, W. (1995). Probabilistic nature of breakdown at freeway merge junctions. Transp Res Rec, 1484, 80-89.

A simulation model for motorway merging behaviour 301 Evans, J., Elefteriadou, L. and Gautam, N. (2001). Probability of breakdown at freeway merges using Markov chains. Transp Res B, 35, 237-254. Gipps, P.G. (1981). A behavioural car-following model for computer simulation. Transp Res B, 15,105-111. Gipps, P.G. (1986). A model for the structure of lane-changing decisions. Transp Res B, 20, 403414. Goodman, P. (2001). The Prediction of Road Traffic Noise in Urban Areas. PhD Thesis, University of Leeds, UK. Hagring, O. (1998). A further generalization of Tanner's formula. Transp Res B, 32, 423-429. Hounsell, N. and McDonald, M. (1992). An Investigation of Flow Breakdown and Merge Capacity on Motorways. Contractor Report 338, Transport Research Laboratory, UK ITE (1999). Transportation and Traffic Engineering Handbook. Institute of Traffic Engineers. Kita, H. (1993). Effects of merging lane length on the merging behaviour at expressway onramps. Proc 12th Int Symp on the Theory of Traffic Flow and Transportation (ed. C.F. Daganzo), Berkeley, 37-51. Kita, H., Tanimoto, K. and Fukuyama, K. (2002). A game theoretic analysis of merging-give way interaction: a joint estimation model. Proc 15th Int Symp on Transportation and Traffic Theory (ed. M.A.P. Taylor), Adelaide, 503-518. Kosonen, I. (1999). HUTSIM- Urban Traffic Simulation and Control Model: Principles and Applications. Helsinki University of Technology, Publication 100. May, A.D. (1990). Traffic Flow Fundamentals. Prentice Hall, New Jersey. Michaels, R. M. and Fazio, J. (1989). Driver behaviour model of merging. Transp Res Rec, 1213, 4-10. Sarvi, M., Ceder, A. and Kuwahara, M. (2002). Modelling of freeway ramp merging process observed during traffic congestion. Proc 15th Int Symp on Transportation and Traffic Theory (ed. M.A.P. Taylor), Adelaide, 483-502. Toledo, T. (2003). Integrated Driving Behaviour Modelling. PhD thesis, MIT, USA. Troutbeck, R. (2002). The performance of uncontrolled merges using a limited priority process. Proc 15th Int Symp on Transportation and Traffic Theory (ed. M.A.P. Taylor), Adelaide, 463-482 Wilson, R.E. (2001). An analysis of Gipps' car-following model of highway traffic, IMA J Applied Mathematics, 66, 509-537. Yang, Q. and Koutsopoulos, H. N. (1996). A microscopic traffic simulator for evaluation of dynamic traffic management systems. Transp Res C, 4, 113-129. Zheng, P. (2002). A Microscopic Simulation Model of Merging Operation at Motorway OnRamps. PhD thesis, University of Southampton, UK.

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FREEWAY RAMP MERGING PROCESS OBSERVED IN CONGESTED TRAFFIC: LAG VEHICLE ACCELERATION MODEL Majid Sarvi, Department of Civil Engineering, Monash University, Clayton Campus, Victoria, Australia; Avishai Ceder, Transportation Research Iinstitute, Civil and Environmental Engineering Faculty, Technion-Israel Institute of Technology, Israel; and Masao Kuwahara, Kuwahara Lab., Institute of Industrial science, University of Tokyo, Japan

INTRODUCTION This paper is the second of a pair of papers discussing the modeling of freeway ramp merging behaviour under congested traffic conditions. The first paper focused on the modeling of the merged vehicle (from the ramp) and was presented in the 15th ISTTT (Sarvi et. al. 2002) while this paper develops a model for the lag vehicle (approaching the ramp area from the freeway) to provide a complete picture of driver interaction behaviours at the time of merging. A four years extensive study has been undertaken to investigate traffic behaviour and characteristics during traffic merging processes under congested traffic flow in order to design a safer and less congested merging points. The overall research approach is illustrated in Figure 1, emphasizing the second component, representing this work. The acceleration and merging process from an entrance ramp to freeway lanes constitutes an important aspect of freeway traffic operations and ramp junction geometric design. Competing traffic demands for space influence this process regarding both the ramp freeway junction and the upstream freeway lanes. A merging driver must make a series of decisions and carry out control tasks, all within the driver capability to process the roadway and traffic information and interpreted it into speed and position control responses.

304 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 1. Overall research flowchart with emphasis on the component of this work It is believed that if an available gap is acceptable, the driver on the ramp accelerates and merges directly (minor interaction with corresponding freeway vehicles). If no gap is available the driver may accelerate to create a merge opportunity, or decelerate and wait for later gap (strong interaction with corresponding freeway vehicles). Influencing this complex driver behaviour are both internal factors, such as driver attitude and vehicle characteristics, and external factors, which may include freeway and ramp speed, lane changing manoeuvres, relative positions of merging vehicles, and proximity of the merging vehicle to the merge end. The acceleration-deceleration characteristics of vehicles involved in merging manoeuvre (freeway leader, freeway lag and ramp vehicles) are essential components in microscopic simulation modeling for simulating the merging freeway entrance ramp. The major objective of this study was to investigate freeway lag vehicle merging behaviour in terms of relative speed and spacing between the lag vehicle and its corresponding ramp and freeway lead vehicles, with respect to various entrance ramp types. Furthermore, to develop a methodology that can be used to model freeway lag vehicle acceleration-deceleration behaviour during freeway merging manoeuvre under congested traffic flow. In doing so it builds on the content of the earlier work in which a general methodology of modeling ramp vehicle acceleration was presented (Sarvi et. al. 2002). For this purpose sets of data capturing a wide range of information were collected using videotape and image processing techniques together with a sophisticated instrumented vehicle. Lag vehicle merging position in conjunction with freeway leader and ramp vehicle was analysed for the both parallel and taper type entrance ramps. Additionally, lag vehicle positions with respect to ramp and freeway leader vehicle speed as well as relative speed and time gap between a lag vehicle and ramp and freeway vehicles were

Lag vehicle acceleration modelfor freeway ramp merging 305 examined. The methodology presented in this work uses the stimuli-response equation as a fundamental rule to model a lag vehicle acceleration-deceleration behaviour. The time-base car following model was modified to a nonlinear stimuli-response equation to capture the characteristics of the collected traffic data. It is believed that the developed acceleration model of lag vehicles, in addition to the ramp driver acceleration model proposed in earlier work (Sarvi et. al. 2002), constitutes a complete picture of driver interaction behaviours at the time of merging under congested traffic conditions. The findings of this work may serve future highway geometric design criteria revision. The developed acceleration-deceleration model can also be used to upgrade the merging vehicles acceleration component of existing microscopic freeway simulation models. Finally the mathematical framework of the developed model may be appropriate to study more driver behavioural aspects of congested freeways.

STUDY APPROACH The study begins with a literature review and continues with observations of vehicle operational characteristics in freeway entrance ramps at several locations to establish a solid overview of the traffic flow characteristics during a freeway merging process. Preliminary methodologies for modeling lag vehicle merging behaviour were then developed conceptually based on knowledge gained from the literature review and on-site observation as well as the objectives to be achieved. In order to best describe the interaction between lag vehicle acceleration-deceleration characteristics under congested situation, traffic data were collected and analysed. During periods of heavy freeway congestion, unstable or stop-and-go traffic flow appears and makes the gap acceptance structure extremely difficult to interpret. In such situations, a ramp driver enters the freeway either by forcing a merge or by accommodating a lag of a freeway vehicle. Consequently, lag vehicle acceleration-deceleration characteristics behaviour cannot be treated as normal and stable behaviour and it is greatly influenced by the ramp and freeway leader vehicles. The collected field data, such as freeway and ramp vehicle speeds, acceleration-deceleration rates, and merging positions were used to identify and quantify key variables for use in the lag vehicle freeway merging model. The field data were applied to the established driver behaviour concepts in order to examine the proposed methodologies for lag vehicle freeway merging behaviour. Verification and validation of the developed acceleration model has been carried out through utilizing an instrumented vehicle (specially designed and developed for obtaining driving behaviour data at merging sections) together with a simulation program.

306 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

FREEWAY MERGING PROCESS ANALYSIS It is believed that lag driver merging behaviour is significantly influenced by the geometric configuration of the entrance ramp and the surrounding freeway and ramp vehicles. Despite the abstractness of ramp driver behaviour, the merge position and gap acceptance can be clearly defined and observed. The data collected in this study is based on the Ichinohashi and Hamazaki-bashi merging points. Figures 2 and 3 demonstrate each entrance ramp.

Figure 2. An image of Ichinohashi merging section.

Figure 3. An image of Hamazaki-bashi merging sections.

Lag vehicle acceleration modelfor freeway ramp merging 307 Under light traffic flow condition the vehicles, which enter a merging section, start to search for a gap considering their relative positions in terms of spacing and relative speeds. Gap searching and acceptance manoeuvre under free flow traffic condition has been extensively studied and documented (Drew et al. 1967, Daganzo 1979, Makigami et al. 1988, Chang et al 1991, Kita 1993, 1998, Ahmed et al. 1996, 1999, Kurian 2000). However, based on a comprehensive macroscopic study and observations in the Tokyo Metropolitan Expressway (Sarvi et al. 1999, 2001) gap searching and acceptance manoeuvres do not occur under heavy traffic flow conditions. According to these macroscopic studies, no significant correlation was found between the acceleration lane length and maximum flow rate of merging sections. In addition, it is found that under heavy traffic situations a squeeze merging can be observed at the end of the merging section. Here this type of merging is defined as a zip merging, which means that the ramp vehicles and those in the freeway shoulder lane merge together one by one regardless of the length of available gap. Based on observed percentage of zip merging at Ichinohashi and Hamazaki-bashi merging sections under congested traffic flow it was found that more than 97% of merging manoeuvres are of zip merging type (Sarvi 2000a). Similar merging behaviour was observed by Sarvi et al. (2004) through Driving Simulator experiments. Therefore, in this study the gap searching and acceptance manoeuvre will not be addressed. The following section examines the relative speed between lag vehicles and the corresponding ramp and freeway leader vehicles, as well as the relative spacing between lag vehicles and ramp and freeway leader vehicles (See Figure 2). Additionally a summary of statistical t tests is given in Table 1.

Relative speed between lag vehicles and Ramp vehicles

Figures 4-a and 4-b show the data between the freeway lag vehicle acceleration and its relative speed for freeway lag vehicles and Ramp vehicle. The relative speed measured with respect to the ramp vehicles is defined as follows: Relative Speed (Vrampflag)=ramp vehicle speed- freeway lag vehicle speed Relative speed is one of the most important variables that affect the freeway merging behaviour. Under free flow situation if a ramp vehicle enters the acceleration ramp with a higher speed than the freeway lag vehicle, a positive relative speed, the ramp driver can then merges smoothly in front with minor interaction with the freeway lag vehicle. In contrast, based on Figures 4-a, and 4-b under congested condition there is a strong interaction between ramp and the freeway lag vehicle in terms of competing for the limited available space. This interaction was also shown through the acceleration behaviour of ramp driver by Sarvi et al (2002). The statistical test results shown in Table 1 agree with that of the graphical presentation in Figure 4.

308

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 4-a. Freeway lag vehicle acceleration rate vs. relative speed for ramp vehicles and freeway lag vehicles at Ichinohashi merging section.

Figure 4-b. Freeway lag vehicle acceleration rate vs. relative speed for ramp vehicles and freeway lag vehicles at Hamazaki-bashi merging section.

RELATIVE SPEED BETWEEN FREEWAY LAG VEHICLE AND ITS CORRESPONDING FREEWAY LEADER VEHICLE Figures 5-a and 5-b show the data between the freeway lag vehicle acceleration and its relative speed for freeway lag vehicles and freeway leader vehicles. The relative speed measured with respect to the freeway lead vehicles is: Relative Speed (Vfleadflag)=freeway lead vehicle speed- freeway lag vehicle speed

Another perspective of the freeway ramp merging behaviour is the acceleration-deceleration characteristics of freeway lag vehicles regarding the behaviour of their corresponding freeway

Lag vehicle acceleration modelfor freeway ramp merging 309 lead vehicles. Based on Figures 5-a and 5-b and similar to the conventional car-following model (where the speed of a freeway lag vehicle is lower than its freeway leader vehicle) a positive relative speed, will accelerate and tries to minimize relative spacing. However, if the speed of a freeway lag vehicle is higher than its freeway leader vehicle (with a negative relative speed) it doesn't need to accelerate and consequently will decelerates in order to avoid collision. Figure 5-c shows a 3D least square fitted surface to the relative speed of the freeway lag vehicle and its freeway leader and ramp vehicles at Ichinohashi merging section. As expected the highest freeway lag acceleration rate is achieved when the freeway leader and ramp vehicles have higher speed than the freeway lag vehicle. The statistical test result shown in Table 1.

Figure 5-a. Freeway lag vehicles acceleration rate vs. relative speed for freeway lag vehicles and freeway lead vehicles at Ichinohashi merging section.

Figure 5-b. Freeway lag vehicles acceleration rate vs. relative speed for freeway lag vehicles and freeway lead vehicles at Hamazaki-bashi merging section.

310 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Freeway lag vehicle acceleration Vs. relative speed between the lag vehicle and its freeway lead and ramp vehicles

Figure 5-c. 3D least squares fitted surface plot of freeway lag vehicle acceleration vs. relative speed between the freeway lag vehicle and its freeway leader and ramp vehicles at Ichinohashi merging section.

SPACING BETWEEN FREEWAY LAG VEHICLE AND ITS CORRESPONDING RAMP AND FREEWAY LEADER VEHICLES Figures 6-a shows the data between the freeway lag vehicle acceleration and its spacing for freeway lag vehicles and ramp vehicles. The spacing measured with respect to ramp vehicles were defined as follows: Spacing (Srflag)=(Spacing between the lag vehicle and ramp vehicle) (Desired spacing as a function of speed) Figure 6-b shows the data between the freeway lag vehicle acceleration and its spacing for freeway lag vehicles and freeway lead vehicles. The spacing measured with respect to freeway lead vehicles were defined as follows: Spacing (Sfleadflag)=(Spacing between the lag vehicle and freeway lead vehicle) (Desired spacing as a function of speed)

Lag vehicle acceleration model for freeway ramp merging

311

Figure 6-a and 6-b represent a spring action of spacing in which the follower accelerates being sucked ahead when the spacing is larger than the desired value. In other words, if the follower feels unsatisfied with his spacing, too long or too short this will inspire him driving faster or slower to keep or recover a comfortable spacing he wants. In this study the desired spacing is referred to as the spacing, which is a function of the speed and is obtained by fitting a parabolic curve to the observed speed of freeway lag vehicles and the space between the freeway lag vehicle and its corresponding leader vehicle (freeway leader or ramp vehicle). The statistical test result shown in Table 1.

Figure 6-a. Freeway lag vehicles acceleration rate vs. spacing between the ramp vehicles and freeway lag vehicles at Ichinohashi merging section.

Figure 6-b. Freeway lag vehicles acceleration rate vs. spacing between the freeway leader vehicles and freeway lag vehicles at Ichinohashi merging section.

312

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Table 1. Summary results of statistical t test Calculated t statistic Parameter Relative speed between freeway lag and ramp vehicle Relative speed between freeway leader and freeway lag vehicle Spacing between ramp and freeway lag vehicle Spacing between freeway leader and freeway lag vehicle

slope 11.55

Critical t value (5% significance level) 1.647

conclusion

intercept 9.47 8.11

16.61

1.647

Dependent

2.51

7.5

1.647

Dependent

2.32

6.51

1.647

Dependent

Dependent

GENERAL BEHAVIOR OF FREEWAY LAG VEHICLES IN THE MERGING AREA The behaviour of ramp driver in the acceleration lane was discussed in earlier work. Similarly a freeway lag driver in the vicinity of merging area performs different tasks in a timesharing mode particularly in interaction with the ramp vehicle. These behaviours vary across drivers and the outcome of these variations could depict the fundamental aspects of the entrance ramp operations. Following is an overview of the observed phenomena during real merging operations emphasizing the behaviour of the freeway lag vehicle. Speed data

The primary data of interest is the speed of freeway lag vehicles negotiating the merging section. It provides the speed change profile during the ramp merging manoeuvre, indicating where and with what magnitude vehicles were accelerating or decelerating in interaction with freeway leader and ramp vehicles. Speed data were calculated by measuring the travel distance between sequential image intervals (0.2 second image interval is used in this study) where a vehicle moves from one image to the next. Figure 7 shows the calculated freeway lag vehicle average speed profile based upon distance from the twenty meters prior to the physical nose. The curve illustrates that on an average sense a freeway lag driver has a higher speed than ramp in the first part of the acceleration lane. This could be interpreted as the result of a competition for the available space between the ramp and the freeway lag vehicles. Then in an average sense either the freeway lag driver decelerates (to accommodate the merging ramp vehicle and avoid collision) or the ramp driver accelerates (to force a merging) on the first step of the freeway ramp merging manoeuvre. Following this step the driver continuously accelerates and follows his leader. For the remaining freeway ramp merging manoeuvre the freeway leader and ramp vehicles have higher speed than the freeway lag vehicle. In addition, this figure shows a clear speed decline due to the interaction between the ramp vehicle and its freeway lag and leader vehicles during the freeway ramp merging manoeuvre. It is interesting to note that similar merging behaviour was observed through the use of a Driving Simulator as it is reported by Sarvi et al. (2004).

Lag vehicle acceleration modelfor freeway ramp merging 313

Figure 7. Average freeway lag vehicle and corresponding freeway leader and ramp vehicles speed profile (Ichinohashi merging section).

METHODOLOGY FOR MODELING FREEWAY LAG DRIVER ACCELERATION BEHAVIOR The movement of a vehicle along a freeway entrance ramp involves successive navigational decisions, pursuit tracking, positional control relative to other vehicles and roadway elements, and steering into the freeway stream. The acceleration-deceleration characteristics for each task are different. At the ramp entrance drivers normally have to adjust their speed in order to accommodate the controlling conditions of the ramp curvature and super elevation and to make a decision of possible lane changing into the adjacent ramp lane. As the ramp vehicle approaches the merging end in a free flow condition, the traffic in the freeway merging lane becomes visible and allows for the ramp driver to begin evaluating available gap. However, in a congested traffic situation the ramp driver stays in the queue of vehicles and waits to force a merging with strong interaction with freeway leader and lag vehicles. This procedure may involve deceleration, acceleration reduction, maintaining a constant speed, maintaining a current acceleration, or acceleration increase according to the existing congested condition as it is observed and analysed by Sarvi and Kuwahara (1999) and Sarvi (2000a). A modified form of the conventional car following models to accommodate the complicated nature of a ramp vehicle acceleration-deceleration performance on acceleration lane was introduced by Sarvi et al (2002). The theoretical framework for modeling lag driver accelerationdeceleration behaviour is built on the content of this work in which a general methodology of modeling ramp vehicle acceleration was presented. It uses the stimuli-response concept as a basic rule and is formulated as a modified form of the conventional car-following models. Essentially, the basis for modeling the freeway lag vehicle acceleration deceleration behaviour is different from that of the conventional car-following model as was identified in

314 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction earlier work. Nevertheless, the fundamental psychophysical concept, Driver Response(t+T) = Sensitivity factors(t) * Stimulus ft) where t is the time and T is the reaction time, of the carfollowing models can still be appropriately adopted as long as the stimuli can be well specified. As it is explained in the precede sections for congested traffic situations, the relative speed between the freeway lag vehicle and its freeway lead and ramp vehicles as well as the spacing between the freeway lag vehicle and its freeway lead and ramp vehicles can be identified as the main stimuli. In general, the manoeuvre of a freeway lag vehicle is mainly influenced by its freeway lead, and ramp vehicle. Significant behavioural differences across different traffic conditions are expected. Four stimuli are considered for evaluating the lag vehicle driver response: relative speed regarding the freeway leader, relative speed regarding the ramp vehicle, spacing regarding the freeway leader, and the spacing regarding the ramp vehicle. The equation for the follow the leader car-following model is expanded linearly to incorporate the influence of both the freeway lead and the ramp vehicles. Herman and Rothery (1963) have proposed a similar concept with regard to a three-car car-following situation. The hypothesized expression of freeway lag vehicle acceleration-deceleration behaviour is given as follows: V" (t + T) ««* (t + T) = a0+ a, ^ \ ' [VFlead (t) - VFlag (t)] lA Flead (') ~

+

+

A

Flag V)i

V,,..(»-'*..(.)]- |S " ) '- /lv " )11

^.(«)-L,(,)f 1S('W1>(')»

(1)

Where: a Fiag (t + T) : Acceleration rate of the freeway lag vehicle at time t+T (m/s2) XR{f) : Location of the ramp vehicle at time t (m) : Xmead (0 Location of the freeway lead vehicle at time t (m) : ^nag (0 Location of the freeway lag vehicle at time t (m) VR (t) : Speed of the ramp vehicle at time t (m/s) : VFiead (0 Speed of the freeway lead vehicle at time t (m/s) VFlag (t) : Speed of the freeway lag vehicle at time t (m/s) S(t), = XFkad (t) - XF]ag it): Spacing between the freeway lag vehicle and the freeway S(t)1

leader vehicle at time t (m) = XR (t) - XFlag (t): Spacing between the freeway lag vehicle and the ramp vehicle at time t (m)

/[v(0]

: Desired spacing as a function of speed (m)

Lag vehicle acceleration model for freeway ramp merging 315 T

: Time lag or driver reaction time (s)

ao,al,a2,a3,a4,m,ll,l2,l3,l4

are parameters to be estimated.

Eq. (1) has a nonlinear form. However, by assigning constant values to some of the parameters, Eq. (1) can be transformed to a linear form. A nonlinear and linear regression technique is performed in order to calibrate the parameters in Eq. (1) for different T values.

ACCELERATION MODELS The data collected in this study is not large enough to cover all the geometric characteristics. Instead, it can demonstrate general phenomena in order to gain more understanding about the freeway merging process under congested traffic conditions. Based on available data at the Hamazaki-bashi and Ichinohashi locations the following scenarios are analysed for calibrating the proposed model (See Figures 2 and 3). Scenario 1: Hamazaki-bashi including inner ramp lane and outer ramp lane Scenario 2: Ichinohashi and inner ramp lane of Hamazaki-bashi Scenario 3: Inner ramp lane of Hamazaki-bashi Scenario 4: Outer ramp lane of Hamazaki-bashi Scenario 5: Ichinohashi Scenario 6: Ichinohashi and Hamazaki-bashi Calibrating linear acceleration model The calibration of linear acceleration-deceleration models for scenarios 1 to 6 are based on Eqs. (2) and (3): xym (t + T)]

»V,,S--r..wr'r'-w-''-M

aFIV(t + T) = a0

+al[VFlead{t)-VFlag{t)-\+a2[VR{t)-VF,ae{t)} + « 3 {5(0, -f[v(t)]} + a4{S(t)2 - / [ v ( 0 ] }

(3)

In this study the magnitude of b in Eq. (2) is chosen as 2, 0 and the value of m is set to 0,1,2 and unfixed parameter. All possible combinations of the explanatory variable components of Eqs. (2) and (3) are examined individually. The statistical analysis software, STATISTICA, was used to solve the linear regression problems. The analysis performed for T= 0 second, 0.33 second, 0.66 second, 1.3 second, 2 second, and 2.67 second. The model that has the largest R value is chosen.

316

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

The best fitted linear acceleration-deceleration model for scenarios 1-6 is given by a n , (t + T) = "0-3 + 0.43[FFta, (0 - VFlag (0] + 0.45[K8 (0 - VFlag (t)] + 0.12{5(0,-/[v(0]} (4) with the R = 0.65 and T= 0.66sec. The evidence that the best model is obtained when T, the reaction time, is equal to 0.66 second supports the intuitive knowledge that there exist a time lag between the time a freeway lag driver detects the stimuli and response time. The positive sign of second and third terms in Eq. (4) illustrates, assuming all other elements remain constant, that the freeway lag vehicle decelerates while approaching its freeway lead and ramp vehicles in order to create sufficient space and avoid collision. Additionally, the statistical F and t test showed the importance of all explanatory variables in Eq. 4. The F value of 83.45 proved the overall significant relationship between the freeway lag acceleration rate and the set of explanatory variables suggested by Eq. 4 (critical F value = 2.60 for %5 significance level). The t values of intercept and explanatory variables are -4.89, 7.32, 7.12, and 2.11 which shows the significant effects of all explanatory variables on freeway lag vehicle acceleration rate (critical / = 1.647 for %5 significance level). Calibrating nonlinear acceleration model

The calibration of the nonlinear acceleration-deceleration model in Eq. (1) is estimated using a nonlinear regression procedure. Similar to the linear models all possible combinations of the explanatory variable components of Eq. (1) are examined individually. The best fitted nonlinear acceleration-deceleration model is: y0M (f +

T\

aFlas (t + T) = -0.24 + 0 . 3 3 2 — - ^ — — — [ F f M (*) - VFMg (/)] L-1^- Flead \ )

Flag V*)\

Vom(t + T) + J

' V,(»'"^ W ]- 1 F -" | -' / "' ( ' > ]

+ 0.16

{S(t)2-f[v(t)]}

(5)

[xR(t)-xFlag(t)V with the R = 0.73 and T= 0.66sec. The signs of the regression coefficients all have reasonable explanations. The positive sign of second and third terms in Eq. (5) illustrates, assuming all other elements remain constant, that the freeway lag vehicle decelerate while approaching its leader vehicle (either freeway lead or ramp vehicle). The magnitude of the third term is bigger than the second term due to the higher interaction between the ramp and freeway lag vehicle compare to the one of the

Lag vehicle acceleration modelfor freeway ramp merging 317 freeway lead vehicle (see Figures 4 and 5). Finally the positive sign of the fourth and fifth terms indicate that the freeway lag drivers always trying to maintain a desirable spacing based on their speed. These results are consistent with the phenomenon demonstrated in Figures 4 to 6. The inclusion of the freeway lag vehicle current speed as one of the explanatory variables is necessary due to its significant regression coefficient. In general, the results of the linear and nonlinear response models show acceptable consistency in both sign and magnitude. For example, the best models for all scenarios are found for T= 0.66sec and the freeway leader and ramp vehicles are present in the best models. The sign of the coefficients of the best linear and nonlinear models are all identical. Based on R-values, the nonlinear models, as expected, are slightly better than the linear models for all scenarios. The differences, however, are not significant. This insignificant difference indicates that the best linear acceleration-deceleration model is a good approximation as well as its capability to reproduce reasonably the interaction between the vehicles as is mentioned by Newell (1999). Verification of the Proposed Acceleration Model Using an Instrumented Vehicle and Simulation Program The validation of the proposed model has been performed utilizing a sophisticated instrumented vehicle and a simulation program. For this purpose the outcome of the proposed acceleration model has been used for the development of the simulation program (Sarvi et al. 2000b). The simulation program has been extensively and precisely calibrated and validated at microscopic and macroscopic levels using observed traffic flows, and lane changing manoeuvres at the Hamazaki-bashi, and Ichinohashi merging points where the traffic demand exceeds the capacity and resulting in upstream queues. The validation of the acceleration model has been carried out through a comparison of the speed of vehicles generated in the simulation program with those from the field data obtained by the instrumented vehicle. The instrumented vehicle is designed and developed to collect deliver behaviour data while driven with the traffic stream at merging sections by an advanced synchronized automatic image processing techniques (Sarvi et al. 2004a). The Video recording using two CCD cameras enabling on-board automatic image processing, utilizing an advanced number plate matching technique (see Figure 8). Employing the microwave radar and user-friendly developed software made it possible to precisely measure the lateral and longitudinal distance and speed between the instrumented vehicle and its corresponding freeway leader and ramp vehicles (running in the adjacent lane). The results presented in Table 2 show that the average speeds of the simulated freeway leader, Freeway lag, and ramp vehicles during the course of merging manoeuvre in two distinct sections, from the physical nose till the end of the zebra marking and from the end of the zebra marking till the end of merging lane, are in good agreement with the one from the instrumented vehicle. Furthermore, a comparison of the trajectories of vehicles generated in

318 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the simulation program with those from the instrumented vehicle is shown in Figure 9. The results show that the simulated and the observed trajectories are in good agreements. In other words, the slopes of the trajectory lines (speeds) of the simulated vehicles before and after the merging process are consistent with the observed ones. Additionally, a significant speed reduction immediately prior to the merging manoeuvre is observed in both the simulated and the observed trajectories. A comparison of average freeway lag vehicle acceleration profile of simulated and observed vehicles while negotiating the merging section at Ichinohashi is shown in Figure 10.

Figure 8. Onboard image processing system using two CCD cameras and number plate matching technique. Table 2. Comparison between simulation and collected data by instrumented vehicle Simulation results Speed (m/sec)

Freeway lag veh. Ramp veh. Freeway leader veh.

Collected by instrumented vehicle Speed (m/sec)

From physical nose till end of the zebra marking

From end of the zebra marking till end of the merging lane

From physical nose till end of the zebra marking

From end of the zebra marking till end of the merging lane

3.86 3.73 3.13

5.80 6.20 6.61

3.84 3.75 3.10

5.82 6.23 6.65

Lag vehicle acceleration modelfor freeway ramp merging 319

Figure 9. Simulated versus instrumented vehicle time-trajectories at Ichinohashi merging section (Each three lines include freeway lag vehicle and its freeway leader and ramp vehicles).

Figure 10. Average freeway lag vehicle acceleration profile of simulated versus instrumented vehicle at Ichinohashi merging section.

CONCLUSION The freeway merging manoeuvres represent a complex process and involve lane changing, acceleration, deceleration, and merging into a gap. This work presents a methodology for

320 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction collecting field data, analysing freeway merging behaviour data and developing and calibrating a freeway lag vehicle (approaching the ramp area from the freeway) acceleration models. Wide ranges of data were collected using videotape and image processing techniques. Comprehensive traffic surveys were conducted at two entrance ramps in the Tokyo Metropolitan Expressway. A theoretical framework for modeling the freeway lag driver acceleration-deceleration behaviour is presented. It is built on the content of the earlier work in which a general methodology of modeling ramp vehicle acceleration was introduced. This methodology uses the stimuli-response psychophysical concept as a fundamental rule, and is formulated as a modified form of the conventional car-following models. Data collected at the two merging points of the Tokyo Metropolitan Expressway are used to calibrate the hypothesized freeway lag vehicle acceleration models. It is found that the surrounding freeway leader vehicle and especially ramp vehicle affect significantly the freeway lag vehicle acceleration behaviour. Furthermore, the average speed of merging vehicles (freeway leader, ramp and freeway lag vehicle) of a simulation program, which has been developed using the outcome of the developed acceleration model, shows reasonable consistency with the collected data from instrumented vehicle. Finally, it is believed that the developed acceleration model of lag vehicles, in addition to the ramp driver acceleration model proposed in earlier work, constitutes a complete picture of driver interaction behaviours at the time of merging under congested traffic conditions.

ACKNOWLEDGMENT The authors would like to acknowledge the useful advice and discussion of Professor Morita from Nihon University and Mr. Nishikawa from Tokyo University for his technical assistance.

REFERENCES Ahmed, K., (1999). Modeling drivers acceleration and lane changing behavior. Ph.D. Thesis, Department of Civil and Environmental Engineering, MIT, Cambridge, MA. Ahmed, K., Ben-Akiva, M., Koutsopoulos, H., Mishalani, R., (1996). "Models of freeway lane changing and gap acceptance behavior". Traffic and Transportation Theory, J.-B. Lesort (ed.), Elsevier Science, 501-515. Chang, G., YM, C , (1991). "An empirical investigation of macroscopic lane changing characteristics on uncongested multilane freeways". Transportation Research 25 A. Daganzo, C , (1979). "Estimation of gap acceptance parameters within and across the population from direct roadside observation". Transportation Research 15B. Drew, D., LaMotte, L., Wattleworth, J., (1967). "Gap acceptance in the freeway merging Process". Highway Research Record. Herman, R and Rothery, RW (1963). "Car following and steady state flow". Proceeding of the Second International symposium on the Theory of Road and Traffic Flow, London, United Kingdom.

Lag vehicle acceleration modelfor freeway ramp merging 321

Kita, H., (1993). "Effects of merging lane length on the merging behavior at expressway onramps". Traffic and Transportation Theory, C. Daganzo (ed.), Elsevier Sci., 37-51. Kita, H., (1998). "A merging-give way interaction model of cars in a merging section: A game theoretic analysis". Transportation Research 33A. Kurian, M., (2000). Calibration of Microscopic Traffic Simulator. Master Thesis, Department of Civil and Environmental Engineering, MIT, Cambridge, MA. Makigami, Y., Adachi, Y., Sueda, M., (1988). "Merging lane length for expressway improvement plan in Japan". Journal of Transportation Engineering 114, No. 6. Newell, GF (1999). A simplified car-following theory: A lower order model. Institute of Transportation Studies, University of California at Berkeley, Research Report UCBITS-RR-99-1. Sarvi, M (2000a). Freeway ramp merging phenomena observed in traffic congestion. Ph.D. dissertation, University of Tokyo. Sarvi, M., Ceder A, Kuwahara M. (2002). "Modeling of freeway ramp merging process observed during traffic congestion". In: Taylor, M. (Editor.) Transportation and Traffic Theory, Elsevier, 483-502. Sarvi, M and Kuwahara, M (1999). "Comparative study on evaluation of merging capacity in Tokyo Metropolitan Expressway". Seisan-Kenkyu 51, 83-86. Sarvi, M., Kuwahara M. (2004a). "Development of an Experimental Vehicle and its Application to Investigate Freeway Ramp Merging Phenomena Observed During Traffic Congestion". Proceedings of the WCTR, Istanbul. Sarvi, M., Kuwahara, M Ceder, A. (2004). "A study on freeway ramp merging phenomena in congested traffic situation by traffic simulation combined with driving simulator". Journal of Computer-Aided Civil and Infrastructure Engineering 19, 351-363. Sarvi, M., Kuwahara, M, Ceder, A and Morita H (2001). "Macroscopic study of freeway ramp merging phenomena observed in traffic congestion". Journal of Eastern Asia Society for Transportation Studies. Sarvi, M., Kuwahara, M and Nishikawa, I (2000b). "A simulation model for evaluation of merging capacity on the Metropolitan Expressway". Seisan-Kenkyu 52, 104-107.

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17

A FIRST-ORDER MACROSCOPIC TRAFFIC FLOW MODEL FOR MIXED TRAFFIC INCLUDING MOVING BOTTLENECK EFFECTS Stephane Chanut, Laboratoire Ingenierie Circulation Transport LICIT (INRETS - ENTPE), Cedex, France

INTRODUCTION The Lighthill-Whitham and Richards (LWR) (Lighthill and Whitham, 1955; Richards, 1956) traffic flow model makes it possible to capture many traffic features. Its simplicity is a major advantage for numerous and various applications and its solution is well known. However, this model describes the traffic as a continuous and homogeneous flow which is assumed to be in equilibrium states. These hypotheses can be too rough for some traffic situations, in particular from a traffic management point of view. For example, the LWR model considers that all vehicles are identical and that all drivers behave in the same way: this can give a rather disappointing description of a heterogeneous traffic flow. Emerging continuum vehicular models attempt to improve the description of the traffic flow given by the LWR model. Two main methods are explored (Zhang and Lin, 2002): 1) disaggregating a heterogeneous traffic flow into more homogeneous classes and/or considering different lane types (multi-class or multi-lane models), 2) considering a dynamic relationship between density and speed instead of a fundamental diagram (non-equilibrium models). The research efforts presented here fall under the first method. Indeed, considering homogenous classes could preserve the efficiency of the LWR model for a homogeneous flow while taking into account the traffic composition. Moreover, trucks are numerous and share a similar behavior, which justifies modeling them as a fluid.

324 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Background A number of multi-lane or multi-class traffic gas-kinetic models have already been considered in the past (see for example (Helbing and Greiner, 1997; Helbing et al, 2001; Hoogendoorn and Bovy, 1998). But the first research on first-order multi-class traffic was carried out by (Daganzo, 1997). In this model, traffic is assumed to be composed of two kinds of vehicles: priority and regular vehicles, and lanes are separated into special and regular lanes. This first modeling approach is based on the use of different lanes: only special vehicles are allowed to use special lanes, and these vehicles use only these lanes while the speed in them is equal to or greater than the speed of the regular lanes. This leads to a solution following a Wardropiantype user equilibrium. This model is extended for vehicle types with different behaviors in (Daganzo, 2002). hi this second model, drivers are separated into fast (rabbits) and slow (slugs) drivers. As in the first model, only fast drivers can use fast lanes, and they use these lanes exclusively unless their speed is lower than the speed of the slugs in the shoulder lanes. Following these studies, multi-class traffic has become a very active research area and many first-order macroscopic models have been proposed to extend the LWR model to two or more vehicle types. Unlike Daganzo's works, these models ((Wong and Wong, 2002; Zhang and Jin, 2002; Zhu, Chang and Wu, 2003; Chanut and Buisson, 2003; Benzoni-Gavage and Colombo, 2003; Logghe, 2003) do not deal with the use of lanes (all vehicles can use all lanes) and describe the different flows from a first-order macroscopic point of view. All these new models capture the following feature of mixed traffic: (a)

the fast vehicles move faster than the slow ones.

Therefore, these models (except Zhang and Jin, 2002) can represent the faster flow of the fast vehicles. However, very few models (Zhang and Jin, 2002; Chanut and Buisson, 2003; Logghe, 2003) deal with a second very common feature: (b) capacity decreases according to the proportion of long vehicles such as trucks. This can be surprising, but more generally, the fundamental diagrams used in these models to describe the phenomenology of a heterogeneous traffic flow are not very strongly linked to empirical evidences. For example, it would sometimes be necessary to include the traffic feature (c): (c) the truck speed is steady in the free-flow regime. Above all, a last phenomenon has to be captured to describe such a mixed flow. Indeed, when the fast traffic overtakes the slow one, the passing flow can be limited due to the restricted road capacity imposed by the presence of slow vehicles. In this case, (d) the slow traffic can be considered as (moving) bottlenecks for the faster ones.

First-order macroscopic model for mixed traffic 325 Until now none of the heterogeneous models directly take into account such moving bottleneck effects for continuous flow. Indeed, only the effect of a single and punctual vehicle has been studied in the literature (see Leclercq, Chanut and Lesort (2004) for an overview of such models). New efforts to model such bottleneck effects still represent slow vehicles exogenously (Daganzo and Laval, 2005) or with a specific car-following model (Laval, 2004), but not as a fluid. In addition, the study of multi-population models is not only interesting for traffic management but also from a mathematical point of view. Indeed, this kind of multi-flow model shows several non-standard mathematical features: they could lead to certain unphysical solutions in the case of specific traffic flows. Main innovations of the proposed model The model proposed here is a generalization of the LWR model for a mixed traffic flow composed of two vehicle types differing in speeds and space occupancies: cars and trucks. It is based on the previous model presented in (Chanut and Buisson, 2003), which already represents the mixed traffic features (a) and (b). The first aim of the paper is to specify the analytical solution of the model: a system of two conservation laws. In particular we will address the problem of the non-standard mathematical features caused by the traffic phenomenology, when we include the empirical assumption (c). The second aim of this paper is to extend the model by taking into account the moving bottleneck effects (d) caused during overtaking, and to show the new capabilities of the model thanks to this extension.

PHENOMENOLOGY OF HETEROGENEOUS TRAFFIC In a first-order macroscopic traffic flow model, fundamental diagrams describe the phenomenology involved. We build two fundamental diagrams for cars and trucks in order to highlight two differences between them: trucks are slower and longer than cars. Heterogeneous traffic and space occupancy It is obvious that a heterogeneous traffic flow composed of trucks and cars occupies more space than a homogeneous traffic flow composed of cars only: it is necessary to take into account the longer size of trucks. In order to express this greater space occupancy, it is possible to weight the trucks' density (Bliemer, 2000). This weighting cannot be identical for all traffic states: space occupancy can depend on vehicle length but also on speed and/or driver behavior (Hoogendoorn and Bovy, 2000). However, it is rather difficult to express such a variation precisely. Therefore, we chose to simply represent the truck effect on space occupancy by defining only variable critical and jam densities of the heterogeneous traffic.

326 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Jam density for mixed traffic In a traffic jam state where all vehicles are stationary, we can consider that the space i , used by each vehicle is composed of its own length /, and of a minimal space d between it and the previous vehicle (we will assume that d is constant). Thus, we formulate the jam density for one lane according to the partial densities k\ andfeas follows (1 is for cars, 2 for trucks): *m.x(*l,*2)=TTr^r k\L\ + K2L2

(1)

Critical density for mixed traffic We assume that the critical density for which traffic becomes congested is a steady ratio of the jam density: kcrit (khk2~) = a kmax (kuk2) (2) Such a definition for jam and critical densities is similar to the model of (Zhang and Jin, 2002). Moreover, equation (2) can be justified by empirical data (see Figure 1 (a)). These data were collected on the French motorway A7; they are based on measurements by dual loop detectors. The ratio of densities was obtained by analyzing the occupancy of the detector; more details can be found in (Chanut, 2004). Figure 1 (a) shows that a can be assumed a constant. Heterogeneous traffic and speeds In general, trucks are slower than cars. The fundamental diagrams must represent this difference in speeds between the two kinds of vehicles. Moreover, speeds must decrease when the total density increases and be nil when the density is the jam density. Otherwise we can imagine that, for the same total density (with a weighted truck density), speeds are lower when the traffic is heterogeneous than when it is homogeneous. This fact is related to the perturbations due to lane changing and overtaking. We decided to model this phenomenon in a specific extension of our first model, and we chose not to include it in a first step in the fundamental diagrams. Congested states Many authors assume that in congested states the speed is the same for trucks and cars (see (Bliemer, 2000; Zhang and Jin, 2002; Logghe, 2003). Indeed, in a congested regime there is no particular reason that cars could go much faster than trucks. Therefore, we assume that in a congested regime all vehicles have the same speed. Free-flow states As for the free-flow regime, we assume that the car speed decreases from the car free speed Vfi to the critical speed Vc, and that the truck speed decreases from the truck free speed Vp. to the same critical speed Vc: we consider that all vehicles are in congested states or that all vehicles are in the free-flow regime. Studying data helps us to define the shape of the curve between zero density and the critical density (see Figure 1 (b)):

First-order macroscopic model for mixed traffic 327

Figure 1: Lessons from data (a) kcrj/kmax ratio versus truck ratio (b) Car and truck speeds versus density

The data presented in Figure 1 (b) were collected on a level section of the French motorway A9 and are based on individual measurements. They show that we can assume a linear decrease of the vehicle speeds versus density (Chanut, 2004). Furthermore, they indicate that another assumption can be made about the truck speed: it can be considered to be steady until traffic becomes congested. However, to avoid a loss of generality, a (slight) linear decrease of the truck speed with the density can be assumed. Multi-class fundamental diagrams Based on the aforementioned heterogeneous traffic features regarding densities and speeds, we propose the following fundamental diagrams for a truck ratio p = -—*—— : k +K

Figure 2: (a) Fundamental relationships: speeds-density for a given truck ratio p (b) Flow-density for heterogeneous traffic

We chose a curve shape for congested states which gives a linear decrease of flows versus density. On Figure 2 (b) we can see the decrease in capacity and in jam density when the truck ratio increases. Thanks to these diagrams the mixed traffic model can represent the lower capacity and the lower jam density due to trucks.

328

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

SOLUTION TO THE MIXED TRAFFIC FLOW MODEL Some properties of the model Presentation By considering two vehicle classes, we have to solve a 2x2 system of conservation laws: one for each vehicle type. We note r, = £,£, the weighted densities and A'the number of lanes. So the system we have to solve is: fl(ftlj) I dp, 8x ^ 8t +

dx

dt

(3)

with speeds defined as presented in Figure 2 (a): v,(P\,P".) =*/i - 3ii»\ + f t ) y( ft. ft ) = rt

"

—

:

1

1 — a \ p[ +

ft

^ ft + /* e[0.o-V]

(fluid)

if p, - ft e laN,N]

(congested)

I

ft are constants. If the truck speed is assumed to be steady for free-flow traffic states, ft is nil and Vf2 is equal to Vc. The model can be written as: ^_ ft | St ft)

dp, 3(ftU;)

dp, fl(fti-;)

3pi

^?ft

p

d_t* ftrtft

(°\ |oj

(5)

J<7>

Solutions to such a system call upon the mathematical theory of the hyperbolic systems of conservation laws (see (LeVeque, 1992) for a complete description of the mathematical features). Study of the eigenvalues The evolution of the system is governed by the eigenvalues of the jacobian matrix, or flux matrix, J(p). From a traffic theory point of view, eigenvalues stand for the speeds of the information: changes in speeds or in traffic composition. They are similar to the characteristics of the LWR model. They can be expressed as: VP> = l U i i + J ; M - I / ( J , I - 7» i : --W.,,7;, | t

X>
+J2i + 4{ Ju - J2, f + 4J, ; Jj,)

where Jij is the term (ij) of the jacobian matrix.

(6)

First-order macroscopic model for mixed traffic 329 Figure 3 shows the evolution of the eigenvalues with the car density, for a given truck density.

Figure 3: Eigenvalues versus car density for a given truck density

In the fluid regime, we made the assumption that the flux is quadratic. For such a flux (see Benzoni-Gavage and Colombo, 2003) the following properties can be proved: *>(p)<«k(p)

™*

\
(7)

This property is a first guarantee for the anisotropic feature of the model (Zhang, 2003): no traffic information can propagate faster than the traffic carrying it. Moreover, in our case it is also possible to show (Chanut, 2004) that: Vi(p)
< MP>

an

<*

0 < \ ( p ) < <MP)

®

This means that all the vehicles travel more quickly than the first information, whereas only the cars travel more quickly than the second information. In the congested regime, the expression of the eigenvalues can be simplified. The first is a negative constant for all congested traffic states and the second is equal to the speed of the traffic flow and therefore is constant for traffic states with the same value of p=p\+P21- a A

r

== -TT-f-Z 1 - a\ ft + P2

(9)

l)=<'>0 I

This means that in the congested regime the first information always travels backwards, whereas the second information travels exactly with the vehicles. Moreover, the eigenvalues are ordered in all cases (fluid and congested regimes): (10) A|(p)
330 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

—A[ + -Jt J> — J-| -t r

'

( P ) =

JL4-J

—\y rt

I

^

(

P

) =

Y

+ Jii

J

-h J\ -t

J

"

A simpler choice is possible for congested states, with: ft ii
1 «

E,ip) =

(12) In the congested regime, this means that the first change in the flow maintains the traffic composition, whereas the second change maintains the density. Mathematical properties: hyperbolicity of the problem In order to build the solution by applying efficient mathematical tools, it is necessary to check if the studied system is strictly hyperbolic. A system is said to be strictly hyperbolic if the jacobian matrix J is diagonalizable and if its eigenvalues are real and distinct. In general this is the case for our model, which maintains good mathematical properties. However, the two eigenvalues can sometimes collide and the system is said to become resonant. This case arises for a free-flow traffic state composed of cars only (located on the axis p2 = 0). This traffic state is called an umbilical point. It can be expressed as (p\umb, 0) with: a

ft —vn ~ V'

(13) Around such a point, and due to the loss of hyperbolicity, the unicity of the solution is not guaranteed. Such a problem must be addressed. Moreover, if we assume that the truck speed is steady in the free-flow regime, an umbilical line, which generalizes the previous problem, appears in the phase plane (pi, pi) between the traffic states (p\umb, 0) and (0, pc). Such a situation is very similar to the critical traffic curve or transition curve presented by (Jin and Zhang, 2003) for an inhomogeneous road. We will see further on that we can guarantee the unicity of the solution even in this case. Finally, a main property of the model is the monotonous variation of the eigenvalues along the integral curve of the associated eigenvector. In our case, this variation is actually negative in the free-flow regime (genuine nonlinearity) and becomes nil for congested states (linear degeneracy). This point ensures the good behavior and use of the rarefaction waves in our model. Solutions to the Riemann problem In this section, we study the Riemann problem with the following jump initial conditions (traffic states are constant on both sides of the discontinuity):

First-order macroscopic model for mixed traffic 331

(ft ftKS 0)=

' ' l ( ^ ) **>*

(14)

Solving such a problem makes it possible to solve any traffic flow problems. General The solution is built with the Lax-curves of each traffic state. These curves are composed of shock waves and rarefaction waves. In the phase plane (pi, P2), shock waves describe the traffic states which can be connected by discontinuities, whereas rarefaction waves describe traffic states which can be connected by a continuous variation of the traffic state. Shock waves The /-shock wave (z=l or 2) of a traffic state pl is the curve linking in the phase plane (pi, pi) traffic states pr verifying the Rankine-Hugoniot condition with p':

(?i<pr)-9i{p;) = * U -A) 3s£K

l * ( p " ) - « ( F ) = «(/>* -A) satisfying the extended - Lax entropy condition:

\s>MPa)

(16)

l

and tangent to the eigenvector r, in p . Two traffic states on the same shock wave can be connected by a simple discontinuity. The shock speed is between the eigenvalues of the two involved traffic states. For the studied system it is possible to prove, in particular for the fluid regime states, that the entropy condition leads to an increase in the value of p. This point is important: the properties (8) make it possible to prove that the second requirement highlighted in (Zhang, 2003) to ensure the anisotropic property of the model is satisfied: shock speeds are lower than traffic speeds of the connected states (see (Chanut, 2004) for more details). Rarefaction waves The /-rarefaction curve (2=1 or 2) of a traffic state p1 is the part of the integral curve of the eigenvector r, along which the eigenvalues are increasing. Two traffic states located on the same rarefaction curve can be connected by a continuous variation of traffic states along the integral curve linking them. Contact waves If the eigenvalues of the two states connected by a rarefaction (or a shock) wave are the same, the wave is called contact wave because the eigenvalue is a constant along the rarefaction wave (or the shock speed is equal to the eigenvalue).Thus, the Lax-curves are particular for a congested traffic state. Indeed, for such a congested state pl, in the congested part of the domain the 1-Lax curve is a 1-contact wave linking all congested states with the same traffic composition as pl (see the eigenvector rl); the 2-Lax curve is a 2-contact wave linking all congested states with the same traffic speed as pl (see the eigenvector r2) (see Figure 5 (b)). Such a feature was already encountered in the model of (Zhang and Jin, 2002).

332 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Solving the Riemann problem hi general, in a system of equations, two traffic states are not directly connected by a shock or a rarefaction wave. Therefore, solving the Riemann problem means finding an intermediate state I-p'"' belonging to a 1-curve of L=p' and such that R=pr belongs to its 2-curve. Figure 4 presents this solution.

Pi

Figure 4: Solving the Riemann problem with an intermediate state I

Using a 1-curve before a 2-curve is related to the fact that eigenvalues can be ordered (see eq. (10)). Lax curves: examples The following figures (Figure 5) show Lax-curves for two traffic states A (free-flow regime) and B (congested state).

Figure 5 : Lax curves for (a) a fluid traffic state and (b) a congested state

A traffic state located on the shock wave (Si or S2) of the traffic state A (or B) can be connected with the state A (or B) by a simple discontinuity. A traffic state located on the rarefaction wave (Ri or R2) of the traffic state A (or B) can be connected with the state A (or B) by a continuous variation of traffic states along this rarefaction curve (Ri or R2) linking them. For a congested state, Lax-curves are contact waves as mentioned before. Figure 5 shows that, in general, the first waves lead rather to a change in densities (and therefore in speeds) whereas the second waves lead rather to a change in truck ratio.

First-order macroscopic model for mixed traffic 333 Typical Riemann problems The properties of eigenvalues and therefore of the system are very different if the traffic state is congested or not. Here we indicate the solution of two typical Riemann problems with an upstream state L congested or not and a downstream state R congested or not: a backward propagation of a jam and a dissipation of a jam. In the case of a jam propagation (see Figure 6), the traffic state R is located on the jam density line, and the traffic state L is in free-flow regime. The analytical solution requires a congested intermediate state I; it is composed of a 1-shock whose speed is negative and of a 2-contact whose speed equals zero. Therefore, the solution represents the extension of the jam (state I) backwards (1-shock) and a change in traffic composition (2-contact). Indeed, inside the jam extension (state I), the truck ratio is lower than in the initial jam (state R) and the initial free-flow traffic (state L). This phenomenon can be explained by the fact that the jam propagation traps the traffic in the composition in which it was flowing before. Therefore, this composition has to be lower in I than in L because of the greater speeds of cars in the freeflow traffic: for the state L, the truck ratio is lower inside the flow than inside the densities.

Figure 6: Jam propagation: analytical solution (a) Phase plan (b) Space-time diagram

Figure 7: Jam dissipation: analytical solution (a) Phase plan (b) Space-time diagram

334 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In the case of a jam dissipation after a traffic light for example (see Figure 7), there are no vehicle in the traffic state R, and the traffic state L is located on the jam density line. The analytical solution gives a traffic state F, which is located on the capacity line p= pc: this stands for the maximal flow we can observe at the traffic light when it turns green. F is connected with L by a 1-contact and has the same truck ratio as L. Then the solution gives a 1-rarefaction wave, which stands for the traffic dispersion, until a state I, in which there are no more trucks: the cars have outdistanced them. Then a 2-rarefaction wave leads to the state R, without any vehicle: this wave stands for the dispersion of the fast car cluster. Particular traffic flows and non standard mathematical features Umbilical points We have seen before that eigenvalues can collide and break the hyperbolicity of the system. Such a situation occurs in our model for the traffic state (p\umb, 0) located on the cars axis and called umbilical point. Around it, the shape of the Lax curves could lead to several solutions and above all to an unexpected and unphysical intermediate state (see Figure 8 (a)). Indeed, we expect that two homogeneous traffic states composed of cars only can be connected by a shock or a rarefaction wave exactly as in the LWR model; this is the case for all couples of two homogeneous traffic states. However, for two fluid states located on both sides of the umbilical point, another solution can be mathematically possible:

Figure 8: (a) An intermediate unphysical state could be possible (the cross stands for the umbilical point) (b) but this traffic state would have no spatial extension

This example shows that it is possible to link the two initial car homogeneous traffic states L and R by a single shock (S2CL)), but also by a heterogeneous intermediate state I with trucks! This unrealistic solution is composed of two shocks: Si(L) then S2(I). In fact, for a quadratic flux such as the one we assume for the fluid regime, (Benzoni-Gavage and Colombo, 2003) obtained the technical proof that the speeds of the two shocks are the same and equal to the speed of the shock S2CL) which directly links the two states L and R.

First-order macroscopic model for mixed traffic 335 Therefore, the intermediate state I has no spatial extent and in the phase plane you cannot see the difference with the "physical" solution. Transition line when introducing a new traffic feature We now assume that a traffic feature our model must reproduce is the fact that the truck speed is steady during fluid traffic; in this case a generalization occurs of the problem pointed out above. Indeed, a transition line appears between the umbilical point (p\"m , 0) and the point (0, pc) (see Figure 9 (a)). On this line, the two eigenvalues are equal and the system is resonant. Otherwise, in free-flow regime, one of the two eigenvalues is a constant and is equal to the truck speed V2. It is Xj on the left of the transition line, \ on the right. As the truck speed is not altered by the vehicle density, it is obvious that two fluid traffic states with the same truck density (and therefore the same truck flow) can be directly linked by a shock or a rarefaction wave (see equation (15)): an example is given in Figure 9 (b). Yet, as around the umbilical point, it is to be feared that another less physical solution could be mathematically possible. However, an overview of the shape of the Lax curves for such points L and R shows that such an intermediate state does not exist (the shock curve S2 (R) is composed of all traffic states which could connect R by a 2-shock) (see Figure 9 (b)). We can see that the shock curve S2*(R) of the state which is closer to the transition line is "included" inside the shape of the shock curve Si(L) of the other state. Therefore, these curves cannot cross each other and the solution remains unique. This result can be generalized (see (Chanut, 2004) for more details).

Figure 9: Truck speed is assumed steady in the free-flow regime (a) Transition curve (b) No other intermediate state can be found

336 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

INCLUDING MOVING BOTLLENECK EFFECTS Perturbation due to overtaking The mixed traffic flow model presented above allows us to reproduce differences in terms of spatial occupancy or kinematics capability for each vehicle category and their consequences on the behavior of the global flow. Nevertheless, it is also necessary to consider the direct interactions between vehicles, such as the perturbations due to overtaking: feature (d) of the introduction part of the paper. Indeed, it is not certain that the passing lanes have a sufficient capacity to allow all cars to flow and to pass trucks with the predicted speed. Therefore, our goal in this part is to extend our model in order to take into account such a phenomenon. Defining a moving bottleneck area in the phase plane Moving bottlenecks have already been studied in the literature (see Gazis and Herman, 1992; Newell, 1998; Lebacque, Lesort and Giorgi, 1998; Munoz and Daganzo, 2002), and grouped together in a common frame by (Leclercq, Chanut and Lesort, 2004). In these models, the obstruction is a single slow vehicle, considered as punctual and exogenously modeled. The models try to extend the LWR model to represent the effect of this slow vehicle on the surrounding flow. In particular, they first describe the traffic states which are disturbed by the slow vehicle, and then model the new flows upstream, downstream and at its level. A specific bottleneck diagram is used to represent the flow at the level of the slow vehicle. In our case, the moving bottleneck effects are caused by a continuous slow flow, which is described endogenously. However, the methodology followed by the models describing the effect of a single slow vehicle can still be used. Thus we first look for a description of the traffic states (pi, p2) for which the car speed and flow are such that the capacity of the passing lanes is not sufficient. This defines in the phase plane (p\, pi) an area we call the "moving bottleneck area", whose definition was given in (Chanut and Leclercq, 2003) in a first approach. Then in a second step we give a new phenomenology for such traffic states. Disturbed states for traffic with a single slow vehicle In the case of a single slow vehicle, it is possible to show that the passing flow is kept on both sides of this moving bottleneck. Graphically, the traffic states upstream, downstream and level with the slow vehicle are linked by a straight line whose slope is the slow vehicle speed V/,. This line is called the capacity line (see Figure 10). When a queue appears, the bottleneck is said to be active and the passing flow is maximal. Therefore the straight line is necessarily tangential to the bottleneck diagram (Newell, 1998; Lebacque, Lesort and Giorgi, 1998). Thus the traffic states which are perturbed by the slow vehicle are those located on the dotted line portion of the fundamental diagram (see Figure 10). Disturbed states for the mixed two-flow model In our case the moving bottlenecks are a continuous flow. To follow the method of the LWR model extensions above, we need a bottleneck diagram to represent the passing flow et the level of the slow flow. So we assume for a given truck density pi that the bottleneck diagram

First-order macroscopic model for mixed traffic 337 represents an (7V-«)-lane road used by passenger cars only: trucks are assumed to stay on the right n lanes. Its capacity g max depends on the number (N-ri) of free passing lanes only. Furthermore, the free-flow part of the bottleneck diagram is assumed to be the same as the free-flow part of the car diagram for JV lanes with a given truck density p2 (see Figure 11 (a)). In this case, to define the traffic states for which perturbations due to overtaking occur, we use the car fundamental diagram for this steady truck density p-i. By extending the results of the LWR bottleneck models for a single slow vehicle, the disturbed states are those located in the A to D part of the car fundamental diagram.

Figure 10: LWR moving bottleneck model for a single slow vehicle whose speed is Vh

Figure 11: (a) Definition of the moving bottleneck area (b) Moving bottleneck area in the phase plane

Note that these states exist only if the car demand is strong enough, i.e. the maximal flow of the fundamental car diagram is higher than the capacity of the bottleneck diagram. Therefore no moving bottleneck effects can occur if the truck density is too high (indeed congestion and equal speeds would occur before overtaking problems). Note also that for the D to C part of the diagram the difference in speeds between cars and trucks is too low to cause overtaking problems (in addition, for the B to C part speeds are equal). Figure 11 (b) gives the resulting area of the disturbed states in the phase plane. This area appears when the car demand is high enough, and disappears when the difference in speeds becomes too low (then for traffic states near the capacity limit).

338 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Fundamental relationships for states in the moving bottleneck area If a traffic state U is located inside the moving bottleneck area, the initial predicted car speed cannot be reached by all cars for such a state. By following the same thought process as Gazis and Herman in (Gazis and Herman, 1992), we can say that only the car density Pi,max associated with the bottleneck diagram capacity can maintain the predicted car speed; the other cars are forced to slow down and stay in the truck lane(s) where their speed is equal to the truck speed \>i. Therefore the mean car speed for such a state is assumed to become the speed of U* (see Figure 11 (a)):

:>l ( A - Pi I =

*

(17)

Consequences on the two-flow model Taking into account overtaking problems in the two-flow model has two main consequences on the traffic flow: first for states inside the moving bottleneck area the car speed is limited as explained above (see equation (17)); then the Lax curves are modified when crossing this area and the intermediate state can change. The following figure gives such an example: for the same two upstream and downstream states L and R, the solution is composed of a 1 -shock then a 2-rarefaction wave. The intermediate state It is more "congested" if an overtaking limitation is assumed (Ii) than if not (Io) (see Figure 12).

Figure 12: Modification of the intermediate state (a) Io: without moving bottleneck effect (the area has been drawn to help the comparison) (b) Ii: with moving bottleneck effect (the former intermediate state Io has been drawn)

First-order macroscopic model for mixed traffic 339

SOME NUMERICAL RESULTS General We conducted some numerical simulations to check the ability of the model to reproduce the features of a two-flow mixed traffic. We used the classical numerical scheme of Godunov, which is well adapted to traffic problems thanks to its upwind feature. However, it is necessary here to adapt it to the features of the two-flow model, as several other authors have already done for some high-order models for example (Zhang, 2001). To use a Godunov scheme, it is only necessary to know which traffic state prevails on the boundary x = xo of the Riemann problem. In our model, the phenomenological hypothesis we used leads to some sign properties for the eigenvalues, and therefore for direction properties of waves in the model. In particular, all 2-waves are propagating towards downstream, due to the positive sign of the second eigenvalue. Moreover, the first eigenvalue is positive for freeflow states (propagation towards downstream) and negative for congested states (propagation towards upstream). All these properties allowed (Chanut and Buisson, 2003) to show that it was possible to precisely determine the state prevailing on the boundary of numerical cells by studying only four typical Riemann states: Upstream stateto1,o?1) Downstream state (o\\ o/) Prevailing state Free-flow Free-flow Upstream state Free-Flow Congested Intermediate or upstream state* Congested Free-flow Capacity state** Congested Congested Intermediate state*** *: similar to the Figure 6. It depends on the 1-shock speed 5. If 5 is positive, the prevailing state on the boundary is the upstream state. If s is negative, the prevailing state is the intermediate state (pi'nt, p2mt) such that: .

{pkv' -/*&).
(18)

rf* =rf+ (A - ft*) **: This capacity state lies on the capacity line and the truck ratio is the same as in the upstream state. ***: In this intermediate state the traffic composition is the same as in the upstream state and the density is the same as in the downstream state. Moreover, (Chanut and Buisson, 2003) showed that it was possible to extend the supply and demand concepts (Daganzo, 1995; Lebacque, 1996) in the case of mixed traffic. The only modification to these concepts results in explicitly computing the intermediate state for the case pointed above: free-flow upstream state and congested downstream state. The sign properties of the eigenvalues are maintained, even with the steady truck speed hypothesis or the moving bottleneck effects. Therefore, there is no need to change the numerical scheme presented above.

340 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Some simulations Simulations are conducted with the following parameters: N=3, Fn=35m/s, Va=Vc=20mJs, a =0.5, ii=5m, i2=15m. Situations with moving bottleneck effects were simulated by assuming that trucks can be located in the middle lane (so n = 2). The space-time discretization was made with a time interval of 0.2s and a space step of 7m. The time interval has to be chosen small enough to limit the dissipation due to the Godunov scheme. The first simulation is devoted to showing the ability of the model to describe a different evolution of the vehicles in a mixed platoon. We assume a mixed cluster with 80% of cars and 20% of trucks located at t=0 on the first ten numerical cells (i.e. 70 m). The initial density is assumed to be 0.05 veh/m. We can see from the Figure 13 that the cluster gradually splits into two smaller clusters: the faster one is composed of cars and the slower of trucks. Figure 14 shows the effects of the perturbation due to overtaking and modeled by the extension of our model. In Figure 14 (a) we can see the slower travel of a car cluster amongst a steady truck flow (density ki). The car cluster is initially located on the first 350m section, and its density is 0.23 cars/m. The steady truck density is assumed to be equal to 0.02 trucks/m. The traffic states are always fluid.

Figure 13: Propagation of a mixed cluster

Figure 14 (b) shows the cumulative number of cars for x = 2.8 km. For a same vehicle, the difference between the two curves represents the delay of this vehicle caused by the moving bottleneck effects. Note that this delay is only due to these effects related to overtaking and not to the effect of truck presence: the first model, without moving bottleneck effects, already takes into account the slower travel of cars due to an additional truck density (equation (4)). The model makes it possible to evaluate the car delay caused by overtaking trucks: in Figure 14 (b) the time wasted by the car n°70 is around 10 s after 3 km.

First-order macroscopic model for mixed traffic 341

Figure 14: Propagation of a car cluster in a continuous truck flow (a) Effect of the moving bottleneck (MB) effects on the cluster (b) Delay related to moving bottleneck effects due to overtaking (x=2.8km)

CONCLUSION AND FURTHER RESEARCH The two-flow LWR model described in this article takes into account the phenomenology of a heterogeneous traffic composed of cars and trucks. The model represents the difference in speeds and in lengths between the two vehicle types and introduces speeds and densities depending on the truck ratio. Its analytical solutions are explained in detail and the paper highlights the fact that a phenomenological hypothesis introduced in the model can lead to non standard mathematical features. However it is shown that it is still possible to guarantee the unicity of the solution and that no unrealistic solution occurs. Moreover an extension of the model is proposed to represent the overtaking problems that cars can encounter. Trucks are then represented as moving bottlenecks. The results of the previous models describing the effects of single slow vehicles like buses are used to model such a phenomenon in the two-flow model. The simulations we carried out show the ability of the model to correctly describe the flow of our two types of vehicles and the additional effects of overtaking. Further research efforts are required to improve the phenomenology and in particular to more accurately model the number of passing lanes which can be used by cars depending on the truck density and therefore the moving bottleneck effects: they required new data. Moreover, new efforts regarding multi-class models could try to implement more than two vehicle classes. For example, in order to represent the effects on the traffic flow of very slow trucks for sections such as grades, it could be useful to implement single moving bottlenecks microscopically described in the two-flow macroscopic model.

342 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

AKNOWLEDGMENTS The author wishes to thank Dr Christine Buisson for supervising his research work daily, Pr Sylvie Benzoni-Gavage for introducing him to the theory of hyperbolic systems, and Dr Ludovic Leclercq for many useful discussions. Helpful comments from the anonymous referees are gratefully acknowledged.

REFERENCES Benzoni-Gavage, S. and R.M. Colombo (2003). An n-populations model for traffic flow. European Journal on Applied Mathematics, 14, 587-612. Bliemer, M.C.J. (2000). Analytical dynamic traffic assignment with interacting user-classes Theoretical advances and applications using a variational inequality approach. PhD report, 184 p. Delft University of Technology, Delft, The Netherlands. Chanut, S. and C. Buisson (2003). A macroscopic model and its numerical solution for a twoflow mixed traffic with different speeds and lengths. Transportation Research Record, 1852,209-219. Chanut, S. and L. Leclercq (2003). Introducing the effects of slow vehicles in a LWR twoflow traffic model. Proceedings of the 5th meeting on Traffic and Granular Flow, to be published, Delft, The Netherlands. Chanut, S. (2004). Diagrammes fondamentaux multi-classes et resolution analytique du modele d'ecoulement de trafic heterogene. Technical note (in French). Daganzo, C.F. (1995). A finite difference approximation of the kinematic wave model of traffic flow. Transportation Research B, 29(4), 261-276. Daganzo, C.F. (1997). A continuum theory of traffic dynamics for freeways with special lanes. Transportation Research B, 31(2), 83-102. Daganzo, C.F. (2002). A behavioral theory of multi-lane traffic flow. Part I: Long homogeneous freeway sections. Transportation Research B, 36(2), 131-158. Daganzo, C.F., J.A. Laval (2005). On the numerical treatment of moving bottlenecks. Transportation Research B, 39(2), 31-46. Gazis, D.C. and R. Herman (1992). The moving and "Phantom" bottlenecks. Transportation Science, 26(3), 223-229. Helbing, D. and A. Greiner (1997). Modeling and simulation of multilane traffic flow. Physical Review E, 55(5), 5498-5508. Helbing, D., A. Hennecke, V. Shvetsov and M. Treiber (2001). MASTER: macroscopic traffic simulation based on a gas-kinetic, non-local traffic model. Transportation Research B, 35(2), 183-211. Hoogendoorn, S.P. and P.H.L. Bovy (1998). Modeling multiple user-class traffic. Transportation Research Record, 1644, 57-70 Hoogendoorn, S.P. and P.H.L. Bovy (2000). Continuum modeling of multiclass traffic flow. Transportation Research B, 34(2), 123-146. Jin, W.-L. and H.M. Zhang (2003). The inhomogeneous kinematic wave traffic flow model as a resonant nonlinear system. Transportation Science, 37(3), 294-311.

First-order macroscopic model for mixed traffic 343 Laval, J. (2004). Hybrid models of traffic flow: impacts of bounded vehicle accelerations. PhD report, 97 p. University of California, Berkeley, The United States. Lebacque, J.-P. (1996). The Godunov scheme and what it means for first order traffic flow models. Proceedings of the 13th International Symposium on Transportation and Traffic Theory (J.-B. Lesort, ed.), 647-678. Pergamon, Oxford. Lebacque, J.-P., J.-B. Lesort and F. Giorgi (1998). Introducing buses into first-order macroscopic traffic flow models. Transportation Research Record, 1644, 70-79. Leclercq, L., S. Chanut and J.-B. Lesort (2004). Moving bottlenecks in the LWR model: a unified theory. Transportation Research Record, accepted publication. LeVeque, R.J. Numerical methods for conservation laws. Second edition, 214 p. Birkhauser Verlag, Basel, Switzerland. Lighthill, M. J. and G. B. Whitham (1955). On kinematic waves II. A theory of traffic flow in long crowded roads. Proceedings of the Royal Society A, 229, 317-345. Logghe, S. (2003). Dynamic modeling of heterogenous vehicular traffic. PhD report, 221 p. Katholieke Universiteit Leuven, Leuven, Belgium. Mufioz, J.C. and C.F. Daganzo (2002). Moving bottlenecks: a theory grounded on experimental observation. Proceedings of the 15th International Symposium on Transportation and Traffic Theory (M.A.P. Taylor, ed.), 441-462. Pergamon, Oxford. Newell, G.F. (1998). A moving bottleneck. Transportation Research B, 32(8), 531-537. Richards, P.I. (1956). Shock waves on the highway. Operations Research, 10(4), 638-667. Wong, G.C.K. and S.C. Wong (2002). A multi-class traffic flow model - an extension of LWR model with heterogeneous drivers. Transportation Research A, 36(9), 827-841. Zhang, H.M. (2001). A finite difference approximation of a non-equilibrium traffic flow model. Transportation Research B, 35(4), 337-365. Zhang, H.M. and W.L. Jin (2002). A kinematic wave traffic flow model for mixed traffic. Transportation Research Record, 1802, 197-204. Zhang, H.M. and W.-H. Lin (2002). Some recent developments in continuum vehicular traffic flow theory. Proceedings of the 15th International Symposium on Transportation and Traffic Theory (M.A.P. Taylor, ed.), 607-624. Pergamon, Oxford. Zhang, H.M. (2003). Anisotropic property revisited - does it hold in multi-lane traffic? Transportation Research B, 37(6), 561-577. Zhu, Z., G.-L. Chang and T. Wu (2003). Numerical analysis of freeway traffic flow dynamics under multiclass drivers. Transportation Research Record, 1852, 201-208.

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18

A VARIATIONAL FORMULATION OF KINEMATIC WAVES: BOTTLENECK PROPERTIES AND EXAMPLES Carlos F. Daganzo and Monica Menendez, Institute of Transportation Studies, University of California, Berkeley, California, USA

INTRODUCTION This paper proposes two algorithms that, based on a variational version of kinematic wave (KW) theory (Daganzo, 2003, 2003a, 2005), can solve complex inhomogeneous KW problems with triangular fundamental diagrams (FDs) with precision and simplicity. Methods for general concave FDs are discussed in Daganzo (2003, 2003a). The proposed algorithms can be used with any combination of gradual, discontinuous, moving and fixed bottlenecks -even if the character of the road changes in space-time; e.g., due to snow-plows. The gradual fixed-bottleneck problem was introduced in Lighthill and Whitham, (1955) and later solved analytically in Newell (1999), but only for some special cases. The discrete moving-bottleneck problem was introduced in Gazis and Herman (1992), and formulated as a KW problem in Newell (1993 and 1998). Mufioz and Daganzo (2002) modeled it in a more general way by treating the moving bottleneck as a special boundary condition. Analytical solutions have only been developed for special cases, however. Existing traffic simulation methods (e.g., Giorgi et al, 2002, and Daganzo and Laval, 2003) only provide first-order approximations for these problems. The new algorithms will be shown to be much better. An upper bound for their numerical error will be presented, along with examples. The paper will also show that real bottlenecks can be approximated by bottlenecks of zero length, and how to formulate proper boundary conditions to represent discrete bottlenecks. This puts moving bottleneck theory and practice on a solid foundation.

346 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The remainder of this section summarizes the essential facts of variational theory. Following sections present: (i) two algorithms to solve a special family of problems (self-similar problems without point bottlenecks); (ii) extensions for problems involving moving point bottlenecks; (iii) applications of these results to examine the effect of bottleneck length; and (iv) a simple and very accurate solution method to solve general problems with any number of moving point bottlenecks. Basic Facts Consider a one-directional road on which vehicles are conserved, and let x, t, q and k respectively denote the distance along the road (increasing in the direction of travel), time, flow and density. The last two variables are functions of time and space q{t,x) and k(t,x). Given is an FD, q = Q(k,t,x), which is assumed to be concave and piecewise differentiable in k, and to satisfy q = 0 for k = 0 and k = K, where K is the jam density; see Figure 1 (a). As proposed in Newell (1993a), we shall describe the KW solution in terms of a vehicle number function, N(t,x), whose partial derivatives, q = dNldt and k = -dN/dx are the flow and density functions. The KW solution satisfies: dN I dt = Q(-dN I dx,t,x).

Figure 1. Basic concepts of Variational Theory: (a) fundamental diagram, and (b) cost function.

The shortest path problem is formulated in terms of the following "cost" function, R, which has units of vehicle number per unit time (i.e., flow):

R(x',t,x) = sup\Q(k,t,x)-kx'}.

0)

Figures 1 (a) and 1 (b) show graphically how Q and R are related; note that R is the LegendreFenchel transform of Q. The cost function is a fundamental property of the highway, just like the fundamental diagram. Physically, it represents the maximum rate at which traffic can pass an observer moving with speed x'. Note that the maximum flow (the highway capacity, qmij is R(0j,x). For traffic problems R is non-negative, non-increasing, convex (if Q is concave), and only defined for the range of valid wave speeds w. (Recall that the wave speed is w(jc,t,x) = dQ/dk .) In the triangular case R turns out to be linear.

Variationalformulation of kinematic waves 347 The cost function gives the cost per unit time of moving along a space-time path 9 with trajectory x(t). We say that a path is "valid" if it is a piecewise differentiable curve with slopes x' in the range of allowable wave-speeds [W(K",?,X), w(0,f,x)]. The cost of traversing a valid path is: d(.r)= \n\x\rxkl'-

(2)

where tB and tp are the times associated with the path endpoints. Variational theory states that if the vehicle number is given along a boundary curve (or curves), D, and we think of these as "start up costs", then the vehicle number at any point in the solution domain is the least cost to reach the point with a valid path from the boundary, including the start up cost; i.e.:

Np =nunJ4f/)+.V <> },

(3)

where Vp is the set of valid paths 9 from B to P, and 5toe D is the start point of 9. Equation (3) is a calculus of variations problem. Its iso-cost contours are the vehicle trajectories. Networks In this paper, a network is a digraph with nodes L embedded in time-space, with directed arcs LL'. Arcs are defined only for node pairs that can be connected by a valid path. We call these "valid node pairs," and say that a network is "validly connected" if all its valid node pairs are connected by a walk. Each arc is assigned a cost, cLU, equal to that of an optimum continuum path between its end nodes, and a duration and distance, tlr > 0 and xlL, consistent with the node coordinates. It is shown in Daganzo (2003, 2003a) that if the FD is piecewise linear there are networks whose shortest "walks" (network paths) between all valid node pairs are shortest continuum paths. These networks are said to be "sufficient" because by solving the shortest path problem on the network one finds the exact least cost on the continuum between every valid node pair. This is advantageous but does not imply exactness when the network is used to solve Equation (3). Prediction errors still arise if a network does not have enough nodes on the boundary (origins) to sample the data with enough frequency. We call these discrepancies "sampling errors." The rest of this section presents some key results. Error Bounds To quantify sampling errors, et, we shall assume (reasonably) that the cost function is bounded, with R(k,t,x)< R, and that the boundary data satisfy a Lipschitz continuity condition; i.e. that there is a ji> 0 such that

NB-NH,\
\/B,B'&D

(4)

348 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

where |-B-5'| = xB -xB. + tB-tB\.

We quantify sampling frequency with two constants, h

and T, as follows. For every valid path to a node L with beginning point BPe D there should be: (i) a network origin B such that B - Bp | < h, and (ii) a short valid path that intercepts the original path from B in time less than t. Figure 2 illustrates the concept. For any h and rwith these properties, sampling errors satisfy: es
Figure 2. Interception of an optimum path. If a network is not sufficient the solution will also include network errors, en. A network where the best walk between any valid node pair is within z cost units of the continuum shortest path (i.e., where en
(5)

Triangular FDs The two basic wave speeds of a triangular FD, whether homogeneous or not, will be denoted w, = w(0,t,x) and w2 = w(K,t,x); w, is the free-flow speed and w2 the "backward" wave speed. We consider first problems with homogeneous FDs, q = Q(k). The following two results, proven in Daganzo (2003a), will turn out to be useful. Result 1 (sufficiency): If the FD is triangular and homogeneous, all valid paths are optimum and all validly connected networks sufficient. Since all valid paths are optimum, a straight path is optimum. Therefore, in this case it is easy to set up networks and calculate their arc costs; the arc cost formula is: cLL. =tLL.R(x'),

where x'= xu, ltLL,.

(6)

This leads to the following: Result 2 (exactness): If (i) the FD is triangular and homogeneous, (ii) the network is validly connected, (iii) the data and the boundary are linear between network origins,

Variationalformulation of kinematic waves 349 and (iv) the network contains straight walks with speeds w\ and W2 from the boundary to every node, then e = 0. Result 2 is true because problem (3) is a linear program, so that its optimum paths must either emerge from vertices or have extreme speeds. Inhomogeneous roads where the FD changes with space by a similarity transformation also have useful properties. In these cases, the FD is q = Q(k,x) = a(t,x)Q0(kIa(t,x)), where a{t, x) > 0 is a proxy for the number of available lanes, which can vary with t and x. The similarity relation implies that the wave speeds are space-independent and that the cost function is self-similar: R(x',t,x) =a(t,x)Ro(x'). If a highway is homogeneous then a(t,x) is constant, a(t,x) = a. If it includes a moving bottleneck then a(t,x)
(a)

(b)

Figure 3. Shortest paths for similar-monotonic cost functions: (a) decreasing dx) (road narrows), and (b) increasing a(x) (road widens).

SELF-SIMILAR PROBLEMS We examine here procedures for solving general self-similar problems, and then analyze their accuracy.

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Solution Methods The procedures in question have two basic features: an efficient network geometry and a fast shortest path algorithm. They are now described. Geometric networks: A geometric digraph formed by two families of parallel equidistant lines with slopes w. and time separations e. (i = 1, 2) is called a geometric network if all its arcs point in the direction of increasing time. Figure 4(a) displays an example. Note that nodes exist at every intersection and that they form an oblique lattice. Note too that any connected subgraph of a geometric network is a validly connected network. In the special case where £, = e2 = e , the lattice includes space-rows, as shown in Figure 4(b). The row spacing is denoted 8. Lopsided networks: Assume now that we delete any number of space rows from a geometric network with space rows, but maintain all the node connections across the deleted rows as shown in Figure 4(c). Assume too that we introduce horizontal links along all the remaining rows, also as shown. It then turns out that the resulting network is validly connected, as the reader can easily verify. We call these networks "lopsided" because the row spacing can be increased as much as desired without changing e by selective deletion of rows. This is of considerable computational advantage when results are only sought at widely spaced locations. Note that every node L of a lopsided network has exactly three "from" nodes. In this case too, the row spacing is denoted 8.

(c) Figure 4. Network types: (a) geometric, (b) geometric with space-rows, and (c) lopsided.

Variationalformulation of kinematic waves 351 Shortest path method: To solve a problem we first assign costs cw to every link, and then use a shortest path algorithm to compute the optimum cost for every node in the solution domain, cL. Costs for origin nodes are known since they are part of the boundary data: c, = N,. Link costs should be computed by minimizing Equation (2). In the homogeneous case the answer is Equation (6). In the self-similar case we can simply evaluate the cost of an extremal path for each arc; see Result 3. If we use a geometric network with short arcs we can also use Equation (6) as an approximation with R(x') replaced by R(x',t,x) and evaluated at the midpoint of the arc. This only introduces errors of order O(e2) as £ —> 0, and can be done even in the general non-self-similar case. Since our networks have translational symmetry, dynamic programming (DP) can be used for the optimization. As a result, the calculation effort to obtain all the cL only increases linearly with the number of arcs. The recursion is: c,.. =/min){c,,+C,.,..},

(7)

where FiL') is the set of "from" nodes for V. When the DP algorithm is used with a geometric (or lopsided) network it will be respectively called the GDP (or LDP) algorithm. Since both network types are sparse (with only 2 or 3 links per node) the calculations of Equation (7) involve a very small number of comparisons and additions per node. The effort per node is comparable to that required by solution methods based on conservation laws. The LDP method requires considerably fewer nodes, though. Accuracy Since a geometric network includes walks with the extremal property of Result 3 for every valid node pair, it follows that a geometric network is sufficient for the self-similar problem if a is non-decreasing (or non-increasing). Consideration shows that this continues to be true for problems with general but time-independent odx) if we add nodes and horizontal links with appropriate costs at all the locations where aix) is a minimum; i.e., at bottlenecks. Therefore, the GDP algorithm avoids network errors in the self-similar case. Note too from Result 1 that both DP algorithms avoid network errors in the homogeneous case. In the above instances only sampling errors remain. According to Equation (5) these sampling errors are bounded by: xR + fih. For a geometric network with space rows the parameters T and h are respectively bounded by e and (e+28). Thus, the sampling error is bounded by £[i?+j3+2/3(<5/£)]. Since 5/e = (w,'- w2')~', we see that the quantity in brackets is a constant. Hence, the error bound is O(e) and uniform; i.e., it is independent of the size of the solution domain. The bound can be tightened in two important cases: (i) homogeneous problems with linear data between origin nodes (when e = 0); and(ii) problems with smooth data (when e = O(e') for .

352 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The exactness of case (i) is a direct consequence of Result 2. The situation arises frequently, since data for typical problems is usually available in piecewise linear (PWL) form. Case (ii) applies when both the data and the boundary can be expressed as smooth functions of a parameter, n, such that the maximum jump in n across consecutive origins is O(e). The result is true because under its conditions the least cost to reach L from a point on the boundary, NL(K), is a smooth function, with a smooth global minimum, NL. Since sampling errors for smooth minima become quadratic in the sampling interval if the sampling intervals are reduced toward zero, and since the sampling intervals are O(e) for e —> 0, it follows that the error is O(e2) for e -» 0. As an illustration of this result, Figure 5 shows how the GDP algorithm performs with one of the examples in Newell (1999). The example involves a self-similar highway with a re-scaled triangular FD that varies with x (w\ = oo, vc2 = -1 and qmax = V* x2), and gradually increasing upstream flow (q = t for t > 0). Note that Newell's figure (and ours as a result) display distance in the reverse direction, using x* = -x instead of x. The example is of interest for its difficulty. In particular note that conventional methods for conservation laws would either introduce infinite errors or use an infinite computation time, since the maximum wave speed is infinite. The GDP algorithm, however, works without a glitch. Part (a) of the figure is a reproduction of the exact wave map in Newell (1999) including the shock-path (q* on the abscissa stands for either flow or time in this reference.) Figure 5(b) is the wave-map produced by the GDP algorithm. It matches Figure 5(a) remarkably well in all respects, including the shock-path and the position where the shock first develops at (/, x*) = (0.5, 1). Figure 5(c) gives the vehicle trajectories produced by the new algorithm. Note how the shock develops. Figures 5(b) and 5(c) were obtained with a very dense network that allowed us to estimate the solution at many relevant points appearing in Figure 5(a). Nevertheless the GDP algorithm is very accurate with large time steps. Figure 5(d) compares the estimated and exact vehicle number across time at three different locations when the time step of the GDP procedure is 0.1 time units. The error bound in this case turns out to be 0.00125 vehicles everywhere in the solution domain - imperceptible to the eye.

MORE GENERAL PROBLEMS WITH POINT BOTTLENECKS This section examines the structure of optimum continuum paths in a very general class of problems involving non-similar highway sections and any number of point bottlenecks. The results are then used in the following section to examine the effect of bottleneck length, and later to show how this class of general problems can be solved. We consider first a homogeneous highway that includes a single bottleneck. We assume that the bottleneck describes a valid path S with trajectory xB(t), and that the unit cost along it rB(t) is less than or equal to R(x'B(t)) for all t. This allows us to state the following; see Figure 6(a):

Variational formulation of kinematic waves

353

Figure 5. Solution of a gradual bottleneck problem: (a) exact wave map, (b) numerical wave map, (c) vehicle trajectories, and (d) normalized vehicle numbers, N*, with N/N*=\, 10, 100 forx*=0.5, 1.5, 2.5, respectively.

Shortcut Theorem: If a problem is homogeneous and two points, P and P', can be connected by a path that touches the bottleneck, then a path from P to P' is optimal if it has the following structure: (i) an access path from P to the earliest possible S e S; (ii) a continuous portion of the shortcut from S to the latest possible S'e S; and (iii) an egress path from S'toP'. Proof. We show first that an optimum path does not have to leave and return to the shortcut. This is true because the cost of any bypass is always matched by the cost of a competing path that is infinitely close to the shortcut (see Result 1), and because the cost of the competing path cannot be less than the cost of a similar path that would use the shortcut - since the shortcut has an equal or smaller unit cost than parallel, neighboring paths. Hence, the shortcut only needs to be left once. It follows that there

354 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction is an optimum path with only three components for: (i) access, (ii) use and (iii) egress from the shortcut. To complete the proof we now show that the access and egress components with the shortest possible durations do the job. This is true for the access part because if the path were to join the shortcut at a later point than S, Sh e S, as shown in Figure 6(a), its cost could not decrease. The reason is that before the change, the cost of reaching St was the sum of the cost of the path from P to S plus the cost of the shortcut from S to Si,. But after the change, the cost is given by any valid path from P to St that does not use the shortcut (as per Result 1); e.g., by a path that shadows the "before" path arbitrarily close to the shortcut without using it. The cost of the new path is obviously higher or equal. The same lines of reasoning also reveal that the egress part should be of the shortest possible duration. Note that the access and egress portions of the optimum paths identified by the Shortcut Theorem are straight and have slopes w, or w2, unless they connect P to the beginning of the shortcut or the end of the shortcut to P'. The result also holds for self-similar highways where oit, x) is non-increasing in x (narrowing) upstream of S, and non-decreasing (widening) downstream. The reason is that under these conditions, paths on the upstream side should be as high as possible and on the downstream side as low as possible (recall Figure 3). When this happens the steps of the proof continue to hold. The steps hold even if the upstream portion of the highway is not similar to the downstream portion, as suggested by Figure 6(b). In this figure, arrows indicate the direction in which ait,x) decreases with x. Cost functions where a is monotonic in x are said to be "similar-monotonic" (SM). The following corollary summarizes these ideas: Corollary 1: The result of the shortcut theorem also holds if the cost function is: (a) homogeneous outside the time range of the bottleneck; and (b) SM upstream and downstream within the range, with the following properties: (i) Upstream: R = ait,x)R0(x'), a(t,x) non-increasing in x for x<xB(t); (ii) Downstream: R = a(t,x)R0(x'), a(t,x) non-decreasing inxfor x>xH(t); (iii) 77,(0 < mmfe(t,xH(t))Ro(x'B

{t)),a(t,xR(t))R,,{x'„ {t))\ '

Note that the corollary applies to shortcuts of both, finite and infinite range. Note too that the access and egress portions of the optimum paths identified by the Shortcut Theorem and this corollary have maximum slopes, unless, as before, they connect P to the beginning of the shortcut or the end of the shortcut to P'. The same reasoning of the shortcut theorem also reveals that paths where oif, x) is a maximum with respect to x, "reverse-bottlenecks", never have to be used.

Variationalformulation of kinematic waves 355

Figure 6. General problems with point bottlenecks: (a) Shortcut Theorem, (b) Corollary 1, (c) Optimum Path Theorem, and (d) composite geometric network. All of the above suggests that problems with multiple bottlenecks can perhaps be analyzed by decomposing their solution domain into regions describable by Corollary 1, as in Figure 6(c)— in this case too, arrows indicate the direction in which o&t, x) decreases with x. To formalize this idea, let us define a quasi-monotonic problem as one whose solution domain can be partitioned by continuous curves into subregions where the cost function is similarmonotonic. Then, we have: Optimum Path Theorem: If a problem is quasi-monotonic, then an optimum path between two points exists that is only composed of one or more pieces of the following two types: (i) continuous sections of bottlenecks, and (ii) straight segments with extremal slopes, w, or w2, appropriate for the subregion containing the segment. Proof: The non-bottleneck portion of an optimum path can be divided into continuous parts, each embedded in a similar-monotonic subregion. Every one of these parts must minimize the cost between its endpoints. Result 3 guarantees that this is achieved for each part by a bilinear sub-path with extremal slopes w, and w,, appropriate for its subregion. The class of quasi-monotone problems is very broad; it seems to encompass all KW problems of practical interest, or at least all those formulated to date. For example, the highway can be

356 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction piecewise similar or piecewise homogeneous with time-dependent features; and any of these, not just the capacity, can be changed by moving bottlenecks. An example of the latter is a snowplow that changes with its passage both, the free-flow speed and the number of available lanes. The theorem underpins the justification given later in this paper for using geometric networks to solve quasi-monotone problems. The basic idea consists in using a geometric network with appropriate slopes and costs within each monotone subregion of the solution domain, and then stitching together these networks by adding along the sub-regional interfaces nodes and links with relevant costs; see Figure 6(d). Nodes and links with reduced costs should also be added along the trajectories of point bottlenecks. If the same composition idea is used with the Shortcut Theorem instead of Corollary 1 we find the following stronger result for problems where bottlenecks and reverse bottlenecks partition the solution domain into homogeneous parts (quasi-homogeneous problems). Corollary 2: If a problem is quasi-homogeneous, then an optimum path to a point exists that is only composed by one or more pieces of the following three types: (i) continuous sections of bottlenecks; (ii) least-duration access and egress sub-paths as in the Shortcut Theorem; and (iii) inter-bottleneck sub-paths, each with either maximal or minimal slopes. Proof: Only (iii) needs proof. Geometrical considerations show that if a connector between bottlenecks were not to have either maximal or minimal slope, we could then find a cheaper connector, of greater or smaller slope, that would shadow one of the bottlenecks for a short time. D Corollary 2 applies to piecewise homogeneous highways. It will be the justification for modeling them with lopsided networks. Before this is done, we examine a question that seems to have generated some confusion in the literature. What effect can the length of a bottleneck have on traffic flow?

THE EFFECT OF BOTTLENECK LENGTH The results of the previous section are now used to answer the question. We will assume in this section that the bottleneck moves with non-negative speed and that its trajectory 5 is contained in a space-time swath of width l(t) < i distance units. Our problem is defined as follows. Inside the swath, R is defined as in Corollary 1. Outside and upstream of the swath, R = (xoRo(x'), with ao >a(t,x). Outside and downstream, We shall denote the upstream and downstream jam R =a,)Rl)(x'), with ao>a{t,x). densities outside the swath K; and K, respectively. Obviously, Corollary 1 applies to the complete problem.

Variationalformulation of kinematic waves 357 We are interested in evaluating the accuracy of a point-bottleneck approximation with the same S and rB(t), but a new cost function R = c^,,R,,(x') f° r x < *«(0 a n d R = cfo/?„(*') for x > xB (t). The new cost function is homogeneous both, upstream and downstream of SCorollary 1 also applies to this approximation. We are now in position to establish the main result of this section. Result 4 (bottleneck length): The difference in vehicle number between the solutions of the original problem and the point-bottleneck approximation is bounded from above by (max(/c, ic) at every point of the solution domain. Proof. Since Corollary 1 applies to both problems, the paths of the Shortcut Theorem are optimum for both problems. Their unit costs can only differ for the access/egress portions internal to the swath. The unit costs vanish for paths with slope w, or w]. Therefore, only access/egress portions with slope w2 or w2 contribute to the difference. If we let t2 and t2 be the time duration of the portions with slopes w, andw 2 , we see that the cost difference cannot exceed R(w2)t_2+R(w2)t2. Note, however, that for homogeneous FDs R(W2) = W2K. Thus, the bound is w2Kt_2+w2Kt2. Since the speed of the swath is non-negative we can write w2t_2 +w2t2 < I. Hence, the bound satisfies w2j£t_2 +w2Kt2 < £max(i£,K). Since the bound defined by the right side of this inequality applies to every pair of points, it follows that the difference in the predicted vehicle numbers for both problems, including the effect of the boundary data, cannot exceed ( max(K, ic). The quantity (max(i£,K) is a uniform error bound when one uses a point-bottleneck approximation. Thus, a simple rule is that the error in vehicle number cannot exceed the maximum number of vehicles that fit alongside the bottleneck. Obviously, the error is negligible for bottlenecks such as traffic signals, trucks, buses, snowplows, convoys of a few vehicles, and other localized restrictions. Figure 7 shows the exact and approximated solution for an example involving a gradual lane drop spanning lA mi. The highway is self-similar with a triangular FD that varies with the upstream distance from the bottleneck: w, = 60 mph w2 = -15 mph i , where y0 = 2000 vph, y, = 32000 vph/mi2, and I = lA mi. The upstream flow is assumed to increase gradually with time: where y2 =2000 vph and y, =10000 vph/hr.

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Figure 7. Error of point-bottleneck approximation. The smooth lines correspond to the exact solution (0.25 mile gradual bottleneck) and the dotted lines to the approximation (point-bottleneck).

The figure shows the cumulative vehicle number at six different locations as a function of time. According to Result 4, the error from an approximation using a point-bottleneck cannot exceed ^max(x;, K) = 83 vehicles. This is consistent with the numerical results of the figure, which show a maximum error of around 40 vehicles. The error grows initially as we move away from the bottleneck, but after a certain point it stabilizes, as expected.

Variationalformulation of kinematic waves 359

GENERAL SOLUTION METHOD We now show and demonstrate with examples how to solve general quasi-monotonic problems with point bottlenecks. General quasi-monotonic problems with multiple point bottlenecks can be handled by superimposing all the point bottleneck shortcuts on a stitched network with geometric components, such as the one in Figure 6(d), adding nodes at the points of intersection, and connecting these with links of proper cost. Consideration shows that any such network contains walks parallel and very close to the relevant paths specified by the Optimum Path Theorem. This is also true of networks composed of lopsided components if the problem is quasi-homogeneous, by virtue of Corollary 2. Accuracy The error introduced by these procedures at a point P is bounded by a quantity that tends to zero as e —» 0 if both the duration and number of sub-paths of the relevant optimum path specified by the Optimum Path Theorem are bounded. Consider any such relevant path. For sufficiently small e there is always a "nearest" walk that differs from the shortest path only in the duration and time-displacement of corresponding sub-paths, and not in their slopes. These discrepancies are of order O(e). Since the number of sub-paths is O(tp), the cost difference caused by all the duration discrepancies should be O(e tp) as e —» 0. The cost difference caused by each displacement can grow at most with the product of displacement and sub-path duration. Therefore, the total displacement error is also of order O(e tp) as e —> 0. Thus, both, the combined network error and the prediction error in Np must be O(e tp) as e —> 0. For piecewise homogeneous problem the displacement error vanishes. Then, if the number of sub-paths in the relevant optimal path to P is uniformly bounded across all points — a usual case in practical applications — the network error is Oie) as e —» 0. This bound is problemdependent, but uniform. Examples We illustrate the accuracy and flexibility of the method with two examples. The first one, adapted from Daganzo and Laval (2003), includes a moving bottleneck on a homogeneous highway. The results of the proposed method are favorably compared to those in Daganzo and Laval (2003). The second example is more complicated; it assumes that the bottleneck changes the character of the road (as a snowplow or a police car would) and that the highway is inhomogeneous. Example 1: Consider a one-mile homogeneous freeway whose FD is an isosceles triangle with free flow speed, w, = 60 mph, backward wave speed, w, - -60 mph, and capacity, qmm = 9000 vph.

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These parameters are not realistic, but were chosen in Daganzo and Laval (2003) to separate the error of their procedure from those of the numerical KW solver—which could only handle isosceles FDs exactly. The freeway is initially in a steady state flowing at capacity when, at t0 = .3 min and x = .3 mi, a truck traveling at vlt = 20 mph enters the road. The truck maintains this speed until t = 2.1 min, when it leaves. It is assumed that the maximum rate at which traffic can pass the truck is rB = 3000 vph. Figure 8(a) shows a time-space diagram with the exact solution of this problem, including all its shocks and interfaces, and the moving bottleneck. The figure also shows by means of solid horizontal lines the trajectories of six detectors. Figure 8(b) shows the (geometric) network we used to solve this problem with a discretization of At =6 sees, including the shortcut for the moving bottleneck. Figure 9(a) shows the iV-curves for both, the solution in Daganzo and Laval (2003), which used At =6 sees (wiggly lines), and the exact solution (smooth lines). Figure 9(b) shows the results from variational theory. They match the exact solution without any error.

Figure 8. Example 1: (a) time-space diagram; (b) network with shortcut

Figure 9. Example 1: (a) N-curves at six locations from Daganzo and Laval (2003); (b) N-curves at six locations from variational theory. The smooth lines are the exact solution.

Variationalformulation of kinematic waves 361 Example 2: This example includes a moving bottleneck that changes the character of the road as it travels through it. We still consider a one-mile homogeneous freeway flowing at capacity, described by an isosceles FD diagram with free flow speed, w, =60mph, backward wave speed, w2 = -60 mph, and capacity, qmm = 9000 vph. In the present case, however, a slow moving police car with vfl = 20 mph enters the highway at t0 = .3 min and x = .3 mi, "reminding" every vehicle that passes it to adopt the speed limit, w, =30mph. (These unrealistic numbers have been chosen to dramatize the effects in our figures, but as we have proven in previous sections the procedure would perform equally well with realistic data.) In essence, the police car changes the FD downstream of its position. The downstream FD is not an isosceles triangle any more: its jam density and the backward wave speed remain unchanged (e.g., w, =w 2 = - 6 0 mph but its free-flow speed becomes w, = 3 0 m p h ^ w , . To complete the formulation we assume that "rubbernecking" restricts the maximum rate at which traffic can pass the police car to ra = 1800 vph. As in the previous example, Figure 10(a) is a time-space diagram with the exact solution, including relevant interfaces and the trajectories for the police car and six detectors. Figure 10(b) shows the composite network for this problem, including the shortcut. Note the different slopes of the "free-flow" arcs, upstream and downstream of the shortcut. Different time increments were also used on each side of the shortcut for convenience. Figure 10(c) shows by means of continuous lines the exact TV-curves for this problem, plotted on an oblique coordinate system. Dots display the results of the procedure. There is no error. The procedure is just as quick for problems that cannot be solved easily by other means. Consider the same police car as before, but assume now that the highway is self-similar and inhomogeneous, including a gradual bottleneck: £, where y0 = 6000 vph, y, = 3000 vph/mi2, and I = 1 mi. The backward wave speed and the free-flow speed are assumed to be the same as before — the latter dropping to 30 mph downstream of the police car. We assume that the maximum bottleneck passing rate depends on location and obeys: rB = qnlaj5. It is still 1800 vph upstream of the bottleneck, i.e., x < 0. To solve this problem we can still use the network of Figure 10(b), but with updated link costs, of course. Figure 10(d) shows the solution. A comparison with the exact solution is not made, but we know from the results of this paper that the numerical solution has a negligible error. In comparing with part (c), note the curvature of the new TV-curves, and their uneven separations.

362 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 10. Example 2: (a) time-space diagram, (b) composite network grid and shortcut, (c) N-curves at six locations in oblique system: homogeneous highway, and (d) N-curves at six locations in oblique system: inhomogeneous highway. Continuous lines in part (c) correspond to the exact solution.

FINAL REMARKS This paper has introduced recipes to solve complex KW problems with precision and simplicity. The techniques are especially useful for inhomogeneous KW problems with multiple moving bottlenecks. They can be used even if the bottlenecks change the character of the road as they move through it. The new methods are now being applied to complex real-life problems with combinations of gradual, moving and time-dependent bottlenecks. They can help improve hybrid (discrete/continuous) models of traffic flow, where trucks and other slow vehicles are modeled as underpowered discrete particles that can both generate queues and be slowed by

Variationalformulation of kinematic waves 363 them; see Laval and Daganzo (2004). Hybrid models that include lane changing are currently being implemented, and initial tests with real data are very encouraging (Laval and Daganzo, 2004a). These models, for example, appear to explain the strange observations of moving bottlenecks reported in Mufioz and Daganzo (2002), and the detailed merge bottleneck phenomena in Cassidy and Rudjanakanoknad (2004). The new methods can also be used to simulate intermittent bus lanes and HOV lanes.

ACKNOWLEDGMENT Research supported by NSF Grant CMS-0313317 to the University of California, Berkeley, and by an NSF Graduate Research Fellowship.

REFERENCES Cassidy, M. and J. Rudjanakanoknad. (2004). Increasing Capacity of an Isolated Merge by Metering its On-ramp. Institute of Transportation Studies Report UCB-ITS-RR-200403, U. of California, Berkeley, CA. Transportation Research B (in press). Daganzo, C. F. (2003). A Variational Formulation for a Class of First Order PDEs. Institute of Transportation Studies Report UCB-ITS-RR-2003-03, U. of California, Berkeley, CA. Daganzo, C. F. (2003a). A Variational Formulation of Kinematic Waves: Solution Methods. Institute of Transportation Studies Report UCB-ITS-RR-2003-07, U. of California, Berkeley, CA. Transportation Research B (in press). Daganzo, C. F. (2005). A Variational Formulation of Kinematic Waves: Basic Theory and Complex Boundary Conditions. Transportation Research B 39(2), 187-196. Daganzo, C. F. and J. A. Laval. (2003). On the Numerical Treatment of Moving Bottlenecks. Institute of Transportation Studies Report UCB-ITS-PWP-2003-10, U. of California, Berkeley, CA. Transportation Research B 39(1), 31-46. Gazis, D. C. and R. Herman. (1992). The Moving and "Phantom" Bottlenecks. Transportation Science 26, 223-229. Giorgi, F., L. Leclercq, and J. B. Lesort. (2002). A Traffic Flow Model for Urban Traffic Analysis: Extensions of the LWR Model for Urban and Environmental Applications. Proceedings of the IS" ISTTT, Oxford, U.K., 393-415. Laval, J. A. and C. F. Daganzo. (2004). A Hybrid Model of Traffic Flow: Impacts of Roadway Geometry on Capacity. Presented at the 83"1 Annual Meeting of the Transportation Research Board, Washington D.C. (submitted for publication). Laval, J. A. and C. F. Daganzo. (2004a). Multi-Lane Traffic Flow Hybrid Model: Quantifying the Impacts of Lane Changing Maneuvers on Traffic Flow" Institute of Transportation Studies Working Paper UCB-ITS-WP-2004-1, U. of California. (Presented at the 84'" Annual Meeting of the Transportation Research Board, Washington D.C.) Lighthill, M. J. and G. B. Whitham. (1955). On Kinematic Waves, i Flow Movement in Long Rivers, ii A Theory of Traffic Flow on Long Crowded Roads. Proceedings of the Royal Society A 229, 281-345.

364 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Mufioz, J. C. and C. F. Daganzo. (2002). Moving Bottlenecks: a Theory Grounded on Experimental Observation. Proceedings of the IS" ISTTT, Oxford, U.K., 441-462. Newell, G. F. (1993). A Moving Bottleneck. Institute of Transportation Studies Report UCBITS-RR-93-3, U. of California, Berkeley, CA. Newell, G. F. (1993a). A Simplified Theory of Kinematic Waves in Highway Traffic: (i) General Theory; (ii) Queuing at Freeway Bottlenecks; (iii) Multi-Dimensional Flows. Transportation Research B 27, 281-313. Newell, G. F. (1998). A Moving Bottleneck. Transportation Research B 32, 531-537. Newell, G. F. (1999). Flows Upstream of a Highway Bottleneck. Proceedings of the 14'" ISTTT, Jerusalem, Israel, 125-146.

19

FIRST-ORDER MACROSCOPIC TRAFFIC FLOW MODELS: INTERSECTION MODELING, NETWORK MODELING Jean-Patrick Lebacque, INRETS-GRETIA2, France, and Megan M. Khoshyaran, ETC Economics Traffic Clinic, France

INTRODUCTION The LWR (Lighthill and Whitham 1955, Richards 1956) model is currently the object of active research efforts, owing to the following facts: it is simple, easily calculable both analytically and numerically, and it recaptures with a reasonable precision many traffic phenomena, and models adequately many traffic situations. It has been implemented in several discretized models, FREFLOW Payne 1971, METANET Messner and Papageorgiou 1990, METACOR Elloumi et al, NETCELL Daganzo 1995, Lo 1999 for instance. Still many improvements are required, among which boundary and intersection modeling are prominent, because they constitute keys to: model identification and calibration using detector data, modeling of large and complex networks, applications to traffic management, such as ramp metering, speed control, dynamic reactive assignment, better understanding of capacity drop, hysteresis. From a methodological point of view, the key to building macroscopic traffic flow models for networks is definition of proper boundary conditions. Since the LWR model results in a system of conservation laws, the analysis of the Riemann problem provides the main tool for defining boundary conditions. The outline of the paper is the following. After a brief review of the literature, the equivalence between supply/demand boundary conditions and classical boundary conditions derived from the vanishing viscosity approach to conservation laws is proven. It is then shown that the simple mathematical intersection models of Holden and Risebro 1995 and Coclite and Piccoli 2002 can be considerably simplified within the supply-demand framework. Not all

366 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction combinations of intersection supply and demand yield consistent intersection models, and a selection criterion is derived, the invariance principle. Two classes of point wise intersection models are introduced. One is based on an optimization principle of integrated supply and demand functions. The second is based on intersection equilibrium models in which the intersection is endowed with basic physical properties: storage capacity, maximum through flow, etc... Both approaches are shown to be equivalent in the case of merges and diverges, and to recapture previous models. A simple merge model is analyzed, and compared with field measurements. Finally, the boundary conditions of the FIFO multicommodity LWR model are analyzed, and the intersection models developed in the paper are combined with this model in order to yield a full network traffic flow model.

A BRIEF REVIEW OF THE LITERATURE The LWR model is expressed by a single conservation law: (1)

^ + ^Qt(K,X) = 0 dt dx with: x, t : the position and time, Q : the flow, K : the density, V : the speed, Qe (K,x) : the equilibrium flow (fundamental diagram). Ve (K,x) denotes the equilibrium speed-density relationship. Link boundary conditions for such hyperbolic systems of conservation laws can be found in the mathematical literature, introduced by the pioneering works of Bardos-LeRoux-Nedelec (BLN) 1979 who used the viscosity method and Dubois-LeFloch (DL) (Dubois and Lefloch 1988) who relied on the Riemann problem approach. These two approaches are known to be equivalent in the scalar case, a result which applies to the LWR model. Both yield existence and unicity of entropy solutions under mild hypotheses in the scalar 1-D case. Further work by Otto 1992 perfected the BLN approach. The reader is referred to Kroner 1997. Mathematicians have been showing some interest in the problem of point wise intersection modeling for the LWR model; see papers by Holden and Risebro 1995, Coclite-Piccoli 2002, Klar and Herty 2004 for instance. The resulting intersection models still lack realism. In the field of transportation relatively little research effort has been devoted to the topic of boundary conditions, and mainly for the LWR model: Lebacque 1996, Lebacque and Khoshyaran 2002, Nelson and Kumar 2004. In order to yield intersection models, link boundary conditions must be combined Buisson et al. 1995-1996, Lebacque and Khoshyaran 2002. Some pointwise intersection models have been proposed in the past (Lebacque 1984, Lebacque 1996, Daganzo 1995, Lebacque and Khoshyaran 2002, Jin and Zhang 2002). The difficulty of specifying boundary conditions has limited the scope of these earlier efforts, mainly limited to discretized models.

Macroscopic traffic models: intersection modeling, network modeling 367

BOUNDARY CONDITIONS AND INTERSECTION MODELING

The supply-demand boundary conditions From the fundamental diagram two equilibrium function can be constructed, the equilibrium supply Se and demand Ae functions. The construction is illustrated by Figure 1 below. At a given point, the local supply and demand are defined as: (2) I.(x,t) = 'Le(K{x+,t),x+) and A(x,t) =

Ae{K(x-,t\x-)

These quantities can be interpreted as the greatest possible inflow resp. outflow at any given location x. The symbols +, - in equation (2) represent right-hand side (resp. left-hand side) limits. Flow must be less than both supply and demand. The entropy solution of (1) is locally flow maximizing, Lebacque 1996. Thus the entropy solution can be characterized by: (3) Q(x,t) = Mm[l.{x,t),A{x,t)] a formula which will be referred to as the min formula in the rest of the paper.

Figure 1: Fundamental diagram, equilibrium supply and demand functions Let us now consider traffic flow on a link. Upstream boundary data is upstream demand AJf), the downstream boundary data is the downstream supply £,(*) Given the link supply iia,i)= I.e(K(a+,t),a)and demand A(b,t)=Ae(K(b-,t\b), we apply the min formula (3) to derive the link inflow Q(a,t) and link outflow Q(b,i) : The upstream demand and downstream supply determine the traffic inside the link. (4) Q(a,t)=Min[Au(t),I.(a,t)] and Q(b,t)=Min[A(b,t),I.d(t))

368 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2: Supply and Demand data for link [ a,b ] The density at the boundaries is given by: V

;

[K(a,t)

if Aa(t)>I.{a,t)

V

' \_K{b,t)

if

S,(?)>AM

BLN (Bardos-LeRoux-Nedelec) boundary conditions. Equivalence with Supply/Demand boundary conditions. The BLN boundary data is the prescription on the boundary of a density-like quantity A. More specifically, Bardos LeRoux Nedelec prove that equation (1) (and more generally scalar conservation equations) admits a unique solution on a link D=[a,b\ given an initial density d'f,

.

on the link D and such that the density at the boundary 8D = \a,b \ relates at all positive times to the boundary data A according to:

{sgn[^,/)-A:]-sga[4C,0-«:]}[e.(^,4c)-e.W]-»(c)>0 \/c&dD = {a,b) and V^>0 The symbol sgn represents the sign function, n(c) is the normal to the domain boundary at c, that is n{a)=-\ and n{b)=\ since (1) is 1-D (Figure 3). D

A(a,t) n(a)

A(b,t)

*

* a

n(b)

b Figure 3: BLN data for a link D = [a,b]

This boundary condition (6) is obtained by analyzing the viscosity solutions of (1), and generalizing standard boundary conditions of the Dirichlet type for parabolic equations. The reader is referred to Kroner 1997 chapter 6, as well as to Otto 1992. The BLN boundary condition (6) at the link entry point can be shown (see Lebacque 2003) to be equivalent to

(7)

K e {A}u [A\K^] ifA< Kcrit , K e [K^K^]

if A > Kcnt

that is to say that the upstream BLN data A is actually equivalent to the demand data A e (^j. The conjugated* of A is such that Qe\A ")=Qe(A)

an(

i A

A'. Proof: see Appendix.

Downstream BLN boundary conditions can be shown (see Lebacque 2003) to be equivalent to

(8)

K e [0,Kcril] if A < Kcrit , K e {A}V

[O,A']

if A > Kcril

Macroscopic traffic models: intersection modeling, network modeling 369

that is to say that the downstream BLN data A is equivalent to the demand data °£C\A)- This result is of course symmetrical to the result for upstream boundary conditions. Pointwise intersections. Both Holden and Risebro 1995 and Coclite and Piccoli 2002 aim at solving a generalized Riemann problem at the intersection. Initial data are the densities Ki0 , Kj0 , which are assumed uniform on semi-infinite upstream links [/] and downstream links [/]. The main problem addressed by Holden and Risebro is: which are the densities Kt, Kj and flows

Q, = Q{K), R; = Qt{Kj), at the node. In both approaches, the following constraints are applied to the Kt, Kj :

' {K.elK^K^} if Ki0>Kicrit t*,6{KroK>[*,,J if Kl0Kiah (9) expresses the BLN boundary conditions between Ku> (boundary data in the BLN sense) and Kt , and between KJO (boundary data in the BLN sense) and Kj respectively. The reader will observe the similitude with (7) and (8). In the supply-demand framework, upstream boundary data is demand Ae(Kl0) and downstream boundary data is supply T,t[KJ0). Thus (9) is equivalent to the simpler condition:

(10)

\ e\

jOJ

^

(

A

R.*lk£°j

K

and K = ^ ) ]

\

V

V

Q 5

/

'- '

if i-T

Rj<^ D

_

^-

Equation (10) expresses that the intersection through flows should be less than upstream demands and downstream supplies respectively that the densities are equal to the initial densities if the flow is not constrained by upstream demands and downstream supplies, and are defined by these constraints otherwise (see Figure 5).

370 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Downstream links (j)

Upstream links (i)

Figure 5: Admissible node states, for downstream and upstream links If traffic states (i)=(KnQ.) and (y)=(^,i? ; ) satisfy the above conditions (9) or (10), it is clear that on each link [z] or [/], the Kto - Kt resp. Kj - Kp discontinuities propagate in the right direction, with <0 resp. >0 speed. This essential fact is demonstrated by the figure 5. Thus the basic variables of the generalized Riemann problem for a point wise intersection are the node inflows 2, and outflows Rj. The constraints that apply to these variables are posivity constraints, conservation constraints, and constraints resulting from (10): ~0
(11)

Ia = I*,

The Holden-Risebro and Coclite-Piccoli approaches resolve the under determinacy of (11) by optimizing a criterion subject to the constraints (11). Holden and Risebro propose to maximize a concave function of the Qt and Rj. A similar idea was developed in Lebacque and Khoshyaran 2002, but based on the zone concept defined for nodes in the STRADA project (Buisson et al. 1995-1996). Coclite and Piccoli propose to maximize the total through flow of the node, ^ Qt = ^ Rj , and to constrain node outflow by assignment constraints. Supplementary constraints can be added to (11). For instance, in an intersection with priority movements conflicting with non priority movements, the non priority flows are constrained by the priority flows. An example of such a constraint is given in Lebacque 1984 (the SSMT model), based on the gap acceptance model of Mahmassam and Sheffi 1981. Other possible constraints are constraints related to the effect of traffic lights, reducing the through-flow. I nvariance principle. In a node model, the through flows (2,) ie; , \R ).

are a function of the node boundary data

(8;.)(E/ , ( o y ) w . If Q,{t)< 5.(/) , then the link exit enters a supply regime and at t+, 8,^) becomes the maximum flow on link [r], i.e. 8 ( (?+)= Qimax Symmetrically, if J? ; (?)
, i.e. <*,(*+) = * , , „ .

Macroscopic traffic models: intersection modeling, network modeling 371

Thus the node through flows (2,),.., , \Rj)

> as functions of the node boundary data (8,).6/ ,

(o, ) w , must be invariant by the following transformation:

(12)

[8.-> Q, if 0 <8. ' if ^
This property will be called invariance principle in the sequel. It must be satisfied, lest the Kw - Kt resp. Kj - KJQ discontinuities propagate into the wrong direction (towards the node). Such models cannot converge as the discretization step becomes vanishingly small. Node models that do not satisfy the invariance principle can only be used as discretized, phenomenological models. The intersection model of STRADA (Buisson et al. 1995-1996) and the distribution scheme of Jin and Zhang 2002 do not satisfy the invariance principle. Let us illustrate this fact on the latter, which is simpler. The idea of the proof is the same for the STRADA model.

Figure 6: A merge model that does not satisfy the mvanance principle Let us consider the merge depicted in Figure 6. The flows into the merge are given by the following distribution scheme: (12)

Q,=S,

if

5>,-<
and

2 , = ^ -

if

^>,><x

k

Let us assume the following values, at a given time t: cr = 3000 vh / h (2 lanes), 8, = 2100 vh / h (1 lane), 82 = 1400 vh / h (1 lane) These values correspond to traffic state («). Traffic states for link 2 are depicted on the right half of Figure 6, relative to the fundamental diagram of that link. Further we assume maximum flow values of Qk,max = 2200 vh / h on both links [1] and [2]. Following (12), the through flows Qi are equal to 2100 3OOO = 2100 + 1400

18OOvh/h

md

Q

lf00 3000 = 1200 vh/h 2100 + 1400

372 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction These values correspond to traffic state (i). This implies that at t+, the supply at the exit of both links is less than demand, hence traffic states (/) are congested and the demand at the node becomes equal to 8* = Qk,max Both through flows now change: =

^25 3000 = 1500 vh/h and Q, = ^ ^ 3000 = 1500 vh/h 2200 + 2200 2200 + 2200

which is traffic states (d) (also congested). Let us illustrate the traffic states for link [2]:

Figure 7 : Traffic states on link [2], at time t+ The velocity of the («) -> (i) Shockwave is less in absolute value than the velocity of the (d) -> (/) rarefaction wave. Thus the state (i) must vanish. As the velocity of the («) -» (d) Shockwave is positive, the state (rf) must vanish as well at t+. Thus the solution (12) must vanish at t+ and cannot apply. The problem is general: the data (upstream demands and downstream supplies) are liable to be instantaneously modified by the through flows, and the modifications must be compatible with the through flows. A sufficient condition for the invariance principle to be satisfied will be given later.

EQUILIBRIUM VERSUS OPTIMIZATION MODELS FOR INTERSECTIONS.

Dynamic node model. A simple dynamic node model, introduced in Lebacque 2003 and Lebacque and Haj-Salem 2004, and which relates to the STRADA exchange zone concept Buisson et al. 1995-1996, can be summarized as follows. The node contains N vehicles and exhibits both supply T.(N) and demand A(N) . These supply and demand functions are assumed to be similar to link supply and demand functions (refer to Figure 1) and express the global behavior of traffic in the node. They imply a storage capacity Nmax, and a maximum through flowgmojt reached for some critical value Ncrit ofN. A partial supply model, supported by empirical data Lebacque and Khoshyaran 2002, is the following linear split model:

Macroscopic traffic models: intersection modeling, network modeling 373

(13)

Z,(^) = P(Z(iV) Vi

The coefficients p. represent the share of available space in the node that is accessible to users from upstream link [/]. Typically, they should be proportional to the number of available def

lanes. It is possible that (3 = ^ ] p \ > 1, if there are more incoming than out coming lanes. The partial demand for downstream link [/] can be assumed proportional to the number NOj of vehicles exiting the node through link [/] : (14)

A (N) =

J

-A(N)

If Ny denotes the number of vehicles in the node, entered through [z], bound to exit the node through [/'] , the following system results, expressing conservation of vehicles [i] -* [/] in the node and the usual Min formula for flows: (15)

N^reQ-^Rj,

with

N = Y,Nu

I : Pointwise intersection model with internal dynamics

Q, = -Mm[S,,fill{N)], Rj = Mini and NO

^-A(N\(TJ\

J =ZX

The coefficients yi}. denote the node turning movement coefficients (the proportion of users of link [z] which use link [/]). The reader is referred to Lebacque and Haj-Salem 2004 for applications of this node concept to traffic hysteresis, capacity drop, speed control, and ramp metering. Equilibrium models. If the model scale is assumed such that iV can be neglected with respect to other vehicle numbers, the dynamics can be neglected and (15) simplifies into the following equilibrium model:

(16)

Q.{ =Mm[Si,fiiZ(N)]

*,--2>*-a=o v/

Vi, Rj = Min [(NOJ IN )A(N ),CT;]

Vy

374 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction in which the N and NOj are unknowns yielding the in and out-flows g, and Rj. This model can be solved because the Qt are decreasing functions of iV, the Rj are increasing functions of TV, and the ratios NOjl N essentially depend on the yjy. In this model, the quantities N and NOj must be considered "abstract", as they are not explicit in the conservation of vehicles through R = the intersection, j ^y,jQi V/

Optimization models. An optimization model, generalizing Holden and Risebro 1995, Coclite and Piccoli 2002 and Lebacque and Khoshyaran 2002, is given by:

Max i> ; (Q.)+2> y (^) j

'

(17)

Qi<Si

X/i, Rj<(Tj

V/

The Coclite-Piccoli model (Coclite and Piccoli 2002) for instance assumes that the total through flow is maximized ($,: identity, ty: null). Here the functions $, and ^/ are assumed increasing and strictly concave, thus the solutions of (17) are unique. The Karush-Kuhn-Tucker optimality conditions for (17) yield: *,-'(&) =

ft*0'

'

Qi^Sn

X,{Si-Q) = Q

Vj

with %i, py, Sy the Karush-Kuhn-Tucker coefficients. The coefficients %,, py, can be eliminated, because of the concavity property of the the functions $, and . Let us consider for instance Qi. Since cD,'1 is decreasing and %i > 0 it follows:

a=« > /" 1 (-Z 5 /*)

if

a«y,i-e. z,=o,

Qi = Si = O,.1"1 - X Vff + ^ P °.-H " E 5 A y v -/ y v J It results:

Q,. =Min S,.,©,.1"1 - ^ y l V j ,

L

.

v / )_

The Karush-Kuhn-Tucker optimality conditions for (17) are

otherwise

Macroscopic traffic models: intersection modeling, network modeling

375

G,. =MmU >
L

V

'

) \

They are necessary and sufficient. The model unknowns are the sj, i.e. the Karush-KuhnTucker coefficients of the flow conservation constraints

The Sj can be determined by a primal Murchland-like algorithm or by solving the dual of (17). The similarity between (16) and (18) is obvious and reflects the fact that both models satisfy the invariance principle. It follows from this principle that upstream demands S; and downstream supplies ff/ cannot be linked directly in order to yield the through flows Qt and Rj, as for instance in Buisson et al. 1995-1996 or Jin and Zhang 2002. They can only be used in conjunction with some node supplies cf)j and demands \pj, yielding the node in- and out- flows: (19) Qi = Min [S,,
(20)

s = N, <6((jS,) = - f r 1 ( 7 / A V 7 ,

X

¥(R)= (^(^drj

In this case the single unknown s is determined by a simple Newton algorithm applied to (18). This result provides an interpretation of the criterion of (18). Case of a diverge. The case of diverges is straightforward in the case of an optimization model. Indeed the constraints applied to the through flows are given by: Q < S , Rj= /jQ < (Tj, hence (21)

Q = Mm[s,MxaJ(crj/rj)\,

Rj=rjQ

376 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The equilibrium model for diverges is not solved as straightforwardly. Nevertheless it can be shown that it yields the solution (21), as shown in Lebacque and Khoshyaran 2002. The values of the ratios NOj IN are given by NOj IN = yj if 8 < Min j (ov ly Under the same condition, N = A"1 (5). If 8 > Min j \(Tj IYj) it is necessary to define j0 = Argy.Miny [tjj Iy

and N is given

1

by N = E" (aM I yh). The ratios NOj IN are given by ^

N

=^

Y

^

O

if

j*jt

J

and ^

w

JO

= 1 -1 ^ = 1 - 1 1 ^ ^ , rr

AT

y

O

Remark The present approach recaptures models such as Holden and Risebro 1995, Daganzo 1995, Lebacque and Khoshyaran 2002, Coclite Piccoli 2002, Klar and Herty 2004, Lebacque and Haj-Salem 2004. Let us consider for instance the merge model in Daganzo 1995. It can be shown (based on the analysis of the following section) that the latter model is equivalent to an optimization model (17) w i t h ^ = 0and <S>, = N ^ [g ; - Qf l{2plQ^)]

STUDY OF MERGES The object of this section is to solve completely the equilibrium model for a simple merge. It will be shown the model predicts a scatter of intersection through-flows. Let us consider the simple merge depicted in Figure 7.

The merge being described by (16), the through flows are given by: e, = Min[«y,,/?,Z(JV)] Vz = l,2

* = Min[A(A0,^] The demands are ordered in the following way:

' n d : *"Ia=0

8 8. —- > ——

P. PP The total inflow, Qj + Q2 is a decreasing function of A'.

Macroscopic traffic models: intersection modeling, network modeling 377

2> ;

if

N<^{8jpn)

0+fi2= Sp+PAN) if ^{SJpn) 5 , + 5 2 , TV = A"1(81+82) If p{8pl pp)< cr<Si+S2, then N = Argv [&p +P n S(v)= crj, i.e. A' = E"1 ((a - Sp )//?„) If a 5,+52,then Qt = 8,., R = 8, + 8 2 = Q1 +Q2 If p{Sp/Pp)
If a
Qp=8p,Q,l=o-8p,R=o

Qi=a(pi/p),R

= Ql+Q2

= CT = Ql+Q2

The through-flows depend on the upstream demands and downstream supply, therefore they cannot be related by a functional relationship, except in the oversaturated case (when a < p{5pl Pp)). Therefore the model predicts a wide scatter of the flow plot. It can be shown that by eliminating the supply and demand from the above expressions, linear constraints result for the through-flows. It follows that the geometrical properties of the scatter plot will not be affected by the sampling interval. Measurements (A4 motorway in the vicinity of Paris, see figure above) and computer simulation data (see figure 8 below) generated according to the equilibrium merge model illustrate this observation. Experimental data and model predictions are compatible. Oltra and Jardin 1996 reported similar experimental scatter of through-flow data.

378 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 8: measurements vs. predictions of the equilibrium merge model

MULTICOMMODITY FIFO LWR MODEL

Introduction In order to complete the description of traffic flow on the network, we introduce the FIFO multicommodity LWR model. In this model, users are disaggregated according to some criterion d, representing the destination, or origin-destination, or path, or user class... The criterion d will be noted as a superscript Thus both density and flow are split according to d. K(x,t)= Yj^iX't)

VX't

d=l...D

^ dt

+

^ : dx

=

o

2(*.0= Y^Q"^) d=\...D

By the conservation of vehicles, (22)

and

W=\...D

Vx r

'

Macroscopic traffic models: intersection modeling, network modeling 379 The specificity of the FIFO multicommodity LWR model is that the velocity of vehicles is assumed to be independent of d. Thus all vehicles have the same velocity

V(x,t)=V(x,t)=Ve{K(x,t\x)

\/x,t, and

\/d=\...D

Let us introduce the composition coefficients rf: Qd=K"V and K" = x"K, Q" = %"Q W Traffic flow is homogeneous, and vehicles leave any link in the same order as they entered it, regardless of their attribute d (hence the name "FIFO"). This model has been studied, especially in its numerical aspects, by several authors, among which Buisson et al. 19951996, Jin and Zhang 2004, Lebacque and Khoshyaran 2001 for instance. The equations of the multicommodity traffic flow are (under conservation form) given by: (23)

^

+

^ ( K ) = 0

dt ox K = ( * ' ) , . , D and F(K)=Ve{K)K dcf

K = ^Kd

= E\K

where £ T = (l...l) and T means transpose

d-\...D

Analysis: Rarefaction waves, contact discontinuities, Shockwaves, Riemann problem After some straightforward algebra, it follows from (22) that the composition coefficients x satisfy the following advection equation and thus are constant along vehicle trajectories, a fact noted by several authors. (24)

^

+ V(K)^~

dt

=0

W =

l...D

8x

The gradient of F, A{K) = VKF{K)=Ve(K)\A + Ve'(K%.E^ identity matrix, admits two eigenvalues, Xf >X_.

with Id referring to the

X_ = Qe '(K), corresponding to a one-dimensional eigenspace X = ( K ) Xt= Ve(K), corresponding to aD-1 - dimensional eigenspace X = {hiE\h = o}. Rarefaction waves. The A. - field is completely non-linear, since VKX = Q"(K)E. choose as representative of X. a vector R- along which X. increases: R_ = - K

thus

VKX_ .R_ = ~Qe"

(K)K>0

We

380 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Integral curves of R. are trivial (straight lines to the origin). If we consider a traffic state Ko , the traffic states K that can be connected to Ko by a rarefaction wave are such that 0 < <j < 1. K = qK0 with g a real number such that: Contact discontinuities. The eigenvalue V is linearly degenerate. Indeed, VKA,T

thus

=VC'{K)E,

VK7,r.h = Ve' (K)E\h = 0 V k l t .

Let us we consider a traffic state Ko . The traffic states K that can be connected to Ko by a contact discontinuity lie in the integral manifold of X+ which contains Ko , that is Ko + X+ . It follows that K = E\K = E\K0=K0. The traffic states K and Ko have the same total density K=Ko and differ only by their composition. Shockwaves. The traffic states K that can be connected to a given traffic state Ko by a Shockwave are such that for some real number s, depending on K and Ko: (25)

F(K)-F(Kj = 5(K-K0)

(Rankine-Hugoniot). It follows that s and K - Ko satisfy 5 ( K - K 0 ) = s ( K - K 0 ) , i.e. are eigenvalue and eigenvector of the operator B (depending on K and Ko) defined as:

B=

^A(K0+t(K-K0))dt

By the Lax entropy condition, when K tends towards Ko and B tends towards A, s should tend towards A. (Ko) and K - Ko should tend towards i?.(Ko) m direction and be opposed to R.(KQ). Some tedious but easy calculations yield the following expression for B: a.

(26)

B = aU + bjr

with

W.(K)-W.(K0) K{K)K\Ko)Ko

K-Ko

We(K)-We(K0),

.

V

J

(K-Kj

°

and We any primitive of Ve. The structure of B is the same as the structure of A, and the eigenelements of B result immediately. B admits two eigenvalues, $

_QXK)-QXK0)

&nd

K-Ka

s

_We(K)-We{K0) K-Ko

corresponding respectively to the one-dimensional eigenspace Y = (b) and to the D-\ - dimensional eigenspace: YT=XT=

{hi'E\h = o}.

If the traffic state K can be connected to the traffic state Ko by a Shockwave, K - Ko is collinear to b, thus K is collinear to Ko . In view of the Lax entropy condition, if K is connected to Ko by a X_ -Shockwave, then K = qK0 with q > 1.

Macroscopic traffic models: intersection modeling, network modeling 381 The Shockwaves and rarefaction waves of the FIFO multicommodity LWR model are the ordinary waves of the underlying LWR model. Inhomogeneous Riemann problem for the FIFO L WR model

Let us consider an inhomogeneous Riemann problem, defined by the traffic states on the left Kf and on the right K r , and by the fundamental diagrams Qe(-^) a n d Qe{-,r) on the left and on the right. Since there are only two eigenspaces which are transversal, and since contact discontinuities propagate at >0 speed, the solution of the Riemann problem will yield one intermediate state Ko which has the same composition as IQ and the same total density as K r : K0=E\K0=E+.Kr=Kr,

(l/A:,)K,=(l/^)K0.

Ko is connected to K r by a contact singularity with velocity Vr = Ve (Kr, r). By multiplying equation (23) on the left by E+, (1) results. The inhomogeneous Riemann problem is solved for the total density K = E+.K in the usual way (as in Lebacque 1996). In the (x,t) plane, the traffic composition is equal to the composition of K? for all points such that x < Vrt and equal to the composition of Kr for all points such that x > Vrt. Network modeling in the FIFO multicommodity LWR model

Flow on links and boundary conditions Flow on links. The total density K follows the LWR model, equation (1). The compositions follow the advection equation (22). The partial flows follow (23). The contact singularity associated to a composition discontinuity always propagates at the speed of the right-hand-side conditions. Thus at a location x at time t , a composition discontinuity propagates at velocity Ve(K{x+,t),x+) (with Ve the equilibrium speed). Discontinuities must be treated as a Riemann problem. Boundary conditions apply notably at entry and exit points of the network. The preceding subsection on the Riemann problem enables us to give the boundary conditions for the FIFO multicommodity LWR model. The total density K follows the LWR model, i.e. equation (1). Therefore the boundary conditions for K are upstream supply and downstream demand, with boundary total flows and densities given by (4) and (5).

382 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction For composition coefficients, the boundary data are upstream composition coefficients in the case of upstream boundary points (entry points). There are no downstream boundary data for composition coefficients.

(a.t)

Q

A

"« ^

V

I^I)'(b)

>

;

Z,(t)

For an entry point (a), the boundary data are the upstream demand Au (t) and upstream composition coefficients %* (t). Thus the total and partial inflows are given by:

GM=Min[A.(4z(a,0] Xd{a,t)=Xdu{t),

Qd(a,t)=Xd(a,t)Q(a,t)

W

The density is given by (5):

[K(a,t)

if

Att(f)>S(a,f)

For an exit point (b), the boundary data are the downstream supply £ s (0- Thus the total and partial inflows are given by:

Q{bj)=Mm[A(bj),Xs{t)l The density is given by (5):

K(b,t+' )J^ [K(b,t) K

lf if

Qd(bjhz"(bj)Q{b,t)

W

*M
I,(f)^A(*,0

Intersections, fixed discontinuities. Depending on the model chosen, the intersection through-flows are calculated according to (16) or (17)-(18). The node demands and supplies are calculated as in (2):

Sl(t)=Ae(K(n-,tj),n;i),


The node turning movement coefficients y,.. (t) are determined using two kinds of information, the composition of traffic exiting links [i] %d (n,t;i), and the assignment coefficients Pjj(?) (fraction of users exiting link [i] with final destination (or class) d, intending to use link [j]). The assignment coefficients are exogenous to the model and must be calculated separately (shortest path, variable message sign, user choice model...).

Macroscopic traffic models: intersection modeling, network modeling 383

Y,-(0=£P;(')X'(».<;0 The densities at the intersection node are calculated according to (5): ,

.v \^(Q(n,t;i\n;i) |_ii:(M;z)

V

'"

if e(n,/;i) <$.(/) if

e(n,r;z) >
rA-;(ij(«, ? ; 7 -) ; «;j)

if

*(„,*; y)<
[AT(»,/;y)

if

R(n,rj)*
Consider now a y?xed discontinuity, i.e. g e is discontinuous at some location. Fixed discontinuities occur at locations where the infrastructure geometry or the traffic regulations (speed limits) change. They can be treated as elementary intersections of the optimization type, in which the total through-flow is maximized (i.e. the min formula (2) and (3)). The usual rules of the LWR model apply to the calculation of total densities. By the RankineHugoniot formula (25), all flows (partial and total) are conserved through the discontinuity, the velocity of which is null. The traffic composition is conserved through the discontinuity.

CONCLUSION Based on the BLN boundary conditions, the paper proves that local traffic supply and demand define the boundary conditions for the LWR model. A rigorous methodology for intersection modeling within the LWR framework is proposed, based on the supply/demand concept and the Riemann problem for nodes. The invariance principle is shown to be necessary for intersection models to be consistent. Two models, the equilibrium and the optimization model, are introduced which satisfy the invariance principle. Analytical solutions are given for the merge and diverge models, and in these special cases optimization and equilibrium model are shown to be equivalent and analytically tractable. The merge model is shown to be compatible with experimental data, and to predict wide scatter of flow plots. Finally, the analysis of the Riemann problem for the multicommodity FIFO LWR model makes it possible to specify composition dynamics and boundary data. Thus the intersection models, the boundary data and the flow models can be combined into a single network traffic flow model. Future research should include modeling of heterogeneous traffic for non FIFO intersection modeling, as well as carrying out experimental tests for intersection models.

REFERENCES Buisson, C , Lebacque, J.P., Lesort, J.B. (1995-1996). 1. Macroscopic modelling of traffic flow and assignment in mixed networks. Proc. of the Berlin ICCCBE Conf. 13671374. (ed. PAHL, P.J., WERNER, H.). 1995. 2. STRADA, a discretized macroscopic model of vehicular flow in complex networks based on the Godunov scheme. Proc. of

384 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the CESA'96 IEEE Conference. 1996. 3. The STRADA model for dynamic assignment. Proceedings of the 1996 ITS Conference. 1996. Bardos, C , LeRoux, A., Nedelec, J.C. (1979). First order quasilinear equations with boundary conditions. Comm. partial Differ. Equations A, 1017-1034. Coclite, G.M., Piccoli, B. (2002). Traffic flow on a road network. Technical Report, SISSA. Daganzo, C.F. (1995). The cell transmission model 2: network traffic simulation. Transportation Research 29B. 2: 79-93. Dubois, F., LeFloch, P. (1988). Boundary conditions for nonlinear hyperbolic systems of conservation laws. J. Differential Equations 71, 93-122. Elloumi, E., Haj-Salem, H., Papageorgiou, M. (1994). METACOR, a macroscopic modelling tool for urban corridors. TRISTAN II Int. Conf, Capri. Holden, H., Risebro, N. (1995). A mathematical model of traffic flow on a network of unidirectional roads. SIAMJ. Math. Anal. 4, 999-1017. Klar, A., Herty, M. (2004). Modeling of traffic flow networks. Proceedings of the Autumn school: "Modelisation mathematique du trafic automobile" (Editors: J.P. Lebacque, J.P. Quadrat, M. Rascle). In press. Kroner, D. (1997). Numerical schemes for conservation laws. Wiley-Teubner. Jin, W.L., Zhang, H.M. (2003). On the distribution schemes for determining flows through a merge. Transportation Research 37B, pp 521-540 . Jin, W.L., Zhang, H.M. (2004). A multicommodity kinematic wave simulation model of network traffic flow. TRB 2004 Annual Meeting. Lebacque, J.P. (1984). Semi-macroscopic simulation of urban traffic. Int. 84 Minneapolis Summer Conference. AMSE. Lebacque, J.P. (1996). The Godunov scheme and what it means for first order traffic flow models. Transportation and traffic flow theory, Proceedings of the 13th ISTTT (J.B. Lesort Editor). Pergamon. Lebacque, J.P. (2003). Intersection modeling, application to macroscopic network traffic flow modeling and traffic management. To be published in the Proceedings of the TGF'03 Traffic Granular Flow Conference, (Delft). Lebacque, J.P., Haj-Salem, H. (2004). Ramp metering, speed management and Braess-like paradoxes. Preprints of the TRISTAN V Symposium. Lebacque, J.P., Khoshyaran, M.M. (2001). Modeling vehicular traffic flow on networks using macroscopic models. In Finite volumes for complex applications II (R. Vilsmeier, F. Benkhaldoun, D. Hanel editors). Hermes Science Publications. Lebacque, J.P., Khoshyaran, M.M. (2002). Macroscopic flow models. Presented at the 6th Meeting of the EURO Working Group on Transportation. Published in "Transportation planning: the state of the art" (Editors: M. Patriksson, M. Labbe), 119-139. Kluwer Academic Press. Lo, H.K. (1999). A dynamic traffic assignment formulation that encapsulates the Celltransmission model. Transportation and traffic flow theory, Proceedings of the 14th ISTTT (A. Ceder Editor). Pergamon. Lighthill, M.H., Whitham, G.B. (1955). On kinematic waves II: A theory of traffic flow on long crowded roads. Proc. Royal Soc. (Lond.) A 229: 317-345. Mahmassani M., Y. Sheffi. (1981). Using gap sequences to estimate gap acceptance functions. Transportation Research 15B, pp 143-148.

Macroscopic traffic models: intersection modeling, network modeling 385 Messner, A., Papageorgiou, M. (1990). METANET: a macroscopic modelling simulation for motorway networks. Technische Universitat Miinchen. Nelson, P. and N. Kumar. (2004). Point constriction, interface and boundary conditions for the kinetic-wave model. TRB 2004 annual meeting. Oltra J., Jardin P. (1996). Etude des ecoulements de trafic sur les voies rapides pendant la congestion. Recherche Transports Securite 54 pp 3-14. Otto, F. (1992). First order equations with boundary conditions. SFB 256. Preprint 234, Bonn. Payne, H.J. (1971). Models of freeway traffic and control. Simulation Council Proceedings 1: ch6. Richards, P.I. (1956). Shock-waves on the highway. Opns. Res. 4: 42-51.

APPENDIX: SKETCH OF THE PROOF OF (9), (10) We shall check in this section, in the case of left-hand-side (upstream boundary conditions), that (6) is equivalent to an upstream demand boundary condition. More precisely the BLN boundary data. A(a,t) is equivalent to a demand boundary data Ae(A(a,t),a). Although some algebra is required, proving the equivalence is relatively straightforward. Symmetrically, in the case of right-hand-side (downstream) boundary conditions, the BLN boundary data A(b,t) is equivalent to a supply boundary data ~Ze(A(b,t),b). The proof is analogue to the proof given below for upstream boundary conditions. First, using the properties of properties of the sgn (sign) function, we rewrite (6) as follows (Al) {sgn[K-K]-sgn[A-K]}[Qe(K)-Qe(>c)}.n>0 \/Kel(a,b) with

I(A,K) = \ V ; \[K,A]

ifK
Note that we skip the time dependency of variables: K means K(a+,t) with a the upstream extremity of the link, andyl stands for A(f). n stands for n(a). The first point is that: / \v 1 \A M \2^[K-A] sgn ^ - x - - s g n ^ - ^ = [0

if

Kel(A,K)

. , yKel{a,b)

if K<£l(A,K)

The second point is that «=-l for upstream boundary conditions, thus (Al) is equivalent to:

sga[K-A][Q.(K)-Q.(K)]£0

VKel{a,b)

which translates into:

ra(^)=MinMkJC]aW a K>A [&(K) = M a x v s M e » if K
386 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

A* denotes the conjugate of A with respect to Qe : ,4* must satisfy both Qe {A) = Qe [A") and A

A*. Let us recall that Kmax, Kcr!t denote respectively the maximum and critical density.

Let us analyze (A2). 1.If K
Msx.rm[KiA] ge(*r) = Qe{A) and it follows from (A2) that K=A.

2. If K < Kcril < A then:

Msx.re[K
3. If Kcril
M a x ^ ^ Qe(ic) = Qe(K) in accordance with (A2).

then:

4. If Kcril < A < K, then:

m

and it follows from (A2) that K=KcrU.

M i n ^ Hge(x:)= QS{K) is satisfied because Qe is decreasing

on [Kcri,,KmJ . 5. If A < Kcrit < K , then Qe (K) A*. 6. If A < K < Kcril, then Qe (K) KcrU:

K^K^K^}

A cannot be prescribed: K is ^A in most cases. This fact is a consequence of the way information propagates in the LWR model. The propagation velocity depends on the traffic state. Let us interpret the cases in which K is ;44. A
:

A >Kcrit and K e [Kcnt,Kmax) :

E e {K) = Qe{K) < Qe(A) = A e (A)

S , W = a W ^ =AeU)

All these cases are characterized by: T.e(K)< ke(A) (suPPfy regime). The case A
A,w=aw<e-B=£.(^) In other words, the boundary data A in the sense of BLN is equivalent to the boundary data Ae(A) in the supply-demand sense. In the case of downstream boundary conditions (i.e. A = A (b,t), K = K (b-,t)), the following admissible values of .K result:

(A4)

A
Ke[0,Kcril]

and A>Kcril:

Ke{A}v[o,A']

20

REAL-TIME ESTIMATION OF TRAVEL TIMES ON SIGNALIZED ARTERIALS Alexander Skabardonis and Nikolaos Geroliminis, Department of Civil and Environmental Engineering & Institute of Transportation Studies, University of California at Berkeley, USA

INTRODUCTION Continuous traffic growth through developed areas and difficulties in building new infrastructure have caused a need for careful monitoring of operating conditions on existing transportation facilities and development of improved strategies to manage the increase in traffic demand. The analysis of existing conditions and evaluation of alternative improvements requires estimates of performance measures for both system operators and users. Travel time is an important measure to assess the existing operating conditions along signalized arterials, evaluate control and management strategies and provide information to travelers. Conventional approaches for estimating travel times on arterial links include speed vs. volume/capacity ratio relationships, mostly used in four-step planning models (BPR, 1964, Akcelik, 1991) or procedures based on the Highway Capacity Manual (HCM) (Transportation Research Board, 2000). The HCM procedure calculates average travel time as the sum of the running time and the intersection delay. The running time is based on the time required to traverse the arterial link based on the arterial design characteristics and the signal density. The intersection delay is calculated based on a deterministic point delay model. Such approaches are not suited for real-time applications especially under congested conditions. Existing real-time performance measurements are typically restricted to the estimation of local intersection-based measures for signal timing parameter tuning or qualitative measures of performance to provide collective traffic information, e.g., through changeable message sign (CMS) displays (Perrin et al, 2002). However, this qualitative information cannot provide to drivers with an accurate representation of the changes in actual traffic conditions, including dynamic travel time variations along urban arterial routes.

388 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Several models have been proposed for estimating travel times from surveillance data. One of the simplest methods estimates the average speed from the loop detector flow and occupancy data, assuming an average vehicle length. These speed estimates however are inaccurate for most traffic environments because they are obtained at the detector location (usually in the middle of the link) and ignore the delay at the intersection, except for short congested links where the detector occupancies reflect the true operating conditions. Other similar approaches use double loop detectors or procedures to estimate the average vehicle length, similar to those techniques used in travel time estimation on freeways (Oda, 1990). Other approaches include statistical models based on regression analysis to estimate link travel times as a function of the site characteristics (spacing, free-flow speeds, saturation flows) and loop detector data. Both linear (Turner et al., 1996), non-linear (Zhang, 1999) and Bayesian (Frechette and Khan, 1998) model formulations have been proposed. Most of these models however, are quite site specific and cannot be readily applied to other environments. Recently, advancements in sensor technology have produced sensors that can provide the magnetic vehicle signatures to re-identify individual vehicles (or vehicle platoons) at a downstream location for travel time estimation (Sun et al, 1998, Lucas et al, 2004). However, the effectiveness of such systems has not been thoroughly tested and their costs are much higher as compared to conventional loop detectors, which make the practical implementation prohibitively expensive especially on large signal systems. A number of approaches employ neural network, fuzzy set theory to develop arterial travel time prediction models that involve the use and fusion of multiple data sources-loop detectors, probe vehicles, license plates (Zhu and Wang, 2002, Cheu et al., 2001, Takaba, 1991). Most of such models have been developed and tested under simulated traffic conditions. The objective of this research is to develop and test a model for real-time estimation of travel times along signalized arterials based on surveillance data (flows, occupancies) from conventional loop detectors and signal settings (cycle length, green times and offsets) from the signal control system. The research is part of a study to develop and implement a performance measurement system for arterial networks (Skabardonis et al, 2004). The next section of the paper describes the formulation of the proposed model. The estimation of the model parameters and the implementation of the model is described in section 3. The application of the model to two arterial networks is presented in Section 4. The last section summarizes the study findings, and outlines ongoing and future work.

PROPOSED MODEL The proposed model estimates the travel time on an arterial link as the sum of the free flow time and the delay at the traffic signal. The model formulation is designed for arterial systems with detectors placed sufficiently upstream from the intersection stopline (system detectors), so the flow and occupancy measurements are not affected by the presence of queues at the traffic signal.

Real-time estimation of travel times on signalized arterials 389 The free flow time is the time a vehicle needs to travel the length of the link without interference from the presence of the signal. This is the time traveling at or close to the freeflow speed and can be readily measured from the loop detectors under low flow conditions. The delay at the traffic signal is calculated as the sum of a) the delay of a single vehicle approaching the traffic signal b) the delay because of the queues formed at the intersection, and c) the oversaturation delay, the additional delay caused when the arrival rate is greater than the service rate at the signal. Figure 1 shows the structure of the proposed model.

Figure 1. Proposed model for delay estimation

Delay of a single vehicle at a traffic signal The delay of a single vehicle ds at a traffic signal, assuming no interactions with other vehicles is defined as the additional time over "free flow speed time" and is the sum of two delays: a) the deceleration plus idle delay dsi, which is the additional time required to decelerate and stop at the stopline plus the waiting time at the stopline, and b) the acceleration delay ds2, which is the additional time required to reach the free flow speed when the signal turns green. Figure 2 shows typical trajectories of vehicles approaching a traffic signal, and illustrates the definition of the delay ds and its components dsl and ds2 for vehicle B. The signal settings at the intersection are known, where g is the effective green time and r is the effective red time. We introduce the following two decision points (Figure 2): 1. Is' decision point: it is located at distance LQ upstream from the intersection stopline where the driver decides whether to decelerate with normal deceleration rate yj because the signal is red or to continue traveling at the same speed. If T is the driver's reaction time and Uf is the free flow speed, then the distance Lg is:

390 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

L0=T-uf+^-

(1)

2. 2nd decision point: it is located at distance Lgm upstream from the stopline where the driver decides whether to decelerate with maximum (emergency) deceleration rate jdm because the signal is red or to continue traveling at the same speed. Lom is also given by the Equation (1) by using the value of y,tm instead of -fa. The first delayed vehicle is vehicle A in Figure 2, which reaches the 2nd decision point at t=0 (start of red time). This vehicle arrives at the 1st decision point at time to before the start of red time:

< , = ^ =^ - - L ]

(2)

Consider a vehicle which reaches the 1st decision point when the signal turns red (t=0). The delay of this vehicle is the sum of the two predefined types of delays for t=0: dsl(0) = r-T—LLJjL=r-T «/ ds2iQ)=^-

! 2yd

(3) (4)

Note that the acceleration delay dS2 is the same for all vehicles that stop at the intersection because each vehicle accelerates from u=0 to u=Uf with normal acceleration rate ya. Vehicle C reaches the stopline when the signal turns green. This is last vehicle to stop at the signal (Figure 2). It arrives at the 1st decision point at time tc before the start of green: tc=T + -L

rd

(5)

Real-time estimation of travel times on signalized arterials 391

Figure 2. Trajectories of vehicles approaching a traffic signal

For the time interval —to < t
ds2(t) = ds2(0)

for -to
for -to
(6)

(7)

If a vehicle arrives at the first decision point at time t>r-tc (vehicle D in Figure 2) it decelerates but the signal turns green before it stops so it accelerates from a speed u ?©. Let t' be the remaining time from the beginning of green when the vehicle reaches the 1st decision \>omt(t' = r-t). The vehicle decelerates from Uf to u in t'-T time units by traveling with deceleration rate %. Then, it accelerates from u to «/by traveling distance 5 with acceleration rate ya- After some manipulations to estimate the distances traveled in the deceleration and acceleration modes, the delays dsi and dS2 are given by the formulae:

dsl(t') =

sl(t')

uf

= 0.5yd{-^uf

rfrt(0=^^=>^(0=^(0

(8)

(9)

392 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The delay ds over the signal cycle as a function of arrival time t at the 1st decision point is given by Equation (10):

( ur -t+\r-T——+—

uf\

d,(t)= 0.5J ^ " + ^ | ( r l T) I 7a J «f 0

' , r-t

-to
(10)

, otherwise

Figure 3 shows the delay of a single vehicle as a function of time t, where to is the time required to decelerate and stop from the first decision point. This function is partly linear and partly quadratic, but continuous and differentiate in its domain.

Figure 3. Delay of a single vehicle arriving at 1st decision point Delay at a single signal because of queue This is defined as the delay because of the queues present at the traffic signal. In the absence of queues all the vehicles would depart by following the trajectory BC in Figure 4, so the queuing delay is the area of the triangle BCD. As it is shown in Figure 4, the delay increases from 0 to a maximum delay at point C, and then decreases to zero for the vehicle that passes through point D in the x-t diagram. The distance Lqm is the distance at which the queuing delay is maximum, and the distance Lq is the maximum back of the queue (the farthest point the queue extends throughout the cycle).

Real-time estimation of travel times on signalized arterials 393 We estimate the queuing delay according to the kinematic wave LWR theory (Lighthill and Whitham, 1955 and Richards, 1956), so to explicitly consider the temporal and spatial formation of queues. We assume a piecewise linear flow-density relationship (fundamental diagram) with parameters Uf (free-flow speed), c (capacity), kj (jam density), and w (congested wave speed) shown in Figure 5. The queuing delay dq of the «-th vehicle arriving at the signal from the beginning of the red time is estimated by considering the following three types of delays (Figure 4): 1. The positive delay dqi which is measured from the vertical axis crossing point B for 0 <x
Figure 4. x-t diagram at a single signal

394 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 5. Assumed flow density diagram

From the geometry of triangles ABC and ABD we obtain: r =^ + Uw

^

^

Jh^L.

L

Uj

(11)

Uf+Un

r = -*-—"— => L=r uw w

2w — uw

(12)

where uw is the speed of the Shockwave shown in Figure 5. The number of stopped vehicles in the distances Lqm andLq are: Nqm= ^f-

and N,= ^-\

(13)

where: [x] : integer part of number x Ls : the effective length of a stopped vehicle, i.e., the reciprocal of the jam density kj The delay of the »-th vehicle for n
(14)

dq2 =-(min(max(«, Nqm), Nq)-Nqm\^-

(15)

d,3=(min(»,7V,)-l)A

(16)

Real-time estimation of travel times on signalized arterials 395 The total delay of the first k vehicles for n
^fmin(K,^J-l

= s

* h{

u,

rmn{ra^{n,Nqm),Nq)-Nqm

Uw

+

mm(n,Nq)-\)

w

)

(17)

The effect of signal offsets in the delay estimation The estimation of intersection delays along an arterial or a network with coordinated signals, must explicitly consider the offsets between adjacent signals. The offsets determine the arrival pattern of vehicle platoons at successive intersections and greatly affect the vehicle delays. Under favorable progression most of the vehicles travel without stops and delays between successive intersections ("green wave"). On the other hand, "bad" offsets cause high delays and may result in spillovers especially for short signal spacing. The green time that can be utilized by the vehicle platoon at the downstream signal depends on the offset o, the distance L between the adjacent signals and the speed of vehicles. Therefore in order to apply the proposed base model for delay estimation we need to modify the phase durations at the downstream intersection. Assuming that the first vehicle departs from the upstream intersection at the beginning of the green with free flow speed Uf, then we have the following two cases as shown in Figure 6. 1. the first vehicle departing from the upstream intersection arrives during the green time at the downstream signal (Figure 6a):

g =8 L

' \ Ar)

as)

r' = r 2. the first vehicle departing from the upstream intersection arrives during the red time at the downstream signal (Figure 6b): g' =g

r

="-{o-yUfj+tq

where tq is the portion of the green time that the vehicles have to stop because of the queue presence. This time interval is calculated by considering the speeds of the Shockwaves (Figure 4), where Nq is the number of delayed vehicles:

r . - n J ^ +^

l

(20)

396 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 6. Offsets and platoon arrivals at adjacent intersections Oversaturation delay Oversaturation delay is defined as the additional delay caused when the arrival rate exceeds the service rate at the traffic signal. If the queue discharge time tq calculated in Equation (20) is equal to the available green time g, then a number of arriving vehicles cannot be served in this green phase, but they will be served in the next cycle (Figure 7). So, the oversaturated delay is estimated as follows: 1. The delay of the vehicles that are not served in the cycle they arrive, originally estimated from Equations (10) and (17), must be increased by the amount of the red interval (r). 2. The green time for the vehicles arriving in the next cycle must be reduced by the time interval required to serve the residual queue from the previous cycle. Under heavy congested conditions this time interval may be larger than the green time, which means that in this cycle only vehicles that arrived in the previous cycle(s) will be served, and some vehicles will be delayed by more than "red time" interval.

Figure 7. Illustration of oversaturation delay

Real-time estimation of travel times on signalized arterials 397 Platoon dispersion Traffic departing a traffic signal initially moves as a tight platoon with short vehicle headways. The platoon spreads out the farther downstream it travels because of different vehicle speeds, lane changes, merging or weaving, and other interferences (parking, pedestrians, and other frictional effects). Thus, the arrival pattern at the downstream signal (and the delays) are affected by the platoon dispersion. Our delay estimation model described above, does not consider platoon dispersion because according to the assumed piecewise linear q-k diagram all vehicles depart from the stopline at the free flow speed and all the flows are at capacity. The platoon dispersion is modeled by applying the kinematic wave theory and assuming a non-linear (concave) q-k relationship (Figure 8). The parameters and form of the non-linear part of the diagram is estimated using data from the 2000 Highway Capacity Manual for undersaturated conditions. The first vehicle departs from the intersection stopline at free flow speed, but because of interfaces between different points of q-k diagram, the following vehicles cannot depart at free flow speed but at a lower speed. The rate of platoon dispersion depends on the curvature of the increasing part of the q-k diagram. We estimate the average platoon dispersion ratio at a distance L, from the intersection stopline using a numerical approach with finite differences. The mathematical formulation and detailed description of the methodology are described elsewhere (Geroliminis and Skabardonis, 2005). Figure 9 shows the average platoon dispersion as a function of distance.

Figure 8. Assumed q-k diagram for modeling platoon dispersion

398 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 9. Mean Platoon ratio as a function of distance L from the intersection stopline

MODEL IMPLEMENTATION Estimation of the model parameters The default values of the parameters needed for the estimation of delays for a single vehicle (Equation (10)) T, ya ya ("fam) are shown in Table 1. These values were obtained from published sources (ITE, 1999). These values also are in close agreement with measured values obtained from the trajectories of floating cars equipped with GPS system. In general, the values of those parameters vary among drivers and depend on vehicle characteristics and roadway conditions. However, the results from sensitivity analysis indicate that the delay ds is fairly insensitive for a wide range of values of these parameters (Table 1). The maximum difference in delay estimates is 5 seconds. Table 1: Model parameters for estimating delay for a single vehicle Model Parameter T (sec) Id (ft/sec2) ya (ft/sec2)

Default Value 1.2 10 3.6

Range

Max Difference in Delay (sec)

0.7 - 2.5 5.0-14 2.4-9.2

5 sec

The parameters for the fundamental diagram uf (free-flow speed), c (capacity), kj (jam density), and w (congested wave speed) are obtained from detector measurements along the links of the arterial network. The loop detectors provide flow and occupancy measurements. The average speed is estimated assuming a vehicle effective length (i.e., the average length of the vehicle plus the detector length), typically 20-24 ft. Direct measurement of vehicle speeds can be obtained from double loop detectors. Figure 10 shows a plot of flow vs. occupancy detector data from the Los Angeles ATS AC (Automated Traffic Surveillance and Control) central traffic control system. The detectors are located on each lane approximately 300 ft upstream of the intersection stopline. Vehicle counts and occupancy from each loop detector are collected every 30 seconds and stored in the ATS AC database. The detector data shown in Figure 10 represent one minute

Real-time estimation of travel times on signalized arterials 399

measurements for the through traffic aggregated across all travel lanes, averaged over multiple link locations. We obtain the model parameters based on the best statistical fit to the field measurements. The typical values obtained from the Los Angeles Lincoln Avenue arterial are c = 1,800 vph/lane, w/ = 35 mph, kj =180 veh/lane-mile and w = 14 mph. These values are in agreement with values for these parameters reported elsewhere.

Figure 10. Flow-density diagram: loop detector data

Implementing the model The implementation of the analytical model to estimate travel times from real-time surveillance data first involves the development of a database with a) signal timing plans at each intersection (cycle length, green times and offsets) for each time period of the day, and b) data from each loop detector in the arterial system. Next, the model parameters should be calibrated from the detector measurements as previously described. Input to the model are the flow q and occupancy data O from detectors at 30 second intervals. The data are stored and verified for accuracy and then processed as follows: 1) The time of vehicle arrivals at the downstream signal is estimated using the speeds estimated at the detector location. The platoon dispersion analysis is engaged for distances longer than 500 ft (platoon ratio smaller than 0.9).

400 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 2) The model adjusts the phase times at the downstream signal considering the signal offsets and the queue discharge time tq from the previous cycle, and determines whether the first vehicle arrives during the green time based on the arrival times from Step 2. The queue discharge time tq = 0 in the first signal cycle. It is also assumed that are no residual queues at the beginning of the analysis period. a. If the first vehicle arrives during the green time, then the number of vehicles crossing the intersection without delay is g' q, where g' is the adjusted effective green time and q is the measured flow in this time interval. If g' > 30 sec then all vehicles are undelayed. If g' < 30 sec, the remaining platoon of vehicles (30 -g')-q is delayed: i. The delay is estimated from Equations (10) and (17) ii. The red time of the next cycle is extended by the amount of queue discharge time tq using equation (20) for Nq equal to the number of delayed vehicles iii. The average delay of all vehicles is the weighted average delay of delayed and uninterrupted vehicles (sec/veh) b. If the first vehicle arrives during the adjusted red phase, the delay is calculated from Equations (10) and (17) for all vehicles. If the number of delayed vehicles is greater than the capacity at the signal g ' c, the delay of the unserved vehicles (residual queue) increases by one red time interval, as described in the estimation of oversaturation delay. Under heavy congestion the queue length may exceed the available link length and blocks the outflow from the upstream intersection. This condition is detected in the model using the flow and occupancy measurements. If the occupancy exceeds 40 percent and the flow (service rate) is less than the capacity, then the service rate at the signal is affected by spillback. The model accounts for the blocking effect by calculating the green time available as the ratio of the measured outflow over the capacity rate.

MODEL APPLICATION The proposed model was applied to two arterial test sites: The M street arterial in Washington DC, and Lincoln Avenue in Los Angeles. The first site is typical of urban arterials in downtown areas; the second test site is typical of multilane major arterials on urban/suburban areas.

Real-time estimation of travel times on signalized arterials 401 M Street, Washington DC The study section consists of seven closely spaced signalized intersections. The average signal spacing is 460 ft. There are two through lanes in each direction of the arterial and the free-flow speed is 30 mph. Most of the signals are two phase fixed-time operating on a common cycle length of 60 seconds. Signal coordination favors the eastbound direction on the arterial. Data on the study network, geometries and traffic volumes were available from previous studies. The study site has also been coded into the CORSIM microscopic simulation model (FHWA, 2003) and the model was calibrated in the previous studies. To apply the model, first loop detectors were coded in each link to obtain the loop count and occupancy for input to the analytical model. Also, a time-varying traffic demand pattern was input to the CORSIM model starting with low flows increasing to higher flows and then decreasing back to lower flows to emulate the growth and decay of traffic demand. Figure 11 shows the model estimated travel times vs. CORSIM predicted values for each signal cycle. The results show that the model predicts travel times per cycle with an error smaller than 5% in most signal cycles. The differences are mostly due to the variability in the driver/vehicle characteristics generated by the CORSIM model. Note also that under high flows the travel time in the westbound direction increases dramatically as opposed to the eastbound direction. This is because the offsets favor signal progression in the eastbound direction which result in high delays and long queues westbound. The proposed model predictions correctly track the pattern of demand variations and delays over the analysis period.

Figure 11. Simulated vs. model predicted travel times on M Street arterial

402 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Lincoln Avenue, Los Angeles The selected test site is 1.42 mile long stretch of a major urban arterial north of the Los Angeles International Airport, between Fiji Way and Venice Boulevard in the cities of Los Angeles and Santa Monica. The study section includes with seven signalized intersections with link lengths varying from 500 to 1,600 feet. The number of lanes for through traffic per link is three lanes per direction for the length of the study area. Additional lanes for turning movements are provided at intersection approaches. The free flow speed is 35 mph. Traffic signals are all multiphase operating as coordinated under traffic responsive control as part of the Los Angeles ATSAC central traffic control system. System loop detectors are located on each lane approximately 300 ft upstream of the intersection stopline. Detectors are also placed on the major cross street approaches. Data (vehicle count and occupancy) from each loop detector are collected every 30 seconds and stored in the ATSAC database. A field study was undertaken obtain a comprehensive database of operating conditions in the study area. First, basic data on intersection geometries, spacing and free-flow speeds were obtain from field surveys. Next, manual turning movement counts at each intersection and floating car studies were undertaken for a four hour period (6-10 am) on Wednesday May 26.thi The study period enabled us to obtain data for a wide range of traffic conditions (from low volume off-peak conditions, peak period conditions and post-peak mid-day flow conditions). Floating cars runs were performed at 7 minute headways. The floating cars were instrumented with laptop computers and GPS units that recorded vehicle location and speed on each second. Finally, the loop detector and signal timing data for the study period were obtained from the ATSAC database. Traffic demand is high especially during the peak hour. Traffic volumes are heavily directional with the higher through and turning volumes in the northbound direction. The average travel speeds on the test section are 25 mph during the offpeak times and drop to about 10 mph during the peak hour in the heavily traveled northbound direction. The average travel speeds are about 25 mph and remain fairly constant in the southbound direction throughout the analysis period. System cycle lengths range from 100 seconds early in the analysis period (6:00 to 6:30 am) to a maximum of 150 sec during the periods of highest traffic volume (7:30 to 8:30 am). Figure 12 shows a comparison between the loop and manual counts on intersection approaches at the test sites (a total of 336 data points). The results verify that the loop detectors are working properly at the study section. The loop counts on the average are about 6% higher than the manual counts, with most of the differences on the high through volume approaches. This "undercounting" of manual counts translates to about 2 veh/lane/signal cycle which could be expected in manual counts at multilane intersection approaches. The proposed model was then applied to the test using as inputs the loop detector and signal timing data from the ATSAC system to estimate the link and total travel times on each travel direction. The test section was also simulated using the CORSIM model. Figure 13 and 14

Real-time estimation of travel times on signalized arterials 403 show for each travel direction the field measured travel times from the floating cars, the predicted travel times from the proposed model, and the CORSM simulated travel times.

Figure 12. Lincoln Avenue: comparison of manual and loop counts

Figure 13. Travel time on Lincoln Avenue northbound direction

404 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 14. Travel Time on Lincoln avenue southbound direction

The results show that the model accurately estimates the arterial travel times on both directions of the arterial over the entire analysis period. The results also show that the CORSIM model can reasonably replicate observed conditions so it can be used as a supplementary data source to test the model predictions under different demand patterns and control scenarios. The evaluation of the differences between field data and model predictions particularly during the peak periods should take into consideration that the model predicts the average travel time per cycle, while the field measurements represent the average of a sample of vehicles throughout the signal cycle.

CONCLUSIONS An analytical model is proposed to estimate the travel times on arterial streets based on data commonly provided by system loop detectors (flow and occupancy) and the signal settings (cycle length, green times and offsets) at each traffic signal. The model is based on kinematic wave theory to model the spatial and temporal queuing at the traffic signals and explicitly considers the signal coordination in estimating traffic arrivals at the intersection. The model is straightforward to implement and unlike other approaches does not depend on site specific parameters or short term traffic flow predictions that make very difficult their transferability to other locations.

Real-time estimation of travel times on signalized arterials 405 The application of the proposed model on two arterial sites and comparisons of the estimated travel times with simulated and field data show that the model accurately predicts travel times on the selected sites. The difference between predicted and measured link travel times per cycle was within 5 percent under a wide range of traffic conditions. For comparison purposes, we estimated the travel times on both test sites using several commonly used travel time estimation methods discussed in the introduction. The results indicate that these methods produced large differences against field and simulated values (Skabardonis, et al, 2004). For example, the method based on the average speeds estimated from the loop detector data (spot speed method) underestimates travel times by 36 percent on M Street arterial and overestimates more than 100 percent on Lincoln Avenue site. These results indicate that the proposed model is superior to existing approaches and can provide estimates that are accurate for both operational analyses and real-time monitoring of traffic conditions. Ongoing work by the research team includes a) the further testing and evaluation of the model on other arterial sites, b) the refinement of the model to estimate additional performance measures commonly used by operating agencies, e.g., number of stops, cycle failures, travel time reliability and c) its implementation in a pilot arterial performance measurement system for the city of Los Angeles (APeMS) similar to the PeMS system for freeways (Choe et al, 2002). The data from loop detectors and signal settings from the ATSAC system will be transferred to UC Berkeley over the internet. The APeMS system will process and verify the data, and calculate the performance measures in real-time. Current and historical information on the performance measures will be provided via a standard web interface accessible via the internet.

REFERENCES Akcelik, R. (1991). Travel time functions for transport planning purposes: Davidson's function, its time-dependent form and an alternative travel time function. Australian Road Research, 21, 49-59. Bureau of Public Roads (1964). Traffic Assignment Manual. U.S. Department of Commerce, Washington, D.C. Cheu, R.L., Lee, D-H, and C. Xie. (2001). An arterial speed estimation model fusing data from stationary and mobile sensors. Proceedings of the IEEE Intelligent Transportation Systems Conference, Oakland, CA, pp. 573-578. Choe, T., A. Skabardonis and P. Varaiya.(2002). Freeway performance measurement system: operational analysis tool. Transportation Research Record, 1811, 67-75. Federal Highway Administration (2003). CORSIM Version 5.1 Users Manual. US Department of Transportation, Washington DC. Frechette, L.A., and A.M. Khan (1998). Bayesian regression-based urban traffic models. Transportation Research Record, 1644, 157-165.

406 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Geroliminis N. and A. Skabardonis (2005). Prediction of arrival profiles and queue lengths along signalized arterials using a Markov decision process, 84th Annual Meeting Transportation Research Board, Washington, D.C. ITE. (1999). Traffic Engineering Handbook. Washington DC. Lighthill, MJ. and J.B. Whitham (1955). On kinematic waves. I. Flow movement in long rivers. II. A theory of traffic flow on long crowded road. Proceedings of Royal Society, A229, 281-345. Lucas, D.F., P.B. Mirchandani, and N. Verma. (2004). On-line travel time estimation without re-identification., 83rd TRB Annual Meeting, Washington DC. Oda, T. (1990). An algorithm for prediction of travel time using vehicle sensor data. IEEE 3rd Conference on Road Traffic Control, London, U.K., pp. 40-44. Perrin, J., Martin, P.T., and B. Coleman (2002). Monitoring commuter congestion on surface streets in real time. Transportation Research Record, 1811, 107-114. Skabardonis, A., N. Geroliminis, and P. Varaiya.(2004), Development of an arterial performance measurement system. Working Paper, Institute of Transportation Studies, University of California, Berkeley. Sun, C , Ritchie, S.G., and K. Tsai. (1998). Algorithm development for derivation of sectionrelated measures of traffic system performance using inductive loop detectors. Transportation Research Record, 1483, 171-178. Takaba, S., T. Morita, T. Hada, T.Usami and M. Yamaguchi. (1991). Estimation and measurement of travel time by vehicle detectors and license plate readers. Proceedings of Vehicle Navigation and Information Systems Conference, Vol. 1, pp. 257-267. Transportation Research Board, (2000). Highway Capacity Manual 2000, National Research Council, Washington, D.C. Turner, S.M., T.J. Lomax and H.S. Levinson. (1996). Measuring and estimating congestion using travel time-based procedures. Transportation Research Record, 1564, 11-19. Zhang, M.H. Link-journey-speed model for arterial traffic (1999). Transportation Research Record, 1676, 109-115. Zhu, Z., and W. Wang. (2000). A travel time estimation model for route guidance systems. Proceedings of the 2nd International Conference on Traffic and Transportation Studies, ASCE, Reston, VA, pp. 547-552

21

CALIBRATION AND VALIDATION OF DYNAMIC TRAFFIC ASSIGNMENT SYSTEMS Ramachandran Balakrishna, Massachusetts Institute of Technology; Haris N. Koutsopoulos, Northeastern University; and Moshe Ben-Akiva, Massachusetts Institute of Technology, USA

INTRODUCTION Intelligent Transportation Systems (ITS) have the potential to improve traffic conditions and reduce travel delays by facilitating better utilization of available capacity. Such systems are quite complex, and employ sophisticated algorithms to achieve their objectives. Several simulation-based dynamic traffic assignment (DTA) systems have been developed for a wide range of ITS applications. These systems, which include CORQ (Yagar, 1975), INTEGRATION (Van Aerde and Yagar, 1988), CONTRAM (Taylor, 2003), DynaMIT (BenAkiva et al. (2001), Federal Highway Administration (2004), DYNASMART (Jayakrishnan et al., 1994; Mahmassani, 2002) and Dynameq (Mahut et al., 2005), target more realistic approaches to short-term transportation planning such as special events and work zones, and address the growing importance of real-time applications such as incident management, route guidance and emergency response. State of the art DTA systems, such as DynaMIT and DYNASMART, integrate demand and supply simulators to estimate and predict network conditions. Estimation and prediction of OD flows, and modeling of route choice, response to information, queuing and spill-back allow DTA systems to effectively capture traffic dynamics, simulate the operation of ATMS/ATIS and estimate and predict traffic conditions. Research on DTA model calibration and validation, integral steps in model evaluation, is limited. This critical aspect is beginning to receive attention in the research community (Doan et al., 1999; Balakrishna, 2002; Kunde, 2002; Chen et al., 2004; Balakrishna et al., 2004).

408 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The objectives of this paper are to (a) develop a systematic methodology for the calibration of a DTA system's main parameters and inputs using multiple days of surveillance data; and (b) validate the estimation and prediction capabilities of a calibrated DTA system (DynaMIT) through a real-world case study with a network in Irvine, CA. The rest of the paper is organized as follows. We first present an overview of the DynaMIT route guidance system. The detailed outline of DynaMIT's model inputs and parameters sets the context for the subsequent discussion of the calibration methodology. Numerical results from a case study are presented next, followed by a discussion of extensions currently underway, along with preliminary results.

OVERVIEW OF DYNAMIT DynaMIT (Dynamic Network Assignment for the Management of Information to Travelers) is a simulation-based, real-time system designed for the generation of consistent, anticipatory route guidance for transportation networks. Anticipatory information (based on predicted network states) is known to be more effective in eliminating driver overreaction (Balakrishna et al., 2005; Ben-Akiva et al, 1996). DynaMIT combines real-time surveillance data with historical data in order to estimate current network state, predict future traffic conditions and generate consistent and unbiased traveler information for ATIS. Consistency ensures that the network conditions experienced by drivers coincide with the predicted conditions on which the information is based. Unbiasedness means that the information provided to any driver is based on the best knowledge of future network conditions. Figure 1 outlines the overall DynaMIT framework. A database of historical data (such as time-dependent OD flows and link travel times) along with a detailed representation of the network references the "average" or "usual" network state. DynaMIT's real-time inputs are traffic counts from the surveillance system, as well as traffic control system settings. While traffic counts provide indirect measurements of the unknown origin-destination (OD) flows, the control system settings affect the capacities of various network components. The state estimation process estimates demand levels and traffic conditions given the set of inputs. Prediction-based information generation involves the generation of consistent and unbiased traveler information. Unlike information based on the current network state, anticipatory information accounts for the temporal evolution of future traffic conditions taking into account drivers' responses to the disseminated information. DynaMIT iterates between the demand, supply and information generation models to ensure consistency and unbiasedness.

Demand simulation Individual travel demand decisions relate to choices of origin, destination, departure time, mode and route. Most of these decisions are pre-trip, made before the beginning of the trip. En-route decisions are usually limited to switching to a more desirable route in response to evolving traffic conditions. DynaMIT employs a disaggregate demand representation in order to model individual drivers' pre-trip and en-route decisions, including response to information. Under this representation, each individual is considered with his/her socio-

Calibration and validation ofDTA systems 409 economic characteristics and access to information. In addition, an aggregate demand representation in the form of time-dependent origin-destination matrices is used for the purpose of estimating and predicting network demand levels. DynaMIT adjusts historical OD flows (which are part of the input database) in order to capture two phenomena. First, driver response to real-time information is captured through disaggregate behavioral models that estimate and predict departure time changes, mode choice, route choice and route switching (Ben-Akiva and Bierlaire, 2003). Secondly, OD estimation and prediction models are used to adjust the OD matrices based on real-time traffic counts, to capture daily demand fluctuations. DynaMIT's demand simulation methodology can be summarized as follows: © = ®kis, + A© in/O + A © ^ + e where © is the actual demand; (Dhist is the historical demand; A(Dinf0 represents the influence of traveler information on drivers' behavior; AtDfaa models systematic yet unobserved modifications in individuals' activity patterns; sis a random error term.

Figure 1. DynaMIT Framework Travel behavior models Driver behavior models predict individual drivers' pre-trip and en-route choices in response to guidance or traveler information. The Path-Size Logit model (PS-Logit, see Ben-Akiva and Bierlaire (2003) and Ramming (2002) for more details) is employed in DynaMIT to compute

410 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction route choice probabilities. The probability P(i) of a driver choosing route i from his/her choice set C is given by: eV,+taPS,

jeC

Vt and Vj are the systematic utilities of routes / andj respectively. The systematic utilities in the route choice model are functions of both driver socio-economic characteristics and trip attributes including path travel times. Depending on the situation, travel times may be habitual or predicted. PSt and PSj are the corresponding path sizes, and capture the correlations between alternatives arising from overlapping path sections. The path size terms capture the effect of overlapping routes on drivers' perceptions of different route alternatives (Ben-Akiva and Bierlaire, 2003; Cascetta et al., 1997). DynaMIT computes path sizes as: aerAAJ J*C\Lj ) where a denotes a link from the set F, of links in path i; la and L, are the lengths of link a and path i respectively; Saj equals 1 if link a is part of route j and 0 otherwise; / i s a parameter. Calibration of DynaMIT's route choice model involves the estimation of parameters j3 (path size parameter y and the various parameters in the systematic utility functions). Timedependent historical (habitual) link travel times (denoted by TT*) are also required. OD estimation and prediction DynaMIT's OD estimation approach combines historical demand patterns with the latest traffic count data to generate real-time OD flow estimates. The OD estimation problem is formulated as a state-space model (see Ashok and Ben-Akiva, 2000) and solved using a Kalman filtering algorithm. State-space modeling requires three definitions that describe the system being modeled: the system's state, a transition equation and a measurement equation. Historical OD flows contain a wealth of information about latent demand patterns and their profiles across the time of day. The state vector for the OD estimation problem is therefore based on deviations of the current (unknown) OD flows from their historical values. Let x* denote the (unknown) vector of OD flows departing their origins during time interval h, and x" the corresponding historical estimate. Let yh represent the vector of link counts measured by the surveillance system during time interval h. Further, let the network have HOD OD pairs and «/ links with sensors. The state of the system is then simply (xA - x"). The transition equation, which describes the evolution of the state from one time interval to the next, is represented as:

p=h-q'

Calibration and validation ofDTA systems 411 where f/ is a matrix of autoregressive coefficients describing the effects o f ( x p - x " ) on (xA+1 -x" + 1 ); w/, is a vector of random errors; q' is the degree of the autoregressive process (the number of past time intervals influencing the current one). It is assumed that the random error terms have zero mean and a covariance matrix Q/,. The measurement equation relates the surveillance observations to the system's state: p=h-p'

where aph is an assignment matrix that maps OD flows xp onto link counts y/,; v/, is a vector of measurement errors; p' is the number of time intervals spanning the longest trip; y" are the link flows obtained by assigning historical OD flows xf . The error terms are assumed to be zero-mean with a covariance matrix Rj. Further, the transition errors w/, and measurement errors v/, are assumed to be uncorrelated with each other. The assignment matrices ap are complex functions of the OD flows, drivers' route choice behavior and traffic dynamics. They play an important role in the OD estimation process. However, their constituent fractions are not known, and need to be obtained through other means such as analytical computation or simulation (see Balakrishna, 2002 and Cascetta et al., 1993) for detailed discussions on the assignment matrix). Estimates of OD flows xA+/ departing their origins during future time interval Qi+1) are obtained by applying equation (/) on the estimated OD flows xph;p
412 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction v0 k-ka\ k j

— - K ) \

if k
(2 >

vo is the free-flow speed for the segment; k is the segment's density; kj is the jam density; ko is the maximum density that sustains free-flow speed; a and fi are model parameters. Queuing phenomena and spillbacks are captured through a lane-based queuing model. The formation and dissipation of queues is controlled by the current demand levels as well as segments' input and output capacities. The calibration of DynaMIT's supply simulator involves the estimation of segment specific speed-density function parameters (as in equation (2)) and capacities. Demand-supply interactions DynaMIT's state estimation and prediction capabilities rely on the accurate replication of the interactions between the demand and supply simulators. The estimation process must ensure that the estimated OD flows are consistent with the surveillance observations. Similarly, the prediction and guidance generation process must achieve consistency between the guidance strategy and the network conditions that drivers will experience when they receive (and react to) the guidance. Both processes require the solution of fixed-point problems. DynaMIT employs an MSA-type algorithm to obtain consistent estimates of current and future states (Cascetta and Postorino, 2001; Bottom et al., 1998; and Bottom, 2000).

CALIBRATION METHODOLOGY A DTA system requires the calibration of both supply and demand model inputs and parameters. The ideal calibration approach should therefore jointly estimate these parameters to capture the complex interactions between the models. The calibration process is complicated, and involves a large set of interacting variables. In this section we focus on the calibration of the parameters of the demand simulator, assuming that the supply parameters can be estimated independently. Extensions to the framework, including a rigorous solution approach, are discussed in a later section (see also Balakrishna et al., 2004). The calibration approach assumes the availability of multiple days of time-dependent, point sensor data such as counts. The archived surveillance dataset contains observations spanning Mdays referenced as {/, 2, ..., m, ..., M). Further, a static matrix of OD flows (denoted by \p) is assumed to be available. Such a matrix can be obtained from planning studies. The variables of interest (whose values will be the outcome of the calibration procedure) are the route choice model parameters /?, time-dependent habitual (historical) travel times TT/, for each interval h, historical OD flows x", autoregressive coefficients making up the matrices f' and error covariance matrices Q^ and R/,.

Calibration and validation ofDTA systems 413 Without loss of generality, a single day type consisting of M days of observations is selected for outlining the calibration approach. Day types may be identified according to a variety of criteria including day of the week, prevailing weather conditions and major special events. It is further assumed that the sensor data for all M days were generated by the same underlying demand patterns and mean OD flows (see Balakrishna et al. (2004) for a more general approach). Consequently, a common time-dependent habitual travel time profile is also assumed. In principle, the general demand calibration problem can be formulated as an optimization problem: # = argmm||y-y(#)| g

where y is a vector of sensor (count) measurements across all days and time intervals; y is a corresponding vector of model outputs; ||«|| is a norm function; 6 is the vector of unknown variables and includes: Historical OD flows xf for all h in {1,2, ...,T), Error covariance matrices Q/, and R/, for all h, Autoregressive matrices i' for all h; for allp, Route choice model parameters fi, Time-dependent habitual (historical) link travel times TT* for all h. The problem is very complicated, not only due to the interactions among the parameters of interest but also because the model output y is obtained from a complex simulation model. Recognizing the complexity and large-scale nature of the problem, the database of historical information and model parameters is developed through a heuristic procedure by sequentially processing the M days of data (Figure 2). The historical database is thus updated incrementally after each day. The OD flows and error covariance matrices for day m are estimated using Feasible Generalized Least Squares (FGLS; Greene, 2000). In this approach, the unknown OD flows and covariances are estimated iteratively. OD flows are first estimated, time period by time period, while maintaining the covariances at their current best available values. The resulting OD flows (and corresponding simulated sensor counts) are used to update the error covariances, after which the OD flows are re-estimated with the new covariance values. Iterations for day m are continued until consistency between the OD flows and the covariances is achieved. Error covariance estimates are typically not available at the beginning of the calibration process. The FGLS procedure therefore initializes both Q/, and R/, to identity matrices (of appropriate dimensions) for the first OD estimation iteration on day 1. Better covariance estimates may be obtained with the new set of OD flows jx j", x™,... i™} and assigned sensor flows, using procedures detailed in a subsequent section. Upon convergence, the final covariance estimates from day 1 may be used to initialize FGLS for day 2. The process is repeated from day to day, so that the outputs generated at the end of day m form inputs to day m+1.

414 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The estimation of OD flows x™ for day m across all time intervals h = 1, 2, ..., T is performed using a sequential procedure outlined in the next section.

Figure 2. Methodology: Within-Day and Day-to-Day Process The autoregressive coefficients, route choice model parameters and habitual link travel times have not been included in the discussion so far. These parameters are re-estimated after every k days of sensor data have been processed (Figure 2), with the value of k based on a priori expectations about the accuracy of the starting parameter values. In the limiting case (when k=l), the autoregressive coefficients, the route choice model and its input travel times are calibrated at the end of every day. The OD flows and error covariances may be re-estimated from day (m-k) to k until consistency between the various parameters is achieved. The overall approach for estimating the OD flows, autoregressive coefficients, error covariances, route choice parameters and habitual travel times is summarized in Figure 3.

Calibration and validation ofDTA systems 415

Figure 3. Overall Estimation Approach Sequential OD estimation The OD estimation problem, solved in each time period h, is formulated as a Generalized Least Squares (GLS) problem (Cascetta et al, 1993):

xmh =argmn[(y; -a*x,) R ^ -ajxj+(x, - X ^ J Q ^ X , - X ^ " ) ]

(3)

where x™ are the best estimates of the historical OD flows for interval h after day m, and x^™ are the corresponding target (or a priori) OD flows. All other symbols are as defined in previous sections. The error covariance matrices Q/, and R/, are the best estimates to that point, obtained through the FGLS technique outlined earlier. The formulation in equation (3) requires two important inputs: the target OD matrix x°h'm and the assignment matrix a* (which maps the unknown OD flows to the corresponding sensor flows). The role of the target OD matrix here is identical to that of the historical OD matrix in the real-time DynaMIT system. The target matrix is usually unknown, since in most cases only a static OD matrix from planning studies is available. It must therefore be derived from the planning matrix x^ until historical OD flows are estimated. The target flows xah'm when h=l and m=l are computed by x"'1 = xP . The target matrices for subsequent intervals h=2, 3, ..., T of the first day are assigned the estimated flows from the immediately preceding time interval:

x"'1 - x 1

416 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Target flows for m=2, 3, ... are obtained through a richer process, by assuming that the ratio of interval-over-interval OD flows is stable on a day-to-day basis (Ashok, 1996): x""1 if h = 1 .. * r . w xT, —^—r X

otherwise

*-l

Assuming that a large enough number of days is available, an initial set of historical flows is estimated by processing a number of days as described earlier. Based on these estimates, the corresponding autoregressive equation is calibrated and used to update the target OD matrix for subsequent days and intervals: x

h

x

«

+

Zu *-l\ P

PI

The assignment matrix &\ is another key input in solving the problem described in equation (3). Each element in the assignment matrix represents the fraction of an OD flow measured by a particular link sensor. Assignment fractions are derived by combining OD flows with a route choice model, and simulating the resulting path flows on the network. Since the estimated OD matrix depends on the assignment matrix, which in turn depends on the OD matrix, OD estimation involves the solution of a fixed-point problem (Cascetta and Postorino, 2001; Brandriss, 2001). Estimating the error covariances Error covariance matrices R/, for the measurement equation are estimated at the end of day m using the residuals (across m days) of observed surveillance counts from the corresponding values output by the DTA system. The residuals for day m and interval h are computed as:

ey - v m - v m where y™ represents the sensor counts recorded on day m and y™ the corresponding model output. Let eyihm represent the residual associated with the i' sensor, so that Then the (ij)'h element of R/, at the end of day m is estimated as: ijh ~ J

m where the prime indicates the transpose operator. The error covariance matrices Q« for the transition equation are estimated at the end of day m using the residuals (across m days) of estimated flows from their corresponding target flow values. The residuals for day m and interval h are computed as: ej m = x™ - xak'm . Let exihm represent the residual associated with the i'h OD pair, so that: The (ij)"1 element of Q^ at the end of day m is estimated as: J

m

Calibration and validation ofDTA systems 417 The above procedure can be applied to estimate the error covariance matrices Q* and R* for every time interval h. However, each day provides exactly one observation for the residual associated with every measurement. Better covariance estimates are obtained by applying reasonable simplifications to the structure of the covariance matrices, so as to increase the number of observations used to estimate each covariance term. Several alternative covariance structures are reasonable. We assume that the covariance terms remain invariant across the time period of interest (a reasonable supposition for short time periods within a day, such as the AM peak). This approach removes the dependence of Qi, and R/, on h, so that each time interval h yields one observation for every error term. Let Q* = Q and R* = R for all h. Let f.x

-\px

p*

...

p*

...

px 1

*L=\?L 4 * - eL - <&,] e?=[ef, e*2 - el] 9y

-[9y

g.y

...

ey

i

Then:

eft.?)1

e?(e*y lJ

lJ

mT

mT

where Tis the number of intervals in a day. A further simplification that estimates only the error variances is obtained by assuming diagonal covariance matrices. Since (mx T) may still be insufficient in many cases to ensure efficient estimates, a strategy of grouping measurements into subsets will augment the size of e' and ef. Such a grouping may be based on the magnitudes of the sensor flows and planning OD flows. Estimating the autoregressive coefficients The autoregressive matrices f/ are estimated periodically, after every k days have been processed. The estimation of autoregressive coefficients involves regressions of the form: X

r,h-Xr,h

~

2-,\Jr,h-AXl,p-\p p=h-q'

)+"-

+

Jr,h-l \Xn0D,p ~ Xn0D,p ))+ Wr,h

where x"h is the OD flow estimated between OD pair r for interval h and day m, and x°™ is the corresponding target flow. wrJt is an error term. It should be noted that noo such regressions are needed to estimate the entire f/_, matrix. Moreover, a matrix must be estimated for each time interval h. Each day of OD flow data would therefore yield exactly one observation for each unknown coefficient. A dataset with M sufficiently large is thus critical to the calibration process. In the absence of a large data set, simplifications to the structure of the autoregressive matrices become necessary. We make the assumption that f/^ depends only on the difference between h and p (and not on the values of h and p), thereby expanding the number of observations available for the estimation of each autoregressive

418 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction coefficient. We also propose the use of a diagonal autoregressive matrix (so that OD flows between an OD pair r are influenced by past flows between OD pair r alone). Reliable target flows are often unavailable in the initial stages of calibration. The elements of f' may be estimated once stable target OD flow estimates are obtained (after several days of sensor data have been processed using equation (4)). Estimating the route choice parameters and habitual travel times The parameters of the route choice model, fi, are estimated periodically (every k days, for example) as the available data is processed. The estimate J3 is determined by solving an optimization problem for a given set of OD flow and error covariance matrices: /? = argminj| y - y | + | 0-/3o || (5) where flo represents initial parameter values obtained from other studies or previous estimations. The model output y is a function of fS. The optimization problem stated in equation (5) involves costly simulation runs for each function evaluation. The Box-complex optimization algorithm (Box, 1965) has been used successfully for the solution of formulations of this type (see Kunde, 2002). Habitual travel times for a given set of route choice model parameters and time-dependent OD matrices i.h,h = 1,2, ..., T are obtained by an iterative weighting process: TT*+1 = X TT* +(1-X) E TT, where TT4* is the habitual travel time matrix for the current iteration k; TT*+1 is the habitual travel time matrix for the next iteration; E TTA* is the matrix of experienced (simulated) travel times at the end of iteration k; X is a smoothing parameter such that 0 < X < 1.

CASE STUDY

Figure 4 The Study Network

The objectives of the case study are to calibrate DynaMIT by applying the approach discussed in the previous section, and validate its performance. Data from Irvine, California was used to perform the evaluation. The study network was comprised of 298 nodes, and 618 freeway and arterial links (Figure 4). The links were further subdivided into 1373 segments based on geometry and other operating characteristics. Examples of such instances included lane drops and acceleration lanes on freeway links, and left-turn lanes at intersections. A total of 5 week days of traffic sensor count data from n\ = 68 sensors was available.

A static planning matrix of OD flows between noo ~ 655 OD pairs covering the morning peak period (7:00 - 10:00 AM) was used to initialize the OD estimation procedure.

Calibration and validation ofDTA systems 419

Figure 5. Day-to-Day Freeway and Arterial Sensor Count Variation Figure 5 shows the variation of sensor counts at typical freeway and arterial locations across all 5 days in the dataset. The lack of significant fluctuations from day to day supports the assumption that all 5 days of observations were generated by the same underlying historical demand patterns. In order to avoid over-fitting and ensure the general applicability of the calibrated model, the available data was partitioned into calibration and validation subsets containing 3 and 2 days of data respectively. Calibration Segment speed-density functions and capacities used by DynaMIT's supply simulator were independently estimated using sensor data. The segments were organized into groups (e.g. freeway, mainline, on-ramp, etc) and data from sensors within the same group were used to calibrate the corresponding speed-density relationships and capacities. For the calibration of the demand parameters using the process discussed in the previous section, the following simplifying assumptions were made. The error covariance matrices were assumed to be diagonal. Consequently, only error variance terms were estimated. The autoregressive matrix was also assumed to possess a diagonal structure. Further, the OD pairs were classified into high, medium and low categories based on their planning OD flow values. A separate autoregressive function of degree q' = 4 was estimated for each category. Q^, R/, and f' were assumed to remain constant across the AM peak period. At the end of the process the historical database was populated with time-dependent historical OD flows x" , error covariance matrices, autoregressive matrices, route choice model parameters and timedependent link travel times. Validation The objective of validation was to test DynaMIT's ability to replicate existing traffic conditions in both estimation and prediction modes. DynaMIT was run in an off-line open loop mode using the database of historical OD flows, autoregressive and error covariance

420 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction matrices, route choice model parameters and habitual travel times estimated using the first three days of surveillance data. Sensor data from two days, not used for calibration, was used for this purpose. Two cases were tested: 1. 2.

Estimation interval of 15 minutes and prediction horizon of 30 minutes Estimation interval of 5 minutes and prediction horizon of one hour

DynaMIT used archived sensor data for the estimation interval to obtain estimates of OD flows. The network state from the end of the estimation interval was used to predict OD flows and traffic conditions for the prediction horizon. Estimation and prediction accuracy with both 15 and 5 minute estimation intervals were evaluated using the RMSN error statistic: RMSN= —t!

5—

where S is the number of observations (S = ni). The RMSN error of counts, for 15-minute estimation intervals, ranged from 0.10 to 0.22 for the calibration data. The validation data had very similar RMSN ranging from 0.11 to 0.22. The two statistics were comparable across time intervals, with the validation RMSN usually slightly higher than its calibration counterpart, as expected. Figure 6 compares observed and estimated counts (aggregated over 15 minutes) for all sensors and for two time periods. The results show a high degree of fit. In addition, the flows measured by freeway and arterial sensors over time were also analyzed (Figure 7). The fit to speeds achieved with the calibrated DynaMIT is illustrated through Figure 8.

Figure 6. Estimated Counts for 7:15 - 7:45 AM The validation of the calibrated DynaMIT's prediction capability with 15-minute estimation intervals is illustrated using the time period from 7:15 AM to 8:00 AM. The network state estimated at 7:30 AM (based on surveillance data collected between 7:15 AM and 7:30 AM) was used to predict OD flows and traffic conditions over a 30-minute period. Comparisons of

Calibration and validation of DTA systems 421 predicted and "observed" counts at all network sensors (Figure 9) indicated RMSN errors of 0.12-0.15, thereby validating DynaMIT's prediction capability. As expected, predictions closer to the current estimation interval replicated observed traffic conditions more accurately than those further into the future. The results reinforce the importance of a rolling horizon implementation for a DTA system that constantly updates predicted traffic conditions as new surveillance data becomes available.

Figure 7. Freeway and Arterial Flows by Time of Day

Figure 8. Calibrated DynaMIT Speeds Tests with 5-minute estimation intervals were performed after re-calibrating the coefficients of the autoregressive process (q' = 4). DynaMIT demonstrated similar prediction accuracy for intervals close to the start of the prediction horizon, with both 15-minute and 5-minute estimation intervals. Some under-prediction of traffic counts with the shorter interval length was noticed towards the end of the prediction horizon (Figure 10). This is attributed to the fact that an autoregressive process of degree 4 was used. Estimating a process of higher degree, possibly 12, is expected to diminish or eliminate this bias.

422 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 9. Evaluation of Prediction Capabilities: 7:30-7:45 and 7:45-8:00

Figure 10. Evaluation of Prediction Capabilities: 7:35-7:40 and 8:20-8:25 In this case study the supply parameters were independently calibrated. Compared to the calibration results the validation of DynaMIT's simulated speeds (Figure 11) indicates that calibration may benefit from both, the use of speed measurements for calibration, and the implementation of approaches that facilitate the simultaneous calibration of the demand and supply parameters. Such approaches can capture explicitly the interactions between the various elements and hence, produce better estimates of the associated parameters. Current research towards such an approach is outlined in the next section.

EXTENSIONS The work presented in previous sections focused on the calibration of DTA model parameters (in particular on the demand side) and the development of the historical database that is essential for the successful implementation of DTA models. A sequential solution methodology was presented that also addresses several key practical issues. The scope of the approach, however, is limited by its sequential nature. In addition it only made use of sensor counts for calibration purposes. The source of this limitation is the use of linear measurement equations that map OD flows to counts. Ongoing research (Balakrishna, 2005) addresses

Calibration and validation ofDTA systems 423 several of these limitations and extends the framework. The extension allows for the simultaneous calibration of all unknown parameters, and the direct use of a wide range of data (counts, speeds, etc) for calibration.

Figure 11. Speed Validation The Box-complex algorithm, when used for the solution of the extended problem, exhibited very slow convergence. Hence its use in large networks is not practical. A recently developed algorithm, SNOBFIT (Huyer and Neumaier, 2004) appropriate for the optimization of noisy objective functions, was tested. Empirical results indicate a far superior performance both, in terms of efficiency and solution quality. Further improvements in solution quality and computational performance were obtained by using the Box-complex algorithm to generate a starting point for SNOBFIT. A test network (Figure 12) was used to test the general framework. The microscopic simulation laboratory MITSIMLab (Yang et al., 2000) was used to generate the data. The sensors supplied data on both speeds and counts. This controlled environment provides great flexibility in testing a number of scenarios, and formulating an experimental design that can support a detailed evaluation of the approach. The calibration variables included 30 OD flows (3 OD pairs for 10 5-min intervals), a route choice model parameter, 24 segment output capacities, 18 speed-density parameters, and an Figure 12: Test Network incident capacity reduction factor. Two cases were examined to assess the impact of using speeds for calibration (in addition to counts), on the accuracy of the calibrated model: a) only counts were used for calibration; and b) both counts and speeds were used.

424 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 1 summarizes the RMSN when only supply parameters are estimated (keeping demand parameters at their true values), and when both demand and supply parameters are estimated jointly. In the latter case, the assumed initial, seed OD matrices had significant error compared to the true values. Despite the poor starting point, the calibration process generated parameters that allowed DynaMIT to replicate the "actual" conditions very well. The use of both speeds and counts significantly improves accuracy of predicted speeds.

Table 1: Error Statistics for Test Network Calibration Data Scenario Counts + Speeds Counts Counts RMSN Speeds RMSN Counts RMSN Speeds RMSN 0.12 0.06 0.11 0.11 Supply calibration (Demand known) 0.06 0.11 Calibration of Supply+Demand 0.11 0.11

CONCLUSION This paper presented a framework for the calibration and validation of DTA systems using standard loop detector data. Detailed methodologies for the estimation of the demand model inputs and parameters within a DTA system were outlined. A case study was performed with data from an actual network using DynaMIT, a simulation-based DTA system. Extensions to the main framework were also presented. The results illustrate the importance of systematic calibration of model parameters and validate the estimation and prediction capabilities of DynaMIT.

REFERENCES Ashok, K. (1996). Estimation and Prediction of Time-Dependent Origin-Destination Flows. PhD thesis, Massachusetts Institute of Technology. Ashok, K. and M. Ben-Akiva (2000). Alternative Approaches for Real-Time Estimation and Prediction of Time-Dependent Origin-Destination Flows. Transportation Science, 34(1), 24. Balakrishna, R. (2002). Calibration of the Demand Simulator in a Dynamic Traffic Assignment System. Master's thesis, Massachusetts Institute of Technology. Balakrishna, R. (2005). Off-line Calibration of Dynamic Traffic Assignment Systems. Working paper, Massachusetts Institute of Technology. Balakrishna, R., M. Ben-Akiva, H. N. Koutsopoulos and T. Toledo (2004). Traffic Simulation Model Calibration Framework Using Aggregate Data. Proc. TRISTAN V. Balakrishna, R., H. N. Koutsopoulos, M. Ben-Akiva, B. M. Fernandez-Ruiz and M. Mehta (2005). A Simulation-Based Evaluation of Advanced Traveler Information Systems. Proc. 84th Annual Meeting of the Transportation Research Board, Washington, D. C.

Calibration and validation ofDTA systems 425 Ben-Akiva, M. and M. Bierlaire (2003). Discrete Choice Models with Applications to Departure Time and Route Choice. In: Handbook of Transportation Science (R. Hall, ed.), 2nd edition, Kluwer. Ben-Akiva, M., M. Bierlaire, D. Burton, H. N. Koutsopoulos and R. Mishalani (2001). Network State Estimation and Prediction for Real-Time Transportation Management Applications. Networks and Spatial Economics, 1(3/4), 293-318. Ben-Akiva, M., M. Bierlaire, H. N. Koutsopoulos and R. Mishalani (2002). Real-time simulation of traffic demand-supply interactions within DynaMIT. In Gendreau, M., and Marcotte, P. (Eds), Transportation and network analysis: current trends. Miscellenea in honor of Michael Florian, Kluwer, pp. 19-36. Ben-Akiva, M., A. de Palma and I. A. Kaysi (1996). The Impact of Predictive Information on Guidance Efficiency: An Analytical Approach. In: Advanced Methods in Transportation Analysis (L. Bianco and P. Toth, ed.s), pp. 413-432. Bottom, J. (2000). Consistent Anticipatory Route Guidance. Ph.D. thesis, Massachusetts Institute of Technology. Bottom, J., M. E. Ben-Akiva, M. Bierlaire and I. Chabini (1998). Generation of Consistent Anticipatory Route Guidance. Proc. TRISTAN III, San Juan. Brandriss, J. (2001). Estimation of Origin-Destination Flows for Dynamic Traffic Assignment. Master's thesis, Massachusetts Institute of Technology. Cascetta, E., D. Inaudi and G. Marquis (1993). Dynamic Estimators of Origin-Destination Matrices using Traffic Counts. Transportation Science, 27(4). Cascetta, E., A. Nuzzolo, F. Russo and A. Vitetta (1997). A Modified Logit Route Choice Model Overcoming Path Overlapping Problems: Specification and Some Calibration Results for Interurban Networks. Transportation and Traffic Theory. Cascetta, E. and M. Postorino (2001). Fixed Point Approaches to the Estimation of O/D Matrices using Traffic Counts on Congested Networks. Transportation Science. Chen, Y.-S., T. Giessen and H. Hendriks and H. S. Mahmassani (2004). Calibrating and Validating a Large-Scale Dynamic Traffic Assignment Model under European Context. Proc. 83rd Annual Meeting of the Transportation Research Board. Doan, D. L., A. Ziliaskopoulos and H. Mahmassani (1999). On-Line Monitoring System for Real-Time Traffic Management Applications. Transportation Research Record, 1678, 142-149. Federal Highway Administration (2004). Dynamic Traffic Assignment Project website. http://www.dvnamictrafficassisnment.ors. Accessed 7th March 2005. Greene, W. (2000). Econometric Analysis. Prentice Hall, 4th edition. Huyer, W and A. Neumaier (2004). SNOBFIT - Stable Noisy Optimization by Branch and Fit. Submitted for publication in A CM Transactions on Mathematical Software. Jayakrishnan, R., H.S. Mahmassani, and T-Y Hu, (1994). An Evaluation Tool for Advanced Traffic Information and Management Systems in Urban Networks. Transportation Research 2C, 129-147. Kunde, K. (2002). Calibration of Mesoscopic Traffic Simulation Models. Master's thesis, Massachusetts Institute of Technology.

426 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Mahmassani, H. S. (2002). Dynamic Network Traffic Assignment and Simulation Methodology for Advanced System Management Applications. Proc. 81s' Annual Meeting of the Transportation Research Board. Mahut, M., M. Florian, D. Florian, S. Velan and N. Tremblay (2005). Equilibrium Dynamic Traffic Assignment for Large, Congested Networks. INRO white paper. Ramming, M. S. (2002). Network Knowledge and Route Choice. Ph.D. thesis, Massachusetts Institute of Technology. Taylor, N. B. (2003). The CONTRAM Dynamic Traffic Assignment Model. Networks and Spatial Economics, 3, 297-322. Van Aerde, M. and S. Yagar (1988). Dynamic Integrated Freeway/Traffic Signal Networks: Problems and Proposed Solutions. Transportation Research, 22A, 444-453. Yagar, S. (1975). CORQ-A Model for Predicting Flows and Queues in a Road Corridor. Transportation Research Record, 533, 77-87. Yang, Q., H. N. Koutsopoulos and M. E. Ben-Akiva (2000). A Simulation Model for Evaluating Dynamic Traffic Management Systems, Transportation Research Record, No. 1710, pp 122-130, 2000.

22

NON-EQUILIBRIUM DYNAMIC TRAFFIC ASSIGNMENT W. Y. Szeto, Department of Civil, Structural, and Environmental Engineering, Trinity College, University of Dublin, Dublin 2, Ireland; and Hong K. Lo, Department of Civil Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, PRC

INTRODUCTION Dynamic Traffic Assignment (DTA) models basically follow two approaches: simulation and analytical. The simulation approach (e.g. Van Aerde and Yagar, 1988; Mahmassani et al., 1992; Peeta and Mahmassani, 1995; Ben-Akiva et al., 1997; Mahmassani and Liu, 1999; Ziliaskopoulos and Rao, 1999) relies on microscopic driving behavior in terms of carfollowing and lane-changing rules to model traffic flow or the mesoscopic approach to move individual vehicles based on the speed-density relationship. As far as the underlying traffic assignment principle is concerned, the approach may adopt elaborate rules to model route choice behavior at the level of individual vehicles. As the emphasis of the simulation approach is not on analytical tractability, strict adherence to the notion of an "equilibrium" route choice principle is often not the primary focus. The end result is that, at the system level, the traffic assignment solutions obtained may not exhibit well-defined properties, in terms of uniqueness or satisfying the notion of an equilibrium principle, such as a dynamic extension of Wardrop's principle or the dynamic user optimal (DUO) principle (Ran and Boyce, 1996). The analytical approach (e.g. Vythoulkas, 1990; Addison and Heydecker, 1993; Bernstein et al., 1993; Friesz et al., 1993; Kuwahara and Akamatsu, 1993; Lam and Huang, 1995; Ran and Boyce, 1996; Chen and Chang, 2000; Carey, 2001; Wie et al., 2002), on the other hand, emphasizes analytical tractability and adherence to the notion of an equilibrium principle. The tradeoff is its limitations in capturing realistic but complex traffic phenomena. Some early studies of the analytical approach simplify the traffic flow modeling by link travel time functions analogous to the static traffic assignment formulations; others use exit flow functions to model traffic flow propagation. These point-queue modeling approaches

428 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction guarantee the existence of solutions under mild conditions (Smith, 1993). Nevertheless, the point-queue modeling approaches are limited in their ability to capture realistic traffic dynamics, as compared with physical queue models. Recently, advancement is introduced by encapsulating a realistic traffic flow model within an analytical DTA framework (e.g. Lo, 1999; Lo and Szeto, 2002a, Szeto and Lo, 2004a). In particular, Lo and Szeto (2002a) formulates the equilibrium route choice principle or a dynamic version of Wardrop's principle in the form of a variational inequality and considers the traffic flow component as a functional mapping. It obviates the need to express traffic flow characteristics as nonlinear, often non-convex constraints. The end result is that this variational inequality (VI) formulation is solvable by projection methods under certain conditions of the mapping function, such as continuity and co-coercivity. This advancement allows the encapsulation of realistic traffic dynamics in the traffic flow model, including such features as Shockwaves, queue spillback, and junction blockage. That is, physical queuing characteristics are introduced to the DTA problem. The functional mapping approach of encapsulating a traffic flow model within an analytical DTA framework opens up new ways of DTA modeling, especially in capturing realistic traffic behaviors. Based on this approach, an important recent finding is that, with the more realistic physical-queue paradigm, the existence of solutions is no longer guaranteed (Szeto and Lo, 2004b). We have found examples to show that under congested conditions, the route travel time functions may become discontinuous, making it not possible in certain cases to find solutions that satisfy the equilibrium route choice principle perfectly. This finding shakes the foundation of the analytical DTA approach. In fact, it is not surprising to find that equilibrium is not a good description of real behavior and systems, as articulated by Goodwin (1998). This theoretical finding may prove to be important in the search of new behavior rules for analytical formulations that are behaviorally sound and consistent with the actual network behavior. Previous research on the use of the notion of "bounded rationality" for traffic assignment (e.g. Mahmassani and Chang, 1987; Jayakrishnan et al., 1994; Zhao, 1994; Hato et al., 1999) inspires our thinking in working with the possibility of non-existence of perfect equilibrium solutions in an analytical DTA framework with physical queues. With reference to DTA, the behavior notion of bounded-rationality, originally proposed by Simon (1955), is used to capture the behavior of individual travelers in the context of departure time choices (e.g. Mahmassani and Chang, 1985, 1987) and of route and departure time choices (e.g. Hu and Mahmassani, 1997; Mahmassani and Liu, 1999). In this present study, we propose the tolerance-based DUO principle, adapted from the bounded rationality notion, to describe the route choice principle at the system level, as consistent with analytical DTA approaches, so that its properties in terms of solution existence and uniqueness can be explored. This tolerance-based DUO principle can be considered as a relaxation of the traditional DUO principle with the traditional DUO principle as a special case. It acknowledges the fact that, in reality, the travel times of the used routes between the same origin-destination pair are rarely exactly the same, and that travelers would stop exploring new routes when they perceive no appreciable differences between their current routes and the candidate ones. This relaxed

Non-equilibrium dynamic traffic assignment 429 DUO principle allows the travel times of the used routes to be slightly unequal, subject to their differences lying within a certain tolerance. Subsequently, this paper introduces the concept of theoretical gap, which is used to analyze the existence and uniqueness of solutions to the tolerance-based DUO assignment problem with physical queues. The tolerance-based DUO principle relaxes the equilibrium requirement from the zero gap (as in the traditional DUO principle) to a nonzero one. The analysis shows that the existence of solutions to the problem depends on the tolerance adopted by the network users. In the case if the tolerance is small (i.e., travelers are sensitive to travel time differences) and the physical layout of the network and demands are such that the theoretical gap never falls within the tolerance, then the system would keep on evolving without converging to equilibrium. In other words, equilibrium traffic assignment is not an intrinsic property to every network. Therefore, the generalized notion of "non-equilibrium dynamic traffic assignment" is necessary to describe certain networks with the more realistic physical queue representation. To obtain solutions, this paper develops a modified route-swapping algorithm, analogous to the day-to-day route-swapping process (e.g. Smith and Wisten, 1995) and the route choice adjustment process based on projected dynamical systems (e.g. Nagurney and Zhang 1997a, b). The advantage of this heuristic algorithm is that it depicts the day-to-day flows as well as the within-day flows; no matter a convergent solution is obtained or not. This paper includes a numerical study to show the effects of the initial solutions and parameters of the swapping algorithm. The rest of paper is organized as follows. The next section depicts the tolerancebased DUO principle. Following that, we formulate the tolerance-based DUO route choice problem and discuss the existence and uniqueness of solutions. The route-swapping heuristic and numerical examples are then described. Finally, some discussions and concluding remarks are given.

TOLERANCE-BASED DYNAMIC USER OPTIMAL PRINCIPLE The traditional DUO principle (e.g. Ran and Boyce, 1996) requires that all used paths between the same OD pair have equal and minimum travel time. As a relaxation, our proposed tolerance-based DUO principle only requires the travel times of all used routes between the same OD pair to be similar, or within an acceptable tolerance errai. from the minimum OD route travel time, where the tolerance level is purely a function of the behavior of the network users. The proposed relaxation recognizes the important fact that it is physically impossible to always fulfill the requirement that all used routes on the same OD pair have exactly the same travel time, as is demonstrated in our recent study with the more realistic physical queue representation (Szeto and Lo, 2004b). This relaxation, adapted from the bounded-rationality behavioral notion, can be expressed as: I f / ; ( 0 > 0 , then i £ ( / ) - * " (*)**.«.

(1)

430 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

^;(?)-^ra(0>0,Vr5,^?,

(2)

where f" (?) and TJ" (t) are respectively the flow between OD pair rs entering route p at time t and the corresponding route travel time; nrs (t) is the shortest OD travel time between OD pair rs for flows departing at time t; ^ is the acceptable tolerance, a non-negative parameter obtained through travel behavior surveys and experiments. In (l)-(2), t is an instant of time; in this paper, t is a time-slice index, as we consider a discrete-time DTA formulation. Condition (2) is included in this principle to ensure n" (t) to be the shortest OD travel time among all the possible routes between OD pair rs for flows departing at time t. By employing the following transformation function: .

.

(0

e{y,u) = \ [y

ifO
(3)

iiy>u,

where u and y are independent non-negative variables, the tolerance-based DUO principle can be alternatively formulated as:

/;

# (i)-x" {t),em) = Oyrs,p,t,

(4)

r

s(?l ;(t)-7r"(t),£tmx)>0,\/rs,p,t.

(5)

Condition (5) implies condition (2) due to the requirement of the non-negative independent variable to the transformation function. Condition (4) implies (1), meaning that the travel time of a used route is greater than the minimum route travel time by not more than an acceptable level ema . According to (4), if route p carries a positive flow at time t (i.e., f" (;) > 0), the transformation

function

s{rfsp {t)-7i" ( ( ) , £ _ )

must

be

equal

to

zero,

implying

rs

0 < 77" (t) - n (t) < smy^ due to the first condition of (3). In other words, if route p carries a positive flow at time t, the travel time of route p is greater than the minimum route travel time by not more than an acceptable level emax. Note that if route p carries zero flow at time t (i.e., f"(t)

= O), s{rfp {t)-nrs (^),fi'ims) must be nonnegative due to (5) and hence the

route travel time 77" (?) must be greater than or equal to the minimum route travel time

As a special case, if s m equals zero, conditions (4)-(5) can be reduced to the following: / ; {t)-s(rjr; {t)-n"

(t),0) = 0yrs,p,t,

e(T%(t)-n"(t),0)>0yrs,p,t.

(6) (7)

Non-equilibrium dynamic traffic assignment 431 According to (3), we have: s(r,"(t)-^(t)0)-\° M ) () j

0
' " k W - - " W if7;(0-* (0>o.

Therefore, equations (6)-(7) are simplified to:

or

f?{t)[T%{t)-x"(t)] = O,Vrs,p,t,

(8)

nZ(t)-x"(t)ZO,Vrs,p,t,

(9)

Iff;s(t)>0,thenr?r;(t)-7rrs(t) r

= 0,

rs

ri ;(t)-7t (t)>0,Vrs,p,t.

(10) (11)

Conditions (8)-(9) are the ideal DUO conditions. That is, if e^ equals zero, equations (4)-(5) can be reduced to the ideal DUO conditions. According to this result, the tolerance-based principle is a generalization of the traditional DUO principle.

TOLERANCE-BASED DUO ROUTE CHOICE PROBLEM

Nonlinear Complementarity Problem (NCP) Formulation For fixed demands, the DTA problem with the tolerance-based DUO route choice principle is to find a route flow vector f * such that: f;s(t)s(Vr;(t)-^(t),s^)

= 0yrs,pJ,

(12)

e(i};(t)-x»(t),em)*0,VrS,p,t,

(13)

£/,"(') = ?"(')

(14)

V«,f,

p

r?r;{t) = ®l,{t),Vrs,p,t, f>0,

(15) (16)

where q" (?) is the demand of OD pair rs at time t ;<E>"r(.) is a unique mapping yielding route travel times for a given route flow vector f. Equations (12)-(13) express the tolerance-based DUO principle. Equations (14) and (16) are the flow conservation and non-negativity conditions. Equation (15) considers the traffic flow component as a unique functional mapping yielding route travel times for given route flows. Many traffic flow models can be used for this mapping. In the numerical study of this paper,

432 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction we adopt the Cell Transmission Model (CTM) (Daganzo, 1994, 1995) as the underlying traffic flow model to capture the effect of realistic physical queuing, such as queue formation and dissipation, and queue spillback. In determining the route travel times in (15), we first use CTM to simulate the resultant traffic pattern, followed by the route travel time extraction procedure developed in Lo and Szeto (2002a). As CTM uses mathematical operations that produce unique traffic patterns for given route flows, and the route travel time extraction procedure produces unique outputs for given cumulative inflow and outflow curves, we establish a unique mapping yielding route travel costs for given route flows. The details of encapsulating CTM for dynamic traffic assignment models can be found in Lo (1999) and Lo and Szeto (2002a, 2002b). The tolerance-based DUO route choice problem (12)-(16) can be reformulated as an NCP by introducing three more conditions. By attaching n" (t) to the flow conservation condition (14), we obtain:

*"(') £/," (')-?" (0 =0' V/V-

(17)

p

As nn(t),

the shortest OD travel time, must be greater than zero, ~^f" {f)-q" {t) = 0 must p

hold at optimality. Adding this to the problem (12)-(16) will not alter the optimality condition. For mathematical completeness, we also introduce two more conditions to the original problem: 7i"{t)>0,\/rs,t,

(18) 19

0 v

S/," (0-?" (0* . «.'-

( )

p

They include 7tr!!(t)>0 and ^f"

(t)-q" (t) = 0 as special cases and therefore do not

p

change the optimality condition of the original problem (12)-(16). The DUO route choice problem with fixed demands can then be considered as the problem of finding a route flow vector f* to satisfy (12)-( 19). By putting (15) into (12) and (13), the DUO route choice problem with fixed demands (12)(19) can be written as an NCP: to find x* such that:

X=

and

'

(20)

x* T -F(x*,o(f*)) = 0,

(21)

F(x*,«>(f*))>0,

(22)

Non-equilibrium dynamic traffic assignment 433

where

x=

{nrs{t)yVrs,t)

,

(23)

J

F(x,«>(f))= (

\

.

(24)

and O ( f ) is a vector of (®J, (f), Vr,s,/>,n ; TE is a vector of shortest OD travel times, and; / ; (/) ,V; (t), n" (t), fmx, q" (t), and
Solution existence and uniqueness For ease of explanation, we first define a non-negative gap function H(i):

H(f) = max[Snp (t) (>£ (/)-*" (t))yrs,p,t],

(25)

where the indicator variable 8™ (t) equals 1 if the flow on route p between OD pair rs at time t is nonzero, and equals zero otherwise. This gap is the largest difference between the travel time of all used routes and the corresponding shortest OD travel times under the route flow pattern f , and measures how far the current solution is from the traditional DUO solution. To analyze the existence and uniqueness of solutions, we define the theoretical gap (TG), which is the minimum of the largest difference between the travel time of all used routes and the corresponding shortest OD travel times of all feasible flow patterns. In other words, TG is the smallest gap of all the feasible solutions. Mathematically, the theoretical gap (TG) is expressed as follows: rG = m i n # ( f ) > 0 ,

(26)

The value of the theoretical gap depends on the network and the demand pattern. In DTA formulations with the more realistic physical-queue representation, the route travel time functions are not always continuous with respect to the route flows, rendering the existence of solutions not always possible (Szeto, 2003; Szeto and Lo, 2004b), or equivalently their theoretical gap never reaches zero. For the tolerance-based DUO route choice problem with physical queues, in a similar manner, solution existence is not guaranteed. A solution exists to the problem if and only if the theoretical gap is less than or equal to s ^ :

434 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction TG=mintf(f)<^x.

(27)

According to (27), the existence of solutions to the physical-queue tolerance-based DUO problem depends on the theoretical gap (which is related to the network topology and demand pattern) and the parameter e^ (which is related to the behavior of the network users). In general, as stmx and hence the constraint (27) are gradually relaxed, solution existence gradually becomes easier. However, we emphasize the importance of specifying the tolerance level from a behavior perspective rather than as a numerical means for obtaining equilibrium solutions. If the tolerance level is specified at a level higher than the actual behavior, then the solution will stop at a premature "equilibrium", even though in reality, travelers are still swapping routes in the search of better ones. In summary, solution existence is both a function of the network topology and travel demand, and the behavioral tolerance of the users on route swapping. On the issue of uniqueness of solution, as one can see, when suaK approaches infinity, all feasible gap values satisfy (27) and hence the corresponding flows are solutions, implying that multiple solutions are possible in this problem. In fact, even if the tolerance is zero, multiple solutions can still be possible, similar to the case of the traditional DUO problem (Lo and Szeto, 2002a; Szeto, 2003).

SOLUTION METHOD Two approaches can be employed to obtain tolerance-based DUO solutions. The first one is to solve the formulation (20)-(24) directly. Since the transformation function is not a continuous function, it is difficult to develop convergent algorithms. One may transform the NCP formulation into a mathematical program through a gap function, and solving the resultant mathematical program through global optimization algorithms such as genetic algorithm (e.g. Lo and Szeto, 2002b) and simulated annealing algorithm. Solution efficiency of this approach is an issue to be addressed. The second approach is to employ existing solution algorithms that maintain feasibility of the traditional DUO route choice problem while using the new stopping rule as: Stop if # ( f ) £ * „ , . That is, the solution procedure of the traditional DUO route choice formulation is repeated until a tolerance-based DUO solution is found. This approach is based on the fact that the traditional DUO solution (if exists) is also a solution to the tolerance-based DUO route choice problem. For this purpose, we employ a search direction developed within the context of traditional DUO problems to solve the tolerance-based DUO problem. The algorithm stops when the gap reaches the tolerance level.

Non-equilibrium dynamic traffic assignment 435 The heuristic algorithm developed in this study is modified from the route-swapping algorithm in Huang and Lam (2002). The proposed iterative algorithm is analogous to the day-to-day route-swapping process (e.g. Smith and Wisten, 1995) and the route choice adjustment process based on projected dynamical systems (e.g. Nagurney and Zhang, 1997a, b). The advantage of this algorithm is that no matter the tolerance-based or traditional DUO solutions exist or not, the algorithm can simulate day-to-day flow patterns in addition to the within-day flow patterns. That is, the transition from disequilibrium to equilibrium or one state of disequilibrium to another can be described. This allows us to study the daily variations in network performance. The basic idea of the algorithm is that a portion of travelers on the unacceptable routes swaps to the acceptable routes. The specific swaps are proportional to three factors: the flows on the current route, the difference in travel time between the current route and the shortest route, and the swapping rate p, which is a parameter representing travelers' sensitivity to the travel time differences. Higher swapping rates indicate travelers' higher sensitivities to the travel time differences, or higher eagerness to swap routes. Moreover, travelers farther from the acceptable travel time and on routes with larger flows are more inclined to changing their routes than otherwise. Due to flow non-negativity, the flows swap out of each unacceptable and f"(t). route are the minimum of pf"(t)x[77"(t)-n"(0] The detailed algorithmic steps are the following: Step 1: Set the iteration counter 1 = 1. Choose the initial route flow f" (t)z. Step 2: Determine the route travel time t]"(t)T through the procedures of network loading and route travel time extraction. Find n" (t)T = min [77™ (t)T, Vp]. Step 3: Update the route flow as below:

/ ; (0r+1 = max [0, / ; (t)T - pf; (t\ [r,; (t\ - n" (t\ ]],peP»/I%,

where

¥"{*),= I

[f?(t\-f?{t)T^;

ptsP"/P,"7

P^={p:r];(t)T-^(t)t<£^}; P" is the set of all feasible path between OD pair rs. Step 4: Stop if H (f ) r < s^ or r = r ^ ,

where

H(i\ =max[<5;! (t\ -(77™ (t)r -n" (t)r)yrs,p,t^

and

rmax is the maximum number of iterations; Otherwise, set r = r + 1 and return to Step 2. In Step 1, the initial solution can be obtained by many methods. In this study, we determine the initial solutions based on either an all-or-nothing or average loading assignment

436 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction procedure. All-or-nothing assignment loads all demands on the shortest time routes, assuming that the travel time is not flow-dependent. Average loading assignment ignores the travel time and sets the flow on a route between an OD pair to be the demand divided by the number of routes between that OD pair. Step 2 does not restrict the choice of traffic flow models to determine the route travel times. In this study, we employ CTM as the dynamic traffic flow model. To determine the route travel times from route flows, one can refer to Lo (1999) and Lo and Szeto (2002a,b) for the details of encapsulating CTM in DTA and the travel time determination procedure. In Step 3, the algorithm first deducts flows from unacceptable routes based on the basic idea described before, and then distributes (induces) these flows equally to the acceptable routes in Pt1. The sum of deduction y/" (t)r should however be equal to the sum of induction to ensure the feasibility of the solution during the adjustment process. Step 4 checks whether the stopping condition is reached. The algorithm terminates when the current gap is not larger than the tolerance or the maximum number of iterations is reached. Otherwise, the algorithm increases the iteration counter by one and repeats the procedure.

NUMERICAL STUDIES In the two numerical studies conducted in this section, we aim to show the effect of the tolerance and swapping rate on the existence of solutions, and the impact of the initial solution and swapping rate on the final solution as well as on the non-equilibrium traffic pattern. The scenario setup for both numerical studies adopts from that in Szeto (2003) wherein the traditional DUO solution is known not to exist. Sensitivity of swapping rate and tolerance The network is as shown in Figure 1, consisting of 10 nodes, 12 links, and 3 OD pairs: (1,6), (1,8), and (9,8). Each OD pair is linked by two routes, as listed in Table 1. The study horizon is discretized into 10-second intervals. A traffic signal is located at node 3, with a cycle time of 80 seconds and green time of 40 seconds. The signal indicates green at the start of the modeling horizon. The traffic signal causes queue formation. The saturation flow rate ss (in vehicles/time interval) and free-flow travel time r» (in seconds) of each link are also shown in Figure 1. Other link characteristics include: Storage capacities: Link (1,6), Link (2,3), Link (3,4), Link (2,10), and Link (10,7) 16.67 vehicles; Link (5,6) - 33.33 vehicles; Link (4,5) and Link (5,8) - 50 vehicles; Link (7,8) - 83.33 vehicles; Link (1,2) - 150 vehicles; Link (9,7) - 200 vehicles; and Link (9,4) - 250 vehicles. Merge priorities: Links (10,7) and (9,4) - 0.9; Links (9,7) and (3,4) - 0.1.

Non-equilibrium dynamic traffic assignment 437 The departure time of traffic leaving an origin is indexed by t. Given that time is discretized at 10-second intervals, the actual departure time is thus 10/. In this example, we fix the route flows at the various departure times as:

r W = { 5 '=U'3 ; /3I8W={5 Ji

W

|0

t>4

J3

yj

f=5 6

'

; yr(O=i" '=' W

\0 otherwise

[0

Figure 1 Table 1 OD pair (1,6) (1,8) (9,8)

Route number 1 6 2 3 4 5

otherwise

The scenario network OD - Route characteristics Node sequence 1-2-3-4-5-6 1-6 1-2-3-4-5-8 1-2-10-7-8 9-7-8 9-4-5-8

Free-flow travel time (seconds) 100 145 110 100 90 110

For each combination of swapping rate and tolerance, the route-swapping algorithm is employed to determine solutions. The initial solution is obtained by an all-or-nothing assignment procedure. The algorithm stops when a tolerance-based DUO solution is found or when the maximum iteration number of 2000 is reached. The number of iterations performed for each combination of swapping rate and tolerance is shown in Table 2. An entry of "2000" indicates that the maximum number of iterations is reached without finding an equilibrium

438 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction solution. This table clearly shows that the tolerance s ^ and swapping rate p are important variables for reaching the tolerance-based DUO solutions (if they are reachable at all). The reason is that, as explained before, to a great extent, the tolerance defines the requirement of obtaining solutions. A higher tolerance means a lower requirement. Moreover, higher swapping rates or larger step sizes often cause the algorithm to hop from one point to another without approaching to the solution location, hi this particular example, the shaded region in Table 2 represents the region wherein the route-swapping algorithm finds a solution. Obviously, this result is case-specific and network-dependent, but generally one can see that a smaller swapping rate and a higher tolerance are generally more capable of stopping at an equilibrium solution. Table 2 Swapping Rate 0.001 0.006 0.011 0.016 0.021 0.026 0.031 0.036 0.041 0.046 0.051 0.056 0.061 0.066 0.071 0.076 0.081 0.086 0.091 0.096

The number of iterations for different combinations of swapping rates and tolerances Tolerance

0.1

0.3

0.5

2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

0.7 228 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

0.9 192 32 20 14 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

1.1 170 29 16 11 10 8 7 6 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000 2000

1.3 151 26 14 10 8 6 5 6 5

1.5 141 24 13 9 7 6 5 4

4

4

1.7 139 23 13 9 7 5 4 4 3 3 4

2000 2000 2000 2000 2000 2000 2000 2000 2000

2000 2000 2000 2000 2000 2000 2000 2000 2000

2000 2000 2000 2000 2000 2000 2000 2000 2000

2000 2000

1.9 137 23 13 9 7 5 4 4 3 3 3 2000 2000 2000 2000 2000 2000 2000 2000 2000

The swapping rate not only has implications on the possibility of finding solutions, but also on the computational time required for finding solutions, as one may expect. For a fixed tolerance, if the solution is reachable, higher swapping rates typically mean faster solution times by having to go through a smaller number of iterations. For example, in Table 2, for the same stmll of 1.1, the number of iterations drops from 170 to 6 when the swapping rate increases from 0.001 to 0.036.

Non-equilibrium dynamic traffic assignment 439 Figure 2 shows the relationship between the minimum attainable gap and the swapping rate. The minimum attainable gap value is the minimum gap value that can be obtained under a specific swapping rate. Note that the minimum attainable gap is different from the theoretical gap, which is network specific and independent of the specific algorithm used. When the minimum attainable gap value is smaller than the tolerance, a solution is said to be obtained or reachable. According to Figure 2, generally, smaller swapping rates result in smaller the minimum attainable gaps. So a small minimum attainable gap favors finding solutions. This observation agrees with the result of the shaded region in Table 2.

Figure 2 Swapping rate versus minimum attainable gap

Sensitivity of non-equilibrium evolution to initial solution and swapping rate In this numerical study, we examine the effects of the initial solution and swapping rate on the network performance from day to day when the tolerance is small. The network setting and demand pattern are the same as in the previous numerical study. Four cases are set up for this purpose, as shown in Table 3. In all four cases, the tolerance is set to 0.1. As discussed earlier, it is known that the traditional DUO solution does not exist for this network and demand pattern. And by setting the tolerance level at a low level, we expect the system to evolve without stopping at an equilibrium pattern. We wish to observe how the system oscillates.

440 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 3 Four testing cases Case 1 2 3 4

Swapping rate 0.001 0.091 0.001 0.091

Initial loading method All-or-nothing (AON) assignment All-or-nothing assignment Average loading (AL) assignment Average loading assignment

a) low swapping rate

b) high swapping rate Figure 3 Gap value versus iteration

Non-equilibrium dynamic traffic assignment 441 Figure 3 shows the gap value over time (or from iteration to iteration). In general, the gap value does not monotonically decrease with time. Instead, after a certain period, the gap increases and decreases periodically with a nearly constant cycle. Examining the detailed traffic results, the large increase in the gap value can be attributed to the effect of junction blockage. The phenomenon of blockage, which is depicted by physical-queue traffic models and happens in reality, can lead to substantial changes in the route travel times even with a small change in the flow pattern (Szeto, 2003). This implies that the current shortest route can become an unacceptable route in the next iteration, leading to a large gap. Based on this limited experiment, it is not clear whether this periodic behavior shall always occur in the absence of a tolerance-based DUO solution. We leave the more exact and detailed treatment of this phenomenon to a future study. Figure 3 shows that the choice of the initial solutions (AON or AL) affects the gap values of the early iterations. As the algorithm proceeds to a substantial number of iterations (1500 iterations for the small swapping rate and 5 iterations for the high swapping rate), essentially the gap values produced by the different initial solutions show no appreciable differences. Both cases follow a similar periodic pattern of gap oscillation. Based on this limited experiment, it appears that the choice of the initial solutions does not have a substantial impact on the final evolving traffic pattern. On the other hand, as seen from Figure 3, the length of the periodicity is related to the swapping rate. For a low swapping rate, it takes around 200+ iterations to complete a cycle; whereas for the case of a high swapping rate, it just takes around 10 iterations. If we interpret the iterations as a day-to-day route adjustment process, more aggressive swapping behaviors lead to more frequent oscillations in network performance. To appreciate the implications of the gap swings on network performance, we plot the total system travel time (TSTT) in Figure 4 by following the same iteration sequence as in Figure 3. Regardless of the swapping rate, the TSTT varies from the high value of 5700 units to the low value of around 5200 units, or a change of around 10%. This level of travel time variation is perhaps well within our experience and expectation. Nevertheless, the change can be abrupt. As shown in Figure 4, the system can jump from the peak TSTT to the bottom TSTT in the next iteration or vice versa. The interesting aspect is that even though we are considering a deterministic network with fixed demands and deterministic route choice principle, the network congestion can still vary substantially.

442 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

a) low swapping rate

b) high swapping rate Figure 4 The total system travel time over iterations

DISCUSSION AND CONCLUDING REMARKS This paper provides some initial thinking on developing a framework to accommodate the possible non-existence of perfect DUO equilibrium solutions for the DTA problem under the physical queue paradigm. This extension or relaxation is actually more reflective of our actual experience, as perfect DUO is an idealization anyway. Specifically, this study developed a route choice principle based on the notion of bounded rationality, namely tolerance-based DUO, and a modified route-swapping heuristic for solutions. The existence of a stable

Non-equilibrium dynamic traffic assignment 443 solution depends on the network topology and demand pattern, and also travelers' behavior on travel time tolerance. This study demonstrated that for a small tolerance, the system can keep on evolving without converging to a stable equilibrium, meaning that equilibrium traffic assignment is not an intrinsic property to every network. Therefore, the generalized notion of "non-equilibrium dynamic traffic assignment" is necessary to describe certain networks when traffic is represented with the more realistic physical queue paradigm. As seen in the numerical study, in the absence of equilibrium, the traffic pattern may keep on evolving. Of particular interest are the stability and periodicity of the changing traffic pattern. The numerical study showed that the change in overall network travel time can be rather abrupt, with its magnitude of change lying within a limit of around 10%, which is perhaps consistent with our experience and expectation. Yet theoretical explorations of the network behavior under non-equilibrium, bounds of the changes in the total system travel time, periodicity of the changes, and their relations to the travelers' aggressiveness in route swapping, etc. are interesting and important questions to be answered in future research. This study only provides an initial step toward this direction.

ACKNOWLEDGEMENT This research is sponsored by the Competitive Earmarked Research Grant HKUST6283/04E from the Hong Kong Research Grant Council. We are grateful to the anonymous referees for constructive comments and Prof. Malachy Carey for helpful suggestions.

REFERENCES Addison, J. D. and B. G. Heydecker (1993). A mathematical model for dynamic traffic assignment. In: Transportation and Traffic Theory (C.F. Daganzo, ed.), pp. 171-183. Elsevier, New York. Ben-Akiva, M., H. N. Koutsopoulos, R. G. Mishalani, and Q. Yang (1997). Simulation laboratory for evaluating dynamic traffic management systems. Journal of Transportation Engineering-ASCE, 123, 283-289. Bernstein, D., T. L. Friesz, R. L. Tobin, and B. W. Wie (1993). A variational control formulation of the simultaneous route and departure time choice equilibrium problem. In: Transportation and Traffic Theory (C.F. Daganzo, ed.), pp. 107-126. Elsevier, New York. Carey, M. (2001). Dynamic traffic assignment with more flexible modelling within links. Networks and Spatial Economics, 1, 349-375. Chen, H. K. and M. S. Chang (2000). Dynamic user-optimal departure time/route choice with hard time-windows. Journal of Transportation Engineering-ASCE, 126, 413-418. Daganzo, C. F. (1994). The cell transmission model: A simple dynamic representation of highway traffic. Transportation Research, 28, 269-287.

444 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Daganzo, C. F. (1995). The cell transmission model, Part II: Network traffic. Transportation Research, 29B, 79-93. Friesz, T. L., D. H. Bernstein, T. E. Smith, R. L. Tobin, and B. W. Wie (1993). A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research, 41, 179-191. Goodwin, P. (1998). The end of equilibrium. In: Theoretical Foundations of Travel Choice Modeling (T. Garling, et al., eds.), 103-132, Elsevier, New York. Hato, E., M. Taniguchi, Y. Sugie, M. Kuwahara and, H. Morita (1999). Incorporating an information acquisition process into a route choice model with multiple information sources. Transportation Research, 7C, 109-129. Hu, T. and H. S. Mahmassani (1997). Day-to-day evolution of network flows under real-time information and reactive signal control. Transportation Research, 5C, 51-69. Huang, H. J. and W.H.K. Lam (2002). Modeling and solving dynamic user equilibrium route and departure time choice problem in network with queues. Transportation Research, 36B, 253-273. Jayakrishnan, R., H. S. Mahmassani, and T.Y. Hu (1994). An evaluation tool for advanced traffic information and management systems in urban networks. Transportation Research, 2C, 129-147. Kuwahara, M. and T. Akamatsu (1993). Dynamic equilibrium assignment with queues for a one-to-many OD pattern. In: Transportation and Traffic Theory (C.F. Daganzo, ed.), pp. 185-204. Elsevier, New York. Lam, W. H. K., and H. J. Huang (1995). Dynamic user optimal traffic assignment model for many to one travel demand. Transportation Research, 29B, 243-259. Lo, H. K. (1999). A dynamic traffic assignment formulation that encapsulates the cell transmission model. In: Transportation and Traffic Theory (A. Ceder, ed.), pp. 327350. Pergamon, Oxford. Lo, H. K. and W. Y. Szeto (2002a). A cell-based variational inequality formulation of the dynamic user optimal assignment problem. Transportation Research, 36B, 421-443. Lo, H. K. and W. Y. Szeto (2002b). A cell-based dynamic traffic assignment model: formulation and properties. Mathematical and Computer Modeling, 35, 849-865. Mahmassani, H. S. and G. L. Chang (1985). Dynamic aspects of departure time choice behavior in a commuting system: theoretical framework and experimental analysis. Transportation Research Record, 1037, 88-101. Mahmassani, H. S. and G. L. Chang (1987). On boundedly rational user equilibrium in transportation systems. Transportation Science, 21, 89-99. Mahmassani, H. S., T. Y. Hu, and R. Jayakrishnan (1992). Dynamic traffic assignment and simulation for advanced network informatics (DYNASMART). Proceedings of the Second International Capri Seminar on Urban Traffic Networks. Capri, Italy. Mahmassani, H. S. and Y. H. Liu (1999). Dynamics of commuting decision behavior under advanced traveler information systems. Transportation Research, 7C, 91-107. Nagurney, A. and D. Zhang (1997a). Projected dynamic systems in the formulation, stability analysis, and computation of fixed-demand traffic network equilibria. Transportation Science, 31, 147-158. Nagurney, A. and D. Zhang (1997b). Projected Dynamical Systems and Variational Inequalities with Applications. Kluwer Academic Publishers, Boston, MA.

Non-equilibrium dynamic traffic assignment 445 Peeta, S. and H. S. Mahmassani (1995). Multiple user classes real time traffic assignment for online operation: a roll horizon solution framework. Transportation Research, 3C, 8398. Ran, B. and D. Boyce (1996). Modeling Dynamic Transportation Networks. An Intelligent Transportation System Oriented Approach. Second Revised Edition. Springer-Verlag, Heidelberg. Simon, H. (1955). A behavioural model of rational choice. The Quarterly Journal of Economics, 69, 99-118. Smith, M. J. (1993). A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks. Transportation Research, 27B, 49-63. Smith, M. J. and M. B. Wisten (1995). A continuous day-to-day traffic assignment model and the existence of a continuous dynamic user equilibrium. Annals of Operations Research, 60, 59-79. Szeto, W. Y. (2003). Dynamic Traffic Assignment: Formulations, Properties, and Extensions. PhD Thesis, The Hong Kong University of Science and Technology, H.K. Szeto, W. Y. and H. K. Lo (2004a). A Cell-based Simultaneous Route and Departure Time Choice Model with Elastic Demand. Transportation Research, 38B, 593-612. Szeto, W. Y. and H. K. Lo (2004b). On the existence of solutions for the dynamic traffic assignment problem with physical queues. Transportation Research B, submitted. Vythoulkas, P. C. (1990). Two models for predicting dynamic stochastic equilibria in urban transportation network. In: Transportation and Traffic Theory (M. Koshi, ed.), pp. 253-272. Elsevier, New York. Van Aerde, M. and S. Yagar (1988). Dynamic integrated freeway/traffic signal networks: A routing-based modeling approach. Transportation Research, 22A, 445-453. Wie, B. W., R. L. Tobin, and M. Carey (2002). The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation. Transportation Research, 36B, 897-918. Ziliaskopoulos, A. K. and L. Rao (1999). A simultaneous route and departure time choice equilibrium model on dynamic networks. International Transactions in Operational Research, 6, 21-37. Zhao, Z. (1994). The £ -equilibrium in transportation networks. Fuzzy Sets and Systems, 68, 195-202.

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PRECISION OF PREDICTED TRAVEL TIME, THE RESPONSES OF TRAVELLERS, AND SATISFACTION IN THE TRAVEL EXPERIENCE Shinya Kikuchi, Sharat Mangalpally, and Anuj Gupta, University of Delaware, Newark, Delaware USA

INTRODUCTION Imagine that people wish to travel from point A to B by auto, and want to arrive at B at the same desired time. The predicted travel time from A to B is given prior to the trip in the form of range of time, and each traveller chooses his/her own departure time. How one chooses a specific departure time, however, depends on one's attitude towards meeting the desired arrival time. The less precise the predicted travel time is (a large the range of predicted travel time), the travellers will have a greater band of time to choose the departure time from. On the other hand, the more precise the perceived travel time is (a narrow range), the narrower the band of possible departure times becomes. The latter case allows the departure times to concentrate, which in turn may make the average travel time of all travellers long due to congestion. There are complicated interplays among the precision of the predicted travel time, the choice of the departure time, the actual travel time experienced, and the difference between the desired and the actual arrival time. What degree of precision will make the travellers most satisfied? We conjecture that there must be a suitable degree of precision in the predicted travel time, if the objective is to minimize the travel time of the individuals and also to minimize the difference between the desired and actual arrival times. This means that a tooprecisely predicted travel time might not maximize the satisfaction of the travellers as it may

448 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction lead to congestion, and at the same time, just as a too imprecise prediction is not acceptable to them. This paper tests this hypothesis. The core issues are the links between the information quality, a person's decision (choice) mechanism, concentration of travellers and travel time. We examine these links by analysing a simple travel setting, in which we model the processes for determining the departure times, calculating the travel times, and measuring the consequence in terms of traveller's satisfaction. This effort is to develop a better understanding about the precision of information and its usefulness, and to suggest further research on the level of precision of information.

THE PROBLEM AND ANALYSIS PROCESS This study, if successful, will suggest that the prevailing view about the precision of predicted travel time, that is, the more precise the better, may be too simplistic. Beyond a certain level, efforts to pursue a higher precision and the benefits from it might become mutually incompatible. The analysis setting is the following. A group of people (travellers) travel from the same origin to the same destination on a single lane roadway. Each travels by auto and has the same desired arrival time. This setting is the same as the one assumed in the seminal papers of Vickrey (1969) and Arnott et al. (1991), in which they dealt with the effects of congestion on travel time and choice of departure time and route. The predicted travel time (PTT) is given to the travellers is in the form of a range. The precision refers to the size of the range in this paper. The first stage of analysis is to identify the time-band in which the travellers are interested in departing, the departure time distribution. The selection of the departure time depends on the attitude of the individual travellers: optimistic, conservative, and in between. The second stage is to calculate the travel time of all travellers based on the distribution of departure times. This process considers the effects of congestion on travel time. The distributions of travel time and arrival time are computed. The final stage is to analyse how the precision of the predicted travel time affects the travel time, and time-missed. This analysis is performed for different volumes of travellers. The analysis process is illustrated in Figure 1.

PREMISES At the onset, we make the following premises. Each will be elaborated on later. 1. A traveller perceives the "predicted" travel time as a "possible" travel time with a range. Although the travel time may be estimated from statistical distributions of the observed travel times of the past and provided to the travellers, the travellers perceive it to be a possibilistic quantity with vaguely defined boundaries. This premise reflects the prevailing view about modelling perception. The early advocates include Shackle (1961, 1979) and Cohen (1970), and more lately Klir and Wierman (1999). The

Predicted travel time precision and response of travellers 449 traveller's perceived travel time is an approximate value, and this value is represented by a fuzzy set. 2. Each traveller's satisfaction is measured by the travel time, the match between the predicted and actual travel time, and the difference between the actual and the desired arrival time. This last item is called time-missed (an absolute value of the difference, either an early arrival or a late arrival) in this study. We introduce this term, because the term "schedule delay" found in the literature, has been used differently among the papers.

Figure 1 Framework of Study 3. Each traveller has a different attitude: optimistic, conservative, and in between, when deciding on the departure time. Given that the PTT is interpreted as possibilistic, there is a band of time in which the travellers can select the departure time depending on their attitude towards risk of missing the desired time. The width of the band depends on the precision of the PTT, the precision of the desired arrival time, and the attitudes of the travellers. Each person selects the departure time according to his or her attitude. 4. The width of the departure time band affects the concentration of travel volume, and this in turn affects the travel time and the time-missed. Thus, the connection, the precision of predicted travel time - the size of the departure time band - the travel time - and the arrival time of the travellers, is the main issue. While predicting the travel time has been one of the most popular research subjects under ITS, the issue of the precision of the predicted travel time has not been rigorously dealt with

450 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction in the literature. The effect of travel time information on traveller behaviour has been examined with respect to departure time decision and route choice in the past. Conquest et fl/.(1993) estimates the proportion of travellers who would adjust their departure time and the route when the information is given to them. Chang and Mahmassani (1988) examines how traveller's choice of the departure time is affected by the past experience, schedule delay, and the availability of possible departure times. Most recently, van Lint (2004) discusses the interplays between the predicted travel time and traffic behaviour. None of the past papers, however, discusses specifically about the relationship between the precision of the predicted travel time and actual travel experience.

IDENTIFICATION OF POSSIBLE DEPARTURE TIMES OF THE TRAVELLERS

Representing predicted travel time in possibility framework It is natural to assume that when one is given a predicted travel time (PTT), say "30 to 35 minutes," then one interprets it to be, "approximately 30 to 35 minutes." The predicted travel time before making the trip is a possibilistic notion from the traveller's standpoint; namely, it is "possible" or "achievable" to travel in 30 to 35 minutes. One's perceived notion of PTT before the trip is not a randomly chosen number, particularly for the auto travel. Although the PTT may be derived from a statistical analysis of prior events, which may be random events, when it is used in the process of determining the departure time in one's mind, it becomes a possibilistic number, because, to some extent, one can control the outcome by driving slowly or fast. This notion of an approximate value is represented by the fuzzy set. To measure the feasibility of arrival at the desired time for a specific departure time, we introduce the possibility theory framework, rather than the probability framework, in which the imprecise PTT forms a possibility distribution. Both the probability and possibility distributions represent the pattern of evidence when determining the truth of a proposition. A probability distribution represents the pattern in which each piece of evidence supports individual crisp sets exclusively. Thus, the distribution is the pattern that represents the propensity of occurrence of disjoint sets. The degree of the truth of a proposition is measured by probability measure, and it is calculated by aggregating the weights of evidence that support the set in question. The additive form does the aggregation. Possibility distribution, on the other hand, represents the evidential pattern in which each piece of evidence supports the nested sets (more than one set at the same time); hence, the distribution represents the degree of agreement among pieces of evidence. The degree of the truth of a proposition, in this case, is calculated by two measures depending on how to weigh the evidence pointing to the set in question. The two measures are possibility and necessity measures. The possibility measure weighs all pieces of evidence either partially or entirely

Predicted travel time precision and response of travellers 451 supporting the set in question. The necessity measure weighs only the evidence that strictly points to the set in question. In our problem, the evidence is imprecise, i.e., an approximate travel time, and we test the truth of a proposition, that is, to arrive at the destination at the desired time, if one departs at a particular time. In this case, the use of possibility theory is consistent with the literature, for example the discussion of possibility theory by Dubois and Prade (1998) in Handbook of Fuzzy Computations. Zimmermann (1996), another renowned person in uncertainty theory, states, "Possibility theory focuses on imprecision, which is intrinsic in natural languages and is assumed to be rather 'possibilistic' than probabilistic." While a large volume of literature is available on possibility theory and probability theory; a good summary is found in Klir and Wierman (1999), Yager and Filev (1994), and Dubois and Prade (2000). Defining possibility and necessity measures for arrival at desired time When a desired arrival time (DAT) is given and also the imprecise PTT is given, the travellers identify a band of possible departure times. If one departs at a departure time (DT), the possibility of arriving at or before DAT is calculated by possibility theory. The possibility of arrival at DAT (if departing at DT) is the possibility that the sum of DT and the travel time (t), is less than DAT. It is measured as follows. Poss(Arrive at DAT ) = Poss(DT + t < DAT) = Max {n (t< DAT-DT)}

(1)

where t is the travel time, and n(t) is the possibility distribution of travel time (t); it is numerically the same as the membership function of the fuzzy set that corresponds to the approximate number for PTT. Max 71 (t< DAT-DT) gives the possibility that the travel time is less than DAT-DT. The operator, Max, in Eq(l) suggests that possibility of arrival is obtained by the maximum weight of evidence that the travel time is smaller than the difference between DAT and DT. The graphical interpretation of this measure is shown in Figure 2. In this case, we assume that the membership function of the fuzzy number for PTT (in an approximate number) is represented by a triangular shaped function. Figure 2 shows that PTT is subtracted from DAT to get the departure time band, and then the possibility distribution of arrival at DAT is obtained for DT; the possibility of arriving before DAT is 1, as long as one departs before time x in the lower graph.

Figure 2 Departure time band and possibility of arrival at DAT

452 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The degree of truth of a proposition, when the evidence is based on the possibility distribution is called the possibility measure. It provides an optimistic view of the truth, because the possibility measure weighs any piece of non-negative evidence that supports the proposition. Use of the Max operator in Eq(l) reflects this concept. It, however, does not indicate the conservative view; that is, the weight of evidence that exclusively supports the proposition. This view is measured by a dual measure of possibility, called the necessity measure. Thus, when the evidential pattern is given by a possibility distribution, two views, possibility and necessity measures, are used to assess the truth of a proposition. When a human perceives a vague quantity and evaluates the truth affected by it, naturally these two views (optimistic and conservative) emerge, and the possibility (Poss) and necessity (Nee) measures correspond to these two views. Poss and Nee measures are related by the following dual relationships; that is, "the Necessity of A is the impossibility of not A (or -.A)," and "the Possibility of A is anything other than necessarily not A ( or -iA)." Nec(A) = 1- Poss(-,A), and Poss(A) = 1- Nec(^A)

(2)

Given these relationships, the necessity of arriving at DAT, if one leaves at DT, is Nec(Arrive at DAT) = 1- Poss( -.Arrive) = 1 - Max n (t> DAT-DT)

(3)

In summary, if one departs at DT, then, the possibility and necessity of Arrive and Not arrive (or Late arrival) are expressed using Eqs(l), (2) and (3) as Poss(Arrive) = Max n (t< DAT-DT), Nee (Arrive) = 1 - Max % (t> DAT-DT) Poss(^ Arrive) = Max % (t> DAT-DT), Nec(^ Arrive) = 1 - Max n (t< DAT-DT)

(4) (5)

The distributions of these measures along the time axis are shown in Figure 3. The lines show the possibility and necessity of Arrival (A), if one leaves at sometime on the x axis. It is seen that Poss(Arrive) extends to the right of Nee (Arrive). At the same time, Poss (Not Arrive) extends to the left of Nee (Not Arrive). Nee (Arrive) is 1, only when the DT is very early, and it starts to decrease even when Poss(Arrive) is still 1. This shows that the value of necessity measure is always less than or equal to the value of possibility measure.

Figure 3 Possibility and Necessity of Arrive (A) and Not Arrive (NA) for different departure times

Predicted travel time precision and response of travellers 453 The attitude of individual travellers when deciding their departure time Within the band of departure times, each traveller chooses a specific departure time according to his/her attitude: optimistic, conservative, and in between. We first model traveller's choice of departure time for two extreme attitudes, optimistic and conservative, and then assume that most travellers' attitudes fall between these two extremes. Along the time axis, a traveller faces the choices, "to leave now" or "not to leave yet." The possibility and necessity measures of arrive or not arrive for a given departure time can be used to model the supports for "to leave now" and "not to leave yet." The support for each choice, however, is different between the optimistic attitude and the conservative attitude. The optimistic person's choices "to leave now" and "not to leave yet," are supported by the following. To Leave now. If I do not leave now, I will necessarily not arrive (I will definitely be late), or it is not possible to arrive on time. The support for "to leave now" is Nee (-lArrive) or Nee (NA). Do not leave yet: If I leave now, it is possible that I arrive on time (therefore, I do not need to leave now). The support for "not to leave yet," is Poss(Arrive) or Poss(A). In essence, for an optimistic person, only a definite chance of being late provokes him to leave, and any slight chance of arriving early provokes him not to leave yet. Based on the values obtained in Figure 3, these two measures can be plotted along the time axis as shown in the upper graph of Figure 4a. In the same manner, the conservative person's choices, "to leave now" and "not to leave yet," are supported by the following. To Leave now. If I do not leave now, it is possible that I will not arrive on time; or there is a possibility that I will be late. The support for "to leave now" is Poss (-lArrive) or Poss(NA). Not to leave yet: If I leave now, I will necessarily arrive on time; therefore, there is no need to leave yet. The support for "not to leave yet" is Nee (Arrive) or Nee (A). In essence, for an conservative person, any slight chance of being late provokes him to eave, and only a definite chance of arriving early provokes him not to leave yet.

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These two measures are plotted along the time axis in the upper graph of Figure 4b. hi this model, at each potential departure time, one faces the two choices, "to leave now" or "not to leave yet", and the support for each choice changes as time passes. Regardless whether one is optimistic or conservative, the support to leave, that is, Nee (Not arrive) in Figure 4a and Pos(Not Arrive) in Figure 4b, increase as time passes, and, at the same time, the support for "not to leave yet," that is, Poss(Arrive) in Figure 4a and Nee (Arrive) in Figure 4b, decrease. In summary, the use of possibility and necessity measures for the support for the two choices, "to leave now" and "not to leave yet," provide the clue for modelling the internal struggle of the decision maker. Measure of Anxiety Different decision making paradigms have been proposed regarding travellers' behaviour. Srinivasan and Mahmassani (1999) assumes that the decision making behaviour of the drivers follows the "satisficing" principle in which the drivers accept an alternative only if it is both satisfactory and sufficient. Senbil and Kitamura (2004) proposes prospect theory for modelling the choice of departure time. According to this theory, choice is made after evaluating the prospect value (defined by the two elements, "value" and "weight") associated with each outcome, e.g., arrival time. Yager and Kikuchi (2004) and Kikuchi and van Zuylen (2003) have used the anxiety-based model of decision making which assumes that a takes a

Predicted travel time precision and response of travellers 455 decision when the anxiety level is maximum. This paper uses the anxiety approach, because it is consistent with the possibility framework. The conflict between "to leave now" or "not to leave yet" generates anxiety in the mind of the traveller. Yager (1982) proposed a way of expressing the decision maker's anxiety when a set of alternatives and the support for each alternative are known. It is a function of the strength of supports for different alternatives. Yager's anxiety model (1982) was simplified for the case of the binary choice situation by Perincherry (1994) and Yager and Kikuchi (2004) as follows. Anx(t) = 1- Max (a1; a2) + 1/2 Min (ai, a2)

(6)

where Anx(t) is the degree of anxiety at time t. ai and a2 correspond to the amount of support for each of the two alternatives. In Eq.(6), the degree of anxiety is the highest (=1) when no support is given to either alternative (ai=a2=0); that is, the decision maker has no information about the support for either alternative. The anxiety is zero when one of the alternatives has the maximum support and the other has no support(e.g. ai=l, a2=0). Anxiety is also high when the supports for both alternatives are equal. Anxiety level is computed at time t, by substituting ai and a2 by the appropriate support for optimistic and conservative travellers. For the optimistic person, Support for "To leave now": ai = Nee (-.Arrive) Support for "Not to leave yet": a2= Poss(Arrive). Anx(t)= 1- Max [Poss(Arrive), Nec(-i Arrive)] + lA Min [Poss(Arrive), Nec(-i Arrive)]

(7)

For the conservative person, Support for "To leave now": al= Poss(-i Arrive) Support for "Not to Leave yet": a2= Nee (Arrive) Anx (t)=l- Max [Poss(-.Arrive), Nec(Arrive)] +1/2 Min [Poss(-.Arrive), Nec(Arrive)].

(8)

Based on Eqs(7) and (8), the anxiety level is presented for the optimistic and conservative travellers in the lower graphs of Figures 4a and 4b. It is seen that, along the time axis, anxiety occurs for the conservative travellers first, and then, the optimistic travellers experience anxiety. This is because the conservative persons want to make sure to arrive at DAT with minimum risk of late arrival; on the other hand, the optimistic persons take the risk to delay the departure time as much as possible. Departure time distribution We consider that the travellers' inclination to depart at a given time is proportional to the anxiety level at that time. The higher the anxiety, the more inclined one is to depart. This

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Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

assumption is supported by the Bellman-Zadeh's principle of optimum decision under a fuzzy environment [Bellman and Zadeh (1970)]. This principle suggests that a decision is taken at the confluence of goals and constraints. In our case, the mutually conflicting alternatives are "to leave now" and "not to leave yet." Because everyone eventually must leave, the support for "to leave" is increasing, and at the same time, the support for "not to leave yet" is decreasing. A decision is taken at the point when the support for "not to leave yet" and "to leave now" are equal. That is the point when anxiety is maximal according to Eq(6). Discussion for this argument is found in Yager and Kikuchi (2004). The profiles shown in Figure 4 are for three attitudes, extreme optimistic, extreme conservative, and normative. We consider a normative attitude to be between these two extremes. Assuming that the travellers consist of a homogeneous mix of the three attitudes and that the propensity to depart is proportional to the anxiety level, the departure time distribution is shown in Figure 5.

Figure 5 Anxiety profiles of three attitudes and departure time distribution

CALCULATION OF TRAVEL TIME Assignment of departure volume in the departure time band We assign the travellers' departure times according to the departure time distribution developed in Figure 5. Then, the area under the departure time distribution represents the total number of travellers. The departure time band is divided into small intervals to calculate the travel time for the travellers. The volume in each time interval is expressed in equivalent hourly flow rate. Figure 6 shows the hourly flow rate corresponding to the case of 500 travellers in the departure time band of 20 minutes (for PTT range of 30-50 minutes).

Predicted travel time precision and response of travellers

457

Figure 6 Distribution of departure volumes

Calculation of travel time The assumed travel setting is the following. All travellers travel 30 miles from origin to destination on a single lane roadway and have the same DAT. The capacity of the lane is 2000 passenger cars per hour. The minimum travel time under free flow speed is 30 minutes. The predicted travel time (PTT) is given in a range starting with 30-35minutes and going until 3060 minutes with 5-minute increments as shown in Table 1. The numbers of travellers considered for each range of PTT are 100, 300, 500 and 700. Table 1 Precision Level and corresponding PTT ranges in minutes Precision Level Range of PTT (min)

1 4 2 5 6 3 30-60 30-55 30-50 30-45 30-40 30-35

The size of the departure time band and the number of travellers in the band dictate the level of the level of congestion on the roadway. The travel time of individual travellers is computed for each combination of the volume and the range of PTT (or the precision level). We use the BPR function to compute the travel time for each combination. The function has the following form.

where

tBpR = t o { l + a ( v / c ) } b tspR = the travel time; to = the travel time under free flow speed; v = the volume for the given time interval; c = the capacity of the link; and a and b = constant parameters.

(9)

The commonly used values for a and b are 0.15 and 4.0, respectively. When the v/c ratio is less than one, the variation in computed travel time is small regardless of the (a, b) values; when the v/c ratio is greater than 1, the computed travel time varies greatly with the selection of the (a, b) values. Singh (1999) reported that Eq(9) tends to underestimate the travel time when v/c ratio is greater than 1 and overestimates the travel time when v/c ratio is less than 1.

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We compute the travel time for the travellers in each small time interval in the departure time band as shown in Figure 6. It is computed separately when v/c ratio is less than 1, and when it is greater than 1, where v is the hourly flow rate in a small time interval in the departure time band, and c is 2000 passenger cars per hour. When the v/c ratio is less than 1, then Travel time: T = tBpR. = 30 {l +0.15 (v/2000)}4

(10)

When the v/c ratio is greater than 1, then T = delay + tBpR (at capacity flow (v/c =1))= delay + 34.5

(11)

where delay in Eq(l 1) incorporates the excess time due to congestion at the entry point of the link, caused either by the flow rate exceeding the capacity in the current time interval or by the backup from the previous time interval carried over to the current time interval. Delay in Eq (11) is modelled in the form of waiting in queue, represented by the cumulative arrivals and departures as shown in Figure 7. This approach is similar to the one used by

Amoti etal. (1991). The travel times calculated from Eqs(lO) and (11) are compared with those obtained by network simulation models, CORSM and VISSIM under the same set-up. We found that the difference in travel time between our model and those from the two simulation models was very small, within 4 minutes. Between the two models, the results from VISSIM are found to be closer than those from CORSM.

Figure 7 Calculation of delay

Predicted travel time precision and response of travellers 459

ANALYSIS OF TRAVELER SATISFACTION ASSOCIATED WITH THE PRECISION OF PREDICTED TRAVEL TIME According to the scheme described above, each traveller's travel time and arrival time are computed for different combinations of precision of PTT and travel volume. Using these data, we analyse the following aspects of travellers' satisfaction with respect to the precision of PTT. They are (1) the average actual travel time, (2) the range of actual travel time and its match with the PTT range, (3) the average time-missed, and (4) the average of the sum of travel time and time-missed. Precision of PTT versus the average actual travel time Figure 8 shows how the average travel time changes with precision. The average travel time is almost constant (equal to the free flow travel time of 30 min) when the travel volume is very small (volume = 100) regardless the precision of PTT. This is reasonable because at a very low volume, even for high precision of PTT all the vehicles can travel at the free flow speed. For a larger travel volume, the average travel time reduces with decreasing precision of PTT. This is consistent with our hypothesis; that is, too-precise PTT may not reduce the actual travel time.

Figure 8 Average travel time vs. PTT for different volumes Precision of PTT versus range of actual travel time Figure 9 shows how the range of actual travel time (minimum to maximum) changes with the precision of PTT for different volumes. It is seen in each graph that as the precision level decreases (range of PTT increases), the range of the actual travel time decreases. At a certain level of precision, the two curves meet; in other words, the travel time that the travellers actually experienced and the information given to them (PTT) match. It is also found that the

460 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction precision levels where the actual travel time range and PTT range match are different for different travel volumes. For example, in the case of 500 travellers the precision level where the match occurs is (30-40) minutes, while for 700 travellers, it is somewhere between (3040) and (30-45) minutes. This finding relates to the reliability of the PTT. Figure 9 shows that when the volume is large, too precise PTT is not useful from a reliability standpoint. However, when the volume is small, too precise PTT is reliable.

Figure 9 Variation in actual travel time range vs. precision of PTT for different volumes

Precision of PTT versus time-missed Figure 10 shows the average values of time-missed (TM) for varying precision levels of PTT under different volumes. Time-missed (TM) is defined as the absolute value of the difference between the actual and the desired arrival times. For a small travel volume (say 100), TM is minimum at 30-35 minutes, and increases as the precision decreases. For a large volume (say 700), on the other hand, the value of TM decreases and incerases with a minimum occurring at a certain precision level, 30-45 minutes.

Predicted travel time precision and response of travellers 461

Figure 10 Time-missed vs. PTT for different volumes Precision of PTT versus sum of travel time and time-missed (TTffiTM) We consider the sum of the actual travel time and time-missed as an indicator of usefulness of PTT. This is a measure of travellers' dissatisfaction, and we denote it by TTffiTM. We recognize that the actual travel time and the time-missed are not strictly additive as a measure of disutility. Time-missed should be divided into late arrival and early arrival with more penalty assigned to the former, as introduced by many, including Vickrey (1969), Small (1982), Hendrickson and Plank (1984), and Arnott et a/.(1991). Nevertheless, in this analysis, for the purpose of measuring the usefulness of PTT to the traveller, we use the sum as an indicator of dissatisfaction. The weights can be incorporated if the values are available for a specific case. Figure 11 shows TTffiTM for different combinations of travel volumes and the precision levels of PTT. It is seen that when the travel volume is small, the more precise PTT results in a lesser value of TT®TM. However, as the volume increases, TTffiTM shows a concave trend in which at a certain precision level it becomes minimum; for example, for 500 travellers, it become minimum at 30-40 minutes. It is interesting to note that for 30-60 minutes PTT, TTffiTM is nearly equal for all volumes. This means that when the travel information is very vague, like, (30-60) minutes, the departure time is dispersed enough so that the vehicles travel at free flow condition for all volumes. Hence beyond this precision level, Travel time is equal to all travellers, but only the time-missed increased due to the large departure time band.

462 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 11 Average of TT©TM vs. PTT for different volumes

DISCUSSION General findings The theme of this paper is to question the effectiveness of pursuing a greater degree of precision of PTT. Our hypothesis is that a greater level of precision will not necessarily yield a higher level of satisfaction to the travellers, measured in terms of travel time, the match between the ranges of actual and predicted travel times, and time-missed. The analysis shows that the travellers' satisfaction does not always improve by increasing the precision of PTT. When the travel volume is small, increasing precision reduces the travel time and time-missed. Thus, the general notion, the more precise the better, is valid. When the travel volume is high, however, this is not the case; a higher precision is not effective for reducing both the travel time and time-missed. Further, a higher level of precision does not necessarily result in a match between the predicted and the actual travel time. It is also interesting to note that the precision levels that result in maximum match between the actual and predicted travel times, and the minimum TT6TM do not coincide. For example, in the case of 700 travellers, the maximum match between the actual and predicted travel time occurs at precision close to 30 - 45 minutes (see Figure 9). On the other hand, the minimum TT©TM for 700 travellers occurs at precision 30 - 50 minutes, (see Figure 11). However, these two precision levels vary if weights are considered for travel time and time missed differently. Depending on the values of weights, these precision levels may match. This observation underscores that the precision of PTT affects travel time, reliability, and TM differently; therefore, it is difficult to find the "optimum" precision level for PTT. Presentation of precision The purpose of this study is to understand the fundamental relationship between the precision of PTT and its consequences on travel experience. One of the critical steps in the modelling process is the distribution of departure times, which depends on the precision of PTT and the

Predicted travel time precision and response of travellers 463 attitudes of individual travellers. Therefore, it is important to study how in practice the different degrees of precision are expressed in the information. Use of words like, "30 min," "approx. 30 min," " 30- 40 min," and "approx. 30-40 min" convey different meanings to different people. Given that, in the case of travel by auto, the travel time is controllable somewhat by the driver, the predicted travel time is perceived as a possibilitic value; in other words, it is possible to travel in approximately 30 minutes. Based on this study, we find that it is not particularly useful to give a very precise PTT when the traffic volume is very high. Thus, how to convey a notion of imprecision in a short message is a question that must be resolved. The selection of language will remain as a subject of future study. It may be the case that use of natural language, such as "more or less," "a little more than," etc. is the way to approach this problem. An important consideration is that the level of precision and the expression to convey it must correspond to the travel volume. Possibility vs. probability framework At the onset, we introduced the framework of possibility theory to represent the process by which travellers choose the departure time. If instead, the probability framework were used, then the imprecision of the PTT needs to be represented by a probability distribution. Theoretically, the probability distribution of departure times needs to be derived as a joint probability distribution made up of different travellers' choices of departure times. Although our reason for choosing the possibility framework is not for ease of computation, the use of the probability framework would overly complicate the calculation and detract from the effort to pursue the objective of the study. The main reason for the use of the possibility framework in this paper, however, is that it fits the nature of uncertainty associated with perception of the PTT, and the decision making process under imprecise information. While we used a triangular shaped possibility distribution for representation of the imprecise PTT in this analysis, different shapes for the possibility distribution can be introduced, e.g., rectangular or bell-shaped distribution, without losing the generality of the conclusion. Different shapes can result in different profiles of anxiety and different departure time distributions, but the main issue is the development of a pattern of concentration of departures, which affects the travel time and time-missed.

CONCLUSIONS This paper is motivated by the question of how precise the predicted travel time (PTT) needs to be in order for it to be meaningful. Based on a simple setting, in which a group of people travel from one origin to one destination on a one-lane roadway, and wish to arrive at the same desired time, we have examined the relationship between the precision of PTT and the satisfaction of travellers, measured by the travel time, time-missed, and the match between the desired and actual arrival times.

464 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The notion that the higher precision of PTT is always better, does not seem to be valid. We need to consider the positive effects of dispersion of departure times that an imprecise prediction may generate. When the travel volume is small, a higher precision results in a shorter travel time, smaller time-missed, and also a better match between the actual and predicted travel time. Thus, the traditional notion is valid. When the travel volume is large, however, a higher precision is not necessarily most useful. Somewhat less precise PTT actually helps reduce congestion and minimize the travel time, and time-missed. Further, the best precision level is difficult to determine as the precision level that results in a match between the predicted and the actual travel time is not the same as the one which minimizes the travel time, the time-missed, and the sum of travel time and time-missed. The findings cast questions about the direction of one of the major ITS efforts, that is, to improve the precision of the predicted travel time. The questions include whether pursuance of precision is always justified from theoretical and practical grounds, how much precision is sufficient to achieve an acceptable level of traveller satisfaction; and what are the appropriate expressions to convey different precision levels to travellers. These questions need to be answered in different contexts, including information compliance levels, networks that involve route choice and mode choice as well as departure time choice, the ways by which PTT is derived, and the environment in which travellers learn and experience the information. We may need to consider the entire problem in the domain of a complex uncertain system, given that there is no perfect way to estimate the travel time; each person's attitude to information is different and difficult to predict; desired arrival time is not always rigid; traffic flow phenomena themselves are random processes; and the boundaries of what is acceptable are subjective and fuzzy. In this larger context rife with uncertainties, it is still worthwhile to pursue what the limits of precision are when predicting the travel time and what the effects of information are in improving the travel experience.

REFERENCES Arnott, R., de Palma, A., and Lindsey, R. (1991). Does providing information to drivers reduce traffic congestion? Transportation Research, 25 A, No.5, 209-228. Bellman, R., and L. A. Zadeh. (1970). Decision-Making in a fuzzy Environment. Management Science, 17B, 141-164. Cohen, I. J. (1970). The Implications of Induction, Methuen, London. Chang, G. L., and H. S. Mahmassani. (1988). Travel time prediction and departure time adjustment behaviour dynamics in traffic system. Transportation Research, 22B, No. 3,217-232. Conquest, L., J. Spyridakis, and M. Haselkorn. (1993). The effect of motorist information on commuter behaviour: Classification of drivers into commuter groups. Transportation Research C, 1, No. 2, 183-201. Dubois, D., and H. Prade. (2000). Fundamentals of Fuzzy Set. Kluwer Academic Publishers, Boston/Dordrecht/London. Dubois, D., and H. Prade. (1998). Basic notions of possibility theory, Handbook of Fuzzy Computations, edited by E. H. Ruspini, P. P. Bonissone, and W. Pedrycz, Institute Of Physics Publishing, Bristol and Philadelphia, B4.1:l-B4.1:10. Hendrickson, C , and E. Plank. (1984). The flexibility of departure times for work trips.

Predicted travel time precision and response of travellers 465 Transportation Research A, 18, No. 1, 25-36. Kikuchi, S., and H. van Zuylen (2003) Modeling of Traveler's Uncertainty and Anxiety, The Network Reliability of Transport, edited by M Bell and Y. lida, Pergamon Press, Oxford, UK, 133-154. Klir, G. J., and M. J. Wierman. (1999) Uncertainty-based information elements of generalized information theory. Physica-Verlag, Heidelberg. Perrincherry, V. (1994). Analysis of large scale systems. Ph.D. dissertation, University of Delaware, Newark, Delaware. Senbil, M., and R. Kitamura. (2004). Heterogeneity in commuter departure time decision: A prospect theoretic approach. Unpublished Manuscript. Shackle, G. L. S. (1961). Decision, order and time in human affairs. Cambridge University Press, Cambridge. Shackle, G. L. S. (1979). Imagination and the nature of choice. Edinburg University Press, Edinburg. Singh, R. (1999). Improved speed-flow relationships: Application to transportation planning models. Paper presented at 7th TRB Conference on Applications of Transportation Planning Methods, Boston, MA. Srinivasan, K. K., and H. S. Mahmassani. (1999). Role of congestion and information in trip-makers' dynamic decision processes. Transportation Research Record, 1676, TRB, National Research Council, Washington, D.C. 44-52. Small, K. (1982). The scheduling of consumer activities: Work trips. The American Economic Review, 72, 467-479. van Lint, J. W. C. (2004). Robust travel time prediction on freeways. Ph.D. dissertation, Delft University of Technology, Delft, The Netherlands. Vickrey, W. S. (1969). Congestion theory and transport investment. The American Economic Review, 59, 251-261. Yager, R. R. (1982). Measuring tranquillity and anxiety in decision making: an application of fuzzy sets. InternationalJournal of General Systems, 8, 139-146. Yager, R. R. and D. P. Filev (1994). Essentials of Fuzzy Modelling and Control. John Wiley and Sons, Inc. New York. Yager, R. R. and S. Kikuchi (2004). On the role of anxiety in decisions under possibilistic uncertainty. IEEE Transactions on Systems, Man, and Cybernetics-Part B: Cybernetics, 34, No.2, 1224-1234. Zimmermann, H. - J. (1996). Fuzzy Set Theory and Its Applications. Kluwer Academic Publishers, Boston/Dordrecht/London.

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BEHAVIORAL DYNAMICS IN ACTIVITY PARTICIPATION, TRAVEL, AND INFORMATION AND COMMUNICATIONS TECHNOLOGY Konstadinos G. Goulias, University of California, Santa Barbara, CA; and Tae-Gyu Kim, The Pennsylvania State University, University Park, PA, USA

INTRODUCTION The unique event of the rapid advance and growing adoption and popularity of Information and Communications Technology (ICT) offers an unprecedented opportunity to study behavioral dynamics. This is unique for activity and travel behavior analysis because of the many claimed relationships between telecommunication and transportation. For example, using telecommunication as a source of information to reach locations and procure goods we can alter our traditional limitations of accessibility and virtually eliminate spatial separation. In addition, we may experience increases in schedule flexibility that may change our activity and travel patterns but may also alter our activity planning, location choice, and variety seeking behavior. On one hand, these developments may increase variability of activity and travel behavior and variability of travel demand potentially decreasing regularity and predictability. On the other hand, they also enable us to perform more interesting studies of travel behavior dynamics that may lead to better policy actions. These claimed substantial impacts of ICT motivate the need for research on the present and future impacts of telecommunication on activity and travel behavior by studying change of behavior. However, studying behavioral dynamics and change requires specific types of data such as panel survey data (i.e., repeated observation of essentially the same persons over time) and specific types of data analysis methods. Using the Puget Sound Transportation Panel (PSTP) data, especially the data collected in Wave 7 (1997) and Wave 9 (2000), this study examines a variety of relationships between

468 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction ICT and activity and travel behavior using a comprehensive conceptual model system that examines correlation patterns from both cross-sectional and longitudinal viewpoints. The data used are from two-day travel diaries of 1480 persons in 866 households that resided in the four-county region surrounding the Seattle metropolitan area. Figure 1 shows the conceptual framework emerging from models developed in the past few years and used here to estimate a structural equations model. A plethora of relationships can be studied using this system. For example, the correlation among the amount of time allocated to in-home, out-of-home subsistence, out-of-home non-subsistence and travel, as well as the total number of activity episodes (in-home, out-of-home subsistence, out-of-home nonsubsistence) can be estimated. The model system is designed to extend and parallel other past studies, and for this reason, variables at the household and person level are used. The information analyzed includes personal and household socio-demographics in the year 2000, changes in socio-demographics and ICT ownership and availability between the years 1997 and 2000, and the activity and travel behavior patterns in 1997. In addition, a new approach to account for the effects of state dependence in activity engagement and time use is proposed to test if behavioral habits of individuals still persist within the 3-year time interval. We first employ a pattern recognition technique known as latent class (LC) clustering analysis to identify relatively homogenous behavioral groups with observed activity engagement and time use variables in 1997. Then we use group membership indicators as explanatory variables in a structural equation model. This way also allows us to avoid serial correlation problems in panel data. In addition, some specific variables to the database in PSTP are also employed to account for other factors such as stratification sampling and potential selfselection bias. The model system depicted by Figure 1 is representative of the plethora of relationships that can be studied using it. For example, the correlation among the amount of time allocated to in-home, out-of-home subsistence, out-of-home non-subsistence and travel, as well as the total number of activity episodes (in-home, out-of-home subsistence, out-ofhome non-subsistence) can be quantified and an estimate of possible substitution effects provided. The model system extends and parallels other past studies, and for this reason, variables at the household and person level that were used in past studies are used here too. A formulation like this allows us to achieve several research objectives at the same time, such as: Identifying complex relationships among in-home and out-of home activities and travel in terms of time use and frequency of activity participation; Checking if there are systematic differences in time use and frequency of specific activities among population segments, while controlling for other exogenous variables; Examining whether or not there are symmetric effects of changes of person and household socio-demographics and ICT ownership and availability on the time use and frequency of specific activities ; and Developing a new technique to account for state dependence (past activity and travel behavior effects) in models that assess the effects of ICT on activity participation and travel.

Behavioral dynamics in travel and activity participation 469

Figure 1 Conceptual Model A key aspect of this model system distinguishing it from other studies is the inclusion of "experience" variables. They are indicators of changes in the social and economic circumstances experienced by each respondent between 1997 and 2000 and they are included in the behaviour equations of the year 2000 as explanatory variables. The definition of these variables is such that allows to study behavioral symmetry when opposite events occur (addition of a vehicle versus subtraction of a vehicle from the household fleet). The second element that characterizes our research is the study of ICT on activity and travel behavior considered jointly with all the other determinants of travel behavior. Moreover, this research also develops a new method to account for state dependence effects in a time allocation model. Using activity engagement and time use indicators in 1997, homogenous groups are identified first through Latent Class cluster analysis and the group membership indicators are included as explanatory variables in the behavioural equations of the year 2000 to account for habit persistence. In the next section, a brief description of the data used is provided first, followed by model formulation and estimation results. The paper ends with summary and conclusions.

470 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

DATA USED The Puget Sound Transportation Panel (PSTP) is the first and the only American generalpurpose transportation urban household panel survey with data from Seattle and its surrounding four-counties in the United States (Murakami & Waterson, 1990). Unlike traditional cross-sectional transportation surveys, the PSTP is a panel or longitudinal survey in which similar measurements (i.e., surveys) are administered on the same households and their members repeatedly over time. Each survey occasion is called a wave and PSTP starting in the fall of 1989 continues until today with sample size of approximately 1700 households per wave with a total often waves. Each survey collects three groups of data. They are household demographics, persons' social and economic information, and reported travel behavior through a household questionnaire and a two-day travel diary. In the travel diary, each household member who is age 15 years or older reports every trip made during two consecutive weekdays, which remained approximately the same throughout the panel years. Each trip is described by trip purpose, type, mode, starting and ending time, origin and destination, and distance. Based on the reported trips, out-of-home activity engagement information is derived from the trip purposes. The durations of activities are computed by the difference between the start time of the next trip and the end time of the current trip giving the sojourn time at an activity location. In the original trip data of PSTP, the trip purpose for each trip has been classified into 9 different types at Wave 7 (work, school, college, shopping, personal business, appointment, visiting, free-time, and home) and 21 different types at Wave 9. The use of these different schemes for trip purposes between the waves makes it difficult to compare the two waves. To make the data for the two waves comparable and this analysis tractable, activity types have been grouped into: In-home Out-of-home Subsistence (work, school, and college), and Out-of-home Non-subsistence (shopping, personal business, appointment, errand/picking-up/dropping-off, free time, recreation/exercise, visiting, etc) Since 1997, PSTP has included additional questions on respondents' traveler information system use and telecommunications and computer ownership to gain some insight on how people use traveler information to make their transportation decisions (Kilgren, 1998). This offered a unique opportunity to correlate ICT use with travel behaviour in a diachronic way. For the analysis in this study, we used the data from 1480 persons in 866 households who participated in both waves 7 and 9, and provided detailed information for all the variables used in this analysis. Table 1 shows the number of persons, households, and a few social and demographic characteristics of the sample. It also shows a descriptive summary of key variables during the two interview days in wave 7 (1997) and wave 9 (2000). Activity duration is the total amount

Behavioral dynamics in travel and activity participation 47 1 of time each person spends in a specific activity in a day. Total travel time is the total amount of time spent by a person traveling during the day. Therefore, the sum of activity durations for various activities and total travel time is equal to 1440 minutes per day for each person. Activity frequency is the number of activity episodes for each activity type each person engaged in a day. However, it should be noted that in this study all activities before the first trip and after the last trip are considered as one activity episode, because travel diaries do not provide information on activities before the first trip and after the last trip (usually at home). Therefore, the total number of trips is the sum of activity episodes minus one. As expected, and as found in earlier studies (Ma and Goulias, 1999'a, 1997b) using PSTP data, survey participants show more similarity in activity and travel patterns between the two days than between the two waves. The relatively large discrepancy between the two waves is most likely a result of genuine change but also the use of different coding schemes for trip purposes in the travel survey between the waves. However, it was found that there has been an increase in in-home activity and out-of-home non-subsistence activity durations between the waves but a decrease in out-of-home subsistence activity and travel. In terms of activity-travel, in the year 2000, the respondents spent an average of 1013.7 minutes in in-home activity, 239.0 minutes in out-of-home subsistence activity, 112.9 minutes in out-of-home non-subsistence activity, and 74.4 minutes in travel per day. The bottom of Table 1 shows the characteristics of technology ownership and use in the sample with focus on the more modern technologies. Computer and Internet at home are the two fastest growing technologies and it shows there is a large proportion of the respondents who use computers in their daily lives. Computers at work appear to be stabilizing at 50% of the sample, and although Internet use at work is increasing, it did not reach the level of market penetration of computers at work. Cellular phone ownership is increasing even faster than computers and the Internet, causing a potential negative impact on pagers. Laptop computers and Personal Digital Assistants (PDA) are used by a small sample segment as reported.

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Table 1 A Selection of Sample Characteristics Wave 7 Characteristics (1997) Number of persons in the sample Number of households in the sample Percent of males in the sample Person & Number of employed persons in the sample Household Number of persons in household Number of cars per household Day 1 Total amount of time in in-home activities (min.) Day 2 Day 1 Total amount of time in out-ofhome subsistence activities (min.) Day 2 Activity & Travel Day 1 Total amount of time in out-ofDuration home non-subsistence activities Day 2 (min.) Day 1 Total amount of time traveling (min.) Day 2 Day 1 Total number of trips per person Day 2 Day 1 Number of in-home activity episodes Day 2 Activity Frequency Day 1 Number of out-of-home Day 2 subsistence activity episodes Day 1 Number of out-of-home nonsubsistence activities episodes Day 2 At home Number of persons who use computers regularly (%) At work At home Number of persons who use the Information & Internet regularly (%) At work Communication Number of persons having cellular phones (%*) Technology** Number of persons having pagers (%) Number of persons having laptops (%) Number of persons having PDAs (%*) * % over total number of person in the sample

Wave 9 (2000)

1480 866 46.9 859 2.5 2.2 965.2 978.5 285.0 274.0 104.9

1480 866 46.9 881 2.5 2.2 1009.7 1017.7 244.9 233.1 110.4

108.2

115.4

84.9 79.4 4.57 4.30 2.43 2.36 0.98 0.93 2.15 2.01 751 (50.7) 700 (47.3) 458 (30.9) 438 (29.6) 419 (28.3) 156 (10.5) 71 (4.8) 6 (0.4)

75.0 73.8 4.08 3.98 2.26 2.23 0.81 0.81 2.01 1.93 982 (66.4) 717 (48.4) 896 (60.5) 544 (36.8) 685 (46.3) 129 (8.7) 79 (5.3) 38 (2.6)

MODEL FORMULATION Variables Used Two groups of activity-travel variables are used as the endogenous variables in the SEM. The first group of the variables is total amount of time dedicated to a specific activity (in-home, out-of-home subsistence and out-of-home non-subsistence) and travel in a day. The second group is the number of activity episodes by the specific activity a person engaged in a day.

Behavioral dynamics in travel and activity participation 473 Table 2 Endogenous Variables Endogenous Variables Hdur Activity and Travel Durations

Sdur Nsdur Ttime Finhome

Number of Activity Episodes

Fsub Fnsub

Descriptions Total in-home activity duration per day (min) Total out-of-home subsistence activity duration per day (min) Total out-of-home non-subsistence activity duration per day (min) Total travel time per day (min) Number of in-home activity episodes per day Number of out-of-home subsistence activity episodes per day Number of out-of-home non-subsistence activity episodes per day

For exogenous variables, five groups of variables were used: household-level variables, person-level variables, time-related variables, ICT variables, and previous activity participation pattern variables. Table 4 provides an inventory of all the exogenous variables used for this study. We used cross-sectional information in the year 2000, as well as longitudinal information between the years 1997 and 2000 about household-level and personlevel social, economic, demographic characteristics and ICT ownership and availability. In addition, to take into account state dependence effects, we also used activity engagement and time use patterns in the year 1997 derived from latent class cluster analysis. To account for task allocation and roles within the household, number of adults, number of children by age group as well as vehicles owned and income representing resource availability are included as exogenous variables, hi addition, in order to account for the sampling stratification of the panel participants, the county of residence and sample indicators (TRANSIT, CARPOOL) are also included as exogenous variables. Indicators for changes in household and person characteristics, such as increase or decrease in car-ownership, changes in household composition, and changes in employment, were used to examine the effect of the changes on time use and mode choice. Two types of time variables are also included. The first type is a diary day indicator to account for a correlation between the first and second diary day. The second type is the set of day of week indicators to account for different activity and travel behaviors among weekdays.

474 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Previous Pattern

Information and Communication Technology

Time related

Person level

Household level

Table 3 Exogenous Variables Used in This Study CARPOOL TRANSIT KITSAP SNOHO TOTADULT TOT6 17 TOT1 5 MIDINC HIGHINC MHINC DK1NC CAR2 CAR3

Indicator, 1— household is sampled from carpool class; O=otherwise Indicator, 1- household is sampled from public transit class; O-otherwise Indicator, 1= living in Kitsap County; O-otherwise Indicator, 1— living in Snohomish County; O=otherwise Number of adults in the household who are 18 years old or older Number of children in the household who are between 6 and 17 years old Number of children in the household who are less than 6 years old Indicator, 1= $35,000 < household income < $75,000; 0=otherwise Indicator, 1- 575,000 < household income; 0=otherwise Indicator, 1= 535,000 < household income; 0-otherwise Indicator, l=household income is unknown; 0=otherwise Indicator, 1= two car household; 0=otherwise Indicator, 1- two car household; 0=otherwise

INADULT DNADULT INKID DNKID INBABY DNBABY INLICEN DNLICEN INBPASS DNBPASS INVEH DNVEH INEMP DNEMP MALE YOUNG MIDAGE PROF MANAG SECRE SALES WK5 DPUPIL DLICEN DBPASS EXPEMP NOVEMP QUITEMP DAY2 TUE WED THU FRI EXPCW NOVCW QUITCW EXPNW NOVNW EXPCH NOVCH NOVKH QUITNH EXPCEL NOVCEL QUITCEL EXPPAG EXPLAP NOVLAP QUITLAP HBOUND WORKER A WORKER_B WORKER C PWORK A PWORK B

Indicator, 1-an increase in the number of adults in the household between waves; 0=otherwise Indicator, 1-a decrease in the number of adults in the household between waves; 0-otherwise Indicator, 1= an increase in the number of kids whose age is 6-17 in the household between waves; 0=otherwise Indicator, 1- a decrease in the number of kids whose age is 6-17 in the household between waves; 0=otherwise Indicator, l=an increase in the number of children < 6 years in the household between waves; 0-otherwise Indicator, 1= a decrease in the number of children < 6 years in the household between waves; 0-otherwise Indicator, 1-an increase in the number of driver's license holders between waves; 0=otherwise Indicator, l=a decrease in the number of driver's license holders between waves; 0=otherwise Indicator, 1-an increase in bus pass holders in the household between waves; 0-otherwise Indicator, l=a decrease in bus pass holders in the household between waves; 0-otherwise Indicator, 1-an increase in the number of cars in the household between waves; 0=otherwise Indicator, l=a decrease in the number of cars in the household between waves; 0-otherwise Indicator, l=an increase in the number of employed persons in household between waves; 0=otherwise Indicator, l=a decrease in the number of employed persons in household between waves; 0=otherwise Indicator, 1-male; 0=female Indicator, 1-18< age < 34; 0-otherwise Indicator, 1—35< age < 64; 0=otherwise Indicator, l=having professional occupation; 0=otherwise Indicator, 1—having managerial occupation; 0-otherwise Indicator, 1-having secretarial occupation; O-otherwise Indicator, 1-having sales occupation; 0=otherwise Indicator, l=working outside of home for 5+ times a week; 0-otherwise Indicator, 1= student; 0-otherwise Indicator, 1-having driver's license; 0=otherwise Indicator, 1= having bus pass; 0=otherwise Indicator, 1- started getting employed outside home in both waves; 0=otherwise Indicator, 1= employed outside home in wave 9 only; 0-otherwise Indicator, 1- employed outside home in wave 7 but not in wave 9;0=otherwise Indicator, l=the second day of diary; 0=otherwise Indicator, l=diary on Tuesday; 0-otherwise Indicator, l=diary on Wednesday; 0=otherwise Indicator, l=diary on Thursday; 0-otherwise Indicator, l=diaryon Friday; 0=otherwise Indicator, 1- using computers at work/school in both waves; 0=otherwise Indicator, 1= started using computers at work/school after wave 7; 0-otherwise Indicator, 1= stopped using computers at work/school after wave 7; 0-otherwise Indicator, 1- using Internet at work/school in both waves; 0-otherwise Indicator, 1= started using Internet at work/school after wave 7; 0-otherwise Indicator, 1- using computers at home in both waves; 0-otherwise Indicator, 1- started using computers at home after wave 7; 0=otherwise Indicator, 1- started using Internet at home after wave 7; 0-otherwise Indicator, 1- stopped using Internet at home after wave 7; 0=otherwise Indicator, 1= using cellular phones in both waves; 0=otherwise Indicator, 1= started using cellular phones after wave 7; 0-otherwise Indicator, \= stopped using cellular phones after wave 7; 0=otherwise Indicator, 1- using pagers in both waves; 0=otherwise Indicator, 1= using laptop computers in both waves; 0-otherwise Indicator, 1- started using laptop computers after wave 7; 0=otherwise Indicator, 1= stopped using laptop computers after wave 7; 0-otherwise Indicator, 1= had home bound pattern in activity & travel behavior at wave 7; 0-otherwise Indicator, 1- had worker_a pattern in activity & travel behavior at wave 7; 0-otherwise Indicator, 1- had worker b pattern in activity & travel behavior at wave 7; 0=otherwise Indicator, 1= had worker c pattern in activity & travel behavior at wave 7; 0=othenvise Indicator, 1= had pwork a pattern in activity & travel behavior at wave 7; 0-otherwise Indicator, 1- had pwork b pattern in activity & travel behavior at wave 7; 0=otherwise

Behavioral dynamics in travel and activity participation 475 To examine the effect of changes in information and telecommunication technology between the years 1997 and 2000, we defined indicator variables for four groups of persons for each ICT: Persons that started using these technologies some time after 1997 and are using them in 2000 (new users); Persons that stopped using these technologies since 1997 (past users); Persons that never used them (non users); and Persons that started some time before 1997 and never stopped (experienced users). We use structural equation modeling techniques here because they can estimate a set of simultaneous equations capturing the interrelationship among a large number of endogenous (the variables considered dependent variables in the system) and exogenous variables (the variables considered given to us outside the model system). This modeling technique is used in the analysis of travel behavior since the middle of the 1970's. A comprehensive and informative review of many transportation research applications using structural equation models (SEM) can be found in Golob (2003).

LONGITUDINAL ANALYSIS Panel data enable us to measure change in a variable and concomitant effect(s) on another. From a behavioral viewpoint it is also interesting and useful to know if there is behavioral symmetry when the changes happen in opposite directions. For example, is the difference in activity participation the same when a person looses a job and how does that compare when a person gains a job? Or what is the effect of socio-demographic changes in the household (that is, an increase or a decrease in the number of children in the household)? This type of effects and relationships were hypothesized some time ago by Goodwin, Kitamura, and Meurs (1990), but they were not studied within a more comprehensive model system until the more recent analysis reported in Kim and Goulias (2004a, 2004b) and Kim (2004). In this section, we first provide a series of descriptive statistics of changes taking place during the 3-year period 1997 to 2000 in PSTP. This is followed by a summary of findings from a SEM application that is described in more detail in Kim (2004). Table 4 shows the average daily number of trips, travel time, and total activity participation for persons that entered the labor force between 1997 and 2000, persons that are no longer in the labor force in 2000 and the entire sample. The newly employed persons make more trips that the average but do not travel more and spend slightly more time in activities. Their variance (standard deviation for each indicator shown on Table 4) is higher for the number of trips and not for the travel time. For the persons that are no longer in the labor force in 2000 we see the opposite. They make less trips, stay for shorter time on the road, and have dramatically lower activity times. The associated variances are also much smaller indicating a lower propensity to heterogeneous behaviour.

476 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 4 Year 2000 Average Activity and Travel Behaviour Characteristics of Persons Entering and Exiting the Labor Force Number of trips per day Employed in Wave 9 but not in wave 7 Mean N Std. Deviation Employed in wave 7 but not in wave 9 Mean N Std. Deviation All the sample Mean Total N Std. Deviation

Total travel time per day (minutes)

Total activity time per day (minutes)

4.61 109 3.63

70.03 109 61.89

368.43 108 267.33

3.63 512 2.93

59.33 512 58.94

189.34 509 193.77

4.08 1480 2.97

74.97 1480 61.33

376.89 1473 258.60

Table 5 shows another aspect of changes in demographics. Persons in households that have experienced an increase in the number of children ages 6 to 17 between the two waves have on average more trips per day and stay on the road longer. On the other hand their counterparts from households that experienced a decrease in the number of kids make on average 4.15 trips per day that is very close to the sample average of 4.08 trips per day. Similarly for their travel time. However, their total activity time is larger than the sample overall average and the average of persons in households that increased their children between the two waves.

Table 5 Year 2000 Average Activity and Travel Behaviour Characteristics of Persons Experiencing Increases and Decreases in the Number of Children at Home Number of trips per day Increase in kids (ages 6 to 17) between waves Mean N Std. Deviation Decrease in kids (ages 6 to 17) between waves Mean N Std. Deviation All the sample together Mean N Std. Deviation

Total travel time per day (minutes)

Total activity time per day (minutes)

5.33 112 3.37

84.52 112 50.33

475.29 112 217.90

4.15 169 2.86

74.50 169 55.71

497.63 168 244.45

4.08 1480 2.97

74.97 1480 61.33

376.89 1473 258.60

Behavioral dynamics in travel and activity participation 477 A third aspect we consider here is ICT ownership and use. Table 6 contains the average travel and activity characteristics for a day in the survey of wave 9 for persons in household that have either acquired internet in their households or stopped using internet in their households. Interestingly "new users" travel more and spend more time in activities. Their variance in activity participation, however, is lower than the persons that no longer have internet at home and than the overall sample. Table 6 Year 2000 Average Activity and Travel Behaviour Characteristics of Persons that Started Using Internet at Home or Stopped Using Internet at Home Number of trips per day Starting Using Internet at Home between 1997 and 2000 Mean 4.65 N 194 2.89 Std. Deviation Stopped using Internet at Home between 1997 and 2000 4.22 Mean N 88 Std. Deviation 3.30 All the sample together Mean 4.08 N 1480 Std. Deviation 2.97

Total travel time per day

Total activity time per day

84.26 194 56.95

500.46 192 213.09

70.00 88 54.70

356.19 88 255.24

74.97 1480 61.33

376.89 1473 258.60

Tables 4, 5, and 6 provide a sample of the types of heterogeneous behaviour we attempt to capture when changes are taking place and we incorporate them into regression models that control for all other kinds of mutual influences among indicators of travel and activity behaviour. Using as guideline the model in Figure 1 and as explanatory variables the list of Table 3 a variety of models were estimated, tested, compared and one was selected as the master model from which to extract inferences. Before proceeding to the presentation of the findings we should mention that a SEM produces three type of effects of one variable on another: direct, indirect, and total effects. The overall model can be written as:

y = By + rx + <;

(1)

where y = pxl vector of observed endogenous variables. x = q x 1 vector of observed exogenous variables. B = P x P matrix of coefficients of the y-variables. F = p x q matrix of coefficients of the x-variables. 4" = p x 1 vector of equation errors. SEM is a covariance-based model, which means the differences between the sample covariances and the covariances predicted by the model are minimized. The underlying

478 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction theory of this estimation procedure is that the population covariance matrix of the observed variables ( £ ) is a function of a set of parameters: I \ - \ c o v a r iance matrix of y covariance matrix of y and xl Z = £(y) = [covariance matrix of x and y covariance matrix of x j =

"(/-B)-1(ror+^)[(/-B)-1]' (/-BrTol
O

j

where O = covariance matrix of x. T = covariance matrix of £". The matrix £(#) basically consists of three covariance matrices. The unknown parameters B, F, , and y are simultaneously estimated by finding the parameters such that the covariance matrix ( £ ) implied by the model is as close as possible to the sample covariance matrix (S). To know when the estimates are as close as possible, a fitting function that is to be minimized is defined. ML estimation method assuming a multivariate normal distribution was employed for this study. ML estimation was found fairly robust to deviation of multivariate normality and sample size commonly used in transportation research (Golob, 2003). The ML fitting function that is minimized is: FML = log|l(0)| + 1KSI-1 ((9)) - log|5] -(p + q)

(3)

The direct effects, which are estimated as B and T, are the influences of one variable on another without the mediation of any intervening variables. The indirect effects are the influences that are mediated by at least one intervening variable. The total effects are the sum of the direct and indirect effects. It should be noted that interpreting a model with the direct effects only provides misleading conclusions when the direct and the total effects are very different. It is the total effects that should be used in interpretation. The model building and analysis using SEM parallels the study reported in Kim and Goulias (2003). In additiion, the relationship among all endogecous variables and the variation explained by cross-sectional variables are reported elsewhere Kim and Goulias (2004a). In this paper we focus on the impact chnages have on behavior measured in the year 2000. The variables reported in Table 7 describe the difference in activity and travel behavior among households and individuals that experienced diametrically opposed events such as increase in employed persons versus a decrease in employed persons, increase in household cars versus a decrease in household cars, as well as personal changes such as change in employment status. As seen in Table 9, in all the events, the effects of personal and household changes on time allocations to activity and travel and the number of activity episodes are not symmetrical and very often are not even in the opposite direction. Consider car ownership change. Persons that experienced either an increase or a decrease in the number of vehicles in the household have on average longer in-home activity duration than

Behavioral dynamics in travel and activity participation 479 the reference group of no change. The travel time, however, reflects the car availability increase and persons spend more time on the road than their counterparts experiencing a decrease in car ownership. Employment changes and children changes as expected explain differences among the participants' behavioural indicators. Table 7 Total and Direct Effects of Household-level & Person-level Change Variables on Endogenous Variables Endo genous Variables ttime Nsdur sdur finhome fsub fnsub Hdur 3.196 0.165 -0.852 0.128 0.1 0.171 Total -2.837 Inadult 0.000 0.000 0.000 0.000 0.000 0.128 Direct -0.330 0.320 -2.697 0.521 11.523 Total -9.342 0.039 0.189 Dnadult 0.000 0.000 0.000 6.682 0.000 0.000 Direct 0.189 -3.686 4.436 -0.857 0.107 0.008-0 0.008 -0.207 Total Inkid 0.000 0.000 0.000 0.000 0.000 0.000 -0.206* Direct 34.928 -6.751 -7.062 0.0 0.031 -0.084 Total -20.819 Dnkid 0.303 0.000 0.000 0.000 0.000 33.127 Direct -7.729 -30.287 32.716 -0.246 -0.047 -0.788 -13.927 Total 11.491 Inbaby 0.000 0.000 34.937* -10.403* -0.246 0.000 -0.456 Direct 0.0 -0.071 5.972 1.524 -0.295 -7.206 0.087 Total Dnbaby 0.000 0.000 0.000 -7.732 0.000 0.111* 0.000 Direct -17.644 30.293 -5.855 -0.292 0.025 -0.396 -6.631 Total Inlicen 0.170 28.227 0.000 0.000 -0.292 0.000 0.000 Direct 0.685 -0.006 -23.684 28.505 0.0 Total -5.509 0.049 Dnlicen 0.000 0.000 0.000 0.000 0.000 0.000 28.387 Direct 3.217 12.606 -0.622 0.150 -0.052 -0.150 Total -15.209 Inbpass 0.000 0.000 0.000 0.000 0.000 0.000 -14.952 Direct -1.601 -0.588 0.114 2.076 -0.101 Total 0.109 0.139 Inemp 0.000 0.000 0.000 0.000 -0.108 0.000 Direct 0.109 -0.013 3.032 1.030 -3.864 -0.155 -0.207 Total -0.199 Dnemp 0.000 0.000 0.000 0.000 -0.155 0.000 0.000 Direct 32.075 1.234 -35.273 2.111 0.024 -0.058 0.0 Total Inveh 0.147 0.000 -35.035 4.606* 0.000 0.000 0.000 Direct -0.096 -0.004 -0.187 21.952 -19.990 -3.855 Total 1.889 Dnveh 0.000 0.000 0.000 -0.096 0.000 0.000 -19.624 Direct -69.568 -123.074 188.917 3.707 0.000 0.612 -0.514 Total Expemp -33.055 177.888 0.000 -0.396 0.000 0.000 0.276 Direct -68.121 0.008 0.554 0.008 -15.282 Total 79.069 4.329 Novemp 0.000 0.000 0.000 0.000 0.408 0.000 79.250 Direct Note: A direct effect value of 0.000 for a variable indicates that the variable was constrained to 0 in the model, because of its insignificance at 90% level. * Significant at 90% level; all others are significant at 95% level. Exogenous Variables

Table 8 reports the effects of ICT on time allocations and activity frequencies giving evidence of a lack of symmetry and linearity in the effects when gaining a technology and losing a technology. Although the effects are not exactly symmetrical in most cases, they often have the opposite patterns. For example, new computer users at work spend more time on subsistence and less time on in-home and non-subsistence while past computer users at work have the opposite behaviors in those activities. One interesting fact is, however, that the

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effects of the past computer users at work are almost symmetrical to those of the experienced computer users, not the new computer users, on in-home, subsistence, and non-subsistence activity durations (i.e., they have different signs but similar coefficients in magnitude). In case of cellular phones, however, the effects of new users and past users are roughly symmetrical on all activity and travel durations. Table 8 Total and Direct Effects of ICT Variables on Endogenous Variables Endogenous Variables finhome fsub fnsub hdur sdur Nsdur ttime -0.193 0.082 -0.517 Total -43.367 58.044 -11.218 -3.466 Expcw -0.193 0.000 -0.248 Direct 0.000 51.205 0.000 0.000 -0.123 0.044 -0.398 Total -23.632 32.152 -6.214 -2.310 Novcw -0.123 0.000 -0.228 Direct 0.000 26.316 0.000 0.000 0.000 -0.336 -0.087 Total 56.910 -56.452 10.911 -11.368 Quitcw 0.000 -0.201 0.000 Direct 0.000 -58.316 0.000 -8.624 0.000 -0.019 0.077 Total -37.708 35.124 -6.789 9.374 Expnw 0.000 -0.111 0.000 8.990 Direct 0.000 36.766 0.000 -16.175 19.468 -3.763 0.468 0.000 0.034 0.145 Total Novnw 0.000 22.569 0.000 0.000 0.000 0.000 0.148* Direct -6.138 -1.567 0.303 7.406 0.000 0.025 0.073 Total Expch 7.281 0.000 0.000 0.000 0.000 0.000 0.000 Direct 0.000 0.069 -23.512 14.110 -2.727 12.133 0.113 Total Novch 0.000 0.000 0.000 0.000 16.523 0.000 11.536 Direct 6.820 1.741 -0.336 -8.228 0.000 -0.028 -0.081 Total Novnh 0.000 0.000 0.000 -8.089 0.000 0.000 0.000 Direct 1.577 0.016 -1.307 -0.334 0.065 0.000 0.272 Total Quitnh 0.000 0.000 0.000 0.000 0.000 0.266 0.000 Direct -46.140 11.381 24.639 10.124 -0.074 0.132 0.073 Total Expcel 0.000 14.574 26.838 8.980 -0.074 0.085 0.000 Direct -21.768 -2.154 17.653 6.273 0.013 0.100 0.000 Total Novcel 0.000 0.000 17.237 4.922 0.000 0.000 0.000 Direct 3.128 -29.385 -6.992 33.245 -0.195 -0.010 -0.346 Total Quitcel 0.000 0.000 -28.781* 0.000 -0.195* 0.000 0.000 Direct 39.070 -46.488 0.000 -0.080 0.009 8.985 -1.118 Total Exppag 0.000 0.000 0.000 0.446 -46.295 0.000 0.000 Direct -0.441 -0.172 -0.486 -71.356 46.193 33.190 -8.034 Total Explap -0.441 -0.202 0.000 0.000 45.486 42.118* 0.000 Direct 0.000 0.025 -0.556 34.995 11.920 -44.073 -2.846 Total Novlap 0.000 0.000 -0.430 0.000 0.000 -41.769 0.000 Direct 0.000 -0.015 0.206 5.119 -71.217 -4.425 70.527 Total Quitlap 0.000 0.000 0.000 0.000 0.000 69.672 0.000 Direct Note: A direct effect value ofO. 000 for a variable indicates that the variable was constrained to 0 in the model, because of its insignificance at 90% level. * Significant at 90% level: all others are significant at 95% level. Exogenous Variables

Experienced and new users of ICT, except for the laptop, generally show a similar pattern in time use and episode frequency by activity. In the case of laptop, the experienced users and new users show quite different effects on all the activity and traveling indicators (different

Behavioral dynamics in travel and activity participation 481

signs or quite different coefficients in magnitude), except for non-subsistence episode frequency. In the period of 1997-2000, we observed a rapid increase in computer and internet users at home. These two rapidly growing technologies seem to have different effects on people's behavior. The new computer users at home spend less time on in-home activity and more time traveling, while the new Internet users at home spend more time on in-home activity and less time in traveling. In addition, these two groups have the opposite effects on the number of subsistence and non-subsistence episodes. Using the coefficients in the Tables here we can perform some additional calculations. Regression coefficients defined for a group of indicators are relative to the excluded group (implicitly assumed to have a zero coefficient). Suppose a woman who did not work in 1997 is newly employed in 2000, thus getting a computer and Internet at work, a cell phone, and a laptop computer. Due to her new employment and ICT use, she spends 139.87 (0.59+79.07+32.15+19.47-2.15+11.92) minutes more on subsistence activity than her previous status, resulting from an increase in the number of employed persons in the household (-0.589 in Table 9), new employment (79.069 in Table 9), using a computer and Internet at work (32.15 and 19.47 in Table 10, respectively), using a cellular phone (-2.15 in Table 10), and using a laptop ( 11.92 in Table 10). Similarly, it turns out that she spends 96.30 minutes less on in-home activity, 51.57 minutes less on non-subsistence activity, and 7.99 minutes more on travel than others. Again, the sum of changes in time allocation is equal to zero (139.87-96.30-51.57+7.99=0) showing how the time reallocation (trade-off among activities and travel) occurs with a limited time budget after her employment. Similar calculations applied to the number of activity episodes lead to 0.57 more subsistence episodes and 0.56 less non-subsistence episodes than her previous status. In addition, men spend on average 34.35 minutes less on in-home activity, 32.32 minutes more on subsistence activity, 6.25 minutes less on non-subsistence activity, and 8.28 minutes on travel, and have 0.08 more subsistence episodes and 0.20 less non-subsistence episodes than women. These are not negligible differences. Dependence on Past Behavior In order to account for habit persistence effects we first derive using latent class cluster analysis a set of relatively homogenous groups of activity participation behavior in 1997. The following clustering variables are used: Total in-home activity duration in dayl and day2 (Hdurl, Hdur2) Total out-of-home subsistence activity duration in dayl and day2 (Sdurl, Sdur2) Total out-of-home non-subsistence activity duration in dayl and day2 (Nsdurl, Nsdur2) Total travel time in dayl and day2 (Ttimel, Ttime2) Number of in-home activity episodes in dayl and day2 (Finhomel, Finhome2) Number of in-home activity episodes in dayl and day2 (Fsubl, Fsub2)

482 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Number of out-of-home non-subsistence activity episodes in dayl and day2 (Fnsubl, Fnsub2) Starting from a one-cluster model we build in a sequence models with more clusters until the cluster sizes become too small to be meaningful and the difference in goodness of fit between successive models is not significant. Table 9 shows the seven types of activity-travel behavior and their average profiles showing wide variability in time allocation and frequency of activity episodes among clusters. The first and largest cluster contains 23.9% of the sample appears to be the non-workers cluster. This group has the second longest time at home (22-22.5 hours per day), no subsistence activity time, and a high frequency of non-subsistence activity episodes in both days. The second, third, and fourth clusters appear to be regular workers/students clusters with 21.1%, 17.01%, and 13.6% of the sample, respectively. Although all these groups have a consistent activity-travel pattern between the two days, they have different levels of activity participation patterns. The second cluster (worker_a) and the fourth cluster (worker_c) show two different ends of activity-travel behavior for regular workers/ students, the third cluster (workerb) is the "middle" between these two behaviors. In other words, among the three groups, worker_a group has the shortest time in subsistence activity but the longest time in non-subsistence activity and traveling in both days. This group is also characterized by high frequency of episodes in all activity types. On the other hand, worker_c group has the longest time in subsistence activity and in-home activity but the shortest time in non-subsistence activity and traveling. Workerc group is characterized by low frequency of all activity episodes. The sixth cluster (hbound) with 7.2% of the sample is characterized by their home bound activity behavior in both days. This group spends the majority of time on in-home activity and a very short time on out-of-home activity and traveling. The other clusters (pwork_a and pwork_b) have different time use patterns in inhome and subsistence activity between the two days.

Behavioral dynamics in travel and activity participation 483

Table 9 Average Profile of Activity and Travel Clusters Variables used to create clusters Cluster Size (%)

1259.59 0.00 120.17 60.23

792.71

775.61

799.82

812.93

1305.55

1143.11

Sdur1 (min.) Nsdur1 (min.) Ttime1 (min.)

443.10 91.32 112.87

524.33 49.66 90.40

564.53 1.48

240.05 280.93

0.00

2.49 0.00

2.69

2.49 1.62

106.09 2.46 1.12

0.28 191.07 105.53

Finhome1 Fsub1

74.17 2.08 1.36

91.69 42.76 1.84 0.00

2.57 0.01

Hdur1(min.)

Cd

Q

(N >i

cd

P

Cluster2 Cluster3 Cluster4 Cluster5 Cluster6 Cluster7 Cluster1 (Nworker) (Worker_a) (Worker_b) (Worker_c) (Pwork_a) (Hbound) (Pwork_b) 21.14 13.75 10.18 6.80 17.01 23.92 7.19

Fnsub1

2.78

1.91 2.24

Hdur2 (min.)

1227.70

783.03

Sdur2 (min.) Nsdur2 (min.) Ttime2 (min.)

0.03 142.08 70.20

Finhome2 Fsub2 Fnsub2

1.81

0.27

2.67

1.94

3.82

810.57

1021.12

1439.73

957.30

417.69 123.76 115.52

794.91 523.41 38.24 83.44

555.88 1.01 72.54

0.03 337.86 80.98

0.00 0.19 0.07

296.25 102.03 84.42

2.57 0.00

2.68 1.74

2.062 1.63

2.06 0.01

2.32 0.001

1.02 0.00

2.60 1.33

2.99

2.41

1.62

0.26

3.11

0.02

2.24

Number of cases : 1480

BIC (based on LL): 160289.54

Number of parameters (Npar): 226

AIC (based on LL): 159091.78

Log-likelihood (LL): -79319.89

CAIC (based on LL): 160515.54

Each cluster is assigned a dummy indicator for cluster membership and then entered as explanatory variable in the SEM. Turning to the effects of these clusters on year 2000 behavior and summarized in Table 10, each group has almost identical patterns of time use and activity frequency to those in the year 1997 (Table 9) derived from the LC cluster analysis. For example, the home-bound group (Hbound) spends more time on in-home and less time on non-subsistence and traveling than any other group. Additionally, this group has the lowest number of activity episodes in all activity types. Among "regular" worker groups (worker_a, worker_b, and worker_c) worker_c group spends the most time on subsistence and the least time on in-home, non-subsistence, and traveling, and it has the least activity episodes in all activity types. On the other hand, workera group has the exact opposite position, except for in-home duration. In other words, this group spends the least time on subsistence and the most time on non-subsistence and traveling among the three groups, and it has the most activity episodes in all activity types. The time use and activity frequency of worker_b group are in the middle of worker_a and worker_c, except for a little less time on in-home than both worker_a and worker_c . Figures 3 and 4 provide a better pictorial comparison of time use and activity frequency patterns. These identical patterns of activity and travel behavior among the groups over the 3 year period of 1997-2000, confirm people's strong habit persistence in activity and travel behavior.

484

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Table 10 Total and Direct Effects of Previous Activity and Travel Patterns on Endogenous Variables Exogenous Variables Hbound worker_a worker_b worker_c pwork_a

Total Direct Total Direct Total Direct Total Direct Total Direct

hdur 28.876 0.000 -80.633 0.000 -69.917 0.000 -82.066 0.000 -58.759 0.000

sdur 9.401 0.000 51.461 50.989 86.376 85.474 104.559 93.186 21.470 23.417

Endoggenous Variables Nsdur finhome ttime -24.783 -0.477 -13.502 -0.477 -22.966* 0.000 14.761 0.361-0 14.413 0.000 24.707 9.723 -0.115 -16.694 0.227 -0.115 0.000 0.000 -0.726 -0.191 -20.209 0.000 0.000 -0.191 1.839 -0.091 35.449 0.000 -0.091* 39.599

fsub -0.026 0.000 0.361 0.232 0.316 0.174 0.162 0.000 0.134 0.107

Fnsub -0.928 -0.221 -0.022 -0.196 -0.160 0.000 -0.726 -0.451 -0.003 0.000

Note: A direct effect value of 0.000 for a variable indicates that the variable was constrained to 0 in the model, because of its insignificance at 90% level. * Significant at 90% level; all others are significant at 95% level.

SUMMARY AND CONCLUSIONS In this paper, using panel survey data and structural equations models complex relationships among the activity and travel indicators were studied by incorporating indicators of change in time. The system of equations defined here includes as dependent variables the amount of time allocated to in-home, out-of-home subsistence, out-of-home non-subsistence, and travel, as well as the total number of episodes by activity type. In order to get a clear trade-off relationship in time use among different activities and travel within a limited 24-hour time budget, in-home duration is extracted and included in the model system. This allows us to take into account a fixed time budget explicitly in the models and detect the existence of very strong substitutional and complementary relationships in time use among different activities and travel. It also enables us to uncover how persons are different in allocating their time when they experience a change in their life and they need to change their activity and travel behavior. The changes we study are changes in socioeconomic, demographic, and technology ownership and availability. Out-of-home activity (subsistence and non-subsistence) duration and travel time have very large substitutional effects (-0.834 to -1.022) on in-home activity duration, while out-of-home activity duration has complementary effects on travel time (0.024 to 0.074). Using information on changes in social and economic circumstances between 1997 and 2000 a number of change variables are defined and included in the model as explanatory variables to test the existence of behavioral symmetry when opposite events take place. In the majority of change effects we found social and economic changes to be non symmetric in influencing activity and travel behavior. In addition, the effects of different levels of ICT ownership and use on activity and travel behavior are also examined jointly with all the other determinants of activity and travel behavior. The overall "technology" effect seems to depend on the location and type of technology. It is also found that the technology effects are differentiated depending on the levels of ICT ownership and use.

Behavioral dynamics in travel and activity participation 485 Through latent class clustering analysis, seven homogenous groups in activity engagement and time use are first identified, and then the group indicators are included as explanatory variables in the model system. This method allows us to test if there is any habit persistence of individuals in activity and travel behavior, and at the same time, avoid many problems resulting from strong serial correlation in panel data. The model results confirm the existence of strong habit persistence in activity engagement and time use even in a relatively long period of time (3 years).

REFERENCES Golob, T.F. (2001), Travelbehaviour.com: Activity Approaches to Modeling the Effects of Information Technology on Personal Travel Behaviour. In Travel Behavior Research, The Leading Edge (edited by D. Hensher), Elsevier Science/ Pergamon: Kidlington, pp. 145-184. Golob, T.F. (2003) Structural Equation modeling. In Transportation Systems Planning: Methods and Applications, (ed. K.G. Goulias). CRC Press, Boca Raton, FL, pp 11.111.23. Golob, T.F. and A.M. Regan (2001) Impacts of Information Technology on Personal Travel and Commercial Vehicle Operations: Research Challenges and Opportunities. Transportation Research Part C, 10, pp.87-121. Goodwin, P.B., R. Kitamura and H. Meurs (1990) Some principles of dynamic analysis of travel demand. In P.M. Jones (ed.), New Developments in Dynamic and Activity-Based Approaches to Travel Analysis, Gower Publishing, Aldershot, England, pp. 56-72. Goulias, K. G., and R. Kitamura (1997) Regional travel demand forecasting with dynamic microsimulation models In Panels for Transportation Planning: Methods and Applications (eds Golob, Kitamura, Long). Kluwer. Chapter 13, pp. 321-348. Goulias, K.G., N. Kilgren, and T. Kim (2003) A decade of longitudinal travel behavior observation in the Puget Sound region: sample composition, summary statistics, and a selection of first order findings. Presented at the 10th International Conference on Travel Behaviour Research, Moving through nets: The physical and social dimensions of travel, Lucerne, 10-14 August 2003. Kasturirangan, Pendyala, and Koppelman (2002) On the role of history in modeling activity type choice and activity duration for commuters. Transportation Research Record 1807, TRB, National Research Council, Washington, D.C., pp. 129-136. Kitamura, R. and M. Kermanshah (1983) Identifying Time and History Dependencies of Activity Choice. Transportation Research Record 944, TRB, National Research Council, Washington, D.C., pp. 22-30. Kitamura, R. A Sequential (1983) History Dependent Approach to Trip Chaining Behavior. Transportation Research Record 944, TRB, National Research Council, Washington, D.C., pp. 13-22.

486 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Kim T. and K. G. Goulias (2003) Cross sectional and longitudinal relationships among information and telecommunication technologies, daily time allocation to activity and travel, and modal split using structural equation modeling. Presented at the 83rd Annual Transportation Research Board Meeting Washington, D.C., January 11-15, 2004. Ma, J. and K. G. Goulias (1997a) A dynamic analysis of activity and travel patterns using data from the Puget Sound transportation panel. Transportation, 24(1), pp.1-23. Ma J. and K. G. Goulias (1997b) An analysis of activity and travel patterns in the Puget Sound transportation panel. Chapter 10 In Activity-Based Approaches to Travel Analysis (Editors D.F. Ettema and H. J.P. Timmermans). Pergamon Elsevier Science, The Netherlands, pp. 189-207. Murakami, E. and C. Ulberg (1997). The Puget Sound transportation panel. In Panels for transportation Planning methods and Applications, (ed. T.F. Golob, R. Kitamura, and L. Long), Kluwer, Boston, pp.159-192. Vermunt, J. K. and J. Magidson (2002) Latent class cluster analysis. In Applied Latent Class Analysis (edited by J. A. Hagenaars and A.L. McCutcheon), Cambridge University Press, Cambridge, UK. pp.89-106.

25

MODELING THE JOINT LABORCOMMUTE ENGAGEMENT DECISIONS OF SAN FRANCISCO BAY AREA RESIDENTS David T. Ory and Patricia L. Mokhtarian, University of California at Davis, Davis, CA USA

INTRODUCTION One of the most fundamental decisions a household member can make is whether or not to work. Assuming one chooses to work, depending on household and personal needs, one may decide to work part-time or full-time, and may subsequently or simultaneously decide to commute five times per week, telecommute some days, work at home full-time or work a compressed schedule. These decisions may be made by the household member or, in some circumstances, the employer, and repeated any number of times as the needs of the household evolve. Such employment status/type decisions are vitally important to travel behavior and forecasting. Predicting travel patterns is greatly aided by first predicting employment type the travel patterns of a fully commuting full-time worker are obviously different than those of a non-worker or telecommuter. As activity-based modeling has evolved in the past years the need for appropriate employment status models has been enhanced. Almost every existing activity-generation scheme starts with an assumed knowledge of employment status (see, e.g. Goulias, 2002); older, trip-based generation models typically require the same assumptions. The purpose of this chapter is to develop a model of the sequential or simultaneous decisions to work or not work; to work full-time or part-time; and to commute each day, telecommute some days, work a compressed schedule, or work exclusively from home. Such a model will both enhance travel behavior modeling and provide insight into the behavior and characteristics driving these work and commute choices. There is an extensive literature on the labor force engagement decision: to work or not. Much of this literature involves the decision of women to join the workforce. Heckman (1974)

488 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction estimated a tobit model that simultaneously captured a woman's decision to work (versus not work), the number of hours spent working, her asking wage, and her offered wage. Cogan (1981) used the premise that working has a cost associated with being away from the family; he then used this idea to model the decision of married women to enter the workforce, and their number of hours worked, using a maximum likelihood estimator. Mallela and WilcoxGox (2003) build upon the Bureau of Labor Statistics' model of work life by adding past work experience as a predictor of current employment; both logit and probit models are estimated. Barkume and Horvath (1995) use gross flows statistics to examine the movements from being unemployed, employed, and not in the labor force. Key findings from this body of work are (not surprisingly) that gender, household assets, and the presence of small children influence the decision to work. There is also a healthy body of literature on the decision to work part-time versus full-time. Williams (1995) used regression modeling and gross flows analysis to model the shifts from unemployment, full-time employment, part-time employment, and not being in the labor force; the end goal of his work was to measure gender differences in these behaviors. To assess gender discrimination in France, Moulin (2003) estimated probit models of the decision to work full-time (as opposed to part-time) separately for men and women, at two points in time (ten years apart). Lane (2004) compared the attitudes and performance of fulltime and part-time nurses in the United Kingdom. Findings germane to the work at hand include the importance of education and gender in the decision to work full-time. The impact of working a compressed schedule (e.g. working 80 hours over 9 days instead of 10) on transportation has been studied by Hung (1996), who found that such a practice can substantially reduce commuting. A much larger body of research involves the study of working flextime (usually defined as the ability to change the start and end time of the work day). Beers (2000) finds that, in 1997, nearly one-fourth of all full-time workers in the United States could vary their work day start and end times (one would expect a much smaller share working a compressed schedule). Golden (2001) finds that those working flexible schedules are often spending more time at work or switch to part-time status; he estimates a probit model of the likelihood of working flextime. In this study, we use "telecommuting" to refer to a salaried employee working at home in lieu of commuting to the conventional work location some or all of the time. We distinguish telecommuters from self-employed home-based business workers. In comparison to other dimensions of the employment type choice, the literature on the decision to telecommute is perhaps somewhat more sparse. However, there are behavioral models of preference for home-based telecommuting (Bernardino and Ben-Akiva, 1996) and between home- and center-based telecommuting (Bagley and Mokhtarian, 1997), choice of home-based telecommuting (Mokhtarian and Salomon, 1996), and frequency of home-based (Yen and Mahmassani, 1997) and center-based (Ho, 1997) telecommuting. Findings relevant to the work here include the importance of commute distance in the telecommute decision. Yeraguntla and Bhat (2005) separately model a handful of these decisions: to work part-time or full-time using a binary logit model; telework decision and frequency using an ordered-

Modeling joint labor-commute engagement decisions 489 response model; and ease of working a flexible schedule, again using ordered response. Their findings fall in line with the work of others. A primary aim of the present chapter is to bring the labor force engagement and travel behavior fields of inquiry together by estimating a single model that jointly predicts each of the discussed choices. When considering the various plausible nesting alternatives for these decisions that are presented later in the chapter, it is important to remember that they need not imply a temporally sequential structure to the decisions - they may simply represent possible correlation patterns among the alternatives in question. Either way, however, to our knowledge, this is the first empirical estimation of these choices jointly. It is also innovative in the use of subjective variables, such as attitudes and personality, to help explain those choices. One limitation of the study is that the labor-commute engagement decisions are modeled at the individual, rather than household, level. Though variables do capture the presence of other income and persons in the household, an ideal model would jointly predict this decision for each member of a household - but the data used here do not support such an estimation. The chapter is organized as follows. The next section will introduce the data used in the modeling. Subsequent sections will discuss how the dependent variable was extracted from the data, the structures considered in the modeling, and the estimation results. A summary section concludes the chapter.

EMPIRICAL SETTING AND AVAILABLE DATA The data analyzed in this study are collected from a 14-page self-administered survey of approximately 2,000 individuals in the San Francisco Bay Area. A total of 8,000 surveys were mailed (leading to a response rate of about 25%) to randomly-selected households in three neighborhoods, namely North San Francisco (half of the surveys), Concord (one-quarter) and Pleasant Hill (one-quarter). North San Francisco is an urban neighborhood, located close to the regional central business district (CBD) and well-served by transit. Concord and Pleasant Hill, in contrast, are suburban cities located across the San Francisco Bay from the regional CBD - reasonably well-served by BART (the regional rapid rail transit system), but with low levels of bus service. Although they are contiguous, they differ in that Pleasant Hill has higher densities and a more fragmented street pattern. Thus, together they represent some diversity in types of suburban development. A summary of key demographic variables is included in Table 1 (note: of the 1,904 total respondents with relatively complete data, only 1,676 are included in the modeling, as discussed later). The table segments the data by the six choice categories used in the modeling, namely: non-worker, home-based worker, part-time worker, telecommuter, fully commuting worker, and compressed-schedule worker. Table 1 indicates that our sample is relatively balanced in terms of gender and neighborhood location. Higher incomes are overrepresented compared to the Census (see Curry, 2000 for further discussion). However, as the focus of the work is to model the impact of income and other variables on the employment

490 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction decision, rather than purely to ascertain the population distribution of such measures, it is more important simply to have a reasonable spread of incomes than that they be exactly representative (Babbie, 1998). The same is true of the dependent variable of interest. Potential Explanatory Variables The potential explanatory variables used in the models can be placed into seven general categories, namely: Commute Characteristics, Travel Liking, Attitudes, Personality, Lifestyle, Mobility Constraints, and Socio-demographics. Each category is described very generally in this section. Commute Characteristics: Measures of one-way commute time and distance are used to possibly explain the decision to telecommute. Travel Liking: Participants rated their liking for travel (segmented by purpose and mode, separately for short-distance and long-distance trips, where long-distance is defined as more than 100 miles one way, for consistency with the American Travel Survey of long-distance travel in use at the time of data collection) on a five-point scale ranging from "strongly dislike" to "strongly like". The purposes considered here are overall, commuting to work or school, and for work/school-related activities. The modes considered are personal vehicle, bus, and train/light rail. Attitudes: Attitudes towards travel, land use, and the environment were captured using responses on a five-point Likert-type scale, to 32 statements. Through factor analysis (see Redmond, 2000 or Mokhtarian, et al, 2001 for details of the factor analyses on these as well as the Personality and Lifestyle variables), the statements were distilled into six basic dimensions, namely: travel dislike, pro-environmental solutions, commute benefit, travel freedom, travel stress, and pro-high density. Selected variables loading heavily on the Attitude, Personality, and Lifestyle factors are summarized in Table 2. Personality: Respondents rated 17 attributes expected to relate to their travel attitudes and/or behavior on a five-point scale (anchored by "hardly at all" to "almost completely"), in terms of how well the attributes described them. Here, the factor analysis revealed four personality types: adventure-seeker, organizer, calm, and loner. Lifestyle: The survey contained 18 statements related to work, family, money, status, and the value of time. Respondents agreed or disagreed with the statements using a five-point Likerttype scale. Four lifestyle factors emerged: status seeker, workaholic, family/community related, and a frustrated factor.

Table 1: Key Socio-demographic Characteristics of Sample (N=l,676) Non-worker

Home-based

Part-time

Telecommuter

Fully-commuting

Compressed

AII(N)

Number (percen t) Concord

62 (37.8)

26(16.5)

50 (23.0)

10(19.6)

245 (24.5)

13(14.8)

406 (24.2)

Pleasant Hill

51(31.1)

35 (22.2)

65 (30.0)

15(29.4)

260(26.1)

28(31.8)

453 (27.0)

North San Francisco

51(31.1)

97(61.4)

102(47.0)

26(51.0)

493 (49.4)

47 (53.4)

817(48.7)

Female8

100(61.0)

83 (52.9)

158(72.8)

25 (49.0)

459 (46.3)

48 (54.5)

873(52.1) 1,644(98.1)

Have a driver's license1* Personal income0

Age'

158(96.3)

152(96.8)

211 (97.2)

51 (100.0)

984 (98.8)

88(100.0)

< $15,000

56 (37.3)

17(11.6)

65 (30.7)

3 (5.9)

23 (2.4)

5 (6.0)

169(10.1)

$15,000- 34,999

38 (25.3)

37 (25.2)

76 (35.8)

2 (3.9)

190(19.5)

13(15.5)

356(21.2)

$35,000-54,999

24(16.0)

32(21.8)

45(21.2)

10(19.6)

325 (33.3)

25 (29.8)

462 (27.6)

$55,000 - 74,999

15(10.0)

24(16.3)

12(5.7)

14(27.5)

193(19.8)

21 (25.0)

279(16.6)

$75,000- 94,999

6 (4.0)

13(8.8)

8 (3.8)

10(19.6)

107(11.0)

6(7.1)

149(8.9)

> $95,000

11 (7.3)

24(16.3)

6(12.8)

12(23.5)

138(14.1)

14(16.7)

205(12.2)

18 23

6(3.7)

1 (0.6)

18 (8.3)

2 (3.9)

23 (2.3)

1 (1.1)

51 (3.0)

24

38 (23.2)

60 (38.5)

73 (33.6)

24(47.1)

451 (45.2)

35 (39.8)

682 (40.7)

41-64

120(73.2)

82 (52.6)

101 (46.5)

25 (49.0)

506 (50.8)

51 (58.0)

885 (52.8)

>65

0 (0.0)

13(7.3)

25(11.5)

0 (0.0)

17(1.7)

1(1.1)

55(3.3)

40

Characteristic

Mean (standard devi:ation)

Total people in househoId (HH)

2.46(1.30)

2.37(1.14)

2.61 (1.29)

2.39(1.02)

2.34(1.22)

2.40(1.10)

2.40(1.22)

Total children under 18 inHH'

0.573(1.03)

0.417(0.857)

0.588 (0.956)

0.353 (0.688)

0.432 (0.827)

0.455(0.801)

0.463 (0.865)

Total workers in HH (fuill/part-time)"

0.911 (0.847)

1.75(0.754)

1.91 (0.826)

1.82 (0.623)

1.75(0.812)

1.70(0.714)

1.69 (0.842)

Number of personal veh icles in HHe

1.82(1.21)

1.85 (0.999)

1.89(1.21)

2.16(1.30)

1.85(1.04)

1.94 (0.975)

1.87(1.08)

Total short distance travel (miles/week/

134.0(119.4)

156.9(162.5)

156.7(128.8)

305.0 (229.7)

231.5(196.8)

188.8(150.8)

205.2(183.2)

b

N=l,673;' N= 1,620; a N= 1,666;' ! N=l,670; f N=l,675

492 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 2: Factor Loadings for Selected Attitude, Personality, and Lifestyle Variables Variable category

Survey variable

Factor name

Travel dislike

Pro-environmental solutions

Commute benefit Attitudes

Factor loading

Traveling is boring.

0.621

I like exploring new places.

-0.537

The only good thing about traveling is arriving at your destination.

0.525

To improve air quality, I am willing to pay a little more to use an electric or other clean-fuel vehicle. We should raise the price of gasoline to reduce congestion and air pollution. We need more public transportation, even if taxes have to pay for a lot of the costs.

,. . . . 0.617 0.612

My commute is a real hassle.

-0.695

My commute trip is a useful transition between home and work.

0.583

The traveling that 1 need to do interferes with doing other things I like.

-0.530

I use my commute time productively.

0.467

In terms of local travel, I have the freedom to go anywhere I want to.

0.511

Travel freedom In terms of long-distance travel, I have the freedom to go anywhere I want to.

0.422

Living in a multiple family unit wouldn't give me enough privacy.

-0.617

I like living in a neighborhood where there is a lot going on.

0.486

Pro-high density

Travel stress

Adventure seeking

Personality

Organizer

Loner

Calm

Frustrated Family/community oriented Status seeking Workaholic Source: Redmond (2000).

I worry about my safety when I travel.

0.544

Traveling makes me nervous.

0.537

Traveling is generally tiring for me.

0.410

I tend to get sick when traveling.

0.318

I am uncomfortable being around people I don't know when I travel.

0.297

Adventurous

0.776

Variety seeking

0.695

Spontaneous

0.574

Risk taking

0.557

Efficient

0.624

On time

0.371

Like being alone

0.935

Like being independent

0.314

Aggressive

-0.599

Patient

0.532

1 often feel like I don't have much control over my life.

0.720

I am generally satisfied with my life.

-0.618

I'd like to spend more time with my family and friends.

0.585

My family and friends are more important to me than my work.

0.472

To me, the car is a status symbol.

0.698

A lot of the fun of having something nice is showing it off.

0.518

I'm pretty much a workaholic.

0.652

I'd like to spend more time on work.

0.373

Modeling joint labor-commute engagement decisions 493 Mobility Constraints: Constraints on one's ability to travel are also expected to affect one's decision to work and commute. Here, participants selected, on a three-point scale ("No limitation", "Limits how often or how long", "Absolutely prevents"), the degree to which physical conditions or anxieties prevented them from engaging in a variety of travel forms, including: "walking", "taking public transportation", and "driving on the freeway". The percentage of time an automobile is available to the participant is also considered to be a Mobility Constraint (oriented in the reverse direction). Socio-demographics: The survey captured an extensive amount of typical socio-demographic data to allow for comparison of our sample with more general populations. The data included measures of age, income, household size, employment type, number of household workers, education level, gender, and make/model of the vehicle driven most often by the respondent. Dependent Variable: Labor Force / Commute Decision Ultimately, each individual in the sample will be placed into one of six categories: nonworker, home-based worker, part-time worker, telecommuter, fully-commuting worker, or compressed-schedule worker. Of course, further variations on these dimensions are certainly possible, such as someone telecommuting but only working part-time, but either the data available do not permit us to make those distinctions (as partially described in this section), or the numbers of cases in those categories are too small to be statistically robust. Unfortunately, the survey instrument used to gather the data did not expressly capture all of these choices. As a result, certain assumptions had to be made to segment the sample, hi this section, we briefly discuss the approach taken in segmenting the sample and highlight some of the difficulties (for a complete discussion of this topic, see Ory and Mokhtarian, 2005b). Table 3 provides the actual survey questions and possible responses for each of the variables used for segmentation. As a first step, the employment status variable was used to segment full-time, part-time, and non-working individuals. Each of these groups will be discussed separately here. Those working full-time were further segmented into the following four choice categories: home-based workers, fully-commuting workers, compressed-schedule workers, and telecommuters. Home-based workers either commute less than once a month (using the commute frequency variable) or stated that their one-way commute time and distance was "not applicable"; fully-commuting workers commute five or more times per week. Those commuting more than once a month but less than five or more times per week are labeled either as telecommuters or compressed schedule workers depending on their past engagement in these activities.

494 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 3: Segmentation Variables and Corresponding Survey Questions Segmentation

Survey Question

Variable

Response Options Full-time Part-time

Employment status

What is your current employment status?

Homemaker Non-employed student Unemployed Retired Never

Commute to work or school frequency

Counting only short-distance trips (100 miles or less one way), please estimate about how often you typically make each of the following types of trips, by any means of travel: commuting to work or school

Less than once a MONTH 1-3 times a MONTH 1-2 times a WEEK 3-4 times a WEEK 5 or more times a WEEK

How long does it usually take you to get to work (one way)?

Write in (mins.), or "Not applicable" check box

How far do you live from work?

Write in (mi.), or "Not applicable" check box

History of telecommuting

We are interested in knowing which of the following you have already done and why... Telecommute (part- or fulltime)

"Not done or not applicable" check box, and "Done: How long ago?" write in (yrs.)

History of working a compressed schedule

We are interested in knowing which of the following you have already done and why... Adopt a compressed work week (such as a "9/80" schedule)

"Not done or not applicable" check box, and "Done: How long ago?" write in (yrs.)

Applicability of home-to-work commute time and distance

23 or younger 24-40 Age category

What is your age?

41-64 65-74 75 or older

Respondents working part-time were placed into two choice categories: part-time worker and home-based business worker. Our initial approach was to label part-time workers who either make less than one commute trip per month or who make more "commute" trips (which could be to school as well as to work) but describe their one-way commute time and distance as "not applicable" as home-based workers (50 such individuals, to whom we refer as "part-time home-based workers", are present in the data); those with a part-time employment status not following these rules were labeled part-time workers. However, in certain instances, models were re-estimated after placing those who work part-time (whether in-home or out-of-home) in the part-time worker category. The final segmentation is made on those whose response to the survey employment status question was homemaker, non-employed student, retired, or unemployed. Those in the homemaker, retired, and unemployed groups who are younger than 65 are placed in the nonworker choice category. Non-employed students are excluded from the analysis, as are those over 65 years old.

Modeling joint labor-commute engagement decisions 495

In sum, 224 respondents are excluded; the six groups of interest have sample sizes as follows: 164 (9.8%) non-workers, 158 (9.4%) home-based workers, 217 (13.0%) part-time workers, 51 (3.0%) telecommuters, 998 (59.6%) fully-commuting workers, and 88 (5.3%) compressedschedule workers.

POTENTIAL MODEL STRUCTURES As discussed above, the goal of the modeling is to predict each individual's decision to not work, work entirely at home, work part-time, work full-time and telecommute, work full-time and commute five times per week, or work a compressed schedule full-time. These decisions could be modeled separately using binary logit, or jointly using multinomial or nested logit. Binary model structures To begin the modeling effort, four simple binary logit models were estimated. Here, we were examining the factors that influence what we see as fundamental decisions: to work or not to work; to work part-time or full-time; to fully commute or partially (or not) commute; and, to work some or completely at home versus completely out-of-home. For each of these models, the six primary choice categories were collapsed into two categories. Multinomial logit model structure The multinomial model considers each of the six primary choices (non-worker, home-based worker, part-time worker, telecommuter, fully-commuting worker, and compressed-schedule worker) individually. Here, we make the potentially restrictive assumption that the random components of utilities for each of the six choices are independent from one another. Two separate multinomial models were considered: the first placed the part-time home-based workers in the home-based worker category, and the second placed these individuals in the part-time worker category. Proposed nested logit model structures As indicated, the assumption of independence across the six alternatives is a strong one. Conceptually, one may expect the unobserved influences on certain choices, such as working a compressed schedule and telecommuting, to be correlated. To account for this correlation, three nesting structures were proposed and estimated, as shown in Figure 1 (a)-(c). In each case, the work/non-work choice constitutes the uppermost nest; a middle nest distinguishes among work choices; and the lowest nest (after which the structure is named) further distinguishes among one of the middle choices. Nesting structure (a) assumes the various options for working full-time are related, regardless of commute frequency. Structure (b) treats the partially-commuting decisions as related, regardless of work frequency. For example, the utility of all three alternatives may be related to a preference for avoiding congestion, or to a

496 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction desire for flexibility to support family needs. Structure (c) combines the non-commuting (home-based work) option into the nest with the partially-commuting options, on the assumption that all ways of avoiding some commuting might be related.

(a) Full-time workers nest

(b) Partial commuters nest

(c) Partial/non-commuters nest

Figure 1: Alternative Nesting Structures

MODEL ESTIMATION RESULTS We focus first and most heavily on our preferred model, the multinomial logit (MNL) structure. We then include some additional insights offered by the four binary and nested logit models estimated. Multinomial logit model The results of the multinomial logit estimation are shown in Table 4. The adjusted A value for the model is 0.110, which is on the low end of the range for the models discussed in the Introduction, but more noteworthy for having been achieved with a six-alternative choice set

Modeling joint labor-commute engagement decisions 497 rather than the binary one treated in most models. First looking at the variables specific to the non-work choice, Table 4 shows expected results, with education decreasing the likelihood of not working, and being female with children (ages 6 to 15) or with other household income increasing the likelihood. Turning to the home-based worker choice, those in production/construction/crafts occupations are more likely to work at home, as are those living in San Francisco (relative to the East Bay suburbs). These results fit the stereotype of small contractors working out of their homes, or of artists and craftspeople working from home (or in studio living/working space), especially near the city center. Three personality and lifestyle factor score variables enter the model specific to home-based workers: calm, frustrated, and workaholic. The coefficients of these variables indicate that home-based workers are likely not to be calm, or frustrated, but do tend to be workaholics. It is certainly possible that those working at home may be less frustrated because they are not commuting, or because they have more work autonomy and7or more time with family - i.e. that frustration level is an effect rather than a cause of one's work status. However (based on the variables loading heavily on the factor, such as those shown in Table 2), we are assuming here that the frustrated factor score represents a general approach towards life, and that those who do not have a difficulty with these frustrations are more likely to tackle the often difficult task of working entirely at home. Of course, the other two variables indicate they aggressively (i.e. not calmly) pursue working at home, which fits their workaholic nature. Variables specific to the part-time choice include the number of children ages 6 to 15 and other household income, both specific to females; the coefficients estimated for these variables carry the same sign (and, thus, the same interpretation) as those specific to the nonwork choice. A Mobility Constraint variable, percentage of time a vehicle is available, enters with a negative coefficient - consistent with the well-known result that having a car to use makes working full-time easier (see, e.g., Wachs and Taylor, 1998). Last, persons older than 65 are more likely to work part-time than their younger counterparts - a logical result.

498 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 4: Multinomial Logit Model Results (N=l,S80) Dependent Variable : Non-worker, home-based worker, part-time worker, telecommuter, fully-cormmuting worker, or compressed-schedule worker Explanatory Variables

Coefficient

t-statistic

-0.277

-3.66

1.15

3.21

0.191

2.38

-1.93

-5.36

1.20

3.15

Variables specific to Non-worker choice Education level [1, .... 6] Number of persons ages 6 to 15 in household (specific to females) [0, 1,...] Household income less personal income {specific to females) [0, 1, .... 5] Variables specific to Home-based worker choice Constant Production/construction employment type [0,1] San Francisco neighborhood [0,1]

0.879

3.74

Calm personality [-2.9,2.4]

-0.470

-3.41

Frustrated personality [-2.0,2.7]

-0.497

-3.28

0.752

3.48

Workaholic lifestyle (specific to females) [-2.1,2.3] Variables specific to Part-time worker choice Constant

0.282

0.739

Number of persons ages 6 to 15 in household (specific to females) [0,1,...]

1.19

3.41

Age 65 to 74 [0,1]

2.79

7.51

3.01

5.33

Age 75 or older [0,1] Household income less personal income (specific to females) [0, 1

5]

Percent of time a vehicle is available [0, 20, ..., 100]

0.449

7.05

-0.0143

-5.81

Variables specific to Telecommuter choice Constant One-way commute distance [ > = 0] Adventure-seeker personality [-2.6,2.7]

-2.26

-6.28

0.0184

2.78

0.577

3.46

Variables specific to Fully commuting choice Constant Number of persons under age 6 in the household (specific to females) [0,1,...] Number ofpersons ages 6 to 15 in household (specific to females) [0,1,...] Age 41 to 64 [0,1] Two-plus adults with children HH status (specific to males) [0,1] Physical or psychological limitation on walking [1,2,3] Liking for commuting to work or school [ 1,... ,5] Family/community lifestyle [-3.9,2.1] Workaholic lifestyle [-2.1,2.3] Travel freedom attitude [-3.0,2.3]

1.67

3.97

-0.955

-.540

0.770

2.25

-0.547

-4.54

0.821

3.42

-0.S18

-3.81

0.194

3.27

0.273

3.39

0.309

4.00

0.244

3.07

-0.316

-3.76

0.316

3.50

Two-plus adults with children HH status (specific to males) [0,1]

-1.62

-4.84

Pro-environmental solutions attitude (specific to males) [-2.3,2.4]

1.37

3.79

-0.598

-3.03

0.609

3.68

Pro-environmental solutions attitude [-2.3,2.4] Pro high-density attitude (specific to females) [-2.5,2.3] Variables specific to Compressed-schedule choice Constant

Pro high-density attitude (specific to females) [-2.5,2.3]

[ ] = range of possible responses Log-likelihood at convergence = -1751.93 Log-likelihood with constants only = -1977.13 Rho-squared = 0.114; Adjusted rho-squared = 0.110

Moving to the telecommuting alternative, individuals choosing this option are more likely to be adventure seekers and tend to live farther from work. The finding that those who live farther from work are more likely to engage in telecommuting agrees with the results of Mokhtarian, et al. (2004) and others. We speculate that adventure seekers would be disproportionately drawn to the still-novel, potentially career-risky option of telecommuting, as a way of achieving or manifesting greater work autonomy, and perhaps as a way to save time

Modeling joint labor-commute engagement decisions 499 (by commuting less) that can be allocated to more adventurous pursuits instead (Cao and Mokhtarian, 2005; Clay and Mokhtarian, 2004). The fully commuting choice holds the largest share of individuals in the sample and also has the largest number of variables entering into the model. Females with young children (less than six years old) are less likely to commute fully, but, interestingly, females with slightly older children (age 6 to 15) are more likely to fully commute. This variable enters the nonworker, part-time worker, and fully commuting worker utility functions with positive coefficients in each case, suggesting that if a woman with children in this age range does choose to work full time, she is likely to do so in a manner that allows for at least some separation from the children by commuting some or all of the time, individuals ages 41 to 64 are less likely to commute fully, perhaps demonstrating the flexibility they've gained through years of working. On the other hand, however, men in households with other adult(s) and children are more likely to commute fully (reinforcing the workaholic variable mentioned below), perhaps in an effort to climb the career ladder to provide for their families, or as an escape from domestic distractions or stresses, or both. Also, individuals who are unable or are limited in their ability to walk are, unsurprisingly, less likely to commute fully. A host of attitude, lifestyle, and personality variables also enter the fully commute utility specification. Not surprisingly, those who enjoy commute travel tend to do so, as do workaholics and those who feel they have freedom to travel when and where they want to. Another expected result is that those with stronger pro-environmental feelings are less likely to commute fully, as they may see doing so as needlessly harming the ecosystem. Interestingly, the family/community lifestyle variable enters with a positive coefficient, meaning that those who value family and friends are more likely to fully commute. This result suggests the paradox that a family-oriented person may feel that the best way to benefit the family is through the financial support that may be maximized by commuting fully. An unexpected result is seen in the pro high-density factor attitude variable (specific to women) entering with a positive coefficient. One interpretation is that those who live in high density areas may not have the space to accommodate working from home, even some of the time. Another is that those individuals tend to have shorter commutes and hence less motivation to reduce them. A third interpretation is that this variable could be a marker for the young, upwardly-mobile professional who values the social and professional opportunities at the workplace, as well as the shopping, entertainment, and other synergistic activities available around a dense urban work location. The compressed-schedule utility function contains three variables, each of which was described in the fully commuting discussion. One minor difference is the pro-environmental solutions attitude entering specific to males. As mentioned previously, the multinomial logit model was estimated in two different ways. The first placed the part-time home-based workers into the home-based worker category, and the second (selected) placed these individuals into the part-time worker category. The goodness-of-fit measure for the first estimation was slightly lower than for the second

500 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

(presented in Table 4Table 4, with an adjusted A value of 0.101. This result seems to indicate that part-time home-based workers are more similar to part-time workers than to full-time home-based workers, which is quite plausible. If working part-time is by choice, these workers may see employment as a secondary drive in their lives, behind going to school, enjoying retirement, or caring for young children. Such a population sits in stark contrast to full-time home-based workers who may be taking enormous risks to start and/or continue operating their own businesses from home. Additional insights from the binary logit models To economize on space, the full binary logit models are not presented here, but are available in Ory and Mokhtarian (2005b). The adjusted A2 goodness-of-fit measures range from 0.041 (for the model of Partial versus Full commute) to 0.179 (for the model of Full-time versus Part-time), at least the higher end of which is in the range of the values of 0.13 to 0.40 found in the literature presented in the Introduction. It is not surprising that our models have lower goodness of fit than models based on more specific populations, such as Heckman's (1974) work with married Caucasian women ages 30 to 44. An interesting finding from the Work versus Don't Work models was the impact of Travel Liking variables. Two Travel Liking variables (overall and commute) enter this model specific to the work choice, with coefficients similar in magnitude, but opposite in sign (overall is negative, commute positive). Those who enjoy commuting more than overall shortdistance travel are more likely to work, whereas those who enjoy overall travel more than commuting are more likely not to work. Previous analysis of these data found similar tradeoffs: the overall enjoyment of short-distance travel is both hindered by having to work (leaving less time for such travel) and reduced by having less enjoyable experiences while traveling to work (i.e. negative feelings may be transferred to non-work travel) (see Ory and Mokhtarian, 2005a, for more discussion of this issue). The average net effect of these two variables on utility is positive, i.e. that affective beliefs about travel are more often a positive than a negative influence on the decision to work. The inclusion of the Travel Liking variables does raise the issue of endogeneity: does a fondness for travel influence the decision to work? Or do those who work grow to enjoy the benefits of travel? Certainly both directions are possible (Ory and Mokhtarian, 2005a), and we believe the modeled direction to be eminently plausible. The Full-time versus Part-time model revealed that variables specific to the part-time choice are largely gender-specific. As expected, females caring for young children are more likely to work part-time as are females in households with other income earners. Also, females who find travel stressful are more likely to work part-time. This result highlights the travel-related aspects of the labor-force engagement decision. An interesting result in that model is the opposing signs of the coefficients on the genderspecific two-plus adults with no children household status variables. Men in such households are more likely to work part-time, whereas women are more likely to work full-time. This finding is consistent with the stereotype of married women supporting their husbands while

Modeling joint labor-commute engagement decisions 501 the husbands return to college in an effort to advance their careers. It probably also represents the stereotypes of women as domestic care-givers and men as financial providers: women are more likely to work full-time when there are no children to physically care for, whereas men are more likely to work full-time when there are children needing financial support. Travel Liking for work/school travel enters the Full versus Partial commute model with a positive coefficient, specific to the full commute choice. Interestingly, the family/community lifestyle score enters with a positive coefficient specific to females. This finding is consistent with the multinomial estimation and again suggests the conflicting desires for time with, and separation from, the family. In the Work some at home versus Work out-of-home binary model, we find behavior consistent with stereotype, with females more likely to work some at home, the effect being more pronounced for women with children under the age of 6. However, the effect of children is partly counteracted by the household status variable. Two-plus adult couples without children are more likely than their single or child-bearing counterparts to work some at home. Although counter to the stereotype that home-based work is attractive as a strategy for balancing family and work obligations, this result is consistent with anecdotal accounts of telecommuters and non-telecommuters, that the home environment is more conducive to working when children are absent. Taken together, these demographic variables point to a multi-faceted role of gender and family in the decision of where to work, with competing influences often involved even within a single household. Additional findings from the nested logit models The three nesting structures described in Figure 1 were estimated. During the estimation process, nesting structures (a) (full-time workers) and (c) (partial/non-commuters) collapsed into an MNL model (i.e. the nesting coefficients were not statistically different from 1.0). The nested model that holds a portion of its structure, interestingly, is structure (b): partial commuters. During estimation, the upper level nest collapsed (due to its inclusive value coefficient not differing significantly from 1.0), leaving the model shown in Figure 2. It is interesting but logical that unobserved variables for the part-time work option are correlated with those for the partial commuting choices of telecommuting and compressed work schedules. These three groups are able to avoid commuting and perhaps add some flexibility to their lives by working in a slightly different way than the norm, while maintaining the relative security of being salaried employees (with fringe benefits). Home-based workers, by contrast, are probably most often self-employed, which is a very different way of gaining flexibility. Such workers may be more interested in having autonomy in their careers than in avoiding commute travel. The final nesting coefficient is 0.643 and is significantly different from 1.0 at the 95% confidence level. This yields an estimated correlation of unobserved variables for alternatives in the nest of 0.357. The overall goodness of fit for this model is slightly lower than for the binary and multinomial models, with an adjusted A2 of 0.089.

502 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2: Collapsed Nesting Structure (b)

The detailed results for the nested model are available in Ory and Mokhtarian (2005b); only two new results are highlighted here. For the non-worker and part-time worker alternatives, the two-plus adults with children in the household status variable specific to females enters with a positive coefficient, indicating the influence of traditional gender roles. In the fully commuting choice, the calm personality variable appears with a negative coefficient specific to females. This finding suggests that a somewhat more aggressive personality is found in women who undertake the historically male-dominated behavior of fully commuting. However, the same variable (gender-neutral) appears in the utility for home-based work, with a similar coefficient (as was seen in the multinomial logit model of Table 4). Thus, homebased work is also more appealing to those who are more aggressive, which is natural in view of the initiative required to succeed at a home-based business.

SUMMARY AND DISCUSSION This chapter presents binary, multinomial, and nested logit models of the decision to not work, work part-time, work full-time and commute fully, work full-time and telecommute, work full-time from home, and work full-time via a compressed schedule. Such an exploration is interesting and useful to both the field of travel demand modeling, which uses employment status as an input to models of activity and trip generation, and to labor force engagement modeling, which explores the driving factors behind these decisions. Using data collected from 1,680 individuals residing in the San Francisco Bay Area, a number of discrete choice models were estimated, with the preferred multinomial logit model discussed in most detail. The binary logit models included the choices to work or not work, work full-time versus part-time, commute fully or partially/not at all, and, for full-time workers, to work completely out-of-home or partially/completely in-home. The multinomial model simultaneously evaluated the choice among all six options, and the nested models included a structure that nested the partial commute choices: part-time worker, telecommuter, and compressed-schedule worker. In general, the model specifications fell in line with traditional models of labor force engagement: gender, education level, and the presence of young children played important roles. Travel variables also were significant in the models, especially variables describing Mobility Limitations, such as the absence of an available automobile and the inability, due to physiological or psychological reasons, to walk.

Modeling joint labor-commute engagement decisions 503 Part of what makes this work unique is the inclusion of Travel Liking, Attitude, Lifestyle and Personality variables; at least one of these measures proved significant in each of the models. Consistent throughout the estimation results was the role the Travel Liking measures played: those who enjoyed commuting tended to do so five or more times per week and those who enjoyed short-distance travel overall tended to commute less, suggesting these individuals value the ability to travel without the hindrance of frequent commuting or the negative psychological impact of traveling in congested traffic (which could lessen the enjoyment of other types of travel). The key Attitude variables were pro-environmental solutions, whose proponents typically chose to commute less frequently; pro-high density (female-specific), which was associated with commuting more often; and travel freedom, which was associated with full commuting. The key Lifestyle variable was the workaholic factor score, associated with both home-based and fully-commuting employment; the adventure-seeker Personality variable played a significant role in predicting the telecommuting choice. One interesting finding was the complex and conflicting roles of family, work, and travel, hi the MNL model, the findings indicate that while women with very young children are less likely to commute full time, women with children ages 6 to 15 tend to prefer the non-worker, part-time worker, or fully commuting worker choices. One interpretation of this latter result is that women with school-aged children who choose to work, do so in a manner to allow for some separation between their children and themselves. In related findings, men in households with other adult(s) and children are more likely to commute fully, as are those who state that family and friends are a primary focus of their lives; in each case, one could speculate that commuting fully offers the best chance to climb the career ladder, which would allow for the individual to better provide for their families. These results are consistent with the findings of Ory, et al. (2004), and suggest a tension between the desire to be with family, the desire to provide for them, and perhaps the desire to escape from them (Hochschild, 1997). Overall, though the estimated coefficients had reasonable and insightful interpretations, the goodness-of-fit measures were on the low end of results typically found in disaggregate models of labor force engagement. There could be several reasons for this: the absence of important exogenous variables, unclear endogenous variables, insufficient data, or a broad population. As most of the models in the literature (see, e.g. Heckman, 1974) use sociodemographic variables similar to ours (such as gender, net assets, and education level), without the additional variables we also have, the problem of missing exogenous variables seems comparatively minor, at worst, in this work. Unclear endogenous variables could definitely be problematic in that the survey did not directly inquire about current telecommuting or flexible scheduling, nor did it distinguish, in some instances, between being unemployed and not in the labor force (i.e. someone on permanent disability may have had difficulty finding a proper employment status option in the survey). Insufficient data could have also reduced the goodness-of-fit measures, as relatively small numbers of telecommuters and compressed-schedule workers (fewer than 100 in both cases) were observed in the sample. Finally, as mentioned previously, this work tackled the task of trying to predict a relatively complicated choice for all individuals age 18 to 64; such a broad population sits in

504 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction stark contrast to the work of others (see, e.g. Heckman, 1974), who focused on much narrower populations (e.g., married Caucasian women ages 30 to 44). It is interesting that the multinomial model fit the data better than any of the nesting structures. Although several groups of alternatives are conceptually correlated (as shown in the hypothesized structures of Figure 1), a violation of the assumption of independent error terms is a function of the model specification, not of similarities among alternatives per se (McFadden, et a!., 1977). Two remedies commonly-suggested for violation of this assumption are (1) to change variables from generic (having the same coefficient across all alternatives) to alternative-specific (allowing the coefficient to vary across alternatives), and (2) to improve the model specification by adding new variables (thereby moving more of the utility function from unobserved to observed and decreasing the correlations across alternatives of the remaining unobserved variables). In our case, remedy (1) is implemented automatically: since none of our variables differ across alternatives, they must necessarily have different coefficients for at least one alternative compared to the others, or they will not be able to influence the choice among alternatives. We believe remedy (2) is in place through the inclusion of our attitudinal and other non-demographic variables. We have seen these variables enter multiple alternatives, and thus when they are unobserved in other studies, they contribute to correlations of the error term across alternatives. The empirical superiority of the conceptually simpler MNL model in our context supports the advice of Horowitz (1991), to improve the specification of the observed portion of utility as much as possible before developing ever more elaborate models of the unobserved error terms. In particular, this work demonstrates again the importance of "internal", subjective variables such as attitudes in determining behavior. As this work represents one of the first efforts to model the joint labor-commute engagement decision, directions for future research are numerous. A first priority is to more accurately capture the dependent variable, both the chosen alternative as well as truly available but nonchosen alternatives (i.e. properly identifying each individual's choice set), both for the respondent and other members in the respondent's household (to facilitate modeling joint decisions among household members). Other viable and potentially interesting enhancements include: explicitly capturing the trade-offs in leisure time and income associated with changing job status; more detailed information about conditions at home (e.g. is the home large enough to hold an office?) and the workplace (e.g. is management receptive to the idea of flexible schedules?); and, capturing more detailed transportation supply variables (i.e. what modes are available for travel? Is walking possible?).

ACKNOWLEDGEMENTS The research was funded by the University of California Transportation Center. This chapter has benefited from the comments of several anonymous referees.

Modeling joint labor-commute engagement decisions 505

REFERENCES Babbie, E. (1998) The Practice of Social Research, 8th ed. Belmont, CA: Wadsworth Publishing Company. Bagley, M. N. and P. L. Mokhtarian (1997) Analyzing the preference for non-exclusive forms of telecommuting: Modeling and policy implications. Transportation, 24(3), 203-226. Barkume, A. J. and F. W. Horvath (1995) Using gross flows to explore movements in the labor force. Monthly Labor Review, 118(4), 28-35. Beers, T. M. (2000) Flexible schedules and shift work: replacing the '9-to-5' workday? Monthly Labor Review, 123(6), 33-40. Bernardino, A. and M. Ben-Akiva (1996) Modeling the process of adoption of telecommuting: A comprehensive framework. Transportation Research Record, 1552, 161-170. Cao, X. and P. L. Mokhtarian (2005) How do individuals adapt their personal travel? Objective and subjective influences on the consideration of travel-related strategies. Transport Policy, 12(4). Clay, M. J. and P. L. Mokhtarian (2004) Personal travel management: The adoption and consideration of travel-related strategies. Transportation Planning and Technology. 27(3). 181-209. Cogan, J. F. (1981) Fixed costs and labor supply. Econometrica, 49(4), 945-963. Curry, R.W. (2000) Attitudes toward Travel: The Relationships among Perceived Mobility, Travel Liking, and Relative Desired Mobility. Master's Thesis, Department of Civil and Environmental Engineering, University of California, Davis, June. Available at http://its.ucdavis.edu/publications/2000/RR-00-06.pdf. Golden, L. (2001) Flexible work schedules: what are we trading off to get them? Monthly Labor Review, 124(3), 50-67. Goulias, K. (2002) Multilevel analysis of daily time use and time allocation to activity types accounting for complex covariance structures using correlated random effects. Transportation, 29, 31-48. Heckman, J. (1974) Shadow prices, market wages, and labor supply. Econometrica, 42(4), 679-694. Ho, C. I. (1997) Modeling the Engagement in Center-based Telecommuting. PhD dissertation, Department of Civil and Environmental Engineering, University of California, Davis, September. Hochschild, A. R. (1997) The Time Bind: When Work Becomes Home and Home Becomes Work. New York: Metropolitan Books. Horowitz, J. L. (1991) Reconsidering the multinomial probit model. Transportation Research B, 25B(6), 433-438. Hung, R. (1996) Using compressed workweeks to reduce work commuting. Transportation Research A, 30(1), 11-19. Lane, N. (2004) Women and part-time work: the careers of part-time NHS nurses. British Journal of Management, 15, 259-272. Mallela, D. and V. Wilcox-G6x (2003) Employment experience as a predictor of current employment. Northern Illinois University Research Paper 2003-15, March.

506 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction McFadden, D., K. Train, and W. B. Tye (1977) An application of diagnostic tests for the independence from irrelevant alternatives property of the multinomial logit model. Transportation Research Record, 637, 39-46. Mokhtarian, P. L. and I. Salomon (1996) Modeling the choice of telecommuting 3: Identifying the choice set and estimating binary choice models for technology-based alternatives. Environment and Planning A, 28, 1877-1894. Mokhtarian, P. L., G. O. Collantes, and C. Gertz (2004) Telecommuting, residential location, and commute-distance traveled: Evidence from State of California employees. Environment and Planning A, 36, 1877-1897. Mokhtarian, P. L., I. Salomon, and L. S. Redmond (2001) Understanding the demand for travel: It's not purely "derived". Innovation: The European Journal of Social Science Research, 14(4), 355-380. Moulin, S. (2003) Decline of the permanent full-time employment model and gender discrimination in France. The 24th Conference of the IWPLMS, IWPLMS International Working Party on Labor Market Segmentation, Rome, 4-6, September, 2003. Ory, D. T. and P. L. Mokhtarian (2005a) When is getting there half the fun? Modeling the liking for travel. Transportation Research A, 39(2/3), 97-124. Ory, D. T. and P. L. Mokhtarian (2005b) Don't Work, Work at Home, or Commute? Discrete Choice Models of the Choices for San Francisco Bay Area Residents. Research Report UCD-ITS-RR-05-XX, Institute of Transportation Studies, University of California, Davis, February. Available at http://its.ucdavis.edu/publications/2005/RR-05-XX.pdf. Ory, D. T., P. L. Mokhtarian, L. S. Redmond, I. Salomon, G. O. Collantes, and S. Choo (2004) When is commuting desirable to the individual? Growth and Change, 35(3), 334-359. Redmond, L. S. (2000) Identifying and Analyzing Travel-related Attitudinal, Personality, and Lifestyle Clusters in the San Francisco Bay Area. Master's Thesis, Transportation Technology and Policy Graduate Group, Institute of Transportation Studies, University of California, Davis, September. Research Report UCD-ITS-RR-00-08. Available at www.its.ucdavis.edu/publications/2000/RR-00-08.pdf. Wachs, M. and B. D. Taylor (1998) Can transportation strategies help to meet the welfare challenge? Journal of the American Planning Association, 64, 13-17. Williams, D. R. (1995) Women's part-time employment: a gross flows analysis. Monthly Labor Review, 118(4), 36-44. Yen, J. R. and H. S. Mahmassani (1997) Telecommuting adoption: Conceptual framework and model estimation. Transportation Research Record, 1606, 95-102. Yeraguntla, A. and C. R. Bhat (2005) A classification taxonomy and empirical analysis of work arrangements. Presented at the 84th Annual Meeting of the Transportation Research Board. Paper 05-1522.

26

A MODEL OF DAILY TIME USE ALLOCATION USING FRACTIONAL LOGIT METHODOLOGY Xin Ye and Ram Pendyala, Department of Civil and Environmental Engineering, University of South Florida, Tampa, Florida, USA

INTRODUCTION Studies in travel demand model and travel behavior are increasingly focusing on modelling people's use of time and the effects of time use patterns on activity and travel characteristics. This tendency mainly arises from the shift in transportation planning emphasis from that of infrastructure construction to transportation systems management, travel demand management (TDM), and transportation control measures (TCM). The traditional trip-based travel demand models focus only on the trips made by individuals. Such a narrow perspective on activitytravel behavior is insufficient to evaluate policies aimed at managing and altering activitytravel patterns and choices. Thus trip-based models are likely to fail in capturing the full implication of various TDM-type policies (Pas, 1993; Schofer, 1993). Policies aimed at discouraging auto use may be considered beneficial from a pure vehicle trip reduction-based perspective. However, forcing solo-driving commuters to take other travel modes may not only suppress vehicle trips but also impose constraints and restrictions on their activity scheduling and travel throughout the day. Congestion may be mitigated, but a greater cost may possibly be imposed on individual commuters (Kitamura, et ah, 1997). People's time use patterns are useful reflections of the relationship between transportation and the quality of their life. The total time available each day, i.e., 24 hours per day, is identical for each person; however, the satisfaction or happiness of a person is likely to depend on how the 24 hour day is spent in terms of time allocation to various activities. For example, commuters may have to sacrifice time with family or at recreational activities when faced with a congested commute. A transportation system enhancement that relieves congestion may afford individuals the ability to spend more time participating in activities that they enjoy

508 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction and derive greater satisfaction. A modelling approach will be a powerful tool for evaluating transportation planning policies if it is capable of estimating individuals' time use allocation to travel and various activities given a set of explanatory variables, such as TDM policy indicators, demographic and socio-economic characteristics, land use characteristics, and level-of-service variables. People's allocation of time to various activity types and to travel can have a profound impact on the frequency, timing, duration, and location of trips. Some people may like to allocate more time to activities outside the home, others may prefer allocating more time to activities inside the home, and still others may enjoy spending time travelling (inside a vehicle). These preferences and resulting time allocation patterns directly impact travel characteristics because time is a finite resource that is bounded and consumed. Thus, people who prefer to spend more time at activities (locations) outside home will have less time available for indoor activities or travel. The remainder of this paper is organized as follows. Following this introductory section, a brief review of previous work on daily time allocation modelling is provided. This is followed by a detailed description of the modelling methodology adopted in this paper. The paper then includes a brief description of the data set used in the study followed by the data description, model estimation results and sample numerical simulation results. Finally, the paper presents conclusions and directions for future research.

MODELLING DAILY TIME ALLOCATION The modelling of daily time allocation to various activity categories and travel has been of much interest to travel behavior researchers in the recent past for the reasons noted in the introductory section of the paper. Covariance-based structural equations modelling methods have been used extensively to model activity and travel time use patterns. For example, Kuppam and Pendyala (2001) applied structural equations models to analyze commuters' daily activity and travel patterns. Commuters' daily time allocation to work and in-home and out-of-home recreation and maintenance activities, as well as travel variables such as trip frequency by purpose, are modelled using a series of structural equations models. Trade-offs in time use allocation among various activity types and travel were explicitly identified using the model results. Golob and McNally (1997) use structural equations to analyze activity and travel participation among household heads in multi-adult households. While structural equations approaches offer powerful methods for simultaneously modelling multiple endogenous variables, they do not guarantee that the time allocation predictions will fall within certain reasonable limits or that they will add up to the available time resource (say, 24 hours). In addition, marginal diminishing effects of exogenous variables are typically not captured in structural equations models. Bhat and Misra (1999) model allocation of weekly discretionary time between in-home and out-of-home locations and between weekdays and weekend days by developing model formulated as a continuous utility-maximizing resource allocation problem. They adopt a

Model of daily time use allocation using the fractional logit methodology 509 utility formulation that makes it possible to model the logarithm of the ratio of two time allocations as a linear regression estimated using the standard ordinary least squares (OLS) approach. Their approach is quite elegant in that it uses a well-defined parametric utility function for time use and classic microeconomic theory of utility maximization. However, the use of the logarithm of the ratio of two time allocations as the dependent variable may entail the exclusion of zero observations from the model estimation process. The exclusion of zero observations may result in bias, particularly in the context of individuals' daily time allocation, where a considerable portion of the sample may not travel outside home and allocate zero time to out-of-home activities on the survey day. This was unlikely to be an issue in the work by Bhat and Misra (1999) because they modelled weekly discretionary time allocation, where the number of zero observations is likely to be minimal. Yamamoto and Kitamura (1999) formulated the time allocation problem as a doubly censored tobit using a dual-state utility function that accommodates both zero and non-zero time allocation observations. They apply the methodology to the modelling of discretionary time allocation to weekdays and weekend days (similar to Bhat and Misra, 1999). The doublycensored tobit approach is used to estimate a model of the ratio of discretionary time allocation between in-home and out-of-home locations and between weekdays and weekend days (also similar to Bhat and Misra, 1999). The doubly-censored tobit model has also been employed by Lee, et ah, (2004) in the analysis of time use patterns within trip chains. In their work, as actual time allocations were modelled directly (as opposed to the logarithm of time allocation ratios), diminishing marginal effects of exogenous variables are not accommodated in the model. Meloni et al. (2004) applied the Nested Tobit (N-Tobit) model, with a hierarchical sequence of two equations, to describe how individuals allocate their discretionary time between inhome and out-of-home activities and between trips and activities. Similar to the work by Bhat and Misra (1999) and Yamamoto and Kitamura (1999), the logarithm of time allocation ratios are treated as dependent variables in the models. The models taking log-ratios as dependent variables have a common disadvantage in that it is computationally difficult to recover the expected value of the dependent variable, which is important for model applications. Consider a log-odds model where ln(y l\-y) = xj3 + v, where By back-calculating y, one two time fractions, y and \ — y, are involved. exp( xB + v) obtains y

, where v is the random error term in the linear model. 1 + exp( xfi + v) Then,E(y \ x) = f e —^— f(v | x)dv. A distributional assumption on v and intensive Lt0 1 + exp(x/? + v) computations are required to approximate the generalized integral for recovering the expected value of y (Papke and Wooldridge, 1996).

510 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction To overcome the limitations noted above, this paper develops and applies the fractional logit modelling methodology in the context of time use analysis. Papke and Wooldridge (1996) proposed this approach to analyze the participation rate in pension plans, where there are only two fractions: a participating fraction and a non-participating fraction where both fractions are bounded in the interval from 0 to 1 and must sum up to 1. By analogy, a binary fractional logit modelling approach can be extended to accommodate multiple fractions as dependent variables. Sivakumar and Bhat (2002) extended the fractional logit modelling approach from its original bi-fractional situation to the multi-fractional situation and applied it to analyze statewide commodity flows. If treated as fractions of individuals' daily time resource, the time spent on travel and various activities can be reasonably modelled within the framework of the fractional logit approach, where all dependent fractional variables are bounded in the interval from 0 to 1 and must sum up to 1. The quasi-maximum likelihood method, as opposed to the traditional non-linear least squares method, is employed to estimate efficient model parameters. In addition, the modelling methodology allows one to easily recover the expected values of dependent variables, accommodate zero observations, and capture the diminishing marginal effects of exogenous variables on the limited time resource. In this paper, two fractional logit models - one for workers and one for non-workers - are estimated to analyze person-level daily time allocation patterns to out-of-home activities, outof-home travel, and in-home activities. This trinomial allocation problem is modelled using person-level data derived from the 2000 Switzerland Microcensus data in which a sample of individuals completed 24 hour travel diaries documenting all of the trips and activities outside home that they pursued during the travel diary day.

MODELLING METHODOLOGY As mentioned in the previous section, time allocation to travel and various activities may be treated as fractions of the total daily time resource that is available to an individual. Modelling such fractional dependent variables can be conveniently done within the framework of the fractional logit modelling methodology. Let 0 < yqi < 1 and ^ yqi = 1, where q is an index that represents the individual and i is an index that represents the activity or travel type (q = 1,2, N where N is the sample size and i= 1,2, I where I is total number of various activity or travel types for each individual). The population model is assumed to be E{yqi\xqi, xq2,..., xqi) = G(/J xqi, xq2,..., xqi), where xqj is a vector of explanatory variables foryqi and /3 are vectors of model parameters. For ensuring E(yqi\xq) to lie between 0 and 1 and sum of yqi across i to be exactly 1, the multinomial logit may be used as a convenient functional form for G( ). Thus, one may have,

E(yql I *„) = G(yqi I xql)=

exp( /

^^)

2>P(*^)

(1)

Model of daily time use allocation using the fractional logit methodology 511 Alternatively, one may have, y«=

,£XP(X" '}

+*„

E{sqi\xqi) = Q

where

(2)

2>xp(^) and x9,/?, represents a time allocation function for activity type i. In this context, it should be noted that Gliebe and Koppelman (2002) developed a proportional shares model of daily time allocation and applied it to analyze the joint activity participation between two household members. Although the proportional shares model appears similar to the fractional logit model, it is not the same. In the proportional shares model, the gumbel-distributed error terms, e,, are a part of the utility functions, Vi = Xfi + e,. In the fractional logit model, the random error term appears solely outside the logit formula and takes no specific probabilistic distributional form other than having a conditional expectation of zero given the set of exogenous variables. The random error term here does not have a behavioral interpretation and serves purely as a random disturbance in the model formulation analogous to Bhat and Misra (1999). The non-linear least squares (NLS) method is one approach that may be used to estimate the model parameters /3. However, Papke and Wooldridge (1996) have shown that NLS estimates are not statistically efficient when applied to the fractional logit model. They state that the quasi-maximum likelihood method, as proposed in Gourieroux, et al. (1984), needs to be applied to estimate efficient model parameters. The Bernoulli log-likelihood function may be given by (Papke and Wooldridge, 1996 and Sivakumar and Bhat, 2002), N LL

i

= LLy«>l°s

exn( x B )

(3) V J=>

)

The log-likelihood function depicted in equation (3) offers a convenient functional form for maximization due to the logit specification used to represent time allocation fractions. In addition, this function is a member of the linear exponential family (LEF) and hence the quasi-maximum likelihood estimator (QMLE) is statistically consistent and asymptotically normally distributed provided equation (1) holds, regardless of the distribution of yqi conditional on xqi. It is worth noting that QMLE differs from conventional maximum likelihood estimates (MLE). MLE procedures entail maximizing the exact log-probability or the log-probability density function (with respect to model parameters), which is usually derived from the distributional assumption on the phenomenon under investigation. For instance, in the traditional multinomial logit model, the exact probability function for random discrete observations is derived from the assumption of gumbel-distributed random utility and the notion of utility maximization. On the other hand, QMLE procedures entail maximizing the

512 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction N

equation]Tlog{l[y q ,f(x q ,/?)]}, where yq = f(xq,P)

+ eq and l(u,m) belongs to the Linear

9-1

Exponential Family (LEF). There is no specific distributional assumption on the random term, eq. If l(u,m) is not a member of the Linear Exponential Family (LEF), QMLE will be invalid because parameters can not be consistently estimated (Gourieroux, et al. 1984). For the QMLE procedures, Gourieroux et al. (1984) show that *jN(p-P) ~ Normal(0,A^BA~l) as N -> =0. On the other hand, the asymptotic property of MLE is given by {PMLE - P) ~ Normal(0, A'1), where, P stands for true value of model parameters, p for quasi-maximum likelihood estimator, A"1 for negative inverse of Hessian matrix, B for outer product of first derivative of the quasi-log-likelihood function, and N for sample size. The log-likelihood function of the fractional logit model is the same as that of the multinomial logit model when dealing with fractional data (continuous values between 0 and 1). The NLOGIT 3.0 (Greene, 2002) program can be utilized to estimate the parameters of the fractional logit model by replacing the discrete dependent variable with fractional data. However, ATxBArx associated with asymptotic variance estimates are not directly provided; instead, NLOGIT 3.0 provides A~l, that can only be used to obtain non-robust variance of the parameter estimates.

A , „ , A dLL A s — - = 2_xqt(yqi -Gqi), OR;

, then,

9=1

8LL8LL'

Ar 2 . _, . , . =2j-Xqii.yqi-GqiY]

OPi

q=\

OPi

82LL N and — = 'y\[-xqiG i(\ - G )]. One can then derive that covariance-variance matrix of/?; N (Q

- y )

is CT,2I"1, where!" 1 and S" 1 are partitions of ATX andiT 1 for^,-, &] =A r ~ 1 Y . "' «='

2

!

G

qiV-Gqi)

and Gqi is estimated expectation of yqi, i.e., G{J3,xq). To obtain the robust variance for/?,, one only need calculate a] and then scale up A7 . Diagonal elements extracted from this scaled matrix are the variances ofP,.

Therefore, in this study, model parameters were

estimated first and the residuals Gqi - yqi were saved. Then the weighted standard errors, a] , were calculated and the non-robust variances were scaled up to finally obtain the robust variances and robust t-statistics.

DATA SET AND DATA PREPARATION The data set used for analysis and model estimation is extracted from the Swiss Travel Microcensus 2000. A very detailed description of the survey and the survey sample can be found in Ye and Pendyala (2003). Only a very brief description of the survey sample is provided in this paper in the interest of brevity. The survey respondent sample consists of 27,918 households from 26 cantons in Switzerland. The survey sample was formed by

,

Model of daily time use allocation using the fractional logit methodology 513 randomly selecting one person over 6 years old from each household with less than 4 household members and two persons over 6 years old from each household with 4 or more members. As a result of this sampling scheme, the person respondent sample consisted of 29,407 persons. All of the persons in the sample were asked to report their travel in a one-day trip diary. The resulting trip data set includes 103,376 trips reported by 29,407 interviewed persons including the possibility of some respondents making zero trips on the survey day. Those zero-trip reporting persons, accounting for 11% of the total sample, are assumed to spend 24 hours on in-home activities. Separate time use expenditure models are estimated for commuters and non-commuters to recognize the differences between these two market segments. In this paper, a commuter is defined as a respondent who reported at least one work trip on the travel survey day while a non-commuter is defined as one who did not report any work trips. In the remainder of this paper, the term commuter and worker will be used synonymously; likewise, the terms noncommuter and non-worker will be used synonymously. Also, for purposes of this analysis, the total time spent at work is considered exogenous. Although one could argue that an individual chooses the nature of his or her employment and may choose to work full time, part time, etc., the amount of time spent at work is generally fixed by the employer once this choice is made by an individual. Thus, for commuters, the available daily time resource is considered to be equal to 24 hours minus the total time spent at work. On the other hand, for non-commuters, the total time resource available is considered to be equal to the full 24 hours of the day. Thus, the three time use fractions are calculated based on 24 hours minus the total time spent at work for workers. For non-workers, the three time use fractions are calculated based on the full 24 hours. As most children do not necessarily have the freedom to allocate their daily time use and often depend on adults for mobility, models were only developed for the adult sample consisting of those 18 years or older. The Swiss Travel Microcensus 2000 is a traditional household travel survey in which respondents were asked to report their trips rather than the full daily time use pattern. However, the time use allocation pattern may be reasonably derived from the trip diary data using information on trip timing and trip purpose. Also, there are instances where the reported trips are not strictly made within a 24-hour daily cycle. The following rules were applied to derive time use fractions from the trip diary data:

514

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Table 1. Person Characteristics of Swiss Travel Microcensus 2000 Characteristic Sample Size

Swiss Sample Non-Workers Workers 17057 7978

Age (in years) Young (18-29) Middle (30-59) Old (260)

41.5 (Mean) 18.4% 75.6% 6.0%

52.8 (Mean) 13.1% 46.1% 40.8%

#Trips/day Work trips Non-work trips

4.68 1.61 3.07

2.92 0.00 2.92

Daily Time Use (min) Travel duration Work School Shopping Recreation Home Other

101.3 465.4 4.2 11.6 88.5 748.2 2.9

93.4 0.0 41.3 23.1 169.9 1100.0 4.0

Notes: Workers are those who made at least one work trip on the travel survey day. All others are non-workers.

1. The trip ending time is considered to be the starting time of the subsequent activity. The trip start time is considered to be the ending time of the previous activity. The activity type is determined by the purpose of the previous trip (i.e., the trip leading up to the activity). A trip to "work" indicates that the subsequent activity is work and trip to "return home" indicates that an in-home activity follows the trip. 2. The daily cycle is defined as the period from 3:00 AM to 3:00 AM on the next day. If travel is reported to start before 3:00 AM or end after 3:00 AM on the next day, it is assumed to start at 3:00 AM or end at 3:00 AM on the next day respectively. The trips or activities (and their respective durations) recorded outside this period are not taken into account for daily time use analysis in this paper. 3. If the purpose of the first trip is not "return home", duration from 3:00 AM to the starting time of the first trip is considered as in-home activity time. If the purpose of the first trip after 3:00 AM is "return home", then the duration from 3:00 AM to the arrival time at home is considered as out-of-home non-work activity. 4. If the purpose of the last trip is not "return home", then the duration from the end time of the last trip to 3:00 AM on the next day is considered as out-of-home activity time.

Model of daily time use allocation using the fractional logit methodology

515

Table 2. Average Fractional Time Allocations for Workers and Non-Workers Variable In-Home Activity Time Fraction Travel Time Fraction Out-of-Home Non-work Activity Time Fraction

Wor kers (N=7 978) Std Dev Mean 0.184 0.777 0.081 0.102 0.154 0.122

Non-Vvrorkers (N=l'7057) Std Dev Mean 0.210 0.777 0.082 0.066 0.179 0.157

Using the rules noted above, activity and travel durations can be computed and aggregated across three categories (travel, in-home activity, and out-of-home activity) for each individual. Household and person demographic and socio-economic characteristics and work-related information for workers (e.g., total work time, working timing, and commute distance) are merged into this person-level dataset for model development and estimation. Table 1 provides a summary of the person characteristics of the adult respondent samples by workers and non-workers that are used for model estimation. In this paper, daily time expenditures are classified into three broad categories: out-of-home travel time, in-home activity time, and out-of-home activity time. In the trinomial fractional logit models presented in this paper, these three categories of time use are considered as three fractions of the total available time resource. Table 2 shows the mean and standard deviation of the three time use fractions for 7978 adult workers and 17057 adult non-workers. It should be noted that the fractions for workers are computed based on a denominator equal to 24 hours minus the total time spent at the work activity. For example, if the total work time is assumed to be 8 hours, the mean in-home activity time is (24 - 8) x 0.777 = 12.43 hours, the mean travel time is (24 - 8) x 0.102 = 1.63 hours, and the mean out-of-home non-work activity time is (24 - 8) x 0.122 = 1.95 hours. Similarly, on average, non-workers spend 24 x 0.777 = 18.65 hours on in-home activity, 24 x 0.066 = 1.58 hours on travel, and 24 x 0.157 = 3.77 hours on out-of-home activity. Interestingly, the fractions of available time resources allocated by workers and non-workers to in-home activities are found to be virtually identical (i.e., 0.777). On average, this may be interpreted as non-workers spending around 6 hours more on in-home activities and around 2 hours more on out-of-home activities when compared with workers. This accounts for the typical 8-hour mandatory daily work activity duration. It is also noteworthy that the average daily travel durations of workers and nonworkers are quite similar.

MODEL ESTIMATION RESULTS This section presents results of the model estimation effort. First, model estimation results are presented for the commuter sample and second, for the non-commuter sample.

516 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Model Estimation Results for Workers The model estimation results for the worker sample are presented in Table 3. For purposes of identification, the out-of-home non-work activity is treated as the base fraction and the other two fractions are determined relative to the base. The estimation results yielded results that are generally consistent with expectations. The constant term in the function for in-home activity time allocation is positive implying that inhome activity time allocation (fraction) is the largest of the three fractions. On the other hand, the constant associated with the travel time fraction function is negative reflecting that travel generally accounts for the smallest fraction of time expenditure among the three categories considered in this paper. The estimated function for in-home activity time allocation fraction indicates that working males are likely to allocate less time to in-home activities relative to working females. This finding may reflect the tendency for female workers to bear a greater share of in-home activities and obligations. Older workers show a positive propensity to allocate larger time fractions to in-home activities and to travel as evidenced by the positive coefficients in these two functions relative to the base alternative. This is consistent with expectations in that older workers may have greater household obligations and hence stay more time in-home and more time travelling to take care of household obligations and chauffeuring. Similarly, married workers and workers with children are found to allocate more time to inhome activities and travel. Auto ownership displays a negative coefficient in both the functions associated with in-home activity time allocation and travel time allocation. In general, greater auto ownership may be associated with higher income levels and the ability to spend more time at out-of-home non-work (shopping and recreational) activities while travelling faster than non-auto owning individuals. Similarly, the variable representing high income shows negative coefficients in both time allocation functions. As expected, Fridays are associated with greater out-of-home non-work activity time allocation as evidenced by the negative coefficients in both equations associated with in-home activity and travel time allocation. The total time spent at work has a positive impact on both in-home activity time allocation and daily travel time allocation. This implies that workers who spend more time at work are likely to spend less time at out-of-home non-work activities as they spend their non-work activity time primarily travelling and at home. Commute distance is found to be negatively impacting in-home activity time allocation and positively impacting travel time allocation. These findings are consistent with expectations as workers with long commutes are likely to spend more time travelling and less time at home. The model also included several variables representing the work schedules of workers. Although these variables were not necessarily yielding significant coefficients, they were retained because they offered plausible interpretations and provided potential policy sensitivity (e.g., flex work hours). In general, these variables offer negative coefficients implying that working full time is associated with shorter in-home activity and travel time allocations. However, it should be noted that the

Model of daily time use allocation using the fractional logit methodology 517 variable representing working hours from 6 AM to 4 PM exhibits the most negative coefficient among the three work schedule variables in both time allocation equations. This appears to imply that working earlier hours in the day may contribute to a greater time allocation to out-of-home non-work activities relative to the other later working hours. Model Estimation Results for Non-Workers Table 3 also provides estimation results for the non-worker sample. Similar to the worker model, the constant term in the function for in-home activity time allocation takes on a positive value whereas the constant term in the function for travel time allocation takes a negative value. Thus, the findings suggest that the greatest fraction of the day is allocated to in-home activities while the smallest fraction is allocated to travel. As in the worker model, out-of-home activity time allocation is set as the base fraction. Several explanatory variables take the same sign as those in the worker model. For example, male non-workers are found to allocate less time to in-home activities relative to female nonworkers. This is once again consistent with expectations that females may be taking on more of the in-home tasks. Fraction of time allocation to in-home activities and travel increases with age, although the fraction allocated to travel decreases beyond a certain peak travel age as evidenced by the negative coefficient associated with the square of the age. Auto availability and high income are associated with a smaller in-home time allocation. Low income individuals are found to allocate less time to travel, possibly because they do not have the disposable income to travel and participate in various activities. The low income may also be representative of low car ownership. Married individuals and those with children are found to allocate more time to in-home stay, possibly due to family obligations. As expected, those who filled the travel diary survey on weekends showed a smaller propensity to allocate time to in-home activities, a finding that is suggestive of the discretionary travel orientation of weekends. The variable representing Friday has a positive coefficient in the daily travel time allocation function. This is consistent with expectations that people will likely engage in more travel on Fridays relative to other weekdays when other household members may be obligated to work and school activities. A few other variables that offered plausible interpretations include education, season of travel, and type of residential location, hi the function representing travel time allocation, higher education is associated with greater travel time allocation. On the other hand, individuals appear to allocate less time to travel in the winter season. This is likely due to the more cumbersome nature of travelling during the colder winter months in Switzerland. Also, those residing in rural country areas show a negative propensity to allocate time to travel. This may be reflective of the fewer destination opportunities that are easily accessible for engaging in out-of-home activities and travel in such areas. This may also be due to lower traffic congestion levels in rural country areas, thus contributing to lower travel times.

Table 3. Model Estimation Results for Workers and Non-workers Time Allocation Function Variable Constant Age in years Age2 Number of vehicles in household / household size Distance between residence and work place (km) Household Size Monthly household income is above Fr 10000 Monthly household income is below Fr 4000 Person lives in rural area Person lives in a household with children Person is highly educated with BA/MS/PhD degree Person is married Person is male Person is retired Person is of Swiss nationality Season is winter Travel is on Friday Travel is on weekend (Saturday or Sunday) Total daily work time in minutes Work starts at 6:00 and ends at 16:00 Work starts at 7:00 and ends at 17:00 Work starts at 8:00 and ends at 18:00 Est. CTi

W o i kers (Sample Size = 7978) 1 In-Home Jactivity Travel Time Function Time Furiction Coeff. Robust T Robust T Coeff. -11.307 11.135 0.8843 -0.7291 8.121 11.180 0.0149 0.0083

-0.2401 -0.0009

-7.092 -1.162b

-0.1740

-4.480

-0.1633 0.0068

-5.930 12.509

-0.1049

-3.311

0.1727

4.432

0.2101

4.129

0.0858 -0.2883

3.146 -12.015

-0.2418

-5.528

-0.1186

-3.364

-0.3018

-8.209

-0.1606

-5.286

0.0009 -0.1395 -0.0594 -0.0258

13.128 -2.418 -1.568b -0.468b

Non-workers (Sample Size= 170 57) In-Home tActivity 'ravel Time: Function Time Fuiiction Coeff. Robust T Robust T Coeff. 1.1752 -1.3893 -16.353 24.316 0.0138 0.0276 8.522 15.591 -0.0003 -8.315 -8.307 -0.1897 -0.0863 0.0841

-2.758 3.091

0.0895

2.638

0.1104 -0.1962 -0.0754 -0.1699

5.186 -10.296 -2.260 -5.967

-0.0353 0.0021 -0.1023 -0.0175 -0.0130

25.053 -1.470b -0.378b -0.191b 0.4185 0.2502 0.4623 0.3902 0.3892

Est. a2 Est. o 3 P2 Adj^ Out-of-Home Non-work Activity Time Function = 0. ' significant at the 0.10 level; b insignificant at the 0.10 level. All the other variables appear significant at the 0.05 level. i

i

i

-1.830"

0.4959 0.3162 0.4893 0.4013 0.4009

-0.0292

-3.248

-0.0674 -0.1024

-2.324 -4.485

0.1279

4.506

-0.1066 0.1509

-4.776 4.863

Model of daily time use allocation using the fractional logit methodology 519 Overall, it is found that the fractional logit models presented in this paper offer very plausible interpretations and high level of sensitivity to a host of socio-economic, demographic, and work-related policy variables. The robust t-tests show that the variables are, for the most part, statistically significant and capable of explaining time use allocation patterns of individuals. The adjusted p2 values are also reasonably consistent with what one might expect from disaggregate models of this type.

SAMPLE NUMERICAL SIMULATION This section presents a small sample numerical example to demonstrate how the fractional logit models presented in this paper can be used to estimate changes in time allocation patterns of individuals in response to changes in explanatory factors and some policy-related variables of interest. Consider a married male who is 40 years old and lives with his spouse and one child. Also, suppose that household income is high and there is one automobile in the household. The auto availability ratio is then 1/3. His expected time use allocation on a work day can be estimated by the fractional logit model presented in the left block of Table 3 including various work-related attributes such as work timing, total work time, and commute distance. The changes in the time use allocation pattern in response to changes in the work schedule may be suggestive of the changes in activity travel patterns that might result from implementation of TDM policies aimed at mitigating peak-period traffic congestion by shifting commuters' work schedules. Table 4 presents the results of the scenario analysis. Base Scenario (0) The base scenario assumes that the individual works for 8 hours (480 minutes) with work starting in the period of 7:00~7:59 and ending in the period of 17:00-17:59. This was found to be the most typical work schedule in the worker sample accounting for 13 percent of the observations. The distance between residence and work place is assumed to be 10 km. In this base scenario, the model indicates that he is expected to spend 99 minutes on travel, 105 minutes on out-of-home non-work activity and 756 minutes on in-home activities. These three figures add up to 960 minutes in accordance with the daily disposable time [1440 - 60 x 8 (work time)]. Time allocations obtained from a series of alternative scenarios are compared with the indications from this base scenario. Scenarios (1) and (2): Effect of Work Schedule In the first scenario, the worker is assumed to start and end work about one hour earlier in the day due to a flexible work hour strategy implemented at the work place. The model developed in this paper can be applied to analyze the policy implications of such an action. Table 4 shows the results of this analysis. As a result of this shift in work time, it is plausible to expect the person to have more discretionary time available in the post-work period (in the evening). This increased time availability may lead to increased level of non-work activity

520 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction participation in that period. Indeed, the application of the fractional logit model shows the person spending eight additional minutes on out-of-home non-work activities. Seven of these minutes come from in-home activity time allocation and one minute from travel time. The total travel time allocation may decrease modestly, possibly due to reduced exposure to congested traffic as a result of the shift in work schedule. While one may argue that the flex work hour strategy did not really yield much in the way of travel time savings (reduction in travel), one could also argue that the person is able to spend more time at non-work activities which presumably provide some satisfaction and benefit to the individual. hi the second scenario, the work schedule is shifted later in the day by about one hour. As expected, this scenario does not yield much change in the total time allocation pattern at all. There is a reduction of two minutes in in-home activity time allocation and a reduction of one minute in out-of-home non-work activity time allocation. The travel time allocation increases by three minutes. Scenarios (3) and (4): Effect of Total Work Time The third and fourth scenarios help identify the potential impacts of total work duration on time allocation patterns. Many flexible work hour strategies provide workers the ability to spend 9 hour or 10 hour work days with the option of getting a part of or the entire Friday off. In the third scenario, the total work time is increased by one hour while in the fourth scenario, the total work time is decreased by one hour. The 60 minutes of additional time spent at work in the third scenario have to naturally come from time spent travelling, in-home, and out-ofhome at non-work activities. In-home activity duration decreases by 33 minutes, travel time decreases by 11 minutes, and non-work activity time decreases by 16 minutes. The fourth scenario offers very similar results in the opposite direction. Of the 60 minutes less spent at work, a person allocates about one-half to in-home activities as evidenced by the 30 minute increase in in-home activity duration. The total travel time increases by 11 minutes and the out-of-home non-work activity time increases by 19 minutes. Scenario (5) and (6): Effect of Commute Distance Analyzing the impacts of commute time reduction that may occur in response to an improvement in the transportation system is of much interest due to the potential for induced out-of-home activity and travel engagement. Ideally, one would like to treat commute time as an exogenous variable in the time use allocation modelling context. However, in most travel surveys, it is difficult to isolate pure commute travel time from travel time to and from other activities because of the prevalence of trip chaining. Then, the home-to-work distance may serve as a useful proxy variable for commute time to measure its influence on commuters' time use allocation behavior as longer commute distances generally entail longer commute times.

Table 4. Sample Numerical Example of Model Application for Scenario Analysis

Scenario Variable AGE AUTO RT DIS_WORK (km) FRIDAY HIGH INC MARRIED PMALE SWISS WITH KID W O R K T M (min) WORK616 WORK7 17 WORK8 18 In-home activity tinne (min) Travel time (min) Out-of-home non-v\'ork activity (min)

BaseO

1

2

3

4

5

6

40 0.33 10 0 1 1 1 1 1 480 0 1 0 756 99

40 0.33 10 0 1 1 1 1 1 480 1 0 0 749 (-7) 98 (-1)

40 0.33 10 0 1 1 1 1 1 480 0 0 1 754 (-2) 101 (+3)

40 0.33 10 0 1 1 1 1 1 480+60 0 1 0 724 (-33) 88 (-11)

40 0.33 10 0 1 1 1 1 1 480-60 0 1 0 786 (+30) 110 (+11)

40 0.33 10+5 0

480 0 1 0 753 (-3) 102 (+3)

40 0.33 10-5 0 1 1 1 1 1 480 0 1 0 760 (+3) 95 (-3)

105

113 (+8)

104 (-1)

89 (-16)

124 (+19)

105 (0)

105(0)

522 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The application of the fractional logit model to these scenarios shows that there is a three minute trade-off between in-home time allocation and travel time due to the 5 km increase or decrease in commute distance. When commute distance increases by 5 km, the individual allocates three additional minutes to travel, all of which are taken from in-home time. Similarly, when commute distance decreases by 5 km, the individual spends 3 minutes less on travel and 3 additional minutes at home. There is no change in out-of-home non-work activity time allocation. Indeed, one might expect the change in commute distance to affect travel time allocation without affecting the actual time spent at non-work activities. Whereas the individual may have to spend 10-20 additional minutes travelling to and from work, the individual may trip chain more efficiently or choose a faster mode so that the total travel time changes only modestly (at about 3 minutes) leaving the non-work activity time allocation largely unchanged. The simple numerical example presented in this section shows that the fractional logit model is capable of serving as a powerful time use allocation analysis tool that is responsive to explanatory factors that may influence time use patterns. Combined with other models of activity and time use such as activity frequency, duration, timing, and type choice models, the fractional logit model would be able to serve as a powerful framework for analyzing activitytravel behavior while truly reflecting daily time resource constraints.

CONCLUSIONS In this paper, the fractional logit modelling methodology has been employed to represent time use allocation fractions and patterns of workers and non-workers. The samples used for the modelling effort have been drawn from the 2000 Swiss Travel Microcensus in which detailed socio-economic, demographic, commute, and one-day travel characteristics were collected. Separate models were estimated for worker and non-worker samples to recognize the differing constraints under which these market segments engage in activities and travel. Time use analysis has been receiving increasing attention in the field due to the increased interest in understanding how transportation investments, enhancements, or policies may affect people's lifestyles and quality of life. A robust time use model is needed to ensure that the policy analysis results are consistent with time constraints under which people operate and make decisions and choices. The fractional logit modelling methodology offers key advantages in the context of time use analysis, as summarized below: 1) Fractional logit model can guarantee predictions of time allocations between 0 and 24 hours or any available time resource. 2) Fractional logit model can guarantee the sum of all time use allocations to be exactly equal to 24 hours or any available total time resource. 3) Fractional logit model accommodates the non-linear relationship between explanatory variables and dependent time use allocations. It is believed that a non-linear relationship is more reasonable than a linear relationship because the marginal effect of an explanatory

Model of daily time use allocation using the fractional logit methodology 523 variable ought to diminish. This property recognizes that time is a finite resource and can not vary infinitely. 4) Fractional logit model parameters can be estimated easily within the simplistic logit formula using the Quasi Maximum Likelihood Method. The iterative process for maximization of the log-likelihood function is fast. 5) Using the fractional logit model, one can include zero observations without additional mathematical transformations. 6) Using the fractional logit model, one can easily recover the expectation of dependent variables without making any distributional assumption on the error terms. Despite the advantages of the fractional logit modelling methodology, there are some notable limitations that should be addressed in future research efforts. The analysis conducted in this paper was also limited by the data available from the Swiss Microcensus travel diary survey. The travel diary survey does not provide any data on specific in-home activities such as sleep, eat, recreation, home-care, child-care, etc. A greater understanding of time use allocation behavior could potentially be obtained by using a true time use diary data set that provides specific information about in-home activities. Such data would allow an explicit identification of in-home activity types and trade-offs that might exist in allocating time between in-home and out-of-home activities. An application of the fractional logit model to a time use data set remains a future research task. The fractional logit modelling methodology is also subject to violations of the well-known IIA (Independence from Irrelevant Alternatives) assumption. The IIA property implies that time use allocations to travel and various activities are mutually independent, hi the models estimated in this paper, the change in the allocation function for one fraction will not influence the ratio of the other two fractions. However, this assumption may be violated when two fractions are correlated with one another. In general, one would expect people to engage in additional travel if they spend more time at activities outside home. If the assumption that additional out-of-home activities result in additional travel, then travel time allocation is inevitably correlated with out-of-home activity time. Then, the use of the multinomial logit formulation is inappropriate. One possible way to solve this problem is to define a nested logit formulation within the fractional logit modelling framework. With the correlated time use fractions specified into respective nests, the fractional nested logit model would serve as a potential method to overcome the IIA limitation. The development and estimation of such a model system remains a future research effort.

ACKNOWLEDGEMENTS The authors thank Dr. Jeffrey S. DeSimone, Assistant Professor in Department of Economics at the University of South Florida for his assistance and guidance in this research. In addition, the authors benefited from discussions with Abdul Rawoof Pinjari who was a graduate research assistant in the Department of Civil and Environmental Engineering at the University

524 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction of South Florida when this research was performed. Finally, the authors thank Dr. Giovanni Gottardi of Jenni + Gottardi, AG, Zurich and the Swiss Ministry of Transport for providing the data set used in this study.

REFERENCES Bhat, C. R. and R. Misra (1999). Discretionary activity time allocation of individuals between in-home and out-of-home and between weekdays and weekends. Transportation, 26, 193-209. Golob, T.F. and G.M. McNally (1997). A model of activity participation and travel interactions between household heads. Transportation Research B, 31, 177-194. Gourieroux, C, A. Monfort and A. Trognon (1984). Pseudo maximum likelihood methods: theory. Econometrica, 52, 681-700. Greene, W.H. (2002). LIMDEP Version 8.0: User's Manual, Econometric Software, Inc. Plainview, NY. Greene, W.H. (2003). Econometric Analysis, Fifth Edition, Pearson Education, Inc., NJ. Gliebe, J.P. and F.S. Koppelman (2002). A model of joint activity participation between household members. Transportation, 29, 49-72. Kitamura, R., S. Fujii and E. I. Pas (1997). Time-use data, analysis and modelling: toward the next generation of transportation planning methodologies. Transport Policy, 4(4), 225235. Lee, Y. and M. Hickman (2004). Household type and structure, time use pattern, and trip chaining behavior. CD-ROM of the 83rd Annual Meeting of the Transportation Research Board. National Research Council, Washington, D.C. Meloni, I., L. Guala and A. Loddo (2004). Time allocation to discretionary in-home, out-ofhome activities and to trips. Transportation, 31, 69-96. Papke, L. E. and J. M. Wooldridge (1996). Econometric methods for fractional response variables with an application to 401(k) plan participation rates. Journal of Applied Econometrics, 11, 619-632. Pas, E. I. (1993). Trip and VMT reduction and travel behavior. Presented at the 72nd Annual Meeting of the Transportation Research Board. National Research Council, Washington, D.C. Pendyala, R. M. and A.R. Kuppam (2001). A structural equation analysis of commuters' activity and travel patterns. Transportation, 28, 33-54. Schofer, J. L. (1993). Distribution effects of clean air amendment trip reduction requirements. Presented at the 72nd Annual Meeting of the Transportation Research Board. National Research Council, Washington, D.C. Sivakumar, A. and C. R. Bhat (2002). A fractional split distribution model for statewide commodity flow analysis. Transportation Research Record, 1790, 80-88. Yamamoto, T. and R. Kitamura (1999). An analysis of time allocation to in-home and out-ofhome discretionary activities across working days and non-working days. Transportation, 26, 211-230. Ye, X. and R. M. Pendyala (2003). Description of the Switzerland Microcensus 2000 Travel Survey Sample. Research Report prepared for Jenni + Gottardi AG, Department of Civil and Environmental Engineering, University of South Florida, Tampa, FL

27

EFFICIENT ESTIMATION OF NESTED LOGIT MODELS USING CHOICE-BASED SAMPLES Laurie A. Garrow, Georgia Institute of Technology; Frank S. Koppelman, Northwestern University; and Barry L. Nelson, Northwestern University, USA

INTRODUCTION Choice-based samples are often used in transportation and other fields to obtain better information about alternatives that are infrequently chosen in the population. For example, when modeling travelers' mode choice in the United States, it is often desirable to oversample transit users due to the low frequency with which this mode is chosen relative to other modes. However, standard maximum likelihood estimation (MLE) techniques do not obtain consistent estimates of the underlying model parameters. The most widely used approach to obtaining consistent estimators is to use weighted exogenous sampling maximum likelihood (WESML) estimators; however, WESML estimators are not asymptotically efficient, i.e., their variance-covariance matrices do not asymptotically attain the Cramer-Rao bound (Manski and Lerman, 1977). Efficient estimators can be obtained using choice-based samples when using the multinomial logit (MNL) model because, when a full set of alternative specific constants (ASCs) are included in the model, the standard (exogenous sample) maximum likelihood (ESML) estimator can be used with adjustment. As shown by McFadden (not published but reported in Manski and Lerman, 1977), ESML gives consistent estimates of all MNL parameters except for the ASCs, in models with a full set of ASCs. Consistent estimators of the constants can be recovered using population choice frequencies. This is a very general result that applies to all MNL models with a full set of ASCs, including models for which the number of alternatives in a choice set varies across individuals. Further, as shown by Cosslett (1981a), the estimates obtained using this method are asymptotically efficient. This paper shows that

526 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the MLE or ESML estimator can also be used for nested logit (NL) models provided that the model has a full set of ASCs and that population share information is known. The ability to recover consistent parameter estimates for NL models using ESML estimators provides practitioners and researchers with a practical method to obtain estimators that are both consistent and asymptotically efficient for NL models using choice-based data. The balance of this paper contains several sections: MNL and NL models are briefly reviewed; common estimators and their relationship to the sampling process are described; McFadden's proof characterizing inconsistency for ESML estimators for choice-based MNL models is summarized; a general procedure for characterizing consistency of ESML estimators is presented; the application of the general procedure to two-level NL models and three-level NL models is illustrated; an explanation for why McFadden's procedure cannot be directly extended to the class of generalized extreme value (GEV) models is presented; and, finally, major conclusions are summarized.

REVIEW OF MULTINOMIAL AND NESTED LOGIT MODELS This section provides a brief review of MNL and NL models to introduce notation and define ASCs. MNL and NL models are random utility maximizing models that describe how individuals choose one alternative among a finite set of mutually exclusive and collectively exhaustive alternatives. We assume the individual chooses the alternative that has the maximum utility, which is known to the individual, but not to the researcher. Thus, from the researcher's perspective, utilities are random variables. The utility function for a MNL model is defined as

Uni=xniff+sm (1) = V*m+Eni where £/„,is the total utility of alternative i for individual n, V*ni is the observed portion of the utility, /?* xni

is the vector of parameters associated with attributes x. Utility is assumed to be linear in parameters, is the vector of observed attributes that varies across individuals and

sni

alternatives, and is the unobserved portion of the utility function.

For notational convenience, we distinguish parameters associated with constants from the other parameters; ft =[a ,0 I where a are parameters associated with ASCs. An ASC is a dummy variable equal to one if alternative i is in the choice set for individual n and zero otherwise. Due to identification requirements, at most J-\ ASCs can be estimated where / i s the number of alternatives in the universal choice set. In practice, the ASC which is normalized to zero is usually the constant associated with the alternative that is chosen most often in the population. Conceptually, ASCs (and, in some cases, alternative specific variables) are included in a model because it has been widely observed that individuals

Estimation of nested logit models using choice-based samples 527 exhibit preferences for alternatives that cannot be explained by the observed attributes of those alternatives. These preferences are described as alternative specific preferences or bias; they measure the average preference of individuals with different characteristics for an alternative relative to a reference alternative (Horowitz, Koppelman, and Lerman, 1986). ASCs can vary across groups of individuals exhibiting similar characteristics {e.g., high income individuals are less likely to choose transit alternatives than low income individuals, all other things being held constant); however, for the purpose of this analysis, we assume ASCs are the same across all individuals. We estimate the utility function of Equation (1) as Vni=xni9 + ai where Vni

(2)

is the estimated value of the observed utility,

9

is the vector of parameter estimates of 9*, and

a

is the vector of parameter estimates of a .

Under the assumption that the error terms are independently and identically distributed Gumbel with mode 0 and variance 1, iid G(0, 1), the probability individual n selects alternative ; is represented by the MNL as Pni = P{i\Xni,9,a)=

(3)

where j e Cn are the alternatives in choice set C specific to individual n. More generally, we can assume a model P with error terms are distributed iid G(0, y) where y is the inverse variance scale. Normalizing P to G (0, 1) changes the probability of Equation (3) to *m\ - \+— \r P

"'=

£

(x (e\ "A '

el

J ~r -j\

(4)

v

Vr) r J

Thus, probability is not affected by scale, and the selection of y is arbitrary; subsequently, we assume y = \ throughout for convenience. The MNL, along with many other choice models, can be derived using a GEV formulation (McFadden, 1978). The GEV family includes all closed-form utility maximizing models that have an extreme value error distribution and impose equal variances across alternatives. GEV models can be generated from any function of Yt for each alternative i, or

G(Y1,Y2,...,YJ),Yl,Y2,...,YjZ0

(5)

528 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where Y is a positive function that depends on attributes x. The function G is non-negative, homogeneous1 of degree #> > 0, goes to infinity with each Yt, and has odd (even) order partial derivatives which are non-negative (non-positive). Under these assumptions, the probability of choosing i is given as

whereGi(Yl,Y2,...,Yj)is

the first derivative of G with respect to Yi. For example, in the MNL j

model the generating function G is given by G(Yl,Y2,...,YJ} = ^Y i t

F, is replaced by

exp(Fj) to ensure positive values, and

£exp'c'/"" I

m=\\ieAm

0


(7)

)

The GEV generating function for a two-level NL model is given as

M(

G(r,,r 2 .-,y/)=I

y* Zif^M

m=l\isAm

,o< Am
(8)

J

The parameter, // m , often called the logsum or inclusive value parameter, is a measure of the degree of correlation among alternatives in nest m. Higher values of /im imply less, and lower values imply more, correlation among alternatives in the nest. A value of //m =1 for all nests is equivalent to a MNL model (no correlation across alternatives).

ESTIMATORS AND THEIR RELATIONSHIP TO THE SAMPLING PROCESS Different sample selection processes are used when collecting data to ensure adequate representation of population groups and chosen alternatives. In general, sampling processes

While McFadden's original proof assumed G was homogeneous of degree one, Ben-Akiva and Francois (1983) showed the assumption can be relaxed to allow for any degree of homogeneity.

Estimation of nested logit models using choice-based samples 529 are characterized as random, exogenous, or choice-based (Manski and Lerman, 1977). All individuals in a population have an equal probability of being selected in random samples; however, in exogenous (stratified) and choice-based samples individuals are selected disproportionately according to observable attributes/segments and choice/purchase decisions, respectively. When discrete choice models are used to represent customer behavior, an appropriate estimator must be used to ensure consistency. The selection of an appropriate estimator depends both on the sampling process and type of choice model. Three estimators for discrete choice models include the maximum likelihood (ML) estimator, the (unweighted) exogenous sampling maximum likelihood estimator (ESML), and the weighted exogenous sampling maximum likelihood estimator (WESML) (Manski and Lerman, 1977). While other, more complex, estimators, such as those based on the method of moments or the method of scores, can be used (e.g., see Train, 2003), the focus of this discussion is on maximum likelihood estimators. This section describes each estimator and its relationship to the sampling process; it closely follows the derivation reported in Lerman and Manski (1979) and summarized in Ben-Akiva and Lerman (1985). Maximum Likelihood Estimators The ML estimator is appropriate to use with any choice model when the estimation data represents a random sample. We define f(i,x) as the joint distribution of choice ;' and attributes x in the population and note that f[i,x)

= p(i\x,P*)p{x)

(9)

where P\i\x,/?

j

is the probability of selecting alternative i given attributes x and

p{x)

parameters /?*, is the probability distribution of attributes x in the population,

P* x

is the vector of parameters associated with attributes x, and is the vector of attributes.

The likelihood function for a random sample is

n-\ ieCn

n=\ i^Cn

where N xn dni

is the number of individuals in the random sample, is the vector of attributes associated with individual n, is an indicator variable equal to 1 if individual n selects alternative i, and 0 otherwise, and other variables are defined as before.

530 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The log likelihood function is LL

(fi)='ZIl dni\nP{i\xj)+J:\np(xn). n~\ /eCw

(11)

n=\

Because the second term in the log likelihood function is independent of /? when observations are selected randomly from the population, the log likelihood function we use to estimate /? is

LLforML = Y,Y,dnMp{i\xn,P)-

(12)

«=1 isCn

Exogenous Sampling Maximum Likelihood Estimators The ESML estimator is appropriate to use with any choice model when data are based on exogenous samples, i.e., samples are defined based on the x attributes or any variables other than the observed/stated choice. The likelihood function for an exogenous sample is S Ns

\

ld"i

Q n

s=\n=\ieCn\_

s_

(13) S Ns

d

Q

ni

=nnn^(;i *«,/?) PM s=\n=\ieCn

where Ns S Hs

n

s

is the number of individuals in strata s, is the number of mutually exclusive strata, is the sample fraction of individuals belonging to 5,

Qs

is the population fraction of individuals belonging to s, and other variables are defined as before.

Conceptually, the weight QSIHS adjusts the sampling rate to the population rate. The corresponding log likelihood function for exogenous samples is

LL(fi) =

izidnilnP(i\xn^)+J:^lnp(xn) (14)

+i»(&)-ixM^)s=l

s=l

The first term in Equation (14) is equivalent to the log likelihood maximized for random samples. The second and third terms do not depend on fi and do not affect the maximum likelihood estimate. The sample fraction Hs can be expressed as

Estimation of nested logit models using choice-based samples 531

Hs= J ZP(i\x,fi)p(x)dx

(15)

Xs lieCs where i e Q are alternatives in choice set for stratified sample 5 and Xs are the attributes used to stratify sample s. For exogenous samples, Hs reduces to: Hs= \\-p(x)dx

(16)

which does not depend on fS and the log likelihood function we use to estimate p for exogenous samples is the same as that for random samples except it is calculated by summing over segments, s, and samples in each segment. Formally,

LL for ESML = £ £ I dni\nP(i \ xn,p).

(17)

s=\ n=\ ieCn

Note the utility function can be defined for both random and exogenous samples so that estimated parameters may be allowed to vary across segments. The motivation for collecting an exogenous sample is to ensure that there is enough data so that /? can be estimated for different segments where appropriate (e.g., different price sensitivities can be modelled for high and low incomes) or to ensure that population groups that are small or are most likely to choose certain alternatives are adequately represented in the sample. Weighted Exogenous Sampling Maximum Likelihood Estimators The use of the WESML estimator for choice-based samples is motivated by the observation that if strata are defined by choices, ;', which depend on choice parameters, p, the log likelihood function does not reduce to the computationally tractable form expressed in Equation (17). Manski and Lerman (1977) proved that parameter estimators obtained by maximizing a different log likelihood function, the WESML, are consistent. Formally, the log likelihood for a WESML estimator is equivalent to that for the ESML estimator except that each observation is weighted by the ratio of the alternative's population share, Qt, to sample share, Hi, or:

LL for WESML = £ £ dni \%\nP{i \ xnj) n=\ieCn

(18)

\Ui)

where JV is the number of individuals in the sample, and other variables are as previously defined. While the WESML estimator is easy to use, it is not asymptotically efficient, i.e., its variancecovariance matrix does not asymptotically attain the Cramer-Rao bound (Manski and Lerman, 1977). From a practical perspective, this can lead to difficulty in determining which attributes are significant for the choice under study. This is empirically demonstrated in Table 1, which

532 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction compares two MNL models using ESML and WESML estimators. The models use alternative-specific variables to estimate air travellers' day-of-departure rescheduling decisions, i.e., whether passengers show, no-show, early standby, or late standby for flights. A standby is defined as a passenger who voluntarily takes a different itinerary with the carrier of interest. Standbys are further divided into those who wait at an airport hoping to take an earlier flight and the late standby who accepts a later flight as a result of missing his/her scheduled flight. The data is from a major network carrier in the United States and is based on booking, ticketing, schedule, operating, and check-in data. Because there are millions of monthly booking transactions and the occurrence of standby choices is relatively small, a choice-based sample was used for model estimation. Sample choice rates are 34%, 24%, 17% and 25% and population rates are 92.7%, 6.3%, 0.89% and 0.13% for show, no-show, early standby and late standby, respectively. The coefficients of the two models are very similar with the exception of the ASCs. Note that in this example, the ASCs associated with the early standby alternative vary according to an alternative characteristic of itinerary duration. (Note that maximum likelihood estimators have the large sample property of consistency, which implies that the estimators approach the true values and therefore, one another, asymptotically; that is, as the sample size approaches infinity. For realistic samples, the estimators are expected to be approximately equal.) However, the extreme weights included in the log likelihood function lead to increased standard errors in the WESML estimates, as seen in the substantially lower t-statistics. Further, the standard errors obtained using WESML are biased. Excluding constants, all late standby and the majority of early standby variables are statistically insignificant at the 0.05 level in the WESML model, making it difficult to discern which variables influence these choices. More efficient estimators exist, but are generally complicated to implement (e.g., see Brownstone (2001) for a discussion) and have not been widely adopted. In contrast, the ESML estimator, which is efficient and easy to implement, can be used for MNL models that have a full set of constants. This is based on McFadden's proof that shows, for MNL models with a full set of ASCs, ESML yields consistent estimators of all parameters except for the constants; moreover, consistent estimators for the constants can be recovered using population share information and subtracting ln(//;. / Qt) from the estimated constants. Note that the consistency proof does not require the choice set to be the same across individuals. The next section reviews the derivation of this result, which is subsequently extended to NL models.

Estimation of nested logit models using choice-based samples 533 Table 1. Empirical Comparison of Weighted and Unweighted Estimators for MNL Model WESML ESML Constants (ref. = show) 1.02 (6.7) 1.33 (11.8) Alternative specific constant for NS -0.42 (2.0) ASC for ESB: Duration <180 mins -4.21 (7.4) -0.55 (2.5) ASC for ESB: 180 < duration <300 mins -4.48 (6.9) -0.87 (3.5) ASC for ESB: Duration > 300 mins -4.94 (6.5) -0.37 (3.4) Alternative specific constant for LSB -6.63 (6.8) 0.26 (2.9) Day of Week (ref Sun-Tues) Wed - Fri ESB 0.36 (1.1) Departure Time (ref. = after 7 pm and for NS 6-9 am) -0.29 (3.3) Depart 9 am - 7 pm NS -0.28 (2.2) -1.46 (7.2) Depart 6 am - 9 am ESB -1.35 (2.1) Depart 9 am - 4 pm ESB -1.25 (2.8) -1.16 (7.3) -0.58 (3.6) Depart 4 pm - 7 pm ESB -0.62 (1.3) Carrier Capacity (100's seats) before scheduled departure for ESB or LSB Arrive 1-90 mins earlier 0.25 (2.2) 0.39 (7.0) Arrive 91-150 mins earlier 0.30 (1.8) 0.27 (4.6) Arrive 151-300 mins earlier 0.15 (1.4) 0.07 (2.0) Arrive 1-90 mins later 0.11 (0.2) 0.08 (1.6) Arrive 91-150 mins later 0.16 (0.4) 0.15 (3.0) Arrive 151-300 mins later 0.11 (0.4) 0.11 (3.6) Schedule Presence (Ratio of total flights for carrier vs. nearest competitor)A0.5* Departure city presence ESB 0.12 (0.9) 0.14 (3.1) Departure city presence LSB 0.33 (8.8) 0.33 (0.9) E-ticket NS -1.78(13.7) -1.81 (20.2) Booking Class (ref. = low yield) First and business NS -0.57 (2.0) -0.74 (3.9) First and business ESB -1.01 (4.2) -0.76 (1.1) First and business LSB -0.95 (0.5) -1.03 (6.4) High yield NS 0.06 (0.4) 0.08 (0.7) High yield ESB -0.05 (0.1) -0.25 (2.1) High yield LSB -0.35 (0.4) -0.46 ((16)) Frequent Flyer (ref. = not a member) General member NS -0.69 (4.1) -0.54 (4.6) General member ESB 0.33 (2.4) 0.27 (0.7) General member LSB -0.63 (0.7) -0.63 (5.6) Elite member NS -0.20 (1.1) -0.13 (1.0) Elite member ESB 0.57 (4.2) 0.50 (1.2) Elite member LSB -0.47 (3.7) -0.49 (0.5) Group Size (ref. = travel alone) Groups of 2-10 individuals NS -0.45 (3.1) -0.60 (5.2) Group of 2-10 individuals ESB -0.74 (1.7) -0.71 (6.0) Group of 2-10 individuals LSB -0.44 (0.5) -0.50 (4.6) Model Fit Statistics LL Zero / LL Constants / LL Model -4980/-1193/-1067 -3575/-3539/-3112 Rho-Squarezero / Rho-Squareconstim, 0.786/0.106 0.129/0.121 KEY: SH=show; ESB= early standby; LSB=late standby. NOTES: Data for March 2001 outbound itineraries for 2,761 observations. Table contains parameter estimate (t-stat). Bold cells are not significant at the 0.05 level. *Applies when carrier of interest has dominant market share.

534 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

CONSISTENCY OF ESML ESTIMATORS FOR MNL MODELS In their 1977 paper, Manski and Lerman show that if the ESML is consistent for choice-based samples, then the following is true

F(p) = l i m £ z dni\nP(i\xnJ)l'N J^n-U.Cn ^ ^ \\nP(i\x,p)p(i\x,p*)p(x)dx.

(19)

Their proof is based on two lemmas, which are summarized below. As their proof is quite involved, we will provide a summary of the key underlying concepts and subsequently use Equation (19) as a starting point for our proof. Conceptually, Lemma 1 uses the Strong Law of Large Numbers to show that the sample average of a sequence of functions converges to the expected value of the function. Lemma 2 shows that the expected value of the log likelihood function is maximized at the true population values of the parameters. Lemma 1 Let £, be a probability measure over a Euclidean space X, let O be a compact subset of a Euclidean space, and let h(x,P)be a continuous function of /? for each x e X and a measurable function of x for each ft e C". Assume that \h(x,p) <<x> for all x and /? and for some finite a. sequence x = xl,x2,...,xN

For any i s l a n d

drawn randomly and independently from X let

N

fN(x,p)

= ^h(xn,p*)lN.

Then, for almost every realized such sequence,

as TV —>TO,

fN(x,/3)^>E((h(xn,fi))

= F(/3)

uniformly for all p e O. Lemma 2 Let h(x,P)be

a real valued function over a spaceXxsuch that h is

integrable w.r.t. a measure 0, all x e X, P e
an element

of

O

such

that

h[x,p*j>0,

for almost

every

x e X and J(h [x, p') - h (x, / ? ) ) ^ > 0, all p e O. Then x

F(p)=\{\nh(x,p)}h{x,pyt x

attains its maximum at P = P" . While Manski and Lerman prove that the ESML estimator is inconsistent for choice-based samples, McFadden characterizes this inconsistency for a single case, i.e., a MNL model that

Estimation of nested logit models using choice-based samples 535 has a full set of ASCs. Inconsistency is demonstrated by showing that the right-hand side of Equation (19) is not maximized at/?*. This proof is outlined below. For notational convenience, we distinguish parameters associated with constants from the other parameters; 0* = [a*,9* 1 where a* are parameters associated with ASCs. For the case of the MNL model with a full set of ASCs, substituting choice probabilities into Equation (19) gives: x e+a

H

i

e

i

exfi'

V JEC

+

<*i

) jeC

Next, define^,. = ln(// ; /£>) and substitute<5; into Equation (20) (also exchanging the order of the summation and integration) to give

\

f

f(

n

«i,?/"

Xi9+

a;

xi9'+ a*

*Y^P(S)*'

Y^

\JeC

J jeC

Using algebraic manipulation, multiply the numerator and denominator of Equation (21) by Xj0*+ [a*j+5j\

2_, e

and rearrange terms to give

jeC x/+(a'+Sj)

F(P)=\&

(

r^^xXln

jcC

N

,+ a KJeC

t

x— J

^

. ,.

.p(x)dx.(22)

JeC

Lemma 2 can now be invoked to show !"!>

3

x

~-—-p(x)dx

(23)

is maximized at 0 = 9 and a, =a* +Si for every ; e C , which implies that F(fi) is also maximized at this point. Intuitively, we apply Lemma 2 only to the expression in Equation (23) by observing that /? does not depend on the distribution of x and that the first term in Equation (22), which is summed over all J alternatives, is a constant when integrated over x. Moreover, the key condition of Lemma 2 is satisfied by observing that, by definition of the likelihood function, its expectation evaluated at /?* is always greater than or equal to its expectation at any other /?. Thus, since the estimated value of the ASC for alternative i is

536 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

given as at = at +St, a consistent estimate for a ; can be recovered using knowledge of population and sample shares, or a* = a ; - 8t = at - In (Ht IQ). Thus, inconsistency is due solely to£ ; . Moreover, as shown by Cosslett (1981a), since this estimator (with adjustment) reduces to the standard maximum likelihood estimator, it is asymptotically efficient. Next, we present a general procedure that is used to characterize inconsistency in ASCs for NL models.

GENERAL PROCEDURE FOR CHARACTERIZING CONSISTENCY OF ESML ESTIMATORS McFadden's proof characterizing inconsistency of ESML estimators for MNL models was published in 1977, which was prior to the developed the NL model (Williams, 1977; McFadden, 1978) and, more generally, the generalized extreme value (GEV) family of models (McFadden, 1978). Subsequent research into the use of estimators for choice-based samples such as that by Cosslett (1981a, 1981b), Manski and McFadden (1981), and Imbens (1992) was focused on developing estimators that are relatively more efficient than the WESML estimator or estimators that did not require information about population shares. However, in general, these estimators are substantially more difficult to implement than the WESML estimator. In this section, we present a generalization of McFadden's proof that characterizes inconsistency of the ESML estimator for MNL models. This generalization procedure is illustrated for other models reported in the literature after 1977. Using the GEV generating function, we now state a general procedure used to characterize inconsistency of ESML estimators for choice-based logit models. (1)

Define the generating function and choice probabilities for a model belonging to the GEV class. Assume this model is specified with a full set of ASCs.

(2)

Substitute probabilities into Equation (19). For ease of algebraic manipulation, it is usually convenient to group terms so the probabilities (for#> =1) can be expressed as: P{i\*,P)=

c(Y

f

YYi) Y\

(24)

/(G;ft,y2,...,y,)) where

f(ji)

includes common terms from Y! and the derivative Gi{Yl,Y2,...,YJ)

/(G i (^,y 2 ,...,y / )) includes terms from the derivative not included in (3)

f(Y,).

Define £,. = In [Hi I Q ) and substitute into the equation from Step 2.

Estimation of nested logit models using choice-based samples 537 (4)

Use algebra to translate the equation from Step 3 into an expression that can be used in Lemma 2. Usually this will require multiplying the numerator and denominator by

(5)

Use Lemma 2 to characterize the inconsistency of ESML estimator (which appears only in the ASCs) and, after verifying that the conditions of Lemma 2 hold, show how consistent estimators can be recovered using population share information.

CONSISTENCY OF ESML ESTIMATORS FOR TWO-LEVEL NL MODELS Using the procedure outlined in the previous section, we can characterize inconsistency of ESML estimators for two-level NL models based on choice-based samples. Figure 1 shows a general representation of a two-level NL model. The set of alternatives j is partitioned into M non-overlapping nests AuA2,...,Am.

Figure 1 Notation for a Two-Level Nested Logit Structure Choice probabilities are derived using a GEV generating function for the J alternatives:

G(Y],Y2,...,YJ)= Y \ Y Y I ' M \ m=\\mAm

,0
(26)

J

where i e Am is the set of alternatives in nest m. Note i can be assigned to only one nest. Yt

is given as exp {Vi) = exp (x,/?).

The probability of choosing alternative i is given as:

r

] Ym~yU™ f—il J

Y\ V LYu>"m i\

P(i\x,P) = ^

i

i-i

^

M

(

m=\lyieAm

A

i

1

u

.

Y"" )

(27)

538 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Combining the first and last terms in the numerator, and replacing Y, with exp{Vt) gives

Pii\x,p) =

l

^

I

"I

=-

;D = —^

I

(28)

lEAm

J

^

V-

\

J

To characterize inconsistency, defineSi = In[Hj /Qf) and substitute ^. and the choice probabilities above into Equation (19) to give:

( (&+*_}) {s^-JUs,} lMm

Mm)

v

y

Mr, j

{fm

y

e\.fm

Mm)

V

)

Next, multiply the numerator and denominator by

m=l i € ^ m

z)(«y) = —^

V and rearrange terms to give

(30)

V

J

K F(fi)= f ' ,Iln ~ xy } JcD{e\a)k D(0,a) V J

D(S)

p(x)dx. >

n

(31)

Estimation of nested logit models using choice-based samples 539 Lemma 2 can now be invoked to show that the expression

[Yi n ^—

Ik

x-

D{p,a) v

p(x)atc

(32)

D(5) )

is maximized at 9 = 9" and a, / /i m = a ; I nm+ St for every ieC, which implies that F(/3) is also maximized at this point. Since the estimated value of the ASC for alternative i is given as at = ctj + /jmSj, a consistent estimate for a ; can be recovered. Specifically, a* =ai-/umSi =at - / u m x l n ( / / , / g i ) . Thus, inconsistency is due to fim x 8t.

CONSISTENCY OF ESML ESTIMATORS FOR MULTI-LEVEL NL MODELS We can also characterize inconsistency of ESML estimators for multi-level NL models based on choice-based samples. Figure 2 shows a general representation of a three-level NL model. Working from the bottom of the tree upwards, the set of alternatives j is partitioned into N non-overlapping nests, BuB2,...,Bn to create the second level. The third level is created by partitioning the set of second-level nests to M non-overlapping n e s t s ^ , ^ , . . . , ^ . An alternative belongs to one two-level nest and one three-level nest. Here, we use N to refer to the number of nests at the second level, not the number of individuals in the sample as per previous derivations.

Nest m Level 3 Nest n Level 2 Alternative i Level 1

Figure 2 Notation for a Two-Level Nested Logit Structure The choice probabilities for the three-level nested logit model are derived using a GEV generating function:

540 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

G(7 1 ,7 2 ,...,y / )=X

X X ^ ""'"""' \

,0
(33)

where 2 e 5 n is the set of alternatives in the level-two nest n, n e Am is the set of level-two nests that belong to the level-three nest m, Hnm

is the logsum coefficient associated with the level-two nest n,

jim

is the logsum coefficient associated with the level-three nest m,

Y

is given as exp (Vl) = exp(xfi).

The probability of choosing alternative i is given as: ul

( Y i

nm-l ( ! ,1 yvftixftm ) J i

V v 1 ZJ i

l

P(iUj),^!iL

1

- — .

m=ll neAm\Ki£Bn

(34)

J

Combining the first and last terms in the numerator, and replacing Yi with exp(V^)gives m><; 1

f ^ V f ^ ")Y ""'""

e

Le

P(i\x,/3) =

J

^ (

(

(

M nE.Am

i&Bn

J

^

\ (

I

M m=\

- -— = \\Mnm\

v-

, where

(35)

'

m=\

{

—n—

V.

J

m

\YnmY

'n€.Am

i
D= ^ ^ y

^-J-. x

eKMm

Mnm)

i<EBn

/ V To characterize inconsistency, defined, = In(i/ ; /Q.) and substitute S, and the choice probabilities above into Equation (19) to give:

Estimation of nested logit models using choice-based samples

iMnXf™

Um*Mnm)

ifm»Mnm

F{0)= fVln

x-

i,fe

^

M^Mnm

, .

D(0,a)

t,

D{e\a)

V

)

^ ( X W , where K

'

)

J [ r-^^-irr m=\

541

nGAm

(36)

i<EBn

D(e,a) =

/ ieBa V

.

/

Next, multiply the numerator and denominator by

p(s)=

l

{

'-

y

(3?)

^ i -

e{MmXMam

+

!

+<J;

MmXfnm

)

n<EAm V

/

and rearrange terms to give

F(6)=

\

K }

, ]

K

>NTln

x-

xD(e\a)tc

D(6,a)

v

p(x)dx.OS)

D(S)

n

!

/

Lemma 2 can now be invoked to show that the expression

J S In ^ /eC

^ 7 ^ D(9,a)

x

^ D(S)

P(x)dx

is maximized at 0 = 6* and a,- / (jum x /i n m ) = a* I (/um x finm) + J- for every ieC,

(39) which

implies that F(/3) is also maximized at this point. Since the estimated value of the ASC for alternative i is given as at = a* + (jum x / i n m ) S i , & consistent estimate for a* can be obtained

542 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

as a* =ai-(^mx t0

x

nnm ) St = at -jumx /unm x In (Hi IQt).

Thus,

inconsistency

is

due

xS

Mm Mnm i-

CONSISTENCY OF ESML ESTIMATORS FOR GNL MODELS The models presented thus far represent models with alternatives allocated to a single nest. The more general class of GEV models allows alternatives to be allocated to more than one nest. However, when alternatives are allocated to different nests, the general procedure for characterizing inconsistency can no longer be applied. Conceptually, this is because the numerator of the GEV probability function is calculated by summing over all nests, which prohibits decomposing the probability function into a form required by Lemma 2. This section illustrates this issue for the GNL models, which are representative of models that have some alternatives allocated to more than one nest. As shown by Koppelman and Wen (2001), the GNL2 includes other models in the GEV class such the paired combinatorial logit (Chu, 1989), cross-nested logit (Vovsha, 1998), ordered generalized extreme value model (Small, 1987), product differentiation model (Bresnahan, Stern and Trajtenberg, 1997), etc. The twolevel nested logit is also a special case of the GNL model; however, because alternatives are allocated only to one nest, we can characterize inconsistency in constants. The choice probabilities for the GNL model are derived using a GEV generating function

G(YuY2,...,YJ)=f1(-Z(TimYi)U^T' m=l\ieAm

(40) )

where zit

are unknown allocation parameter which characterizes the portion of alternative i assigned to nest m. Allocation parameters are non-negative, i.e., Tim > 0 and ^ r l m = 1 Vz. Other variables are as defined for the two-level NL. m

The probability of choosing alternative i is given as:

P«\ *,*>) = -

—

^

si Mr

m=l\ieAm

Z ^ -

(41)

J

Note that the allocation of alternatives to different nests results in the summation over all nests in the numerator. It is this summation which prohibits decomposing the probability function into a form that can be used to isolate the inconsistency based on the approach 2

In the literature, the use of CNL and GNL are often interchanged. We follow the textbook description of Train (2003) to distinguish between Vosvha's original CNL model and subsequent generalizations of his model.

Estimation of nested logit models using choice-based samples 543 outlined in this paper. Thus, McFadden's proof cannot be directly extended to models in the GEV class that have alternatives allocated to multiple nests.

SUMMARY In this paper we showed that McFadden's proof characterizing inconsistency in constants for MNL models using choice-based samples can be extended to NL models. This paper developed a general procedure for characterizing inconsistency in ESML estimators for choice-based samples. Consistent estimators can be recovered by specifying models with a full set of ASCs and using population share information. Specifically, we show that consistent estimators for the constants of a two-level NL model can be recovered by subtracting ^m\n(HiIQj) from the estimated constants where jum is the logsum coefficient associated with nest m, 0 < fim < 1 and //,- and Qi are the sample and population fractions of individuals choosing alternative i, respectively. Similarly, consistent estimators for a threelevel NL model can be recovered by subtracting/^ x/inm In(Hl IQt) from the estimated constants where jj.m and finm are logsum coefficients associated with the third-level nest and second-level nest, respectively. Similar results apply for four and higher order NL models. Finally, we described why McFadden's proof cannot be directly extended to other models in the GEV family such as the GNL that has alternatives allocated to more than one nest. Thus, we have extended the characterization of inconsistency of ESML estimators for MNL models with a full set of constants to NL models of any order with a full set of constants. This provides an easy to apply method to obtain consistent and efficient (at least with respect to all non-constant) estimators for NL models. Since all known alternative approaches are complex and not widely known, this provides practitioners with a realistic approach to use choice-based samples in estimation of nested logit models of any order. Empirical examples of how to apply the result using actual airline and mode choice datasets are given in a parallel paper by Koppelman and Garrow (2005).

REFERENCES Ben-Akiva, M. and B. Francois (1983). Mu-homogeneous generalized extreme value model. Working Paper, Department of Civil Engineering, MIT. Ben-Akiva, M. and S. Lerman (1985). Discrete Choice Analysis: Theory and Application to Travel Demand. MIT Press, Cambridge, Massachusetts. Bresnahan, T., S. Stern and M. Trajtenberg (1997). Market segmentation and the sources of rent from innovation: Personal computers in the late 1980s. Rand J. of Economics, 28, Special Issue, SI7-S44. Brownstone, D. (2001). Discrete choice modeling for transportation. In: Travel Behavior Research: The Leading Edge (D. Hensher, ed.), pp. 97-124. Elsevier, Oxford, United Kingdom.

544 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Chu, C. (1989). A paired combinatorial logit model for travel demand analysis. Proceedings of the Fifth World Conference on Transportation Research, Vol. 4, pp. 295-309. Ventura, CA. Cosslett, S. (1981a). Efficient estimation of discrete-choice methods. In: C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Choice Data with Econometric Applications pp. 51-111. MIT Press, Cambridge. Cosslett, S. (1981b). Maximum likelihood estimation for choice-based samples. Econometrica, 49, 1289-1316. Horowitz, J., F. Koppelman and S. Lerman (1986). A Self-Instructing Course in Disaggregate Mode Choice Modeling. United States Department of Transportation, Federal Transit Administration. Report DOT-T-93-18. Imbens, G. (1992). An efficient method of moments estimator for discrete choice models with choice-based samples. Econometrica, 60(5), 1187-1214. Koppelman, F. and C.-H.Wen (2001). The generalized nested logit models. Transportation Research Part B, 35B(7), 627-641. Koppelman, F. and L. Garrow (2005). Estimation of nested logit models using choice-based samples: Example applications. Paper presented at the 84th Annual Transportation Research Board, Washington, D.C. Lerman, S. and C. Manksi (1979). Sample design for discrete choice analysis of travel behavior: the state of the art. Transportation Research, 13 A, 29-44. Manski, C. and S. Lerman (1977). The estimation of choice probabilities from choice-based samples. Econometrica, 45, 1977-1988. Manski, C. and D. McFadden (1981). Alternative estimators and sample designs for discrete choice analysis. In: C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Choice Data with Econometric Applications pp. 2-50. MIT Press, Cambridge. McFadden, D. (1978). Modeling the choice of residential location. In: A. Karlquist et al. (eds.), Spatial Interaction Theory and Residential Location pp. 75-96. Amsterdam. Small, K. (1987). A discrete choice model for ordered alternatives. Econometrica, 55, 409424. Train, K. (2003). Discrete Choice Methods with Simulation. University Press, Cambridge, United Kingdom. Vovsha, P. (1998). Application of cross-nested logit model to mode choice in Tel Aviv, Israel, metropolitan area. Transportation Research Record, 1607, 6-15. Williams, H. (1977). On the formulation of travel demand models and economic evaluation measures of user benefits. Environment and Planning A, 9, 285-344.

28

FUNCTIONAL APPROXIMATIONS TO ALTERNATIVE-SPECIFIC CONSTANTS IN TIME-PERIOD CHOICE-MODELLING Stephane Hess and John W. Polak, Centre for Transport Studies, Imperial College London, UK; and Michel Bierlaire, Institute of Mathematics, Ecole Polytechnique Federate de Lausanne, Switzerland

INTRODUCTION AND BACKGROUND Alternative Specific Constants (ASCs) play an important role in the use of random utility models (see for example Train, 2003). Indeed, they capture the mean of the error terms of the utility functions. As such, they are critical in order to properly reproduce market shares when the models are used for forecasting. The estimated values of ASCs are affected by sampling bias (except when using exogenous sampling strategies with Generalized Extreme Value models), and by data collection methods such as stated preference (SP) surveys. Also, when the number of alternatives is high, the model may be saturated by ASCs, suffer from overfitting and, consequently, lose its forecasting power. Another source for biased ASCs is the estimation of models based on sampling of alternatives. One area of research in which the number of constants can be very high, with high levels of correlation between related constants, is the modelling of time period choice. This has become increasingly important over recent years. Indeed, since both congestion and many of the consequent policy responses to it have strong time of day related components, as the salience of congestion as a policy concern has grown in recent years, so has the importance of developing theoretically sound yet practically applicable approaches to modelling the timing of travel, particularly for departure times of trips during congested peak periods. A number of different modelling approaches have been proposed in the literature. Building on ideas originally proposed by Vickrey (1969), several authors have presented frameworks in which the choice of departure time is modelled deterministically and as a continuous quantity

546 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction (e.g., Arnott et al, 1990; de Palma et al, 1997; Hyman, 1997; van Vuren et al, 1999). A related set of studies have modelled departure time jointly with route choice using model systems in which a continuously variable choice of departure time is linked to a discrete choice of route (e.g., Mannering et al., 1990; Mahmassani and Chang; 1985, Mahmassani et al., 1991). More recently, interest has also developed in formulating continuous models of departure time choice within the framework of hazard based duration models (e.g., Bhat and Steed, 2002; Wang, 1996). In addition, models of individual trip departure time are also increasingly being embedded in more general models of activity choice and scheduling behaviour (see, e.g., Ashiru et al., 2003 and the work reviewed therein). However, the dominant approach to modelling departure time, both in the academic literature and in practice, involves re-formulating the underlying continuous departure time choice problem as a choice problem involving a finite number of discrete time periods and modelling the choice between these periods within the framework of discrete choice analysis. This approach, first proposed in the work of Cosslett (1977) and Small (1982), has been widely applied using both revealed preference data (e.g., Abkowitz, 1981; Athanassiou and Polak, 2001; Bhat 1998; Bradley et al, 1998; Chin, 1990; Hendrickson and Plank, 1984; McCafferty and Hall, 1982; Small, 1982,1987) and stated preference data (e.g., Bates et al, 1990; de Jong et al., 2003; Polak et al, 1991; Polak and Jones, 1994). Models based on this time period choice approach are typically formulated in terms of a trade-off between, on the one hand, the time of day varying travel times and costs experienced by travellers and on the other hand, travellers' inherent preferences for undertaking certain activities at certain times of day. The characterisation of these temporal preferences is challenging and the most commonly used approach is to use a version of the concept of schedule delay (Vickrey, 1969) to quantify the loss in utility associated with shifting a departure earlier or later relative to the preferred (or, more often, the actual) time of departure of the existing trip. This approach works well in diagnostic and exploratory modelling but when time period choice models are being used in long term forecasting applications, the embedding of this sample type information in the utility function can be highly problematic, since the notion of 'existing trip' is not meaningful in a forecasting context and, moreover, the characteristics of forecast trips are typically not available at the same level of temporal precision as that available in the estimation sample dataset. Hence, when time period choice models are developed for forecasting applications, temporal preferences of travellers are captured by the use of a set of constants, associated with the different time-periods. Depending on the degree of granularity employed in the definition of the time periods, this approach can lead to significant problems with identification and interpretation, as well as heightened computational cost. These problems are compounded when time period choice models are specified in terms of tours rather than trips (e.g., de Jong et al, 2003; Polak and Jones, 1994) since the number of possible time period constants increases substantially. Similar issues apply in many other areas of research where a large number of constants are to be identified, such as in the analysis of spatial choice or route choice processes. In this paper, we propose to address these problems by using an explicit modelling of the ASCs, based on the idea of a functional approximation to the distribution of the set of constants across alternatives. Such functional approaches not only reduce the number of parameters to be

Functional approximations to alternative-specific constants 547 identified, but also avoid some of the issues of identification, and facilitate interpretation of the results. Furthermore, they potentially reduce the influence of SP errors on the estimation of ASCs when using data collected through SP surveys. The remainder of this paper is organised as follows. In the next section, we briefly describe the data, before moving on to model specification in section 3. Section 4 presents the results of the modelling analysis, with model validation and conclusions being the topics of section 5 and 6 respectively.

DATA This paper makes use of data collected in 2000 for the development of the Dutch National Model System (c.f. de Jong et al, 2003). An initial Revealed Preference (RP) survey was conducted to select a sample of respondents for the follow-up SP survey. The RP survey included rail-travellers as well as car-travellers, contacted at a selection of sites across The Netherlands, concentrating on areas where road and rail congestion was encountered in peakperiod journeys. In the ensuing SP survey, respondents were presented with four alternatives in each choice-situation, three alternatives using their current mode of travel, and one alternative involving a mode-change. The choice thus involves the re-timing of an existing tour or the switch of this tour to an alternative mode. The three alternatives on the current mode are described as "retimed earlier", "base", and "retimed later", with the "base" alternative being close in departure time to the actual observed tour. Except for non-homebased business travellers, the travel alternatives were presented to respondents in the form of complete tours, comprising explicit and linked outbound and return legs. Respondents were presented with 14 SP replications, spread over two games. The travel purposes distinguished were commuting, business, education, and other. This dataset has recently been used in the estimation of Error-Components Logit models by Hess et al. (2004). In the present analysis, we concentrate on car-commuters, using a division into commuters with flexible working hours, and commuters with inflexible working-hours.

MODEL SPECIFICATION In models of time period choice destined for the use in forecasting systems, the aggregate scheduling information is generally captured through the use of a set of constants, associated with the different time-periods. The usual specification uses a separate constant for each possible combination of outbound and return time period, hence indirectly also capturing effects associated with activity duration. However, with N separate time periods, this leads to a requirement for N(N+l)/2 constants, of which N(N+l)/2-l can be identified. With the typically used 24 1-hour time periods, this thus leads to a total of 299 constants that need to be estimated just to represent the choice of time period. This number increases dramatically with finer specifications of the time periods, compounding problems of computational cost and parameter identification. An alternative approach consists of the use of two separate sets of constants for outbound and return time periods. The independence between the two sets

548 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction however leads to the requirement of an additional set of constants representing activity duration, hi the 1-hour example, this thus leads to three sets of 24 constants, where 23 constants are estimated in each set. Despite the fact that this leads to a reduction in the (maximum) number of estimated constants from 299 to 69, the approach is still computationally expensive and problems of identification can remain, especially in the presence of time periods never chosen, or time periods always chosen when available. In this paper, we discuss an alternative approach, based on the idea of a functional form for the constants. These functional approaches will inevitably lead to poorer model fit than an approach using a full set of constants, but fewer problems with identification and significance levels should occur. In a properly specified time period choice-model, the constants associated with the different time periods will reflect the distribution of travellers across time periods. With an appropriate size for the time periods, it can be expected that these distributions follow some form of systematic pattern. As an example, for commuting, it can be expected that there is a peak in the distribution of the outbound constants sometime between 7am and 9am, with a corresponding peak in the distribution of the return constants between 5pm and 7pm. The differences in the shape of the distribution to the left and right of these base periods can give important insights into travel and more specifically rescheduling behaviour, and can be seen as a reflection of the flexibility of the working-hour arrangements. With any approach using constants to capture time period choice, it is crucial to reduce the impact of artefacts of the SP survey on these constants. In the present analysis, three additional ASCs were hence defined on the basis of the SP design; associated with the retimed-early, retimed-late and mode-shift alternatives; such that the constant for the base alternative is normalised to zero. Finally, in the presence of quota-based data, and in the absence of a reliable re-weighting strategy, it is crucial to acknowledge that the results produced in this analysis relate to the sample used and should not be generalised; this does however not translate to the theoretical findings and recommendations. Similarly, the results are affected by the fact that not all time periods are available to all respondents. In this paper, we make use of two groups of functional forms for the distribution of the constants. In the first group, we make an a priori assumption that there is a single peak in the distribution, with (possibly asymmetrical) slopes to either side, while, in the second group, we allow for the possibility of multiple peaks. Both sets of approaches ignore the potential interaction between the distributions along the three dimensions; the analysis of such interactions is an important avenue for further research. With time-period choice captured exclusively with the help of the constants, it is possible to reduce the number of alternatives from the number of time-periods (which in this case, in conjunction with the SP design, would have led to a total of 1,200 alternatives) to the number of SP alternatives (in this case four), with the help of an appropriate functional form, or the use of dummy variables. While this approach eases computational costs, it prevents a GEVstyle treatment of correlation in the errors between alternatives in adjacent time periods. A preliminary analysis however revealed that such correlation levels are relatively low in the

Functional approximations to alternative-specific constants 549 present dataset, which could be a reflection of the specific SP design. The theoretical findings of the paper clearly also apply to the case where a full set of time-period alternatives is used. The free estimation package Biogeme (Bierlaire, 2003, roso.epfl.ch/biogeme) was used for the calibration of the models. This allows the specification of nonlinear utility functions in free form, as required in the present analysis. Exponential formulation The first of our proposed approaches makes use of an exponential transformation. Looking at the distribution of the constants referring to the outbound period, we define a base period Bout, and, for a given outbound period x, we set: 50_ = max(B0Ut-x, 0)

and

50+ = max(x-B0Ut, 0).

The value of the function at time period x is then given by: f[50.80+) = exp(/30--50-+ /3O+-5O+), where 50_ and 50+ are defined as above, and where the two shape parameters /30_ and /30+ characterise the slope to either side of the peak. The formulation can be adapted for return time periods as well as for durations, where the duration was in our example calculated as the difference between the time period containing the return departure time and the time period containing the outbound arrival time. As such, the possible values along this dimension range from 0 to 23, instead of 1 to 24. Aside from the issue of allowing only for a single peak, the main problem with this approach is that of finding an appropriate choice for the base period. While the parameters ft,, and /30+ are relatively straightforward to estimate, the estimation of the base parameters (e.g. Bout) is significantly more complicated. Indeed, these parameters interact in such a way with the arguments of the distribution (and its form) that significant problems with local optima are almost inevitably encountered in their estimation. As such, the base parameters generally need to be fixed a priori, with multiple runs used to determine the optimal location for the peak. Power formulation Even though the estimation of separate shape parameters j30. and /30+ allows for different slopes to the left and the right of the peak, the use of the exponential transform imposes a strict a priori constraint on the behaviour of the functional form. An alternative approach uses a power function formulation, where the value of the function at time period x is given by: f[5o-5o+) = ft--(S0-)XD- + & + -(5 o + ) X o + ,

550 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where \ . and \, + need to be estimated in addition to j30. and /30+. Aside from being more flexible than the exponential formulation, the \>. and \,+ have a specific interpretation, in that they are elasticities of J{50.,80+) with respect to 50_ (for xB0Ut) respectively. Indeed, withx
and we have: Xo. =

[dA80.)M0.]-[50Jfi5o-)l

A similar reasoning applies in the case where x>B0Ut. The estimation of the base parameters is hampered by similar problems to those faced in the case of the exponential formulation. Empirical formulation While the exponential and power formulations have advantages in terms of identification and parsimony over the approach using a full set of constants, they lack flexibility, by imposing a single-peak constraint. This formed the motivation behind the development of an alternative approach, where a middle-way between the parametric and the constants-only approach was chosen, based on the use of empirical distributions. Formally, let N give the total number of time periods, with As denoting the lowest represented time period, and Ae giving the highest represented time period, where these may differ from the first, respectively last time period. With Kintermediary support points at values At (k^],...JT) defined such that 1
/(*)= A-! +(ft-A-i)-fl-/*~* 1

,Ak_x<x
I A-A-J A+(A-A)-fi-^~-j

,XZAK

where ft gives the value of the function at time period k, ft gives the value of the function at the start-point and ft gives the value of the function at the end-point. Intermediary values are found via interpolation, using linear segments between support points. With K=N-2, the formulation reduces to the approach using a full set of constants. In this empirical formulation, the parameters A* (k=l, ...,K) play an analogous role to the base period parameters in the exponential and power formulations discussed earlier and are subject to similar estimation difficulties such that it is of interest to keep these parameters fixed. Furthermore, problems can be caused by time periods that are always or never chosen, just as was the case with the approach using a full set of constants. The easiest way to estimate a model with this formulation seems to be to use an iterative approach starting with the biggest possible set of identifiable constants, and to drop constants in favour of interpolation if the

Functional approximations to alternative-specific constants 551 results support such action. This approach can also be seen as a useful tool for testing the validity of a single peak assumption, and where this is supported, in the subsequent search of an appropriate base specification for the exponential and power formulations.

MODEL RESULTS Basic MNL models The first model fitted to the data was a simple Multinomial Logit (MNL) structure (c.f. Train, 2003), using no explicit treatment of the choice of time period, and fitted with the aim of finding an appropriate specification of utility and providing a base model against which to compare the more advanced models. Separate models were estimated for commuters with flexible and inflexible working hours. The main explanatory factors used were travel-time and travel-cost. Attempts were also made to include a frequency coefficient for the public transport alternative, but no significant effect could be identified, which can at least partly be explained by the use of only three levels of frequency in the survey. No gains in model fit could be made by the use of non-linear transforms. A number of interactions with sociodemographic attributes were explored, notably in the form of dummy variables reflecting the preference of certain population segments for the various SP alternatives. Attempts were also made to estimate separate rail-cost coefficients for commuters receiving full, partial, and no employer compensation for public transport travel. Finally, a continuous formulation was used where appropriate to account for the effect of income on certain marginal utilities. Specifically, letting yt denote the household-income of individual i, with the average household-income in the population defined asy, the utility-term associated with a given cost-variable x for this decision-maker is given by /? {yi jy)'" x, where X represents the elasticity of cost-sensitivity with respect to the relative income of individual i. The results of the MNL analysis are summarised in Table 1. The fact that the population-level dummies for the retimed alternatives and the mode-shift alternatives are all negative reveals that, ceteris paribus, respondents prefer the base alternative, and hence, by implication, the observed (RP) alternative. The results show a slightly higher ratio between the constants for late and early departure in the case of commuters with inflexible working hours than for commuters with flexible working hours, supporting results by Hess et al. (2004) that reveal far higher sensitivities to late arrival for commuters with non-flexible working hours than for commuters with flexible working hours. For travel cost, a significant coefficient for commuters with flexible working hours could only be identified for car-cost and uncompensated rail-travel (at the 71% level), while, for commuters with inflexible working hours, an additional coefficient was identified for semi-compensated rail-travel (at the 83% level, with an 80% level for car-cost). An income effect could only be identified for car-travel in the flexible commuters group, and for semi-compensated rail-travel in the inflexible commuters group, both times indicating decreasing sensitivity with increased income.

552 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 1: MNL estimation results

Estimated parameters: Sample size: Final log-likelihood: Adjusted p1: Constants Earlier departure All travellers Low education Part-time Working from home Under sixty Later departure All travellers Low education Part-time Working from home Under sixty Mode-change All travellers Low education Part-time Under sixty Common rescheduling Under sixty 7>«ve/-time(min) Car PT Travel-cost (guilders) Car

XA

FLEXIBLE

INFLEXIBLE

16 :2092 -1979.33 0.2566

18 2101 -1814.06 0.321;>

Estimate

T-statistic

Estimate

-1.4592 -0.6469 -0.5332 -0.3223

-15.13 -3.00 -2.64 -2.20

-1.8327 -0.2765 0.2885 0.3261

-1.81 1.47 2.59

-16.53 -4.30 -1.86 3.27

-2.3044 -0.3274 -1.0000

-13.74 -1.65 -3.95

-1.6991 -1.2649 -0.3797 0.4398 -4.82

-2.0959 -

0.4144

2.36

-2.3587 1.5449 0.8387 -0.6524

-6.80 6.43 3.28 -3.14

1.19

0.1529

1.59

-0.0233 -0.0308

-9.40 -9.82

-0.0191 -0.0141

-7.96 -5.77

-0.0161 -0.3345

-6.85 -8.13

-0.0054

-1.27

-0.0082 -2.0263

-1.36 -4.33

-

XB

-

V

-16.82 -

0.3593

Semi-compensated PT Uncompensated PT

T-statistic

-0.0571

-1.05

-

-0.0242

-2.32 -

MNL models with constants The next step consisted of explicitly accounting for the time period choice dimension by using the approach based on three disjoint sets of 24 constants. To permit identification, a single constant was set to zero along each dimension, where the eight outbound constant (7am-8am) was selected for this, along with the seventeenth return constant (4pm-5pm), and duration constant number eight (8-9 hours). This is consistent with the average outbound journey time of around 1 hour, which would imply that commuters leaving in time period 8 arrive in time period 9, which together, with a return departure in time-period 17, would imply a duration between eight and nine hours. This is forthwith referred to as the 8-17-8 normalisation.

Functional approximations to alternative-specific constants 553 As expected, this approach led to identification problems, with a number of constants receiving very large positive (for alternatives always chosen when available) or negative values (for alternatives never chosen). Additionally, out of 69 estimated constants, 24 reduced to zero in the model for flexible commuters, with a corresponding figure of 16 in the model for commuters with inflexible working hours. These constants are associated with outbound periods, return periods, or durations that are not included in any of the SP choice-sets.

eturn eriod

exible

lexible

<< «.

E _

s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-

-3.2 (-5.89) -1.68 (-5.11) -0.94 (-4.54) -0.38 (-3.09) 0 -0.38 (-2.55) -0.61 (-2.61) -0.23 (-0.68) 0.81 (1.32) 1.49(2.2) 1.34(1.89) -0.23 (-0.18) 2.77(1.52) 3.04(1.92) 3.75 (2.32) 4.65 (2.94) 3.47(2.19)

-

-

-3.23 (-4.41) -1.05 (-4.91) -0.29 (-2.5) 0 -0.3 (-2.52) -0.95 (-4.81) -0.99 (-3.5) -2.53 (-4) -2.15 (-3.46) -3.99 (-3.15) -

-

1 2 3 4 5 6 7 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

-

-3.72 (-2.68) -2.59 (-2.57) -1.71 (-2.23) -1.38 (-2) -2.3 (-4.09) -0.86 (-2.02) -0.34 (-1.25) 0 (-0.01) 0 -0.02 (-0.12) -0.21 (-1.03) -0.81 (-2.62) -1.35 (-2.88) -1.24 (-1.82) -

2.33(1.44) -0.71 (-0.76) 0.03 (0.03) 0.55 (0.79) -0.07 (-0.13) -0.21 (-0.52) 0.22 (0.9) -0.21 (-1.42) 0 -0.28 (-1.93) -0.98 (-3.76) -1.22 (-3.47) -1.52 (-3.14) -2.03 (-3.31) -2.1 (-2.85) -2.15 (-2.65)

•3

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

lexible

lexible c

exible

exible

s*

E _ -

'.ration

tbound eriod

Table 2: Estimates of constants for fiilly-specified model with extra availability conditions

c 0.58 (0.63) 0.65 (0.86) 0.52 (0.84) -0.12 (-0.23) -0.13 (-0.31) -0.89 (-2.79) -0.71 (-3.1) -0.38 (-2.55) 0 -0.19 (-1.6) -0.72 (-3.79) -0.95 (-3.24) -1.22 (-2.66) 0.24 (0.33) -

-4.6 (-4.1) -1.37 (-1.87) -1.55 (-2.53) -1.19 (-2.23) -1.36 (-3.08) -1.39 (-3.81) -0.83 (-3.2) -0.15 (-0.91) 0 -0.04 (-0.31) -0.14 (-0.7) -0.01 (-0.04) -0.33 (-0.55) -2 (-2.38) 2.27(1.81) -

-

-

-

-

A set of availability conditions was then introduced to deal with the identification issues. In the flexible model, alternatives associated with outbound constants 4, 15, 16, and 17, return constants 1, 6, 9, 10, and 23, and duration constant 14 were made unavailable. In the inflexible model, the unavailability constraint was applied to alternatives associated with outbound constants 1,16 and 22, return constants 7 and 10, and duration constants 14, 15 and 17. The sample size for inflexible commuters remained unaffected, while for flexible commuters, the unavailability conditions meant a reduction by 9 observations to 2,083. This approach led to final LL of -1794.86 for the flexible model and -1709.33 for the inflexible model, such that in both cases, the use of constants clearly leads to very important gains in

554 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction model fit over the basic MNL model, even after taking into account the difference in sample size in the flexible case. Table 2 summarises the results of this estimation, where only the time period constants are reproduced; the estimates of the other coefficients and constants remained largely unaffected. The results suggest that a single-peak approach would be appropriate in most cases (e.g. outbound and return for flexible commuters), while there are some cases in which multiple peaks should possibly be allowed for, notably along the duration dimension for flexible commuters, where the low significance levels should however be borne in mind. It should also be noted that the results along the outbound dimension for inflexible commuters are inconclusive. Overall, this analysis has shown that significant problems with identification can arise when using a full set of time-period constants. Nevertheless, the analysis has also shown that the approach leads to very significant gains in model performance, showing the importance of acknowledging the time period choice dimension. These findings act as the motivation for our search for a functional form to represent the distribution of the constants.

MNL model with exponential formulation Given the results from the fully-specified model, it was decided to continue using the 8-17-8 normalisation in the models based on the exponential formulation. This decision was supported by several experiments which showed significant drops in model performance when using alternative specifications, confirming 8-17-8 as the most appropriate single-peak normalisation, in both population segments. Although not strictly necessary, the extra availability conditions were used again, to enable comparison with the models with constants. The final model fits, along with the estimates of the shape parameters, are summarised in Table 3. Again, the remaining coefficients remained largely unaffected by the treatment of time period choice. In the model for flexible commuters, the use of the exponential formulation leads to an improvement in log likelihood by 124.15 units over the simple MNL model, with 6 additional parameters, which, despite the use of 9 fewer observations (due to the extra availability conditions), is clearly significant. Similarly, in the model for inflexible commuters, the use of the exponential formulation leads to significant gains over the MNL model, by 65.51 units, with 6 additional parameters. All shape parameters are negative, implying a decreasing slope to either side of the base, and, with a few exceptions, the estimates are all significant at high levels of confidence. An interesting observation can be made for the outbound shape parameters in the inflexible model. Unlike in the flexible model, the right-side parameter is significantly more negative than the left-side one, implying a steeper slope to the right of the base, which can be seen as an effect of the inflexible working hour arrangement, and the accompanying low tendency to accept shifts to a later departure time. A similar observation can be made for the return constants, where the slope is steeper to the right for flexible workers, while it is steeper to the left for inflexible workers. For the duration constants, it is more difficult to find an easy interpretation. To illustrate the effects of the shape parameter, a brief graphical analysis was conducted, with plots shown in Figure 1.

Functional approximations to alternative-specific constants

555

Table 3: Results for models using exponential formulation

Final LL

Par.

Adj. p 2 :

A,-

A*

-1855.18

22

0.3006

-0.51 (-6.44)

-0.54 (-5.53)

-0.24 (-2.35)

-0.33 (-5.42)

-0.74 (-3.36)

-0.38 (-4.29)

-1748.55

24

0.3436

(-3.44)

(-3.55)

(-2.81)

(-1.56)

(-1.73)

(-2.37)

Flexible

fti-

Inflexible

Figure 1: Implied shape of distribution of constants using exponential formulation

The results presented in this section have shown that important gains in model fit can be obtained over the simple MNL model by using a functional form for the constants representing the time period choice dimensions. These improvements were not as important as those obtained when using a full set of constants (124.15 vs 184.47, and 65.51 vs 104.73), but come at a lower cost in terms of the number of additional parameters (6 vs 35, and 6 vs 45). Furthermore, although the additional availability approach was used, this is not strictly necessary with these functional forms.

556 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction MNL model with power formulation Given previous results, the 8-17-8 normalisation was used again in the power formulation. As was the case in the other structures, the results in terms of time and cost coefficients, as well as general dummy variables, remained largely unaffected by the treatment of the time period choice; the results in terms of model fit and shape parameters are summarised in Table 4. Some problems were encountered with the right-bound duration terms for inflexible commuters, where inconsistent results were already observed with the constants approach. The shape parameter (&+) took on a value essentially equal to zero (albeit with a significant tstatistic), while the associated exponent (\j+) took on a very large value. Imposing bounds on these parameters led to unsatisfactory results, such that these values were retained. In terms of model fit, the power formulation, just like the exponential formulation, leads to very significant gains in performance over the simple MNL model, with increases by 153.27 and 69.83 units respectively, at the cost of 12 additional parameters. For flexible commuters, the power formulation leads to gains when compared to the exponential formulation (looking at the adjusted p2 measure), while this is not the case for inflexible commuters. In terms of implied shapes of the distribution, the results still suggest decreasing shapes in the majority of cases, but with notable exceptions, especially for inflexible commuters. This is partly reflected in the findings for the models using a full set of constants. Table 4: Results for models using power formulation

Flexible commuters Modelfit: -1826.06 Parameters: 28 Adjusted p : 0.3092 Est.

ft. ft+ ft. ft+ ft. ft+

-0.1108 -0.4907 -0.2718 -0.0245 0.0089 -0.3146

T-stat. -1.15 -5.90 -2.58 -1.09 1.45 -3.72

AnAn+

ArAr+ ArfAd+

Est. 2.9033 1.1015 1.1784 2.3522 2.6136 1.2162

Inflexible commuters

Modelfit: -1744.23 Parameters: 30 Adjusted p : 0.3430 T-stat. 3.15 9.15 8.16 4.24 9.26 7.05

ft. ft+ ft. ft+ ft. ft>

Est. -0.4084 0.0371 0.0453 -0.4971 -0.5211 10"12

T-stat.

Est.

-3.86

V

2.36

Ao+

0.85

V

-5.34 -4.95 13.92

Ar+ Ad. Ad+

1.4175 2.0059 1.4976 0.9681 0.9202 14.5252

T-stat. 7.00 13.95 3.39 8.55 8.17 121.47

The final step of this part of the analysis consisted of a graphical illustration of the results obtained with the power formulation, as shown in Figure 2, where only the represented part of the ranges of time periods are shown in these figures. The graphs show the increased degree of flexibility of the power formulation when compared to the exponential formulation. They also show that, unlike the exponential formulation, the power formulation is able to pick up the non-strictly increasing slopes to the left of the peak for duration constants in the flexible model, and return constants in the inflexible model, as well as non-strictly decreasing slopes to the right of the peak for outbound and duration constants in the inflexible models. Although these results are less easy to reconcile with the intuitive meaning of the constants than in the case of the exponential formulation, they reflect the results from the model using a full set of constants, and are indicative of the problems of interpretation faced in models calibrated on

Functional approximations to alternative-specific constants 557 SP data. Along with the better fit for this approach, at least in the flexible model, these results show a higher degree of flexibility than was the case with the exponential formulation.

Empirical distribution The next step in the analysis consisted of using the empirical distribution approach. For this, an iterative procedure was used. The models were first estimated with the full set of identifiable support points, leading to the equivalent of the model using a full set of constants. Interpolation was then used where appropriate to replace intermediate support points. For ease of analysis, the normalisation ft=0 was used in all three dimensions. Only the results for the final iteration step are presented here. For flexible commuters, the base model had a log likelihood of-1794.86, with 51 estimated parameters, which is the equivalent of the model using constants, with availability constraints applied. Progressive iterations led to the replacement by interpolation of outbound constants 6, 12 and 13, return constants 11, 15, and 20, and duration constants 2, 6, 7, 9, 10 and 11. The removal of these 12 intermediary support points led a statistically significant reduction (at the

558 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 92% level) in log likelihood by 9.67 units. No further interpolations were possible at this stage without major reductions in model fit. The shape of the distributions in the final model is reproduced in Figure 3. The plots show that in the final model, the three curves are fairly smooth, although the duration curve presented more than one peak. The main exception is the value of return constant 13. Its removal however leads to a drop in LL by a very significant 6.15 units, which is not acceptable. The estimates of the final model are shown in Table 5.

figure 3: Shape ot empirical distribution tor flexible commuters

For inflexible commuters, the base model had a LL of -1709.33, with 63 estimated parameters. Progressive iterations led to the replacement by interpolation of outbound constants 6, 7, 9, 11, 12, 18 and 19, return constants 11, 13, 18, 20, 21, and 23, and duration constants 4, 6, 9 and 10. The removal of these 17 intermediary support points led a drop in LL by 4.21 units to -1713.54, where the associated likelihood-ratio test-value of 8.42 has a U p value of 0.96, showing that the drop in LL is not statistically significant. The shape of the distributions in the final model is illustrated in Figure 4. The plots reflect the issue of multiple peaks for outbound and return constants, which cannot be accommodated in the exponential and power formulations. The estimates of the final model are shown in Table 6. The discussion in this section has shown that the empirical approach reduces the number of estimated parameters compared to the full set of constants approach, without major drops in model fit, while avoiding the inflexibility of the parametric approaches. However, some issues of applicability remain, and the potential influence of the SP design on the estimation of the constants is possibly greater than with the parametric approaches. To conclude, it should be noted that the iterative parameter reduction process was conducted manually; the implementation of this approach in estimation software is an important avenue for further research, incidentally also in the investigation of random taste heterogeneity.

SYNTHETIC DATA EXTENSION AND MODEL VALIDATION The analysis conducted in the earlier parts of this paper has highlighted the issues of identification arising with the use of a full set of constants. The analysis has further shown that the use of interpolation between a pair of estimated constants can significantly reduce the number of estimated parameters and increase parameter significance, without major reductions in model fit. This suggests that some simplification of the treatment of constants is

Functional approximations to alternative-specific constants 559 possible, and even preferable. The results using the exponential and power formulation support these findings; indeed, even though these approaches lead to significant drops in model fit, these drops are still at an acceptable level, when one considers the important gains in terms of ease of estimation and applicability. Table 5: Estimates for flexible commuters with empirical formulation

As = 5 Al = 7 A2 = 8 A3 = 9 A4= 10 AS = 11 Ae= 14

Outbound Estimate 0 0s 1.90 01 (32 2.19 1.91 03 1.27 04 1.23 05 -0.64 0e

T-stat 6.03 6.10 4.56 2.65 2.25 -0.67

As = 7 Al = 8 A2 = 12 A3 = 13 A4 = 14 A5 = 16 A6 = 17 A7 = 18 A8 = 19 A9 = 21 Ae = 22

Return Estimate 0 0s 1.61 01 02 2.58 (33 1.65 3.11 04 3.91 05 3.90 06 3.86 (37 3.67 08 2.51 09 2.63 0e

T-stat 1.56 2.01 1.26 2.38 2.76 2.67 2.58 2.38 1.52 1.50

As = 0 Al = 1 A2 = 3 A3 = 4 A4 = 5 A5 = 8 A6 = 12 Ae = 13

Duration Estimate 0 0s 0.00 01 -0.84 02 -0.93 03 -1.80 04 -0.75 05 (36 -2.00 -0.71 0e

T-stat 0.01 -1.37 -1.36 -2.50 -0.89 -1.89 -0.57

Ta ble 6: Estimates for inflexible commuters with empiric il formulat on

Return

Outbound As = 4 Al = 5 A2 = 8

0s 01 02

A3 = 10 A4= 13

(33

A5= 14 A6= 15 A7=17 A8=20 Ae=21

04 05 06 07 08 0e

Estimate 0

T-stat

1.66 3.07 2.34 4.10 4.01 2.45 5.12

4.01 5.81 3.62

7.13 5.99

4.61 4.19 1.73 2.83 3.91 3.23

As = 8 Al = 9 A2 = 12 A3 = 14 A4 = 15 A5 = 16 A6 = 17 A7 = 19 A8 = 22 Ae = 24

0s 01 02 03 04 05 06 07

Duration

Estimate 0

T-stat -

-3.10 -1.94

-2.57

(38

-2.59 -2.10 -2.49 -2.27 -3.11 -4.10

0e

-4.20

-1.38 -1.74 -1.38 -1.59 -1.42 -1.84 -2.20 -2.12

As = 0 Al = 1 A2 = 2 A3 = 3 A4 = 5 A5 = 7 A6 = 8 A7 = 11 A8 = 12 A9 = 13 Ae = 16

Final model: outbound

Final model: return

0s 01 02 03 04 05 (36

07

ft 09 0e

Estimate 0

T-stat -

3.13 2.93 3.27 3.07 4.33 4.48 4.35

3.74

4.03 2.34 6.43

Final model: duration

Figure 4: Shape of empirical distribution for inflexible commuters

3.16 3.42 3.08 4.00 4.01 3.48 2.90 1.52 3.51

560 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 7: Model fits on synthetic data

Model MNL MNL with constants Empirical formulation Exponential formulation Power formulation

Final LL -19625.9 -18973.5 -18996.3 -19208.3 -19029.5

Parameters 15 28 24 17 19

Adjusted p1 0.2344 0.2593 0.2586 0.2506 0.2575

It should be stressed again that the calibration of constants on SP is potentially influenced by a host of factors, notably the design of the SP experiment, and the reaction of respondents to the nature of the survey. This could also partly explain the mixed success with the highly flexible power formulation. As such, the use of RP data is more promising. In the absence of RP data, the decision was taken to generate a synthetic dataset, free of SP errors, for further evaluation of model performance, as well as for model validation. The construction of the synthetic dataset is based on set of 2,092 observations for commuters with flexible working hours. From this, a sample of 20,920 observations was produced, using small percentage changes on the original attributes for the 2,092 observations (between -20% and +20%). From this sample, 20,000 observations were selected at random to form the estimation sample, with the remaining 920 observations serving as a validation sample. The true model was assumed to be one using a full set of constants. However, only the outbound time period dimension was considered in this application, for reasons of simplicity. The aim here is to show the applicability of the functional forms to a clean dataset, and the extension to a model using constants along the three dimensions of outbound, return and duration, is straightforward. The values used for the coefficients in the data generation process are those found in the estimation of the model using a full set of constants, where the constants 4, 15, 16 and 17 (for which issues with identification existed) were replaced by linear interpolation. Five separate models were estimated on this dataset, a simple MNL model, an MNL model using a full set of constants, a model using the empirical distribution approach, a model using the exponential formulation, and a model using the power formulation. The final fits of the different models are shown in Table 7. The model using the full set of constants was estimated to validate the data; the estimates of this model are virtually indistinguishable from the parameter values used in the data generation process, they are all significant at high levels of confidence, and the model has a final LL of-18,973.5 units, with 28 estimated parameters. The remaining four models also perform very well in terms of recovering the 15 parameters not related to the choice of time period (time and cost coefficients, and socio-demographic and SP constants). However, significant differences in model fit exist between the models. As such, the simple MNL model, which does not allow for any treatment of the outbound constants, has a final LL of -19,625.9 equating to a drop in LL by 652.4 units when compared to the model with constants. In the model using the empirical formulation, only 4 parameters could be replaced by interpolation, which nevertheless led to a statistically significant drop by 22.8 units to a final log-likelihood of-18996.3. The model using the exponential formulation yields a final LL of-19,208.3 units, with 17 parameters, thus comfortably outperforming the simple MNL

Functional approximations to alternative-specific constants 561 model. Finally, the model using the power formulation obtains a final LL of-19,029.5 with 19 estimated parameters, equation to a drop in LL by a mere 56 units when compared to the model with constants, which although significant, is relatively small, illustrating the flexibility of this approach. The model also clearly gives a better fit than the one using the exponential formulation.

Figure 5: Graphical analysis of models estimated on synthetic data

To conclude this analysis, we conduct a brief graphical experiment (Figure 5), looking at the shape of the implied distribution of the constants in the true model, in the empirical formulation, and in the two functional approximations, using a lower bound at 4, and an upper bound at 17. The results show that the shape assumptions made by the exponential formulation are too strict to allow it to approximate the true shape. On the other side, the power formulation is able to very closely approximate the true shape of the distribution. As a final step in the analysis, the four models estimated above were applied to the validation sample of 920 observations. The results of the validation exercise are summarised in Table 8, which gives, for each model, the average probability of correct prediction, in addition to the log-likelihood of obtaining the observed choices, and the associated adjusted p2 measure. As expected on the basis of the relative differences in model fit, the differences in performance on the validation sample are relatively small. Nevertheless, the exercise shows that any model using a treatment of time period choice obtains better performance than the MNL model. As would be expected, the performance on the validation sample is poorer than

562

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

that obtained on the estimation sample, but within a margin that would suggest that the models have not been overfitted to the estimation data. Although, as mentioned above, the differences in performance are too small to draw any overly reliable conclusions, the results clearly show very good performance by the two parametric approaches (exponential and power), especially when taking into account the lower cost in terms of parameters (as in the adjusted p2 measure). This validation exercise thus suggests that, in the presence of a large number of inter-related constants, these approaches present a viable alternative to the use of a full set of constants. The development of more flexible formulations, notably ones allowing for multiple peaks, is an important avenue for further research. Table 8: Prediction performance on validation sample

Model MNL MNL with constants Empirical formulation Exponential formulation Power formulation

Correct prediction

Final LL

42.29% 43.69% 43.67% 43.01% 43.53%

-949.61 -933.50 -934.32 -940.80 -938.43

Parameters 15 28 24 17 19

Adjusted p2 0.1878 0.1904 0.1931 0.1935 0.1938

SUMMARY AND CONCLUSIONS In this paper, we have discussed issues researchers are faced with in the case of models using a high number of constants that have a high level of interdependency. The example used in our analysis is that of time-period choice, where constants are used to capture period-specific unobservables along three dimensions; outbound, return and duration. The potentially high number of constants to be estimated in a fully specified time-period choice model can lead to issues of identification, high computational cost, and problems with interpretation. Due to the heightened correlation between adjacent time-periods, some form of interdependency is clearly to be expected. This, in conjunction with the above discussed problems with a fullyspecified model, was the motivation for replacing the full set of constants by a functional approximation. In this paper, we have proposed several alternative approaches, based on parametric as well as non-parametric functional approximations. The analysis has shown that the different approaches all have advantages as well as disadvantages, but that important gains in identification can in general be obtained with the use of a functional approximation. Although the discussion in this paper has centred on the application to time-period choicemodelling, it should be noted that there are potentially other applications in which the use of a functional form can be preferable to the use of a full set of constants. One example is that of route choice. Another area where the use of a functional form is very promising is that of spatial choice, such as the choice of residential or work location. Here, there are likely to be factors that cannot be captured in full in the observed part of utility, but whose mean varies across space in a clear pattern, for example as a function of the distance to the central unit, the border unit, or the nearest street. Several avenues for further research can be identified. In terms of the application presented in this paper, these relate to the extension to models allowing for an explicit modelling of the

Functional approximations to alternative-specific constants 563 correlation between adjacent time periods, and the representation of random taste heterogeneity across agents. Another important area for future work is the development of parametric functional forms that allow for multiple peaks, and approaches that enable researchers to estimate the location of the peak, rather than fix it a priori. Finally, it is of interest to allow for differences in the shape of the distribution of constants across decisionmakers, with one possibility being the use of a random distribution of the shape parameters across agents in the exponential and power formulations.

ACKNOWLEDGEMENTS The authors would like to thank Frank Hofman for making the Dutch National model data available for the present research. The authors would also like to acknowledge the involvement of Andrew Daly and Charlene Rohr in earlier stages of this research.

REFERENCES Abkowitz, M.D. (1981b) An analysis of the commuter departure time decision, Transportation 10, 283-297. Arnott, R., de Palma, A., Lindsey, R., 1990. Departure time and route choice for the morning commute. Transportation Research 24 (3), 209-228. Ashiru, 0., Polak, J.W. and Noland, R.B. (2003) The utility of schedules: A model of departure time and activity time allocation with application to individual activity scheduling, 10th International Conference on Travel Behaviour Research, Lucerne. Athanassiou, S. and Polak, J.W. (2001), Generalised extreme value models of time of travel choice, Proceedings of the European Transport Conference, AET, London. Bates, J.J., Shepherd, N.R., Roberts, M., van der Hoorn, A.I.J., Pol, H.D.P., 1990. A model of departure time choice in the presence of road pricing surcharges. Proceeding of the 18th Summer Annual Meeting, Seminar H, PTRC, London, 215-226. Bhat, C. (1998), Analysis of travel mode and departure time choice for urban shopping trips, Transportation Research 32B(6), 361-371. Bhat, C.R., and Steed, J.L., (2002), A continuous-time model of departure time choice for urban shopping trips, Transportation Research 36B(3), 207-224. Bierlaire, M. (2003). BIOGEME: a free package for the estimation of discrete choice models, Proceedings of the 3rd Swiss Transportation Research Conference, Ascona, Switzerland. Bradley, M.A., Bowman, J.L., Shiftan, Y., Lawton K., Ben-Akiva, M.E., 1998. A system of activity-based models for Portland, Oregon. Report prepared for the Federal Highway Administration Travel Model Improvement Program, Washington, DC. Chin, A.T.H., 1990. Influences on commuter trip departure time decisions in Singapore, Transportation Research 24A(5), 321-333. Cosslett, S. (1977), The trip timing decision for travel to work by automobile, in "Demand Model Estimation and Validation, The Urban Travel Demand Forecasting Project",

564 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction McFadden,D., Talvitie.A., Cosslett,S., Hasan,L, Johnson,M., Reid,F., and Train,K. (eds), Chapter III-3, pp.201-221, University of California, Berkeley, CA. de Palma, A., Khattak, A.J., Gupta, D., 1997. Commuters^ departure time decisions in Brussels. Transportation Research Record 1607. de Jong, G., Daly, A., Pieters, M., Vellay, C. and Hofman, F. (2003) A model for time of day and mode choice using error components logit, Transportation Research 29E(3), 246268. Hendrickson, C. and Plank, E. (1984) The flexibility of departure times for work trips, Transportation Research 18A(1), 25-36. Hess, S., Polak, J.W., Daly, A.J., and Hyman, G. (2004) Flexible substitution patterns in models of mode and time-of-day choice: New evidence from the UK and the Netherlands, Proceedings of the European Transport Conference. AET London. Hyman, G., 1997. The development of operational models for time period choice. Department of the Environment, Transport and the Regions, HETA Division, London. Mahmassani, H.S., Hatcher, S.G., Caplice, C.G., 1991. Daily variation of trip chaining, scheduling and path selection behaviour of commuters. Proceedings of 4th IATB Conference, Quebec. Mahmassani, H.S. and Chang, G.L. (1985) "Dynamic aspects of departure time choice behaviour in a commuting system: Theoretical framework and experimental results', Transportation Research Record 1037, 88-101. Mannering, F.L., Abu-Eisheh, S.A. and Arnadottir, A.T. (1990) Dynamic traffic equilibrium with discrete/continuous econometric models, Transportation Science 24(2), 105-116. McCafferty, D. and Hall, F.L. (1982) "The use of multinomial logit analysis to model the choice of time of travel' Economic Geography 58(3), 236-246. Polak, J.W., Jones, P.M., Vythoulkas, P.C., Meland, S. and Tretvik, T. (1991) 'The Trondheim Toll Ring: Results of a Stated Preference Study of Travellers' Responses', EURONETT Deliverable 17, Transport Studies Unit, University of Oxford. Polak, J.W. and Jones, P. (1994), "Travellers' choice of time of travel under road pricing", 73rd Annual Meeting of the Transportation Research Board, Washington D.C. Small K. A. (1982), "The scheduling of consumer activities: Work trips", American Economic Review 72(3), 467-479. Small, K.A. (1987), "A discrete choice model for ordered alternatives", Econemetrica 55(2), 409-24. Train, K. (2003). Discrete Choice Methods with Simulation, Cambridge University Press, Cambridge, MA. van Vuren, T., Carmichael, S., Polak, J.W., Hyman, G., Cross, S., 1999. Modelling peak spreading in continuous time. Proceedings of the European Transport Conference, AET, London. Vickrey, W.S., 1969. Congestion theory and transport investment. American Economic Review (Papers and Proceedings) 59, 251-261. Wang, J.J., 1996. Timing utility of daily activities and its impact on travel. Transportation Research 30A(3), 189-206.

29

PROJECT-BASED ACTIVITY SCHEDULING FOR PERSON AGENTS Eric J. Miller, Department of Civil Engineering, University of Toronto, Toronto, Canada

INTRODUCTION This is the third is a series of papers that are attempting to cumulatively develop an integrated conceptual framework for modelling both short- and long-run decision-making of persons within a household framework (Miller, 2005a, 2005b). This is an agent-based, microsimulation framework for modelling in a comprehensive and internally consistent manner both short-run (daily/weekly) activity/travel scheduling decisions and longer-run "resource management" decisions (change vehicle holdings, change residential location, change employment status/location, etc.) that form the core of land use modelling systems. Both persons and households are viewed as decision-making units (DMUs), where a DMU is defined as an intelligent agent that is capable of: perceiving the world around it; acquiring and using resources; scheduling its activities; and acting into the world. Figure 1 defines the key elements of a DMU, which consist of a perceptual interface that communicates with collaborating agents (e.g., person-household interactions) and the world in general; a knowledge base in which perceived information, agent tastes and preferences, agent memory, etc. are stored; three sub-classes of decision management agents (DMA), where a DMA is a specialized agent capable of making specific decisions. Specific DMAs within a DMU consist of: a set of project DMAs, in which each project consists of a logically connected set of activities directed towards a common goal (Axhausen, 1998); an activity scheduler that constructs provisional schedules for the DMU to execute, given activity episodes generated by the projects; and a stress/resource manager that allocates available resources (money, cars, houses, etc.) to projects and modifies these resources over time in response to stresses experienced by the DMU (Salvini and Miller, 2005; Miller, 2005b).

566 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 1: The Decision-Making Unit (DMU) Agent

The objective of this paper is to develop the specification of the project and scheduler DMAs, and the project-schedule interface. The paper begins with definitions of key terms, design issues and assumptions. Data to support model estimation are briefly reviewed. The bulk of the paper then deals with the specification of the project and scheduler agents and their interaction. Major issues include: the overall scheduling process; generating activity episodes; determining episode attributes; and the role of utility within the scheduling process. The paper concludes with discussion of future work required to develop an operational model.

DEFINITIONS A number of key terms that are used throughout this paper are defined below. Activity: Something we do. In this model, "activity" is an ill-defined term that does not require a formal, technical definition (e.g., "activity" does not exist as a class in the model). Project: A logically connected set of activities with a common goal or purpose (Axhausen, 1998). It is argued at length in Miller (2005a, 2005b) that all personal and household activities can be usefully grouped into a relatively small number of projects. Episode: A single, specific occurrence of an activity. Planning Period: The period of chronological time for which activities are currently being planned/scheduled. It is assumed that the planning period is at least one day long and, generally speaking will be multi-day in length Provisional Schedule: At any time t, the provisional schedule consists of the set of episodes that have been scheduled for execution during the current planning period.

Project-based activity scheduling for person agents 567 Event: Something that actually occurs. Events include physical ("in the world") events such as completion of an episode, communications between two agents, and events generated by other agents/processes of which the agent becomes aware (e.g., a traffic jam), and scheduling/decision-making ("in the mind") events such as the generation of an episode. World: The entire physical and psychological system being simulated. That is, the World consists of all the agents being modelled, their behaviours and their interactions. Flexibility: Loose, relatively non-operational term that summarises the extent to which a given episode is constrained along one or more of its dimensions. It says nothing about how "important" a given activity is. Information about episode flexibility can be contained within the project, or may be "externally" determined by other agents acting within the World. Precedence: Refers to the chronological ordering of episodes within the provisional schedule. That is, one episode has precedence over another if it must logically occur first. Utility: The intrinsic benefit associated with participation in a given episode or project. Says nothing about flexibility (or precedence). Priority: Refers to the ordering in which episodes are considered for scheduling. Priority is potentially a function of episode flexibility, utility and precedence. Skeleton Schedule: The component of the provisional schedule that is relatively stable from one planning period to the next. The skeleton schedule consists of "routine" activities that are scheduled over longer time periods than the current planning period being modelled. The skeleton schedule is the initial (and known) provisional schedule at the start of the planning period (Doherty, et al, 2001; Arentze and Timmermans, 2000). Agenda: A list of candidate activity episodes for possible scheduling.

KEY DESIGN ELEMENTS, ISSUES & ASSUMPTIONS A key design concept in this model is that it is project-based. As discussed at far greater length in Miller (2005a, 2005b), the project approach allows the modeller to encapsulate detailed knowledge^ehaviour within each project. We do not have to burden the scheduler with this. Thus, as projects come and go (e.g., a woman quits work to stay home with a newborn baby), the scheduler remains unchanged in its function. Projects also allow for polymorphic behaviour. Different projects (work, social, etc.) can generate episodes and their attributes in different ways. If one wishes, projects can be viewed as largely a modelling/programming convenience. The behavioural plausibility of the approach, however, is also very high (even if people actually don't "think in projects" much of the time). To "take on a project" involves undertaking a conscious commitment or contract. Much of our daily activity involves tasks tied to such commitments (jobs, children, teams, community and religious organisations, etc.). The concept of the project aligns closely with Maslow's theory of motivated behaviour and the hierarchy of physiological and cognitive needs (Maslow, 1970). And clearly different aspects of our lives do "decompose" into fairly independent "projects" that to a large extent "have a life of their own" (work, school, family, etc.). The second key design concept is that we are trying to build a model that allows for true scheduling dynamics. That is, we are not trying to build a "reduced form", "static" model of

568 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction an activity pattern, but rather to build a model of the process by which an activity pattern emerges of the dynamics of daily life (Doherty, et al., 2001). Agent-based microsimulation is fast becoming the standard approach to activity-based travel modelling of all types (Arentze, et al, 2003; Arentze and Timmermans, 2000; Bhat and Koppleman, 1999; Bhat, et al., 2001; Bowman, et al., 1998; Charypar and Nagel, 2003; Fellendorf, et al., 1997; Fosgerau, 1998; Fujii, et al., 1998; Meister, et al., 2005; Miller and Roorda, 2003), and represents a natural, attractive and, indeed, virtually mandatory approach to modelling activity scheduling processes. Thus, this model is based on the representation of the person as an intelligent agent whose behaviour is simulated at a high level of fidelity over a specified period of time. As soon as one begins to inspect the way in which we plan and execute our daily lives, it quickly becomes apparent that this is an event-driven process, in which a schedule is incrementally built, altered and executed in response to a wide variety of events in the physical world (telephone calls from clients, a foreman requests that you work overtime, etc.), as well as "psychological events" (e.g., a decision to work at home tomorrow may trigger subsequent changes to other planned events) (Litwin and Miller, 2004). In this model, a purely event-driven simulation (EDS) framework is adopted in the attempt to maintain as much fidelity with the real-world process as possible. The final key criterion is flexibility in model design, to provide the ability to test alternative hypotheses/models, and to extend the model in terms of detail, scope, etc. as data, need, understanding, etc. grow, without having to restructure the entire formulation/software (although this does not mean that one set of code can be written that can encompass/anticipate all possible models). The simulation software ideally should be as much a framework within which alternative behavioural hypotheses can be tested, as it is a modeller se.

EMPIRICAL DATA TO SUPPORT ACTIVITY SCHEDULING PROCESS MODELLING Activity-based models typically rely on activity diaries of realised activity patterns for empirical model development. Use of such data essentially restricts one to modelling observed outcomes. To develop models of scheduling process generally requires "observing" this process in some fashion. This either involves laboratory experiments, in which subjects' decision-making can be observed in a controlled, but hypothetical/synthetic, environment, or appropriate survey instruments that can capture at least some elements of scheduling dynamics within a real-world context. Both approaches have merit. In Canada, parallel small-sample household panel surveys are underway in Toronto and Quebec City that are attempting to collect "process data" (Doherty, et al., 2004; Roorda and Miller, 2004). Each panel involves three waves over three years in which realised activity patterns are gathered for each household member in each wave for either two consecutive days or a full week. In addition, various forms of "process data" are being gathered. In the case of the Toronto panel, these include the following.

Project-based activity scheduling for person agents 569 Wave 1: A full week's worth of CHASE data was obtained for at least one person in each household. CHASE (Computerised Household Activity SchEduling) is a computer-based survey instrument that not only records executed activity events but also records portions of the planning process leading to these realised events. At least once a day each respondent must not only enter what they have done during the day, but what they have planned to do during the remainder of the survey period. In this way, CHASE provides a reasonably detailed view of the dynamic evolution of the provisional schedule over time, including the addition, modification and deletion of activity episodes over time (Doherty and Miller, 2000). Wave 2: Stated adaptation experiments were conducted with respondents to explore how they would have dealt with various event-driven changes in their planned activities (e.g., "if you ran an hour late for one activity, what would you do?"). Wave 3: Prior to completing their activity diary, respondents were asked to fill out a large one-page sheet, with "routine" activities that are performed each week. The days, times, purposes and locations were all sketched in. If something is done routinely but has variable location and/or time, this was also indicated. Follow-up interviews further probed to explore the extent of variability that occurs in the execution of these "routine" activities.

SCHEDULING EPISODES I - GENERAL CONSIDERATIONS The focus of this paper is on specifying the short-run (day-to-day and within-day) workings and interactions of a person's project and scheduler agents, as illustrated in Figure 1. The household-level context and set of interactions within which personal decisions are made are discussed in detail in Miller (2005a, 2005b). In this paper, household resource levels are assumed to be known and fixed within the activity scheduling planning period under consideration and the stress-based processes by which these resources change over time are not dealt with. This paper also assumes that the skeleton schedule is known at the beginning of the planning period being modelled. An event-driven model of activity scheduling is deceptively simple in the abstract. At each point in time, a provisional schedule exists which consists of activity and travel episodes (trips) that have been scheduled up to this point in time. The provisional schedule is assumed to always be "feasible" in the sense that planned modes of travel are currently expected to be available, one expects to be able to get to activity locations on time, episodes do not overlap in time, etc. At any point in time a "trigger" (event) can occur that causes a new episode to be generated. When this happens, the scheduler attempts to schedule the episode, given the current provisional schedule, possibly including some rescheduling of existing provisional episodes. That is, all assignments within the provisional schedule are literally provisional (changeable) up to the time at which the next episode must actually be undertaken. Episode triggering events can be externally generated (i.e., generated by another agent within the World). The trigger/event, however, can also be internally generated by the agent itself.

570 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction A key characteristic of this process that must be stressed is that it implies that all decisions are made one episode at a time. Modifications to a limited number of existing episodes can be considered when trying to schedule this episode, but the entire schedule is never "globally" assessed or "optimised" at any point in the process. That is, while the scheduler will try to make the "best" insertion of the new episode that it can (given whatever rules/objectives are built into the model); it will not, in general, re-assess or re-schedule the entire schedule. Thus, event-driven scheduling is inherently myopic and (probably) "greedy" in nature. To convert this very loose conceptualisation into an operational model, one must specify the process that actually generates episodes, how the salient attributes of these episodes are to be determined, and how the scheduler will actually use these attributes to schedule episodes. Wrapped within these considerations are the roles that precedence, utility and priority play in episode generation and scheduling. These issues are dealt with in the following sections.

GENERATING EPISODES In Miller (2005a, 2005b) it is assumed that the primary task of each project is to generate an agenda of potential activity episodes. The scheduler would then in some manner select from these lists of episodes to build the provisional schedule over time. This is in many respects a plausible formulation. Certainly for many projects one can imagine a "to do" list of one-time and recurring activities that require scheduling over some planning period. This approach allows for episodes that might not be immediately schedulable, but which are retained "in the system" until they can be scheduled. It also facilitates dealing with precedence constraints, in which some episodes within a project need to be executed before others. Significant practical problems exist, however, in trying to implement this conceptual model, largely due to the fact that this hypothesised agenda is not observable in any practical manner. Thus, the construction of an operational model of the agenda appears to be a very difficult proposition. A more viable alternative is to "side-step" the agenda formulation problem altogether and simply generate episodes at various points in time for immediate consideration for scheduling. If the generated episode cannot be scheduled, it is "deleted" from further consideration. This too is a plausible process for many activities (a client calls and you agree "on the spot" to a meeting). Such a model of episode generation is potentially estimable from CHASE-type process data. Various modelling approaches are also readily conceivable (hazard-type models, random utility models, etc.) that can be empirically tested.

EPISODE ATTRIBUTES In addition to activity type, episode attributes include duration, start time, end time, location, and the participants involved. The determination of episode attributes is initially and primarily the function of the project, but as is discussed in detail in the following subsections, dynamic scheduling and execution elements exist as well. Appropriate methods for determining each episode attribute are discussed in the following sub-sections.

Project-based activity scheduling for person agents 571 Episode Duration Duration is the most important episode since it is the "container" within which the episode occurs: without duration, episodes literally would not exist. Episodes typically "need" a certain amount of time for their execution; and the benefit/utility derived from the episode may often depend upon its length. Observed episode durations can reasonably be assumed to result from a three-part process. First, a nominal or desired duration is specified at the time the episode is generated ("I need to go shopping and it is likely to take about 2 hours"). Second, this nominal duration may be modified during the scheduling/re-scheduling process due to scheduling constraints/conflicts. Finally, the dynamics of actually executing the planned schedule may result in a realised duration that differs from what was planned. The setting of the nominal duration resides within the project, since it is here that the "understanding" of the episode resides. The scheduler controls the dynamic adjustments to this nominal duration during the scheduling process; while the overall simulation of the agent's execution of planned activities within the World determines the realised duration. The modelling challenge is to operationalise the concept of a "nominal duration" (and how it is determined within the project), as well as to determine the decision rules used by the scheduler to produce scheduled durations. Many models of this process are conceivable and should be tested. One such model involves, for each episode, the project specifying a range of feasible nominal durations {MinDuration, MaxDuratiori). In scheduling this episode the scheduler will assign the maximum duration possible, given scheduling constraints (i.e., given maximum available gaps). If a gap of at least MinDuration in length cannot be found, then the episode is rejected. Another possibility is to implement a utility-based approach in which duration is selected based on (myopic) utility maximisation principles. Episode Start and End Times Episode start and end times are determined by the interaction of three factors. Intrinsic constraints associated with the activity in general and/or the specific episode being scheduled (e.g., job start/end times and store opening/closing hours). Schedule constraints are imposed on the start and/or end time of an episode by previously scheduled episodes (e.g., a 8:30 a.m. work start time will constrain the end time for pre-work in-home morning activities). Personal preferences for episode start/end times also may exist. For example, some people may prefer to work out at the gym before the workday starts, while others may prefer to exercise after work. In the absence of pre-emptive scheduling constraints either time period will generally be feasible, and the choice between the two time slots will be preference-based. Schedule constraints obviously dynamically emerge from the scheduling process itself. The key modelling issue here is the determination of the extent to which previously scheduled episodes define "hard" constraints (i.e., their start/end times and/or associate durations cannot be modified), as opposed to "soft" constraints (i.e., they might be modified within a rescheduling process). This may involve utility-based calculations to determine trade-offs in schedule time allocations, but other models are also plausible. Personal preferences imply the

572 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction need for a utility function that is start/end-time sensitive. This issue is discussed under "Episode Utility", below. Intrinsic constraints clearly are determined within the project. It is the project that contains the detailed, project-specific information and/or methods required to determine these constraints for a given episode. The work project, for example, knows about job start/end time requirements. All the scheduler needs to know is the actual values of any intrinsic constraints that exist, not the source of this information or the process by which it may have been generated. These constraints can be simply expressed as the earliest and latest start and end times for the given episode: (MinStart, MaxStart) and (MinEnd, MaxEnd) Travel Episodes and Travel Mode Choice The trips required for a person to move from one activity location to another are travel episodes that must be scheduled in much the same way as the activity episodes served by the trip. From a modelling convenience perspective it is desirable to schedule each travel episode at the time its associated activity episode is scheduled, so as to ensure feasibility of the provisional schedule. Considerable discussion exists in the literature, however, concerning when travel/mode choices are actually made within a dynamic scheduling process. In particular, it is argued that at least some episodes might be provisionally scheduled without explicit consideration of how the participant is going to get there, and/or that the "there" may not yet be determined. For example, one might agree to meet friends on Friday evening to see a movie without, in the first instance, agreeing on what movie, where will be seen. Of the two, location choice is probably the more fundamental than mode choice. Location is much more directly tied to the activity episode per se; i.e., travel mode is just a means to the end of participating in the activity, the utility of which can be affected by its location (i.e., one may prefer to see the movie showing at theatre X rather than the one showing at theatre Y). Further, if the episode location is, in fact, known, then it is strongly arguable that travel mode choice is implicit in the location choice decision. That is, it is illogical to choose an episode location that is infeasible to travel to. Determination of feasibility, in turn, implies assessment of the alternative means (modes) of getting there. Thus, for example, the movie theatre ultimately chosen by a teenager and his friends might be one that is transit-accessible, since no-one in the group has access to a family car for the evening. Thus, while not arguing for perfect rationality or information in decision-making, it is certainly the case that all realised episodes must prove to be feasible, and so at some point in the scheduling process feasibility considerations must come into play. From a model design perspective it is reasonable to assume that, once a location is specified, a provisional travel mode to that location is chosen, so that the provisional schedule associated with this location choice is feasible. As discussed in detail elsewhere (Miller, 2005a; Miller and Roorda, 2003; Miller, et al., 2005), once locations are known, travel mode must be computed on a tour (trip chain) basis in order to deal with coupling constraints associated with the auto-drive mode. It is argued that each time that a trip-chain is altered through a scheduling decision the tour-based mode choices for the travel episodes within the tour are re-evaluated.

Project-based activity scheduling for person agents 573 Given the discussion to this point it is clear that travel episodes are fundamentally different than activity episodes in terms of how they and their attributes are generated. The "need" for a trip is triggered by the scheduling of a particular activity episode, not by a "travel project" per se. The end time for a trip "going to" an activity is determined by the start time of this destination activity, while the start time for a trip "coming from" an activity is determined by this origin activity's end time. And the trip's travel time (duration) is determined by the travel mode choice decision (in combination with, of course, the expected demand-supply interactions occurring within the transportation network at the time the trip is scheduled to occur). That is, the travel episode "emerges" out of the scheduling process, rather than a project-based process, and thus the generation of travel episodes and the determination of travel mode are methods that most logically reside within the scheduler itself. The one project-based element that might exist with respect to travel mode choice is that it is possible that certain activity episodes might have intrinsic mode-specific constraints. If such intrinsic constraints exist, then they must be passed from the project to the scheduler as attributes of the episode. It is likely, however, that in practical applications such examples will be rare. What may be a more likely case is that the episode activity type will be a useful attribute within the mode choice probability calculations (e.g., a grocery shopping trip may be more likely to require a car given that it may involve needing to transport many bags of groceries, relative to a trip to work which typically involves, at most transporting a brief case, etc.), and so will need to be included in the episode attributes that are passed to the scheduler. Episode Location Episode locations are determined in a variety of ways. "Usual" episode locations for activities such as work and school are determined by longer-term decision processes that lie outside the current discussion. This information clearly resides within the project, which is the manifestation of the longer-term decision (i.e., to hold a job or to go to school). Many other activity locations are also fixed by longer-term decision processes. These include most locations for doctors, places of worship, clubs, etc. In most practical models, however, these locations either are not known a priori and/or the activity type definitions may not be sufficiently precise to fully specify the detailed type of location being visited (e.g., "personal business" might involve a trip to a bank, a dentist, or many other service establishments). Thus, it may be necessary to develop a "location choice" model for such episodes. Such a model may include "gravity-type" factors (i.e., ceteris paribus one may go to a dentist whose office is closer to one's home and/or work place than further away), but it does not depend upon day-to-day scheduling dynamics. Such a model is a surrogate for the longer-term, unobserved, not explicitly modelled choice of the dentist, etc. It clearly resides within the project. Depending on the project (and, perhaps, the activity type in question), such a model might be run once (or at discrete intervals, as determined by the stress/resource manager controlling the project) to determine the episode location, which then is treated as a "fixed" location that can be passed to the activity scheduler as an episode attribute.

574 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Social networks of family and friends are also the outcome of longer-term processes that may or may not be endogenously represented within the model. In either case, the "anchor points" (home, school workplace, etc.) within these networks are also "fixed" within the short run scheduling process. As with the location of one's doctor, this spatial distribution of the social network is determined within the social project (and, perhaps, a "family project"). Truly dynamic location choices, in which location is based on trip-chain considerations (e.g., stop at this store, rather than some other, because it is "on the way" to some other previously planned activity), are probably relatively rare in practice and probably largely "shopping" related. In particular, it can be argued that this schedule-dependent location choice will most typically occur in situations in which one is relatively indifferent among a set of feasible locations (any hardware store will do to purchase some nails; all fast-food burger outlets are more or less the same; etc.), and so trip-chaining efficiencies or "scheduling convenience" may dominate the location choice. Thus, the potential for multiple feasible locations needs to be allowed for. It is assumed that the project will generate for each episode a non-empty set of feasible locations, LocationSet. In the majority of cases, it is expected that this set will consist of exactly one location, which either is fixed within the scheduling period (e.g., one's place of work or school) or is determined within the project through a suitable location choice model. For some activities, however, multiple locations may be identified. In such cases, the scheduler will evaluate episode insertion for each feasible location and choose one insertion location combination, based on the schedule insertion rules under which it is operating. In the previous section, the issue of when a location is chosen was raised. For single location cases, this clearly occurs at the time of episode generation (if I am going to the doctor's, I know where the doctor is located). For multiple locations, as a practical matter it makes sense to also make the location decision at the time of scheduling, since: we do not observe "real" location choice timings; it is likely in most cases that intervening scheduling dynamics will not change the validity of this choice; and, if dynamics do intervene, a new location can always be selected as part of the rescheduling process. Episode Participants Many episodes involve multiple participants. These include joint household activities, social episodes with non-household members, serve-dependent activities, etc. All such episodes require collaboration among the persons involved to determine the joint scheduling of the episode. Intra- and inter-household joint scheduling is not considered in this paper, except to note that inter-household social episodes are assumed to be determined by the "social project", while intra-household joint episodes are assumed to be dealt with by various household-level projects within the household DMU.

EPISODE UTILITY It is a reasonable proposition to assume that people act in ways that bring them benefit or utility. Most behaviour is motivated by the desire to fulfil various needs (Maslow, 1970),

Project-based activity scheduling for person agents 575 especially, arguably, actions that involve out-of-home travel (since travel is a wilful act requiring the conscious expenditure of resources; presumably there is expectation of some "reward" for this effort). This is not to say that all activity/travel decisions are perfectly rational in nature. But in constructing our daily lives we are faced with decisions: we can do this or that today, but not both; we can spend more time on this or less time on that; we can do this in the morning or in the afternoon. As a practical matter, it seems virtually inevitable that a utility-based approach is required to build operational models of these trade-offs. In microeconomic theory utility is clearly associated with the consumption of goods (and services). In the case of activity episodes this concept seems to extend to the notion that we "consume" time and derive utility thereby (Jara-Diaz, 1998). This implies that activity episode utility is a function of episode duration, and, given usual rational decision-making assumptions, that observed durations of competing activities result from trade-offs of their marginal utilities. Up to a point this is plausible. But it is also arguable that the situation is more complicated than this, for a variety of interconnecting reasons. First, many activity episodes are of essentially fixed duration (e.g., formal school hours) and/or are not under the direct control of the person engaged in the activity (delay in waiting in a doctor's office). Duration is determined in such cases by the process (project) generating the episode, not by any explicit utility-based calculation. Episodes possess utility, and this utility may even vary with duration (e.g., additional income from working the overtime shift), but this utility does not result from a utility optimisation process. Further, in many cases, utility is largely derived from the accomplishment of the act, rather than its duration. Whether it takes one or two hours to see my doctor, the benefit I receive from the visit is the same. Many activities are essentially utilitarian in nature: I paint the house because it needs painting, not because I enjoy house painting. In all such cases, utility fundamentally derives from achieving the objectives of the project rather than from the time spent within the individual episode per se. Activity scheduling is a dynamic, constrained process in which prior decisions impose schedule constraints on subsequent decisions. To the extent that prior decisions impose hard constraints (i.e., the person is unable to modify a previous decision), scheduling subsequent episodes into "available gaps" will involve discrete decisions about durations (there is a 1.5 hour gap available for shopping this afternoon, take it or leave it) rather than marginal optimising trade-offs (the "optimal duration" for this shopping trip is 1.75 hours). Further, it is strongly arguable (and empirically supportable from CHASE data) that activity scheduling is a fairly myopic process. There is little reason to believe that people generally "globally optimise" their schedules, especially during the day-to-day. Scheduling conflicts tend to be resolved by very "local", incremental adjustments to the existing provisional schedule; they do not, in general, completely re-schedule their day or week to accommodate a new episode. At the same time, "flexibility" does exist in the scheduling process. Not all schedule constraints are hard: (limited) re-scheduling of previously scheduled episodes does occur, involving modifications to durations, start times, etc. This can even include the deletion of a

576 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction previously scheduled episode in favour of a new one. "Utilitarian" activities like shopping, house painting, travel, etc. may well also have positive duration utility (i.e., I may like to shop or paint houses). So clearly utility concerns enter into dynamic scheduling decisions. Further, it is likely that utility in general derives from two components: the act itself and what it "accomplishes" (sometimes referred to as goal utility) and the participation in the act, which is a function of episode attributes such as duration and start time and which can be referred to as process utility (Winston, 1987; Thomas, 1990; Axhausen and Garling, 1992). Finally, the notion that we "consume" time - i.e., that it is a "good" like food or gasoline - is also not the whole story. It is equally true that time is a resource in the same way as money is a resource that we "spend" in order to "purchase" goods and services (including activities) that provide utility. This view is actually explicit in conventional economic models, in which a "time budget" is added to the usual "money budget" and the utility from consumption (activity?) is maximised subject to these budget constraints. (Jara-Diaz, 1998) Utility-based activity scheduling models are beginning to emerge in the literature (Ashiru, et al, 2003; Bhat, 2005; Charypar and Nagel, 2003; Golob, 1981; Meister, et al., 2005; JaraDiaz, 1998; Joh, 2004; Kraan, 1997; Timmermans, et al., 2001). No consensus currently exists concerning how to specify these activity utility functions, nor how to apply them within the scheduling process. The intent of the foregoing discussion is not to resolve these very deep issues, but rather to simply contribute to their debate, with emphasis on their relevance within an event-driven activity/travel simulator. Resolution of how to "handle" utility calculations is, in fact, central to the project-scheduler interface. Utility calculations would logically seem to "belong" within the project, since it is the project that is the repository for the detailed understanding of individual activities. To the extent, however, that episode scheduling requires utility-based trade-offs on the part of the scheduler, this implies that the scheduler must also have "access" to utility functions for each activity. A pertinent analogy exists between the scheduler agent and various forms of "market agent" that often are used in microsimulation models of buy-sell market interactions (see, for example, Abraham and Hunt (2003) or Salvini and Miller (2005)). The market agent is an abstract, high-level agent that typically collects buy and sell offers from active agents in the market and determines which buy bids will be accepted or matched with which sell offers. In such a formulation, buyers and sellers do not directly interact or require explicit, detailed knowledge of one another. A scheduler can be thought of as a sub-class of sorts of the market agent in that it is a higher-level agent that accepts "bids" from lower-level projects for available time slots in the provisional schedule. The parallels of the market process to activity scheduling are very direct. In this case, the "good" is an activity episode that is "bid" by a project for "acceptance" by the scheduler, which "offers" time slots for use. The key episode attributes in this "transaction" are the episode duration, start time and location. The "bid price" being offered by the project is the episode utility. The resource unit being "exchanged" is effectively the time being offered to spend on the episode. What is really being "exchanged", however, is utility: the project is "offering" a certain amount of utility. The project qua "independent" agent wants to maximise

Project-based activity scheduling for person agents 577 its "internal" utility by having the episode offer accepted. The scheduler as a representative for the person agent as a whole will accept this offer if it is (a) feasible and (b) contributes to the maximisation (or at least enhancement) of the person's overall utility. Nominal episode attributes (or feasible ranges of these attributes) are generated by the project as part of the episode generation (offer triggering) process. Nominal time slot attributes are automatically and continuously "offered" by the scheduler in terms of current gaps in the provisional schedule (i.e., time periods within the schedule for which nothing is currently planned). The scheduling process involves searching over the set of "time slot offers" for possible "matches" with the offered episode. A successful "match" involves successfully inserting the offered episode into the provisional schedule (thereby changing the set of "time slot offers" available to the next episode that is generated by the projects). The scheduled episode attributes that emerge from this matching process can be different from the nominal attribute values originally generated by the project, as determined by the scheduling process. Given the event-driven nature of the proposed model, in which activity episodes are generated and considered for scheduling one at a time, projects compete with each other for time slots in two ways. First, given overall "scheduling pressure" (e.g., a person may be "very busy" and have limited scope for accepting new episodes) or other "strategic" scheduling concerns (e.g., blocks of time might be protected in an anticipation of the generation of subsequent "important" activities) that episode utilities may have to exceed some "threshold" to be considered for scheduling. This threshold might well be context dependent. It represents an implicit form of inter-project competition, since it means that only more highly valued episodes will be scheduled. Second, rescheduling involves the reconsideration of previously assigned episodes within the provisional schedule. This means that "time slot offers" can include overlaps with currently scheduled episodes in additional to currently free gaps in the schedule. From the point of view of a previously scheduled episode, this involves reevaluating the original offer in the light of the new, competing episode's offer. The outcome of this is a consistent, feasible set of time slot offers to the new episode and the previously scheduled episodes that are being rescheduled. These time slot offers will presumably be "accepted" by the projects/episodes involved as long as they still meet feasibility constraints, otherwise the offer will be "rejected" and the episode will be deleted from the schedule. If more than one feasible location exists for the new episode, the impact of choosing each such location on overall episode feasibility and utility is evaluated for each time slot considered. A "best" location will be selected for each time slot (where this "best" location may well vary by time slot). Location choices for rescheduled episodes can also be reevaluated, if need be. Similarly, for each episode insertion option considered it is proposed to run the tour-based mode choice model to determine the travel episodes associated with the proposed insertion, both to ensure the feasibility of the proposed insertion and to compute the utility contribution of the travel episodes to the overall activity schedule. The process sketched above is explicitly utility-based in nature and requires that the scheduler be able to undertake utility calculations in the assessment of scheduling options. To support

578 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction this, each episode object that is instantiated by a project will include within its methods its own utility function. This utility will, in general, have three elements: a "goal" component, as defined by the generating project (EpisodeGoalUtility); a "process" component, which is a function of duration and start time; and a location-specific component (EpisodeLocationUtility(i) for the ith location in LocationSei). To determine episode process utility for a given time slot option, the scheduler will have a method of the general form GetEpisodeUtility(Duration, StartTime, Location(i)), which will request from the given episode its utility given the specified start time, duration and location.

SCHEDULING EPISODES - PUTTING THE PIECES TOGETHER Figure 2 summarises the project-based, event-driven episode scheduling process developed in this paper. Key elements of this process are: 1. A set of P projects. These projects are instantiated by longer-term processes within the person and household. They are fixed and given within a single planning period. 2. A provisional schedule. At the beginning of the planning period, the provisional schedule is initialised to the person's known skeleton schedule. At all times during the planning period a feasible, "current" provisional schedule exists. 3. At any time t within the planning period a project can generate an activity episode. 4. The scheduler agent will attempt to schedule this episode, giving the schedule as it currently exists at the time of episode generation. 5. The act of scheduling this new episode may result in the rescheduling of one or more existing episodes within the provisional schedule. This can include deleting existing episodes from the schedule. 6. If the new episode cannot be successfully scheduled, it is deleted from the system.

Project-based activity scheduling for person agents 579

t = noon, Day 2

In object-oriented programming terms, the generation of an episode involves the instantiation of an activity episode object by the project. Attributes of this episode object include: ActivityType: The episode activity type (work, shopping, etc.). This attribute may affect trip mode choice utility calculations, as well as, possibly, other scheduler-based decisions. It is also useful to be able to track episode generation/scheduling/execution by activity type for model testing and validation purposes, as well as for general model output/reporting purposes. {MinDuration, MaxDuration): Minimum and maximum nominal durations. The duration assigned to the episode by the scheduler must be at least MinDuration long. If not, the episode is infeasible within the current schedule and is deleted from the system. (MinStart, MaxStart): Earliest and latest feasible episode start times. If a start time cannot be assigned to the episode within these bounds, the episode is deleted. (MinEnd, MaxEnd): Earliest and latest feasible episode end times. If an end time cannot be assigned to the episode within these bounds, the episode is deleted. LocationSet: Set of feasible locations to be considered for this episode (includes home for inhome activities). ParticipantSet: Set of participants (in addition to the person doing the scheduling. ModeSet: Set of intrinsically feasible modes. Intrinsic mode constraints are a function of the person (e.g., persons without a driver's licence cannot use the auto-drive mode) and, possibly,

580 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the activity. They do not include spatial/network constraints (e.g., too far to walk) or scheduling constraints (e.g., not enough time to get there by transit), both of which are determined during tour-based mode choice calculations within the scheduler. EpisodeGoalUtility: Expected utility to be derived from the episode if it is undertaken in terms of its satisfaction of personal needs/wants/goals. Excludes any duration or start time based "process" utility components. Also excludes any location-specific utility components. EpisodeLocationUtility(i): Location-specific utility component for location i in LocationSet. Excludes any travel-based accessibility (travel time, parking availability, etc.) components. Once a project has generated an episode, it is passed to the scheduler for possible scheduling. Many scheduler processes/rule-sets are conceivable, but the general process involves searching over the time interval [MinStart, MaxStari] for an episode start time (EpisodeStart) that has a feasible duration (EpisodeDuration) and end time (EpisodeEnd). If more than one feasible start/end/duration option exists, the one that maximises the person's utility will be chosen, where this utility is the sum of the utility from the new episode being scheduled, less any utility loss due to rescheduling of existing episodes in the schedule. These utility calculations include travel mode and location effects (see below). Ceteris paribus, rescheduling will always result in utility loss (or at least no utility gain) for the rescheduled episode, since it will generally involve shortening durations and/or shifts in episode starts/ends to, at best, equally preferred times. Preference, therefore, will generally be given to solutions that are able to maximise the new episode's utility that do not require rescheduling existing episodes (i.e., insertions of the new episode into an previously unscheduled "gap" in the provisional schedule). Each discrete insertion considered (i.e., each insertion into a previously unassigned gap or between two previously scheduled episodes) must be evaluated in terms of the travel episodes associated with this activity episode in terms of: determination of their travel times, modes and contributions to the overall utility of the episode insertion. As previously discussed, this evaluation is performed at the level of the tour in which the insertion is being made. If multiple locations are specified by the project as being feasible for the new episode, then each insertion (and associated travel mode choices) must be evaluated for each feasible location, with a location being chosen for each insertion that maximises overall utility. The single biggest challenge in constructing the scheduler agent is determining the rules for rescheduling; that is, determining to what lengths the scheduler is allowed to go in order to fit a new episode into the provisional schedule. As already discussed, it is reasonable to argue that rescheduling is a localised, limited activity (i.e., that people do not re-evaluate their entire schedule every time a new episode is generated), since it is generally unlikely that significant utility gains will occur thereby (i.e., "global re-optimisation" will rarely be "worth the effort"). This is particularly the case given the highly constrained nature of many activities (I must go to work; I must pick my child up from day-care; etc.). Also, to the extent that

Project-based activity scheduling for person agents 581 episode generation reflects priority, it is arguable that earlier scheduled episodes should be given preference over ones that are generated later, again reinforcing the constrained nature of the process and the limited potential for global re-optimisation. The modelling challenge, however, is to deduce an appropriate rule set from observed behaviour to control this process. A key strength of the CHASE survey method is that it does provide us with a view of "add, delete, modify" scheduling activities, and the available CHASE datasets provide us with a relatively rich information base upon which to build a scheduling/rescheduling model. A key feature of the model design that has not yet been explicitly discussed is that unspecified in-home activities represent "default" episodes within the provisional schedule. That is, any unscheduled gap at any point in the schedule will be filled by unspecified in-home activities. This seems to be reasonable, since indeed, if something more "concrete" (go to work/school, visit friends, go shopping, etc.) is not planned, the "default" presumably is doing "something" at home. Thus, the focus of the model is on travel-generating out-of-home activities, which presumably are the ones of primary policy analysis interest and for which we tend to have more detailed information. This approach also relieves some of the burden on the project formulation of the problem to "explain" all human activity. Projects are only required to explain the generation of "major" activities that are of direct interest within the model. Note that trip-chains naturally emerge out of this formulation. The insertion of a new episode into a previous gap in the schedule automatically becomes a home-episode-home tour. Insertion of a new episode either immediately before or after a previously scheduled episode changes the previously scheduled home-episode-home tour into a home-episode-episodehome chain. And so on. In this formulation there is no need to define either "primary stops" or "primary modes" for tours. More fundamentally, the tour is literally emergent out of the scheduling process, as opposed to most current models in which tour type is a structural, organising principle of the model (i.e., "choose primary tour or activity pattern type" tends to be a common model formulation).

SUMMARY AND FUTURE WORK This paper has presented a framework for project-based, event-driven activity scheduling that is consistent with a larger unified model of short- and long-run household-based decisionmaking. Two critical agents are identified and discussed within this model: the project that generates activity episodes in the pursuit of fulfilling personal needs, wants and goals; and the scheduler that deals with the insertion of these episodes within a provisional schedule that constantly evolves over time. It has defined the attributes of episodes that the project must generate so that these episodes can be scheduled, and it has discussed key design issues, requirements and options for the methods (sub-models) to be used by the project and scheduler in episode generation and scheduling, respectively. In particular, it has discussed at some length the motivation for utility-based scheduling, hi addition, travel and location choices are integrated in a holistic way within the activity generation/scheduling process.

582 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction A prototype, rule-based activity/travel scheduler currently exists that incorporates some (but not all) of the features discussed in this paper: TASHA (Travel/Activity Scheduler for Household Agents) (Miller and Roorda, 2003). Much work is required to convert TASHA into a model that more closely approximates the conceptual model presented in this paper. Undoubtedly the two key challenges in doing this are the development of appropriate, dynamic models of episode generation and practical models of activity scheduling that incorporate appropriate combinations of utility-based and rule-based calculations. The data available to support the development of such models consists of three types. First, CHASE and other "process" survey data obtained from the on-going Toronto and Quebec City household panel surveys provide direct information concerning episode generation and scheduling dynamics and will be used extensively in the development of the models of these processes. Second, activity diary and time-use surveys provide valuable information about realised activity patterns. While not providing insights into scheduling dynamics per se, they do provide useful information about (successful) episode generation rates and realised episode attribute distributions that can be used to augment our more detailed but much smaller sample panel data and to help validate models developed with the more detailed, but limited sample data. Activity diary data also provide valuable information for the development of location and mode choice models. Finally, very large, repeated cross-section travel surveys exist for the Greater Toronto Area for the years 1986, 1991, 1996 and 2001. Each survey includes trip information for up to 100,000 households. Thus, although only trip-based and possessing other limitations, these surveys provide an enormous, detailed and statistically sound set of snapshots of realised travel behaviour within the GTA over a 15 year time period. These data will contribute to the modelling of episode generation rates, location choices and travel mode choice. In addition, they provide a comprehensive base for model validation.

ACKNOWLEDGEMENTS This research was funded by a Major Collaborative Project Initiative (MCRI) grant from the Social Sciences and Humanities Research Council (Canada). The significant contributions to the concepts presented in this paper by Juan Antonio Carrasco, K.M. Nurul Habib and Matthew Roorda are gratefully acknowledged.

REFERENCES Abraham, J. and J.D. Hunt (2003) "Dynamic Submodel Integration Using an Offer Accept Discrete Event Simulation", presented at the Iff1 International Conference on Travel Behaviour Research, Lucerne, Switzerland, August 10-15. Arentze, T., F. Hofman and H.J.P. Timmermans (2003) "Re-Induction of Albatross' Decision Rules Using Pooled Activity-Travel Diary Data and an Extended Set of Land Use and Costs-Related Condition States", presented at the 82nd Annual Meeting of the Transportation Research Board, Washington, D.C., January.

Project-based activity scheduling for person agents 583 Arentze, T. and H.J.P. Timmermans (2000) Albatross: A Learning-Based Transportation Oriented Simulation System, European Institute of Retailing and Services Studies, Eindhoven, The Netherlands. Axhausen, K. (1998) "Can We Ever Obtain the Data We Would Like to Have?", in T. Girling, T. Laitila and K. Westin (eds.) Theoretical Foundations of Travel Choice Modelling, Oxford: Pergamon Press, pp. 305-324. Axhausen, K. and T. Girling (1992) "Activity-Based Approaches to Travel Analysis: Conceptual Frameworks, Models and Research Problems", Transport Reviews, Vol. 12, No. 4, pp. 323-341. Bhat, C.R. (2005) "A Multiple Discrete-Continuous Extreme Value Model: Formulation and Application to Discretionary Time-Use Decisions", to be presented at the 84th Annual Meeting of the Transportation Research Board, Washington, D.C., January 9-13. Bhat, C.R. and F.S. Koppelman (1999) "Activity-Based Modeling of Travel Demand" in R. Hall (ed.) Handbook of Transportation Science, Boston: Kluwer, 1999. Bhat, C.R., S. Srinivasan and J. Guo (2001) Activity Based Travel Demand Analysis for Metropolitan Areas in Texas: Model Components and Mathematical Formulations, Product 4080-2, prepared for the Texas Department of Transportation. Bowman, J.L., M. Bradley, Y. Shiftan, T.K. Lawton and M.E. Ben-Akiva (1998) "Demonstration of an Activity-Based Model System for Portland", presented at the 8th World Conference on Transport Research, Antwerp. Charypar, D. and K. Nagel (2003) "Generating Complete All-Day Activity Plans with Genetic Algorithms", presented at the Iff1 International Conference on Travel Behaviour Research, Lucerne, Switzerland, August 10-15. Doherty, S.T. and E.J. Miller (2000) "Tracing the Household Activity Scheduling Process Using One-Week Computer-Based Survey", Transportation, Vol. 27, pp. 15-91. Doherty, S.T., E.J. Miller, K.W. Axhausen and T. Girling (2001) "A Conceptual Model of the Weekly Household Activity-Travel Scheduling Process", in E. Stern, I. Salomon and P. Bovy (eds) Travel Behaviour: Patterns, Implications and Modelling, Cheltenham U.K.: Elgar Publishing Ltd., pp. 148-165. Doherty, S.T., E. Nemeth, M.J. Roorda and E.J. Miller (2005) "Design and Assessment of the Toronto Area Computerized Household Activity Scheduling Survey", forthcoming, Transportation Research Records, Journal of the Transportation Research Board. Fellendorf, T.H., U. Heidi and W. Scherr (1997) "PTV Vision: Activity-Based Demand Forecasting in Daily Practice", in D.G. Ettema and H.J.P. Timmermans (eds.) ActivityBased Approaches to Activity Analysis, Oxford: Pergammon Press, pp. 55-72. Fosgerau, M. (1998) "PETRA: An Activity Based Approach to Travel Demand Analysis", presented at the 8th World Conference on Transport Research, Antwerp. Fujii, S., R. Kitamura and T. Monma (1998) "A Utility-Based Micro-Simulation Model System of Individuals' Activity-Travel Patterns", presented at the 77th Annual Meeting of the Transportation Research Board, Washigton, D.C., January 11-15. Jara-Diaz, S.R. (1998) "Time and Income in Travel Demand, Towards a Microeconomic Activity Framework", in T. Girling, T. Laitila and K. Westin (eds.) Theoretical Foundations of Travel Choice Modelling, Oxford: Pergamon Press, pp. 51-73.

584 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Joh, C-H (2004) Measuring and Predicting Adaptation in Multidimensional Activity-Travel Patterns, Ph.D. thesis, Eindhoven: Technische Universiteit Eindhoven. Lancaster, K.j. (1966) "A New Approach to Consumer Theory", Journal of Political Economy, Vol. 34, pp. 132-157. Litwin, M.S. and EJ. Miller (2004) "Event-Driven Time-Series Based Dynamic Model of Decision Making Processes", presented at the 82nd Annual Meeting of the Transportation Research Board, Washington, D.C., January. Maslow, A.H. (1970) Motivation and Personality, Second Ed,, Harper & Row, New York. Meister, K., M. Frick and K.W. Axhausen (2005) "A GA-Based Household Scheduler", presented at the 84' Annual Meeting of the Transportation Research Board, Washington, D.C., January 9-13. Miller, EJ. (2005a) "Propositions for Modelling Household Decision-Making", accepted for publication in Proceedings of the International Colloquium on The Behavioural Foundations of Integrated Land-use and Transportation Models: Assumptions and New Conceptual Frameworks, M. Lee-Gosselin and S.T. Doherty (eds). Miller, EJ. (2005b) "An Integrated Framework for Modelling Short- and Long-Run Household Decision-Making", presented at the EIRASS Conference on Progress in Activity-Based Analysis, Maastricht, The Netherlands May 28-31. Miller, EJ. and M.J. Roorda (2003) "A Prototype Model of Household Activity/Travel Scheduling", Transportation Research Record, Journal of the Transportation Research Board,~No. 1831, pp. 114-121. Miller, E.J., M.J. Roorda and J.A. Carrasco (2005) "A Tour-Based Model of Travel Mode Choice", forthcoming Transportation. Roorda, M.J. and EJ. Miller (2004) "Toronto Activity Panel Survey - Demonstrating the Benefits of a Multiple Instrument Survey Panel", presented at the 7th International Conference on Travel Survey Methods", Costa Rica, August. Salvini, P.A. and EJ. Miller (2005) "ILUTE: An Operational Prototype of a Comprehensive Microsimulation Model of Urban Systems", forthcoming, Network and Spatial Economics. Timmermans, H.J.P., T. Arentze and C-H Joh (2001) "Modeling Effects of Anticipated Time Pressure on Execution of Activity Programs", Transportation Research Record, Journal of the Transportation Research Board, No. 1752, pp. 8-15. Thomas, J.F. (1990) "Rethinking Utility Theory", Journal of Behavioral Economics, 19, 2. Winston, G.C. (1987) "Activity Choice: A New Approach to Economic Behaviour", Journal of Economic Behaviour and Organization, Vol. 8, pp. 567-588.

30

MODELLING COMMERCIAL VEHICLE EMPTY TRIPS: THEORY AND APPLICATION Jose Holguin-Veras, Juan C. Zorrilla, and Ellen Thorson, Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute

INTRODUCTION The 21st century has been characterized by the increasing role of information technology in everyday life. Modern computer systems allow for information to be transferred faster and safer through the Internet, thus making it convenient for consumers to shop online in the comfort of their homes. As this trend continues, the demand for lighter, higher-value goods increases, since they are likely to have a lower cost to the consumer on the Internet. This increase in demand, combined with population growth and other factors, is resulting in an increase in the amount of freight that has to be transported, particularly by truck. The Oregon State Department of Transportation, as well as other ports and institutions all over the United States, expect their freight volumes to more than double by the year 2040, a rate higher than the projected population growth for that state (Metro, 2001). According to the Federal Highway Administration, about 70% of small package freight is transported by trucks; resulting in a contribution by the trucking industry of 48 billion dollars to the national GDP in the year 1998, a 130% increase from 1990 (USDOT, 1999). In light of these trends, it is clear that better, more efficient forms freight transportation systems are essential. A hindering factor to efficient ground transportation is the larger commercial vehicle traffic that increases congestion on the roads. This results in a significant increase in air and noise pollution as well as the number of dangerous traffic accidents involving trucks. A study by the Organization for Economic Co-Operation and Development revealed that in some European countries noise from truck traffic resulted in a property loss equivalent to nearly .4% of the national GDP. The study also showed that individuals (aggregated) are willing to pay up to .65% of the GDP for a significant reduction in noise (Hecht, 1997). The increase in truck traffic, along with the accompanying increase in externalities, puts pressure on the trucking industry and on Metropolitan Planning Organizations (MPOs). The

586 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction trucking industry will need to handle larger delivery volumes, facing lower revenues due to the level of competition and increasingly stringent regulations for externalities. MPOs will have to improve their planning processes in order to accommodate the ever increasing demand for goods transportation. In order to do this, the organizations need more efficient demand models than the ones currently in use. In many cases, MPOs use adaptations of passenger car models to estimate freight demand in the area. Although these simplistic approaches can sometimes provide rough estimates, they are fundamentally flawed since they do not capture the key dynamics of freight phenomena. One recurrent misconception involves equating commercial vehicle traffic to freight transportation demand, when in fact commercial vehicle traffic is an expression of how the trucking industry organizes itself to satisfy the demand, i.e., the commodity flows. As a result, a significant number of freight demand models focus on modelling vehicle-trips, which is analogous to attempting to model passenger demand for a transit system by modelling the flow of buses and disregarding the passenger flows. Currently there are two major platforms for modelling freight transportation demand: vehicletrip and commodity based models (Ogden, 1978). Vehicle-trip models focus on modelling the actual number of vehicle trips, which has some practical advantages. Among them are the relative ease and high-quality with which data can be obtained due to an increasing number of Intelligent Transportation Systems. Also, since the model focuses on vehicle trips it has no problem generating the number of empty trips between regions. However, these models have two fundamental limitations. The first one is that these models cannot be applied to multimodal transportation because the vehicle-trip is in itself the result of a mode choice and the selection process is not represented in the data (Holguin-Veras, 2000, 2002). Furthermore, since the models assume that the vehicle is the unit of demand, as opposed to the commodity being transported, the model neglects the economic characteristics of the shipment that have been found to play a significant role in the majority of choice processes in the trucking industry (e.g., Holguin-Veras, 2002). Commodity based models, as the name points out, focus on modelling the flow of goods from one region to the other (measured in a unit of weight). Since the commodities are the unit of demand, the modeler can capture the underlying factors that determine freight movement, such as value, weight, and volume. In this platform, the loaded trips are estimated by dividing the total flow from one region to the other by an average payload from all loaded trucks. The problem with commodity-based models is that they are unable to model empty trips, which can make up about 30 to 50 percent of the total trips in a region. This occurs since the commodity flow in one direction determines the loaded trips, but does not bear a relationship to the number of the empty trips in the same direction. To resolve this, some complementary models have been developed, such as Hautzinger's (1984), Noortman and van Es' (1978) and Holguin-Veras and Thorson (2003a). The complementary models developed by Noortman and van Es and Holguin-Veras and Thorson will be described and compared in this paper. It is important to highlight that comprehensive tour based formulations in which the commodity flows and the corresponding truck trips are jointly estimated as part of a game theoretic spatial price equilibrium framework (e.g., Holguin-Veras, 2000; Thorson, Holguin-

Modelling commercial vehicle empty trips 587 Veras and Mitchell, 2004) completely eliminate the need for a sequential estimation process involving a commodity based model and empty trip models. This sequential estimation, in which a model produce estimates of commodity flows and another model estimates the corresponding empty trips, places a heavy burden on the commodity based model because the accuracy of the entire exercise depends on the correct estimation of the commodity flows. Furthermore, sequential estimation is statistically less efficient because of the use of two sets of models. However, since the state of the development of tour based formulations based on spatial price equilibrium principles is still far from being standard practice, there shall be no doubt that sequential modelling of commodity flows and empty trips is likely to be used for years to come. The empty trip models mentioned before are some of the approaches for modelling empty commercial vehicle traffic that are discussed in the paper, which also proposes some new empty trip models. The models range from simple nal've formulations to some more complex ones involving trip chains, probabilities and memory components. The performance of the alternative formulations to model empty trips is assessed by applying these models to a sample data set from the Dominican Republic (see description in Holguin-Veras, 2003), as well as a data set from Guatemala City (see Holguin-Veras and Thorson, 2003 a). The paper starts with some background information on the subject, followed by a brief description of previous developments in the area, a description of the test cases for the model, the methodology, and finally the results and conclusions.

THEORY The relationship between commercial vehicle empty trips and the commodity flows in the opposite direction seems to have been first identified by Noortman and Van Es (1978) who included it as part of the Dutch Freight Transport Model (Hautzinger, 1984). Since the original manuscripts have long been lost, few details of Noortman and van Es' work have survived besides the reference by Hautzinger (1984). The key features of these early models, as well as the others developed by the authors, are explained in this section. The notation used in the paper follows the tradition of Noortman and van Es (1978), Hautzinger (1984) and Holguin-Veras and Thorson (2003a). Notation: m,j = commodity flow between origin i and destination j dtj = distance (or any measure of trip impedance) between origin i and destination j a

ij

=

Sidy)

=

average payload (tons/trip) for loaded trips between i and j

m

ii

x,. = — = estimated number of loaded trips between i and j yij = f(xtj, dtj) = estimated number of empty trips between i and j z

ij

=x

a

+

y«

=

estimated total number of trips (loaded + empty) between i and j

588 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Naive approaches to modeling empty trips It is important to discuss, as part of the theory section of this paper, the various nai've approaches that have been and are being used to model empty trips. Among other things, this may call to the attention of both academicians and practitioners the magnitude of the errors— which are further discussed in the Empirical Evidence section—associated with these schemes. Three different approaches are discussed here. The first one represents the case in which empty trips are completely disregarded, i.e., the total flows are estimated using only the loaded trips. The second and third cases represent variants of what Hautzinger (1984) termed a "nai've proportionality model" in which the empty trips are estimated as a direct function of the commodity flows simply as: m,, Z

,=T

(i)

Where a is a suitable constant, to be determined empirically, that is intended to estimate both empty and loaded trips from the commodity flows. There are, at least, two different ways to select the value of the constant a. The first way is by minimizing the directional errors, as shown in equation (2). The second way is selecting a so that the resulting total number of trips matches the control totals, as outlined in equations (3) and (4).

e = Il[z;-^J

(2)

Ym 2 f A e= Y.z;-^-

(3)

u a" \ ) Which implies that: a"=^ Yz"

(4)

>j

Needless to say, these alternative ways to select the parameter a lead to diametrically opposite results. In the case of Equation (2), the directional error is minimized though the resulting total number of empty trips may be different to control totals. Equation (3), on the other hand, implies matching of control totals, which lead to potentially high directional errors. Modeling empty trips as a commodity Another approach that has been used is to calibrate trip distribution models to represent the empty trip flows (e.g., Tamin and Willumsen, 1984; Fernandez et al. 2003) effectively treating empty tips as a (rather unique) commodity. Although a pragmatic decision, these techniques may turn out to be quite problematic for the following reasons. First, there is no way to ensure consistency between the estimated empty and loaded trips. This is very important because the percentages of empty trips have been found to be quite stable. As a result, it is unlikely that forecasts made by completely independent models would maintain a reasonable level of consistency between the directional flows and the overall magnitude of loaded and empty trips. Second, it is unclear what kind of production and attraction explanatory variables could be used to explain empty trip flows. This is conceptually

Modelling commercial vehicle empty trips 589 important because, since empty trips are usually correlated with the commodity flows in the opposite direction, users of these techniques may find themselves using variables traditionally associated with production as attraction variables, and vice versa, in the trip distribution models. Early formulations According to Hautzinger (1984), the first empty trip model was formulated by Noortman and van Es (1978). This model assumes that the number of empty trips between i and j is a function of the commodity flow in the opposing direction, j to i, multiplied by a constant p0 that is determined empirically. This assumption along with the assumption that the average payload for trips from i to j is equal to that for trips from j to i (ay = a,-,-) leads to: z

,j =— {miJ+Pomji)

(5)

Note: This model will be referred to as NVE

Hautzinger highlighted what he considered a limitation of Noortman and van Es' model. He pointed out that the Noortman and van Es's model implies that the difference between the flow of commercial vehicles is a function of the difference between the commodity flows (which can be found by subtracting z,,- from zy). According to Hautzinger, the empirical evidence indicates that even in cases of extreme differences in the commodity flows, the vehicle flows zy and z,, tend to be equal (the data gathered by the authors and discussed elsewhere in the paper suggests otherwise). Hautzinger then developed a formulation that takes into account the base location of the commercial vehicles, as shown in equation (6). Using relative simple assumptions, Hautzinger arrived at: (Plml+pin,JI) " a ( l - ( l - / > , XI-/>,)) The parameters p, and pj could be interpreted as the probabilities of returning empty for vehicles based in i and j respectively. This model, by construction, implies that the total number of trips in both directions are equal regardless of the commodity flows. As indicated above, this model structure may be problematic in cases where its underlying assumption does not hold. Hautzinger, expanding on Noortman and van Es' ideas, hypothesized that the parameters of the empty trip models should be a function of distance and the commodity flows as shown in equation 4. Although he did not provide details, he suggested that its parameter, /t,lies in the interval [0.1, 2.3]. Pi = f(mji Ima>dij) = exp(-2(myi /m9)2) (7) The authors attempted the estimation of the parameter of Equation (7) using two different origin-destination data sets but were largely unsuccessful. The chief reason is that, in these data, there were significant imbalances in the directional commodity flows which lead to very

590 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction large values of the ratio mji/mij that, when squared, could not be computed by the exponential function. This prevented using this model in the paper. Trip chain models The next stage in the development of empty trip models involved the use of probability and spatial interaction concepts to model empty trips as trip chains (Holguin-Veras and Thorson, 2003a). These formulations attempt to model the trip chain formation process, as opposed to Noortman and van Es' and Hautzinger's that only focus on modelling empty trips back to the home base, as part of a simplified trip chain model (from the home base i to a destination j , and back to i). Holguin-Veras and Thorson (2003a) defined order of a trip chain model as the number of destinations -in addition to the primary trip- that the model is effectively able to consider. In the context of such definition, both Noortman and van Es' and Hautzinger's models are zero-order trip chain models. The models developed by Holguin-Veras and Thorson (2003a) are based on a simplified depiction of commercial vehicle trip chains as part of a first order trip chain model. As the name implies these formulations are able to consider the flow of commercial vehicles, and specifically empty trips, as part of trip chains with one destination in addition to the primary trip. Although far from being realistic depictions of the higher order trip chains that take place in real life, these models are undoubtedly a step forward with respect to current methodologies. For a general framework able to consider higher order trip chains as part of a game theoretic spatial price equilibrium framework, see Holguin-Veras (2000) and Thorson, Holguin-Veras and Mitchell (2005). The flow of loaded trips from i to j can be represented as depicted in Figure la. Similarly, the flow from j to i is shown in Figure lb. The total number of trips between i and j , zy, can be assumed to be a random variable with two components: loaded trips, xy, and empty trips, yy, as shown in equation (8): E(zij) = E(xij) + E(yiJ) (8)

Modelling commercial vehicle empty trips 591

a) Loaded trips between i and j

b) Loaded trips between j and i

Legend: 0 Alternative destinations after completion of primary trip

Figure 1: Schematic of commercial vehicle trip chains The expected value of the loaded trips is equal to the commodity flow, my, divided by the average payload, ay. It is important to note that some of these trips are going to be made by commercial vehicle operators residing at zone i, while others will be made by operators that, though residing in other locations, are able to pick up loads in i as part of a trip chain. The basic formulation is shown in equation (9). Holguin-Veras and Thorson (2003a) demonstrated that, with proper assumptions, equation (9) reduces to the Noortman and van Es' model. E(ziJ) = n^ + E(yij)

(9)

Defining: x"j = Number of loaded trips from i to j as part of an nth order trip chain y~ = Number of empty trips from i to j as part of an nth order trip chain It can be shown that the total number of commercial vehicle trips between i and j , equation (9), can be transformed into equations (10) and (11) (Holguin-Veras and Thorson, 2003a). The second term of equation (11) represents the expected value of the zero-order empty trip chains, while the third term takes into account higher order trip chains.

£(z,) = ££(*," ) + i > ( ^ )

(10)

E(zu) = ^ + E(y°) + E(yl) + J^E(y;) = ^ + E(y°) + E(y\) l+ a>1 a

U

»>i

a

ij

(11) E(yv)

Assuming that: (1) the term within brackets is equal to a constant y; (2) the zero order empty trips are a function of the opposing commodity flow, as in the Noortman and van Es' model, and a probability, p, that represents the probability of a zero order trip chain; (3) empty trips between i and j depend upon the total number of vehicles arriving at the origin zone i; and (4) the likelihood of selecting j as the next destination depends upon the distance traveled; equations (12) thru (14) may be found (after Holguin-Veras and Thorson, 2003a).

592 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

E(zi]) =n^ + E{yl) +r'E{y\) a

(12)

ij

i40=—+/>—+ (i-rt/I^My> M (£//>)

(13)

or alternatively:

E{zs)=^

+P

as

^

+ rI.xu{p»U)P»(E/j))

an

(14)

h*j

Where: p = probability of a zero order trip chain / = (1 - p)y" - parameter to be determined empirically xhj = number of loaded trips from h to i P1" (/)= probability that a vehicle that came from h to i chooses j as the next destination Ph'{EI j) = conditional probability that a vehicle following the tour h-i-j does not get cargo toj P1" (/)P*' (EI j) = probability that vehicles traveling in h-i-j go empty to j A number of different specifications of the destination choice probability function were tested by Holguin-Veras and Thorson (2003a). In general terms, they assume that the probability that a vehicle that came from h to i chooses j as the destination, F*"(j), is a function of the commodity flow from j to i, /«/,-, and the trip impedance between i and j . This suggests estimating P^'Q) as a function of spatial interaction models. A number of possibilities exist with various degrees of plausibility. Some of them express the choice probabilities as a function of the previous origin h, while others only consider the interaction between zones i and j (in this case the superscript is only i). The most promising ones are shown in Table 1. The assumption in equation (15) is that the destination choice probabilities are a function of the commodity flows from zone i (my). Equations (16) and (17) take into account both the commodity flows and the trip impedances between zones (dv). Equation (18) additionally takes into account the trip impedance of any travel done previous to the trip from i to j (d/,,). Table 1: Destination choice probability functions

P'(j) = ^r-

(15)

^lie-"

(16)

P!(j)=

(17)

M

'(''r'

T.rnai.du) " i

mJd,, +dhX11 P'*'(/)Tm.id.+dJ fi« ... .

i

(18)

Modelling commercial vehicle empty trips 593 Equation (15) has the drawback that it does not take into account the impedance to the trip. For that reason, it would not be able to properly consider the fact that trip chains with short trips are more likely than with long trips. Equations (16) and (17) do take into account the trip impedance, though they are unable to consider the effect of the distance already travelled during the trip chain. The specification deemed most complete is the one in equation (18), that takes into account both the trip impedance, as well as the distance already travelled by the truck. Not surprisingly, equation (18) was found to outperform all alternative models in several test cases (Holguin-Veras and Thorson, 2003a). These assumptions lead to the models shown in Table 2. Table 2: Trip chain formulations mn

m

m

ij

P

ji

m

i1 1-V vx

ij

1 fc. M ym

a

hi

*'

e-fXdd)

(19)

P*'Vp*' J A

*-

>

(20)

.n

m

«..>-"*

P

mij{dijYli

ji

a

ii

a..

hi

(21)

i

l " "^ x

iJ

u

Pu (E' j)

(22)

Note: These models will be referred to as HVT models 1,2,3 and 4 respectively.

These trip chain formulations lead to an interesting result pertaining to the relationship between the opposing total number of vehicles between i and j . As discussed before, the Noortman and van Es' model implies that the difference between the opposing total number of vehicles, zy, and z,,-, is a function of the opposing commodity flows my and z,,- (which Hautzinger deems unrealistic); while in Hautzinger's model the total number of vehicles zy and Zjj are, by construction, always equal regardless of the commodity flows. The latter assumption, though plausible for certain cases, is far from being a general result, because there may be cases in which this assumption does not hold. By taking the difference between the expected values of opposing total number of vehicles, and assuming ay = ajj, from equation (14):

E{ziyE{zl)J~^{mi]-mji)

U U + rWxh{p%j)P%Elj))-TxpXP ii)P {Elii\

(23)

Equation (23) is an intermediate result between the Noortman and van Es' and Hautzinger's models. Its first term corresponds to the Noortman and van Es' model. The second term, representing the contribution of higher order trip chains, could be either positive or negative, thus increasing or reducing the result with respect to the Noortman and van Es' model. Because of their flexibility, these models are able to represent a wide spectrum of situations without being constrained by their mathematical construction.

594 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Further developments This section discusses the new formulations introduced in this paper, which, for the most part, are intended to relax one of the assumptions made in the original models developed by the authors (Holguin-Veras and Thorson, 2003a). As the reader may recall, the equations (19) through (22) were derived assuming that the probability of a zero order trip chain, i.e., p, was constant for all trip interchanges. However, there is compelling evidence and theoretical support to argue that p should be a function of a number of variables including trip distance and the magnitude of the opposing commodity flow m,-,-. In this section, a number of alternative formulations for p are hypothesized and tested with real life data from the two origin-destination samples available to the authors. Two cases are considered in this section, which are described next. Case I: p is a function of the opposing commodity flow A situation that may lead to a variable p involves taking into account that large values of the opposing commodity flow niji may increase the probability of undertaking a zero order trip chain. In this context, a large value of m,,- could make it more attractive to undertake a zero order trip chain because it increases the likelihood of getting a load for the return trip. A formulation that could be used to test this hypothesis is the one in equation (24): p=pytm'(24) As shown, it has two parameters po and pi and is a function of the opposing commodity flow my,. Substituting equation (24) in equation (13) leads to equation (25), which is a more general form of equation (13), to which it reduces ifp; is equal to zero. E{Zij)=^L+Py-'r^ a

+ ii-py^)y' a

ij

ji

zHhL(p»(j)p»(E/j))

(25)

h

*iahi

Case II: p is a function of trip length In real life, one would expect that—industry wide—the probability of a zero order trip chain for long trips would be smaller than for shorter trips. (It seems that both Noortman and van Es (1978) and Hautzinger (1984) realized the role of distance and hypothesized that the parameters of their models are a function of the trip length, though neither suggested formulae.) This is because of two factors: a highly competitive environment that puts downward pressure on rates and increasing marginal costs associated with longer trips. The combined effect of these forces pushes trucking companies to reduce the zero order trip chains in long trips. As a result, the less likely that empty trips will be generated. This would imply that/) should also be a function of trip length, as outlined in equation (26): (26) p = pa(\ + p\dij) This hypothesis could be captured in the model outlined in Equation (27). z z z

(,-=^+Po(l + M K v x

x

d x

(27)

,j = ij + Po ji + PoP\ ij n

(28)

a = xij + Poxji + Pi wiji

(29)

Where: dy is the distance between i and j , or any other measure of trip length; p[ and p, are parameters; and wLJi\s, the product of *_,-, and dy (equal to vehicle-miles traveled while loaded, if dy is a distance measure).

Modelling commercial vehicle empty trips 595 As shown in Equation (29), this model is an enhanced version of Noortman and van Es' model in which its parameter has been assumed to be a linear function of distance. Because of its linear form, the model could be estimated by statistical procedures (e.g., ordinary least squares, Poisson regression, Maximum Likelihood) in addition to numerical procedures. However, as shall be seen in the next section, this model has some unique features that suggest exploiting its mathematical structure as part of constrained parameter estimation procedures.

PARAMETER ESTIMATION PROCEDURES A number of different procedures have been developed to compute the optimal parameters of the trip chain models outlined in Table 2. These procedures could broadly be classified as unconstrained and constrained parameter searches. The former represents the case in which the search process is allowed to freely find the set of parameters that minimize an error function, e.g., the summation of the squared differences between observed and estimated values. In the latter case, external considerations such as making the sure the number of estimated empty trips is consistent with areawide estimates, are used to bound the parameter values. In the typical unconstrained parameter search, a numerical algorithm, e.g., golden search, is used to iteratively find the optimal values of the parameters. In some cases, closed form solutions have already been found to estimate the optimal parameters associated with unconstrained searches. Equation (30) could be used to estimate the optimal value of the parameter of the Noortman and van Es' model (Holguin-Veras and Thorson, 2003b) if the calibration empty trip matrix is known.

2>4

Where: y~ is the actual empty trip matrix, and x~ the estimated loaded trip matrix. The fundamental limitation of the unconstrained parameter procedures is that, although they are likely to yield the lowest estimation errors, they do not provide any guarantees that the resulting total number of empty trips is consistent with area wide estimates. To overcome this limitation, the authors suggest the use of constrained parameter search procedures. The concept behind a constrained search is relatively simple and is based on the observation that the percentage of empty trips is fairly stable across the world. Numerous studies (e.g., Ogden, 1992; Holguin-Veras and Thorson, 2003a, Wilbur Smith, 1969) have found out that empty trips represent between 30-40% of commercial vehicle trips in urban areas, and close to 50% in intercity transportation. This suggests that a good way to ensure the quality of the model's estimates is to constrain its parameters to match the total number of empty trips produced. Such constrained parameters can be found by analyzing the total number of trips, both loaded and empty, estimated by the different models. The total number of trips from the Noortman and van Es' model is:

596 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

z

' =2X =2X+2>* =2X+PTXM i.j

i.j

i.j

i,j

=X' +PX' =

x +r

'

(31)

'J

Z*, X* and Y* represent the total number of trips in the area, the total number of loaded trips, and the total number of empty trips respectively. If the percentage of empty trips, Pe, is known, the parameter/? of Noortman and van Es' model can be found from equation (31) as:

P

e

= ^ = FX' . - ^ - ^ p - ^ X +Y

X +pX

\+p

(32)

\-Pe

Closed form solutions for the constrained parameters of the models presented in Table 1 exist only for the simplest cases. Among them, it is worthwhile to highlight the following case. If the probabilities P(E/j) are constant and equal to P(E , a closed form solution can be found from equation (14) by taking summations over i and j . 2' =LE{zlJ)=z\^L

+ p'^L + yI.xlu{p''i(j)PM(E/j))\

i.j [ CL;j

dji

>>*i

Z" =X'+pX'+yX*P(EI») Thus: Y' pX"+yX'P{EI.) p = " X'+Y" X' +pX' +yX'P(EI»)

(33)

J

(34) p + yPJE/*) \ + p + yP(EI»)

Equation (35) has some interesting implications. The first one is that the percentage of empty trips depends solely on the values of the parameters p, and the product yP(E/»). The parameter p, as it does in equation (32), determines the amount of empty trips that are estimated as part of zero-order trip chains. It is also clear that a successful calibration would result in the parameter y capturing any errors in the previous estimation of P(E . The parameters of the destination choice models play no role whatsoever in the amount of trips produced by the model. Their key role is to determine the split of the empty trips according to the potential destinations. Constraining the parameter search reduces the number of parameters by one, and it simplifies the parameter search process. As indicated before, if the model only has one parameter, e.g., Noortman and van Es', and the percentage of empty trips is known or could be estimated, a reasonable estimate of the parameter could be made using equation (32). The limited empirical evidence available indicates that, provided the percentage of empty trips is accurately known, the difference between a model with the optimal (unconstrained) parameters and the Noortman and van Es' model with parameter estimated using equation (32) is relatively small. Holguin-Veras and Thorson (2003a) estimated that the RMSE is within 3% (intercity) and 6% (suburban) of the optimal values. The parameter values were found to be within 25% (intercity) and 8% (suburban) of the optimal values. The model shown in Equation (29) also leads to interesting developments. From Equation (29), one can find the estimated empty trips as: >>ij = Poxji

+ P\

w

(36)

i/«

Which implies that: Y' =ZyiJ=PoI.xiJ +PXdijxjl=pX+PlW'L 'J

ij

i,j

(37)

Modelling commercial vehicle empty trips 597 and Z*=Zzij=ZxiJ+Zyij=X'+Y'=X'

+p0X'+PlW'L

(38)

>J ij

i'j

Where: W*L is the total vehicle-miles traveled by the trucks while loaded with cargoes (if distance is used as the trip length measure) Since estimates of vehicle-miles traveled are routinely produced by public agencies, e.g., U.S. Census Bureau, (1997), Equation (38) opens the door for new avenues of constrained parameter estimation. Assuming that the percentage of empty trips, and the vehicle-miles traveled are known for both loaded and empty runs (labeled W*L and W'E respectively) the parameters of the model could be estimated by solving the following system of equations. Derivation of the first equation: From Equation (32): Y' pnX" + p,W',

P=

—r = —P-

.

L

,

(39)

X +Y X +p0X +plWL The following could be obtained: Po(l-Pe)X'+Pl(\-PeW;=PeX' and

(40)

P^X'+P^W:-X'

(41)

p Substituting p = — — from Equation (32) results in: ^X' + L=X' P P Or, alternatively:

(42)

( ^ - \)X" +^W"L=Q P P Derivation of the second equation: Since:

(43)

W'E = Z d9ys = I d9 (PoXji +Pl dJiXji) = E (PodijXji +Pl dlXji) p0W'L+PlS'=W; Where: S'

(44) (45)

=T.d2jiXji

The system of equations formed by equations (42) and (45) becomes: ^-X" L=X' P P PoW'L+PlS'=W; Solving it using Cramer's Rule yields:

(46) (47)

598 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

pX'S'-W'W* ft=

w

Pl

(48)

_W'EX'-pX'W'L _{WE-pW'L)x' X'S'-(W'Lf X'S'-(W"Lf

p Where p = —e— 1-^ It is interesting to observe that the optimal (constrained) parameters po and pi are a function of the parameter p associated with the constrained estimation of the Noortman and van Es' model, shown in equation (32). As shown in Equations (48) and (49), increasing values of p (produced by increasing values of Pe) increase the values of po while reducing;?/. This reflects the fact that the chief role of po is to capture the relationship between empty trips and opposing commodity flows; while that of pi is to capture the contribution of vehicle-miles to empty trips (which necessitates that, for a fixed number of vehicle miles, the more trips there are the less the contribution of the vehicle-miles to empty trips).

EMPIRICAL EVIDENCE This section provides a summary of the key empirical results pertaining to the basic assumptions underlying empty trip models, and the overall performance of the various approaches described in this paper. The data used in this section correspond to two major origin-destination (OD) samples carried out by the first author in Dominican Republic (1981) and Guatemala (1999), in which estimates of commodity flows, vehicle trips (loaded and empty) were produced. Descriptions of these datasets can be found at Holguin-Veras (2003) and Holguin-Veras and Thorson (2003a). hi the Guatemala project, roadside OD interviews were conducted along with classified traffic counts. The questionnaire obtained information about the time of the interview, vehicle type, origin, destination, commodity type, load factor, type of trucking company, and shipment size. The traffic counts were used to expand the sample of 5276 observations by time of day and vehicle type. The overall expansion factor was 6.476. The authors were interested in including a data set representing U.S. conditions but were not successful in finding one with both commodity flow and vehicle-trip (loaded and empty) estimates. About the underlying assumptions Most empty trip models are working under the assumption that empty trips are correlated, to a certain degree, with the opposing commodity flows. Although conceptually acceptable, this idea has never been statistically tested. The analyses conducted by the authors of two different origin-destination samples indicate the presence of a significant correlation (i.e., .78 and .82 for Dominican Republic and Guatemala respectively) between empty trips and the opposing commodity flows. Another assumption that up to now has lacked empirical evidence is related to the relationship between the absolute value of the difference between the opposing total vehicle flows, i.e., \ztj-

Modelling commercial vehicle empty trips 599 Zji\ vis a vis the difference between the opposing commodity flows i.e., \my-mji\. Figure 2 shows the results corresponding to the data available to the authors. As the reader may recall, Hautzinger (1984) criticized the Noortman and van Es' model because the latter model implies that the difference between the opposing (total) commercial vehicle is a function of the opposing commodity flows. Figures 2 and 3 show scatterplots of the absolute values of the differences of total traffic, i.e., \zirzy\ and the corresponding absolute values of the differences in commodity flows, i.e., \my-mji\. As shown, there is a significant correlation between them (i.e., 0.85 for the Guatemala data and 0.78 for the Dominican Republic; 0.75 and 0.68 respectively, after removing outliers) which suggests that indeed the original assumption of Noortman and van Es is a fairly good assumption. It is worthwhile to mention that the variance of \zij-Zji\ increases with \my-niji\ suggesting a heteroscedastic phenomenon, and the need to use models that explicitly deal with this problem.

Figure 2: Absolute Difference in Opposing Traffic Flow vs. Absolute Difference in Opposing Commodity Flow (|Zij-Zji| vs |mij-mji|)- Guatemala (1999)

600 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 3: Absolute Difference in Opposing Traffic Flow vs. Absolute Difference in Opposing Commodity Flow (|Zij-Zji| vs |mij-mji|)- Dominican Republic (1981) Model performance The performance implications of the alternative empty trip formulations were analyzed by Holguin-Veras and Thorson (2003a, 2003b) using a real life data set from Guatemala City and experimental economics data. Two different performance measures are of interest: the total error (estimated by the Root Mean Squared Errors, or RMSE, of the total flow) and the directional error (the error in the estimate for a given origin-destination traffic). Their findings suggest that not modelling empty trips increases the RMSE of the total flows by an average of 57.24% for intercity movements (with 31.59% empty trips) to 83.58% for suburban movements (with 35.94% empty trips), with respect to the best empty trip model. An analysis of the statistical distributions of the relative directional errors for the non-zero OD pairs in the experimental economics data indicated that the Naive approaches had higher probabilities of producing large errors, while the more sophisticated HVT models had higher probabilities of producing accurate results. Figure 4 shows the error distributions for the second Naive approach (equation (3) in this paper) and the most complex HVT model developed in Holguin-Veras and Thorson (2003a) (HVT 5 which is described by equation (22) in this paper). This figure shows that the second Naive approach had errors larger than |50%| in 27% of the cases, while HVT 5 had errors this large in only 5% of the cases. With the HVT 5 model, almost 80% of the cases had errors less than |10%|, while the Naive approach had errors this small in only about 45% of the cases. In addition, no case had an error larger than |100%| with the HVT 5 model, while, with the naive approach, 8% of the cases had errors this large, including some approaching 290%.

Modelling commercial vehicle empty trips 601

Figure 4: Distribution of Directional Estimation Error (Experimental Data) The remaining naive approaches do not fare much better. Both naive approaches involving the use of empirical constants that adjust the commodity flows to compensate for the missing empty trips were found to lead to significant errors in the estimation of directional flows, with an average for all origin-destination interchanges in the range of 13-30%. Similar errors were found for the estimates of total flows. It is troublesome that, because of their mathematical construction, these formulations overestimate one directional flow while underestimating the opposite flow. These findings clearly highlight that it is important: (1) to model empty trips; and, (2) in doing so, that proper modelling methodologies be used. Tables 3 and 4 show a comparison of the different models. Four different models are considered, the model proposed by Noortman and van Es (equation #5), and models 1-3 by Holguin-Veras and Thorson (equations 19, 20 and 21). The first row of each model shows the results obtained when maintaining the parameter p as a constant, the second row shows the results of assuming p is a function of the commodity flow (equation 24), and the third row presents the statistics after assuming the parameter is a function of the trip length (equation 26). The fourth and fifth columns (pO and pi) are the parameters of the p function, note that when p is fixed pO = p; Gamma and Beta are the other parameters involved in the HolguinVeras and Thorsons' formulations.

602 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction The error presented is the Sum of Squared Differences (SSD), that is, the square of the difference between the actual number of trips and the estimated ones. The Root Mean Squared Error (RMSE) is also presented, this value should give an idea of the error on a zonal basis. The percent difference between RMSE is also presented, this should be used as a comparison between models. The percents are estimated in relation to the best model, it is the difference between the error of the model in that row and the best model, divided by the error corresponding to the best model. Table 3 corresponds to the data from Guatemala (1999), while Table 4 presents the results for Dominican Republic (1981). Table 3: Model comparison with Guatemala data (1999) Model characteristics: Model parameters Performance measures pO pi Gamma, Beta Error RMSE % Diff Model Dest. Choice p function 0.3880 Constant 0.082 18.75% 4,799 None NVE 0.3080 8.00E-04 Commodity 0.079 14.35% 4,450 None — — 0.3980 -1.40E-04 Distance 0.082 18.64% 4,790 None — 0.2935 0.7171 0.077 10.79% 4,177 HVT#1 mij Constant 0.1607 1.69E-03 0.8027 0.070 3,465 0.90% Commodity mij 0.076 10.06% 0.3173 -1.30E-03 0.7450 4,122 mij Distance 0.2885 0.7424 0.0030 4,148 0.077 10.40% HVT#2 mij exp(bdij) Constant 0.1561 1.72E-03 0.8237 0.0025 3,433 0.070 mij exp(bdij) Commodity 0.44% 0.076 9.99% 0.3112 1.11E-03 0.7528 0.0013 4,117 mij exp(bdij) Distance 3,992 Constant 0.075 0.2706 8.31% HVT#3 mij dijAb 0.9158 0.4967 A 3,403 Commodity 0.00% 0.069 0.1544 1.69E-03 0.8905 0.2405 mij dij b Distance 8.28% 3,990 0.075 0.2759 -2.78E-04 0.9119 0.4778 mij dijAb Note: the optimal model for each group is shown in bold letters. The percentage difference in the error (% Diff) is computed with respect to the overall best model. NVE refers to the model proposed by Noortman and van Es (equation 5) HVT refers to the models proposed by Holguin-Veras and Thorson (equations 19, 20 and 21)

Table 4: Model comparison with Dominican Republic data (1981) Model characteristics: Model Dest. Choice p function None NVE Constant None Commodity None Distance HVT#1 mij 1 Constant mij Commodity mij Distance HVT #2 mij exp(pdij) Constant mij exp(Pdij) Commodity mij exp(Pdij) Distance HVT #3 mij dijAP Constant mij dijAP Commodity mij dijAP Distance

pO 0.5647 0.4593 0.7337 0.4840 0.3540 0.7088 0.4624 0.3241 0.6777 0.3319 0.1963 0.4710

Model parameters pi : Gamma 1.94E-04 -2.40E-03

—

115.2381 121.7443 144.3705 126.7188 3.13E-04 134.4919 -4.70E-03 146.3436 418.1770 4.47E-04 429.0625 -3.87E-03 381.1522

2.80E-04 -4.87E-03

Performance measures Error RMSE % Diff 0.832 17.54% 425,708 0.827 16.81% 420,381 0.817 15.35% 409,960 — ' 393,117 0.800 12.96% 384,222 0.791 11.67% 8.40% 0.767 — i 362,023 0.0107 383,651 0.790 11.59% 0.779 10.07% 0.0109 373,285 0.0057 358,429 7.86% 0.764 0.721 0.6872 319,295 1.80% 0.6814 308,110 0.708 0.00% 0.70% 0.713 0.6106 312,444

Beta

Note: the optimal model for each group is shown in bold letters The percentage difference in the error (% Diff) is computed with respect to the overall best model. NVE refers to the model proposed by Noortman and van Es (equation 5) HVT refers to the models proposed by Holguin-Veras and Thorson (equations 19, 20 and 21)

As shown in Tables 3 and 4, the empty trip models with a variable p consistently outperform those with a constant p by significant margins. Interestingly enough, neither of the variable p models seemed to consistently outperform the other; as shown, the model with p as a function

Modelling commercial vehicle empty trips 603 of the opposing commodity flow worked better with the Guatemala data, while the model with p as a function of distance worked better with the Dominican Republic data. It is also worthwhile to mention that the rank order of the alternative formulations is exactly the same in both cases, which adds empirical evidence about the relative merit of the alternative models. Figures 5 and 6 show the Root Mean Squared Error values for the different models and their variants.

Note: NVE stands for the model proposed by Noortman and van Es, while HVT refer to the models proposed by Holguin-Veras and Thorson (see equations 19, 20 and 21 in table 2)

Figure 5: Root Mean Squared Error of Alternative Formulations (Guatemala)

Note: in figure 6, NVE stands for the model proposed by Noortman and van Es, while HVT refer to the models proposed by Holguin-Veras and Thorson (see equations 19, 20 and 21 in table 2)

Figure 6: Root Mean Squared Error of Alternative Formulations (Dominican Republic)

604 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Econometric results Although most of the models discussed in this paper are complex non linear formulations that cannot be linearized, the models in equations (5) and (28) can be estimated using ordinary least squares (OLS) techniques. Under the typical assumptions of the OLS model, the level of significance for the parameters of equation (5) and (28) could be estimated. At the very least, this provides an indication of the significance of the various processes captured by the corresponding parameters. Guatemala data: zg = xtJ + 0.3878*;,.

R2 = 0.73

(50)

R2 = 0.73

(51)

(47.14) z.. = x.. + 0.3976*;, - 0.0001402rf,y*;; (34.84)

(-1.24)

Alternatively, z.. = *.. + 0.3976(l - 0.000353rf,y )c;,.

(52)

Dominican Republic data: z.. = *.. + 0.5647*,,.

R2 = 0.48

(53)

R2 = 0.51

(54)

(26.69) z.. = Xij + 0.7335*,,. - 0.0024037^..*,,. (19.80)

(-5.56)

Or,

zff = *,, + 0.7335(l - 0.003277rf.. jxj,

(55)

Equations (50) thru (55) have some interesting implications. The first one is that the basic assumption of empty trip models, i.e., that empty trips are correlated with the opposing commodity flow, is a strong assumption from the econometric standpoint. This corroborates the correlation analyses discussed earlier in the paper. The second implication is that, as originally assumed by Hautzinger (1984) and Noortman and Van Es (1978), distance seems to play a role, at least in some cases. As shown, the parameter of the loaded vehicle miles (dyXji) is negative in both cases, though of weak significance (around 20%) in the case of the Guatemala data.

Modelling commercial vehicle empty trips 605

PRACTICAL IMPLICATIONS In practical terms, it is obvious that explicit modelling of empty trips is a must. As seen in the previous section, the directional errors introduced by either not modelling empty trips or by using simplistic approaches, e.g., naive methodologies, are simply too large to be ignored. This section outlines a number of recommendations about when to use the different methodological approaches discussed in the paper. Case A: OD matrices of empty and loaded (or commodity flows) trips are available This case, which represents the ideal situation, enables the modeller to take advantage of the state of the art of empty trips modelling. Depending on the characteristics of the application, specifically, on the importance of trip chains, there is a fairly wide range of possibilities. If trip chains are important, which could be the case of urban goods modelling and some intercity modelling projects, the models outlined in Equations (19) to (22) are undoubtedly the best alternative, particularly Equation (22). If trip chains are not that common, the estimates produced by the models in Equations (19) thru (22) will tend to the Noortman and van Es' model, hi such a case, the modeller could either use these models or use the Noortman and van Es' model directly by computing its optimal parameter using Equation (30). Case B: No OD matrix of empty trips is available The absence of data about the actual patterns of empty trip flows is a major limitation. However, as indicated before, the empirical evidence indicates that estimating the parameter of the Noortman and van Es' model from a known (or estimated) percentage of empty trips could lead to acceptable results. Although far from being the ideal situation, this approach would be, at least, better than not modelling empty trips or using any of the naive approaches. In this context, the parameter of the Noortman and van Es' model would be estimated from Equation (32) and the resulting model applied to a known commodity flow matrix (or a loaded OD matrix) to compute empty trip flows.

CONCLUSIONS This paper attempts to provide a comprehensive discussion of the theoretical developments pertaining to commercial vehicle empty trip models, the corresponding estimation procedures, empirical evidence, and practical implications of modelling empty trips. It also describes new mathematical formulations to model commercial vehicle empty trips. In doing so, the paper expands and synthesizes the previous research on the subject. The paper introduced two novel formulations to estimate the probability p of zero order trip chains. These formulations relax the assumption of p constant used by the authors in their previous works. One model assumes that p is a function of the opposing commodity flow and

606 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the other assumes that p is a function of the trip impedance. In all the cases tested, these new models outperformed previous empty trip models by up to 10%. Formulae to compute the optimal parameters for the model in which p is a function of distance were derived in the paper. The derivations show that the optimal parameters are a function of the vehicle-miles travelled (empty and loaded) and of the optimal parameter for the Noortman and van Es' model obtained by the authors previously. The paper analyzes the empirical support for the fundamental assumptions used in almost all empty trip models. On the basis of the data available to the authors from two origindestination samples, the authors concluded that assuming that empty trips are a function of the opposing commodity flows is indeed a fairly good assumption (with correlation coefficients in the range of 0.78-0.82). The empirical analyses also indicate that the absolute value of the difference between the opposing commercial vehicle empty trips is statistically correlated with the absolute value of the opposing commodity flows. These results support the basic assumptions of the Noortman and van Es' model and those developed by the authors. In spite of this paper's contributions, there shall be no doubt that the state of the art of freight demand modelling and specifically of commercial vehicle empty trips models still needs substantial improvements. As they stand now, the current methodologies and the new ones discussed in the paper are nothing more than the beginning of the long march toward new paradigms of freight transportation modelling.

ACKNOWLEDGMENT This research was supported by the National Science Foundation CAREER Award programs grant CAREER-CMS-0092938, and by an Eisenhower Fellowship to the third author. This support is both acknowledged and appreciated.

REFERENCES Fernandez, J.E., de Cea, J. and A. Soto O. (2003) A Multi-Modal Supply-Demand Equilibrium Model for Predicting Intercity Freight Flows. Transportation Research Part B, 37, 615-640. Hautzinger, H. (1984). The Prediction of Interregional Goods Vehicle Flows: Some New Modeling Concepts. Ninth International Symposium on Transportation and Traffic Theory, VNU Science Press, 375-396. Hecht, J., (1997) "The Environmental Effects of Trade." Published by the Organization for Economic Cooperation and Development, Paris, France. Holguin-Veras, J. (2000). A Framework for an Integrative Freight Market Simulation. Invited Paper. Published by the IEEE 3rd Annual Intelligent Transportation Systems Conference ITSC-2000, Dearborn Michigan October 2000: 476-481. Holguin-Veras, J. (2002). Revealed Preference Analysis of the Commercial Vehicle Choice Process. Journal of Transportation Engineering, 128(4), 336-346.

Modelling commercial vehicle empty trips 607 Holguin-Veras, J. (2004). On the Estimation of the Maximum Efficiency of the Trucking Industry: Implications for City Logistics. In: City Logistics III (E. Taniguchi and R. Thomson, eds), pp. 123-134, Elsevier, Amsterdam. . Holguin-Veras, J. and E. Thorson. (2003a). Modeling Commercial Vehicle Empty Trips with a First Order Trip Chain Model. Transportation Research Part B, 37(2), 129-148. Holguin-Veras, J. and E. Thorson (2003b). Practical Implications of Modeling Commercial Vehicle Empty Trips. Transportation Research Record, 1833, 87-94. Metro Regional Center, 2001. Implementing the Regional Transportation Plan. Published by the Metro Council for the Portland, OR region. Noortman, HJ. and J. van Es (1978) Traffic model. Manuscript for the Dutch Freight Transport Model. Ogden, K.W. (1978). The Distribution of Truck Trips and Commodity Flow in Urban Areas: A Gravity Model Analysis, Transportation Research, Vol. 12, pp. 131-137 Ogden, K.W. (1992). Urban Goods Movement. Ashgate Publishing Limited ISBN 1-85742029-2, England. Tamin, O.Z. and L.G. Willumsen (1988). Freight Demand Model Estimation from Traffic Counts. PTRC Annual Meeting, University of Bath, England. Reprinted in Simplified Transport Demand Modeling, Juan de Dios Ortuzar, Ed. PTRC, 1992. Thorson, E., J. Holguin-Veras and J. Mitchell (2004) "An Approach for Solving the Integrative Freight Market Simulation" published by the XIII Pan American Conference of Traffic and Transportation Engineering (CD ROM) Albany NY. U.S. Census Bureau (2000) Vehicle Inventory and Use Survey: Microdata File. Issued January 2000 as CD-EC97-VIUS. USDOT (1999) "Some Broad Themes Related to US Freight Shipments." United States Department of Transportation, Office of Freight Management, Federal Highway Administration. Washington, DC. Wilbur Smith and Associates (1969) Motor Trucks in the Metropolis. Prepared by Wilbur Smith and Associates, New Haven Connecticut under commission from the Automobile Manufacturers Association, August 1969.

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31

RATIONALITY AND HETEROGENEITY IN TAXI DRIVER DECISIONS: APPLICATION OF A STOCHASTIC-PROCESS MODEL OF TAXI BEHAVIOR Ryuichi Kitamura, Department of Urban Management, Kyoto University, Kyoto, Japan, and Department of Civil and Environmental Engineering, University of California, Davis, USA; and Toshio Yoshii, Department of Urban Management, Kyoto University, Kyoto, Japan

INTRODUCTION Taxis play important roles in densely developed metropolitan areas, supplementing mass transit by providing door-to-door service with unrestricted operating hours. Taxis cruising on urban streets to pick up passengers, however, add to the road traffic, sometimes quite substantially. Wong et al. (2001) report that taxis overall constitute 25%, and in some places as much as 50 to 60%, of traffic in Hong Kong. Taxis waiting for customers at a taxi bay may cause problems as well when the capacity of the bay is inadequate and the queue of taxis extends onto the street and obstructs road traffic, a phenomenon often observed in Japanese large cities. These problems call for the development of infrastructure, regulations and strategies for taxi operation. This in turn calls for a thorough understanding of the behavior of taxi drivers and the nature of the demand for taxi trips. Little is known on either of them at this point. This study addresses the former. It can be reasonably assumed that taxi drivers attempt to maximize the fraction of occupied periods, i.e., the period when they are carrying paying passengers. When passengers are delivered to their destinations, the taxi drivers have the choice of whether to start cruising to be hailed by a new passenger, or to head for a taxi bay to wait for a passenger. Spatial dimensions are added to the decision, i.e., to which direction to cruise, or at which taxi bay to wait. Obviously these decisions involve great degrees of uncertainty. The degrees of uncertainty are further compounded because taxis compete with each other for passengers,

610 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction and taxi drivers cannot predetermine how many other taxis are cruising on the same street ahead of them, or how many taxis are already in the queue at the taxi bays to which they are heading. Availability of passengers is also uncertain. Previous studies have examined pricing and subsidy issues involving taxis (e.g., Arnott, 1996), or the distribution of taxi trips in urban networks (e.g., Yang and Wong, 1998; Wong et ah, 2001) with focus on demand-supply equilibrium. In this present study, a descriptive model of taxi operation is developed with emphasis on taxi behavior, and with taxi passenger demand treated as externally given. The focus of this study is on taxi drivers' choice between cruising to find a passenger, or waiting at a taxi bay, and how this choice influences system performance. The interplay between supply and demand for taxi service, on the other hand, is placed outside the scope of this study. This is partly because adequate demand data for empirical analysis are not available for this study. Network equilibrium is another aspect to be incorporated into the study in the future. The intent of this study is to establish a discretestate, stochastic process model framework and apply it to probe into certain aspects of taxi operation. In the stochastic process model of taxi behavior proposed in this study, the time it takes for a taxi until it loads the next passenger, whether after cruising on streets or waiting at a taxi bay, is assumed to be a function of both the intensity of passenger demand and the number of taxis cruising or waiting for passengers. Upon delivering passengers to their destinations, taxi drivers are assumed to choose between cruising and waiting at a taxi bay. This choice is assumed to be dependent on the expected excess times of the respective options (excess time is defined as the time elapsed between the unloading of a passenger and the loading of the next passenger). The validity of this assumption is examined in the study using empirical data obtained from an experiment in the Nagoya metropolitan area of Japan, in which taxis were equipped as probes to provide data on their operation. Another question addressed in the study is how the system as a whole operates when taxi drivers rationally choose between cruising and waiting so as to minimize their respective expected excess times. System performance is replicated with the discrete-state, stochastic process model assuming fixed state-to-state transition probabilities, using parameter values obtained from the probe data. This analysis has led to the finding that when individual taxi drivers operate rationally, system performance is suboptimal; i.e., when each driver rationally chooses that option, either cruising or waiting, which has a smaller expected excess time, the system-wide expected excess time is not minimized. In this sense, system performance is suboptimal when individual taxi drivers optimize their expectations. In addition, statistical analysis of the probe data indicates that taxi drivers are indeed rational and tend to choose the option with a smaller expected excess time, are biased against waiting at a taxi bay, and are heterogeneous in their propensities in the choice between cruising and waiting. This paper is organized as follows. In the next section, the structure of the discrete-state, stochastic process model system of taxi behavior is described in its general form. In the following section, taxi drivers' choice between cruising and waiting after delivering passengers is formulated as a discrete choice model. A simplified stochastic process model is then formulated based on the probe data from Nagoya, and its properties are discussed.

Rationality and heterogeneity in taxi driver decisions 611 Results of a numerical analysis are presented in the following section to show that drivers' individualistic rational choice and system optimality are not consistent with each other. Finally, taxi drivers' choice is analyzed empirically to demonstrate that their choices are indeed rational, and act against system optimality.

A STOCHASTIC PROCESS MODEL OF TAXI BEHAVIOR The stochastic process model of taxi behavior developed in this study has a state space comprising: {occupied, cruising to be hailed, transit to taxi bay, waiting at taxi bay}. "Occupied" (O) is the state where a taxi is carrying a passenger to the destination. "Cruising to be hailed" ( Q is where the taxi travels through street networks looking for a passenger to pick up. "Transit to taxi bay" (7) is where the taxi is heading to a taxi bay to wait for a passenger there. And "waiting at taxi bay" (W) is the state where the taxi is in the queue of taxis at a taxi bay, waiting to load a passenger. The taxi bay is assumed to have a capacity of K taxis. The state-to-state transition matrix associated with this state space may be defined as

or o p=c T

W

Pco

Poc PoT

o

PcT

o"

o

Pro

PTC

°

Prw

1

0

0

0

(1)

where pAB is the transition probability from state A to state B. Note that pCT applies to the situation where the driver gives up on cruising and heads for a taxi bay to wait there. Likewise, pT0 and pTC apply to presumably rare situations where the taxi picks up a passenger, or starts cruising, on its way to a taxi bay, respectively. A transition from T to O also occurs when a taxi finds the queue at a taxi bay empty and loads a passenger immediately. In the empirical analysis presented later in this paper, transitions from C to T and ones from T to C are assumed not to occur, largely because the probe data offer no information on the driver's intention and it is impossible to identify these transitions. Ignoring spatial aspects and assuming homogeneity among drivers for illustrative simplicity, the sojourn duration in state O is determined by the distribution of passengers' trip durations. The sojourn duration in state C is determined by: the intensity of taxi passenger demand, and the number of cruising taxis, which is in turn determined by the distribution of the four states. The sojourn duration in W is determined by the taxi passenger demand at the bay and the number of taxis waiting, which is again a function of the distribution of states. The behavior of the system thus involves simultaneity that the distribution of states depends on the distribution of sojourn durations in the respective states, and at the same time sojourn durations in the states depend on the distribution of states. Before discussing this simultaneity, the nature of a sojourn in each state in the system is discussed below.

612 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction A sojourn duration in state O equals the duration of the trip made to carry a passenger to its destination. Noting that the straight-line distance from a point to a randomly chosen point on a featureless plane has a gamma distribution when the impedance of spatial separation is taken into account, it is reasonable to adopt an Erlang distribution for passenger-occupied taxi trips (hereafter "loaded trips"), i.e., for the distribution of sojourn durations in state O. The rate with which a transition is made out of state O is denoted by/l 0 . Noting that an Erlang random variable is a sum of negative-exponentially distributed random variables, state O is represented in the numerical analysis of this study by a series of states which have a common negative exponential sojourn distribution. In other words, an Erlangdistributed sojourn in state O is represented as the sum of negative-exponentially distributed sojourns in the newly introduced series of states. This facilitates representation of Erlangdistributed sojourns while retaining the computational ease offered by the Markov process. For cruising, suppose the process is in steady state and the number of taxis cruising is Nc, which is the product of the total number of taxis in the system, N, and the steady-state probability of state C, nc. Suppose passengers arrive at roadside to hail a taxi according to a negative exponential distribution with parameter fj.c. Then the arrival rate of passengers for a given taxi is /Jc/Nc , yielding the mean waiting time of NC/JJC that the taxi spends cruising before finding a passenger. This equals the mean sojourn duration at state C. Therefore, the sojourn duration in C has a negative exponential distribution with parameter Z-c Me The travel time to reach a taxi bay is determined by the location where the taxi drops off its passenger and the location of the taxi bay. As is the case for state O, it would be reasonable to apply an Erlang distribution to the travel time to a taxi bay and therefore for the distribution of sojourn durations in state T. The queueing system of taxis and passengers at a taxi bay is now analyzed to determine transition probabilities, pT0, pTC, and pTW, and the distribution of sojourn durations in state W. The situation at the taxi bay can be represented as a double queue (Kendall, 1951; Sasieni, 1961) which involves the queue of taxis and the queue of passengers. A steady state solution has been derived with the assumptions that: a passenger boards a taxi instantaneously and therefore either a queue of taxis or a queue of passengers, but not both, can exist at any time point; the taxi bay has a capacity of K, i.e., no more than K taxis can wait at the bay at any time, and a taxi that arrives at the bay when K taxis are already waiting is turned away; and the arrival rate of taxis is greater than that of passengers at the bay. The arrival rate of taxis at the taxi bay is now determined using the parameters so far defined. For simplicity, it is for the moment assumed that there is only one taxi bay in the system. When there are NT taxis heading for the taxi bay, the rate of arrival at the bay is NjX,.. Let the arrival rate of passengers at the taxi bay be /iw and let \ = NT?^ be the arrival rate of

Rationality and heterogeneity in taxi driver decisions 613 taxis. It can be reasonably assumed that the arrival of passengers and that of taxis both form Poisson processes. Then, the probability that there will be k taxis waiting at the taxi bay is given as pk=(l-p)pK-",

0
(3)

where p = JJ.W /Xt (< 1). The probability that there will be m passengers waiting is given as qm={\-p)pK*m,

m>\.

(4)

Note that, by the assumption of instantaneous boarding, k > 0 implies that m = 0, and m > 0 implies k = 0. The probability, p0 = ( 1 - p)pK, implied by Eq. (3) represents the state of the process where neither a taxi nor a passenger is in the system. The probability that a taxi does not have to wait for a passenger equals the probability that m > 1, which is ?r[Tg=0] = fj(l-p)pK+"'=pK+\

(5)

The probability that a taxi is turned away because the taxi bay is full is given as pK = 1 - p. Therefore, the transition probabilities pT0, pTC, and pJW can be written as Pro = PK+l> Prc = ! " A and pm =p(\-pK).

(6)

Now, the distribution of sojourn durations in state W equals the distribution of waiting times in the queue at the taxi bay. The waiting time is zero when, at the time a taxi arrives at the taxi bay, the taxi queue is empty and the passenger queue is not. Otherwise there will be a positive waiting time. Then the conditional distribution of the waiting time of a taxi which has just joined the queue, given that it has to wait, is obtained from Pr[r?>/|J,>0]-Pr[r,>0] K-\

= ^ (Pr[less than n completions in t j arrival found n taxis in system] pn)

K-1 ( . t\m K-\

=

K-\ / .. f\rr>

(l-p)pV*'Z^Xp--=(l-p) / ,V"£ ( ^ ) ZA m\ t^m

P

tA ml

I

^

nK-m

P

}

,

\-p

K-\ I .. t\m

Recall that the passenger queue is empty when n = 0. Thus, the conditional distribution of T9 is given as

Wq(t)=Vr[Tq0]

= l^e-g^f(l-p-)/(l-^).

(8)

614 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction This result suggests that an Erlang distribution will be adequate for the sojourn durations in state W. The mean waiting time can be derived from this distribution. Alternatively, using the (\-pK),

result that the mean queue length of taxis is K

the mean waiting time of a

\-p taxi, given that it has to wait, is given as W = MK--^{1-P«)\

(9)

DRIVER DECISION This section addresses parameters poc and pOT which pertain to the decision of what action to take after delivering passengers to their destinations. The behavioral principle is introduced that taxi drivers adopt values of poc and p0T such that the expected fraction of time spent in state O will be maximized. Namely, they will cruise to pick up passengers with a certain probability, and will head for a taxi bay for the rest of the time, such that their individual load factors are maximized. With the assumption that trips from a taxi bay and those originating on street after cruising share the same distribution, maximizing the load factor is equivalent to minimizing the expected excess time. Note that excess time comprises the durations of unloaded trips and equals the time until the next transition back to state O after leaving it. Letting SA be the mean sojourn duration in state A, the expected excess trip durations are jCruising

Voc=pcoSc+pCTVOT

[Transit to Taxi Bay

V0T =ST+

PTWSW

+pTO-0 + pTCV0C = ST+ PTWSW

+ pTCVoc

which assumes that neither the distribution of cruising trip durations nor that of trip durations to a taxi bay depend on where the trip starts. Solving this simultaneous equations system, _ Pco^c

+

PCT^T

+

PCTPTW^W

1 — prT pTr V

-J ? T+ rn

"or ~ r

J

cT i PTC(PCOSC+PCTST

.

P rw w

1

+ PCTPTWSW)

~PCTPTC

It would be reasonable to assume that taxi drivers take into account a number of transient factors—e.g., weather, traffic accidents—in addition to Voc and VOT which are based on expected sojourn durations and constant transition probabilities. Let the effect of these factors be collectively expressed by a random error term for each alternative action, and suppose the attractiveness of each action can be expressed as the sum of the expected excess trip duration and the random variable as y-'oc=~'oc+eoc [UOT=~VOT+EOT

where soc and EQT are the random error terms.

,,~\

Rationality and heterogeneity in taxi driver decisions 615 Furthermore, suppose each taxi driver has a certain inclination toward or against one of the two options, and let this inclination be represented by a random error component, T' (-00 0) be the weight driver i assigns to the expected excess trip duration. The attractiveness of each alternative action is now expressed as \Uoc=-M%c+s'oc {U'OT=-M'V0T+FX'+Ti

(13)

+ el)T

Assuming that driver i chooses the option which has a larger U', and that (e'oc -e'0T) has a standard normal distribution, por = Pr[C4>[/ic] = Pr[4 c - 4 r < -Ml(V0T - Voc) + p'X+ r'] =

i

(14)

l

®{-fj
where (D) is the standard cumulative normal distribution function. The optimal probabilities in fact will depend on the location of the driver, as well as on the time of day. In the analysis of this study, the spatial dimension is suppressed and absorbed in the distribution of trip durations. Its incorporation remains as a future subject.

FORMULATING THE MODEL SYSTEM WITH PROBE DATA In this section, the taxi system is formulated as a Markov process based on probe data. The data are from an experiment that involved 1,570 taxicabs in the Nagoya area, Japan. These taxicabs were equipped with GPS and communications devices to function as probes. They transmitted information on their status (including location, speed, acceleration, and passenger loading) every 300 meters or 550 seconds, whichever came first; at every stop and start; and every time when a passenger boarded or alighted. The data thus facilitate the representation of taxi behavior as the stochastic process described above. The experiment lasted for a period of over 3 months in 2002 (see Miwa and Morikawa, 2003). A difficulty that emerged during the analysis was determining whether a taxi picked up a passenger while cruising (a transition from C to O) or after waiting at a taxi bay (a transition from Wto O). Although the locations of designated taxi bays are known, there are numerous de facto taxi bays in urban areas of Japan where taxis queue waiting for passengers. It is thus difficult to distinguish whether a taxi had been cruising or waiting solely on the basis of the location where a passenger was picked up. In this study, therefore, the average speed of a taxi before a passenger boarded is used to discriminate between the two cases. The distribution of

616

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

averages is assumed to comprise two distributions: one for taxis that were cruising, and the other for taxis that were waiting at a taxi bay. To discriminate between the two, a threshold value of 10 km/h is established so as to maximize the between-group variance of the two latent distributions. A span of 6 minutes before passenger boarding is adopted because it yielded the largest fraction of between-group variance. Data from a random sub-sample of 99 taxis over a period of 10 days are used in the analysis. Altogether nearly 47,000 sojourns in the four states defined earlier are recorded over the period. The mean number of sojourns is 47.3 per taxi per day. After some manipulations to eliminate logically inconsistent transitions that appear in the probe data (e.g., a transition from O to W, which implies that a taxi drops off a passenger, then immediately starts waiting at a taxi bay), the transition frequency matrix of Table 1 is obtained. Although the table is not consistent internally, it presents overall tendencies in transitions among the four states. Table 1. Observed Transition Frequency Matrix "From" State Occupied (0) Cruising (C) Transit (T) Waiting (W) Total

0 12,053 24 8,180 20,257

C 10,713

"To" State T 7,716

W

8,180 10,713

7,716

8,180

Total 18,429 12,053 8,204 8,180 46,866

The frequencies of sojourns in the sample are shown in Table 2 by state and by period of the day when the sojourns started. The periods adopted here are set long enough such that most sojourns will be contained within one period. Temporal variations in passenger demand in the downtown area show that passenger boarding peaks between 11:00 pm and 1:00 am, alighting peaks between 6:00 and 8:00 pm, and the demand subsides between 5:00 and 7:00 am. The four periods shown in Table 2 are established based on these observations. The number of sojourns between 3:00 and 8:59 am is about or less than one-half of those of the other periods for all states. Particularly, sojourns in T and W are much less frequent compared with the other periods; taxis tend to cruise rather than wait at taxi bays in this period. The ratio of the number of sojourns in Wto that in C is 0.465 in this period, while it is 0.591 between 9:00 am and 2:59 pm, 0.716 between 3:00 pm and 8:59 pm, and 0.873 between 9:00 pm and 2:59 am. Taxi behavior thus varies quite substantially across these periods. Sample distributions of sojourn durations are shown in Figures 1 and 2 for states O and C. The distributions are approximately negative exponential, except for those for state O. Although the distributions for state W (not shown) conform to the negative exponential distribution to a lesser extent, it fits well for the third time period (3:00 to 8:59 pm). Overall, the negative exponential distribution appears to be an acceptable approximation except for state O, for which an Erlang-2 distribution is applied in the next section.1

1 The Kolmogorov-Smimov test yielded a Z value of 1.46. For the other distributions for states C, T and W, the negative exponential distribution is rejected by both chi-square test and Kolmogorov-Smimov test, in part because of the extremely large sample size.

Rationality and heterogeneity in taxi driver decisions 617 Table 2. Frequencies of Sojourns by State by Period State Occupied (O) Cruising (C) Transit (7) and Waiting (W) Total

3:00 a m 8:59 am 2,602 1,889

9:00 am 2:59 pm 5,377 3,734

Period of Day 3:00 p m - 9:00 pm 8:59 pm 2:59 am 5,124 5,325 3,304 3,126

Total 18,428 12,053

879

2,206

2,365

2,730

8,180

5,370

11,317

10,994

10,980

38,661

Although all transitions to O were made from C or W, the number of sojourns in O does not agree with the sum of those in C and W, largely because this tabulation includes only those sojourns made between transitions among the four states and excludes transitions with which taxis entered or exited from the system at the beginning and end of the drivers' shifts or breaks.

Figure 1. Distribution of sojourn durations in state O (occupied)

Figure 2. Distribution of sojourn durations in state C (cruising)

618 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Given the above discussions and tabulation results on sojourn durations, the model system described in the previous sections reduces to one that can be conveniently manipulated. The transition matrix of Eq. (1) is now simplified as

P=

OR)

Poc

POT

CX

°

°

° ,

T q w\_\

0 §

0 §

\-q o

°

(15)

where q is the probability that the queue of taxis at a taxi bay is empty and the taxi loads a passenger immediately upon arrival. Now, suppose sojourn durations in state O can be represented by an Erlang-2 distribution whose mean is \jX0. Further, suppose sojourn durations in states C, T, and FFhave negative exponential distributions with parameters Ac, XT, and/l^, respectively. The process is now expressed as a Markov process with

o, To l o o 2 o o Poc P= C 1 0 T q 0 W[\ 0

o PoT

0 0 0

0 0 0

o o 0 \-q 0

(15')

where state O is represented by two states, O\ and O2, and A = (21v,2Ao,Ac,lr,Zl,)-

(16)

A sojourn in state O\ and that in O2 together form a sojourn that has an Erlang-2 distribution. Before proceeding further, precise definition of the Markov process (Qinlar, 1975) is due. Let E be the state space and Y(t) be the state of the process at time ( E Q+ = [0,oo]. Then, Y(t) e E for any t > 0. The stochastic process, Y = {Y(t); t e D + } , is said to be a Markov process with state space E if, for any t, s > 0 andj s E, ?r[Yl+s=j\Yu;u
= -pr[YM=j\Y,].

(17)

Now, let Xn be the state after the n-th transition, Tn be the time when the «-th transition is made, and Xo be the initial state and To = 0. Then Xn& E and 0 = To < T\ < T2 < < Tn < for any n e C = {0,1,2,...}. Then, for any n e ,j e E, and « s C t Pr[^ n+1 =j,Tn+l-Tn >u\X0,...,Xn;T0,...,Tn]

= Pse-z"

(18)

when yis a Markov process. In Eq. (18), Xt > 0 and

4=limeOJ,0, Q(i,j,t) = Pr[Xn+1 = j,Tn+1 -Tn
(19) i}.

(20)

Rationality and heterogeneity in taxi driver decisions 619 Py is the probability that a transition will be eventually made from state i to state/, given the process is currently in state i. The probability that a transition will be made from state i to state/ within time u, given the past history of the process, can be expressed as a product of the state-to-state transition probability, Py, and the probability that the sojourn in state i will terminate within time u, irrespective of the past history of the process and the current time. Returning to the process characterized by the transition matrix of Eq. (15') and the parameters of Eq. (16), sojourn durations in state O (or, O\ and Oi) are determined by the distribution of passenger trips and network condition which are both exogenous and can be considered stable at least for an appropriately defined time interval. Likewise, sojourn durations in state T are determined by the trip length to a taxi bay from the location where a passenger is dropped. It can therefore be assumed that the distributions of sojourn durations in states O and T are exogenously determined and invariant for an appropriately defined time interval. Thus the parameters for these states may be defined as *r>=^->*T=^

(21)

where So and ST are observed mean sojourn durations in state O and state T, respectively. Sojourn durations in states C and W, on the other hand, depend on the state of the process as discussed earlier. A sojourn in state C begins when a taxi drops off a passenger and ends when it boards a new passenger. The sojourn duration thus equals the waiting time till a new passenger is found, which depends on the intensity of passenger demand and also on the number of taxis cruising. Then, as noted earlier, k

c

M(L = JhL_ Nc

(22)

Nnc

where, as before, juc is the arrival rate of passengers to hail cruising taxicabs and Nc is the number of cruising taxis. Sojourn durations in state W also depend on the intensity of passenger demand and the number of taxicabs waiting at taxi bays. Observed distributions have indicated that it is appropriate to assumed that K = 1 when representing taxi bay queues as the double queue system described earlier. Let the number of taxi bays in the system be nB, and let jiw be the system-wide arrival rate of passengers at taxi bays as before. In equilibrium it can be assumed that the expected waiting time is invariant across the taxi bays. Then, from Eq. (9) the mean waiting time of a taxicab, given it has to wait, is obtained as W = - ^ - ( l - p ) = HA

—|

(23)

where NT is the number of taxis in state Tin steady state (=NnT), and N^ is the systemwide arrival rate of taxis at taxi bays. Recall that both /ic and /iw are assumed to be

620 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

exogenous. Also note that, for condition W>0 to hold, the relation, AjNT ^ / v > must be satisfied. Parameter A^ for state Wean be determined by letting Sw= — = nR\—

—

(24)

where Sw is the observed mean sojourn duration in state W. This relation can also be used to determine nB. The probability that there is no queue of taxis, q in Eq. (15'), is obtained from Eq.(5)as? = p 2 = ( ^ / ^ ^ ) 2 . The most important feature of the model system is the simultaneity that the distribution of states affects sojourn durations, which in turn affect the distribution of states. Namely, the sojourn duration in state C is influenced by Nc which is determined by the steady-state distribution of states; the probability that a taxi does not have to wait at a bay, q, is a function of NT, and so is the mean waiting time at a bay. Thus two of the sojourn duration parameters, Ac and Ay, and a parameter in the transition matrix, q, are determined by the steady-state distribution of states, which in turn is influenced by these parameters.

NORMATIVE SYSTEM BEHAVIOR Because of the structure of the transition matrix, the steady-state distribution of states can be determined analytically by finding a unique (up to multiplication by a constant) and strictly positive solution to v = vP where v = {v0 ,vo^,vc,vT,vw).

(25)

The ratio vtjv. can be interpreted as the expected number of

visits to state i between two successive visits to state/ Since the expected sojourn duration in state_/ is \j Aj , v./A.

is the proportion of time when the process is in state j when the process is in steady state, i.e., 7rj=\im?r[Y,=j], VjeE. and Nj = NTTJ ,jeE,

where as before TV is the total number of taxicabs in the system.

For the transition probability matrix of Eq. (15'), one obtains

(27)

Rationality and heterogeneity in taxi driver decisions 621

Vc=P

°
(28)

vw={\-p2)PoTr where r =

-2 . Once the N, are given, Ac and X^ can be evaluated, and Eq. (26) 3 + ( l - p )pOT can be applied with Eq. (28) to obtain a steady-state distribution of states. This distribution in general does not agree with the one used to obtain Xc and \,. Thus an iterative procedure is adopted in this study to obtain the N} and the Xj that are consistent with each other and determine steady state of the taxi system. The key parameters of the analysis in the rest of this section are poc and p0T, namely, the probability of cruising and that of heading for a taxi bay after a passenger alights. The analysis explores how these two parameters are related to system performance. Of particular concern is the efficiency of the system, which is represented by the fraction of loaded trips, i.e., the fraction of time when a taxicab is carrying passengers. Also of interest are the expected excess times associated with the two options of cruising and waiting at a taxi bay. The numerical example below shows stalk discrepancies between socially motivated system optimization and individual-based optimal behavior; user equilibrium does lead to very inefficient system performance with too many cruising taxis competing for passengers. The parameters used in the following numerical exercise are taken from the second period (9:00 am to 2:59 pm): fic =1.059, fiy =0.649, So = 10.70 ->X0 =0.0935, ST = 10.50 -+XT = 0.0952, and Sw = 32.4 and N° = 9.91 -> nB = 67.3, where N° is the observed fraction of taxis in state T, and nB is calibrated using Eq. (24). Steady-state probabilities, the jtj, are evaluated for values of poc and p0T to assess how driver decision influences system performance. The results are summarized in Figures 3 and 4. As Figure 3 indicates, the steady-state distribution of states is very sensitive to transition probability p0T, and therefore to poc. As the figure shows, the steady-state probability that a taxi will be cruising at a random time point (7rc) increases with poc and the probability that a taxi will be waiting (nw) increases with p0T. The former probability is below 1.0~5 at pOT= 0.629. As pOT increases and poc decreases, the number of cruising taxis (Nc) decreases, and the few cruising taxis find passengers instantaneously as Xc approaches

622 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction infinity (see Eq. (21)). Thus nc diminishes as shown in the figure. At the same time, p decreases as the number of taxis waiting at taxi bays (Nw) increases, resulting in increased waiting time and therefore increased nw. On the other hand, p approaches 1.0 as p0T approaches 0.380 from above; all passengers cannot be served at taxi bays when p0T decreases below this point.

Figure 3. The Limiting Distribution of States as a Function of POT

The fraction of occupied taxis, or the probability that a taxi will be occupied (n0), takes on a maximum value of 0.257 when p0T = 0.569. As discussed earlier, n0 can be adopted as a measure of system performance as the taxi fleet is most effectively utilized when n0 becomes maximal. Quite important is the result shown in Figure 4 that the expected excess time involved in waiting at a taxi bay by far exceeds that of cruising when p0T = 0.569. Thus drivers who are waiting at taxi bays would tend to switch to cruising at this optimal point, i.e., p0T would tend to decrease. Consequently the system would divert from the optimal point and shift toward the user-optimized point where the excess times of the two options are equal to each other. This happens when pOT =0.481, which yields n0 =0.218. At this user equilibrium point, the total duration of loaded trips is more than 15% smaller than at the system-optimized point. The numerical analysis of this section clearly indicates that social optimum and individualistic user optimum are not compatible with each other in this taxi system where taxi drivers have the option of either cruising to catch a new passenger or wait at a taxi bay for one. This is, as is the case with road congestion, due to the externality that the excess time of a taxi depends on what other taxis do. In this particular case, system optimization favors waiting at taxi bays because the excess time of cruising decreases more rapidly than that of waiting increases as

Rationality and heterogeneity in taxi driver decisions 623 p0T increases.2 Yet, the difference in excess time between the two options tends to dictate each individualistic driver to adopt the option of cruising. Furthermore, the empirical analysis of driver behavior presented in the next section indicates that drivers tend to be biased against waiting at a taxi bay.

Figure 4. Mean Excess Time by Driver Action

ANALYSIS OF DRIVER DECISION From the above numerical example, it is evident that the transition probabilities pertaining to driver decision, poc and pOT, are the key parameters that determine system performance. As discussed earlier, these transition probabilities can be formulated as a discrete choice model with the assumption that driver decisions are rational. This rationality, however, does not necessarily imply that the system achieves a social optimal. The effectiveness of any countermeasure to correct the suboptimal situation must be assessed with the knowledge of how taxi drivers behave in the system and react to the countermeasures. The analysis of this section attempts to achieve this by characterizing driver decision of whether to cruise or to wait at a taxi bay. The model of rational behavior presented earlier can be simplified greatly with the assumptions that pCT = 0 and pTC = 0, and the excess trip durations of Eqs. (11) simplify to \Voc=S_c

_

\v - <> + <> ['or ~ ° r + "V

2 Note that the cost of passengers waiting for taxis is not incorporated into this analysis; the analysis here is strictly from the viewpoints of taxi drivers and operators. In the socially optimum solution shown here, passengers hailing for cruising taxis will be incurring a large amount of cost of waiting for one.

624 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Representing driver idiosyncrasy by the individual-specific random error component, r', Eqs. (13) are now rewritten, for the n-th transition of taxi i, as3 U'oc = ft'Voc +soc = MX'c") + thSDTg* + s'onc V'£ = M%T +P'X'-"+ r' + e>£ = y + MXT"

}

+ MASDT™

(29) +n ^

+ M6SDT^

+ T< + e%

where at(n) = the zone and time period in which the n-th transition from state O to either Cor Ttakes place, TQ'J^ = the mean sojourn duration of state d (= C or T) starting in at(n), SDT^n) = the standard deviation of sojourn durations of state d starting in at(n), T^'M = the mean sojourn duration in state Wstarting in at(n), SDT^"-n) = the standard deviation of sojourn durations in state W starting in atin), Y = a constant term associated with the alternative to wait at a taxi bay. Terms are introduced into the model to represent the variations in sojourn durations with the intent of evaluating the drivers' attitude toward risk in their decision. Note that mean sojourn durations and standard deviations are evaluated by geographical zone and by period of the day. The coefficient, the fi., are assumed to be invariant across taxis in this formulation. The error terms are all assumed to be i.i.d. normal, i.e., " = s'oc-s'ol T'U N(0,
N(0,a2s) = JV(O,l), Mi, V« , Mi,

E(s'-",Tr) = 0, Vi,i', Vn,and E(£I:

= 0 V/,/', V«,«'; i=tV or

n*n',

for i = 1,2,...,/ and n = 1,2,...,«.., where / is the number of taxis in the data set, and ni is the number of transitions from O observed for i. Note that, in this binary choice formulation, the distribution of the difference, e''" = E'QC - E'QT , is relevant and identifiable, but not the individual distribution of s'g"c or s'g"T. The estimated random-effect model (Hsiao, 1986) is presented in Table 3. Parameter p in the table represents the serial correlation of the sum of the error terms, «''" =T' +£''". As expected, the coefficient estimates of T°'CM, 7^"', and T^'M are all negative and significant; the attractiveness of either cruising or waiting at a taxi bay decreases as the expected excess time increases.

3

As the earlier discussions implied, it is anticipated that the decision to cruise or wait can be explained to some extent by measured variables that represent the characteristics of the driver, e.g., years of experience as a taxi driver. Unfortunately no information on the driver is available in the data set, and the term, fi'X', is not incorporated into the model.

Rationality and heterogeneity in taxi driver decisions 625 Importantly, the coefficients of T$n) and T^'w are more than twice as larger as that of T°'CM. Evidently taxi drivers view the time to travel to a taxi bay and that to wait there for a passenger, to be more onerous than the same amount of time spent cruising to pick up a passenger. Consequently, an increase in the excess time involved in waiting affects the probability that waiting will be chosen more substantially than would the same amount of increase in the excess time involved in cruising affect the probability of cruising. Moreover, the negative constant term, y, implies that, ceteris paribus, drivers are biased against waiting at a taxi bay. Parameter p is significant, indicating that drivers are heterogeneous in their choices between waiting and cruising. A note is due here on the treatment of the effect of time of day on the choice. As noted earlier, mean sojourn durations and standard deviations of sojourn durations are evaluated by time period for model estimation. Yet the model coefficients are assumed to be invariant across the periods, time of day is not introduced into the model as an explanatory variable, and the error component, r', is assumed to be independent of time of day. Exploring alternative representations of time of day effects remains as a future task. It is also noted that the model is estimated with the assumption that all the explanatory variables are exogenous, i.e., they are not influenced by the driver's choice. This, however, does not adequately reflect the simultaneity noted earlier between driver decision and system status. This also remains as a subject for future research. The empirical analysis of this section has thus revealed that taxi drivers have tendencies of favoring to cruise for a passenger in their choice between cruising and waiting. The numerical analysis of the previous section has indicated that too many taxis tend to cruise for optimum system performance. This tendency, the statistical results of this section indicate, stems from taxi drivers' inclinations toward cruising which have two aspects: first, there is bias toward cruising and against waiting, and second, drivers react to an increase in the expected excess time involved in waiting more strongly than to that involved in cruising. Consequently there exists a mechanism of stronger negative feedback for waiting at a taxi bay. Estimation results also offer implications about taxi drivers' risk attitudes. The coefficient estimates of SDT^ are negative; taxi drivers tend to avoid cruising if the variance of excess times becomes larger. In this sense, drivers are risk averse about cruising. On the other hand, the coefficients of SDT£}") and SDT£'M are positive and significant, especially that of SDT£"-n). Larger variations in excess times, i.e., larger degrees of uncertainty, do not discourage drivers from heading for a taxi bay. This counterintuitive result may represent selectivity bias rather than drivers' risk attitudes, however; if drivers tend to join the queue at a taxi bay when the bay is very close or after cruising some time in vain, then their sojourn durations in state T may tend to have larger variations in areas or periods when they tend to choose to wait. With this potential bias in mind, it is tentatively concluded that cruising is more attractive and tends to be chosen by taxi drivers when it is less uncertain, while waiting tends to be chosen when it involves more uncertainty.

626

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Table 3. Random-effect Model of Driver Decision Variable 1

OC

SDT£M Tal(n) 1

OT

P

Coef.

t

-0.0164

-2.88

-0.0092

-3.15

0.0016

-0.0390

-6.41

< 0.0001

0.0040

0.0129

3.54

0.0004

-0.0372

-6.22

< 0.0001

0.0212 -0.4560 Constant P =
6.07 -2.78

< 0.0001 0.0053

16.43 13,792

< 0.0001

rpat(n)

V SDT«W

N 1(0)

-9559.9

HC)

-9357.7

L(C)

-9278.2

-2[L(0)-L(fi)] -2[L(C)-L(/3)] R2 Adjusted R2

563.4 158.9 0.0295 0.0290

1(0): Log-likelihood value with no model parameters L(C): Log-likelihood value with the constant term alone L{C): Log-likelihood value with all model parameters -2[ 1(0) - L(P) ]: Chi-square distributed with degrees of freedom (df) -2[ L(C) - L(j3) ]:Chi-square distributed with degrees of freedom (df) = 7

An important implication of the analysis is that taxi drivers are heterogeneous; the transition probabilities vary across them as well as across situations, or, system status. The significance of the fij implies that transition probabilities poc and p0T vary depending on the expected excess time of waiting or cruising, forming a mechanism of negative feedback. It must also be noted that the variation in the excess time also influences the transition probabilities. Moreover, the significance of the heterogeneity parameter, p, implies that the transition probabilities vary from driver to driver, possibly reflecting their risk attitudes.

CONCLUSION A Markov process model of taxi behavior has been proposed in this study. Although passenger demand is treated as inelastic, the model system has offered several important insights. In particular, it has demonstrated that the taxi system, just like traffic flow on road networks, contains a discrepancy between the social optimum and user equilibrium; when individual taxi drivers operate while being motivated to maximize their own individual gains, the system as a whole settles at a suboptimal point. This is a case of social dilemma, i.e., as a result of individuals' pursuit of their own gains, the total gain of the system as a whole declines, and eventually the individuals' gains decrease below the level that can be attained if

Rationality and heterogeneity in taxi driver decisions 627 they cooperated toward a social optimum. Kitamura et al. (1999) illustrated this for the case of public transit ridership based on a model of commute travel in an abstract city; if commuters can agree on a rule to restrict their car use, then every commuter's gain can improve. The presence of social dilemmas in taxi operation implies that improved system performance must be based on some modification of driver behavior. The study results suggest that drivers' inclinations toward cruising to pick up passengers must be restrained by rules, regulations, or incentives. By estimating a random-effect discrete choice model of drivers' choice between cruising and waiting at a taxi bay, the study has shown that taxi drivers behave rationally, motivated by the reduction of excess time. Furthermore, they are biased against waiting at a taxi bay, and react more strongly to an increase in the excess time involved in waiting than to that involved in cruising. This propensity toward cruising contributes to the above discrepancy between social optimum and equilibrium because the system must have a much larger fraction of taxis waiting at taxi bays to attain a social optimum. The empirical analysis of driver behavior has also shown that drivers are heterogeneous and their preferences toward cruising or waiting vary from driver to driver; some drivers consistently prefer to cruise, or to wait at a taxi bay, than others. It has been indicated that cruising becomes more attractive when it involves less uncertainty, while waiting becomes more attractive when it is more uncertain. The study results suggest certain directions toward the improvement of taxi operation in large cities. Yet the validity and generality of the study results must be rigorously examined and the model must be extended to represent taxi operation more realistically. In particular, more rigorous treatment of time of day in the choice model is desired. Also desired is representing the effects of other drivers' choices in the discrete choice model formulation. Subjects for future research also include: testing the validity of the Markovian assumptions—especially history independence and negative-exponential and Erlang sojourn durations; incorporating elastic passenger demand and representing supply-demand equilibrium; examining the nature of taxi drivers' attitude toward risk; and developing efficient procedures to simulate the system with heterogeneous transition probabilities that account for heterogeneity as well as the spatial dimension.

ACKNOWLEDGMENTS The authors are grateful to anonymous referees for their constructive comments to an earlier version of this paper, which aided in improving the quality of the paper. Yasuhiro Shiomi and Daisuke Fujita at Kyoto University processed the probe data set for the analysis, and Yusak O. Susilo, also at Kyoto University, estimated the random-effect model of drivers' choice. The data were provided to the authors by the Internet ITS Consortium.

628 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

REFERENCES Arnott, R. (1996). Taxi travel should be subsidized. J. of Urban Economics, 40, 316-33. Cinlar, E. (1975). Introduction to Stochastic Processes, Prentice-Hall, Englewood Cliffs. Hsiao, C. (1986). Analysis of Panel Data. Cambridge University Press, Cambridge. Kendall, D.G. (1951). Some problems in the theory of queues. J. of the Royal Statistical Society B, 13(2), 151-85. Kitamura, R., S. Nakayama and T. Yamamoto (1999). Self-reinforcing motorization: Can TDM take us out of the social trap? J. of Transport Policy, 6(3), 135-45. Miwa T. and T. Morikawa (2003). Analysis on route choice behavior based on probe-car data. Proceedings of the 10th World Congress and Exhibition on Intelligent Transport Systems and Services, Madrid, Novemberl6-20, CD-ROM. Sasieni, M.W. (1961). Double queues and impatient customers with an application to inventory theory. Operations Research, 771-81. Wong, K.I., S.C. Wong and H.Yang (2001). Modeling urban taxi services in congested road networks with elastic demand. Transportation Research B, 25, 819-41. Yang, H. and S.C. Wong (1998). A network model of urban taxi services. Transportation Research B, 32(4), 235-46.

32

A ROLLING HORIZON APPROACH TO THE OPTIMAL DISPATCHING OF TAXIS Michael G H Bell and KI Wong, Centre for Transport Studies, Imperial College London, UK, and Alan J Nicholson, Dept of Civil Engineering, University of Canterbury, NZ

INTRODUCTION In many towns and cities, taxis are requested by telephone. There is then a process by which each request is assigned to a taxi. The nature of that process depends on the structure of the industry, which varies from city to city. For example, in some cases taxi drivers are selfemployed, the call centre offers the job to all drivers, and the driver who responds first is assigned the job. In other cases, the drivers work for a taxi company and a dispatcher in a control centre assigns the job to the best placed taxi within the company. In all cases, taxis are in radio contact with some form of call centre from where information about each job (location of the request and any other relevant details) emanates. Often in operations of this type, the jobs are queued on a display in front of the driver, with the driver reporting back to the control centre upon completion of each job. Perhaps the simplest and most practical algorithm is to send the taxi that can reach each request first. This requires reliable information about travel times. In some cases, this may lead to the assignment of a taxi that is at the time of the request already occupied even though there are unoccupied taxis, if an occupied taxi can complete its current job and reach the request before any unoccupied taxi can do so. For this reason, unoccupied taxis sometimes reposition themselves to areas generating more requests, particularly when taxi drivers earn some form of commission from each job. The assumption made in this paper is that taxis are assigned jobs by a central dispatcher with the simple objective of minimizing passenger waiting time. This objective can be regarded as

630 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction synonymous with the maximisation of fleet utilisation. However, dispatching the taxi that can reach a request first is at best a heuristic. It is conceivable that the taxi that can reach the request first might actually be better assigned to a subsequent request. A better heuristic might therefore compare the passenger waiting time incurred by assigning a taxi to the current request with the expected passenger waiting time if the taxi were instead assigned to the next request, and assign the taxi with the largest comparative advantage. The principal of looking ahead to the next request could clearly be extended to looking ahead to the next two, three or more requests, probably with rapidly diminishing returns. An efficient method is required for finding the assignment of the next N taxis that minimizes the total expected customer waiting time based on a time-dependent probability for the origin and destination locations (zones) of future requests and a mean headway between requests. This method could then be built into a rolling horizon approach to taxi assignment. Unfortunately, finding the taxi that minimizes the expected total wait over a rolling horizon is computationally very demanding, and for large fleets is not feasible in real time. Consequently heuristics have been devised which assign the taxi that maximizes a measure of comparative advantage with respect to the next request (HI) or to the next N requests (H2). The measures of comparative advantage imply that the chosen taxi is not dispatched to the next request (HI) or the next N requests (H2). This use of a tabu rule helps achieve great computational savings by not needing to update delivery times and locations while searching for the optimum.

PREVIOUS WORK ON VEHICLE ROUTING PROBLEMS The vehicle routing problem (VRP) has been studied for decades, and different approaches and algorithms have been developed to solve specific problems (Toth and Vigo, 2002). The traditional VRP seeks the optimal pick up and drop off schedules for a fleet of vehicles with finite capacities subject to time constraints of various kinds. A common objective is to minimize the total fleet operating cost, taking into account the delay of a customer waiting to be served. Types of VRP range from routing and scheduling, distribution problems, to fleet management and vehicle allocation. In many applications, for example the replenishment of stock, requests for emergency services, taxi services, etc., the wait for service, rather than the total travel time, is a more suitable objective, especially when the demand pattern is both dynamic and stochastic (Bertsimas and Van Ryzin, 1991). Many studies have been devoted to deterministic and static problems. These problems assume the input is known (is not subject to uncertainty) and is not subject to change during the planning horizon (is not dependent on time). By the end of the eighties, with the advance in information and telecommunication technologies, attention shifted to the stochastic VRP (SVRP) and dynamic VRP (DVRP). Sources of stochasticity could be the uncertainty in future demands (forecasts), network travel times, and service times involved in meeting the demand. Gendreau et al. (1996) provides a good summary of the SVRP problem. Larsen (2001) distinguishes vehicle routing and scheduling problems according to their degree of dynamism, reflecting roughly the proportion of immediate to advanced requests for service

Optimal dispatching of taxis 631 (the higher this proportion the more dynamic the system). With increasing dynamism, the objective shifts away from travel cost minimization toward response time minimization. Psaraftis (1988) identifies the characteristics of dynamic vehicle routing problems and describes early studies in this area. The stochastic and dynamic problem (also known as the real-time routing and dispatching problem) requires both stochastic and dynamic issues to be considered (Ghiani et al., 2003). In such a problem the uncertain data are gradually revealed over the time of operation, and the dispatching decisions are made in an on-going fashion as new data arrives. The solutions to the VRP problem and all its variants have long been regarded as NP-hard. For the simple variants, combinatorial optimization was often used to solve the problems, and exact solution could be obtained by direct enumeration for tiny-sized problems. It is not possible to describe every problem variation, as some practical constraints (e.g. time window constraints and fleet capacity constraints) normally exist for specific problems, and the problem structure and formulation can be very different. Toth and Vigo (2002) give a good review of recent research and applications. While there is an extensive literature on vehicle routing and scheduling where the locations and times of demands are known beforehand, there is relatively little on real-time vehicle dispatching (see Taniguchi et al., 2002). Ichoua et al. (2000) review recent research on this topic, and identify two approaches. The first is based on the notion of a rolling horizon. As time unfolds, static problems are solved repeatedly over the events found within the horizon. When the horizon is short, a myopic strategy emerges, which under certain conditions can provide near-optimal solutions. Adaptation of static vehicle routing and scheduling procedures can take the form of either reoptimisation, which tends to be computationally intensive, or fast local operations (such as insertion or ^-interchange), perhaps combined with periodic re-optimisation. Much recent research on algorithms in this category is devoted to the use of meta-heuristics. These are iterative procedures aimed at finding better near-optimal solutions for large-scale combinatorial optimization problems. Examples include simulated annealing, genetic algorithms, neural networks, tabu search, etc. Ghiani et al. (2003) provide a recent review of real-time vehicle routing problems and algorithms, with particular emphasis on parallel computing strategies. The second and less popular approach involves the use of stochastic methods, such as Markov Decision Processes (Minkoff, 1993) or stochastic programming (Powell, 1996). Some other strategies do exist. Different diversion strategies have been suggested and studied by Regan et al. (1995) and Ichoua et al. (2000), in the context of a truck-load carrier. The basic underlying idea is to consider diverting a vehicle away from its current destination to serve a request that has just occurred near its current position. On the other hand, the benefits of analytical analysis of VRPs have been emphasized by Bertsimas and Simchi-Levi (1996). Heuristics are often sensitive to changes in data, so one that works well on a set of standard problems does not necessary perform well on others. They surveyed the development of theoretical analysis for some classic problems and variations in algorithms, leading to qualitative insights into the algorithmic structure and better understanding of the models.

632 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Another concern with VRPs is variation in congestion and travel time. This has received little attention and nearly all models proposed in the literature adopt the assumption of constant travel time. Two reasons for this were pointed out by Fleischmann et al. (2004). It adds complexity to the problem formulation which may require essential structural modifications in the algorithms, and the estimation and data collection for time-varying travel times are difficult so the reliability of the results are open to question. Two notable approaches to handling variations in travel time during vehicle routing found in the literature are the following. In one travel time is treated as a random variable following a known distribution (Kenyon and Morton, 2003), and in the other the problem is formulated with time-varying travel time (Fleischmann et al., 2004). However, these concepts have not yet been exploited in dynamic situations. It is noted that VRP models with time-varying travel times are not dynamic as long as the information is assumed to be known for the whole study period at the time of making a decision. In real time systems, planning decisions are frequent and the network condition and congestion level can always be updated at the moment of planning, which minimizes the offset of the decision from the optimal. This justifies the use of constant travel times in DVRPs. Taxi dispatching as considered in this paper is a form of real-time dispatching problem. Customer requests are dynamic, the requested pickup time is usually "as soon as possible", and the delivery to the drop off point should be without deviation. The problem is therefore highly dynamic. As real-time VRP problems are NP-hard, exact algorithms are not yet capable of handling the typically large practical problems. This justifies the use of heuristics in real-time environments. Most solution algorithms for real-time VRPs do not take historical information into account. For some of the emergency services, like fire and ambulance, which are also highly dynamic, the quality of a priori information is generally poor in terms of the locations of future requests. Gendreau et al. (1997, 2001) studied the allocation of ambulances for emergency medical services. This problem consists of determining which ambulance should be sent to answer a call and relocating the available ambulances to locations, which provide good coverage. Historical information was not taken into account. Concerning other door-to-door demand responsive transportation problems, the dial-a-ride problem (DARP) has been studied extensively (see Cordeau and Laporte, 2003), but relatively little research on the dynamic case is reported. Most of studies on DARP assumed the advance booking of requests so the operator plans the routes with full information. One case of a dynamic DARP was studied by Madsen et al. (1995), who solved the dynamic DARP a problem for elderly and disabled people. Requests arrive dynamically throughout the day and are inserted in vehicle routes, taking into account the difficulty of insertion into the existing route. In the case of taxi dispatching, however, the spatial distribution of requests can be expected to follow a regular and predictable pattern which varies over time, although the actual arrival of requests is stochastic. Special events are also associated with regular patterns of requests. Taking this regularity into account by looking ahead with a rolling horizon should significantly improve dispatching performance. The essentials of heuristics with a look-ahead capability are discussed in Ghiani et al. (2003).

Optimal dispatching of taxis 633

RECENT STUDIES OF TAXI DISPATCHING In recent years, there has been an explosion in the number of studies of VRP problems, due to advances in information and communication technologies (ICT), in particular satellite locationing, cellular communication and geographic information systems, which make more efficient dispatching of vehicles possible. In the context of taxi dispatching, the importance of Global Positioning Systems (GPS) has been addressed by Liao (2001), who surveyed three taxi companies in Singapore comparing the improvements in system accuracy and efficiency achieved when their taxi dispatching systems were updated from a radio-paging system to a satellite-based one. A recent study of an existing taxi dispatching system has been discussed by Lee et al. (2003). A common practice adopted in the GPS based dispatching systems employed by some taxi operators is the assignment of the empty taxi nearest to the location of a request. Lee et al. suggested an alternative dispatching policy involving assigning the empty taxi with the shortest travel time to the location of request, as determined by real-time traffic conditions. Lee et al. (2004) also proposed a trip chaining dispatching strategy for pre-booked requests. Assuming all demands are pre-booked (encouraged by a lower fare, say), they suggested a strategy that chains jobs for each taxi, such that the drop-off time and location of one job fits the pick-up time and location of the next job. They formulated the problem as a Pickup and Delivery Problem with Time Windows, and showed the potential to reduce the taxi fleet size, but did not consider uncertainty in the timing and location of requests. While they assumed that exact information on requests is known in advance, this is not in general the case. The requests are subject to uncertainty and arrive dynamically in narrow time windows. Our paper focuses on real-time dispatching but takes expected future requests into account.

METHODOLOGY Assume a request with a certain pick up and delivery point is generated at a particular time. In the case of taxi assignment without looking ahead, a simple dispatching rule is to assign the taxi with the minimum passenger wait time, referred to as a Nearest Neighbour (NN) policy. In practice, many taxi dispatchers would pre-position their idle taxis to be closer to where future requested pickups are likely to arise. While this dispatching strategy is likely to be reasonably efficient, a better assignment could be made if the dispatcher were to anticipate future requests.

634 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

(a) Dispatching without look-ahead

(b) Dispatching with look-ahead Figure 1. Benefits of looking-ahead in taxi dispatching An example of the benefits of looking ahead is illustrated in Fig 1. When the demands are revealed over time, from t = 0 to t — 3, the NN dispatching policy leads to the assignment shown in Fig. la. If the dispatcher were able to correctly anticipate future demands, he would send the vehicle in the upper left region, rather than the one in the centre, to the first request because he could then dispatch the taxi in the centre to the second request, thereby significantly reducing the distance to the second request at the expense of a small increase in distance to the first request. As shown in Fig. lb, if the dispatcher can correctly predict future requests, substantial savings in vacant vehicle kilometres can be achieved, which will in turn reduce passenger waiting times. The passenger requests are handled on a First-In-First-Out basis regarding assignment, though it is noted that the actual pick-up times may not follow a FIFO sequence. Suppose the dispatcher knows the probabilistic request profile (i.e. the probability of each possible origindestination pair being the next request) and the average headway between requests. Suppose also the dispatcher looks ahead to the next N requests and seeks to assign taxis to minimize the total expected waiting time. The expected wait for the wth request can be determined given the preceding n-1 assignments. The reason for the conditionality is that after each assignment

Optimal dispatching of taxis 635 the scheduled delivery time and corresponding delivery point for the taxi just assigned needs to be updated. Given this recursive structure, Bellman's decomposition may be applied. Variable definitions Assume a customer request with a defined pick-up point and delivery point is generated at a particular time. Let / and J be the set of origin and destination zones, and K be the taxi fleet. Let Pij{i) = Probability of a taxi trip from i toy at time t a(k) = Scheduled delivery time of taxi k d(k) = Scheduled delivery point of taxi k c(i,j) = Travel time from ('toy h = Expected headway between requests Nearest Neighbour taxi assignment In the case of taxi assignment without looking ahead, a simple dispatching rule is to assign the taxi which minimizes the wait time of the current requester. This is a form of Nearest Neighbour (NN) policy. Suppose a request with pick up point i and delivery point j is generated at time t. The customer wait time if taxi k is assigned is (1) \^t,a(k\d{k\i) = m&x{Q,a(k)-t) + c(d(k\i), k = l,2,...,K where the maximum is of the two terms in the outer brackets. The dispatcher assigns the taxi that can reach the requester first, so there will be for one or some values of k* a minimum value of the customer wait w (2) k* = argmm{w(t,a{k),d(k\i)), k = l,2,...,K k

Here the function "argmin" represents the argument of the minimum, i.e., the value of k for which the value of the given expression attains its minimum value. When the minimum of the function is reached at more than one value of k, the smaller value is arbitrarily chosen for k*. In practice, a rule of equal opportunity between drivers could also be applied, where the dispatcher seeks to balance the work load across the taxis subject to no or only a small deviation from minimum waiting times. If the taxi arrived at its scheduled delivery point before this job is assigned to it, we assume it remained idle at that location. The idle time for taxi k* is therefore l{t,a{k*)) = max(O,t-a(k*)) (3) In practice, many taxi dispatchers would reposition their idle taxis to be closer to where future requested pickups are likely to arise. The scheduled delivery time and delivery point of taxi k* are then updated as follows a(k *) = t + w(t, a(k *), d(k *), i)+c{i, j)

d{k*)=j

(4)

(5)

636 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction While this dispatching strategy is likely to be reasonably efficient, and is equivalent in some respects to a greedy algorithm, a better assignment could be made if the dispatcher were to anticipate future requests and consider these when assigning a taxi to the current request. Taxi assignment with look-ahead Suppose the dispatcher knows the probabilistic request profile (i.e. the probability the next request will correspond to a particular origin-destination pair) and the average headway between requests. The expected time of the next request is t+h, where h is the average headway between requests. The probability that this request has pickup point ;' and delivery point j is p~(t + h). Suppose the dispatcher looks ahead to the next N assignments and seeks to minimize the total expected waiting time. The problem can be formulated as a Dynamic Programming (DP) problem with a finite horizon. In the terminology of dynamic programming, let the dispatching policy be it = {ko,...,kll,...,kN}, where kn is the taxi dispatched at stage n. In the iV-stage problem, the expected cost of a policy K, given the initial customer pick-up location i0, is 7, (/„) = w(t, a{k0), d(k0), / 0 ) + Xn=LJV Wit + nh, a(kn), d{kn))

(6)

where cost is measured in terms of expected waiting time, and i//{t + nh,a{kll\d{kn)) = £

(max(0, a(kn )-t-nh)

+ c(d{ka), /))ptf (t + nh)

If kn e n then a(jtn+|) = t + nh + w(t, a(kn), d(ka ), i) + c{i, j)

(7) (8)

and d{ka+l)=j otherwise

(9)

«(O=«(*.)

do)

d{K+]) = d{kn)

(11)

and The optimal policy is /„. {i)=mm{jM

(12)

Note that as a consequence of (8) and (9), which update the delivery time and point for the taxi just assigned, the expected wait for the «th request, yAj+nh, a(kn), d(kn)), depends on the preceding n-1 assignments and corresponding requests. Consequently the calculation of JK(i) is computationally very demanding. If the taxi fleet consists of K taxis then the total number of possible assignments is KN+I. For each assignment after the first, all possible requests need to be considered. Consequently computational complexity is O(KN*'Olf), where OD is the number of feasible origin-destination pairs. Given its recursive structure, Bellman's decomposition may be applied. However, finding the taxi that minimizes the expected total wait over the rolling horizon remains computationally very demanding, and for large fleets is impractical in real time due to Bellman's "curse of

Optimal dispatching of taxis 637 dimensionality". Consequently heuristics have been devised which assign the taxi that maximizes a measure of comparative advantage with respect to the next request (HI) or to the next N requests (H2). The measures of comparative advantage imply that the chosen taxi is not dispatched to the next request (HI) or the next N requests (H2). This use of a tabu rule helps achieve great computational savings by not needing to update delivery times and locations within the search process. Two heuristic algorithms In this section two heuristic algorithms are proposed. The concept of the first heuristic is to search for a taxi which minimizes the difference between the wait incurred if the taxi is assigned to the current request at point i and a times the expected wait if the taxi is assigned instead to the next request. HI (a): k* = arg min(w(f, a(k), d(k), i) - a ^ w{t + K a(k), d{k\ i)Plj (t + h)), k = l,2,...,K

(13)

where a is a parameter which discounts future waiting time. Each taxi is looked at once, so the computational complexity is only O(K* OD). The second heuristic is similar to the first, but takes into account the location of the other taxis. It searches for a taxi to minimize the wait for the current request plus the expected wait for subsequent requests, which will be serviced by other taxis, weighted by the parameter a. H2 {N, a): k* = argmin(vv(f, a(k), d(k\ i)+a max^] pQ(t +

min(w(f + nh, a(k'\ d(k'\ <))), k = l,2,...,K,

w = l,2,...,JV (14)

The second heuristic tries to approximate the solution to problem (12) but in a myopic way. It seeks the minimum of the sum of the current customer wait when assigning the taxi k and the weighted maximum minimum expected customer wait arising over the next n steps in the absence of taxi k. That weighted maximum minimum expected wait represents the disadvantage in the future when taxi k is assigned now and is therefore not available for the next N steps. The validity of this tabu is highest when the fleet size is large and the request rate is high, so a taxi is less likely to be optimally assigned to more than one request in a short period. Equation (14) therefore considers whether taxi k should be dispatched to the current call or should be used for one (at most) in the future. This approximation allows the heuristic not to update the locations of assigned taxis during the search and hence it saves the computing steps when compared with an exhaustive search for the optimal policy. Parameter N is the number of steps looked ahead and a is a parameter which discounts future waiting time, because of the uncertainty associated with future demand. Note that when N equals 1

638 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction and a equals 1.0 the heuristic is the same as the one-step look-ahead method in (12). For looking ahead N future requests, the computational complexity is O(K2* OD* N), which is linear in the size of the rolling horizon and so gives an acceptable computing time. As the following simulation results demonstrate, the heuristics are able to reduce total waiting time considerably, particularly during periods when the demand for taxis is high.

Figure 2. A 4 x 4 grid network

SIMULATION EXPERIMENTS In order to demonstrate the effectiveness of the proposed methodology, simulation is used to verify the improvement offered by looking ahead periods of different durations. Monte Carlo simulation is adopted to generate the customer requests, and taxi assignment follows one of the methods presented above. In this section, the effects of various factors on the performance of the algorithms are examined. A grid network with 16 zones (nodes) as shown in Fig. 2 is considered. The travel times between each pair of zones are assumed to be constant. In reality, taxis do not always locate at zone centres, adding to travel time variability. The travel time for each link is set to be 10 minutes and there is an intra-zonal travel and pick up time of 10 minutes, so the origindestination travel times range from 10 minutes (intra-zone) to 70 minutes (for travelling between the vertices on the diagonal). The trip rate pattern is set such that there are more trips leaving from the zones in the centre to zones near the boundary (see Table 1, which shows the origin trip rates (O, = Hj py) and destination trip rates (Dj = Z; Py) corresponding to the 4 by 4 grid network in Fig. 1). The trip rate pattern is assumed to follow a gravity type distribution, ^=A,.fl y O,D,exp(-0c(i,;)) (15) where At and Bj are the coefficients related to the constraints of 0, and Dj, and 6 is the dispersion parameter of the model which is set to be 0.1. The A, and Bj, and thus py, can be easily solved using the standard iterative balancing procedure.

Optimal dispatching of taxis 639 Table 1. Origin and destination flows corresponding to the 4 x 4 grid network Origin flow rate (%) (O, = I ; ptj) 3.85 3.85 7.69 3.85 15.38 3.85 3.85 3.85

3.85 11.54 19.23 3.85

3.85 3.85 3.85 3.85

Destination flow rate (%) (Dj = X, p,y) 5.77 5.77 5.77 5.77 11.54 7.69 5.77 7.69 3.85 5.77 5.77 5.77

5.77 5.77 5.77 5.77

Taking the trip rate pattern of (15), the requests are generated by Monte Carlo simulation. With a given fleet size of 20 taxis, trip generation follows a Poisson process with a mean headway derived from the arrival rate, varied for different cases, for a total study period of 10 hours. Random variations in arrival times are added to the simulation, and serve to test the robustness of the heuristics. For each request, a taxi is assigned using the dispatching algorithm. Once assigned, the taxi cannot be substituted with another, and no requests will be rejected by the dispatcher. How busy the system is will be indicated by the incoming demand intensity, which is related to the proportion of taxi service time required to fulfil all requests. Performance for different strategies The dispatching algorithms are tested for different demand intensities. When the demand is relatively low, one would expect all requests to be met while the waiting times could depend on the dispatching policy. For heavy demand (i.e. in the peak period), the system may not be able to keep up with the demand, so delay may increase continuously and without limit. The simulation is performed with 10 different seed numbers and the results are averaged to account for the effect of seeds. For the same comparison between heuristics, the demand sequences are kept the same for each of the seed numbers. The average customer wait for different incoming demand rates are shown in Fig. 3a. NN represents the Nearest Neighbour policy, 1SLA represents the optimal one-step look ahead policy (the optimal policy for N = 1), HI (0.5) and HI (0.7) stand for heuristic 1 with discount factors of 0.5 and 0.7 respectively, H2 (5, 0.5) and H2 (5, 0.7) stand for heuristic 2 with a look ahead of 5 steps and discount factors of 0.5 and 0.7 respectively. Each simulation run is executed for a period of 10 hours. Experiments show that for heuristic HI the best value of a lies between 0.5 and 0.75, for heuristic H2 the best value of a also lies between 0.5 and 0.75, and that the best value of N is related to the taxi fleet size (but this has less effect than a). In the following, the results of a few parameter values are shown for both heuristics.

640 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction It can be seen (Fig. 3a) that for all algorithms, the passenger waiting time is insensitive to changes in the arrival rate up to about 30 passengers per hour, but as the arrival rate increases further, the passenger waiting time increases rapidly, as the taxis become busier and have a lower availability. To display the results in detail, numerical values of passenger wait for demand with increments of 1 psg/hr are shown in Table 2. For each heuristic the value immediately smaller than 50 is highlighted. In all cases, the 1-step look-ahead (1SLA) outperforms the Nearest Neighbour (NN) policy. The performances of the heuristics depend on a and the demand intensity. With a fixed taxi fleet and when the system gets busy (over 35 psg/hr), heuristic HI with an a value of 0.7 performs the best of all the algorithms tested. It saved around 15% of the passenger wait when compared with the algorithm NN. However, when the demand rate is below 20, it is poorer than NN. The heuristic HI with a of 0.5 is still better than with 1SLA, but with fewer savings in passenger waiting time when the system is busy. Heuristic H2 for both cases gives comparable results to those for 1SLA. The heuristics with larger a values generally perform well when the demand is high, but it generates higher waiting time when the demand is low, when compared with the NN case. This could be explained by the fact that the strategy with a higher a value tends to reserve the capacity for future demand. When compared to the heuristic HI with an a of 0.7, HI with a of 0.5 tends to be more stable and closer to the look-ahead results for lower demand intensities. This suggests that the performance of the heuristic depends on the chosen discount factor (a), and the best a depends on the demand intensity. This dependency is worth further investigation. One way to do this could possibly be to relate a to the arrival rate and/or the proportion of time the taxi fleet is occupied. The efficiency of the dispatching algorithm determines the response time to requests and/or the number of taxis needed. Fig. 3b shows the total distance travelled by the vehicles in order to satisfy all of the requests. It can be seen that the distanced travelled is less for the heuristics than the NN case, and the saving in customer wait time actually comes from reducing vehicle kilometres, which in turn makes the overall system more efficient, and decreases the amount of traffic in the network. With HI the vehicle distance travelled is reduced by 5 - 10 % overall. The above results are the average of 10 trials with different seed numbers. Although the detailed results obtained do depend on the selection of the seed number for the random number generator, there was little variation in the rankings across the trials. The seed number chosen therefore had little effect on the conclusions.

Optimal dispatching of taxis 641

(a)

Average customer waiting times for different demands

Figure 3. Comparison of heuristics for different demand arrival rate

642 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 2. Average customer waiting times in minutes for different demands Demand (Psg/hour) 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

NN

1SLA

22.5 23.6 25.4 28.3 31.7 36.0 40.9 48.4 55.4 62.9 71.6 81.1 89.3 98.0 108.0 116.2

22.0 23.3 25.0 27.3 30.1 33.5 37.8 43.7 50.3 56.9 65.8 73.8 81.7 90.8 101.2 110.6

HI, a=0.5 23.8 24.5 25.8 27.0 28.5 31.4 34.5 38.6 43.2 48.7 57.0 62.3 70.6 79.9 88.0 96.8

HI, «=0.7 26.9 28.5 29.2 31.1 32.7 34.1 36.5 39.9 43.8 47.9 54.7 61.1 68.3 75.4 83.4 91.6

H2, N=5, a=0.5 22.4 23.6 25.2 27.5 31.0 34.9 39.4 44.3 51.6 60.1 67.6 77.9 86.8 96.1 104.9 114.8

H2, N=5, «=0.7 22.3 23.5 25.3 27.5 30.6 34.0 39.4 43.3 50.6 58.9 67.3 76.1 84.3 95.5 103.2 112.6

Stability of the system over time The above analysis shows that when the demand is low the NN policy could perform better than the other strategies. However, as the demand increases, the dispatching system may not be able to handle all the requests and queues may grow over time. Although the busy period of taxi operation normally extends over 10 to 12 daytime hours, short term over-capacity periods will exist within this period, so it is interesting to investigate the maximum capacity of different dispatching strategies that are compatible with steady state queues. For each dispatching strategy, there is a maximum demand for which the number of un-served passengers and/or passenger waiting times remain stable and do not tend to grow over long simulation runs. This is referred to as the critical point of the system. We investigate this point for the different heuristics by simulation. Stability is tested over a longer period of simulation (25 hours), and we compare the average wait for each period of 5 hours (i.e. from 0 to 5, 5 to 10, 10 to 15, 15 to 20 and 20 to 25 hours). Fig. 4 compares the different strategies for the final period of 20 to 25 hours. The waiting time remains more-or-less constant for all heuristics for demand below 25. With 30 requests arriving in an hour, the average wait in the final period grows to around 140 minutes with the NN policy but remains around 50 minutes with HI. The system is unstable in the sense that the waiting time grows over time. The growth rate of the waiting time is lower for the heuristic approaches compared with the NN policy, when the arrival just exceeds the critical point. The critical point is higher for HI than for the NN policy. Overall, HI can cope with demand around 5 - 1 0 percent greater than other strategies, with much lower waiting times if the system is operating above its critical point.

Optimal dispatching of taxis 643

Figure 4. Stability of different strategies over time in the final period Waiting time and fleet size It has been found in other studies that, under a dispatching system, the expected waiting time is inversely proportional to the square root of the density of idle taxis (Arnott, 1996; Bertisimas and van Ryzin, 1991), while under a cruising taxi system, waiting time is inversely proportional to the density of vacant taxis (Wong et al., 2001). It could be expected that when both the customer demand and taxi fleet size is doubled, the density of idle taxis would remain about the same or increase a bit. This is because the spatial mismatch between taxis and passengers would be decreased by having more idle vehicles in the network. As a result, in a dispatching system, by increasing the demand and taxi fleet simultaneously, the expected waiting time decreases but at a diminishing rate as the demand and fleet size increase. This was also investigated with our model. Using the heuristic HI (0.7) the model is solved for different taxi fleet sizes and the average passenger wait against arrival rate is shown in Fig. 5. This shows that for a taxi fleet size of 10 the maximum acceptable wait (assumed here to be 30 minutes) is reached for a demand of 10 psg/hr, while it is 27 psg/hr and 59 psg/hr for fleet sizes of 20 and 40 respectively. For a higher fleet size the waiting time is generally lower when the demand is low, as the dispatcher can always respond more quickly to requests with widely dispersed vacant vehicles in the network. It can be seen from these results that a dispatcher with more vehicles is able to do more jobs with each taxi, which increases the efficiency of the whole system, although the benefit is diminishing as the fleet size expands. It can also be seen from Fig. 5 that, with a relatively low demand and a large fleet size, the

644 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction passenger wait could approach the theoretical minimum value, which is the intra-zonal travel and pick up time.

Figure 5. Average passenger waiting time against the demand, for different fleet sizes Performance and dwell time variations The simulation settings assume that the travel times in the network are constant. In reality, the vehicle travel times and dwell times (time spent picking up and dropping off passengers) would be subject to variation, which may affect the rest of the itinerary assigned to the vehicle. We capture this uncertainty as a request dependent variation, and a simulation run is designed to investigate the robustness of different heuristics against the stochastic variations in dwell time. This extra element of randomness allows to some extent for traffic congestion and makes the problem stochastic as well as dynamic. The constant intra-zonal travel and pick up time of 10 minutes defined in the travel time matrix is now modelled as a stochastic variable, and we assume that the dwell time for each request can be determined by d=d + e (16) where d is the (average) dwell time of 10 minutes for a request, d is the stochastic dwell time, £ is the stochastic variation, which is assumed to follow a normal distribution of e ~ N(0,j3d), where ji is the coefficient of variation. In the simulation, the dispatcher has no information about this variation so adopts the average dwell value in the dispatching decision.

Optimal dispatching of taxis 645

Figure 6. Average passenger waiting time against coefficient of variation for dwell time Fig. 6 portrays that average passenger waiting against the coefficient of variation /? for the dwell time. For a 10-hour simulation, the passenger arrival rate is set at 30 psg/hr, and the coefficient of variation of travel time is varied between 0.0 and 0.6. When /? is zero, there is no variation and the differences in the performances of the heuristics are similar to those shown in Fig. 3. While the NN policy generates the highest average wait, the other heuristics are better, with HI giving the least average wait. When the variation is increased, HI always performs the best among all heuristics, 1SLA is better than NN except for one case where ft equals 0.5, and H2 is better than NN for p less than 0.3 but may be worse than NN for a larger p value. The heuristics capturing future demand distribution outperform the NN policy, even with random dwell times. Fig. 6 also suggests that the impact of variation in travel time and dwell time in taxi dispatching is worth further investigation. Performances and time varying travel time Another consequence of traffic congestion is the time dependency of travel time. The robustness of the heuristics was also tested for the case where travel time varies systematically over time, and the heuristics take this dependency into account. Travel time is assumed to vary by y during the day, where y lies between 1.0 (no congestion) and 2.0 (heavy congestion), as shown in Fig. 7. For y equals to 1.0 (no peak period), the results are broadly as before. When y increases, the system becomes busy and passenger wait increases. Nevertheless, it can be seen that, for every case, the heuristics generally perform better than

646 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction NN, with the margin of improvement increasing with the peak period congestion factor, because the heuristics anticipate the peak period.

Figure 7. Average passenger waiting time against peak congestion factor

CONCLUSIONS AND FUTURE RESEARCH OPPORTUNITIES The real-time dispatching of taxis in response to incoming requests is studied, with the objective of minimizing the total wait experienced by requesters over time. Several dispatching rules are examined. Given the distribution of future requests, which may be obtained from historical trip origin and destination data, a rolling horizon approach is formulated. As finding the taxi that minimizes the expected total wait over a rolling horizon is computationally very demanding, simple heuristics have been devised. A simulation model is developed and the properties of several dispatching algorithms are explored. The performance of both the rolling horizon and the heuristic approaches is promising, but depends on the demand for taxi travel. The performance of the rolling horizon approach improves with the number of look-ahead stages, but the "curse of dimensionality" is the main impediment to the practical application of a multi-stage look-ahead period. The heuristics presented in this paper look particularly promising when demand is high, but further investigation is required. There are a few future research opportunities identified by this study. The potential of the proposed heuristics can be further investigated, taking the possibilities offered by recent developments in ICT into account. On the other hand, by adopting some simplifying assumptions, the heuristics can be evaluated via analytical approaches. Operational issues to be considered in the investigation could include "repositioning" (a practice whereby an idle taxi heads for a taxi rank rather than waits at the destination of the last customer) and "diversion" (the assignment of a job to an idle taxi en route to a taxi rank). The network congestion issue and its impact, which have been ignored in almost all previous studies, may

Optimal dispatching of taxis 647 affect the overall outcome. In so far as congestion leads to random variation and dependency in travel time, the simulation results suggest the heuristics tested still offer significant benefits.

ACKNOWLEDGEMENT The second author gratefully acknowledges the financial support of the Fellowship scheme of the Croucher Foundation of Hong Kong. We would also like to thank the participants of the BiNs workshop in Soeul, Korea, for suggestions following the presentation of an early draft of this paper.

REFERENCES Arnott, R. (1996) Taxi travel should be subsidized. Journal of urban economics, 40, 316-333. Bertisimas, D.J. and G. van Ryzin (1991) A stochastic and dynamic vehicle routing problem in the Euclidean plane. Operations Research, 39, 601-615. Bertsimas, D. J. and D. Simchi-Levi. (1996) A New generation of Vehicle Routing Research: Robust Algorithms, Addressing Uncertainty. Operations Research, 44, 286-304. Cordeau, J.F. and Laporte G. (2003) A tabu search heuristic for the static multi-vehicle dial-aride problem. Transportation Research, 37B, 579-594. Fleischmann, B., M. Gietz and S. Gnutzmann (2004) Time-varying travel times in vehicle routing. Transportation Science, 38(2), 160-173. Gendreau, M., G. Laporte and R. Seguin (1996) Stochastic vehicle routing. European Journal of Operational Research, 88, 3-12. Gendreau, M., G. Laporte and F. Semet (1997) Solving an ambulance location model by tabu search, Location Science, 2, 75-88. Gendreau, M., G. Laporte and F. Semet (2001) A dynamic model and parallel tabu search heuristic for real-time ambulance relocation, Parallel Computing, 27, 1641-1653. Ghiani, G., F. Guerriero, G. Laporte and R. Musmanno (2003) Real-time vehicle routing: Solution concepts, algorithms and parallel computing strategies. European Journal of Operational Research, 151, 1-11. Ichoua, S, M. Gendreau and J-Y. Potvin (2000) Diversion issues in real-time vehicle dispatching. Transportation Science, 34, 426-438. Kenyon, A. S. and D. P. Morton (2003) Stochastic vehicle routing with random travel times. Transportation Science, 37, 69-82. Larsen, A. (2000) The dynamic vehicle routing problem. Ph.D. thesis, Department of Mathematical Modelling (IMM) at the Technical University of Denmark (DTU). Available on the Internet. Liao, Z.Q. (2001) Taxi dispatching via global positioning systems. IEEE Transactions on engineering management. 48(3), 342-347. Lee, D.H., H. Wang, R.L. Cheu and S.H. Teo (2003) A taxi dispatch system based on current demands and real-time traffic conditions. 82nd Transportation Research Board Annual Meeting, 12 - 16 Jan 2003, Washington DC, USA.

648 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Lee, D.H., H. Wang and R.L. Cheu (2004) Trip-chaining for taxi advance bookings: A strategy to reduce cost of taxi operations. Proceedings of the 83rd annual meeting of the Transportation Research Board, Washington, D.C., U.S., January 11-15, 2004. Madsen, O.B.G., Ravn, H.F. and Rygaard, J.M. (1995) A heuristic algorithm for a dial-a-ride problem with time windows, multiple capacities, and multiple objectives. Annals of Operations Research, 60, 193-208. Minkoff A.S. (1993) A Markov decision model and decomposition heuristic for dynamic vehicle dispatching. Operations Research, 41, 77-90. Powell W.B. (1996) A stochastic formulation of the dynamic assignment problem, with an application to truckload motor carriers. Transportation Science, 30(3), 195-219. Psaraftis, H.R. (1988) Dynamic vehicle routing problems. In: Vehicle Routing, Methods and Studies (B.L. Golden and A.A. Assad, ed.), pp. 223-248. Elsevier Science, Amsterdam. Regan, A.C., H.S. Mahmassani and P. Jaillet (1995) Dynamic decision making for commercial fleet operations using real-time information. Transportation Research Record, 1537, 91-97. Taniguchi, E., T. Yamada, H. Hiroshima and M. Tamaishi (2002) Modelling dynamic vehicle routing and scheduling with real time information on travel times. In: Transportation and Traffic Theory in the 21s' Century (M.A.P. Taylor ed.), pp.329-347. Pergamon, Elsevier Science, Netherlands. Toth, P. and D. Vigo (2002) The vehicle routing problem. SIAM Monographs on Discrete Mathematics and Applications, Philadelphia. Wong, K.I., S.C. Wong and H. Yang (2001) Modeling urban taxi services in congested road networks with elastic demand. Transportation Research, 35B, 819-842.

33

CAPACITATED ARC ROUTING PROBLEM WITH EXTENSIONS Haiying Qiao, The Robert H. Smith School of Business, and AH Haghani, Department of Civil and Environmental Engineering, University of Maryland

BACKGROUND The objective of Arc Routing Problem (ARP) is to find the most efficient route to traverse the arcs of the network. ARP has many practical applications such as street sweepers, snow plowing and salting, postal delivery, meter reading, etc. Capacitated Arc Routing Problem is a capacitated version of Arc Routing Problem that has vehicle capacity constraint associated with satisfying arc demands. Golden and Wong (1981) developed the original formulation of CARP. According to that formulation, Golden and Wong proved that CARP is an NP-hard problem. CARP is a very useful extension to ARP, however, there are several other extensions that are useful and reflect real-world operations in many applications. These extensions include dealing with mixed networks, multiple depots and service route continuity. The field operational constraints often require that the routes should be operable. One simple operability requirement is that the operator can easily service the network without confusion. Service route continuity is a requirement that stipulates that the links in the network that are serviced by one vehicle should be connected to each other. The solution for traditional CARP does not impose this constraint and therefore, the service route for a vehicle can be discontinuous. That means the vehicle operator sometimes has to skip some links before servicing a link. In this case, it may be confusing for the operator to memorize which links require service and which ones do not. Therefore, even when the

650 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction route is optimal in the CARP context, it may not be easily operable. Service route continuity is a very practical requirement. Many real-world applications of arc routing problem have such constraints, but are ignored in the literature. In this paper, the service route continuity extension is also introduced in the original CARP formulation.

CARP FORMULATION WITH MIXED NETWORK, MULTIDEPOTS, AND SERVICE ROUTE CONTINUITY EXTENSIONS With slight modifications, Golden and Wong's formulation (1981) can be extended to mixed networks. We can deal with a mixed network as directed network and add constraints for the undirected links that ensure that the sum of the service of both directions should be one. The multi-depot extension is more difficult, because the original illegal sub-tour breaking constraints will not work in this case directly. A network transformation is designed to transform the multi-depot network to a single-depot network to satisfy the sub-tour breaking constraint. In this section, we will present the formulation for the Mixed CARP (MCARP) first and then discuss the network transformation necessary to extend to the multi-depot situation. The following notation is defined: G V D

(V, A) a connected directed graph. {l,....n} the vertex set {m + l,....n} Set of depots

V A A, <^A A2 c A Xijk Yijk

V\D {(i,j): i,j e V and i * j} is the arc set Undirected links Directed links 1 if and only if edge (i, j) is traversed from vertex i to vertex y on route k 0 otherwise 1 if and only if edge (i, j) is serviced from vertex z to vertex j on route k

Cy

0 otherwise Cost associated with arc (i, j)

Q qtj

Capacity of vehicle Arc (i, j) demand

k

(1,...,K) number of vehicle

The problem formulation is as follows:

Capacitated arc routing problem with extensions 651

Minimize

£ £C,X, t

(1)

k {i,j)zA

Subject to: Z^*-I^=°

zeF, foralU

Z(Ym

(iJ),(jJ)£A

+

YJit) = l

(2) (3)

k

IX =1

('J)^A

(4)

For all k, and (z, j) e 4

(5)

k

Y

uk ~ Yj* = 0

X

#

^ e {0,1}

For all k and (z, y) e ^

The objective function (1) seeks to minimize the total travel distance or cost for all vehicles. Constraints (2) ensure the route continuity. Constraints (3) and (4) ensure that each arc should be serviced as required. Constraints (5) require that a link must be serviced in both directions by the same truck. Constraints (6) are the capacity constraints. Golden and Wong (1981) suggested sub-tour breaking constraints through network flow. Define J m as a flow variable which can take on positive values only if tour elimination constraints are defined as follows:

Je

f.r
"

J ijk

For ieV,Vk,V(',J)eA

iJt

=1. The sub-

(8)

ijk

Note that the above constraints can break the illegal sub-tours in a single depot network, but they cannot be applied to the multi-depot network. For this purpose, we can add one more set of constraints to limit that the depot should only be passed once and should be the ending node of a trip: J^XJik=l

For n\\k,ieD

(9)

jeD

In reality, this constraint is too strict, since a route may pass by depot(s) several times. To solve the issue, we can transform the original network so that the optimal solution will not change by adding constraints (9). The transformation is as follows. For each depot, we add a duplicate node and duplicate all arcs that start from the depot or end at the

652 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction depot for the duplicate node. By this transformation of the original network, the path through the depot is equal to the path though its duplicate node. Therefore adding constraint (9) will not change the solution. But we should replace constraints (3) and (4) with following constraints for the links that are duplicated:

I X + yM+Yiik +YJik)=l

' c/.o e A

k

i is the duplicate node for i

(10)

i is the duplicate node for i

(11)

k

Service Continuity Extension As Haghani and Qiao (2002) suggested, the MST formulation can address the continuity very well by a network transformation. We can formulate the problem directly by integrating the Spanning Tree constraints with the CARP problem. The basic idea of this integration is that, for each route, which is a sub-network of the original network, if the links serviced in the route can be constructed as a spanning tree in the transformed network then the serviced links are connected. The difficulty of the implementation is that the spanning tree formulation is for a subset of the network that is the route assigned to a truck. However, the routes are not known in advance, so several binding constraints are used to formulate this problem. To integrate the MST and CARP together, the following notation is used: (y',A~)

Transformed network as discussed in Haghani and Qiao (2002).

v'

Node set corresponding to the link set in the original network.

A

Arc set which stands for the connectivity between the two links in the original network.

rijU

= 1, if the arcs {i,j),(k,l)

Nk

=1, n e V' if either Y? =1 or Y~ =1; 0 otherwise.

rnm

= 1 if the links denoted by n and m is connected,

k

k

is connected; 0 otherwise.

n,meV'.

k

Rm

=1 if N = 1 , N m = 1 and rnn =1 n e V', m e V', 0 otherwise.

/?*,

= 1 , which means the d u m m y node in the transformed network has a link to each node, 0 otherwise.

tkm

A decision variable for the spanning tree formulation and can be 1 only when Rknm = 1.

Capacitated arc routing problem with extensions 653 eknm

Set of variables that specifies the flow on link (n,m).

dn

Demand of the node n in the transformed network.

Note that in the transformed network n e K'is the node corresponding to the link (ij) and (j,i) in the original network. Nk =1 means that the node is serviced by vehicle k. The constraints for the service continuity extension are as follows: Y* + y* - Nkn >= 0

for

all (ij) e A and vehicle k

(12)

Yk + Yj, - KN* <= 0

for all (ij)

and vehicle k

(13)

K(N* + N"m - 2) + (1 - Rkm) < 0

for all n e V , m e V', rnm = 1, and vehicle A

(14)

(N* +N* _2) + A:(l-Rn*m)>0

for all « E F ' , SIEC',

r nm = 1, and vehicle^ (15)

<*„<«*„

for all neV , meV ,

rnm = 1, and vehicle A: (16)

y;»

for all n e K \l and vehicle A

(17)

y - t <j

for all « E K' \land vehicle k

(18)

e

f° r a ^ (n'm)e ^

l vehicle k

(19)

f° H (n,m)eA and vehicle k

(20)

L

s

=jV»

'L6

Ye* - Y e ' =/v'

ra

GA

an(

Constraints (12) mean that if y^+y*=o then Af*=O. Constraints (13) state that if y*+r* >i then Af* = 1. Constraints (12) and (13) together mean that if either or both directions of a link in the original network are serviced by a vehicle, then the corresponding node in the transformed network is also "serviced" by the vehicle (selected in the corresponding sub-tree). Constraints (14) enforce that if JV* =1, AT* = 1, and rnm = 1 then R^ =\. Constraints (15) state that if either Nk = 0 or JV* = 0, then Rkm =o- These binding constraints allow us to obtain the links that connect the nodes in the transformed network. Based on the above discussions, we know that the node set {AT* =1, neV'} is serviced by vehicle k. The link set {Rknm = 1, n,m e V'} is the set of links that connect these nodes. These node and link sets construct a sub-network of the transformed network. If the sub-network is connected, then a spanning tree must exist. Or if there

654 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction exists a spanning tree, then the sub-network (Nk,Rkm)

is connected. This means that the

links serviced by vehicle k are connected. Variables tknm are the decision variables for the spanning tree formulation and can be 1 only when links in the sub-network that are defined above are selected in the spanning tree. Constraints (16) ensure that the decision variables tknm

for spanning tree can only be 1 when Rkm = 1

. Constraints (17) - (20)

are the constraints for the Minimum Spanning Tree that force service route continuity. These constraints have already been discussed in the CMST formulation. With a minor modification, we can move the capacity constraint in the Arc Routing Problem to the Minimum Spanning Tree Problem. We can simply modify constraints (20) to constraints (21) and delete constraints (6): T V - T V =Nkd for all (n, m) e .4 and vehicle & (21) The proposed formulation was used to generate lower bounds for the MCARP. Several methods for finding lower bounds have been suggested for the traditional CARP in the literature. However, no procedure appears to have been proposed for finding lower bounds for the MCARP, let alone the MCARP with the above extensions. To generate lower bounds for the case study problem in this paper, we applied the Lagrangian Relaxation (LR) method and relaxed the capacity constraints in the above formulation as suggested by Pearn (1984). However, The lower bounds obtained from LR are not tight. The reason is that the total requirement of a route can hardly be close to the capacity of the truck. We also developed several other methods such as generating cuts and reducing the problem size based on its characteristics. Further development of procedures to obtain very tight lower bounds is an important area for future research. There are few algorithms in the literature dealing with the MCARP, let alone with the CARP with the extensions that we discussed in the previous section. However, Many algorithms have been proposed for undirected version of the CARP. The well-known algorithms are the Path Scanning Algorithm (Golden et al., 1983), the Modified Path Scanning Algorithm (Pearn 1989), the Augment and Merge Algorithm (Golden et al., 1983), the Construction and Strike Algorithm (Christofides, 1973), the Modified Construction and Strike Algorithm (Pearn, 1989), the Augment and Insert (Pearn 1989), and the Parallel Inserting Algorithm (Chapleau, 1984).

THE ALGORITHM In recent years, more and more meta-heuristic algorithms have been developed for the undirected CARP. These algorithms include: algorithms based on Genetic Evolution (Lacomme et al., 2001), algorithms based on Tabu Search (Hertz et al. 2000, Amberg et

Capacitated arc routing problem with extensions 655 al. 2000, Greistorfer 2003), algorithms based on Neighborhood search (Hertz et al, 2001, Beullens 2003), and algorithms based on Simulated Annealing (Lacomme et al., 2001). The underlying network in this research is a mixed network, since directed and undirected networks are special cases of the mixed network, the proposed algorithm should be applicable to both types of networks. The problem formulation extensions also make the algorithm more complicated and less efficient in solving undirected CARP than the algorithms in the literature. The improvement methods described in this section are based on the deletion and insertion operations. We design the algorithm for the Mixed CARP while keeping different extensions to the CARP formulation in mind so that the algorithm is applicable for solving extended formulations of the CARP. For mixed networks before any deletion and insertion operations for each non-directed link, the improvement methods check which direction of the link should be deleted from or inserted into a route according to the cost first, and then the respective operation is performed. If service route continuity constraints exist, before any deletion and insertion, we will check whether the operation will violate the service continuity constraints or not, if yes, the operation is not performed. Otherwise, we will continue the operation or check other constraints. Multi-depot extension requires the heuristic to determine to which depot the vehicle should return. To implement this, we will compare the costs of assigning the links that require service to different depots at runtime and choose the depot with the lowest cost. This check and assignment of new depot is done whenever an operation is completed. The proposed algorithm is a combination of several route construction and improvement heuristics that are based on link deletion, link insertion, link exchange, and route merging as well as Tabu search strategies. The detailed descriptions of these procedures are given in Qiao (2002) and are summarized below. Arc Deletion and Insertion Operations Arc Deletion Operation This operation deletes one serviced link from a specific route. When a link that is serviced by a vehicle is deleted, the neighboring links that are not serviced by the route are also deleted. This is because these links only serve as means of connection for the link that is serviced in the vehicle's route. When the serviced link is deleted, there is no need to traverse these links that are not serviced.

656 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction As an example, consider the route 0 15 8 2 3 0 shown in Figure 1, in which links 5-8 and 2-3 are serviced (solid line). Assume that we want to delete link 5-8. After deleting link 5-8, we need to delete link 1-5 and 0-1 because the route does not service these links. The remaining link to be serviced is 2-3. If the shortest path from depot to node 2 is link 0-2 then the reestablished route after deleting link 5-8 will be: 0 2 3 0. Before deleting link 5-8

Figure 1. Example 1 for deletion operation There is another scenario as well. A route traverses a link twice, but the link is serviced just once. In this case, we may swap the service and possibly delete the link originally serviced by the route. For example, consider the route 0 1 5 8 2 3 4 3 2 0 shown in Figure 2. Assume that we want to delete link 5-8. After deleting link 5-8, we may also delete link 2-3 and consider servicing the link in the opposite direction if link 2-3 is un-directed. This may result in route 0 4 3 2 0 as shown in Figure 2. Arc deletion operation works according to the following steps: 1. Find the place of the link to be deleted in the route. 2. Delete the links that are not serviced before the link and after the link. 3. Check whether the next link that is serviced by the route can be serviced elsewhere in the route. If yes, delete the link. 4. Reconnect the route by adding the shortest paths in between. Before deleting link 5-8

Figure 2. Example 2 for deletion operation

Capacitated arc routing problem with extensions 657 Arc Insertion Operation This operation tries to find the best position to insert a link into an existing route. Two cases need to be considered. The first case is when there are link(s) in the route that are not serviced by the route. In this case, we will try to delete all continuous un-serviced links, insert the link under consideration for insertion in these gaps and choose the insertion with the lowest cost. As an example, consider the route 0 2 5 8 9 0 shown in Figure 3. Link 5-8 is the only link to be serviced in this route. Now consider link 2-3 for insertion. Assume that the shortest distance from 0 to 2 is 3 and from 3 to 5 is 4. The distance 0 2 5 is 5. After inserting link 2-3, the route will be 0 2 3 5 8 9 0 (please note that link 2-3 is not in the route). Therefore, the insertion cost is 3+4-5 = 2. The procedure is shown in Figure 3. Before inserting link 2-3

Figure 3. Example 1 for insertion operation The second case is when all links in the route are serviced by it. In this case, we have to find the best position based on the nodes instead of the gaps. As an example, consider the route 0 12 5 8 9 0 shown in Figure 4. All links are serviced in this route and we are considering inserting link 2-3 in the route. The nearest node is 5, which means that the distance of link 3-5 is less than that of link 3-2. So the final route is 0 1 2 5 2 3 5 8 9 0 . The algorithm allows a loop in the route. The insertion operation is accomplished by performing the following steps: 1. For consecutive un-serviced links, find the start nodes and end nodes. 2. Delete these consecutive un-serviced links. 3. Insert the link under consideration into the route by connecting the link with the shortest paths. 4. Compute the insertion cost (The insertion cost is the cost of the route after insertion minus the cost of the route before insertion.) 5. Repeat steps 1-4 and find the insertion place with most savings.

658 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 6. For every node in the original route, insert the link into the route by connecting the link with the node via the shortest path. 7. Compute the insertion cost. 8. Repeat steps 6-7 and find the insertion place with least cost. 9. Compare the costs obtained from steps 5 and 8. Realize the insertion by inserting the link into the place with the smallest cost. Before inserting link 2-3

Figure 4. Example 2 for insertion operation Initialization Heuristic Methods In this section, we will introduce two different initialization methods. The first one is a simple method, which assigns a truck for every link that needs service. The second one is based on a greedy insertion algorithm. Initialization Method 1: Simple Method This initialization method is to assign each link to be serviced a dedicate truck. The algorithm for this initialization method can be described as: 1. Select an un-serviced link in the network. Find the shortest path from the depot to the start node of this link and the shortest path from the end node to the depot. 2. Construct a route by linking the two shortest paths and the link, set the link as serviced. 3. Repeat steps 1-2 until all links are serviced.

Capacitated arc routing problem with extensions 659 Initialization Method 2: Nearest Link Insertion Method The second method is a little more complicated. The idea behind this is that we want the links that are serviced by a route to be close to each other. Instead of inserting a link that has the least insertion cost in a route, in all routes we insert links that are closest to all serviced links in those routes. The procedure is described below: 1. Select the furthest un-serviced link from the depot in the network. 2. Connect the link to the depot with the shortest paths to construct a route, set the link as serviced. 3. Find the nearest connected link to the links currently serviced by this route, insert the link in the route. 4. Repeat step 3 until the capacity constraint is reached. 5. Repeat steps 1 through 4 until all links are serviced. 6. The advantage of this initialization is that it may provide good solutions quickly and some merits of good solutions are already obtained at the very beginning. Improvement Heuristic Methods Few improvement methods are suggested and implemented in the literature. In this research, we focus on developing improvement methods. The two heuristic methods to obtain the initial solution described above are very simple. The proposed improvement methods can be applied to all solutions obtained. Improvement Method 1: Merge Heuristic This improvement method is similar to the Golden and Wong's merge algorithm (1981) to merge two routes into one route. One limitation of the original merge algorithm is that the original sequence of the service in each route should be kept and the other limitation is that the algorithm is designed for undirected networks. By using insertion and deletion operations, the Merge Heuristic does not necessarily have to keep the original sequence of service, and therefore, more efficient routes can be found. Since the underlying network is a mixed-network, the direction of service should be kept for directed links. This is different from the Golden and Wong's merge algorithm. The Merge heuristic procedure can be described as follows: 1. Select two routes. 2. Select all links serviced by one route. 3. Check capacity constraints before inserting the links into the other route. If the check fails, go to step 5.

660 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 4. If the cost of the resulting route is less than the original cost, add the resulting route to the solution and delete the original routes from the solution. 5. Loop over all pair-wise routes, until no further improvements can be made. The Merge Heuristic reduces the number of routes very quickly, so it should be used at the very beginning of the search. It is seldom applied in the refinement stages since all routes are close to the capacity. Improvement Method 2: Delete and Insert Heuristic This improvement method borrows the idea from the Or-opt heuristics for the VRP as most researchers have suggested. This method deletes one link from one route and inserts the deleted link into another route. The procedure attempts to find the largest cost saving after the deletion and insertion. Delete and insert algorithm can also reduce the number of the routes if the link to be deleted is the only link serviced in a route. The procedure is as follows: 1. Select two routes (i.e. routes A and B) 2. For each link serviced by route A, check all constraints before deleting the link from the route. If all checks succeed, delete the link. Otherwise, go to step 4. 3. Check capacity constraints before inserting the deleted link into route B. If the constraints are not violated, insert the link into route B and compute the cost savings, otherwise go to step 2. 4. Loop until all links in Route A have been checked. 5. Choose the resulting route(s) with the largest savings, add these route(s) to the solution, and delete the original two routes (A, B) from the solution. 6. Loop over all pair-wise routes, until no further improvement can be made. The limitation of this algorithm is that it deletes and inserts link(s) between two routes. It is possible to improve the solution by deleting more than one link from a route and inserting these links into different routes. Investigating these possibilities is left for future research. Improvement Method 3: Link Exchange Heuristic Link exchange method also borrows the idea from the crossing heuristics for the VRP. The idea of this method is to improve the solution by exchanging two links between two routes. Since we have already defined the deletion and insertion operation, the procedures of this heuristic is very simple. The implementation of improvement method is described as follows. 1. Select two routes (i.e. routes A and B). 2. For each link serviced by route A, check the capacity constraints before deleting the link from the route. If all checks succeed, delete the link from the route. Otherwise, check the next serviced link in route A.

Capacitated arc routing problem with extensions 661 3. For each link serviced by route B, check capacity constraints before deleting the link from route B. If all checks succeed, delete the link. Otherwise, check the next serviced link in route B. 4. Check capacity constraints before inserting the link deleted from route B into the route A. If all checks succeed, apply the insertion operation. Otherwise, check the next serviced link in route B. 5. Check capacity constraints before inserting the link deleted from route A into route B. If all checks succeed, apply the insertion operation. Otherwise, check the next link in route A. 6. Compute the cost savings of the two resulting routes. 7. Loop until all links in Route B have been checked. 8. Loop until all links in Route A have been checked. 9. Choose the resulting routes with largest saving, add the resulting routes to the solution, and delete the original two routes from the solution. 10. Loop over all pair-wise routes, until no further improvements can be made. Link Exchange heuristic is a very useful improvement method when the number of routes is equal or close to the minimum number of trucks. One constraint of the algorithm comes from that it must exchange the links, so the number of routes is the same. Usually this procedure is applied in the refinement stages. Improvement Method 4: Multiple-link Exchange Heuristic We implement this heuristic as a complement of the Link Exchange Heuristic. The basic procedure is the same as the Link Exchange Heuristic. The difference is that the number of links deleted from a route or inserted into a route can be more than one and the links to be deleted or inserted are also randomly selected. The number of exchanged links can be 2, 3 or more. The implementation of the heuristic is similar to improvement method 3. The implementation of the improvement method is described as follows (we use exchange of 2 links as an example. For exchange of 3 links, we need to choose two links randomly in steps 2 and 4): 1. Select two routes (i.e. routes A and B). 2. Select a link and randomly choose another link serviced by route A. 3. Check capacity constraints before deleting the links from the route. If check succeeds, delete the selected links from the route. Otherwise, stop. 4. Select a link and randomly choose another link serviced by route B. 5. Check capacity constraints before deleting the links from route B. If check succeeds, delete the links. Otherwise, stop.

662 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 6. Check capacity constraints before inserting the links deleted from route B into route A. If capacity checks succeed, apply the insertion operation. Otherwise, go to step 10. 7. Check all constraints before inserting the links deleted from route A into route B. If check succeeds, apply the insertion operation. Otherwise, go to step 9. 8. Compute the cost savings of the two resulting routes. 9. Loop until all links in Route B have been checked. 10. Loop until all links in Route A have been checked. 11. If there is a saving, then add the resulting routes to the solution and delete the original two routes from the solution. It is possible that the total number of serviced links in a route is less than the number of the links to be exchanged. For example, route A just services two links, while improvement method is a 3-link exchange. In this case, we just delete the two links from route A and insert three links into the route. That is the same as the Merge Heuristic. Improvement Method 5: Random-link Exchange Heuristic We implement this also as a complement of the Link Exchange Heuristic. The basic procedure is the same as the Link Exchange Heuristic. The difference is that the number of links deleted from a route or inserted into a route is randomly generated. Since this procedure is also very similar to the Multiple Link Exchange Heuristic, the detailed description of the procedure is not included. Depending on the random numbers that are generated, this improvement method includes all previous improvement methods exchange possibilities. Since it includes a random process, it is difficult to predict the performance of this method. This method is applied in the route refinement stages also.

Avoiding Local Inferior Area It is very common that most heuristics get trapped in local inferior area. For example, if a link is inserted into a route and an improvement is achieved, we may obtain further improvement by deleting the link from the route. A route can also be so "good" that no achievement can be made by the heuristic while other routes are all "bad". Tabu search is a meta-heuristic strategy that forbids revisiting the visited area in a certain amount of time or within a certain number of iterations (Tabu tenure). By deferring revisiting these areas, this strategy helps us search larger solution space while keeping the memory of possible good areas in a repository called Tabu List, hi the proposed algorithm, some basic Tabu Search strategies are applied to avoid local inferior areas. These strategies are described in detail in Qiao (2000) and summarized below.

Capacitated arc routing problem with extensions 663 Tabu Types To avoid local inferior areas, we need to know what types of these areas (or Tabu types) exist. Different Tabu types mean different local inferior areas that should be avoided. First, the Tabu types are defined according to the two basic operations described above, namely the insertion and the deletion operations. For example, if a link is inserted into a route, in the next several iterations, deleting the link from the route again is forbidden. This will help avoid cycling. Typical Tabu types to avoid cycling are suggested by Glovers (1993). Local inferior solutions are not only caused by simple cycling, but also by other factors. For example, an inferior solution can happen when the link exchange algorithm is applied several times to improve two routes and the two routes are optimized just within the scope of the two routes. It is possible that we cannot exchange the links in these routes with any other routes to improve the solution since the two routes are already too good to be changed. Considering that the purpose of the Tabu Search is to avoid local inferior solutions, we expand the Tabu types to our improvement methods. By applying Tabu Search, we try to avoid reusing or revisiting an improvement method in a certain time (Tabu tenure). As discussed in the previous section, there are several improvement methods we designed. A Tabu for one improvement method is not necessary a Tabu for another improvement method. For each improvement method and each route pair, we define a Tabu to restrict the application of the improvement method during the Tabu tenure. When a Tabu is defined, it should not be kept in the Tabu area all the time, after certain number of iterations, the Tabu must be removed and the algorithm should be allowed to search the Tabu areas and apply the improvement methods again. A mechanism is needed for Tabu expiration.

Tabu Expiration All current Tabu will be stored in a Tabu list. A Tabu may expire in certain time or after certain number of iterations that is called the Tabu tenure. Tabu tenure is an important parameter in a Tabu search algorithm. A tenure duration that is too large or too small may not provide good solution. Some scholars also use variable Tabu tenures in their algorithms. To simplify our algorithm, we just use a fixed Tabu tenure.

664 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In the proposed improvement methods, the length of the Tabu tenure is related to the number of routes. We approximate the number of routes using the following formula: L ' -I. Where T is a measure of the total service requirements for the whole network and W is a measure of the capacity of the truck to provide the service. Find a good Tabu expiration mechanism is an interesting area for future research. Elite Route Pool (ERP) In the previous section, we defined the Tabu area that the next solution tries to avoid. We may also define an area that is attractive. We define this area as Elite Route Pool (ERP). Elite Route Pool is a pool of routes in good solutions. Every route in the pool has a weight, which is the frequency with which the route appears in the solutions. The Elite Route Pool is where we save the elite route information. The next initial solution for improvement can be generated from the Elite Route Pool. The procedure is described below: 1. Randomly select a route in ERP according to the frequency (weight) of the route. 2. Delete the links that have already been serviced by other routes in the initial solution. 3. Add the route to the initial solution. 4. Repeat the steps 1 through 3 until all required links in the network are serviced. By keeping the weight of the routes in the ERP, the routes with high weights have a higher probability of being chosen in the next initial solution. Any improvement methods designed for the Arc Routing Problem can also be applied to improve the next initial solution. This enables us to apply a more flexible and strategic search. The Proposed Algorithm For different problems or different purposes, we may use different combinations of above initialization methods, improvement methods and Meta heuristic methods. The final heuristic procedure for a problem depends on the characteristics of the problem such as time requirements. It is not necessary to utilize all above improvement methods or metaheuristics in all heuristic procedures. For example, a random link-exchange heuristic normally takes longer and may not improve the solution in quite a long time. For some time contingent applications or real time applications, we may use the quick initialization algorithms and delete and insert algorithms to find an acceptable solution in a short time. In the next section, we will solve some benchmark problems in the literature. The results are compared with those of some existing algorithms. The following heuristic procedure is used for solving the benchmark problems: 1. Apply the initialization method 1 or 2. 2. Apply the improvement method 1 - Merge Heuristic. 3. Apply the improvement method 2 - Delete and Insert Heuristic.

Capacitated arc routing problem with extensions 665 4. Apply the improvement method 3 - Exchange Heuristic. 5. Apply the improvement method 4 - 2 Links Exchange Heuristic, 3 Links Exchange Heuristic successively. 6. Apply the improvement method 5 - Random Link Exchange Heuristic. 7. Add the current best solution to the Elite Route Pool (ERP) and get a new initial solution from ERP. 8. Repeat steps 2 and 7 until a certain number of iterations reached. Note that although the above procedure does not explicitly mention the Tabu search strategy, in every step we check whether or not the operation or the improvement method is an active Tabu. Therefore, the Tabu search strategy is implicitly applied throughout the procedure.

TEST RESULTS The problem set contains 23 problems described in DeArmon (1981). The original networks have between 7 to 27 vertices and between 11 to 55 links. Since our algorithm is designed for the MCARP, we replace each undirected link with two directed ones but require only one of them be serviced. The proposed test algorithms are compared with eight other existing algorithms, these algorithms are: Pearn's algorithm (Pearn), Pearn (1984). Augment and Merge algorithm (Aug-Mge), Pearn (1984). Path Scan algorithm (Path Scan), Golden et al. (1983). Construction and Strike algorithm (Con-Ske), Christofides (1973). Modified Path Scanning (MPS), Pearn (1989). Belenguer algorithm (Bel), Belenguer and Benavent (1997). Hertz Tabu search based algorithm (Carpet), Hertz et al. (2000) Hertz neighborhood descent algorithm (VND), Hertz (2001). The above algorithms are designed only for the CCPP. There are some other results reported in the literature. It appears that VND and Carpet are currently the best. Details of these algorithms have been discussed in the literature. Our algorithms are originally designed to solve the mixed CARP, which is more general than the CCPP. The lower bounds and results for existing algorithms are from Golden and Wong (1981), Belenguer (2002), and Amberg (2002). For convenience, we will use a table to compare the results. Table 1 shows the results for different algorithms. From the table, we can find that the proposed algorithm can out-perform many algorithms such as Pearn's algorithm, path scanning, construction and strike algorithm, but it is not as good as Carpet and VND. This means the proposed algorithm still has more room to improve.

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Capacitated arc routing problem with extensions 667 Since the proposed algorithm is designed for the MCARP with other extensions and we transformed the undirected networks to directed ones which doubles the size the network, the proposed algorithm is certainly more complicated and less efficient than the algorithms designed only for undirected network. Unfortunately no comparable algorithm for MCARP exists to compare with the proposed algorithm.

CASE STUDY

Background This case study is based on a real-world snow maintenance operation in Maryland. In Maryland, the snow routes are separated into smaller segments according to the counties. These salt domes are located strategically to make sure that all routes in the county can be serviced within two hours. If a large snowstorm comes, more trucks will be needed to service the same area and the operation team rents trucks according to contracts that are normally signed in advance with local providers to make sure trucks are always available during a storm. Maryland State Highway Administration (MD SHA) requires that all roads should be serviced at least once every two hours and the same route be serviced by the same truck during a storm. This means that a truck will not service other roads during the same storm. Given the above discussions, the following is a list of the key characteristics of the MD SHA snow operation. 1. The entire highway network is separated along the county lines. Each county is served by its own maintenance crew. 2. Each county has one or more salt domes. 3. A truck should service the same roads each time. 4. All roads should be serviced at least one time by the trucks without reloading salt. 5. Service routes should be balanced. 6. Service routes should be continuous. 7. One pass of the truck can service two lanes. That means both sides of a two-lane highway can be serviced in one pass. 8. Both sides of a road are serviced by the same truck. 9. The same route assignments are used for salting and plowing operation. The data for case study is from Calvert County, MD. The entire network of service has 42 nodes and 104 links and is divided into four sections (Sections I, II, III, and IV). Some roads are 4-lane highways and some are 2-lane highways. All 2-lane highways can be serviced by one pass from either direction. Currently, there are a total of 14 trucks that are servicing the network. The total travel distance for current assignment is 431.7 miles and the deadhead distance is about 267.1 miles. The route structure is designed by the maintenance personnel based on their many years of experience considering many practical limitations. Currently, all trucks used in Calvert County have the same capacity of 16,000 lbs. These trucks can service

668 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 32 lane-miles in a medium storm. In this case study, we will apply the proposed algorithm to design salting routes for the combined Sections I and II, combined Sections III and IV and the entire service network as a whole. Results We applied the algorithm described above to solve the case study problem. We also used 16 different Tabu tenures and chose the best solution among them. Our results indicate that the heuristic algorithm can save 2 trucks, about 8.89% in total travel distance, and about 34.6% in total deadhead distance for the entire service network when the entire network is considered simultaneously for route design. When we consider smaller service networks of Sections I and II and Sections III and IV, good lower bounds can be obtained. The gap between the lower bounds and the heuristics results are 3.96% for Sections I and II and 6.41% for Sections III and IV. The lower bound gap for the entire service network is not as good (27.16%). The number of trucks needed is 5 for Sections I and II and 7 for Sections III and IV that are the same as the numbers we obtained through the CMST formulation (Haghani and Qiao 2002). A simple truck capacity and service requirements matching analysis indicates that at least 11 trucks (164.62/16=10.29) should be used to service the network (not considering any other operational constraints). The result of 12 trucks is very close to this lower bound. Since the total travel distance for the entire network is almost equal to the sum of the results for Sections I and II and Sections III and IV, we may also conclude that Sections I and II and Sections III and IV can be separated. This also conforms to the topological map of the network. Sections I and II are the upper part of the network and Sections III and IV are the lower part of the network. The two parts are connected by one node and no other link connects these two parts. We also applied the procedures mentioned in Section 2 to obtain lower bounds for small networks, i.e. section I & II (14 nodes and 34 links) and section III & IV (28 nodes and 70 links), the gaps between the lower bounds and the proposed algorithm solutions are 3.96% and 6.41% respectively. For the whole network (42 nodes and 104 links), the procedures for obtaining lower bounds did not produce very good results (the gap is 27.20%). As mentioned before, developing methodologies for obtaining good lower bounds is an important area for future research. Sensitivity Analysis We can analyze the impact of changes in values of the important parameters in the problem formulation using the heuristic algorithm. The first parameter we analyzed is the truck capacity constraint. We tested different capacities: 28 lane-miles, 32 lane-miles, 36 lane-miles and 40 lane-miles. The results show that different truck capacities have a significant impact on the final deadhead distances. The deadhead decreases with increase in capacity. The reason is that with larger capacity, a truck can service more area, so the total number of trucks is reduced. Therefore, the deadhead is

Capacitated arc routing problem with extensions 669 also reduced. Another reason is that a truck can service more "remote" areas, so other trucks do not need to travel a long distance to service those areas again. According to this analysis, considering different salting routes for different weather conditions may reduce deadhead and improve efficiency. The second parameter we analyzed is the service route balancmg parameter. The service route balancing requirement is very important for the operability of the routes. However, a very strict service route balancing requirement may or may not increase the we test different service route balancing parameters such as, no service route balancing requirement, a relaxed requirement where the demand is between 16 and 32 lane miles, a moderate requirement where the demand is between 20 and 32 lane miles, and a strict requirement where the demand is between 24 and 32 lane miles. The results show that the service route balancing constraints do not have a significant impact on the final solutions. The main reason for this is that the truck capacities and the network structure are such that an inherent balance will be created by the designing of the routes to satisfy other constraints. The last parameter we analyzed is the location of salt domes. The results indicate other possible locations such as nodes 32 and 7 may perform better in terms of the total deadhead distance. Other node pairs such as 30 andl5, 32 andl4 and 28 and 7 are also preferable to the dome locations when considering deadhead distances. One should note that there are perhaps other operational constraints in locating the domes that are ignored in our analysis. This can also be observed from the map easily. We also tested different locations for the main depot. The results indicate that the original location of depot (node 0) is the best location if only the deadhead distance is considered.

CONCLUSIONS AND FUTURE RESEARCH In the paper, several improvement methods for solving the CARP suggested and embedded in a heuristic procedure. The results of implementation of this procedure on benchmark test problems look very promising as compared to other existing algorithms. The case study application indicates that the proposed procedure has the capability of solving real world problems and obtaining good solutions. Sensitivity analyses with respect to various important parameters in the case study show that the procedure performs as expected with respect to changes in these parameters and can be applied to analyze depot locations. In this research, a fixed Tabu tenure is used and different Tabu tenures are tested. This method is not efficient since we do not know what Tabu tenure is good. A better way to design the Tabu tenure is desirable. Viable Tabu length can be used according to previous search experiences.

670 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction A more advanced idea to improve the Elite Route Pool strategy is that instead of constructing the next initial solution from the routes in the ERP, the next initial solution can be constructed from the patterns. So instead of saving elite routes into the pool, we can save the common patterns of the Elite Pattern Pool (EPP). For example, a pattern can be defined that links A and B are serviced in one route or links A and B are not serviced together. These patterns are obtained from previous computing experiences. By using patterns, the information about meritorious solutions is stored in a smaller set of links instead of the whole route, so route construction is more flexible and more accurate. When an elite route is added to the pool, the information of the elite route is digested into patterns and stored in the EPP. Defining the patterns and finding them will be a challenging task. As discussed before, more improvement methods can be developed. For example, link exchange improvement can be performed among 3 or more routes. All the improvement methods we suggested are pair-wise improvements. We can expand the scope to whole sets of routes instead of only two routes.

REFERENCES Amberg, A., and S. Voss, 2002, "A Hierarchical Relaxations Lower Bound for the Capacitated Arc Routing Problem," Proceedings of the 35th Hawaii International Conference on System Sciences. Amberg, A., W. Domschke, and S. Voss, 2000, "Multiple center capacitated arc routing problems: a Tabu search algorithm using capacitated trees," European journal of operational research, 124, 2, 360-376. Belenguer, J. M., and E. Benavent, 2002, "A cutting plane algorithm for the capacitated arc routing problem", Computers & operations research. 30, 5. Beullens P., L. Muyldermans, D. Cattrysse, and D. Van Oudheusden. 2003, "A Guided Local Search Heuristic for the Capacitated Arc Routing Problem," European Journal of Operational Research, 147, 3,629-643. Chapleau, L., J. A. Ferland, G. Lapaleme and J. Rousseau, 1984, "A Parallel Insert Method for the Capacitated Arc Routing Problem," Operations Research Letters, 3, 95-99. Christorfides, N., 1973, "The Optimum Traversal of a Graph," Omega 1, 461-470. Glover, F., and M. Laguna, 1993, "Tabu Search," in Modern Heuristic Techniques for Combinatorial Problems, edited by C. Reeves, Blackwell, Oxford, 77-150. Golden, B. L., J. S. Dearmon and E. K. Baker, 1983, "Computational Experiments with Algorithms for a Class of Routing Problem," Computers Operations Research, 47-59. Golden, B. L. and R. T. Wong, 1981, "Capacitated Arc Routing Problems," Networks, SOSSIS. Greistorfer, P., 2002. "A Tabu Scatter Search Metaheuristic for the Arc Routing Problem," Computer and Industrial Engineering, 3, 129-135. Haghani, A., and H. Qiao, 2002, "Snow Emergency Vehicle Routing with Route Continuity Constraints," Journal of Transportation Research Board 1783, 119-124. Hertz, A., and M. Mittaz, 2001, "A Variable Neighborhood Descent Algorithm for the Undirected Capacitated Arc Routing Problem," Transportation Science, 35, 4, 425434.

Capacitated arc routing problem with extensions 671 Hertz, A. G. Laporte, and M. Mittaz, 2000, "A Tabu search heuristic for the capacitated arc routing problem," Operations Research, 48, 1, 129-135. Lacomme, P., C. Prins, W. Ramdane-Cherif, 2001, "A Genetic Algorithm for the Capacitated Arc Routing Problem and Its Extensions," EvoWorkshops, 473-483. Pearn, W. L., 1984, "The Capacitated Chinese Postman Problem," Ph.D. Dissertation, University of Maryland, College Park, MD. Pearn, W. L., 1989, "Approximate Solutions for the Capacitated Arc Routing Problem," Computers Operations Research, 589-600. Qiao, H., 2002, "Capacitated Arc Routing Problem: Formulations, Algorithms and Application," Ph.D. Dissertation, University of Maryland, College Park, MD.

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34

USER-EQUILIBRIUM ROUTE SET ANALYSIS OF A LARGE ROAD NETWORK Hillel Bar-Gem, Ben-Gurion University ofthe Negev, Be'er Sheva, Israel, and David Boyce, Northwestern University, Evanston, Illinois, USA

INTRODUCTION By 1953 Martin Beckmann had formulated and analyzed the static, deterministic userequilibrium (UE) route choice model with variable origin-destination flows (demand) (Beckmann et al, 1956; Boyce, 2004). Twenty years later, Suzanne Evans (1973, 1976) proposed and analyzed an algorithm for solving a version of Beckmann's model that corresponded to practitioner models with origin-destination (OD) constraints and transit as well as road modes. The effectiveness of her algorithm for large networks was studied by several researchers; see Boyce and Bar-Gera (2004) for a recent review. Other young researchers at that same time proposed and tested algorithms for solving the user-equilibrium problem with fixed OD flows (LeBlanc et al, 1975; Nguyen, 1974; Florian and Nguyen, 1976), building upon the pioneering theoretical research of Jorgensen (1963), Dafermos (1968) and Brynooghe et al (1969). More recently, Patriksson (1994) thoroughly synthesized the accumulated literature from this 40-year research effort, and proposed his own userequilibrium algorithms for fixed and variable flow problems (Larsson and Patriksson, 1992; Lundgren and Patriksson, 1998). Nagurney (1999) synthesized a related literature on the variational inequality approach to a family of problems, including user-equilibrium; Florian and Hearn (1999) offered their review, which includes recent developments on road pricing. Despite these highly creative and sustained efforts over a 50-year period, as well as steady progress in applying some of these findings to practical problems of urban travel forecasting, effectively nothing is known about the characteristics of the route sets that comprise the solution of the UE model for a large road network. Basic questions, such as the following, remain unanswered:

674 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction 1. What are the number, range and frequency of routes per OD pair in the UE solution, that is a solution in which all used routes have equal travel costs, and no unused route has a lower travel cost? 2. How does the number of routes per OD pair vary with OD generalized travel cost, OD flow and direction from each origin to destination with respect to the road network? These questions are the focus of this chapter. Such questions are important, not only because they probe the basic properties of this commonly applied route choice model; they also help us to assess whether the model is a sound choice for forecasting future travel. Moreover, knowing the answers to these questions could stimulate empirical research on corresponding questions concerning actually used routes, which would provide for the first time an effective basis for route choice model validation. See Bovy and Stern (1990) for an extensive review and analysis of route choice behaviour from many viewpoints. Several additional questions should be addressed eventually, but are not considered here: 1. How different are the routes (sequences of links) that comprise multiple-route sets between OD pairs? Harris (2002) made an initial exploration of such route sets using data similar to those analyzed here. 2. As the route flows in the deterministic UE solution are not unique (Patriksson 1994, p.44), what is the range and distribution of UE route flows among alternative solutions; how do they compare with most likely UE route flows? 3. How does the UE route set vary with the definition and parameter values of both the generalized cost function and the deterrence function in the UE model, which partially determine the level of congestion in the road network? 4. How is the route set affected by the topology, detail and coding conventions of the road network, such as with respect to turns and turn penalties, and the structure and level of service of the transit network? 5. How is the route set affected by the pattern of origin-destination flows? 6. For all of the questions listed above, we are interested in answers for a highly converged solution. Moreover, the answers should not be affected by limitations of the solution algorithm, such as on the number of routes in the solution, for example. The capability to solve the static, user-equilibrium problem precisely for both small and large networks became available for fixed OD flows only in 1999 with the publication of the Origin-Based Assignment (OBA) algorithm by Bar-Gera (1999, 2002). An extension to the case of variable OD flows over multimodal networks was presented by Bar-Gera and Boyce (2003). To our knowledge, only the OBA algorithm is capable of producing solutions with the detail and precision necessary to answer the questions listed above. Our approach to answering these questions may be described as exploratory and intuitive. Given a single, highly-converged solution of a single-class, two-mode, user-equilibrium model with negative exponential deterrence functions for the Chicago Region, we explore the first two questions listed above. We emphasize that our findings pertain to this one example. Our tools of analysis are spreadsheets and graphs, relying heavily on visual expressions of the results to formulate research hypotheses and reach tentative conclusions. The analyses were initially more detailed, and then aggregated for this synthesis.

User-equilibrium route set analysis 675 The presentation of the results is organized as follows. First, a brief, technical description of the model is offered, together with parameter values and overall summary measures such as mean generalized costs by mode, total regional flow by mode, etc. Certain properties of the route set and the convergence of the solution are also explained. Next, we present aggregate results concerning the route pattern on the road network, such as the number of OD pairs classified by the number of routes per OD pair. With this background, we then present more detailed analyses to consider how the number of OD pairs, and the total number of routes, in each class of routes per OD pair varies by OD generalized cost, OD flow and direction from the origin to the destination. For each analysis, selected graphs are presented to support our observations and conclusions. Interpretations of the findings and a brief conclusion complete the chapter.

MODEL DESCRIPTION We consider a study area that is divided into a set of zones, Z, connected by transit services and by a road network. The road network consists of a set of nodes, N, and a set of directional links, A. In the Chicago regional model there are 1790 zones, 12,982 nodes and 39,018 links. A route is a sequence of nodes, [ v ^ . . . ^ ] , such that [v;,vl+1]e A . The set of available routes from origin peZ to destination qeZ is Rpq, and the set of all routes is R. The purpose of the model is to predict: (a) the mode-origin-destination (MOD) flow dmpq, in persons per hour, for every origin psZ, destination qeZ, and mode me{a=auto, t=transit}; (b) the distribution of auto OD flows to route flows hr, for every route r s R . Auto OD flows are the sum of person-trips divided by a constant auto occupancy factor (aof= 1.2) and the given OD truck flows dpqck, in equivalent passenger cars per hour: dp"q'° = dapq/aof + dpqct; hence J^hr = d"pq'°. Total link flows are the result of route flow aggregation, fa= ^hr. r&k

i>1

A

refiner

solution is feasible if it respects the constraints on total origin flows, ^dmpq =d.p. , and on mq

total destination flows, ^dmpq =d^q, where d.p. and d..q are given inputs. In the model mp

implemented for the Chicago Region, total MOD flows amount to 1.5 million (1,513,211) persons per hour, while total truck flows amount to 0.4 million (445,184) passenger-carequivalents per hour, representative of the morning peak period (6:30 - 8:30 am) in 1990. The remaining model inputs refer to transit and road levels of service. Transit data are invehicle travel time c"pq, out-of-vehicle travel time c°pq", and fare c{pqe, for travelling from origin peZ to destination qeZ by transit. These are fixed regardless of flows. Origindestination generalized cost by transit, um, is a weighted sum of the three components plus a constant bias; in the results reported here the transit bias is zero, and the weights are 0.25, 0.90, and 0.08 respectively.

676 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Travel

time

by

auto

on

link

a,

tta,

is

a

function

of

total

link

flow,

4

tta(fa) = tt° (l + 0.15-(/a/^) ), where ff°and ka are respectively the free-flow travel time and capacity of the link. Link generalized cost is ta = tta(fa)+O.\5-la,

where la is the link

length and the coefficient, 0.15, reflects a combination of both the direct effect of distance on generalized cost and the indirect effect of fuel consumption. Fixed additional auto costs, acapq, account for the parking fee and out-of-vehicle travel time at the origin and destination. The route generalized cost is cr = acapq + Z?o The minimum OD generalized cost by auto is uapq = min{cr: r e Rpq \. For every route r e Rpq, define the excess cost as: ecr = cr -uapq. The user-equilibrium assumption is that the excess cost of every used route is zero. Approximate UE solutions are evaluated by the maximum excess cost over all used routes. MOD flows have the doubly-constrained Logit form, dmpq =Ap-Bq -exp^-^-w^), with cost sensitivity /J = 0.2, and balancing factors, Ap, Bq, that ensure the constraints hold on total origin and destination flows. Approximate solutions of this model are evaluated by total misplaced MOD flow, X dmpq - Ap Bq exp(- fi umpq) I . mpq

The combined model of user-equilibrium and mode-origin-destination choice with the specific structure described above can be formulated mathematically either as a fixed point problem, or as a convex optimization problem (Bar-Gera and Boyce, 2003). For the generalized link cost function stated above (separable, monotonically increasing), the equilibrium, or optimal, solution uniquely determines total link flows. Total link flows in turn uniquely determine link costs, route costs, and the set of minimum cost routes, referred to here as the set of UE routes. Of course, the route flows on this set of UE routes are not unique. The model is solved by the OBA algorithm; route flow solutions are described by a set of restricting a-cyclic subnetworks Ap for each origin, and origin-based approach proportions a

efo.ll, such that a pG

L

7

J'

=0 for all a <£ A , and pQ

p

J

Va ^^^

pd

=1

V p e Z , v e N,v^ p . The ~

'

A

implicit set of routes from origin p is the set of all routes within Ap, that is Rpq[Ap\={r&Rpq

:acr=>a<sAp\.

The implicit route flows are given by hr = dapq X\apa .

For the OBA algorithm, the number of UE routes per OD pair is unrelated to the number of iterations, unlike algorithms based on the method of Frank and Wolfe (1956). The availability of route flows and approach proportions allows (a) adjustments of modeorigin-destination flows while retaining the current route proportions, (b) adjustments of restricting subnetworks to accommodate more routes, and (c) efficient adjustments to approach proportions that utilize second order derivatives of the objective function. The resulting algorithm offers precise convergence for large-scale networks. For the specific model of the Chicago Region presented here, the algorithm produced a solution with maximum excess cost of IE-13 equivalent auto in-vehicle minutes of travel time and total misplaced MOD flow of IE-10 person-trips per hour.

User-equilibrium route set analysis 677 By itself, a precisely converged solution does not necessarily guarantee a set of routes that is similar to the true, unique set of UE routes. As link flows and link costs converge towards their equilibrium values, so should excess costs. Therefore, excess costs of UE routes should decrease continuously towards zero, while the excess cost of any non-UE route should converge to a strictly positive value. The minimum equilibrium excess cost of all non-UE routes, considered as the rejection gap, is strictly positive as well. In principle, if a threshold below the equilibrium rejection gap is chosen, then at a certain finite level of convergence, the excess cost of all UE routes will be below the threshold, and the excess cost of all non-UE routes will be above the threshold. We chose to include all routes with excess cost below a threshold of 2E-12. The smallest excess cost of a rejected route is at least 30E-12, which determines the estimated rejection gap. Therefore, there are no routes with 2E-12 < excess cost < 30E-12. There are several reasons to believe that the chosen set of routes is probably similar and perhaps identical to the true set of UE routes. One reason is the stability of the set of included routes in the final iterations of the origin-based assignment. Another reason is the order of magnitude difference between the chosen threshold and the estimated rejection gap. Additional reasons are discussed in Bar-Gera (2004), particularly the fact that the chosen set of routes maintains consistent consideration of alternative route segments, a fundamental property of sets of minimum cost routes in general and the set of UE routes in particular. Summary of the model of the Chicago Region To provide a general background for interpreting the findings for the route pattern, we present selected overall measures of the model solution for travel by auto, transit and truck in Table 1. The table gives the total person flows by mode and vehicle flows by auto and truck between zone pairs (interzonal) and within zones (intrazonal), and the total generalized costs for the same categories. The mean values shown are the ratio of total costs to total flows. These generalized costs include the added costs of parking and out-of-vehicle time, whereas the costs shown in Figs. 5-8 below are based only on in-vehicle travel times and distances travelled. Therefore, the costs in the figures are somewhat lower than shown in Table 1.

678 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Table 1. Summary Measures for the Chicago Region Model for the Morning Peak Period Hourly flows, 6:30-8:30 am, 1990 Total flow (_/hour) Interzonal Intrazonal Regional Total costs (hours) Interzonal Intrazonal Regional Mean costs (min's) Interzonal Intrazonal Regional

auto

Persons total transit

%*

auto

Vehicles truck

total

1,116,689 64,971

331,028 523

1,447,717 65,494

23 1

930,574 54,142

418,507 26,678

1,349,081

1,181,660

331,551

1,513,211

22

984,717

445,184

1,429,901

363,314

147,549 214

510,862

29

302,762

210,387

513,148

13,896

14,109

2

11,580

6,116

17,696

377,209

147,762

524,972

28

314,341

216,503

530,844

19.5

26.7

21.2

19.5

30.2

22.8

12.8

24.5

12.9

12.8

13.8

13.1

26.7

20.8

19.2

29.2

22.3

19.2 *percent transit = 100 x (transit/total)

80,820

Regional value = Interzonal value + Intrazonal value

DESCRIPTION OF THE ROUTE SET Using the procedure described above, we determined that the solution for the user-equilibrium route set has a total of 8,602,026 routes connecting 3,202,310 zone pairs, or an average of 2.686 routes per OD pair; intrazonal travel is excluded from all calculations reported here. Of these zone pairs, 1,781,131 or 56% are connected by only one route. The maximum number of routes between an OD pair in the solution is 1,920 from zone 628 to zone 32. The locations of these zones are marked on Fig. 1. The centroid of zone 628 is located 9.6 miles west of the Chicago Central Business District (CBD), somewhat west of Oak Park, Illinois. The centroid of zone 32 is located 11.2 miles north of the Chicago CBD in Evanston, Illinois. The airline interzonal distance is 12.8 miles. The direction of zone 32 from zone 628 is 32 degrees east of North. Effectively, no freeways or tollways serve this zone pair, as shown in Fig. 2. The numbers of routes depend to a substantial extent on the level of congestion in the network. In two other solutions for the same model implementation with deterrence parameter values of 0.14 and 0.1 instead of 0.2, the total numbers of routes are 11.5 million and 20.6 million, as compared with 8.6 million for this case; the maximum numbers of routes for a single zone pair are 21,360 and 5,088, respectively. With no congestion (fixed link costs), the UE solution is one route per OD pair, or 3.2 million routes, except in the unlikely event of exactly equal fixed route costs.

User-equilibrium route set analysis

Fig. 1 Zone System of the Chicago Region

679

680 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Fig. 2 Roadway Network of the Chicago Region

User-equilibrium route set analysis 681 Overall description Next, consider Fig. 3 showing the number of OD pairs with a specified number of routes, shown on the x-axis ranging from 1 to 10,000. Two plots are shown on the figure: to the left is the number of OD pairs with each specified number of routes per OD pair; to the right is the cumulative proportion of OD pairs. For example, at the upper left, we can see there are nearly 1.8 million OD pairs with one route per OD pair, as noted above. The second point to the right and downwards indicates over 0.7 million OD pairs with two routes per OD pair. Continuing downwards to the right, we see decreasing numbers of OD pairs corresponding to increasing numbers of routes per OD pair, until we reach the rightmost point on the x-axis with 1,920 routes per OD pair. By examining the figure closely, one may be surprised to see so many different numbers of routes per OD pair. The pattern is quite regular, with a wider range for the smaller number of OD pair counts. Turning to the right axis, the cumulative proportion of OD pairs begins with 3.1E-7 in the lower right corner for zone pair (628-32) and continues upwards and to the left ending with a cumulative proportion of 1.0 for one route per OD pair. By closely examining this aspect of Fig. 3, we can determine, for example, that about 90% of all OD pairs have five or fewer routes. This is not surprising, since 56% of all OD pairs have only one route. As noted above, this distribution depends to a substantial extent on the level of congestion in the network; the overall pattern shown in Fig. 3 represents a somewhat congested solution. Another way of examining the same relationship is to consider the total number of routes with each number of routes per OD pair, as shown in Fig. 4. Clearly, the point in the upper left corner corresponding to OD pairs with only one route remains unchanged. However, the route count for OD pairs with two routes is doubled to 1.4 million, the number with three routes is tripled to 0.6 million and so forth. Those cases with only one OD pair, shown on the x-axis of Fig. 3, now form a diagonal line rising to the right. The minimum total number of routes is now 47, for the case of 47 routes per OD pair with only one occurrence. As with Fig. 3, we also show the cumulative proportions of total routes in Fig. 4, beginning on the lower right with a value of 2.2E-4 for the case of 1,920 routes, rising to the left, and ending at 1.0 for one route per pair. From this figure we see that 90% of all routes connect OD pairs with 40 or fewer routes. Therefore, when examined from the viewpoint of total routes, OD pairs with as many as 40 routes become more important. We suggest that these two figures convey interesting and important information about the route set on the Chicago road network. Naturally, these graphs raise further questions of how the number of OD pairs and the total number of routes vary with respect to the length, flow and direction of travel between OD pairs for various numbers of routes per OD pair. For example, are the origins and destinations connected by a single route primarily nearby? Or, are they OD pairs with very low flow? And what about the direction from origin to destination in relation to the grid-like arterial network of Chicago? We take up these interesting questions in the following subsections.

682 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Fig. 3. Number of OD Pairs by Number of Routes per OD Pair Cumulative Proportion of OD Pairs

Fig. 4. Total Number of Routes by Number of Routes per OD Pair Cumulative Proportion of Total Routes

User-equilibrium route set analysis 683 Description of the route pattern with respect to cost To investigate the first question posed above, we present Figs. 5-8 relating the number of OD pairs and total number of routes to the generalized cost of travel between OD pairs. In each figure the x-axis is the generalized cost throughout the network, which ranges from 0 to 200 auto in-vehicle minutes, the unit of cost defined by the model formulation. In Figs. 5 and 7, the y-axis is the number of OD pairs and the total number of routes, respectively. In Figs. 6 and 8, these numbers are converted to proportions within each unit increment of generalized cost. Note the labelling scheme in the figures: values for 1, 2 and 3-5 routes are shown in black with diamond, triangle and square symbols; this sequence is repeated in gray for 6-9, 10-99 and 100-1,999 routes, hi each figure six classes of routes per OD pair are shown, which aggregate the route set based on more detailed analyses; see Table 2 for additional details. Table 2. Number of OD Pairs and Total Routes by Classes of Routes per OD Pair Routes per OD pair Number of OD pairs Total number of routes

1

2

3 to 5

6 to 9

10 to 99

100 to 1,999

1,781,131

724,829

412,279

150,810

131,306

1,955

1,781,131

1,449,658

1,517,572

1,023,664

2,460,034

369,967

Turning to Fig. 5, we observe that OD pairs in substantial numbers are present in the first five classes of routes over the full range of costs, although the numbers vary widely. The sixth class of routes (100-1,999) has a smaller range (16 to 139 minutes) with smaller numbers of OD pairs (1 to 63 pairs, the largest occurring at a cost of 64 minutes). We find it remarkable that OD pairs with one route are found over the entire range of generalized costs from 1 to 184 minutes, with a few occurrences for even higher costs. The maximum count per unit cost value for 1-route OD pairs is 29,099 with a cost of 39 minutes. Similarly, substantial counts are found for 2 routes and the higher route classes. The general shape of these distributions is quite similar. The generalized cost for OD pair (628-32) is 44.4 auto in-vehicle minutes. To provide information on how OD flows are related to OD costs, we show the proportion of total flow for each unit cost as a solid line extending from the lower right to the upper left. As expected, higher flows occur at lower costs. The scale for these flow proportions is given on the right y-axis. The highest flow proportion is 0.06 (6%) of total flow at a cost of 8 minutes. Fig. 6 offers a different interpretation of these OD counts. The sum of OD pairs within each unit cost value is normalized to 1.0, in order to judge the relative size and importance of the route classes by cost. OD pairs with only one route dominate at lower costs, beginning with 98% of the total at a cost of 1 minute, but decline steadily to about 32% at 150 minutes where the pattern becomes rather chaotic. The next four classes (2; 3-5; 6-9; 10-99) increase very quickly at first, from being practically negligible at 1 minute to 19, 9, 3 and 2% respectively at 20 minutes. Then they continue to increase gradually to 26, 26, 5 and 11% respectively at 150 minutes where all patterns become chaotic. Figs. 7 and 8 offer a comparable analysis in terms

684 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Fig. 5. Number of OD Pairs by Cost and Routes per OD Pair Proportion of Total Flow

Fig. 6. Proportion of OD Pairs by Cost and Routes per OD Pair

User-equilibrium route set analysis 685 Fig. 7. Number of Routes by Cost and Routes per OD Pair Proportion of Total Flow

Fig. 8. Proportion of Total Routes by Cost and Routes

686 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction of the total number of routes. As shown in Fig. 7, the six route classes are similar in size and pattern. Total routes for OD pairs with 10 to 99 routes are the largest class. Description of the route pattern with respect to flow To investigate the second question posed above regarding the effect of OD flow on the number of routes, we present two figures relating the number of OD pairs, and total number of routes, to the flow of vehicles between OD pairs. In each figure the x-axis is the OD flow per zone pair. The flows are shown in terms of orders of magnitude. For example, the value, -1, represents flows from 0.1 (1E-1) up to but not including 1.0 (IE 0). The value, +3, represents flows from 1,000 (1E+3) up to 10,000; the maximum interzonal OD flow determined by the model is 2,186.1 vehicles per hour. There is one exception to this rule. At the left end of the x-axis, the value, -14, includes values from zero (0) up to 1E-13. Hence, the range of the figure is from 0 up to 10,000. The definitions of the y-axis and classes of routes per OD pair are the same as Figs. 5-8.1 Since OD flows in the model are functions of cost, as defined in the Model Description, and small flows result from large costs, it is not surprising to find that Figs. 9 and 10 are essentially mirror images of Figs. 5 and 7. As with Figs. 5 and 7, the classes of routes extend across the full range of OD vehicle flows. However, the magnitude of the numbers of OD pairs and total routes are different because the level of aggregation on the x-axis is 18 intervals of flow instead of 200 units of cost. Also shown in Figs. 9 and 10 on the right y-axis, and by a dashed line, is the proportion of total flow, from the same data as in Figs. 5 and 7. Now the higher flows lie on the right, and decrease linearly on a log-log scale to the left, as flows decrease in magnitude. The number of OD pairs and total number of routes increases with flow, both as represented by order of magnitude and by total flow. In considering the magnitude of the increase, keep in mind the log scales. The reason for the larger number of OD pairs and total routes between zero and 1E-13, as shown on the left axis, is that this flow interval includes zero OD flows. The model does determine the number of routes between OD pairs with zero flow; as can be seen, the number of routes is quite large in some cases. The substantial number of OD pairs with flows of less than one vehicle per hour results from the large number of zones (1,790). Description of the route pattern with respect to direction As a final exploratory analysis, we consider the question of why many OD pairs have very few routes over the whole range of generalized cost and OD flow, while others have many routes, ranging in this case as high as 1,920 for one OD pair. As shown in Fig. 2, the arterial road network in the Chicago Region is relatively grid-like, especially in the city and inner suburbs where zones are small (one square mile or less) and the majority of zones are located. As an initial conjecture, we consider whether those zone pairs with only a few routes may be

User-equilibrium route set analysis 687 Fig. 9. Number of OD Pairs by Flow and Routes per OD Pair Proportion of Total Flow

Fig. 10. Total Number of Routes by Flow and Routes per OD Pair Proportion of Total Flow

688 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction aligned with the grid (e.g. north to south or east to west), whereas zone pairs with larger numbers of routes maybe aligned diagonally across the grid (e.g. southwest to northeast). To examine this conjecture, we computed the direction of each destination from the origin, as measured from North. That is, if the destination lies directly north of the origin, its direction is 0 degrees. Sixteen directions were defined, each containing 22.5 degrees. Thus the direction representing south to north actually ranges from 348.75 degrees to 11.25 degrees. In Figs. 11 and 12, these 16 directions are shown on the x-axis by their midpoints, rounded to whole degrees. The 0 degrees direction is shown at the extreme left, and repeated at the extreme right as 360 degrees, to facilitate comparison with adjoining directions to the northwest. The principal compass points of North (0, 360), East (90), South (180) and West (270) may be readily identified, together with NE (45), SE (135), SW (225) and NW (315); the values at the next level of detail are also represented, e.g., NNE (22.5). On the y-axis, we show the proportions of OD pairs and total number of routes by direction. These proportions are computed separately for each direction, as in Figs. 6 and 8. The definitions of the number of OD pairs and the total number of routes, as well as route classes, is identical to Figs. 3-10. Examining the proportion of OD pairs by direction, we observe there does appear to be some support for our conjecture, especially for route classes 6-9 and 10-99. Class 100-1,999 supports the conjecture strongly for southbound travel (180 degrees) and westbound travel (270 degrees with no OD pairs), and somewhat for northbound (0 degrees) and eastbound travel (90 degrees). The pattern is emphasized more strongly in Fig. 12, showing the total number of routes. In this case, the values for 10-99 routes tend to support the conjecture, especially for travel towards the southeast and northwest directions. As noted earlier, OD pair (628, 32) with the largest number of routes (1,920) has a direction of 32 degrees east of North, placing it in the second direction labelled 23 degrees; one of 103 OD pairs with 100 or more routes in the second direction, it represents about 7% of the routes in that class. Clearly, additional studies are required to reach more definitive conclusions, if warranted.

INTERPRETATIONS As noted in the Introduction, this initial analysis of the model solution leaves many questions unanswered. Without repeating each of those questions here, we suggest that examining how these findings vary with road network congestion should have high priority. A larger question concerns how one should interpret the OD, mode and route flow patterns and the route set underlying the user-equilibrium model, both with regard to the model formulation itself and with regard to reality. Given the large numbers of user-equilibrium routes in the solution, some readers may question the plausibility of the solution. Moreover, what is the meaning of route, mode and OD flows that are so very small? From a computational viewpoint, these values are a result of the form of the doublyconstrained Logit function as well as the route choice model's user-equilibrium property.

User-equilibrium route set analysis

Fig. 11. Proportion of OD Pairs by Direction and Routes

Fiq. 12. Proportion of Total Routes by Direction and Routes

689

690 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction While the model could be truncated so as to exclude values less than some threshold, this restriction would change the properties of the model itself. Since our objective here is to understand the model's properties, such truncation would seem to be counterproductive. We suggest that the model solution should be viewed as an average or expected value, recognizing at the same time that flows on the order of 10~6 would require an enormously large number of replications to occur. To achieve an impression of how the route flows might appear on a given day requires the application of a sampling procedure. That is, a random sample equal in size to the total number of trips per hour could be drawn from the MOD flow proportions and the route proportions. This sampling procedure would result in integer-valued MOD matrices as well as route flows, which could be summed to obtain link flows. Repeating this procedure perhaps 100 to 500 times would generate frequency distributions of link flows for each link, which could be examined to determine their means, range and variance and their correspondence to the deterministic link flows. This procedure would also answer the question of how many routes actually occur in a sampled realization for OD pairs with many routes. In examining these sampled realizations, we would need to keep in mind that no large-scale data set presently exists on route choice. In the absence of such data, we should be cautious in interpreting these results. Beyond these immediate questions of assessing the model's validity is the relation of the model to travel forecasting practice. The model studied here is closely related to the sequential procedure, which remains the main paradigm of urban transportation planning practice. A difference from practitioner models is the assumption of a single class of travellers, but this restriction simply represents the current status of this research. Another shortcoming is the lack of a choice function for the time period of travel. A more general model could represent several time periods during the morning, and include a choice among them, based on relative levels of congestion as well as the cost of arriving too early or too late at the destination. Such a model would be more advanced than most practitioner models which typically apply survey-based factors to allocate 24-hour travel forecasts to shorter periods of time. The model is also static, with all of the well-recognized limitations of that paradigm: continuous OD, mode and route flows from origin to destination; and non-decreasing link travel times in relation to link flows increasing without limit. Dynamic models of route, departure time, mode and OD choice have been the subject of ongoing research for nearly 30 years, and especially the last 15 years. Although dynamic models seek to represent the properties of road networks more realistically than their static counterparts, to our knowledge no dynamic model comes close to depicting the range of choices described by static, integrated models. For a recent state of the art review, see Peeta and Ziliaskopoulos (2001).

CONCLUSIONS Our exploratory research has addressed for the first time the question of the characteristics of route sets in the solution to an integrated user-equilibrium model. Using graphical methods we have displayed the relationship of OD pairs and total routes in terms of generalized cost, OD flow and direction of travel, as well as the more aggregate relationships. We find the results surprising with respect to the broad range of values over which the classes of routes

User-equilibrium route set analysis 691 occur, and stimulating for gaining an improved understanding of the user-equilibrium model's properties.

ACKNOWLEDGEMENT The authors wish to thank Sanya Niumpradit for contribution to this chapter through his initial exploration of these questions in a term paper at the University of Illinois at Chicago in 2001, and Frederik Noth for assistance with computations on which that term paper was based. Helpful questions and comments of four referees are gratefully acknowledged.

REFERENCES Bar-Gera, H. (1999). Origin-Based Algorithms for Transportation Network Modeling. Technical Report 103, National Institute of Statistical Sciences, October, (Ph.D. thesis, Civil Engineering, University of Illinois at Chicago); downloadable file available at: http://www.niss.org/downloadabletechreports.html Bar-Gera, H. (2002). Origin-based algorithm for the traffic assignment problem. Transportation Science, 36, 398-417. Bar-Gera, H. (2004). Primal Method for Determining the Most Likely Route Flows in Large Road Networks. Submitted for publication in Transportation Science, September. Bar-Gera, H. and D. Boyce (2003). Origin-based algorithms for combined travel forecasting models. Transportation Research, 37B, 405-422. Beckmann, M., C. B. McGuire and C. B. Winsten (1956). Studies in the Economics of Transportation. Yale University Press, New Haven. Bovy, P. H. L. and E. Stern (1990). Route Choice: Wayfinding in Transport Networks. Kluwer, Dordrecht. Boyce, D. (2004). Forecasting Travel on Congested Urban Transportation Networks: Review and Prospects for Network Equilibrium Models. Presented at TRISTAN V, Guadeloupe. Boyce, D. and H. Bar-Gera (2004). Multiclass combined models for urban travel forecasting. Network and Spatial Economics, 4, 115-124. Brynooghe, M., A. Gilbert and M. Sakarovitch (1969). Une methode d'affectation du traffic. In: Proceedings of the 4th International Symposium on the Theory of Road Traffic Flow (W. Leutzbach and P. Baron, eds.), Beitrage zur Theorie des Verkehrsflusses Strassenbau und Strassenverkerhrstecknik, Heft 86, Herausgegeben von Bundesminister fur Verkehr, pp. 198-204. Abteilung Strassenbau, Bonn, Germany. Dafermos, S. C. (1968). Traffic Assignment and Resource Allocation in Transportation Networks. Ph.D. thesis, Operations Research, Johns Hopkins University, Baltimore. Evans, S. P. (1973). Some Applications of Optimisation Theory in Transport Planning. Ph.D. thesis, Civil Engineering, University College London, London. Evans, S. P. (1976). Derivation and analysis of some models for combining trip distribution and assignment. Transportation Research, 10, 37-57.

692 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Florian, M. and S. Nguyen (1976). An application and validation of equilibrium trip assignment methods. Transportation Science, 10, 374-389. Florian, M. and D. Hearn (1999). Network equilibrium and pricing. In: Handbook of Transportation Science (R. W. Hall, ed.), pp. 361-393. Kluwer, Boston. Frank, M. and P. Wolfe (1956). An algorithm for quadratic programming. Naval Research Logistics Quarterly, 3, 95-110. Harris, B. (2002). Multiple paths through a network. Presented at the 42nd Congress of the European Regional Science Association, Dortmund, Germany. Jorgensen, N. O. (1963). Some Aspects of the Urban Traffic Assignment Problem. ITTE Graduate Report #1963:9 (M.S. thesis, Civil Engineering), University of California, Berkeley. Larsson, T. and M. Patriksson (1992). Simplicial decomposition with disaggregated representation for the traffic assignment problem. Transportation Science, 26, 4-17. LeBlanc, L. J., E. K. Morlok and W. P. Pierskalla (1975). An efficient approach to solving the road network equilibrium traffic assignment problem. Transportation Research, 9, 309-318. Lundgren, J. T. and M. Patriksson (1998). An algorithm for the combined distribution and assignment model. In: Transportation Networks: Recent Methodological Advances (M. G. H. Bell, ed.), pp. 239-253. Pergamon, Oxford. Nagurney, A. (1999). Network Economics: A Variational Inequality Approach. Revised second edition. Kluwer, Boston. Nguyen, S. (1974). An algorithm for the traffic assignment problem. Transportation Science, 8, 203-216. Patriksson, M. (1994). The Traffic Assignment Problem - Models and Methods. VSP, Utrecht. Peeta, S. and A. Ziliaskopoulos (2001). Special Issue on Dynamic Traffic Assignment (Part I). Networks and Spatial Economics, 1, 231-390.

35

COMPARISON OF STATIC MAXIMUM LIKELIHOOD ORIGIN-DESTINATION FORMULATIONS H. Rakha, Virginia Tech, Charles Via Jr. Dept. of Civil and Environmental Engineering, H. Paramahamsan, Virginia Tech, Blacksburg, Virginia USA; andM. VanAerde, posthumously.

INTRODUCTION Operational types of analyses typically require evaluating the impact of alternative traffic control schemes on a number of Measures of Effectiveness (MOEs) including delay and throughput, energy and emissions, and accident risk. Because traffic networks are dynamic environments, the traffic flow composition may alter in response to changes in traffic control. For example, installing a ramp meter on a freeway on-ramp could potentially result in some drivers altering their routing pattern in order to avoid incurring delays at a specific on-ramp by altering their on-ramp. In order to capture the traffic routing patterns in response to varying traffic conditions, an O-D matrix is required. Partial or complete O-D demands can be observed directly in the field through some form of survey or vehicle tracking, in which the actual origin and destination of a vehicle or trip maker are obtained. The surveys involve selecting a subset of drivers/vehicles that is representative of the entire population in order to estimate the O-D demand. In the absence of direct observation, O-D demands are typically estimated based on land-use data. Typically, within the traditional four-step transportation planning process, trip productions and attractions are estimated based on zonal and land-use data such as population, employment opportunities, and measures of affluence. Subsequently, the O-D demand is computed using a Trip Distribution gravity model. A third approach attempts to infer the unknown O-D demand from observed traffic flow counts. The third method circumvents the need to estimate zonal trip productions and attractions from aggregate zonal characteristics. Instead, this method uses readily available loop detector counts to estimate an O-D demand that replicates the observed

694 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction flows as closely as possible. Van Aerde et al. (2003) demonstrated that the Trip Distribution gravity model is a sub-problem of the more generalized synthetic O-D problem (flows only observed on the zone connectors). Specifically, by only providing link counts at the network entrance and exit links and considering a gravity model impedance function seed matrix, the synthetic O-D problem reverts to a Trip Distribution gravity model. Over the last four decades, extensive work and research has been devoted to develop systematic approaches to estimate O-D demands reliably. Initially, these efforts were limited to longer term planning purposes, however, due to the need to manage existing transportation infrastructure more efficiently; the current emphasis is geared towards estimating O-D demands for more short-term traffic operational analyses. The synthetic O-D approaches include mathematical and heuristic approaches, static and dynamic O-D approaches, and approaches that estimate the routes in addition to the O-D demands. These approaches are described in detail by Van Aerde et al. (2003). This paper will focus on the mathematical approaches for estimating static O-D demands assuming that the routes are known a priori. A subsequent paper will analyze the combined routing and O-D estimation problem, which introduces a level of complexity and error into the problem. Several theoretical and practical problems are encountered when estimating network demands synthetically from link flows. The first of these problems is that it is theoretically possible that multiple O-D demands exist that exactly replicate the observed flows. The use of a seed matrix can be utilized in order to select a unique solution, which resembles the seed matrix most closely, from the multiple solutions that satisfy the link flow constraints. The second of these problems is the fact that in practice, link flow continuity at nodes rarely exists within a set of observed data because of errors in the data collection and because data are gathered at different points in time; with the result that often no O-D demand exists that exactly matches the observed link flows. Again, multiple solutions may exist that provide equal link flow errors. Van Aerde et al. (2003) developed a generalized formulation that resolves the flow continuity issue as will be discussed later in the paper. A third problem that exists is that for most practical problems, the routes that are utilized by the O-D demands are usually unknown. Consequently, a level of complication is added to the problem in order to compute O-D demands in addition to routes. The final problem is that the true O-D is rarely known, thus making it extremely difficult to evaluate how well the O-D demand was estimated. This paper will apply the synthetic O-D formulations to problems for which the routes are known a priori. The paper uses some simple examples to identify the implications of various simplifying assumptions that are quoted in the literature on the final O-D estimates. It should be noted that the results presented here are likely to represent a 'best case' view of how the models perform given that they do not include errors associated with establishing the driver route choice behavior. It should be noted however, that a sample application to a scenario in which the O-D demands and routes are computed simultaneously is presented for illustration purposes in the last section of the paper. Initially, the two basic formulations for estimating the most likely O-D demand (as defined by the maximum entropy/minimum information) are presented and discussed in terms of the logic behind each formulation. These formulations include the maximum entropy/minimum

Comparison of static maximum likelihood origin-destination formulations 695 information associated with the O-D matrix (termed the trip formulation) and the maximum entropy/minimum information associated with the link flows (termed the volume formulation). These formulations are described in terms of how they were derived and any simplifying assumptions that were made in order to analytically solve the basic formulation. After presenting the alternative formulations for solving the synthetic O-D problem and the associated simplifying assumptions, the following section investigates the consistency of the two formulations (trip and volume formulation) for two simple hypothetical networks, one that ensures a constant total demand and another that presents a variable total demand. The reason for selecting these simple networks is to be able to enumerate the solution space in order to characterize how the optimum solution varies as a function of the problem formulation. Furthermore, the various formulations are tested for a number of cases including (a) the impact of seed O-D scaling on the final solution; (b) the impact of the seed O-D feasibility on the final O-D table estimate; and (c) the impact of lack of flow continuity on the final solution.

STATIC O-D FORMULATIONS Entropy maximization and information minimization techniques have been used to solve a number of transportation problems as discussed by Wilson (1970). The application of the entropy maximization principles to the static O-D estimation problem was initially proposed by Willumsen (1978) who demonstrated that by maximizing the entropy, the most likely trip matrix could be estimated subject to a set of constraints. This section describes two formulations that were proposed by Van Zuylen et al. (1980) to solve the static O-D problem together with later enhancements (Willumsen, 1981; Willumsen, 1984; and Ortuzar and Willumsen, 2001). In addition, enhancements to these formulations as reported by Van Aerde et al. (2003) are presented. This section describes the various simplifying assumptions that were made in order to solve the problem analytically. These simplifying assumptions are described in detail because their impact on the final solution is systematically tested in an attempt to identify the errors associated with various formulations. Consequently, the contribution of this paper to the state of knowledge is to identify the shortcomings of various formulations that are described in the literature and the domain of application of each formulation.

Trip Formulation The trip-based approach to defining maximum likelihood considers that the overall trip matrix is made up of uniquely identifiable individual trip makers. Let the total number of O-D trips equal T and the number of trips traveling between origin i and destination j equal Tv; as defined in Table 1. Then the number of ways (defined as entropy) in which T trips can be divided into groups of Ty trips without replication can be computed as Z, (Ty ) = T !/XI-7V ! The objective then becomes to identify the solution that maximizes the entropy function as

696

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

[1 1]

Max. Z,(7V)= Y^T)-

"

The above formulation does not take into account any prior information, from for example a previous survey (seed matrix ty). While the seed matrix does not necessarily have to satisfy the observed link flows, the seed matrix can be utilized to expand the maximum likelihood function as shown in Equation 1-2 to solve for Ty given a seed matrix ty as input. It can be noted that the likelihood of an individual trip from i toy is %/X r 'i/ > based o n the above seed matrix. Consequently, the probability of Ty trips being drawn from i toy is (t^

ij

'i

..ty ) " .

)

Table 1: Variable Description. Variable

Description Actual number of trips from origin i to destination j.

t.J

Seed number of trips from origin i to destination j (Seed matrix).

T

= E ff *v

Total number of trips traversing the network.

' = 'Zij'a

Total number of trips traversing the network for the seed matrix.

A

The fraction of trips from origin i to destination j that utilize link a.

". =

ly
Estimated volume on link a associated with the seed O-D matrix. Observed volume on link a. Volume on link 'a' that is closest to Vd while satisfying flow continuity. Lagrange multiplier for O-D pair ij.

The above formulations of objective functions for expressing likelihood require additional constraints in order to be complete (Van Aerde et al., 2003; Willumsen, 1978; and Van Zuylen, 1980). The simplest of these constraints indicate that the sum of all trips crossing a given link must be equal to the link flow on that link, as indicated in Equation 2. As will be shown later, the simplest mechanism, for including the above constraints in the earlier objective functions, is to utilize Lagrange multipliers. These multipliers permit an objective function with equality constraints to be transformed into an equivalent unconstrained objective function.

V.=YJ^

Va

[2]

u Because link flow continuity is typically not observed in field data because they are usually collected at different time points and involve errors, Van Aerde et al. (2003) proposed that the flow constraints be minimized, rather than eliminated. In other words, rather than finding the most likely O-D matrix that exactly replicates the observed link flows, the problem is reformulated as finding the most likely O-D matrix from among all of those that come equally close to matching the link flows. The expression that was proposed to capture the error to be minimized is shown in Equation 3, and is subject to the flow continuity constraints Va' = ^rTijpfj . These complementary flows {va') are those which deviate the least from the observed link flows, while satisfying link flow continuity. Given that these complementary

Comparison of static maximum likelihood origin-destination formulations 697 link flows do satisfy flow continuity, they can now be added in as rigid equality constraints to the objective function of Equations 1-1 or 1-2, and be guaranteed to yield a feasible solution. Min. Y (Ty ) = ^(Va - V' f = jjva

- £?>£ j

[3]

Alternatively, one can solve Equation 3 and Equations 1-1 or 1-2 simultaneously. Unfortunately, it is not easy to combine one expression that desires to maximize likelihood with another that desires to minimize link flow error, as Lagrange multipliers can only add equality constraints to a constrained objective function. Consequently, Van Aerde et al. (2003) proposed taking the partial derivatives of Equation 3 with respect to each of the trip cells that are to be estimated as

af-I^.-Evs) =2[E(^-P5)-[lPg-[i;^^jjj = o

vu,

M

which yields as many equations as there are trip cells. The equalities of Equation 4 can be added to Equation 1-2 to yield

Max

- ^P(*J"-?M? (K '-*KMp^))))-

[5]

The net result, of the above process, is to suggest that most synthetic O-D generation problems consist of two sub-problems. One of these involves finding a new set of complementary link flows that do produce link flow continuity, at which point the maximum likelihood problem can be solved as before. Alternatively, one can compute the partial derivatives, that will yield link flow continuity, while deviating by the least amount from the observed link flows, and then utilize these flows directly in the maximum likelihood formulation using Lagrange multipliers. Both solution approaches will be compared in the following section. A first challenge with maximizing Equation 5 is that it yields very large numbers that are difficult to deal with from a computational standpoint. Furthermore, as it is common to maximize objective functions by taking their derivatives, and as it is more difficult to contemplate the derivative of a discontinuous expression, such as those including factorials, a simple approximation is proposed in the literature (Van Aerde et al., 2003; Willumsen, 1978, 1981, and 1984; Van Zuylen and Willumsen, 1980; and Ortuzar and Willumsen, 2001). This approximation involves taking the natural logarithm of either objective function (Equation 1-1 or 1-2). Taking the natural logarithm of the objective function both makes the output easier to handle and allows us to use Stirling's approximation as a convenient continuous equivalent to the term ln(x!), as ln(r !) =T In7" -T . [6] The resulting converted objective function using the Stirling approximation on the original objective function of Equation 5 can be derived as

Max. T-ln^j-^In^j-^^^C.-PjO-^d:^^))))-

^

This equation, when solved, yields the most likely O-D matrix that matches the observed link flows as closely as possible (Van Aerde et al., 2003). It should be noted that the objective

698 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction function of Equation 7 is composed of two components. The first being the likelihood of an O-D matrix table and the second accounting for the minimum error between the field observed flows and the flows that satisfy flow continuity, hi the case that the seed matrix is the optimum matrix the likelihood component resolves to zero. While Van Aerde et al. (2003) demonstrated that the formulation of Equation 8 can be used to solve the O-D problem, and has been incorporated in the QUEENSOD software (M. Van Aerde & Associates, 2002), others have made further approximations. Some of these, such as those proposed by Willumsen (1978, 1981, and 1984) and Ortuzar & Willumsen (2001), are widely utilized in the literature. The most common approximation that is applied, after Stirling's approximation is invoked, is to consider that the total number of trips in the network T is constant, and to drop this term yielding

Max. z,(r ff ,, ff )=-£v»(r-]-

[9]

V Hj )

ij

Equation 9 is then revised by subtracting T and adding t (total demand associated with the seed matrix) to the objective function as

Max. z4(r!/,^) = - X j r ^ « ^ j - ^ + r , J j ,

[10-1]

where both terms (T and t) are equal based on the assumption that the total number of trips remains constant. A further approximation is then applied using equation 10-2 to yield Equation 10-3, and eventually 10-4 (Willumsen, 1978, 1981, and 1984; Van Zuylen and Willumsen 1980; and Ortuzar and Willumsen, 2003). Ortuzar and Willumsen (2003) make a simple substitution of % for Ty in Equation 10-4 to yield Equation 10-5, which can then be solved using simple weighted linear regression. It should be noted that the formulations of Equations 10-4 and 10-5 can, at times, yield negative values for the number of trips between a given O-D pair; however, this problem can be resolved by adding non-negativity constraints on the variables Ty within the problem formulation (i.e. Ty >0). lx-i\

Lnoo=

l/x-n3

iix-i\2

+

[l0 2]

(—Hi—) s(—) '

Max.

,,„.,

+

( -I)

f^-ii T + f

Zs(Ts,t9) = -'£ TV ^— ,,

v

'i/

\— V

)

1

Min. Z6{Tii,tij) = Yi^r{Tii-tiJ) }

l

'j

2

-Ty+tv )

[10-3] ,

[10-4]

ij \n'J

Min. Z1{TiJ,tij) = Y[^-{TiJ

-

TJL

)

2

-,, y ) ]

[10-5]

Volume Formulation An alternative maximum likelihood formulation considers that the basic unit of observation is a volume count not an O-D count. If one now considers a potential solution matrix, one can show that the number of ways in which these volume counts can be drawn from the original

Comparison of static maximum likelihood origin-destination formulations 699

V i

set of counts is computed as

f l r r i"

(t-p* a

\|II

J?

^

')Ti/P*

Consequently, the maximum

entropy formulation can be cast as

Max

- ^^^-nn^n^r-

[11]

Taking the natural logarithm of both sides of Equation 11 and applying Stirling's approximation, as was done earlier with the trip formulation, the literature derives Min. Z9(rff,*ff) = I I r * * S i » ( ^ ) [12]

VALIDATION OF SYNTHETIC O-D FORMULATIONS FOR FEASIBLE SEEDS As described earlier, two types of formulations have been proposed in the literature for solving the static O-D problem from observed link flows. The first of these formulations uses the O-D trips in the basic formulation (trip formulation), while the second uses the observed flows in the basic formulation (volume formulation). As demonstrated in the previous section, numerous approximations have been made to each of the formulations in order to solve the problem analytically. This section investigates the impact of various approximations on the optimum solution. The objective of this section is to compare both formulations (trip and volume) using two simple networks, that reflect conditions in which the total demand is constant versus variable. Further comparisons are made for conditions where flow continuity is not satisfied and the seed matrix is infeasible. Given that O-D trips can only be integers (e.g. we cannot have 100.2 trips) the problem space is confined to integer values. It should be noted that this constraint can be relaxed if one considers a real value to represent a probability function of generating an O-D demand as is done in the INTEGRATION software (Rakha and Ann, 2004). Specifically, an O-D demand of 52.31 trips can be modeled by generating an OD demand of 52 vehicles with a probability of 69% (1.00 - 0.31) and an O-D demand of 53 vehicles with a probability of 31% (0.31) so that on average 52.31 vehicles are generated. For purposes of this study, however, we will only consider integer solutions. With this in mind, a feasible solution refers to an integer solution that satisfies the non-negativity, the flow continuity, and the total demand constraints, in the case that the total demand remains constant. It should be noted that we do not solve each of the formulations; instead we enumerate all possible solutions to construct a solution surface. The optimum solution is then derived from the solution space as either the maximum or minimum depending on the objective function that is being considered. Consequently, the use of small and simple networks is essential to achieve these objectives. It should be noted, however, that the conclusions that are derived are general and applicable to large networks, as is demonstrated in the last section of the paper.

700

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Test Network Description Two networks are used to test the various formulations. The first network is a 5-link network, as shown in Figure 1. It has two origin nodes A and B, two destination nodes C and D, and two intermediate nodes E and F. The network has four O-D pairs, TAc, TAo, TBC, and TBD- The five one-way links are numbered from 1 through 5 with observed link volumes, as demonstrated in Figure 1.

Figure 1: 5-link Network Configuration.

Comparison of static maximum likelihood origin-destination formulations

701

Table 2: Solution Set for 5-Link Network. Demand Scenario 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

O-D Demand (Ts) Toe

TaJ

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

Tfc 60 59 58 57 56 55 54 53 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36

35 34 33 32 31 30

Tu 0 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

There are a total of 31 feasible demand scenarios that are identified by specifying one of the O-D demands to compute the other three demands, as demonstrated in Table 2 (Given TAc TAD=40-TAC, TBC=10-TAC, and TB£r=3Q-TAD). These feasible demands satisfy the non-negativity constraints (TAc, TAD, TBC, and TBD ^ 0 ) , the flow continuity constraints {Vi+V2=Vj and Vj=V4+Vs), and the total demand constraint (T = 100). It should be noted, however, that the first and last feasible demand scenarios (scenarios 0 and 30) result in zero flows for one of the origin/destination combinations (TBD and TAo, respectively). These extreme scenarios cause problems for mathematical formulations where the Ty term is in the denominator. Consequently, the two extreme demand scenarios are not considered any further in the analysis and we only consider solution scenarios 1 through 29 throughout the remainder of the paper. The second network is a 2-link network, as shown in Figure 2 with 3 nodes A, B, and C. Node B is an origin and a destination node, A is an origin only node, and C is a destination only

702 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction node. The network permits 3 O-D pairs, TAB, TAC, and TBc and each O-D pair has only one feasible path available. If for an observed set of flows, one of the O-D demands is specified, then the other two demands can be computed, as demonstrated in Table 3. It should be noted that, unlike the 5-link network, the total demand varies across the alternative feasible solutions. Hence this network provides an opportunity to test the various formulations when the total demand is not constant.

Figure 2: 2-link Network Configuration. Table 3: Solution Set for 2-Link Network. Demand Scenario

TAB

TAC

TBC

0

0

10

1

1

2 3

2 3

4

4

5

5 6 7 8 9 10

9 8 7 6 5

5 6 7 8 9 10 11 12 13 14 15

6 7

8 9 10

4

3 2 1 0

Total trips (7) 15 16 17 18 19 20 21 22

23 24 25

Formulations Analyzed As part of the study, nine objective functions are compared considering the two test networks that were described earlier. The nine objective functions that are considered are summarized in Equations 13-1 through 13-9. The objective function of 13-1 and 13-7 do not incorporate any simplifying assumptions while the formulations of 13-2, 13-8, and 13-9 only incorporate the Stirling approximation. It should be noted that when the seed matrices are feasible formulations 13-2 and 13-9 become identical because the flow term reverts to zero. The formulations of 13-3, 13-4, and 13-5 in addition to the Stirling approximation incorporate the additional assumption of constant total demand (i.e. T »t). Finally, the formulation of Equation 13-6 includes a final additional assumption that ty =Ty. It should be noted that each formulation includes a non-negativity constraint on the Ty variables.

Max:

[13 1]

- rfenife V ij

V

"

)

Max.:7-ln(^)-X7'i,-lnf^-l ^ ' '

ij

V 'ij

J

[13-2]

Comparison of static maximum likelihood origin-destination formulations 703

I 13 " 3 !

Max.:-YiTiJLn['r-\-Tii] tj V

V. 'ij

}

)

Max, - £ [jyLn [^- j - T,? +% j

[1*4]

[13-6i

Min :

- Z(^-(^-^)2)

Max, n

V.<- n f ^ r

[13-7]

U

Max, T ln(TT) - YJ9 ln^J j - ^

2^(Va -pj.) - ^ P | ^ r ^ « , j j j j

[13-9]

Consistency of Trip and Volume Formulations This section compares the optimum solution for Equation 13-1 to the optimum solution for Equation 13-7 considering a feasible seed matrix, as defined earlier. Infeasible seed matrices are discussed later. Figure 3 illustrates the variation in the trip formulation objective function solution space for a constant total demand (5-link network) as a function of the initial seed/feasible solution combination. The x-axis represents all possible feasible scenarios (solutions 1 through 29 in Table 2), while the y-axis represents all possible feasible seed scenarios (solutions 1 through 29 in Table 2), and the z-axis represents the objective function that corresponds to the solution/seed matrix combination. A slice along the y-axis illustrates how, for a specific seed, the objective function varies for all possible feasible solutions. The solution that corresponds to the maximum objective function represents the optimum solution. The surface plot illustrates that the objective function is maximum when the solution scenario is identical to the seed scenario (i.e. along the diagonal of the x-y plane). Consequently, Figure 3 illustrates that for a feasible seed matrix, the trip formulation selects a matrix, from the set of feasible solutions that is identical to the seed matrix (maximum objective function value). Similar results are observed for the case in which the total demand is not constant, as illustrated in Figure 4 (solutions 1 through 9 in Table 3).

704

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 3: Variation in Objective Function for Equation 13-1 for 5-Link Network for Feasible Seed.

Figure 4: Variation in Objective Function for Equation 13-1 for 2-Link Network for Feasible Seed. Figure 5 and Figure 6 illustrate a similar trend for the volume formulation. Specifically, the optimum solution is identical to the seed solution if the seed is a feasible solution. Comparing

Comparison of static maximum likelihood origin-destination formulations

705

Figure 3 and Figure 4 to Figure 5 and Figure 6 respectively, it is evident that although both formulations produce different objective function values, they are consistent in finding the optimum O-D matrix. Furthermore, both formulations, when provided with a feasible seed solution, select the seed matrix (maximum objective function).

Figure 5: Variation in Objective Function for Equation 13-7 for 5-Link Network for Feasible Seed.

Figure 6: Variation in Objective Function for Equation 13-7 for 2-Link Network for Feasible Seed.

706

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Consistency of Formulations The next step in the analysis was to compare the various formulations considering a feasible seed matrix. Once the seed matrix was fixed the optimal solution was obtained by enumerating all possible feasible solutions, this was repeated so that the seed matrix covered the entire spectrum of feasible solutions. This process was repeated for all the formulations using both networks. Surface and contour plots were obtained for each case with the x-y plane representing the feasible demand scenario (solutions 1 through 29 in Table 2 in the case of the 5-link network and solutions 1 through 9 in Table 3 in the case of the 2-link network) and seed demand scenario (solutions 1 through 29 in Table 2 in the case of the 5-link network and solutions 1 through 9 in Table 3 in the case of the 2-link network) and the z-axis representing the objective function value. The results indicate that when the total demand is constant, all formulations converge to the seed solution. Alternatively, when the total demand varies all objective functions except 13-4, 13-5, and 13-6 produce an optimal solution that corresponds to the feasible seed matrix, as illustrated in Figure 7. This finding clearly demonstrates that the formulations that are presented in the literature do not guarantee that the optimal solution converges to the seed solution even when the seed matrix is feasible when the total demand varies across the various feasible solutions, which is typical of practical applications.

Figure 7: Variation in Equation 13-4 Objective Function for 2-link Network for a Feasible Seed Combination.

Comparison of static maximum likelihood origin-destination formulations 707 Impact of Seed Matrix Scaling on Optimal Solution Considering the objective function of Equation 13-1, given that the seed matrix appears in the function as a ratio of a particular origin destination pair to the summation of all the seed values which, is in fact a probability estimate. Hence any scaling of the seed matrix does not impact the optimal solution regardless of the value of the scaling factor, which demonstrates that the structure of the seed matrix, and not its feasibility, is critical in estimating the optimum solution. With regard to the objective function of Equation 13-2, if we replace all ty's by k multiplied by ttj, where k is a positive non-zero scale scale factor and expand we derive the same Equation 13-2. Similarly, Equation 13-3 can be simplified using the same procedures to derive z T

( u>>«) = - l ¥ » (rv) + E V - » (>u) + ZT'iLn ']

(k}

U

It is clear that the last term will be a constant when the total number of trips is a constant as is the case for the 5-link network. In such a case the optimal O-D matrix will remain the same even though the value of the optimal objective function changes, as illustrated in Figure 8. Alternatively, in the case that the total number of trips across the various solutions vary (as is the case for the 2-link network) the scale factor affects the optimal solution, as illustrated in Fiaure 8.

Figure 8: Optimal Solution vs. Scale Factor using Equation 13-3

Equation 13-4 has an additional seed term that is not included in the objective function of Equation 13-3, which scales the objective function up or down depending on the value of the scaling factor. As the objective function in Equation 13-4 includes the objective function of Equation 13-3 it suffers from the same problem that were defined for the objective function 13-3.

708 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Considering the objective function of Equation 13-5, where we replace all tj/s by ktj/s, we can derive

z(Tii,tii) = y^-

+ Y—-Ykta =k2Y^~ + YT

ii.

One can see that, if the total number of trips is a constant the last two terms of the above expression is a constant and effectively the first term defines the optimal solution. Hence scaling of the seed matrix will not affect the optimal solution if the total trips remain constant; however this is not the case when the total number of trips is not constant, as is the case with the 2-link network. Carrying out a similar procedure for Equation 13-6, we derive

z (7-,,%) = -Y^- + kY'-0- - Yju , demonstrating that the optimal solution is independent of the scale of the seed matrix as long as the total number of trips remains constant. For the objective functions that are presented in Equations 13-7 and 13-8 the seed matrix is a ratio and hence any scaling of the seed matrix has no impact on the final optimal solution outcome. With regard to Equation 13-9 or the single step approach, any scale factor cancels out due to the difference of the logarithms of the seed values. Consequently, the scale of the seed matrix has no bearing on the final solution. In summary, the study demonstrates that the assumptions that were proposed in the literature can result in sub-optimal solutions when the total number of trips does not remain constant over the full range of potential solutions. Impact of Uniform Seed on Optimal Solution The next step in the analysis is to study the impact of different uniform seeds on the various objective functions. Typically, a uniform seed is assumed when prior information is not available. From the previous discussion it is clear that for some objective functions, scaling of the seed matrix can impact the optimal solution. Consequently, the seed matrix was held constant at 10 for each of the O-D demand values. The optimal solutions that are computed through exhaustive enumeration of all feasible solutions are presented in Table 4. Alternatively, the objective function for Equation 13-9 is solved using the QUEENSOD software.

Comparison of static maximum likelihood origin-destination formulations

709

Table 4: Comparison of Optimal Solution Scenario Considering a Uniform Seed. Equation Eq. 13-1 Eq. 13-2 Eq. 13-3 Eq. 13-4 Eq. 13-5 Eq. 13-6 Eq. 13-7 Eq. 13-8 Eq. 13-9

5-Link 18 18 18 18 16 20 18 18 18

2-Link 4 4 5 4 4 4

Table 4 demonstrates that when the total number of trips is constant (as in the case of the 5link network) all formulations with the exception of Equations 13-5 and 13-6, give the same optimal solution. When the total number of trips vary, as in the case of the 2-link network, Equation 13-3 and 13-4 along with Equations 13-5 and 13-6 deviate from the optimal solution. Further, the extent to which the solution shifts is dependent upon the magnitude of the uniform seed matrix because these four formulations are sensitive to the scale factor of the seed matrix, as was demonstrated earlier. Impact of Infeasible Seed on Optimal Solution For large networks an a priori matrix may not form a feasible solution. In fact, in most of practical applications a seed matrix does is not a feasible solution, hence it is important to study the properties of the formulations when the seed matrix is infeasible. This not only helps us in understanding the effects of infeasibility on the performance of the various formulations, but also assists in identifying the best formulation for computation purposes. Approximately 10,000 sets of seed matrices were obtained by filling the seed matrix with uniformly distributed random numbers. This process was repeated for both networks. For formulations, which were affected by the total demand of the seed matrix, the seed matrix was scaled to ensure that the total number of trips remained constant. The solutions for all the formulations and for every seed matrix were then obtained, for both networks by enumerating all feasible solutions. Given that the formulation 13-1 does not involve any simplifying assumptions, the solution obtained by it was used as a reference for comparison purposes. An error was estimated by comparing the solution scenario obtained by a respective algorithm relative to the 13-1 formulation solution scenario. Given that we had a total of 29 feasible solutions for the 5-link network the maximum error would be 29 while the maximum error would be 9 in the case of the 2-link network. Frequency plots of the error were then created and analyzed. In the case of the 5-link network, where the total demand remains constant, the formulations of Equations 13-1, 13-2 and 13-7 always produce identical optimal solutions. Alternatively, the formulations 13-3, 13-4, 13-8 and 13-9 produce solutions that are fairly close to the results obtained using the formulation of Equation 13-1, as illustrated in Figure 9a (error within a single solution). This finding is evident from the small deviation about '0' in the figure.

710 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Finally, the objective functions of Equations 13-5 and 13-6 operate poorly, as demonstrated in Figure 9b. Consequently, the objective functions of Equations 13-5 and 13-6 are limited in terms of their applicability. In the case of a variable total demand (2-link network), the formulations 13-1 and 13-2 produce identical results while the formulations 13-3, 13-7, 13-8, and 13-9 produce minimal errors, as demonstrated in Figure 9. Alternatively, the formulations 13-4, 13-5, and 13-6 produce significant errors. Impact of Link Flow Continuity As was discussed earlier, the single step and double step approaches estimate O-D trip table, even when flow continuity is not satisfied. The reason being, the link volumes are converted to equivalent link volumes that satisfy link continuity while minimizing the link flow error. In the double step approach, the flow continuity problem is resolved first and then based on maximum entropy techniques; the final O-D trip table is obtained. In the single step approach, the flow continuity and maximum entropy problems are solved simultaneously in a single pass. Both approaches were tested on the sample networks when flow continuity was not satisfied. Using the MATLAB and EXCEL solvers together with the QUEENSOD software, which solved the single step problem (formulation 13-9) was then implemented and applied. The results obtained from the three packages were similar and were identical to solutions obtained when one implemented the double step approach. However, the single step approach was found to be computationally more efficient.

Comparison of static maximum likelihood origin-destination formulations

(a) Equation 13-3 for 5-Link Network

711

(b) Equation 13-5 for 5-Link Network

(c) Equation 13-9 for 2-Link Network

Figure 9: Error versus Frequency Plot.

SAMPLE MODEL APPLICATION The QUEENSOD software estimates the maximum likelihood O-D matrix by solving Equation 13-9 numerically. A full description of the numerical approach is beyond the scope of this paper; however the solution algorithm is described briefly. The numerical solution begins by building a minimum path tree and performing an all-or-nothing traffic assignment of the seed matrix. A relative or absolute link flow error is computed depending on user input. Using the link flow errors O-D adjustment factors are computed and utilized to modify the seed O-D matrix. The adjustment of the O-D matrix continues until one of two criteria are met, namely

712

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

the change in 0-D error reaches a user-specified minimum or the number of iterations criterion is met. If additional trees are to be considered, the model builds a new set of minimum path trees and shifts traffic gradually to the second minimum path tree. The minimum objective function for two trees is computed in a similar fashion as described for the single tree scenario. The process of building trees and finding the optimum solution continues until all possible trees have been explored. The proposed numerical solution ensures that in the case that the seed matrix is optimum no changes are made to the matrix, as was the case in the formulation of Equation 13-9. In addition, the use of the seed matrix as a starting point for the search algorithm ensures that the optimum solution resembles the seed matrix as closely as possible while minimizing the link flow error. In other words, the seed matrix biases the solution towards the seed matrix. As a sample application, the QUEENSOD software is executed on a 3500-link network of the Bellevue area in Seattle, WA, as illustrated in Figure 10. The network is composed of 260 zones with a total demand of approximately 110,000 vehicles during the a.m. peak period. The O-D demand was calibrated to a.m. peak Single Occupancy Vehicle (SOV) and High Occupancy Vehicle (HOV) flows. The seed matrix was created using the standard four-step transportation planning process by applying the EMME/2 model and converted from EMME/2 format to INTEGRATION format. Other applications of the QUEENSOD software can be found in the literature (Rakha et al., 1998 and Rakha et ah, 2005).

Figure 10: Bellevue Network Illustration.

The calibration of the O-D demand to tube and turning movement counts was conducted using the QUEENSOD software using the planning trip distribution O-D matrix as the seed solution. A total of 882 counts were available for calibration purposes. These counts were gathered from a number of sources covering a time span of approximately 2 years. The calibration resulted in a high level of consistency between estimated and field observed link flow counts (coefficient of determination of 0.98 between the estimated and observed flows),

Comparison of static maximum likelihood origin-destination formulations

713

as illustrated in Figure 11. Figure 11 demonstrates that in calibrating the O-D table to observed traffic counts, the trip distribution O-D matrix (seed matrix) was modified significantly (coefficient of determination of 0.56 between trip distribution and synthetic O-D table). Consequently, it is evident that a modification of the trip distribution matrix was required in order to better match observed link and turning movement counts. It should be noted however, that the total number of trips was increased by only 4% as a result of the synthetic O-D calibration effort. Consequently, the illustrated calibration effort resulted in a significant modification of the trip table with minor modification to the total number of trips. Figure 12 illustrates that the total demand varied in the range of 15% across the various iterations of the algorithm demonstrating that the assumption of constant demand does not hold for practical applications. Finally, it should be noted that the estimation of the O-D demand involved estimating the routes of travel in addition to the O-D table (i.e. routes not know a priori). The results of Figure 11 demonstrate that the model was capable of matching the observed flows while estimating the routes of travel of individual trips.

Figure 11: Comparison of Estimated Flows and O-D Demand to Observed Flows and Seed Demand.

Figure 12: Variation in Total Demand across Various Iterations

CONCLUSIONS The paper presented the various maximum likelihood static O-D estimation formulations that are reported in the literature. The various formulations and solution algorithms for solving the synthetic O-D problem were compared considering that routes are known a priori under a number of conditions, (a) when the total number of trips is constant versus variable, (b) when

714 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction the seed matrix is feasible versus infeasible, and (c) when flow continuity is or is not satisfied. The conclusions of the study are: 1. There is no difference between the optimal solutions obtained from the trip and volume formulations regardless of whether the total demand remains constant or varies. 2. The proposed single step approach (Equation 13-9) yields results that are consistent with the results obtained from applying exhaustive enumeration to the trip formulation. 3. Willumsen's primary formulation (Equation 13-3) cannot be applied to networks, where the total number of trips is a variable, without introducing considerable error in the final matrix estimate. 4. Willumsen's subsequent formulation (Equation 13-5) and Ortuzar and Willumsen's formulation (Equation 13-6) can be applied if and only if the total trips is a constant and TijItij^L, which severely curtails the utility of the formulation. 5. The solutions obtained from the trip formulation, volume formulation and the single step approach are independent of the scale of the seed matrix whereas, Willumsen's formulation and its subsequent derivatives are influenced by the application of a scale factor to the seed matrix. 6. In the case where the solution search is initiated with an infeasible seed Willumsen's formulation (Equation 13-5 and 13-6) can produce errors as high as 20%. 7. In the case of an infeasible seed and a variable total demand the error produced by Van Aerde et a/.'s single-step approach (Equation 13-9) is less than 5%. 8. The numerical solution of Equation 13-9 that is incorporated in the QUEENSOD software estimates O-D tables efficiently even when routes are not known a priori.

ACKNOWLEDGEMENTS This research effort was funded by the Mid-Atlantic University Transportation Center (MAUTC) and the Virginia Department of Transportation (VDOT).

REFERENCES M. Van Aerde & Assoc, Ltd. (2002), QueensOD© Rel. 2.10 User's Guide: Estimating Origin Destination Traffic Demands from Link Flows. M. Van Aerde & Assoc, Ltd. Ortuzar J. and Willumsen L. G. (2001), Modelling Transport: Third Edition, Wiley. Rakha H., Van Aerde M., Bloomberg L., and Huang X. (1998), Construction and Calibration of a Large-Scale Micro-Simulation Model of the Salt Lake Area, Transportation Research Record, No. 1644, pp. 93-102. Rakha H. and Ahn K. (2004), INTEGRATION Modeling Framework for Estimating Mobile Source Emissions. Journal of Transportation Engineering, Vol. 130(2), March/April, pp. 183-193. Rakha H., Medina A., Ahn K., El-Shawarby I., and Arafeh M. (2005), Evaluating Alternative Lane Management Strategies along 1-81, Transportation Research Board 84th Annual Meeting, Washington D.C., CD-ROM [Paper 05-2086].

Comparison of static maximum likelihood origin-destination formulations 715 Van Aerde M., Rakha H., and Paramahamsan H. (2003), Estimation of O-D Matrices: The Relationship between Practical and Theoretical Considerations. Transportation Research Record, No. 1831, pp. 122-130. Van Zuylen J. H and Willumsen L. G. (1980), The most likely trip matrix estimated from traffic counts, Transportation Research, 14B, 281-293. Wilson, A. G. (1970), Entropy in Urban and Regional Modelling, Pion: London. Willumsen L. G. (1978), Estimation of an O-D matrix from traffic counts: A review, Institute for Transport Studies, Working paper no. 99, Leeds University. Willumsen L. G. (1981), Simplified transport models based on traffic counts, Transportation 10, 257-278. Willumsen L. G. (1984), Estimating time-dependent trip matrices from traffic counts, Ninth International Symposium on Transportation and Traffic Theory, pp. 397-411. Utrecht: VNU Science Press.

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36

A TIME-DEPENDENT ACTIVITY AND TRAVEL CHOICE MODEL WITH MULTIPLE PARKING OPTIONS Hai-Jun Huang and Zhi-Chun Li, School of Management, Beijing University of Aeronautics and Astronautics, China; William H. K. Lam, Department of Civil and Structural Engineering, Hong Kong Polytechnic University, China; S. C Wong, Department of Civil Engineering, The University of Hong Kong, China

INTRODUCTION In recent years, parking has become an increasingly serious problem in most densely populated cities due to the dramatic increase in the number of vehicles. Consequently, traffic conditions, particularly in peak periods, have tended to deteriorate in many urban areas, and the search time for parking has substantially increased in central business districts. Parking is closely tied to road usage (Glazer and Niskanen, 1992), and every car has to park at the end of a trip. Hence, parking policy offers a potentially strong instrument for influencing traffic flows (Verhoef et al., 1995). It has widely been recognized that parking policy plays an important role in the planning and management of transportation systems in densely developed urban areas (Young et al., 1991; Wong et al., 2000; Tong et al., 2004). It has been observed that drivers spend a significant percentage of their total trip time searching for a parking space (Axhausen and Polak, 1991). This can significantly affect the level of traffic congestion and environmental quality within the urban area (Feeney, 1989). Parking, as an integral component of urban transportation systems, influences not only the parking system itself, but also the whole transportation system, and even the socio-economic system (Bifulco, 1993). To make the urban transportation system work more

718 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction efficiently, it is necessary to understand and predict the parking choice behaviour of travellers with regard to various parking policies, and also their travel choice behaviour in relation to changes in parking supply and parking charges. There is a body of literature on parking and travel choice models. Generally, these models can be classified into three categories: (1) the discrete choice approach (Polak et al., 1990; Hunt and Teply, 1993; Thompson and Richardson, 1998; Hensher and King, 2001; Hess and Polak, 2004); (2) the probability based approach (Wong et al., 2000; Dell' Oreo et al., 2003); and (3) the network based approach (Florian and Los, 1980; Bifulco, 1993; Lam et al., 2002). However, these studies mainly focus on time-stationary (static) equilibrium analysis, and little attention has been paid to time-dependent analysis. The static model cannot reveal the temporal and spatial interaction between road traffic and parking congestion, and hence may not be able to capture the congestion effects on travel choices and parking behaviour that can be carried over time during the day. All the aforementioned studies belong to the traditional trip-based approach, which treats the trip as an independent unit of analysis (Kitamura, 1984). The trip-based approach fails to recognize the existence of linkages among trips, and between trips and activity participation (Jones et al., 1990). In some instances, the forecasts of trip-based models have proved to be inaccurate due to their misrepresentation of travel behaviour (Recker et al., 1986). Therefore, the trip-based approach cannot predict many individual responses to various transport policies and leads to biased predictions of the effects of these transport policies (Pas, 1983). As a matter of fact, the travel demand that is generated within a city can be regarded as the result of the many decisions that people make about participating in activities in various locations, constrained in time and space. In other words, travel is usually undertaken not for its own sake, but rather for participation in something at the end of a trip (Damm and Lerman, 1981). Thus, travel demand forecasting must be based on a reasonable theory that takes account of human activities. In recent years, it has been widely recognized that travel is a derived demand that results from the needs and desires of individuals for participation in various activities (Mahmassani, 1988). Interaction between travel and activity appears to be a key to understanding travel-related choices (Supernak, 1992). Activity participation is the end of travel that such participation produces, whereas travel that connects the activities in the time-space continuum is the means of activity participation (Recker et al., 1986). Therefore, to better understand the travel behaviour, it is necessary to incorporate activity participation into the traditional travel demand analysis approach. In the past three decades, activity-based analyses have received widespread attention and have made substantial contributions to the better understanding of travel behaviour. This approach studies travel patterns in the context of the structure of activities, of the individual or household, with a framework that emphasizes the importance of time and space constraints (Goodwin, 1983).

Time-dependent activity and travel choice model with parking options 719 Damm and Lerman (1981) formulated a theory of activity scheduling for urban workers to investigate whether they participated in a non-work activity on the way from home to work or from work to home, and if so, for how long. Kitamura (1983) and Kitamura and Kermanshah (1984) employed the sequential modelling approach to analyze the time-, history-, or futuredependency of the activity and destination choices in a trip chain. Pas (1983) employed a methodology of the systematic identification of class to analyze daily travel-activity behaviour. Horowitz (1980) developed a structural model for multi-destination non-work travel based on the principle of utility maximization. Kitamura (1984) derived a model of daily time allocation from the concept of random utility maximization. Axhausen (1990) combined an activity chain simulation model with a traffic flow simulation model to achieve a simultaneous simulation of activity chains and traffic flows. Hamed and Mannering (1993) and Mannering et al. (1994) developed a methodology capable of modelling daily activity-travel patterns using discrete/continuous econometric models and hazard duration models with survival distributions. Bhat (1998a) examined travellers' activity behaviour after returning home at the end of the work tour using data from a Boston region household activity survey. Comprehensive reviews of the activity-based approach can be found in Kitamura (1988), Jones et al. (1990), and Bowman and Ben-Akiva (2000). Recently, Lam and Yin (2001) applied the activity-based approach to modelling time-dependent traffic assignment. Lam and Huang (2002, 2003) developed a combined activity/travel choice model for hunting for the dynamic user equilibrium solution on congested road network with queues. Huang et al. (2003) explored the activity and travel schedule behaviour under advanced traveller information system. Abdelghany and Mahmassani (2003) presented a stochastic temporal-spatial micro-assignment and activity sequencing model for activity/trip chains in a dynamic framework. However, the combination of the parking problem with activity-based analyses in a unified framework has not been explored. As a matter of fact, there is a close relation between activitytravel scheduling and parking choice. An activity-travel schedule may significantly interact with the choice of parking location. For instance, travellers who depart from home to perform a work or non-work activity during rush hours may spend a lot of time searching for an available parking space because of high parking occupancy. This may also give rise to an increase in cruising time on the road network, and hence create additional traffic congestion and pollution problems, hi contrast, the characteristics of parking locations such as parking availability, the search time delay for parking, and parking charges, may plausibly be expected to vary according to the activity-travel patterns of travellers across the time of day. Therefore, travellers may change their choices of parking location and parking duration according to the characteristics of different parking locations at different times of the day. In this paper, we propose a time-dependent equilibrium model for studying the parking behaviour of travellers in the activity-based paradigm. To capture the temporal and spatial constraints of the activity-travel schedule, we introduce the concept of time windows for each activity in a traveller's itinerary. The travellers who start to perform the activity outside the time windows will

720 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction undergo a large penalty. Therefore, to some extent the time windows of the activities implicitly determine the order of the activities and even the activity patterns. We assume that the likelihood that a traveller selects a particular activity is based on utility maximization, where the traveller selects the activity that maximizes his/her perceived utility. We use the nested logit formulae to describe the choices on activity type, activity location, and activity duration, and employ the deterministic equilibrium conditions to govern the choices on parking location and travel route. All these choices follow a hierarchical structure and are finally integrated by a variational inequality problem. The proposed modelling approach can be used to assess the effects of parking policies on activity-travel scheduling behaviour and to make long-term strategic plans. The remainder of this paper is organized as follows. In the next section, we introduce the basic considerations about the proposed modelling approach. Then, a time-dependent equilibrium model is presented and a solution algorithm for solving this model is given. Furthermore, we provide a numerical example to test the proposed model and solution algorithm. Recommendations for future researches are given in the last section.

BASIC CONSIDERATIONS Assumptions To facilitate the presentation of the essential ideas of this paper, the following assumptions are made. Al The whole study horizon [0, 7] is discretized into equally spaced intervals, sequentially numbered / e T = {0,1, , T}, and 8 is the length of a period so that T8 = T. A2 All commuters are homogenous concerning their values of time and utility perception; however, their activities are differentiated by the activity type, activity location and activity duration, hi addition, the parking facilities are characterized by type (e.g., on-street and off-street) and time-varying parking charges. A3 The decision-making on choices of activity, route and parking location is done before entering the network, i.e., any en route shifting is not allowed. This assumption is appropriate for assessing transportation policies at a strategic level. A4 The trip-chain pattern of a person is not pre-specified before having a trip for an activity but caused by his or her choices on a series of decision-makings within a day. A person always makes decision on his or her next activity once he or she completes the current one. A5 The decision-making process on activity, travel and parking choices follows a hierarchical structure, i.e., travellers first decide what to do (activity type choice) and where (activity location choice), then decide how long for it (activity duration choice), and finally where to park their cars

Time-dependent activity and travel choice model with parking options 721 (parking location choice) and the path by which to reach the chosen parking location (travel route choice). A6 The travellers have full information about traffic and parking conditions, so that they can find parking locations with minimal disutility and travel routes with minimal travel times. A7 Each activity is associated with a specific time window, outside which a traveller starting to perform the activity will suffer a large penalty. A8 The parking charge depends on both the time that a traveller enters the parking location and the parking duration. Walking time is dependent on the walking speed and the distance between the parking site and the final destination (Bifulco, 1993). A9 One or more activities may be performed in a zone. If two or more activities are performed in a zone, travellers first perform the primary activity and then other ones (Bowman and Ben-Akiva, 2000). Travel and activity utilities Consider a city network G = (N, A), where TV is the set of all nodes, including origin nodes, intersection nodes, destination nodes, and parking locations, A is the set of all directed links, including centroid connectors from origin to road network, road links, parking access links from road network to parking location, and pedestrian or walk links from parking location to final destination. Let r denote an origin node, r e N, and 5 be a destination node, s e N. An origin node, which generates trips, may also be a destination node for an activity, and vice versa. Let J denote the set of all activities j , j ej , and Dj the set of all feasible activity durations ij associated with activity j , tj e Dj . Let Irs be the set of all feasible parking locations serving the travellers between OD pair (r, s). In the activity-based travel demand analysis, activity utility, defined as the importance or the degree of need for an activity, plays an important role because the daily activity schedules of individuals directly influence travel demand (Supernak, 1992). When deciding whether or not to participate in an activity, one makes a trade-off between the utility of activity and the disutility of associated travel-parking. In this paper, we assume that every traveller attempts to make an activity-travel schedule that has the maximum overall utility. Define UJrs(t,%j) as the comprehensive utility gained by a traveller who departs from origin r for destination 5 in period t and performs activity_/ for duration Tj in the destination zone s. This comprehensive utility can be expressed as UJn(t,Tj) = -CJrs(t,Tj) + a[Vj(t,TJ) + Vs+Vsj], VreN,seN,tsT,XjeD,jeJ

(1)

722 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction where CJrs(t,%j) is the travel disutility from r to 5 departing in period * for performing activity/ for duration Xj, Vj{t,Xj) is the total utility of performing activity/ by a traveller, who leaves origin r for j- in period t and performs the activity / for duration x j , Vs is the systematic component of utility common to all elements with destination s (e.g., these caused by area, population and employment), VSJ is the systematic component of utility varying across both activity and destination (e.g., these caused by business conditions around the destination if j represents shopping activity), a is a conversion factor between travel disutility and activity utility. In general, the utility that is gained from an activity depends on the starting time of the activity and the duration that it lasts. When a traveller's departure time from home and activity duration is determined, the beginning and ending times for his/her activity can be calculated. Therefore, the activity and travel schedule of a traveller is governed by departure time and activity duration. Let a(t) be the starting time of an activity for these travellers who depart from origin r in period t, which is also the time arrival at the activity location (i.e. destination). Let Tri(t) be the in-vehicle travel time for those who depart from r in period t and park their cars at location i, dt (t + Tri (t)) be the searching time delay for finding an available parking space for those who depart from r in period t and arrive at location i at time t+Tri(t), and wis be the walking time from i to 5. Then, a(t) is given by CT(0 = t + Tn(0 + d, (t + Tri(0) + wis,-iteT, The utility Vj(t,Xj)

(2)

in Equation (1) is the sum of all utilities gained from performing activity/

from time a(?) to time a(t) + Tj.lt can be expressed as V

iCTJ) = Z

V

J(*>.

Vf e T T

' J eDJeJ

(3)

where Vj (t) is the marginal activity utility in time interval t. Define C'rs ,-(f,Xy) as the travel disutility caused by leaving r for 5 during period ; for performing activity/ for duration Tj and parking in location /', formulated as C/Sji(/,Tj) = Trl(t) + C{s(t + Tri(t),tj),VieIn,reN,seN,teT,XjeD,jeJ

(4)

J

where C is{t + Tri (t),ij) is the travel disutility by traversing parking location i for destination s, and performing activity/ for duration %j there, expressed as Ci(t + Tn(t),xj) = a\di(t + Tri(t)) + a2zi(t + Tri(t),x]) + aiWls+as<~>js[c:(t)-\, VJ e Irs,r e N,s e N,t e T,Xj s D,j s J

(5)

Time-dependent activity and travel choice model with parking options 723 where z; {t + Tri(i),Xj) is the time-varying parking charge, QJs[a(t)] is the schedule delay penalty cost of early or late arrival at destination s for performing activity j , a{ , cc3 are the reciprocal transformation coefficient between searching time, walk time and travel disutility, and a2 , as are the conversion factors between parking charge, schedule delay cost and travel disutility. The parking charge, which depends on the arrival time and desired duration for parking, can be formulated as (Tong et al., 2004) z(. (t + Tri(t),zj) = ei xh(t + Tri(t),Zj)xPi(t + Tri(t)), Vi£lrs,reN,seN,teT,Tj eDJsJ (6) where 6 ; is the parking charge discount given by location z's policy (in general, a smaller 6, corresponds to a longer duration for parking), h(t + Tri(t),xj) is the parking duration related to activity duration and walking time from parking site to activity location, and pt (t + Tri (/)) is the hourly parking fee which is related to the arrival time interval at parking location. The parking duration, hit + Tri(t),%j)or h for short, equals the activity duration plus the time needed for the round trip on foot between parking location and activity location, i.e., h = h(t + Tti(t),Tj) = Tj+2w!s, VieIrs,r<=N,seN,teT,jeJ

(7)

The walking time wis, which depends only on the distance from parking location to destination, can be defined as wh = V(i, s)/
^j[tf-^s-a(t)l ®{[<s(t)] = Xj[G(t)-tf

-

if tf-Ai>a(t)

AJS],

if tf + AJS < a(?),Vs e N,t eTJeJ

(9)

0, otherwise where [tj. -AJs,tJs +A^] is the desired time window for starting activity j at destination s without any schedule delay penalty, tj is the middle point of the time window, and ^ and Xj are the unit costs of early and late arrivals, respectively. Given by Equation (2), a(t) is the start time of having an activity for travellers who depart from origin r in period t.

724 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction As in Lam et al. (2002), the BPR (Bureau of Public Roads) type function for searching time is adopted in this study, i.e., d, (t + TH(t)) = d° +0.15(£>. (f + 7\(0)/K;-)4, V; eIrs,re

N,s eN,tzT

(10)

where df is the free-flow access time for parking a car in location i, Dt (t + Tri (t)) is the parking accumulation or occupancy in location / when vehicles arrive at time (t + Trj (t)), and K, is the capacity of location i for parking. The parking accumulation in location i can be computed from path inflows. For OD pair (r, s), the cumulative arrivals, UJri(t), for performing activity j and parking their vehicles in location i by the period t (not including period t) can be represented by 0 = Z Z TTtiJ^j')' .f-l

k

PtPr,

VieIa,reN,seN,teT,jeJ

(11)

T

where Pri is the set of all paths between origin r and parking location i, ft

(k,Xj) is the flow

rate of travellers who depart from r to i in period k by path p and perform activity _/ for duration Xj around destination s. Then, the total cumulative arrivals in any parking location i till period / from all OD pairs for performing activity j , U{ (t), can be computed by t//(0 = 2X^(0, Vi<£ln,t£T,jeJ

(12)

For a certain OD pair, the travellers with parking duration h, who arrive at parking location ;' before period (t-h) should leave this location till period t. So, the cumulative departures from location / by period t are ^(0 = 'i; i X Z X 4 f t ^ ) ' VieIa,reN,seN,teT,jeJ

(13)

Considering all OD pairs, the total cumulative departures from any parking location i, are *f(0 = I > M VieIn,teT,jeJ (14) Therefore, the parking accumulation in location i during period t, for having activity j , can be computed by D({t) = U{{t)-Vi'{f), VieIrs,reN,seN,teT,jeJ (15) The total parking accumulation in location i during period; is Di(t) = Y,D'i{t), VieljeT (16) j

where / denotes the set of all parking lots in the network.

Time-dependent activity and travel choice model with parking options 725 Behaviour modelling According to A5 made previously, the activity and travel choices follow a hierarchical decisionmaking process. Specifically, before making a trip, a traveller first determines his or her activity type and activity location, and then selects the desirable activity duration. We now describe the hierarchical choice process using the nested logit (NL) model, which is not subject to the independence of irrelevant alternatives (IIA) property as in the multinomial logit (MNL) model. The NL model is often used to characterize the travel and activity choices in the literature (Horowitz, 1980; Kitamura and Kermanshah, 1984; Mannering et al., 1994). In addition, the NL model was used by Ben-Akiva et al. (1986) for studying the departure time and route choices, by Bhat (1998b) for investigating the travel mode and departure time choices, and by Huang and Lam (2003) for formulating the departure time and route choices under advanced traveller information systems. Let Nr(t-\) be the number of potential travellers within the origin zone r at the end of period t1, these people will make decisions on travel and activity schedules in the next time period t. Let qJrs(t,Xj) be the travel demand departing from r for s in period t and performing activity j for duration %j . Using the nested logit model, we have explUi (t, x ; ) / B] qUt,zi) = NM-\)*

HnKt J)

A

J

n

J

exv\UJ (t *) I vl expire>/. )H1

Xexp[t/;;(,,T,)/p]Zexp[^,*)/y]'

yreM,seNjeT,xi

eDhjeJ

(17)

J J J

'

where UJrs(t*) = pin^exp[[// s (f,-c y )/S], which reflects the accessibility of performing activity j at destination 5 by the travellers originating from r at period t. 6 and y are the scaling parameters that measure the variability in choice behaviour among the travellers. To be consistent with random utility theory and reasonable signs of cross-elasticity, the mixed discrete/continuous nested logit model requires thaty > p > 0. Let qJrs (t) be the marginal travel demand departing from r for s at period t for performing activity j . From Equation (17), one obtains qJa(t) = Nr(t-l) " g g y V y ] , VreN,seN,teT,jeJ Combining Equation (17) and (18), we get

(18)

726

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

qJrs(t,tJ)

= qJrs(t) —

^—t

, VreN,SeN,teT,%jeD,jeJ

(19)

V Now we formulate the parking location choices for travellers departing from origin r at time slice t and performing activity j for duration Xj at destination s. Following A6, travellers park their cars in such location that has the minimum disutility from origin to final destination. This results in a deterministic parking location choice equilibrium: for each OD pair, travellers who depart in the same time interval and perform activity j for duration Xj, experience identical and minimal generalized travel cost; no traveller would be better off by unilaterally changing the parking location. In other words, the travel cost of any unused parking location for those who perform activity j for duration Xj , is greater than or equal to the minimal travel cost. The parking location choice equilibrium can mathematically be expressed in a complementary form as [QU ? ' T ,)- c «('> T j)]9i.<>> T /) Ci/t^^^Ciit^j),

=0

' V'£lrs,reN,seN,teT,TjeDJeJ

(20)

yieIrs,reN,seN,teT,TjeD,jeJ

(21)

qij(t,Tj)>0, \fieIrs,reN,seN,teT,Tj£D,jeJ Ci(t,Tj) = mmCij(t,Tj), VreN,seN,teT,TjeD,jeJ where q^sj(t,Xj)

(22) (23)

is the travel demand departing from origin r for destination 5 via parking

locationi in period t and performing activity y for duration Xj . According to A6, the route choice of all travellers follows a deterministic time-dependent equilibrium: for each OD pair, the actual travel times experienced by travellers departing in the same time period are equal and minimal (Ran and Boyce, 1996). Let Trip(t) be the path travel time for travellers departing from origin r to parking location i in period / via path p, where p e Pri and Pri is the set of all paths between origin r and parking location i. This timedependent route choice equilibrium can be mathematically expressed as lTri,p(t)-Tri(t)]flJip(t,Tj) = O, VpePri,ieIn,reN,seN,teT,TJeD,jeJ Trip{t)>Tri(t), VpePri,ieIn,reN,seN,teT

(24) (25)

fiP(t,T)>0, Vp<EPri,i<Elrs,reN,seN,teT,TjeD,jeJ

(26)

r.(/) = minr. it), Vie I ,reN,seN,teT

(27)

ptp,<

"p

The path travel time in Equations (24)-(25) can be represented by the sum of all link travel times along the path (Chen and Hsueh, 1998)

rn-.,(0 = S Z c . <*>&(*) VpePH,ieIa,reN,seN,teT a kit

(28)

Time-dependent activity and travel choice model with parking options 727 where b"pt (k), an indicator variable, is equal to 1 if the flow on path/? departing in period t from origin r to parking location i arrives at link a in period k, and 0 otherwise, ca (k) is the link travel time when entering link a in period k. Similar to the work by Chen and Hsueh (1998), the link travel time can be formulated as a function of the inflows entering that link before period k, i.e., ca(k) = f(uji),uj2),---,ua{k)), \?azA,keT (29) where ua (k) is the inflow entering link a in period k, and

«,(*) = ZZ£4<(*>' Va^A,kzT r,i p

(30)

t

where u"pt (k) is the inflow to link a in period k for these travellers who departs from origin r in period t and select path p toward parking location i. The link inflow, u"pt(k), is composed of path inflows as follows: <„, (*) = YZfip(!'TjK.

(*) yaeA,pePrl,ieIn,rsN,seN,teT,keT.

(31)

So far, we have defined and formulated all behaviour choices to be investigated in this paper. In summary, both the path choice and parking location choice follow the time-dependent deterministic user equilibrium, whereas the combined choices of destinations, activities and activity durations are governed by a nested logit model. We now define the time-dependent network equilibrium with simultaneous choices on activity type, activity destination, activity duration, parking location and travel route as follows. Definition 1 A flow pattern \j'jip(t,Xj),qJrsi(t,Xj),qJr*(t,Xj),qJr*(t)}

is a time-dependent

equilibrium of the joint choices on activity type, activity location, activity duration, parking location and travel route, if it satisfies the conditions (18)-(31) simultaneously.

EQUIVALENT VARIATIONAL INEQUALITY FORMULATION In this section, we present an equivalent variational inequality (VI) formulation for the problem given by Definition 1. First, we list all constraints for the problem. Constraints Let Q denote the set of path flows and OD demands that satisfy the following constraints: £ft P ('.^) = <.C*j). VieIn,reN,seN,leT,TJ eDJeJ (32) P

YiqJrs,At,T]) = qJrs(t,Tj), VreN,seN,teT,TjeDJeJ

(33)

728 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

T,qUt>*j) = 0,

(34)

VreN,teT

(35)

Vp(EPri,i<Elrs,r£N,seN,teT,Tj<ED,j£j

(36)

<,(f,Tj),q'n(t,Tj),q'rs(t)>0, Vi e /„, r e tf, s e JV, f e T,rj eDJeJ (37) where Equation (32) states the conservation of path flows. Equation (33) is the parking demand conservation. Equation (34) is the demand conservation for activity duration. Equation (35) is the conservation for activity type and location combination. Equations (36) and (37) are the usual non-negativity constraints on path flows and demands, respectively. In addition, the conservation of flows at origin node r should be considered

Nr(t) = Nr(t-l)+ X "aC-i-CaM)- X "aC-1). VreN,teT

(38)

SEB(I-)

aeA(r)

where A{r) is the set of links with head node r and B(r) is the set of links with tail node r. And, the following definitional constraints should also be included T *.,(?) = HHc°(kK (*) V ^s pn.is hs £N,S£N,teT (39) a k>t

»a(k) = YsYYJ.fip^jKW> rj

p

VaeA,keT.

(40)

j t,Zj

VI formulation The VI formulation equivalent to conditions (18)-(31) can be stated as follows: Theorem 1 A time-dependent flow pattern in the context of daily activity-travel schedule reaches an equilibrium state if and only if it satisfies the following VI requirement:

Z S S C o \YLuipW-f£P(f,xj)-\ +Sli:xcf(/,x,)(^,,a,xi)-^,,a,x,)) +SIlfpln^^-a(F y (/,x y .) + n+K/)](^(/)xy)-^(My)) r,s j 1,1 j \

Irs (')

J

-YTTy^ r,s

j

t

q rs{t)

'

r \L

subject to {flp(t,-ij),qJrStl(t,Zj),qJrs(t,Zj),qJrs(t))eCl,

L

(qiAt)-qj;(t))>o )

where Cl is given by (32)-(40). The

variables with asterisks in the above VI are the optimal solutions. The proof of Theorem 1 is similar to that in Lam et al. (2004). It is omitted here due to the paper length constraint, but will be provided if readers ask for. It should be pointed out that the VI (41)

Time-dependent activity and travel choice model with parking options 729 is route based, hence route enumeration is required in both model formulation and solution algorithm. It should also be noted that the indicator variable 5"pt(k) in (28) depends on link travel times which in turn depend on link inflows. Consequently, the path travel times are essentially non-linear and non-convex. It can be proven that all functions (given or endogenous) are continuous, hence there exists at least one solution (Huang and Lam, 2002).

SOLUTION ALGORITHM In this section, a route-based heuristic algorithm is developed to solve the model proposed in this paper. The basic logic of the algorithm follows the method of successive averages (MSA), but in each of the steps a time-dependent version of the Gauss-Seidel decomposition approach is used. This decomposition approach is one of the partial linearization methods (Patriksson, 1993). The step-by-step procedure of the algorithm is given below. Step 0 Let the iteration count n = 1. For each OD pair and time period, on the base of Nr(0), set the initial demand pattern ^Jrs(n)(t)}, {jJ^n)(t,Xj)} and ^f{t,Xj)\, then compute the time-dependent path flow pattern

and

\f/^p\t,Xj)j.

Step 1 Compute the link inflows uS"\k) by (30) and (31), and then the link travel time c««by(29). Step 2 Calculate the path travel time T^"p(t) by (28), let T}"}(t) = min T%p(t) and

P^={p:T${t) = T^\t),pzPn). Step 3 Compute the generalized travel cost c£f(t,ij)

by (4)-(16), let C*"\t,xj) =

CJr}"\t,Xj)Jelrs).

rm*Cif(t,Zj) and /£> = [i:CJ$\t,ij) =

Step 4 Compute the comprehensive utility UJrs(t,Xj) by (l)-(3), update Nr(t-\) J

J

by (38), and

n

then get the travel demand g r^"\t) and g r^ \t,Xj) by (18) and (19), respectively. Step 5 Assign g^"\t,Xj) demand g £ f

to the parking sizes in /^"' defined in Step 3, get the parking (t,Xj).

Step 6 Assign g^f(t,ij)

inflows

onto the paths in P^ defined in Step 2, get the auxiliary path

j

h r$(t,xj).

Step 7 Update the path departure flows by MSA, fj.{n+l)(t

x 1= fj{n)(t

x ) + -
I V)

730 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

StepS

If \f^it,xJ)-f^(t,xj)^

, \q^f\t^j)-^"\t^j)\<^

.

\q'^n(t,xj)-

giW(t,ij)\<s and \qJrs(n+l)(t)-qJrs(n)(t)\<e , stop; set n = n+.l and go to Step 1, otherwise. The parameter s is a pre-specified precision, taking 0.01 in this paper. In Step 0, the initial time-dependent path flows can be determined by assigning the initial demand patterns onto an empty network. The operations in Steps 1 to 6 directly follow the equations presented in previous sections. In Step 7, the MSA is used to update the path departure flows with predetermined step size.

NUMERICAL EXAMPLE In this section, a numerical example is given to illustrate the model and algorithm in this paper, and investigate the effect of various parking schemes on activity and travel schedules. Input data To illustrate the application of the proposed model and algorithm clearly, we apply them to a simple network consisting of 3 zones, as shown in Figure 1. Four types of activities, i.e. staying at home (H), working (W), having dinner (E), and joining social activity (S) are considered. Each link is two-way link with identical free-flow travel time and capacity in two directions, as given in the Table 1. There exist two categories of parking locations around work zone and social activity zone, i.e. the on-street parking locations (c, d, e, g, i) and the off-street parking locations (b,f, h). There is only one off-street parking location (garage) a around residential zone. Both the W and E activities can take place in work zone, but W is the primary one. The marginal temporal utility profiles of these four activities are shown in Figure 2. The study horizon is from 00:00 to 24:00 and is discretized into 96 time periods each with 15 minutes. Table 2 gives the unit costs of the schedule delay penalty, the desired start time windows and the feasible durations for these four activities. It is assumed that there are 3000 seed travellers in the residential zone, i.e. N(0) = 3000. The parking access (and egress) times for all on-street parking sites and all off-street parking locations are assumed to be 0.05 hour and 0.1 hour, respectively. The charges over the time of day for on-street and off-street parking follow a step function, as shown in Figure 3. However, the off-street parking (garage) in home zone and the on-street parking around restaurant are free. Moreover, it is assumed that the parking for working activity is subsidized by the employer's company. The distance from each off-street parking point to any nearby activity location is 0.5 km, while the distance from each on-street parking points to any nearby activity location is 0.25 km. The walking velocity is co = 5.0 km/h. The discount coefficients 9, for all parking locations are assumed to be 1.0. The capacities of nine parking locations are: tfa=3000, £s=2600, Kc=K
Time-dependent activity and travel choice model with parking options 731 Other parameters are:a = 0.025, ax =1.4, a , =0.1, a 3 =1.8, as =0.3, (3 = 0.1, y = 0.3. The case with the above inputs is called "base case".

Figure 1 The example network

Table 1 Input data associated with link travel time functions Link

(he) (c,2)

a.(i, o3)

c°a (h) 0.45 0.05 0.35 0.05 0.05

Cfl(veh/h)

Link

1500 1000 1500 1000 500

(g. d)

(d,2)

(W, e) (e,E) —

Cfl°

(h)

Ca(veh/h)

0.15 0.05 0.15 0.03 —

1000 500 1000 500 —

Table 2 Input parameters associated with activities Activity type H W E S

Arrival early penalty Arrival late penalty Xj (HK$/h) C,j (HK$/h) 0.0 8.0 5.0 3.0

0.0 25.0 20.0 15.0

Time window (o'clock)

Activity duration (h)

[00:00, 24:00] [08:45,09:15] [12:30, 13:00] [18:00,21:00]

[8.0, 12.0] [5.0, 12.0] [0.5, 02.0] [1.0,03.0]

732 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2 Temporal utility profiles for the four activities

Figure 3 Step charges designed for on-street and off-street parking locations

Numerical results We firstly examine the convergence of the iterative algorithm. Figure 4 plots the value of the equilibrium criterion against iteration number. This equilibrium criterion is defined as

Time-dependent activity and travel choice model with parking options 733

G=IIE^>(^,)(^«-^)(o)+EEE^;)(^J)(C(/,ti)-e>(^)) r,i j,p t,Xj

r,s j,i t,tj

+YYY^(t

x w-\t x )-g*»(Q «*U™W®

+YYYam(t\\am(t)-NM(t-n

ex

i

P(t/"°)(f'*)/y) i

This is a gap function which can measure how closely the outputs generated at n-th iteration satisfy the time-dependent user equilibrium condition (19)-(32). Figure 4 shows that the algorithm converges well, although some oscillations exist in the first 150 iterations.

Figure 4 Convergence of the solution algorithm

Figure 5 displays the parking occupancies in nine locations for parking. It can be seen that before 6:00 all people still stay at their homes, thus location a around H are full. After 6:00, some people start to leave their homes for working; at about 9:15, almost all people arrive at their offices due to the rigid work start time windows. Starting from this time, location a is idle and the parking locations around W become busy. We can see that the off-street parking location b is first occupied, due to its limited capacity then, the overflows go to the on-street parking locations c and d. After 11:00, some people leave their offices for having lunch; they first occupy the onstreet parking location e around restaurants as it is free for parking; at about 12:30, location e is

734 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction fully occupied and some people turn to the off-street location/; at 13:30, the occupancy rate of location/reaches 82.3%. At about 14:00, most people return to offices for working. After 16:00, some people leave their offices for performing social activities. They first park their cars at the on-street parking location g, then at the off-street parking location h when g is congested. After about 21:45, all people end their daily activities and go back home. Nobody uses the parking location / because the chain "home-social activity-work" hardly occurs under the requirement of rigid work start time windows.

Figure 5 Parking occupancies at different sites over the time of day Figure 6 gives the population against day time in the three zones. It indicates that most people stay at home before 7:45 and after 21:15, work and have lunch from about 8:45 to 17:45, and join social activities from about 18:00 and 21:00. Figure 7 depicts the departure rates for working against day time and duration length. It is found that the highest departure rate for working reaches 200.4 vehicles per hour at around 8:15, these people work for 8.25 hours long. To examine the model's sensitivities subject to different parking policies, we design the following four scenarios for comparison, i.e., Scenario I-base case, Scenario II-the capacity of location b is increased to ^=3000, Scenario III-the charge of location h is decreased by 2 HK$, and Scenario IV-the combination of II and III. Table 3 summarizes the results generated in these scenarios. It can be seen that in Scenario II the average time for work increases by 0.23 hour and the average time for staying at home increases by 0.08 hour over the Scenario I, but the average time for social activity decreases by 0.12 hour. This is because the capacity increasing of location b reduces the difficulty of parking cars around W, so that people can postpone their

Time-dependent activity and travel choice model with parking options 735 departure times from home and work longer in offices. Some people leave their offices late and have to cancel their social activities. The alleviation of parking congestion improves relevant things in other aspects, e.g., the average travel time and parking delay time drop 0.1 hour and 0.09 hour, respectively.

Figure 6 Population in the three zones over the time of day

Figure 7 Departure rates for working against time and duration length

736 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In Scenario III, the charge reduction in parking location h attracts more people to participate in social activities after work, but not to go home directly. More people leave home for office ahead of time so as to win a slacker time budget for performing social activity, and thus the traffic congestion from home to workplace in the morning rush hour is alleviated. As a result, the average time for social activity gets an increase of 0.21 hour, the average time for staying at home drops 0.13 hour, and the average travel time decreases by 0.05 hour. Table 3 Effects of different parking policies

Scenario I Scenario II Difference (II-I) Scenario III Difference (III-I) Scenario IV Difference (IV-I)

Travel 1.38 1.28 -0.10 1.33 -0.05 1.30 -0.08

Average time spent for Parking Walking H 0.61 0.44 11.19 0.52 0.41 11.27 -0.09 -0.03 0.08 0.65 0.48 11.06 0.04 0.04 -0.13 0.56 0.46 11.13 -0.05 0.02 -0.06

(hour) W 8.17 8.40 0.23 8.08 -0.09 8.26 0.09

E 0.64 0.67 0.03 0.62 -0.02 0.65 0.01

S 1.57 1.45 -0.12 1.78 0.21 1.64 0.07

It should be noted that the results generated in Scenario IV are not simply the sum of those in Scenarios II and III. This implies that the systematic changes caused by parking capacity improvement and parking fee adjustment are much more complicated than expected. This numerical example shows that the proposed modelling approach provides a powerful tool for better understanding and assessment of the complex activity/parking/travel-related behaviours in various scenarios. By introducing the unified framework, we offer a new avenue of comprehensively evaluating parking schemes from strategic viewpoint.

CONCLUSIONS In this paper, we employed the concept of the temporal utility profile of activities to model the activity choice and parking behaviours, and developed a time-dependent equilibrium model in the framework of the activity-based travel analysis. The model simultaneously considers traveller choices of activity type, activity location, activity duration, parking location, and traffic route over the time of day. The decision-making process for these choices follows a hierarchical structure. The model is formulated as an equivalent variational inequality problem and is solved by a heuristic solution algorithm. The numerical results show that the proposed model can truly reveal people's choices on the travel and activity behaviour over the time of day, and the induced occupancies in various parking locations. The findings also show that parking schemes such as increasing parking capacity, decreasing parking charges and the combination of both, will greatly influence people's travel, parking and activity patterns, and thus the associated time allocations. Our model can be considered as a coherent activity-choice parking demand analysis tool for long

Time-dependent activity and travel choice model with parking options 737 term strategy planning. By introducing the activity-based approach, we offer a new avenue of parking demand analysis for transportation planning and policy evaluation. Future research work may focus on the extension of the proposed model to multiple behaviourally heterogeneous groups and more explicit consideration of activity duration. The extension of incorporating modal choices into trip chains may also be conducted. The combined activity/parking choice models will also undoubtedly benefit from the attempt to empirically measure the utility functions of various activities. Although the numerical results that are presented in this paper are self-consistent, we recognize that various experimentations on larger networks are necessary for further validation of the proposed model.

ACKNOWLEDGEMENTS The work described in this paper was supported by grants from the National Natural Science Foundation of China (70429001), and the Research Grant Council of the Hong Kong Special Administrative Region (PolyU 5040/01E, PolyU 5143/03E and HKU 7134/03E).

REFERENCES Abdelghany, A.F. and H.F. Mahmassani (2003). Temporal-spatial micro-assignment and sequencing of travel demand with activity/trip chains. Paper presented at the 82nd Annual Meeting of the Transportation Research Board, Washington, D.C. Axhausen, K. W. and J. Polak (1991). Choice of parking: stated preference approach. Transportation, 18, 59-81. Axhausen, K.W. (1990). A simultaneous simulation of activity chains and traffic flow. In: Development in Dynamic and Activity-Based Approaches to Travel Analysis (P. Jones, ed.), pp.206-225. Avebury, Aldershot, England. Ben-Akiva, M., A. De Palma and P. Kanaroglou (1986). Dynamic model of peak period traffic congestion with elastic arrival rates. Transpn. Sci, 20, 164-181. Bhat, C.R. (1998a). A model of post home-arrival activity participation behaviour. Transpn. Res., 32B, 387- 400. Bhat, C.R. (1998b). Analysis of travel mode and departure time choice for urban shopping trips. Transpn. Res., 32B, 219-234. Bifulco, G.N. (1993). A stochastic user equilibrium assignment model for the evaluation of parking policies. European Journal of Operational Research, 71, 269-287. Bowman, J.L. and M.E. Ben-Akiva (2000). Activity-based disaggregate travel demand model system with activity schedules. Transportation Research, 35A, 1-28. Chen, H.K. and C.F. Hsueh (1998). A model and an algorithm for the dynamic user-optimal route choice problem. Transpn. Res., 32B, 219-234.

738 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Damm, D. and S.R. Lerman (1981). A theory of activity scheduling behaviour. Environment and Planning, 13, 703-718. Dell'Orco, M., M. Ottomanelli and D. Sassanelli (2003). Modelling uncertainty in parking choice behaviour. Presented at The 82nd Annual Meeting of the Transportation Research Board, Washington D.C. Feeney, B.P. (1989). A review of the impact of parking policy measures on travel demand. Transportation Planning and Technology, 13, 229-244. Florian, M. and M. Los (1980). Impact of the supply of parking spaces on parking lot choice. Transpn. Res., 14B, 155-163. Glazer, A. and E.Niskanen (1992). Parking fees and congestion. Regional Science and Urban Economics, 22, 123-132. Goodwin, P.B. (1983). Some problems in activity approaches to travel demand, In: Recent Advances in Travel Demand Analysis (Carpenter, S. and P. Jones, eds.), pp.470-474. Gower, Aldershot, England. Hamed, M. M. and F. L. Mannering (1993). Modelling travellers' postwork activity involvement: toward anew methodology. Transportation Science, 27, 381-394. Hensher, D. A. and J. King (2001). Parking demand and responsiveness to supply, pricing and location in the Sydney central business district. Transpn. Res., 35A, 177-196. Hess, S. and J. Polak (2004). Mixed logit estimation of parking type choice. Paper presented at the 83rd Annual Meeting of the Transportation Research Board, Washington, DC. Horowitz, J. (1980). A utility maximizing model of the demand for multi-destination non-work travel. Transpn. Res., 14B, 369-386. Huang, H.J. and W.H.K. Lam (2002). Modelling and solving the dynamic user equilibrium route and departure time choice problem in network with queues. Transpn. Res., 36B, 253-273. Huang H.J., W.H.K. Lam and K.S. Chan (2003). Modelling commuter behaviour in networks with ATIS for combined activity, destination and route choice problem. Presented at The 10th Int. Conf. on Travel Behaviour Research, Lucerne, August, 10-15. Hunt, J.D. and S. Teply (1993). A nested logit model of parking location choice. Transpn. Res., 27B, 253-265. Jones, P. M., F. S. Koppelman and J. P. Orfeuil (1990). Activity analysis: state of the art and future directions. In: Developments in Dynamic and Activity-Based Approaches to Travel Analysis (P. Jones, ed.), pp.34-55. Avebury, Aldershot, England. Kitamura, R. (1983). Sequential, history-dependent approach to trip-chaining behaviour. Transportation Research Record, 944, 13-22. Kitamura, R. and M. Kermanshah (1984). Sequential model of interdependent activity and destination choices. Transportation Research Record, 987, 81-89. Kitamura, R. (1984). A model of daily time allocation to discretionary out of home activities and trips. Transpn. Res., 18B, 255-266. Kitamura, R. (1988). An evaluation of activity-based travel analysis. Transpn., 15 (1), 9-34. Lam, W.H.K., M.L. Tarn and M.G.H. Bell (2002). Optimal road tolls and parking charges for balancing the demand and supply of road transport facilities. Proceedings of the 15th

Time-dependent activity and travel choice model with parking options 739 International Symposium on Transportation and Traffic Theory (M.A.P. Taylor, ed.), pp.561-582. Elsevier, Oxford. Lam, W.H.K. and Y. Yin (2001). An activity-based time-dependent traffic assignment model. Transpn. Res., 35B, 549-574. Lam, W.H.K. and H.J. Huang (2002). A combined activity/travel choice model for congested road networks with queues. Transportation, 29, 5-29. Lam, W.H.K. and H.J. Huang (2003). Combined activity/travel choice models: time-dependent and dynamic versions. Networks and Spatial Economics, 3, 323-347. Lam, W.H.K., Z.C. Li, H.J. Huang and S.C. Wong (2004). Modelling time-dependent travel choice problems in road networks with multiple user classes and multiple parking facilities. Working paper. Mahmassani, H.S. (1988). Some comments on activity-based approaches to analysis and prediction of travel behaviour. Transportation, 15, 35-40. Mannering, F., E. Murakami and S.G. Kim (1994). Temporal stability of travellers' activity choice and home-stay duration: some empirical evidence. Transportation, 21, 371-392. Patriksson, M. (1993). A unified description of iterative algorithms for traffic equilibria. European Journal of Operational Research, 71, 154-176. Pas, E.I. (1983). A flexible and integrated methodology for analytical classification of daily travel-activity behaviour. Transportation Science, 17(4), 405-429. Polak, J.W., K.W. Axhausen, and T. Errington (1990). The application of CLAMP to the analysis of parking policy in Birmingham city center. PTRC Summer Annual Meeting, Brighton. Ran, B. and D.E. Boyce (1996). Modelling Dynamic Transportation Networks: An Intelligent Transportation System Oriented Approach, Second Revised Edition, Springer, Berlin. Recker, W.W., M.G. McNally and G.S. Root (1986). A model of complex travel behaviour: Part I. Theoretical development. Part II. An operational model. Transpn. Res., 20A, 307-330. Supernak, J. (1992). Temporal utility profiles of activities and travel: Uncertainty and decision making. Transpn. Res., 26B, 60-76. Thompson, R.G. and A.J. Richardson (1998). A parking search model. Transpn. Res., 32A, 159170. Tong, CO., S.C. Wong and W.W.T. Lau (2004). A demand-supply equilibrium model for parking services in Hong Kong. Hong Kong Inst. of Engineers Transactions, 11, 48-53. Verhoef, E., P. Nijkamp and P. Rietveld (1995). The economics of regulatory parking policies: The possibilities of parking policies in traffic regulation. Transpn. Res., 29 A, 141-156. Wong, S.C, CO. Tong, W.C.H. Lam and R.Y.C Fung (2000). The development of parking demand models in Hong Kong. ASCE J. Urban Planning and Development, 126, 55-74. Young, W., R.G. Thompson and M.A.P. Taylor (1991). A review of urban parking models. Transport Reviews, 11, 63-84.

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37

DOUBLY DYNAMIC EQUILIBRIUM DISTRIBUTION APPROXIMATION MODEL FOR DYNAMIC TRAFFIC ASSIGNMENT N.C.Balijepalli and D.P. Watling, Institute for Transport Studies, University of Leeds, Leeds, England

INTRODUCTION The dominant research theme in traffic network modelling for more than a decade has been the effort to formulate and solve more complex and realistic models, especially to support applications of ITS (Intelligent Transport System) measures. The objective of this theme of research has been to support both "on-line" problems, such as short-term forecasting, deployment of real-time traffic information and en route diversion advice, as well as "offline" problems, concerned with medium-term planning and appraisal of systems. The present paper focuses on such off-line, planning applications. In this context, while many limitations have been identified in established network equilibrium theory in being able to reflect the impacts of such systems, two particular features have attracted much attention. These are namely: - the representation of dynamic traffic flow phenomena and their impact on departuretime-dependent route choice ("within-day dynamics"); and - the representation of behavioural adaptation, learning and information acquisition processes ("between-day dynamics"). A range of techniques has been proposed to address each of these limitations, yet two broad philosophical approaches have been adopted. The first stream of work is based on the attempt to generalise existing network equilibrium theory to the within-day dynamic case, utilising formulations and solution methods from optimal control theory, variational inequalities and mathematical programming. Such a stream provides a range of potentially efficient solution methods, which can be applied and monitored in a well-controlled environment, based on an

742 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction explicit and well-researched theoretical principle. This theory allows properties of existence and uniqueness of the model prediction and convergence of solution algorithms to be established, and to be linked to the specific assumptions of the underlying model. The second stream is based on an explicit representation of the emergent dynamical processes involved, rather than a direct characterisation of some stationary/equilibrium state of the process. When implemented using an explicit simulation of the system dynamics, this stream of work provides the opportunity to model behaviours that are well beyond those permitted in the well-controlled environment of equilibrium theory, yet without reference to a well-researched theoretical principle describing the system trajectory, and without the control on model outputs one can exert within an equilibrium framework. The price paid for the flexibility of this approach is the greatly added complexity in interpretation of model outputs (as well as in their use for an overall economic evaluation), with predictions that may be transient, stochastic, chaotic, periodic, auto-correlated and depend on the initial conditions. Thus, each of the two streams offers its own advantages, but has its drawbacks. In the present paper, we shall present research that for the first time unifies the advantages of these competing philosophies, by bringing together three elements, namely: - the representation of both within- and between-day dynamics within the overall framework of a discrete-time stochastic process, as proposed in the unifying framework of Cantarella and Cascetta (1995); - the representation of within-day traffic flow dynamics by a continuous-time, wholelink, dynamic network loading model of the kind popularly used in the dynamic traffic assignment literature; and - the solution and estimation of the equilibrium properties of the resulting stochastic process using a combination of conventional within-day dynamic network equilibrium theory and a theoretical approximation result. In doing so, as well as providing a practical solution technique, we are able to place the diverse subjects of network equilibrium theory, simulation and stochastic process models within a common theoretical framework, whereby advances in one area may be used to the advantage of other areas. Specifically, the modelling approach adopted is based on modelling the day-by-day evolution of driver decisions as a discrete-time markov process, and the evolution of traffic flow dynamics within the day by a continuous-time whole-link model (hence it is termed 'doubly dynamic'). While it is known that the stationary output of such a model, in the form of an equilibrium probability distribution of time-dependent network flows, may in principle be estimated by Monte Carlo simulation, there are many potential drawbacks of such an approach, such as the difficulties of detecting stationarity and the effects of multiple attractors, and of dealing with Monte Carlo error in policy tests (Cantarella & Cascetta, 1995; Watling, 1996; Watling, 2002). With such considerations in mind, Hazelton & Watling (2004) proposed an analytic approximation method for directly estimating the equilibrium probability distribution of such a model, without the need for simulation, and requiring only knowledge of a conventional Stochastic User Equilibrium (SUE) solution. Utilising a result previous established by Davis

Dynamic equilibrium distribution approximation model 743 and Nihan (1993), namely that this equilibrium distribution is asymptotically multivariate normal, with mean the SUE flow vector, then all that is required to complete the approximation process is the variance-covariance matrix of the flows, as provided by the approximation result of Hazelton and Watling (2004). However, a significant restriction of their work is that it was restricted to within-day static models. The contribution of the present paper is to make the important step of extending this work to the within-day dynamic case, in which the origin-destination demand flows are time-sliced by departure time (to any desired resolution), and in which the interactions on the network are handled through a continuoustime, dynamic network loading model (estimated in practice by an arbitrarily fine discretisation). The paper is structured as follows. In the following section, the underlying model is explicitly stated in quite general terms, and an approximation derived for the moments of the equilibrium probability distribution of the process. This approximation in turn requires the computation of various equilibria and Jacobian matrices, and so the next sections are devoted to explaining techniques for performing these computations. In particular, one complete section is devoted to describe the derivation of the route travel time Jacobian for a particular form of popular whole-link, dynamic network loading model. Two simple numerical examples are then considered, whereby the approximation method is compared with a simulation-based approach. Finally, conclusions and areas for future research are identified.

MODEL FORMULATION AND APPROXIMATION METHOD The model we consider examines the general case of multiple origin-destination movements, served by a network of links with overlapping path-flows on which there are time-varying travel time-flow relationships defined in continuous time. On the demand side, the origindestination flows are assumed to be specified (as is typically the case in practice) in discrete departure periods, though these periods can be made as fine as the modeller desires. Though not a necessary restriction, for notational simplicity we assume that all origins are discretised into the same periods. On the network side we shall consider the continuous-time dynamic network loading map as a relationship between a given vector of path flows (by departure interval and origin-destination movement) and the vector of resulting mean path travel times (by departure interval and origin-destination movement). Broadly speaking, the model and derivation of the approximation result in this section re-interprets that given in Hazelton and Watling (2004), with origin-destination movements instead interpreted here more generally as commodities. In this case, a commodity consists of a triple (origin, destination, departure period), so that the number of commodities will be the product of the number of origins, number of destinations and number of departure periods. (It is noted in passing that this definition of a 'commodity' is readily extended to consider 'user class' as a fourth dimension, at some additional cost of notational complexity in labelling perceived costs by commodity. Since already our notation is rather complex, this obvious generalisation is not given here, we shall focus only on the single user class case.) For the fine details and proofs of the results employed from Hazelton and Watling (2004), the reader is referred to the source paper. It will

744

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

suffice to say that the thrust of the approximation method is an asymptotic (law of large numbers) argument, as the demand levels become large, but assuming that the network capacity grows in relation to the demand. This allows a large sample approximation to be made of any arbitrary network problem, whether or not the original problem has demands that are "large" in any sense. Below, an outline is provided of the key assumptions and results. We begin, then, by introducing the notation and assumptions of the model adopted. The commodity demand flows (i.e. origin-destination demands for each departure time period) are held in a vector q of dimension K with elements ^ k ' served by a set of routes

k

with

k

_ Each commodity k is

elements; the full route set across all commodities thus

p=ixi k has dimension -' . Note that this notation includes some duplication, since for each OD movement, each entry time period will be duplicated, yet this substantially eases the derivations below. If the P -vector of commodity route flows (across all commodities) is contained in the vector f, then c*- ' denotes the P -vector of mean commodity route costs as a , where c ^ denotes the function of the commodity route flows. We presume c^ ' ~ + ^ mean commodity route travel times (obtained from a suitable dynamic network loading model), ^ denotes the value-of-time, and where b is a P -vector representing the composite effect on commodity route costs of other flow-independent attributes (tolls, distance, etc.). We suppose that c ^ > is sufficiently smooth to be at least piecewise differentiable in f.

It is assumed that all the trip makers of commodity k are rational in their behaviour when choosing their route, in an attempt to minimise their perceived cost of travel. For each p(n)k r GR k , the perceived travel cost r at the start of day k is given by commodity k and route U

r

-*-r

where

+T

r(n-l)k

r

lr

(1)

is the population-mean perceived cost for commodity k and route r at the end

of day n - 1 , and lr is a random variable describing unobserved attributes contributing to the population-dispersion of the perceived attractiveness of route r by commodity k. The P vector C(n-l) represents the collection of population-mean perceived costs across all routes and commodities. The probability of choosing route r on day n is then given by: pJ(C(n-1)) = Prob(c ( r "- 1)k +T 1 « k
k' for each commodity k, resulting (for example) in a logit model if independent I1 it Gumbel distributions are assumed, and a probit model for a multivariate normal distribution. While the behavioural choice-side of the model is quite conventional, a simple linear learning filter is used to replicate drivers building up their experience of travel costs on a day-by-day

Dynamic equilibrium distribution approximation model 745 basis following the completion of each day's trip. Although several authors including Horowitz (1984), Cascetta (1989), Ben-Akiva et al (1991), Iida et al (1992), Nakayama et al (1999) etc used similar perception updating models, very few e.g., Iida et al (1992) presented some analysis of the model parameters. While there are other approaches for updating the perceptions such as Bayesian approach (Jha et al 1998), we restrict ourselves to the weighted average approach in order to derive analytic results though all others used learning models in simulation experiments. Thus following the completion of trips on any day n, the populationmean perceived costs are updated based on a weighted average of costs actually incurred in a finite number of previous days m, using the form: m

c'-^sWJrxr'cCF0"'') i-1

s(X) = fjX

O<X<1

(3)

m

H

=(\-X )/(l-X)

J=1 is simply a scaling factor to make the weights sum to where unity and c(.) is the commodity route cost-flow function as defined above, and where F ° is a vector random variable of dimension P denoting the network path flows by commodity on day n. Assuming that for any day n and for each commodity k, all qk drivers wishing to travel make their travel choices independently, conditional on their experiences in past days, then the number of drivers taking each possible route on day n by each commodity k, conditional on the costs (3) experienced in the past, is obtained as: F ( " )k |c (n - 1) ~Multinomial(q k) p k (C ( "- 1) )) . , , . , , „ v^k.F v » independently fork =1,2,...K (4) | where F(n)k is the vector of route flows on day n by the commodity k. The route flows on day n are then given conditionally by the partitioned vector as below: ' F(n)l c (n-l) > F(n) c (n-l) =

F Wy c (n--iy L F(n)-Ki cTn-lT

\

\

)

(5)

p(n)k Q(n-l)

where

is given by (4).

Then the expectation and variance-covariance matrix would follow as: 'E[F(n)lc(n-iy E [ F (n) c ( n - l ) ] =

I I El(n)k

EiF(T)2"[C("n-T)"]

(6) c(n-l)

k

(n-1)

J * where, L and the conditional covariance matrix has a block-diagonal form:

(7)

746 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

fvarCF^jc^ioj n

(n 1}

Var(F C - ) =

0

0

j"4

0

V ) where by standard results for the multinomial distribution:

Var (Fk C ^ ) =q k [diag(p V ^ V p

(8)

V ^ V ^ V ] (9)

where the superscript T denotes the transposition operator. Note that the moments above are all obtained conditionally on the perceived route costs; however, for any sensible prediction we are interested in the unconditional moments. Based on standard results, the unconditional first moment is given as: E r F (n)l

L

E r E r F (n) c (n-l)T| J LL JJ

(10)

Now, applying the work of Davis and Nihan (1993) to the case of multiple departure periods (as opposed to multiple origin-destination movements, their application), the mean of the multinomial distribution in (10) converges asymptotically to the solution of the within-day dynamic stochastic user equilibrium (SUE) problem F , as the demand grows to infinity in tandem with the capacities. Since Davis & Nihan also establish asymptotic convergence in distribution of the process to a Multivariate normal, the only remaining piece of information thus required to characterise the full equilibrium distribution is the covariance matrix. Now, the unconditional second moment is given as (again by standard statistical identities): Var(F) = E[var(F(n) C ^ )1 + Var(E[F(n) C ^ 1)

L

J

L

J

( 1 1

)

The first term on the right hand side of (11) is given by n

n

fEtvV^lc^HJoi

o

E[Var(F C^ ~^)]= I I I I I I I oIIIIIII_]J\_jIIIIIIIIo I I I I I I 0

I

! 0 !E[Var(F{"n)Klc(n"^)]

' '

EVar(F (n)k C ' ° " ) —» 0* k where for each commodity k, J

I

)

(12)


w /..1SUE

covariance matrix evaluated at SUE path flow proportions yv 0' k =q k [diag(p^ E )-p^ E (p k s -) T ]

2SUE

'v

K S U E

n

'"''*

\

': (13)

kSUE

where v denotes the SUE path flow proportions for the commodity k. Based on applying the results established in Hazelton and Watling (2004) to this new application area, the second term on the right hand side of (11) can be shown to be in the limit Var(E[F-s(Xr2[QDB0''(QDB)T +QDMB0"(QDMB) T ] where ® is the collection of the conditional covariance matrices (13) across all commodities; \i - uiagl. 1) [s a diagonalised matrix of the demand flow vector q where r is the commodity/route incidence matrix; D is the Jacobian of commodity/route choice probabilities

Dynamic equilibrium distribution approximation model 747 p(C) with respect to commodity/route costs C; and B is the Jacobian of commodity/route travel costs c^ ' with respect to commodity/route flows f. The matrix M is an intermediate matrix, with

~s"'

+

where I represents an identity matrix of appropriate dimension.

The implementation of the approximation method above requires the calculation of the two terms on the right hand side of (11), namely (13) and (14). Expression (13) is computed by finding the time-dependent SUE by the Method of Successive Averages, with the SUE route flow proportions by commodity then input to (13) along with the time-varying demand profile. Expression (14) requires as input the memory length, memory weights and demand profile, together with the Jacobians B and D evaluated at SUE. Computing D is straightforward for a logit model, and is possible for other approaches such as probit. Computing B is potentially more problematic, partially because the route travel time depends on link travel times at the appropriate link entry times for a vehicle following that route, and these entry times entry times themselves depend on the network flows and the travel times on previous links in the route. In the following section we show how this Jacobian may be deduced for a common form of dynamic network loading model from the literature.

DYNAMIC NETWORK LOADING: IMPLEMENTATION AND COMPUTATION OF JACOBIAN The key remaining element to determine for the approximation presented is the Jacobian B of the route cost flow function c(f), which in turn is equivalent (under the stated assumptions) to the Jacobian of the route travel time function (or dynamic network loading map) c(f). In this section, a method for computing this Jacobian is presented based on a form of whole-link travel time model popular in the dynamic traffic assignment literature. Link travel time model and notation Specifically, following Friesz et al (1993), we assume that the travel time Ta(t) for vehicles entering any link a (with links indexed a = 1,2,..., A) at any continuous time t is related to the number of vehicles xa(t) on the link at that entry time, given by the relationship *.(t) = v|/,(x,(t)) (t>0) (15) for some non-negative, differentiable function y a (.) with derivative \\i[ (.). Several researchers including Astarita (1996) and Xu et al (1999) also investigated the properties of (15) either in linear or non-linear form. Astarita (1996) proves that the linear form of (15) limits the outflows to a maximum of the capacity of the link without additional constraints, while Xu et al (1999) show that it satisfies FIFO if the inflows are bounded from above in the linear case, or if the gradient of i|/ a (x a (t)) ' s bounded from above in the non-linear case. However, as (15) assumes that the travel time is independent of the relative position of the vehicles on the link, it would be in general more appropriate to use on congested links.

748 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In order to implement the given model, a fine discretisation is performed, yet we shall be careful in mapping this to the underlying continuous time axis, and ultimately in aggregating to the typically coarser departure time periods over which route choices are made (for the application of the theory in the previous section). In practice, our method makes no premise about the level of discretisation involved, this is under the control of the modeller. We denote by 5 the time increment of this discretisation, and denote the complete analysis period by (0, N5] for some positive integer N. The time increments are thus the intervals (t-5, t] for t = 5, 25, ..., N5, which are referred to as minor time steps. Below, when we refer to a time step (or interval) t, it is to be understood that we are referring to the period (t-5, t]. We assume that 8 is chosen so as to be smaller than the free flow time to traverse any link, i.e. 8 < v|/a (0) for all a = 1,2,..., A. This is an assumption we shall use implicitly on a number of occasions - implying that a vehicle could not enter and exit a link in the same increment of time. Quite separately to the issue of how to discretise the travel time model, a coarser timediscretisation is assumed for specifying the origin-destination demand rates. In practice, the level of time-discretisation for the OD demands will be controlled by data availability; in order to get a sensible approximation to the underlying continuous time dynamic network loading model, it is then sensible that the modeller chooses a somewhat finer discretisation for implementing the travel time model. The OD demand rates are assumed (for notational convenience) to be specified over a common discretisation of the whole analysis period (0,N5], divided it into L major time periods, also referred to as departure periods (w;_i, Wj] (for i = 1,2,,..L) such t h a t ( w o , w 1 ] u ( w 1 , w 2 ] u u(w L _,,w L ] = (0,N5]. These match exactly the departure periods defined in the previous section, and for convenience are assumed to be of the same duration, i.e. w ; - Wj_, = K for all i = 1,2,..., L and some given K . Now, unlike conventional network equilibrium theory, a distinction arises here in the flow variables we will consider. In the stochastic process approach, the commodity route flow vector f is an absolute flow (measured in units of number of vehicles or drivers). In order to translate to the flow units more conventionally used for dynamic network loading models, we shall consider the flow rates u, given by a simple scaling of f as u = K-1f. Thus, we shall compute the route commodity travel time Jacobian as a function of u, namely J = Jac(c,u), but then it is simple to recover the required route travel time Jaocobian (of ? with respect to f) as K~'J . Then finally the route cost Jacobian is given by B = yic"1 J . In the previous section it turned out to be convenient to merge the notion of an origindestination movement and departure period into a single entity, referred to as a commodity. In the present section, on the other hand, a slight change of notation will considerably ease the presentation. In particular, we now suppose that all R routes across all origin-destination movements (but neglecting departure periods) are indexed r = 1,2,...,R, making the origindestination movements implicit in the routes. We then write our commodity route flow rate vector u with the route and departure period explicit: thus, for each route and time period referred to in u we identify the corresponding route label r (in our new route labelling system) and departure time period label i, and will thus henceforth refer to the departure time specific route flow rates as u^ for r = l,2,...,R and i = 1,2,...,L. It will also be convenient to move between time defined on a continuous axis and time defined in terms of the discrete departure

Dynamic equilibrium distribution approximation model 749 intervals. Thus we introduce the indicator function I(t) (0 < t < NS) which takes the value i if continuous time t refers to departure interval i (for i = 1,2,.. .,L). Preliminaries for computation of derivatives A key element of the description below is the consideration of the cumulative outflows from each link. In particular, we define Va(t) to be the cumulative outflow from link a (a = 1,2,.. .,A) at any time t (t > 0) which is natural to associate with the end of each simulation time increment (t-5, t] since it is a cumulative flow, hi particular, we write Va(t) as the sum of its constituent outflows corresponding to each (minor) inflow period, t = 5, 25, ..., N5, as:

Va(t) = £ i > a s r ( t )

(a = l,2,...,A;t = 8,28,...,N6)

(16)

r=l s=l

where Vasr(t) denotes the cumulative outflow from link a at time t arising from flows on route r entering the network before the end of the minor time increment sS (i.e., before the end of the interval (s5 - S, s5]). An important point to note is that we are disaggregating by the minor time increments, not by the major time increments at which the OD demands are defined. Before considering the outflows further, let us turn attention to the cumulative inflows to each link. There are two kinds of contribution, those from vehicles starting their journey on this link (contribution U a (t)) and those entering from incident links (contribution \j\ (t)), thus we write the cumulative in-flow to any link a in the form: U a (t) = UUt) + Uf(t) (a = l,2,...,A;t = 5,25,...,N5). (17) If sar is a 0/1 indicator variable, equal to 1 only if link a is the first link on route r (a = 1,2,..., A; r = 1,2,...,R), then based on the notation introduced above, the contribution to the cumulative in-flow to any link a from vehicles starting their journey is:

UUt) = i s i s i ( f t r ( t - w 1 ( t H ) + x ' s i r ( w , - w M ) ] r-i

v.

;=i

(t = 5,28,...,N5).

(18)

;

Then, if we define the 0/1 indicator variable Eabr to be 1 only if link a follows link b on path r, then the contribution from flows incident to link a is:

Ua(t) = i l X b r V b ( t )

(a = l,2,...,A;t = 5,25,...,N5).

(19)

r=l b=l

Note that this notation automatically deals with traffic reaching its destination at the end of a link b that is incident to link a (for a particular route r), since the corresponding Eabr would then be zero. We then have, by conservation of flow, the number of vehicles on the link at time t as the difference between the cumulative inflow and cumulative outflow: xa(t) = Ua(t) - Va(t) (a = l,2,...,A;t = 5,2S,...,N5). (20) Combining equations (16)-(20), we may then express the number of vehicles on the link at any minor time increment as a linear combination of the route in-flow rates starting on that

750 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction link in the current time period, and the exit flows from incident links which are decomposed according to (16). Relationships between route and link travel time derivatives

Now, let us turn attention to the travel times. Recall that the travel time on any link a for vehicles entering at time t is denoted x a (t). Now, our ultimate interest is in the path travel times from the dynamic network loading model. Supposing that we already knew the link travel times at any continuous time t, then we would simply trace along the links of the route in the relevant time trajectory, following the notion of a nested cost operator introduced by Friesz (1993). Thus if the n(r) links used in sequence by any route r have the link indices alr —> a2r —> ... —> a n(r)I , then we shall define gkIj as the time that drivers departing on route r exit linka^ if they begin their journey at timejS for j= 1,2,...,N. These intermediate link exit times are built up recursively according to: Si, =J5 + T air (j8); g ^ g ^ + T ^ t e , . , , , . ) fork = 2,3,...,n(r) (21) with the desired route travel time for the complete journey given by the difference between the departure time j8 and the final exit time at the end of recursion (21), namely a travel time Of

gn(r),rj-JS-

However, there is a difficulty with (21) in that practical implementation of the dynamic network loading model requires a time discretisation, and therefore we will not generally know link travel times at any continuous time. Therefore some interpolation is required; in fact, we shall aim to differentiate this interpolation directly (rather than its underlying continuous time system) and so there is a need to specify this more precisely in the notation. Specifically, if we suppose now that x a (t) as previously defined refers to any continuous entry time t > 0, and x a (t) denotes the travel times known only at the discrete entry periods t = 0,5,25,...,N5, then if for any real number x, the notation <x> denotes the integer part of x, a linear interpolation yields: Xa(t)-^(<^>-5)+t"<^>'8^((<X>+l)-S) o

( t > 0 ; a = l,2,...,A).

(22)

Combining (21) and (22) thus allows the intermediate link exit times on any route (and the complete route travel time) to be calculated for any departure time, based on knowledge of link travel times at the minor time increments. Clearly, xa (t) can be related to the number of vehicles on the link for discrete time intervals through (15), in just the same way as Ta(t) would be in continuous time. Our ultimate interest is in the mean route travel time during each of the L major entry time periods (each of which consists of % minor time increments, in terms of the notation already defined), namely: £„ ="!>»(,),,-J5)I(JS)

(23)

Dynamic equilibrium distribution approximation model 751 Having now built up an expression for the route travel time by departure period, from (23) and earlier expressions, we may now start with the differentiation (with respect to the route flow rates by departure time). It is trivial to see from (23) that: dCjr 95 h s

S

y

=

dgn(D,rj

* j,,

Q4)

*hS

St. w^^jS^Wj

Now, some care is needed in differentiating the path travel times above derived from the nested cost operator (21). To understand this point, consider a network consisting of a single path (path r = 1) made up of two links in series (n(l) = 2), the path using first link 1 then link 2 (i.e. a,, =1, a21 = 2), and suppose that 5 = 1. For a vehicle departing at minor time step j , the nested operator (21) would give a complete path travel time of g2lj " j = j + t, (j) + t 2 (j + T, (j)) - j = T, (j) + X2 (j + T, (j)) . (25) Now suppose that we made an infinitesimal perturbation to the flow rate u hl on route 1 for some major time period h that is before or includes our current minor time step j (i.e. h < I(j)). Then the time-profile of the number of vehicles on each link will be perturbed, and clearly through (15), there will be a direct impact on the time-profile of travel times on each link. However, since the argument of x2(.) in (25) is itself a function of xx{.) there will also be an impact on the 'census time' at which we pick out the relevant travel time on link 2 that contributes to this path travel time. Returning to the general case, the interpolation (22) in fact allows us to decompose these two effects. Since the case k = 1 in (21) is straightforward (the first link on a path does not have the 'census time' problem), we restrict attention to the cases k = 2,3,...,n(r), substituting (22) into (21): gy = g*-.* + V «

%

"< > -5) + g k ^ ~ < § t

8

x a k , ( « St^X > +1).S) .

(26)

Now an infinitesimal perturbation to any of the (earlier) route flow rates will directly impact on the two travel time terms involved, but they are now evaluated at discretised census times that will not be affected by a small perturbation. The impact on the census times is captured, on the other hand, through the interpolation term, which is a function of the continuous exit time gk_,^ from the previous link. Thus differentiating (26) yields: dg^

5gk-,.,

^hs

^hs

f

axakr(<8t-^>.S)

|

SShs

g k _ 1 ,,-< e k -'% / >.sgt a t r ((< e t -^>+i).S) 5

Su h s

(2?)

+

i r % ^ a k «< 8 t l ^>+l).5) (k = 2,3,...,n(r)). 8 5uhs Expression (27) therefore defines a recursive method for computing the link exit time derivatives along any path, with the recursion initiated by the first link on the path (k = 1) for which: dg, d _^a lr (J5) ~zz— - — z z dUhS

3uhs

v-°)

752 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Thus the required derivatives on the right hand side of (24) may be obtained as the limit of the recursive process (27)/(28), as the link exit times are traced through to the path's destination. It may be seen that (27) assumes prior knowledge of the link travel time profile derivatives at any given discrete entry to a link. An important point that we will exploit later is that these link travel time derivatives (at given entry times) can be independently derived link-to-link, without the concern for the 'census time' impacts, since the latter impacts are subsequently captured by tracing recursion (27) along the relevant paths. We turn attention, then, to the process by which the link travel time derivatives may be computed. Now, combining equations (15), (17), (19) and (20), and then differentiating by the chain rule, yields for any given t = 5,28,...,N5 :

^ =V:Ul(t)+ ttBitaVb(t)-V.(4fM« + tiB r t l B f«-^«].( 2 9) *k

I

tltt

J I S5hs

^£r

Su hs

duhs )

It is straightforward to deduce from (18) that: ft- w «.H i f s a s = l a n d h = I(t) a u , m —lii = w r w H if6 a s =landh
(30)

otherwise

Then the only unknowns in (29) are the link exit flow derivatives, which by differentiating (16) are decomposed according to:

9V^t)=yy5\^(t) 3u hs

r=1 w

( a = U ,..., A ;t

= 8,28,...,N5;h = l,2,...,L;s = l,2,...,R).

(31)

9u hs

Computational process for determining decomposed link exit flow derivatives The process to determine all the relevant V air (t),x a (t),

%~ and

Ac- terms in

(29) and (31) operates chronologically, advancing all links/paths across the network by one (minor) time increment (t-5, t] before moving on to the next time increment. At the same time, the link exit times gklj for each path and departure increment are computed from (21) and (22) as they become available. Importantly, the fact that all these terms (exit flows and travel times, their derivatives, and the link exit times) are calculated in time order means that there values are all known for all links at all time increments earlier than the current one. The steps followed for time increment (t-5, t] (for each t = 8,28,..., N8) are as follows: For each link a = 1,2,. ..,A, each major period h = 1,2,..., L, and each route r = 1,2,..., N that uses link a (for routes not using link a the flows and derivatives are clearly all zero): Step 1: In this step we project the path flows entering the network in earlier time increments into outflows exiting link a in the current increment (t-5, t] (recall the assumption made earlier, that 8 is sufficiently small that any vehicles exiting a link in one time increment could not have entered the link in the same time increment). Suppose that our current link a is the kth link of route r. For each such earlier minor time increment i8 for i = 1,2,...,/s - 1 , we first

Dynamic equilibrium distribution approximation model 753 determine the entry time to link a, the kth link along route r, which will simply be the exit time from the (k-l) th link, namely gk_ln (extending the definition (21) so that gori =i5). These link exit times will be known from the application of this procedure in previous iterations. For each i = 1,2,..., X - 1 there are then three possible cases (assuming FIFO to hold): Case A: No flow on route r entering the network in period ((i -1)8,i8] exits link a before the end of the current increment (t-<5, t]. This occurs if the first vehicle from the entry interval arrives after the end of the current increment, i.e. if gk_1r;_j + t a (gk_, r j_,) > t, and then: Vair(t) = 0 and ^"^/C

= 0 (for s = 1,2,...,N).

Case B: All the flow on route r entering the network in period ((i -1)8, i8] exits link a before the end of the current increment (t-5, t]. This occurs if the last vehicle from the entry interval arrives before the end of the current increment, i.e. if gk_, r ; + Ta(gk_, r i ) < t , and then: Vair (t) = 8uI(i)r

and 5V™ W^

= 8 if h = I(i) and r = s, and is 0 otherwise (s = 1,2,..,N).

Case C: Some of the flow on route r entering the network in period ((i -1)8, i8] exits link a before the end of the current increment (t-5, t], but not all. This occurs if the first vehicle from the entry interval arrives before the end of the current increment, and if the last entering vehicle arrives after the end of the current increment, i.e. if g k _ lrW + Ta(gk_lri_,) < t and gk-u,i +Ta(gk-i,r,i) >l m this case > the path r inflow 8uI(i)r over the entry period ((i-l)S,iS] is translated to an outflow from link a spread (uniformly, it is assumed) over a period from the earliest to the latest exit from this link, namely ( g ^ , ^ , +x a (g k _ lri _ 1 ), g k _ lri + t a ( g k _ l r i ) ] . However, the end of this interval is after the end-time t of our current interval, and so only a proportion of this translated flow actually exits by the end of the current increment, namely: y

_

m

(t ~ (gk-U.i-. + Ta (gk-l.r.i-. ))) Su I ( i ) r gk-I,r,i "'"'''avSk-l.r.i/

l-St-l.r.i-l

+

^a lSk-l,r,i-l ))

Now, when differentiating the expression above, we treat the exit times from the previous link, the g terms, as constants (recall the discussion following equation (28)). Thus for s = 1,2,...,N, it follows that:

Ji'

t

= (g k -l,r,,+ta(g.c- 1 ,r,i)-(gk-,,r,- 1 +ta(g k - 1 ,r,i-,)))" 2

X

9Uhs

|(gk-l.r.i + T ,(8n,,,i)-(8k-l,,,H

+T

.(gk-l,r,,-l))) X

"(t-(gk- 1 , r , 1 - 1 + xa(g l - 1 ,^)))§^8-^^8u 1 ( i ) ; 9^h S

Su h s

754 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

" (t " (gk-l,r,i-l + t a (gk-l,r,i-l ))) 5u i(i)r

—

—

(33)

w h e r e ^ J 1 * h = «i) and r = B _ 3uhs [0 otherwise Note, however, that the link travel time derivatives in the expression above are evaluated at a continuous time instant that may not match with the time discretisation chosen, and so they must be interpolated: differentiating (22) at a given continuous time g yields ft. (g) ,, ^ a (< % > -S) , g - < % > § ^ a ((< % > +1).5) (35) 9uhs dShs 8 du^ and the discrete time derivatives above are known from the application of this procedure to earlier time increments. Step 2: Using equations (15)-(20) and Vair(t) computed in Step 1, hence compute the travel times x a (t) on all links a = 1,2,..., A for the current time step t. Using the derivatives air

^ y~~ /

for the current time increment t, for all links a, all routes r and s, and all entry hs

periods h, calculate the derivatives —-—- by (29) for all a, s and h at the current time dUhs

increment t. Calculate link exit times from (21) and (22) as they become available from the information computed so far. Having carried out the computations above for the all time increments, the path travel time derivatives are then computed from recursion (27)/(28), and the results substituted into (24) to give the required Jacobian of the dynamic network loading model.

NUMERICAL EXAMPLES Example 1: In this simple numerical example, we demonstrate the theory described in the previous sections by considering a two-link parallel route network servicing one O-D pair. Travel time functions are assumed to be linear functions of the number of vehicles on the link, of the form t a ( t ) = a a + P a x a (t), with parameters as, = 12 minutes and |3a = 0.025 minutes/vehicle for route 1 and <\ = 9 minutes and /3a = 0.035 minutes/vehicle for route 2. A minor time increment of 5 = 1 minute was used for implementing the travel time model over a 15 minute period (N = 15 minor time increments). A single demand period is modelled (L = 1 major time intervals) with a demand of 400 drivers over the single within-day time period of K = 15 minutes (so the OD demand matrices q and Q reduce to the scalar 400). Drivers' dispersion in travel cost perceptions in choosing their routes on one day, conditionally on past experience, are assumed to be explained by a logit model: p r (C) = e 9C' < ^ e ec ' > where d>0 is the logit choice scaling parameter. The value of time

I s

J

Dynamic equilibrium distribution approximation model 755 is assumed to be y = 1, with travel time the only component of travel cost. Within-day dynamic Stochastic User Equilibrium flows are obtained using the Method of Successive Averages based on 25000 iterations. This module is run iteratively, feeding the dynamic link loading model with the flows and receiving the updated travel times, until the equilibrium criteria are met. m the drivers' learning model, we assumed m = 5 days and X = 0.5. We compare here the means and variances by three methods viz., simulation, nai've and the approximation, as explained below. Simulation: The assignment process was simulated over a period of 40000 days and a period of 4000 days initial period was discarded as burn-in time to allow the network to reach sufficient levels of loading (i.e. discarding the transient states of the process). Summary statistics are computed from the non-discarded days. The results of this long simulation are treated as a benchmark against which to compare our approximation method. Naive Method: In this method we assume that the conditional covariance matrix (13) is sufficient to explain the variance in flows, thus ignoring day-to-day variability in choice probabilities and travel costs. Approximation: The stochastic process approximation method is applied, giving an estimate of variance using the expression (14). In each case, when applying the approximation method, appropriate Jacobian matrices of choice probability and travel times during the single major time period must be computed; for example, in the case of 5=0.1, these are given by: _[*- 0.0248 0.0248] _ fo.0129 0 1 ~|_ 0.0248 -0.0248J ~ [_ 0 0.0172 j ' Table 1 compares the means and variances for Route 1 by the above three methods for various values of the logit choice parameter 0. In this two-link case, computing the mean and variance of flows on the other route is trivial, and so is not shown here. Table 1

Comparison of Estimates of Means and Variances on Route 1

e= 0.005

Method Mean

e= 1

$ = 0.1

Variance

Mean

Variance

Mean

Variance 26964

Simulation

198.88

99.66

182.49

102.84

170.13

Nai've

198.88

100.00

182.52

99.24

143.71

92.08

Approximation

198.88

100.00

182.52

101.89

143.71

451.70

When the logit choice parameter is relatively low (5=0.005), the random dispersion element in drivers' travel costs is very high making them choose routes almost at random regardless of the measured (mean) cost, thus approximately equal numbers of drivers are expected to select each route, and our variation is approximately binomial. In this case as would be expected, the nai've approximation is sufficient to explain the variance in traffic flows. As expected, the approximation method adds little to the conditional covariance matrix in this case. It is noted also that the mean flows by all the three methods are identical to two digits.

756 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In case of 0=0.1, the mean flows by all the three methods are again very close. However, the naive variance underestimates the benchmark (simulation) variance. This underestimation of the variance is expected because the naive method assumes fixed multinomial choice probabilities as opposed to the real case of day-to-day varying choice probabilities, i.e. it neglects one source of variability. Our approximation successfully corrects this underestimation by inflating the variance towards the benchmark simulation variance. While the magnitude of the correction required is not large, we cannot attach any significance to this as this is intended purely as an artificial, illustrative example. When the logit parameter is relatively large such as 8 = 1, the dispersion in perceived costs is very small and hence on any one day all drivers tend to behave in a similar manner. In this case, the statistics presented in the table are produced by a rather different trajectory of the process, in which a large number of drivers choose one route on any given day and the following day they almost all shift their choice towards the other route, having experienced congestion on the first route. This en masse behaviour causes a large variance in the system and the probability mass becomes concentrated towards the two extremes (all-or-nothing solutions), resulting in a bimodal distribution. Thus, the variance is measuring some kind of system instability (in the same sense that one could calculate a variance for a deterministic periodic system) rather than random variation of a stationary variable. The naive variance is not expected to explain such behaviour and hence is too small. Indeed, neither is the approximation method designed to explain such variation, and the corrected variance is seen to be far lower than the simulation variance. This is not surprising as we are estimating the variance of a bimodal distribution with an approximate Normal distribution, which is clearly not an ideal choice; nevertheless, it is partially successful in correcting the variance in the appropriate direction. Example 2: This numerical example illustrates an application of the theory formulated in the earlier section to the case of a network with multi-link routes. Consider a five link network with links numbered as shown in Figure 1. It is worth noting that links 2 and 4 are shared by two paths hence require the flows to diverge and merge in space and time which is a common requirement in dynamic traffic assignment over networks serving multiple O-D pairs. Thus, even though this example deals with a single O-D pair, the principles described here can easily be extended to networks serving multiple O-D pairs.

Figure 1: Test Network Topology

Dynamic equilibrium distribution approximation model 757 On each of the five links whole-link type dynamic travel time functions of the general form xa (t) = a a + P a x a ' (t) are defined with parameter values as shown in Table 2. Table 2 Network Parameters Link

Qi (minutes)

ft, (minutes/vehicle)

7a

Functional Form

1 2 3 4 5

12 9 10 12 9

0.025

Linear

0.00015

1 1 2

Quadratic

0.025

1

Linear

0.035

1

Linear

0.035

Linear

We note here that FIFO compliance is one of the important features of any dynamic traffic assignment model, as well as being an assumption of our approximation method, and so we confirm that FIFO has been monitored closely during the execution of the program and has been found satisfactory. On the demand side, we have modelled the demand spread over L = 4 departure periods each of K = 15 minutes duration with a varying profile thus indicated by, q k =[400 700 100 100]. For the dynamic network loading purposes, we assumed a minor time step of 5 = 1 minute. In addition, we also retained a memory length of m = 5 days with a memory weighting of X =0.5. The route choice is assumed to be based on a logit principle, and dynamic Stochastic User Equilibrium (SUE) flows are solved for using the Method of Successive Averages together with the dynamic network loading method as before. Figure 2 indicates the SUE profiles of inflow, travel time and outflow from various links on the network. We compare here the variance of the route flows by a similar set of alternative methods, viz., simulation, naive and the method of approximation run to similar specifications as indicated previously.

758 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 2: Dynamic Network Loading Results (9 = 0.1) Figures 3 and 4 compare the variance of route flows obtained by the simulation method with that obtained by the method of approximation. In the case of 8 = 0.01, the drivers are quite insensitive to the costs and choose routes almost completely at random. In this case, the naive variance is sufficient to explain the simulation variance (Figure 3), and hence the approximation method leaves them almost unaltered. In contrast, in the case of 8 = 0.1 (Figure 4), we notice that the naive variance is insufficient to explain the simulation variance and the approximation method adds a correction term and lifts the flow variances close to the simulation variance.

Dynamic equilibrium distribution approximation model 759

CONCLUSIONS In the present paper we have successfully made a first step at linking together the fields of network equilibrium theory, whole-link dynamic travel time models, and stochastic process models of the within- and between-day evolution of traffic flows over a network. In making this link, we have devised a computational procedure for estimating the equilibrium distribution of such a doubly dynamic stochastic process model, utilising information from a conventional stochastic user equilibrium model. A key element of the approximation is the computation of the Jacobian of the whole-link travel time mapping, which may have many applications beyond that discussed here. This research opens up many avenues for further research, including analysis of the stability of the dynamical process, application of the method to alternative forms of whole-link travel time model, and travel behaviour research investigating the impact of the various learning and decision parameters on the network flows.

ACKNOWLEDGEMENTS The first author sincerely thanks UniversitiesUK and the University of Leeds for funding his research studies in England and ITS Leeds for the studentship being provided. The authors jointly thank Ronghui Liu for commenting on an earlier draft, and Malachy Carey, Richard Connors, Agachai Sumalee and Martin Hazelton for several thought provoking discussions and suggestions on this research. The authors also thank the anonymous referees for their useful comments and helpful suggestions.

REFERENCES Astarita, V. (1996) A continuous time link model for dynamic network loading based on travel time function, J.B.Lesort ed. Proc. of 13th International Symposium on Transportation and Traffic Theory Elsevier, Oxford, U.K.,79-102. Ben-Akiva, M., De Palma, A. and Kaysi, I. (1991) Dynamic network models and driver information systems, Transportation Research A 25(5), 251-266. Cascetta, E. (1989) A stochastic process approach to the analysis of temporal dynamics in transportation networks, Transportation Research B, 23(1), 1-17. Cantarella, G.E. and Cascetta, E. (1995) Dynamic Processes and Equilibrium in Transportation Networks: Towards a Unifying Theory, Transportation Science 29(4), 305-329. Davis, G.A. and Nihan, N.L. (1993) Large Population Approximations of a General Stochastic Traffic Assignment Model, Operations Research 41(1), 169-178. Friesz, T.L., Bernstein, D., Smith, T.E., Tobin, R.L. and Wie, B.W. (1993) A Variational Inequality Formulation of the Dynamic Network User Equilibrium Problem, Operations Research 41(1), 179-191. Hazelton, M. and Watling, D. (2004) Computation of Equilibrium Distributions of Markov Traffic Assignment Models, Transportation Science 38(3), 331-342.

760 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Horowitz, J.L. (1984) The stability of stochastic equilibrium in a two-link transportation network, Transportation Research B 18(1), 13-28. Iida, Y., Akiyama, T., and Uchida, T. (1992) Experimental analysis of dynamic route choice behaviour Transportation Research B 26(1), 17-32. Jha, M., Madanat, S. and Peeta, S. (1998) Perception updating and day-to-day travel choice dynamics in traffic networks with information provision, Transportation Research C 6(3), 189-212. Nakayama, S., Kitamura, R., and Fujii, S. (1999) Driver's learning and network behaviour: dynamic analysis of the driver-network system as a complex system, Transportation Research Record 1676, 30-36. Watling, D.P. (1996). Asymmetric problems and stochastic process models of traffic assignment, Transportation Research B 30(5), 339-357. Watling, D.P. (2002). A Second Order Stochastic Network Equilibrium Model. II: Solution Method and Numerical Experiments, Transportation Science 36(2), 167-183. Xu,Y.W., Wu,J.H., Florian,M., Marcotte,P., Zhu,D.L. (1999) Advances in the continuous dynamic network loading problem, Transportation Science 33(4), 341-353.

38

A COMBINED MODEL OF HOUSING LOCATION AND TRAFFIC EQUILIBRIUM PROBLEMS IN A CONTINUOUS TRANSPORTATION SYSTEM H.W. Ho and S.C. Wong, Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China

INTRODUCTION The modeling of traffic equilibrium problems in a transportation system is classified into two general approaches in the literature: discrete modeling and continuous modeling. The discrete modeling approach is commonly adopted for the detailed planning and analysis of a transportation system (Patriksson, 1994; Gendreau and Marcotte, 2002; Lee, 2003). The continuous modeling approach is used for the initial phase of planning and modeling in broadscale regional studies, in which the focus is on the general trend and pattern of the distribution and travel choice of users, and the changes in these two factors in response to policy changes in the transportation system at the macroscopic, rather than at the detailed, level. During this initial phase of study, there is usually insufficient data on the system to set up a dense network for detailed analysis using the discrete modeling approach, because the collection of such data is time consuming and labor intensive, and sufficient resources are generally not available in this phase of a study. Moreover, at this stage, planners are more concerned about the philosophy and concept of the system design, such as the catchment regions of competing facilities, the locations of cordons for congestion charging, the location of an additional central business district (CBD), and the expansion of the transportation system in different parts of the city. In such cases, it may be premature and ineffective to form a detailed transportation network for analysis. For example, the future population centers in the city are so uncertain that it may be difficult to define the location and density of the zone centroids in the system. Even if a hypothetical scenario for the demand location and network condition is set up, the results that are obtained from a discrete analysis are so detailed that they may prevent a proper interpretation of the system.

762 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction In an attempt to overcome these difficulties in regional planning and modeling, the continuous modeling approach has recently received much attention. The continuous modeling approach is by no means a substitution for the discrete modeling approach, but can be complementary to it, especially for the macroscopic modeling of a very dense transportation system. This approach facilitates the interpretation of results during the initial stage of planning and system design, and helps to clarify the macroscopic behavior of a system in response to major policy changes. With the continuous approach, a dense network is approximated as a continuum in which users are free to choose their paths or routes in a two-dimensional space. The fundamental assumption is that the difference between adjacent areas within a network is relatively small compared with the variation over the entire network. Hence, the characteristics of a network, such as the flow intensity, demand, and travel cost, can be represented by smooth mathematical functions (Vaugham, 1987). A promising extension has recently been made of modeling an arbitrary city shape, and a solution algorithm that adopts the finite element method (FEM) to solve the resultant continuum model (Zienkiewicz and Taylor, 1989) has been advanced (Sasaki et al, 1990; Wong et al., 1998; Wong, 1998; Yang and Wong, 2000) It is well recognized that land use and transportation systems interact strongly with each other, and the consideration of only one of these systems will not fully reveal a system's responses to major infrastructure developments and policy changes. Significant efforts have recently been made to develop analytical models to describe the effects of this interaction. Arnott (1995) simultaneously considered housing rent and travel patterns for an idealized city. For a more complex network system, Boyce and Matsson (1999) adopted the network equilibrium approach, which divided the problem into two levels. At the lower level, they incorporated housing rents and home origins into the traditional discrete network equilibrium model. Housing rents serve as one of the main decision factors in the choice of home location. At the upper level, the social benefits that are achievable by the users of the system are maximized with respect to the allocation of housing provision, which in turn affects housing rents at the lower level. Consequently, this bi-level model provides transportation system planners with a useful tool for the optimal allocation of housing units to minimize overall travel disutility that takes into account the corresponding changes in land use patterns. We propose a continuous model for the study of the macroscopic role and impact of housing rent and the interaction between housing and travel patterns in land use and transportation systems on a regional scale. In this paper, housing rent is incorporated into the model of a continuous transportation system as one of the major criteria, along with travel cost, that affects a user's choice of home location. This housing location problem is then formulated as a set of differential equations in the two-dimensional plane. We first use a special type of angular symmetrical circular city to illustrate the solution properties of the model. We then derive a generalized model for a city of arbitrary shape, which is solved by the finite element method. Numerical examples are also given to demonstrate the effectiveness of the proposed methodology.

Housing location and traffic equilibrium 763

ANGULAR SYMMETRICAL MODEL To describe the modeling features and solution properties of the problem, we first consider an angular symmetrical circular city, which is used to illustrate the consistence of the governing equations of flow conservation, travel cost potential, and demand distribution. This circular city is bounded by two concentric circles of radii i?o and R in the inner and outer boundaries, respectively, where 0 < RQ < R. We assume that no commuters pass through the outer boundary. The inner boundary embraces the CBD of the modeled city, in which activities such as work and school are concentrated. During the morning peak, residents in the suburb commute to the CBD and from the CBD in the evening. For broad-scale planning, we assume that the housing locations are continuously distributed in the suburbs, and that the activities are concentrated in the CBD. The model describes the interaction between the housing locations of commuters and the traffic equilibrium pattern. For simplicity, we assume that the problem is angularly symmetrical. This means that all of the variables and functions in the model are independent of the angular position, and thus the two-dimensional problem is reduced to a one-dimensional problem that is represented by the radial variable r, or the distance of a location from the center of the CBD. Several classes of commuters are considered in the model of this angular symmetrical problem (and also in the generalized model to be discussed later). Different classes of commuters have different perceptions of travel costs, housing rent, and congestion costs. The incorporation of multiple classes of commuters adds realism to the model representation, because commuters with different perspectives and behavior compete for resources such as road space and housing units. Mathematical Formulation This section describes the mathematical formulation of the problem in a circular city. The formulation comprises the flow conservation equation, travel cost potential, and the demand distribution function. According to the flow conservation principle, there is no loss of commuters in the model region. Let fj(r) be the flow intensity (expressed as the number of commuters per unit time crossing a unit width) of class i commuters at location r away from the CBD. The flow direction points toward the center of the CBD along the straight-line trajectory due to the angular symmetry of the problem. Furthermore, let qi(r) be the density of demand (expressed as the number of commuters per unit area) of class i commuters at location r. We assume that each commuter travels to the CBD during the morning peak hour, and returns home along the reversed route during the evening peak hour, which means that the morning and evening travel patterns are symmetrical. Consider a ring of width 8r with an inner radius of r and an outer radius of r + 8r. The flow conservation principle requires that in the morning peak the following relationship is satisfied, 2n(r + Srjfj (r + Sr) + 2nrq[ (rpr = 2nrfj (r). Rearranging and assuming the limit 5V —> 0,

r^j^

+ fl(r) + rqi(r) = 0,or

764 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

-£-{rfi(r))+rq,{r) = O.

(1)

dr We now define the travel cost potential. Let ct{r) be the local transportation cost (expressed in dollars per unit length of travel) of class i commuters at distance r toward and away from the CBD. As we assume that the morning and evening travel patterns are symmetrical, and that commuters consider the daily travel cost in their choice of home location, the local transportation cost here represents the total cost in both periods. The cost impedance function is defined as ci(r) = ai{r) + bi(r)Jjjf](r), (2) where a,(r) is the free-flow travel cost and bt(r) the congestion sensitivity parameter that are perceived by class i commuters at location r. The different perceptions of these classes of commuters are due to the different values that they assign to travel time and their tolerance of congestion. Furthermore, let ut(r) be the travel cost potential that class i commuters experience between their home location and the CBD. Consider a short radial length 5r between locations r and r + hr. The cost potential is related to the local transportation cost by Uj (r) + c,(r)o> = ut (r + hr). When 8r —> 0, we have

^ - * , M = o.

(3)

The interaction between housing location choice and traffic equilibrium is governed by a demand distribution function, which is used to represent the way in which commuters choose their home location in the city. Consider the demand of class i commuters in an infinitesimally small area that is bounded by r and r + 8r at a subtended angle 89 from the center. We assume that the choice of home location depends on the housing opportunities in the area and the travel impedance between the home location and the CBD, which is specified as dQi =kexp(-aiDi(q(r),ui(r)))rdrdQ = q^rdrdB , (4) where a ; is a positive scalar parameter that measures the sensitivity of class i commuters to the disutility that is associated with location r, k is the proportional constant, Qt denotes the total demand of class i commuters in the city, d Qt is the demand that is distributed to the infinitesimally small area, and

A(gW.«iW)=y,-wfi+A(O

* ( r ) , }+Mr)

(5)

I, H(r)-q(r)J is the disutility function that is perceived by class i commuters at location r. This disutility function consists of two components. The first is the housing rent component, hi(q{r)) = y;(r)(l + /?,(r)q(r)/(H(r)-q(r))) , which depends on the total housing supply density H(r) (expressed as the number of housing units per unit area) and the total demand density q(r) = '^i.qi(r) in that location, and where y,(r) and P,(r) are positive scalar parameters that represent the fixed and demand-dependent components, respectively, of the rent function at location r. The fixed rent component represents the minimum rent that a resident has to pay to occupy a housing unit, and the demand-dependent component represents the additional rent that residents have to spend to secure a housing unit in a

Housing location and traffic equilibrium 765 competitive market in which housing rents increase with demand. In this model, when the demand approaches the housing supply, the rent increases to a very large value, which ensures that the demand does not exceed the housing provision. The second term in equation (5) is the total transportation cost that class i commuters experience between their home location and the CBD. This disutility function assumes that commuters decide upon their home location according to two criteria: housing rent and total transportation cost. The first equality of equation (4) shows that the housing demand that is attracted to an area is directly proportional to the negative exponential of the perceived disutility that commuters experience and the size of the area. We integrate the first equality of equation (4) over the whole modeled region, which gives

jf" fRadg; = jf" ^kexpi-a^iqC-lu^y^dQ.

(6)

Hence, we have k = ^\*aexp(-alDi{q(r),ui(r))>dr']j

.

Substituting k into the second equality of equation (4), we write (g(r),«,-(r))) q =Qi 271

(7)

(g)

f exp(-a/JDI.(9W,«I-(r)))rdr

Equations (1), (3), and (8) form the combined model of housing location and the traffic equilibrium problem for the angular symmetrical circular city. In the following subsection, we study the solution properties of this model. Solution Properties of the Model For a given «,(r), equation (8) constitutes a fixed-point problem in <7,-(r),Vz. However, as the denominator is an integral function, it is very difficult to solve this equation explicitly, and thus we use a diagonalization approach, in which we first assume that the denominator is a constant. Denote K, = f exp(- afii (q(r),«,- (r))> dr .

(9)

Equation (8) can then be simplified as a

(r) = Qi exP(-g

)

(l0)

2nKi ' Summing for all of the classes of commuters and rearranging, we require that for a given G(q(r))=2nq(r)-YJlQi

e x p ^ a ^ f e W . W / W ) ) / ^ = 0,

where q(r) = ^ .?,('") is the total demand density.

(11)

766 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Proposition. For a given set of constants Kt > 0 and ut(r), Vz, there is a unique solution q (r) e (0,H(r)) that satisfies equation (11). Proof. We can show that at any location r (for brevity, the location variable r is omitted in the expressions in this proof), G(o) = - ^ \ g , - exp(-a,(y,- +ui))lKt <0 , and G{H)=2%H >0 because Di(0,ui) = yi+ui when q = 0, Di(H,ui)^><x> when q -> / / , and a,->0,Vz. As G(<7) is continuous in the interval (0,H), there is at least one solution within this interval. Differentiating G(q) with respect to q, we have

G'(q) = 2n + j:^'Qf

^(-aiDl)>0.

(12)

t \H-q) Kt As G(q) is a strictly increasing function in q, there is one and only one solution q* e (0,H) such that G{q*) - 0. This completes the proof. This proposition implies that for a given set of constants Kt>0 and M,(r) , Vz , the endogenous variable q is uniquely defined. Therefore, according to equation (8), the set of qh\fi , is also uniquely defined. Note that this uniqueness is only affirmed for a diagonalization step in which Kt > 0 and ut(r), Vz, are assumed to be fixed. Intuitively, and also verified by the numerical experiments in the following discussion, the set of demand variables 0 and Uj(r), Vz', are assumed to be constants. From the proposition, we know that there is one and only one solution q(r)e(0,H(r)) to equation (11), and thus we can apply the bisection method to determine the value of this variable, which is obtained by finding the root of the equation G(q{r)) = 0 . The uniqueness property that is given in the proposition guarantees that the bisection method always converges to the unique solution. Hence, we can specify the following implicit functional relationship, q(r) = F(u(r)), (13) where q(r) is a function of u(r) = \ui (r), Vz'), because the travel cost is the only variable that is included in the disutility functions. From equations (2) and (3), we have Y/(r) YJ

=

J_d!fi«_^W. bt{r) d r bt{r)

Summing equation (1) over all of the commuter classes, we can show that

From equation (13), we have

Substituting equation (14) into equation (16) and rearranging gives us

(14) K

Housing location and traffic equilibrium 767

Af_^^W_^W] dr{bi(r)

dr

+ rF(ll(r) ) = o.

(17)

b,(r) )

Equation (17) defines a set of second-order ordinary differential equations (ODEs) in u(r), which can be easily solved by the standard methods. We now define the boundary conditions of the ODEs. First, we define the following natural boundary condition at the inner boundary (or the CBD boundary), «/(*o) = O, (18) for all classes. Commuters at location RQ are already at the CBD boundary and incur no transportation cost, and thus the travel cost potential vanishes. Because this is an angular symmetrical problem, and the total flow of class i commuters who cross the CBD boundary must be equal to the total demand of class i commuters in the city, we have fi(R0) = ~

r

-

(19)

2-HRQ

Substituting the flows of all classes into equation (2) and using equation (3), we can show that

^M

=ai+J^YQ

(20)

dr r=Ro 2nR0^J Equation (17), together with boundary conditions (18) and (20), constitutes an initial value problem (IVP) for the travel cost potential, in which for a given u(r) the function q(r) = F(u(r)) is determined by the bisection method within the range (0,H(r)) every time that it is needed in the solution process. From the solution that is obtained, the demand of each class i can be determined by equation (10). We can then derive the traffic flow by solving equation (1), which gives us 1R fi(r) = -kqi(Qd^. rJ

(21)

r

For a given set of Kt, Vz, the aforementioned diagonalization step determines the demand function q (r) that is consistent with all of the functional relationships in the model, except for equation (9). Substituting q (r) into equation (9), we obtain

K* = f exp(- afit (q\r), u, (r))) r dr.

(22)

If Kt =Kt ,Vz, then q (r) is the solution to the problem, which involves equations (1), (2), (3), (5), and (8). Hence, if we treat the diagonalization step (solving equations (17) for u(r) with boundary conditions (18) and (20) for a fixed Kt,\/i) as a black-box module, then the problem becomes one of finding a fixed-point solution to K* = Kt ,\/i. As Kt,\/i are scalar quantities, we propose a simple but efficient hierarchical line search (HLS) method to find the fixed-point solution. We number the classes as 1, 2, ..., M in a hierarchical manner, where M is the number of classes, and class Mis the top level and class 1 the bottom level. Let K ; =(Ki,Ki_\,...,K^)

768 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction be the subset of the diagonalization variables from level i down to the bottom level 1. We now consider the hierarchical problems (LM)

KM{KM,KM_1\KM_]jj=KM,

(L,)

<(^,
(Li)

<(£,)=£,.

(23)

At each level i in the hierarchy, we perform a line search on variable Kt to find the solution to the fixed-point problem L,-, but at each of the trial points in the line search the algorithm solves the sub-problems at all of the lower levels in a sequential manner. The HLS method is described by the following pseudo-code. Find aKM£ (K$* , K^) that satisfies K*M (KM ) = KM; Given each trial point KM from the upper-level, find a Ku_x e \K^_X, K™-\) that satisfies K*M_X(KM_X) = KM_X ; Given each combination of trial points (KM,...,Ki+x)from

the upper-levels, find

x

a Kt e (K™™ , K™ ) that satisfies K* (Kt) = Kt ; Given each combination of trial points (KM,...,K.2) from the upper-levels, find a I , £ (xf", ^ m a x ) that satisfies K*X(KX ) = KX; Endfind Endfind Endfind In this HLS algorithm, K™n and J^,max are the minimum and maximum values of the search range for class i. With the fixed-point solution of Kt - Kt, \/i, the demand, flow, and travel cost can be solved. Numerical Example A numerical example is given to demonstrate the characteristics of the housing location and traffic equilibrium problem. The radius of the inner boundary of the modeled city is 0.5 km and that of the outer boundary is 13.8 km. The housing supply density in the city is 350 units/km2. We consider two classes of commuters. The total demand of the class 1 commuters is 60,000 units, and that of the class 2 commuters is 80,000 units. We assume that each commuter takes up one housing unit and travels to the CBD during the morning peak hour and returns home along the reversed route during the evening peak hour. Hence, we only need to consider the morning peak in the traffic equilibrium model, and the travel cost represents

Housing location and traffic equilibrium 769 the sum of the costs in both periods. The sensitivity parameters for the housing choice functions in equation (8), a! and a2, for the class 1 and 2 commuters are 0.0015 and 0.0020, respectively. The local transportation cost functions are (Class 1 commuters) cx = 0.50 + 0.002(/, + f2) and (Class 2 commuters) c 2 = 0.75 + 0.003{/, + / 2 ) . The housing rent functions are (Class 1 commuters) hx = 80(l + \0q /(350 - q)) and (Class 2 commuters) h2 = 80(l + q/(350 - q ) ) . Class 1 commuters are more sensitive to housing rent than class 2 commuters, which means that class 1 commuters place a greater value on housing rent when making decisions on the choice of home location. In contrast, class 2 commuters are more sensitive to travel cost when choosing their home location. The problem is solved by the HLS method. Figure 1 depicts the typical Kx - Kx curves for different values of K2 at the upper level. Numerically, it shows that each Kx - Kx curve intersects with the 45 "-line at a single point. Hence, the HLS method is efficient in identifying the intersection point in these lower-level sub-problems. Figure 2 illustrates the typical K2 - K2 curve at the upper level. Each point on the curve satisfies the fixed-point constraint on Kx at the lower level, which is an intersection point in Figure 1. From the HLS method, we find that the fixed-point value of Kx is 6.2068, and that of K2 is 41.0486. Although we cannot rigorously prove the existence of the solution to the problem, we describe the solution properties of the angular symmetrical problem in detail in this section. From the response curves that are shown in Figures 1 and 2, we show that, at least numerically, the solution to the problem exists and is well posed. Intuitively, if one assumes a very small Ki in equation (10), then the resultant demand will be overestimated. Thus, it is likely that the updated value Kt (Kt) will be greater than Kt. In contrast, if one assumes a very large Kt, then the demand will be underestimated and the updated value Kt {Kt) will become smaller than Kt. There may therefore be an intermediate value of Kt that satisfies the fixed-point problem Kt (Kj) = Kj. We then apply the same argument from the bottom level upwards (analogous to the mathematical induction), and thus it is quite reasonable to argue that the solution may exist, as revealed in Figures 1 and 2. Nonetheless, due to the complexity of the problem it is extremely hard, if not impossible, to rigorously prove the existence and uniqueness of the solution.

770 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 1. Kt versus K^ for different values of K2

Figure 2. K2 versus K2, in which the fixed-point solution to K} is endogenously determined. Figure 3 shows the distributions of the demand of the two classes against the distance from the centre of the CBD. Class 1 commuters prefer to live in the suburbs away from the CBD, whereas class 2 commuters prefer to live closer to the CBD. This is because class 1 commuters are more concerned about housing rent, and thus choose a housing location that is far away from the CBD to take advantage of the lower housing rent there, although the travel cost to and from the CBD is higher (as revealed by Figure 4). In contrast, class 2 commuters are more concerned about their travel cost, and so choose to live as close to the CBD as possible, even though they have to pay higher housing rents Overall, the total demand of the two classes diminishes with the distance from the CBD.

Housing location and traffic equilibrium 771

Figure 3. Demand of the different classes. Figure 4 shows the perceived travel costs and flow intensities of the two classes against the distance from the CBD. The travel costs of both classes increase with the distance from the CBD. The sharp increase in travel costs in the proximity of the CBD is due to the congestion effect, as is revealed by the large flow intensities there. Note that the flow intensity of the class 2 commuters is higher than that of the class 1 commuters due to the greater demand and larger parameters that are specified in the cost-flow relationship of the class 2 commuters.

Figure 4. Flow intensity and total travel costs of the different classes.

THE GENERALIZED MODEL In the previous section, we studied an angular symmetrical circular city and illustrated the solution properties of this simple case. This serves an important step toward the development of a generalized model, because it is useful to demonstrate the consistency of the set of differential equations. With the support of the results for the simple angular symmetrical city, we are now more confident about extending the formulation to a general case. In the generalized model, we consider a modeled city of arbitrary shape, which does not carry any symmetrical properties. This greatly enhances the applicability of the developed housing

772

Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

location model to real-life systems. The modeled city with a CBD is shown in Figure 5. Commuters travel between their demand locations in the city region and the CBD along the least costly route, which is not necessarily a straight line as in the angular symmetrical case. We denote the city region (outside the CBD) as Q, the outer boundary of the city as F (measured in the counterclockwise direction) with F c Q , and the location of the CBD as O, which is encircled by the inner boundary of the city Tc (measured in the clockwise direction) with Fc
Figure 5. The modeled city with an arbitrary shape. Mathematical Formulation The generalized model is governed by the differential equations c, (*, y) 4 ~ 4 + V«,- (x, y) = 0, V(*. y) e Q, Vi,

(24)

\Wx,y)\

. ( ^ ) -J ^ Q V 4 ^ r f f ^ = °- V(,,,)eaV/,and (25) exp(- a,-D,- (?(x, y),«,- (x, >>))) dfl V f, (JC, >>) + qt (x, y) = 0, V(;c, y) e Q, Vi, (26) where all of the variables - except for the flow variables and the local transportation cost function - have the same definitions as those that were defined in the angular symmetrical model. In this generalized model, the flow-related variable of class ; commuters is defined by a vector ti(x,y) = (fix(x,y),fiy(x,y)), and the flow intensity is defined as the norm of the flow vector, i.e., \ti(x,y)\ = ^jfix(x,y)2

+ fiy(x,y)2

, where fix(x,y)

and fiy(x,y)

are the

flow flux of class i commuters in the x- and _y-directions, respectively. In addition to the governing equations (24)-(26), we define the following boundary conditions for the generalized model: K , - = 0 , V(x,^)er c ,Vz

(27)

f,= O , V ( i , y ) e r , V i . (28) Equation (27) is equivalent to boundary condition (18) in the angular symmetrical model. Equation (28) ensures that there is no flow across the outer boundary F of the modeled city.

Housing location and traffic equilibrium 773 This boundary condition is needed in the generalized model, but not in the angular symmetrical case, because the angular symmetry implicitly enforces that all of the flow vectors along the outer boundary of the circular city point straight to the CBD, and hence are normal to the outer boundary. Consequently, there is naturally no flow across the outer boundary. However, in the generalized model, the flow vectors are not restricted to being normal to the outer boundary, and thus condition (28) is needed. Furthermore, for the solution procedure of this generalized problem, we use a mixed finite element formulation in which the flow and cost potential variables are interpolated separately. Hence, boundary condition (20), which is needed in the angular symmetrical problem, can be omitted in the generalized model. Equation (25) is the demand distribution and equation (26) the flow conservation relationship of the generalized model, and both equations should be satisfied at every point in the modeled city region Q. These equations share similar meanings with those that are discussed in the angular symmetrical model, and thus the discussion is not repeated here. Equation (24) specifies the relationship between the cost potential function and the local transportation cost c, (x,y), which is defined by c,-(x,y) = at (x,y) + bt(x, y)\^

f, (*,y) j . where the parameter

y is used to account for the nonlinear effect of the total flow on the local transportation cost. In equation (24), the flow vector is the direct opposite of the gradient of the cost potential function, i.e., - f, (x, y) II VM,- (x,y) , wherever f * 0, (29) where '//' means that the two vectors are in the same direction, and the norm of this gradient equals the local transportation cost, i.e., (30) C; (x,y) = |V«!.(^>')| . For any used pathp from the home location H e Q of a class i commuter to the CBD, the total travel cost Cpi(H) that is incurred by the commuter can be obtained as C,,.(H)= \pCi d , = J ) C ,.|Lds = -^V«,..ds = -(«1.(O)-«I-(H)) = «1.(H)>

(31)

using equation (24) and the fact that f, /|f,| is a unit vector that is parallel to ds along path/). Therefore, the total travel cost is independent of the used paths and Ui(x,y) can be interpreted as the total travel cost that is perceived by the class i commuters. In contrast, for any unused path p between the home location H e Q of a class i commuter and the CBD, the total travel cost C-pi (H) that is incurred by the commuter is

C?1.(H)= lpc, ds> J^c I .ids = -^V« I .-ds = -(«1.(O)-«1.(H)) = «1.(H),

(32)

which is also based on equation (24). Hence, C ^ ; ( H ) > « ; ( H ) . The inequality in the above derivation occurs because for some segments of the path p the normal vectors f, /|f, and ds are not parallel, and hence ds > (f; /[f, |) ds for these segments. Therefore, the total travel cost for any unused paths is greater than that for the used paths. The model thus guarantees that

774 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction commuters of the same class choose the least costly route in the city region in a user-optimal manner. Solution Algorithm The finite element method (FEM) is used to approximate the continuous variables in the modeled city (Zienkiewicz and Taylor, 1989). As there is no explicit objective function for this housing location and the traffic equilibrium problem, the mixed finite element procedure that was developed by Wong et al. (1998) cannot be applied directly. Hence, we adopt the Galerkin formulation of the weighted residual technique (Cheung et al., 1996; Zienkiewicz and Taylor, 1989), in which the differential equations (24)-(26) are transformed into the following equivalent integral expressions: \[jcl{x,y)^^

+ Vul(x,y)\V(x,y)dQ = O, Vz,¥(x,y),

(33)

&L*.y)-Q, ff e x fa '»f f ' y f ' l o W ) < m =0- ™Mx-y)'(34) U

[

J^exp(- a,D, (q(x, y),ut (x, y))) da J

\[i{V-fi(x,y) + qi(x,y)'yV(x,y)dn = O, VI,V|/(JC,^), (35) where \\j(x,y) is the trial (or weight) function in the weighted residual technique. Boundary conditions (27) and (28) are enforced by taking a zero weight function (Cheung et al., 1996). In the Galerkin formulation, the local interpolation function of the finite element is used as the trial function. The modeling area is first discreteized into a finite element mesh, in which the Galerkin formulation is applied at the element level. The governing equations at a particular finite element node s are given as follows:

U/sJ

ee£ J

* ««[

J{iexp(-a;Z>,.(?(x,;)O,M,.(*>.y)))
(36) where Q.e denotes the domain of the finite element e, Es is the set of finite elements that connects node s, Ns(x,y) is the local interpolation function of the finite element that corresponds to node s, rsi is the nodal residual vector for class i commuters at node s, which represents how far the governing equations (24)-(26) are locally satisfied around node s, and *P is the solution vector of the problem. For the global satisfaction of the governing equations, we require that {R(«P)} = Col{rs,(«P)}={0}, (37) which defines a system of non-linear equations. We apply the Newton-Raphson algorithm with a line search to solve the problem, in which we derive the iterative equation {^IM^MJ*]-"^},

(38)

Housing location and traffic equilibrium 775 where [j^ ] is the Jacobian matrix of vector {Rk} in iteration k, and K is the step size (to be determined by the golden section method). The solution procedure is summarized as follows. Solution algorithm Step 1: Find an initial solution {*P0}. Set k = 0. Step 2: Evaluate {R(yVk)} and[j(*¥k)]. Step 3: If the relative error |R('5Ki)j/| 1*^ |, is less then an acceptable error s, then terminate, and {"P^} is the solution. Step 4: Otherwise, apply the golden section method to determine the step size K , which minimizes the norm of the residual vector R ^ - K J ^ ^ ' R ^ ) ) . Then, set

Step 5: Replace {fk} by {Vk+l }. Set k = k + 1, and go to Step 2. Numerical Examples We present two numerical examples for this generalized model. The first example is the same as the example for the angular symmetrical city. However, we use the FEM as given in the previous subsection to solve the problem, in which the acceptable error s is taken as 10~7. The solution converges in five iterations. The numerical values are identical to those that are obtained from the specially designed HLS scheme (see Figures 3 and 4). Although the flow vectors in the generalized model are free to rotate in the two-dimensional space, the solution shows that they always point straight to the CBD, or that commuters always minimize their total travel cost by choosing a straight-line route to the CBD because of the angular symmetry of the problem. This not only reaffirms the solution that is obtained by the HLS scheme, but also demonstrates the accuracy and efficiency of the FEM in solving the general twodimensional problem. The second example considers a city of arbitrary shape, as shown in Figure 5, in which we use the same set of parameters as in the first example, except those that are related to the local transportation cost function. The local transportation cost functions that are taken in this example are (Class 1 commuters) cl(x,y) = 0.50v(x,y) + 0.0004v(x,y)(\f](x,y)\ + \f2(x,y)\)12 and (Class 2 commuters) c2(x,y) = 0.75v(x,y) + 0.0006v(x,yX\{i(x,y)\ + \{2(x,y}f-2, where v(x,>>) = 1.10-0.005V(*-14) 2 +(y-20)2 is the factor that accounts for the variation in the location-dependent parameters in the local transportation cost function. This factor increases when the distance to the CBD decreases, which represents the characteristic of the network that the junctions are more closely spaced in the proximity of the CBD, and thus the parameters in the local transportation cost function increase. This second city spans about 35

776 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction km from east to west and 25 km from north to south. Its CBD is located at the southwestern corner of the city. Figure 6 shows the finite element mesh that is used to solve the problem. Similar to the first example, the acceptable error s is taken as 10~7 in the FEM.

Figure 6. The finite element mesh. Figure 7 shows the convergence curve of this numerical example. The solution converges exponentially and meets the acceptable error in eight iterations. Similar to the first example, the total travel cost increases (Figure 8) and the flow intensity decreases (Figure 9) with the distance from the CBD. Moreover, the demand density patterns behave differently for the two classes of commuters, but in a similar manner as in the first example. However, the flow vectors no longer point straight to the CBD, and the flow trajectories become curves that bend around the congestion areas as shown in Figure 10. This is because the angular symmetry no longer holds in this example, and the commuters choose their direction of travel in the twodimensional space to minimize their traveling cost on a user-optimal route.

Figure 7. Convergence of the second example for the generalized model.

Housing location and traffic equilibrium 777

Figure 8. Total travel cost of the class 1 commuters.

Figure 9. Flow intensity of the class 1 commuters.

778 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction

Figure 10. Flow pattern of the second example for the generalized model. Figure 11 shows the demand patterns of the two classes for this example. Although the changes in the demand level are generally on a par with the results in the first example for each class, the iso-contours of the demand (housing) levels are no longer concentric, and are distorted in response to the change in the geometrical constraint of the city, which cannot be addressed in the angular symmetrical formulation. Hence, the generalized model provides a better realization of the combined housing location and traffic equilibrium problem without the need to adhere to unnecessary restrictions as a way of simplifying the geometry of the city.

Figure 11. The demand contours of commuters.

CONCLUSIONS A combined housing location and traffic equilibrium problem in a continuous transportation system is introduced in this paper. Our main objective has been to study the effects of congestion externality and housing rent in the disutility function on the choice of housing location in a city. Two models are formulated to address the problem. The first model is developed for a circular city with angular symmetrical properties, in which the problem is specified as an IVP for a system of ODEs, and a specially designed but effective HLS scheme

Housing location and traffic equilibrium 779 is used to solve the problem. The solution properties of this model are thoroughly investigated, and form the basis of the development of the second model, which is a generalized model for a city of arbitrary shape that does not have symmetrical properties. The formulation of the generalized model is based on the Galerkin method in the weighted residual technique and the finite element method. The problem is solved by the NewtonRaphson algorithm with a line search. Two numerical examples are used to demonstrate the effectiveness of the methodology. The first example is an angular symmetrical circular city, to which both the HLS and FEM are applied, with identical solutions being obtained. The second example is a city of arbitrary shape, which shows the effectiveness of the generalized model in addressing the problem of irregular geometry and unsymmetrical properties in practice. Future research will include the evaluation of the economic aspects of the housing location problem, the formulation of a bi-level problem that maximizes the social benefit of housing distribution, and the investigation of polycentric urban forms.

ACKNOWLEDGMENTS The work that is described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7134/03E).

REFERENCES Arnott, R. (1995). Time for revisionism on rent control. Journal of Economic Perspectives, 9, 99-120. Boyce, D. and Matsson, L.-G. (1999). Modeling residential location choice in relation to housing location and road tolls on congested urban highway networks. Transportation Research, 33B, 581-591. Cheung, Y.K., Lo, S.H., and Leung, A.Y.T. (1996). Finite Element Implementation. Blackwell Science, England. Gendreau, M. and Marcotte, P. (2002). Transportation and Network Analysis - Current Trends: Miscellanea in Honor of Michael Florian. Kluwer Academic, Dordrecht, the Netherlands. Lee, D.H. (2003). Urban and Regional Transportation Modeling: Essays in Honor of David Boyce. Edward Elgar Publishing Inc., Northampton, U.S.A. Patriksson, M. (1994). The Traffic Assignment Problem: Models and Methods. VSP, the Netherlands. Sasaki, T., Iida, Y. and Yang, H. (1990). User equilibrium traffic assignment by continuum approximation of network flow. Proceedings of the 11th International Symposium on Transportation and Traffic Theory, Japan, Yokohama, July, 233-252. Vaughan, R.J. (1987). Urban Spatial Traffic Patterns. Pion: London. Wong, S.C. (1998). Multi-commodity traffic assignment by continuum approximation of network flow with variable demand. Transportation Research, 32B, 567-581.

780 Transportation and Traffic Theory: Flow, Dynamics and Human Interaction Wong, S.C., Lee, C.K. and Tong, CO. (1998). Finite element solution for the continuum traffic equilibrium problems. International Journal for Numerical Method in Engineering, 43, 1253-1273. Yang, H. and Wong, S.C. (2000). A continuous equilibrium model for estimating market areas by competitive facilities with elastic demand and market externality. Transportation Science, 34, 216-227. Zienkiewicz, O.C. and Taylor, R.L. (1989). The Finite Element Method. McGraw-Hill International Editions.