First Course in Bulk Queues

A FIRST COURSE IN BULK QUEUES M. L. CHAUDHRY Royal Military College of Canada

J. G. C. TEMPLETON University of Toronto

A Wiley-Interscience Publication

JOHN WilEY New York

•

t SONS Chichester

-

Brisbane

0

Toronto

..

Singapore

-

----------------------------~--~----~""~,9

To our parents

Copyright © 1983 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Library of Congress Cataloging in Publication Data

Chaudhry, M. L. A first course in bulk queues. "A Wiley-Interscience publication." Bibliography: p. Includes index. 1. Queuing theory. I. Templeton, James G. C. II. Title. T57.9.C46 1983 519.8'2 82-21764 ISBN 0-471-86260-6 Printed in the United States of America 10

9

8

7

6

5

4

3

2

;;

PREFACE

The theory of queues had its origin early in the present century with the work Markov and Erlang on stochastic systems. Markov chains and processes ren among the principal analytical tools of the theory of queues, while the teleph systems studied by Erlang constitute one of the principal 2-reas of applicatiOl queuing models. Queuing theory and its applications have expanded greatly since the days of pioneers. The theory has attracted the interest of some very capable app mathematicians, some of whom (Neuts, Pollaczek, and Takacs, for example) known mainly for their work on queues and closely related topics, while ot (such as Kendall, Khintchine, and Lindley) have done important work on que but are probably more widely known for work in other fields. Published applications of queuing models have expanded beyond telephone include ambulance dispatching, aircraft and automotive traffic, computers, mili operations of various kinds, medical appointment scheduling, machine rep inventory studies, water reservoirs, processing of steel ingots, and many ot: Other applications, such as the fairly recent replacement of parallel single-sE queues of bank customers by a single multiserver queue, require no new theory are therefore unlikely to be described in research journals. The introduction public acceptance of the single multiserver queue, in banks and elsewhere, see] depend on a fairly widespread appreCiation of some of the simpler facts al queues, and on a much more widespread demand for "equal treatment for all." Queuing theory has sometimes been dismissed by mathematicians as trivial by engineers as inapplicable to practical problems. Nevertheless, some of the n ematics done in queuing theory has been and continues to be of very good qm most of the mathematics is competent, and many applications of old and theories are made. In this book we discuss bulk queuing models in which customers arrive in gr or are served in groups. Both the theory of bulk queues and its application to tical problems can be said to have begun in 1954 with a paper in which N Bailey obtained a steady-state distribution of queue size for a particular bulk-se queue, and applied the results to a practical problem in scheduling medical app ments. Since Bailey's pioneer work, hundreds of papers have been publishe bulk queues. Some of these papers are of substantial practical value or mathem. interest or both: some are not. Many techniques have been developed or exte to handle the additional analytical complexities which result from the introdu of bulk arrival, bulk service, or both. Bulk queues exist in the real world,

PREFACE

have an extensive mathematical theory, and they often cannot be satisfactorily approximated by simpler queues. It has been difficult to learn more about bulk queues than is available in the short sections devoted to this topic in most introductory textbooks in queuing theory. A student or practitioner, seeking either a general knowledge of bulk queues or a model appropriate to a particular problem, has been faced with a search through a large and scattered literature, consisting of research papers together with a few sets of lecture notes on special topics. This literature lacks consistency of notation, sometimes uses methods not well adapted to the problem at hand, has some surprising gaps, and contains a substantial number of uncorrected errors. We therefore conclude that the theory of bulk queues is a large and useful body of knowledge which needs and deserves the extended unified account which we have tried to provide here. This book is neither a formal treatise for the mathematician nor a collection of case histories for the practitioner. We try to follow a middle path by discussing a variety of models and techniques, thus giving the reader the means of constructing other models best suited to a particular problem. In so dOing, we have filled in some gaps in the theory and have corrected some errors in the literature. Results are not always presented in the most general known form, but in some places we have generalized the published work on which we have drawn. We do not give a systematic discussion of the art of modeling or of specific applications, but have chosen models and techniques of solution with a view to application. We have tried to make the work accessible by keeping the mathematical prerequisites to a reasonable minimum. We expect that most readers will have made a study of the theory of queues as set forth in one of the well-known introductory textbooks, such as those by Conolly, Cooper, Giffin, Gross and Harris, Kleinrock, Kosten, and Page. Study of these books implies some knowledge of elementary probability theory and linear algebra, Markov chains, differential equations, Laplace transforms, and some rudiments of complex analYSiS, in particular Rouche's theorem. Except in Section 5.3 and Chapter 6, the reader who is prepared to have faith in the stated properties of complex power series will require very little knowledge of complex analysis. Since we could not give complete coverage to the existing literature in the field, we have tried to give fairly detailed coverage to the methods and models we thought most important and useful, and to provide a fairly full list of references to other material not covered. While we have tried to give proper credit for all the material we have presented or adapted from the work of other authors, we apologize to any authors whose work we may have slighted inadvertently, either by omission or by inadequate citation. We hope that readers will notify use of any remaining errors or omissions that they may find. At our chosen mathematical level, we were unable to include important work by Bhat, Borovkov,Dagsvik, DeSmit, Keilson,Kingman, Loris-Teghem, Neuts, Pollaczek, Takacs, and other authors. We have also excludp,d work that seemed to be of purely mathematical interest, some work included in eXist~ng monographs, and most work on time-dependent solutions. In Appendix B we gIve references to some publications on queues which combine bulk arrival and

PREFACE

bulk service, but have not given a detailed discussion of this work because of complexity. One of the most important features of the book is the provision of a large c lection of problems and complements. Few of these are routine drill problems, l: many are fairly easy with the aid of the text. Some problems invite the reader obtain alternative derivations for results already derived by other methods in t text. Some of the problems represent substantial extensions to the content of t book, and are likely to be found challenging by most readers. This is particula true of the Miscellaneous Problems and Complements. Almost every probl, includes a statement of the required answer, and many problem stateme] include hints or outline solutions to be completed by the reader. Chapters 1 and 2 are introductory, and do not discuss bulk queues in any det: Chapter 1 gives results in analysis and probability which are flsed in later chapte Chapter 2 is a self-contained introduction to (non bulk) queuing theory, in particu to the M/G!1 and GI/M/1 queues, with emphasis on the techniques that may used to investigate them. This chapter could be used as a textbook for ash· introductory course on queues, but its main purpose is to introduce methods wh are used repeatedly in later chapters. We expect that most readers will have SE some of the material in Chapters 1 and 2, and many readers will have seen most it. Even well-prepared readers, however, may find it useful to have this mate] readily available, in a notation consistent with that used in the rest of the bo. when studying later chapters. In Chapters 3, 4, 5, and 6 we discuss bulk queues: Single-server bulk-arri queues in Chapter 3, single-server bulk-service queues in Chapter 4, and mu channel queues with bulk arrival or bulk service in Chapter 5. Chapters 3, 4, an, may be read in any order. Chapter 6 discusses relations among different queu systems, and expected values for busy periods and idle periods. Some import: mathematical results have been collected in Appendix A, and a guide to the notal of the book is given in Appendix C. References to publications cited in the t, are listed at the end of each chapter. There is also a list of additional publicati. cited only in the Overview section at the end of each of Chapters 3 to 6. This book is addressed to several (overlapping) classes of readers. Students us the book as a text should be well versed in advanced calculus and element probability, at a level to be found in a good undergraduate program in engineeri computer science, or statistics, preferably supplemented by an introduction queues. Chapter 2 could be used by itself as a text or reference for an unortho( short introductory course on queues at the senior undergraduate or first-y graduate level. Chapters 3 to 6 could be used, together with the first two chap or separately, as a graduate textbook on queues, emphasizing techniques. Fc student who has studied an introductory book on queues such as Kleinrock Gross and Harris, Chapters 1 and 2 could be used for reference and Chapters 3 t would be almost entirely new material. For practitioners who construct quet models of real systems in the course of their work (management consulta members of industrial operational research groups, and so on), we offer a 1: collection of queuing models, most of which are amenable to computation. We :

x

PREFACE

offer some guidance in constructing other models. For professionals who read the journal literature on queues, we offer a gUide to that literature and a unified discussion of a large and significant part of it. Finally, we wish our readers a pleasant journey through the collection of queuing models presented here, and success in applying or improving them.

ACKNOWLEDGMENTS

M.L. CHAUDHRY J.G.c. TEMPLETON Kingston, Canada Toronto, Canada March 1983

We take great pleasure in acknowledging the generous support provided by many our professional colleagues who contributed to this book in one way or anoth Our special thanks go to Professor J. Gani, University of Kentucky, who read pa of the first draft and encouraged us to approach a publisher. Others who w. contacted during the process of writing this book, or who read parts of the bo and made suggestions for improvements, are Professors U. N. Bhat, South, Methodist University;P. Buckholtz, Royal Military College of Canada; B. W. Conol University of London; A. Ghosal, Council of Scientific and Industrial Resear Delhi; W. K. Grassmann, University of Saskatchewan; N. S. Kambo, Indian Instit of Technology, Delhi; J. Medhi, Gauhati University; M. J. M. Posner, University Toronto. Special thanks are due to our former and current students G. East D. F. Holman, B. Madill, and S. H. Sim for finding typographical (and other) err< Remaining errors of omission and commission are, however, the responsibility the authors, and we look forward to receiving comments and suggestions fr readers. We are most grateful to the following granting authorities for their sUPF for many years to our research work: Chief Research and Development, Can Council, Natural Sciences and Engineering Research Council, Canada. ' work for this book started when one of us (M. L. C.) was awarded a Canada Cou Leave Fellowship. We are indebted to the Department of Mathematics Computer Science, Royal Military College of Canada, and the Department Industrial Engineering, University of Toronto, for their support of this project. also thank the Indian Institute of Management, Calcutta, and the Departmen Operations Research, Stanford University, where part of our work on this book done during academic leaves. Thanks are also due to our secretaries and typi Charlene Chamberlain and Elisabeth Engelhardt, Royal Military College of Can: Sandra Hampshire, Sherry Harris, and Monika Heptner, University of Toronto; Antonietta Rotundo for their constant help in typing and retyping the mated; needed. We are intellectually indebted to many of the authors of queuing pa and books through whose work we learned immensely. Finally, we wish to express our enormous and sincere appreciation to our faJ members - Parphulat, Van dana, Amit, and Ajit (M. L. C.), and Elizabeth Marybeth (J. G. C. T.) for their understanding, constant support, and encoUJ ment to finish this book. Vandana, Amit, and Ajit also helped prepare the t script. M. L. C. J. G. C. 1

CONTENTS Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6

Lattice Random Variables Addition of Random Variables: Convolutions Other Useful Distributions Laplace Transform On Certain Functions Used in Queuing Theory Partial Differential Equations Problems and Complements References

Chapter 2 2.1 2.2 2.3 2.4

Some Techniques of Queuing Theory

3

Basic Material in Queues Techniques and History of Analysis Basic Renewal Theory Overview Problems and Complements Miscellaneous Problems and Complements References

3

Chapter 3 3.1 3.2 3.3 3.4 3.5 3.6

3.7

Preliminaries

Bulk-Arrival Queues

The System MX/EJ/l The System Gr/Mjl The System M r jM/1 The System M X /G n ll The System E?! IErl 1 with Generalized Erlang Input The SystemMf· ,M;' IG 1 , G2 /1 Overview Problems and Complements Miscellaneous Problems and Complements References

4 7

8 8 9 10

10 10 11 12 12 13

14 15 15 16 17.

xii

xiv

CONTENTS

Chapter 4

4.1 4.2 4.3 4.4 4.5 4.6

Bulk-Service Queues

TheSystemMIGBll The System MIGBII/M with Customers Served in Batches of Variable Capacity The System MIG h 11 with Customers Served in Batches of Exactly Size k The System GEOMIG B 11 The System MIMB 111MB Overview Problems and Complements Miscellaneous Problems and Complements References

179 180 186

191 193 198 202 203 216 232

CONTENTS

Generalized Argument Principle Rouche's Theorem Lagrange's Theorem The Riemann-Stieltjes Integral and the Laplace-Stieltjes Transform Some Theorems on Laplace-Stieltjes Transform and Generating Functions Linear Difference Equations with Constant and Variable Coefficients A.9 A.10 Inversion of a Discrete Transform References Bulk·Arrival, Bulk-Service Queues B References Glossary of Symbols C

A.4 A.5 A.6 A.7 A.8

AUTHOR INDEX Chapter 5

Multichannel Bulk Queues

237 SUBJECT INDEX

5.1 5.2 5.3 5.4 5.5 5.6 5.7

The System MX(t)IM/= with the Input Rate ACt) The System MXIMle The System MXID/e The System DS/M/e TheSystemM/MB/e The System GI/M B /e Overview Problems and Complements References

Chapter 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

Relations Among Queuing Systems

The System Gr/E/! The System GI/Er/l Relationship between GI/Er/l and Gr/Ell The System EIG h /1 The System Eh /G/l Relationship between EhIG/! and E/G h11 Expected Busy and Idle Periods Overview Problems and Complements Miscellaneous Problems and Complements References

237 240 252 258 261 267 275 275 292 295 296 302 304

309 316

317 322 325 326 334 340

Appendices A.l A.2 A.3

Differentiation of a Definite Integral with Respect to a Parameter Fundamental Lemma of Branching Process Theory Transform Inversion and Residue Calculus

342 342 343

A FIRST COURSE IN BULK QUEUES

Chapter 1

PRELIMINARIES Queues, or waiting lines, in which customers arrive, wait for service, are sef1 and then leave the system are a familiar feature of daily life. While the prese of waiting lines may be predictable, their detailed behavior usually is not. In ( structing mathematical models applicable to real queuing systems, therefore, we stochastic models characterized by random variation over time. Mathematically, queuing theory is the study of certain stochastic processes t( discussed later. The study of these processes deals with the probabilistic deve ment of the states of the process with changes in the value of the parameter, usu time. Thus to study queuing theory, we need to know something about the th{ of stochastic processes as a part of probability theory. It will be assumed, for purpose of this section, that the reader has a preliminary knowledge of probab theory. In particular, at least familiarity with the following is assumed: 1 2

3 4

Random variable (r.v.), discrete, continuous, or mixed. The cumulative distribution function (c.d.f.), the probability mass func (p.mJ.), or the probability density function (p.d.f.) of an r.v. X. Where t seems to be little danger of confusion, we may refer to these function: the single word 'distribution'. Moments of X, and moments of functions of X. The probability laws dealing with the calculation of probabilities of un of events, intersections of events, and so on.

Though some knowledge of the above is assumed, it is still felt that some SI should be devoted to rudimentary ideas concerning the various tools. This will help us in establishing the notation used for the rest of the book. The first t with which we start is the probability generating function (p.gJ.). Since one oj problems in queuing theory is to solve a set of difference equations which; from the consideration of the limiting behavior of the queuing system, this ele; and powerful technique is often helpful in expressing solutions to these problen closed forms. Means, variances, and higher-order moments can easily be comp' from these functions. We do not claim that we are giving a full account of t functions in the next section. For a fuller treatment of this, we refer the read! an introductory textbook on probability theory, such as Blake (1979), or a s( what more advanced treatise (without using measure theory), such as Feller (19 The discussion of probability theory from a measure theoretic point of view rna

PRELIMINARIES

2

found in the books by Loeve (1977, 1978), Neveu (1965), and Kingman and Taylor (1966), among others. It is also assumed that the reader has some basic knowledge of the theory of complex variables. However, some of the theorems of complex variables that are needed in certain sections are given in Appendix A. For a more complete treatment of these and some other related topics, the reader is referred to books mentioned there.

1.1

lATTICE RANDOM VARIABLES

Among the discrete r.v.'s those assuming only values of the form nd, where n is a nonnegative integer and d is a real constant ;;;'0, play an important role. In other words, if X is such an LV., then P(X = x) = 0 ifx nd, d;;;' 0, andn = 0, 1,2, .... The set of integer multiples of a real constant is called a lattice (in a geometric sense, not to be confused with the algebraic system called a lattice, nor with the statistical design called a lattice square). R.v.'s whose values are confined (with probability 1) to such a lattice are called lattice LV.'S. In view of this, binomial, Poisson, hypergeometric, and other nonnegative integer·valued LV.'S may be considered to fall under the broader class of lattice LV.'S. R.v.'s that do not satisfy the above property may be called nonlattice and the corresponding distribUtions, nonlattice distributions. Examples are absolutely continuous r.v.'s and mixed LV.'S.

*

1.1

LATTICE RANDOM VARIABLES

IP(z) I';;; 1,

if Izl .;;; 1.

Thus P(z) is absolutely convergent at least for I z I .;;; 1. Polya (1954) compares a p.g.f. to a bag. Instead of carrying several little separately, it may be convenient to carry a single bag which contains all til pieces. Quite similarly, instead of handling each term of the probability sec Px(O), Px(l), Px(2), ... , px(n), ... individually, we put them all in ~ :E~=opx(n)zn =P(z), and then we have only one mathematical piece to 1 the p.gJ. It should, however, be made clear that taking the pieces out of the bag m be as difficult a problem as getting the probabilities out of the p.gJ. In simpl, one can invert the p.gJ. and get the explicit probabilities. However, when th! becomes more complicated, the problem does not remain that simple, as below. The p.g.f. Px(z) = :E px(n)zn is a special case of the z-transform discus Jury (1964) and other authors, and used in the solution of difference equ In the z-transform :E anz n or :E anz- n the coefficients {an} are not nece~ probabilities. Among the useful quantities obtained from p.g.f.'s, and from other J functions introduced later, are the mean and the variance of the probability bution {p x(n)}. They are easy to derive and are recorded here in terms of tht of the p.g.f. of X and its derivatives, evaluated at z = 1. The mean is =

mx

Probability Generating Function

= E(X) = I

npx(n)

= P(l)(I)

n=O

Let X be a nonnegative lattice LV. assuming integral values 0, 1 ,2, ... , such that the probability that X equals n is, say, P(X = 11) == px(n). The sequence of probabilities {px(n)}~ = 0 or {px(n), n ;;;, O} (henceforth called the probability distribution of X) is said to be proper ifpx(n);;;' 0 and :E~=o px(n) = 1. In this book we shall be concerned only with proper distributions, and hence the adjective proper will not be repeated. Thus if {px(n)} is the distribution of X, then

Px(z) =

I

where

p(r)(l) == Drp(z)lz=l r

= d P(z) 1

dz r

E(X(X-l»

is called the p.g.f. of the sequence {p x(n)}~ =0 or more frequently, the p.g.f. of the r.v. X. Since Px(z) is an analytic function of the complex variable z, we may use the theory of analytic functions to obtain results concerning Px(z) and hence {px(n)}. The subscript X of the function Px(z) will not normally be used as long as there is no confusion regarding the LV. involved. Also px(n) is very often written as pen), Pn , or Pn . We shall be using all these notations in the subsequent chapters when the context of the r.v.'s involved is cleaL As P(I) = I, it is easy to show that

r = 1,2,3, ....

As

I

n(n -l)px(n) =P(2)(1),

n=O

px(n)zn = E(zX)

n=o

z= 1 '

the variance

ok is given by 01: =

p(2)(l)+pOJ(1)-{P(t)(IW·

Higher order moments about the origin, E(Xr), or about the mean, E«X - r.

r = 1,2,3, ... , may be obtained similarly. It is a simple matter to consider the p.g.f.'s of some of the standard discf( tributions such as Poisson, geometric, and binomial. However, problems in q theory frequently involve finding the p.gJ.'s of sequences such as {px(n {px(n + 2)}, {nPx(n)}. For a short table of z-transforms we refer the rea Beightler et al. (1961), and for further details to Jury (I964) or Cadzow (I97

PRE UMINAR I ES

4

1.2

1.2

ADDITION OF RANDOM VARIABLES: CONVOLUTIONS

ADDITION OF RANDOM VARIABLES: CONVOLUTIONS

{PWj(I)} = {Px1(1)} * {Px,{l)} *

In queuing theory we are frequently concerned with sums of LV.'S. In particular, we often have to consider the distribution of an r.v. which is the sum of N independently and identically distributed r.v.'s (henceforth frequently abbreviated as Li.d. r.v.'s). When the r.v.'s in the sum are Li.d., much simpler formulas and techniques are available than in the general non-i.i.d. case. In this section we shall consider separately two cases of sums of r.v.'s: (1) the sum of discrete Li.d. r.v.'s; and (2) the sum of continuous LLd. r.v.'s.

In practice there are many problems in which the LV.'S involved are not independent but identically distributed. Let now the LV.'S Xi be i.i.d. witJ common distribution {px(n)} and the p.g.f. Px(z). Then equation (1.2.3) c: replaced by the much simpler formula

In the Li.d. case we write the convolution of Xl, X 2, ... , Xj as +-- j

(1.2.1)

Wo = 0

where the X's are nonnegative, independently distributed discrete r.v.'s and N is an LV. independent of the X;'s. We discuss first the particular case of eq. (1.2.1) when N is fixed and equal to 2 and Xl and X 2 are integer valued. Thus if PX 1(n) = P(XI = n), Px, (m) = P(X2 = m), n, m > 0, then considering various mutually exclusive cases for the event W2 = I and using the condition of independence, it can easily be shown that I

Pw, (I)

=

P(W2

I

= l) =

j=O

PX 1(/- j)px,

(n

(J

PWj(z) = {Px(z)}i.

{px(n)} * {px(n)} * ...

Case 1. Sum of Discrete i.i.d. r.v .'s. Consider the partial sum

'" * {PX/l)}.

* {px(n)}

= {px(n)}i*

or

factors - - - - -.....

in case there is no confusion between the

LV.'S

and

{px(nW* = {px(n)}

involved, wit,!1 {px(n)}o* = oon.

Since by ~efinition ~e sequence {px(n)}o* is {I, 0, O, ... }, its p.g.f. isPx(z) The notatIOn p(j)(n) IS used for convenience. It may, however, be remarked t1 the subscriptj in p(j)(n) is without the parentheses (), the meaning is different If N instead of being a constant is an LV. independent of Xi, then WN is the of a random number of r.v.'s. If P(X; = n) = px(n), for every i, peN = j) = PJ P(WN = I) = PWN(I), and the corresponding generating functions are Px(z) P andPwN(z),it follows that ,1 PWN(Z) = E(ZWN)

I

I

=

Px1U)Px,(l-j)·

(1.2.2)

j=O

=

=

I

E(ZWNjN=j)P(N=j)

j=O

When Xl and X 2 are not independent, then ~

PW,(l) = ;:'Px"x,(l-j,j) = ;:'Px1,x,U,l-j)

where PXj>x, (Xl' X2) is the joint probability mass function defined by Px"x, (n, m) = P(XI = n, X 2 = m).

The sequence {Pw,(/)}i=o is called the convolution of the sequences {Px1(1)}, {Px, (l)} and is denoted by {Px 1(In * {Px, (I)} or{px, (I)} * {Px 1(I)}. In terms of

the p.g.f.'s, this can be written as

PX1 (z)Px,(z) = Pw,(z)

if Xl and X 2 are independent. This is easy to prove by considering E(ZXl + x,). If N = j > 1, the whole concept can be generalized, and we have in the case of independent r.v.'s j

Pw/z) =

The sequence in this case is denoted by

mf! / x m (z).

=

I

{Px(z)}ipNU)} = PN(PX(z»).

(1

j=O

The distribution ~PwN(l)} whose p.g.f. is given by equation (L2.4) is callec compound distriBution. If j = 0, the value of WN , by convention, is taken t zero. The concept of univariate p.gJ. can easily be extended to bivariate and IT variate p.g.f.'s. In the case of two discrete nonnegative LV.'S Xl and X 2 , we hav( bivariate p.g.f. Pxpx,(y,z) = E(yXIZXZ)

n

m

where pX1,x,(n, m) is a joint probability and stands for P(X 1 = n, X 2 = m). It be remarked that sometimes one LV. is discrete and the other is continuous. ~ cases are dealt with through the Laplace transform (L.T.) and the p.g.f. The p.g.f.'s PXp x, (y, 1) and PXpx , (1, z) are the marginal p.gJ.'s of the d:

PRELIMINARIES

6

butions {px,(n)} and {px,(m)}, resp~ctively. This concept may be generalized to more than two LV.'S, but the mathematics involved, in general, will become cumbersome. However, certain results connected with the sum (1.2.1) are easy to derive, whether N is a fixed constant or an r.v. For example, if XI, X z , ... , Xn (n fixed) are independently Poisson distributed with parameters AI> ... , An' the p.g.f. of Wn is P w (z) = exp[u(z - 1)], where u = Li =1 Ai' Since the p.g.f. of a Poisson r.v. X with nparameter A is exp [A(z - 1)], one can see that the sum of n independent Poisson distributed r.v.'s is itself a Poisson r.v. Case 2. Sum of Continuous LLd. LV.'S. Let N be fixed and equal to 2, and let Xl and X z be two nonnegative continuous independent r.v.'s. Then the distribution function (d.f.) F x, +x,(t) [with p.d.f. fx, +x,(t)] of the sum Wz = Xl + X z is given by F w, (t) = P(W z ~ t)

Sot P(XI + X z ~ tlXz = y)fx ' (y) dy J. tFx (t-y)fx (y)dy 0'

,

where the last result has been obtained by using the independence of Xl and X z · We use the more common definition F x(x) =P(X~x) of the dJ. of an r.v. X, rather than the alternative definition Fx(x) =P(X <x). Since limh ->0+ Fx(x +h) = Iimh ->o+P(X ~x + /1) = P(X ~x) = Fx(x), the d.f. Fx(;x) is continuous from the right (but not in general from the left). By conditioning on the value of Xl, we could have obtained

J fx (y)Fx (t-Y)dy. t

Fw(t) = 2

0

1

2

The corresponding p.d.f. is

1.2

ADDITION OF RANDOM VARiABLES: CONVOLUTIONS

fw,(t) =

fo fx"x,(t-y,y)dy = It fx"x,(y,t-y)dy t

0

where fx x (x, y) is the jOint p.d.f. of the LV.'S XI and X z . Convoi~ti'ons of the above type are handled easily with the help of L.T. are discussed in Section 1.4. We have restricted the r.v.'s to nonnegative value limits of integration (or summation) are to be adjusted appropriately when th( take values over the whole real line from - 0 0 to 00. As in the discrete case, the concept of convolutions in the continuous cas, be extended to N r.v.'s, where N may be either a constant or an r.v., as following example, which occurs in queuing theory. Consider the sum (1.2.1). Let first N be fixed and equal to n. For the case' N is an r.v., see the Problems and Complements. Let Xi'S be Li.d. exponentia: with the common p.d.f. /1>0,

t;;;'O.

The p.dJ. (1.2.5) is called the (negative) exponential p.d.f. However, the tive 'negative' will not be repeated since we shall be concerned with only thi: of exponential distribution. Now if we let fwn(t) be the p.d.f. of Wn ,

It is easy to show by induction that

fwn(t) =

/1nt n - 1 e- Ilt (n-I)!

n;;;'I,

Wn has mean n//1 and standard ~ViatiOn Yn!/1. This p.d.f. is used in various: by changing the parameters. For example, if /1 is replaced by An, the p.d.f. bec. (il.n)n t n - l e- Ant

fw,(t)

(fxJt -y)fx,(y)dy =

ft fx (Y)fx (t-y)dy. 0'

,

t;;;'O.

(n-I)!

(I

and consequently the mean now is l/iI.. The c.dJ. corresponding to the (1.2.6) is

We call the p.d.f.fw, (t) the convolution of fx, (t) andfx, (t) and write it as

fw, (t) = fx, (t)

* fx, (t)

= fx,(t)

* fx, (t).

If X I and X z are also identically distributed as X, then

fw,(t) = {tx(tW* or f(z)(t) if there is no confusion with the r.v.'s involved. Furthermore, if Xl and Xz are not necessarily either independent or identically distributed, then we have

The distribution (1.2.6), (1.2.6a), or (1.2.7) is called the Erlangian distrib (or n-Erlang) and will be denoted by En. In particular, it may be noted that an exponential distribution, one of whose properties is that its mean is equal standard deviation. The letter E in En is the first letter of the name of A. K. Erlang, who w( for the Copenhagen Telephone Company during 1909-1920 and did pioUE

PRELIMINARIES

8

work in the development of the theory of queues and its applications to telephony. For a study of Erlang's work, one may see Brockmeyer et al. (1948). In queuing theory the subscript n in En is called the number of phases. The phases may have a physical meaning, or they may be just an artifice to allow us to obtain a wide class of distributions. We have seen that the exponential distribution is a particular case of the Erlangian distribution. Another important distribution, which can be derived from the Erlangian by letting n -+ 00, is known as the constant or deterministic distribution in which P(X = c) = I for some constant c. This is denoted by D and is easy to derive with the help of an L.T. or a characteristic function (c.f.). For the procedure, see Problems and Complements. The Erlangian distribution may be generalized to the gamma distribution with p.d.f.

a>O,

fJ.>0,

1.3

OTHER USEFUL DISTRIBUTIONS

convergence of sums of r.v.'s to a limiting Gaussian r.v. Any of these theorems 1 in fact, be called a central limit theorem. These theorems are discussed in n books on probability, but one particularly good reference is Gnedenko and mogorov (1954).

1.3

OTHER USEFUL DISTRIBUTIONS

Hyperexponential Distribution

Just as the Erlang distribution (1.2.6) is obtained by considering the sum of. pendent exponential LV.'S, so we may obtain another useful family of LV: considering mixtures of exponential r.v.'s. Consider an LV . .s;.with p.d.f.

t;;;'O n

n

fs(t) =

where a instead of being a positive integer is a positive real number and rCa) is the gamma function defined in Section 1.5. The p.d.f. of the n-Erlang distribution is identical (except for a scale factor) with that of the X2 distribution with 2n degrees of freedom, since t(n/2) -1

fx~(t) =

e-t/2

(1.2.8)

2n/2r(n/2) ,

Since the X2 distribution is important in statistics and its c.d.f. is extensively tabulated, it is fairly easy to obtain values of n-Erlang c.d.f.'s. limiting Results Two of the most important limiting results connected with the sums of LV.'S are stated below without proof:

1

2

The strong law of large numbers. If Wn = r.v.'s with the common mean fJ. < 00, then

2.:['=1

Xi' where the X;'s are Li.d.

The central limit theorem. If Wn = 2.:['=1 Xi' where the X;'s are i.i.d. LV.'S with the common mean fJ. and common variance 0 2 , fJ. < 00 and < 0 2 < 00, respectively, then ( x2 2 e- / limP Wn-nfJ. ""

avn

l.

°

=(Y

-00

..;2rr

This is one of the simplest (though it will be sufficient for our needs) of a large number of different but related theorems which have been proved concerning the

I

arfJ. r exp(-fJ.rt),

ar>O,

r=1

I Or r=1

= 1.

Then S is said to have a hyperexponential or HEn distribution. The HEn ( butions are useful in queuing theory and are complementary to the En distribu of Section 1.2, as discussed in Problem 1O(b). The HEn distribution may be generated as follows. Consider a queue \\ service channel consists of n branches in parallel, working in such a way that customer entering service is independently assigned to one of the branche, probability of assignment to branch r being Or' r = 1, 2, ... , n. The service distribution in the rth branch is exponential with rate fJ.r' and a customer con ing service in any branch)eaVes the system at once. While the customer occ any branch, the service channel is considered completely occupied and no bral available to serve another customer. Then the customer's total service time ~ have an HEn distribution with the p.d.f. given above. If all fJ.r = fJ., then the distribution reduces to the exponential El . Truncated Distributions

In several important applications in the theories of queues, reliability, life-te and so on, one is interested in the truncated distributions of r.v.'s, truncated ( on the left of a given point or on the right of another given point, or botl illustrate this concept by means of two simple cases, (1) when the LV. is c< uous, and (2) when it is discrete. The concept, however, may be extended obvious way to other distributions. Case 1. fJ.e-JJ.x,

Truncated Exponential

LV.

Let X be an exponential r.v. with fx 7, 7 > 0, we \

x;;;. 0. Then for such an r.v. truncated on the left at X =

have the p.d.f.

f x (x) = 7

0, ( A fJ.e-JJ.x ,

X<7

PRELIMINARIES

HI

1.4

LAPLACE TRANSFORM

(J

The constant A is determined from' the normalizing condition fooo f x/.;x) dx = 1, which gives A = e™, and

.

jf

X;;;'7.

It may also be noted that when that of X, as it should.

7

-7

0, the distribution of Xr becomes the same as

If we are interested in the r.v. X, truncated on the right at X = should have if

x>e

if

o';;;x.;;;e

e, e > 0, then we

and

j=0,1,2, ... ,i

~ ~

• k =0

Once A is known, one can easily determine the moments of the new r.v. X r . In particular, one may note that E(Xr) = 7 + 1/11.

k

k!

and zero elsewhere. As an application of this, let us consider the case of particles which are er from a radioactive source following the Poisson distribution with parame A counting mechanism recording these emissions functions only if not morE two particles arrive. If more than two particles arrive during a specified time val, the counting mechanism gets locked. Thus if X is the number of pal recorded during a specified time interval, then proceeding as above it can eas seen that X has the distribution j=0,1,2.

For further applications, see Problems and Complements.

Note again that as (J --)- 00, Xe -7 X (in distribution). Furthermore, if we are interested in an LV. X, truncated both to the left of X = 7, 7 > 0 and also to the right of X== 7 + e, (J > 0, then we would have

otherwise with 1

E(Xr,e) ==

7

+ -11 -

ee- Ile 1 _e- Ile

Note that when e --)- 00 with 7 a finite quantity, or when 7 discussed earlier.

. --)-

0, we get the two cases,

Case 2. Truncated Poisson LV. Truncated discrete r.v.'s are treated in a manner similar to the one used for truncated continuous r.v.'s. For instance, if a Poissondistributed LV. X having parameter A is truncated on the right at X = i, then X has the following distribution:

Pj

=P(X ==j) == (~Aje-" J'1•

Using the normaliZing condition

2:} = 0 Pj

if j;;;'i + 1 '

if j == 0, 1,2, ... , i.

= 1, we find C and consequently

1.4

LAPLACE TRANSFORM

The Laplace transform (L.T.), in the form of the moment generating functiol long been used in elementary probability theory. In the terminology of SectiOl the L.T. provides a "bag" in which to carry an r.v. in much the same way < p.g.f. provides a "bag" i~;vhich to carry an integer-valued LV. As happens also the p.g.f., taking the r.v. out of the bag (inverting the L.T. to obtain the formed distribution) may be difficult. For this reason, the use of L.T.'s in qu theory, and especially the reporting of final results in L. T. form, has been J criticized. In some cases, results using L.T.'s have been superseded by more ex results not requiring inversion. Nevertheless, the L.T. still provides useful i mation in a fairly simple form, and it is constantly used in many parts of qw theory. We will most frequently use the L.T. in the slightly more general form 0 Lap!ace-StieJtjes transform (L.-S.T.), which is applicable to a wider class of than the simple L.T. Let X (continuous, discrete, or mixed) be a nonnegative LV. with the c bution function Fx(x). The L.-S.T.!x(a) of Fx(x) is given by lx,(Ci.) = EVax) = fooe- ax dFx(x) o '

(I

The integral on the right-hand side of equation (1.4.1) is a Stieltjes integral.

~

lx(Ci.) is the L.-S.T. of Fx(x), we call Fx(x) the inverse L.-S.T. of tx(a). I

brief sketch of this integral (transform), see Appendix A.7, but for more det discussion, see the references quoted therein. A transform such as lx(a) not

PRELIMINARIES

12

affords a substantial simplification, but permits us to discuss the discrete and continuous cases simultaneously. However, it may be said that a deep knowledge of such transforms is not required to study this book. A look at Appendix A.7 would enable the reader not already familiar with such transforms to handle them without much difficulty. It may be mentioned here that when a is complex, say a = g + if, g and f real, the integral converges for Re 0: == g;;;' o. Furthermore, if X is continuous and its p.d.f.fx(x) = dFx(x)/dx exists, we have

which is the usual L.T. When X is discrete, the integral needs to be replaced by a summation. Some of the properties of the L.T. that we shall be needing are dis· cussed below. Similar properties for the L.-S.T. can be derived, but we shall not discuss them here. One of the important properties is the convolution property. The convolution of two independent continuous nonnegative LV.'S XI and X 2 , which we discussed in Section 1.2, isfw,(t) = fxJt) * fx,(t). It can be shown that the L.T. of fw, (t) is

1.4

LAPLACE TRANSFORM

L{h(t)} = o:nix(a)-{an-lfx(0)+an-2f~)(0)+ ...

+ afJt- 2 )(O) + fi"-l)(O)}. Characteristic Function

Another important transform, which avoids the convergence problems of the I and the L.T., is the characteristic function (c.f.) Let Fxex) be the distribt function of an LV. X. The c.f. X(w), where w is a real variable defined ove] whole real line and i = y'=I, is defined by Xx(w) = E(e iWX ) = L== e iwx dFx(x). (1 If Fx(x) has a p.d.f.fx(x), the c.f. becomes xx(w) =

f~=eiWXfx(X)dx.

(l

When Fx(x) has a p.m.f. {Px{k)} k=O, equation (1.4.2) becomes a series express

[w,(a) = ix..(a) ix,(a).

Further, if the

LV.'S

(I

are identically distributed wtih common p.d.f. fx(x), The moments about the Origin may be calculated from fw,(a) = {ix(aW· =

These important results, which may be easily generalized to the case of n r.v.'s, n ;;;, 2, are used repeatedly in later chapters. When the indicated limits exist, lim a[x(a) = 0:""","'0+

lim fx(t),

L

J.1~(iwy/r!.

r=O

/h . . .IS The rth moment about t e ongIn J.1~

=

E(XT)

=

coefficient of (iwY/r!.

t-+oo

a -+ 0 + (also written as a -I- 0) means a -+ 0 through values greater than zero. The moments about the origin of the r.v. X having the p.d.f.fxCx), if they exist, can be obtained directly from its L.T. For example,

Since

Xx(w) =

I

(-lyo:rll~

r =0

r!

higr.er order moments about the origin are given by J.1~ = E(Xr) = coefficient of (-aY/r! in the power series expansion of [x(a). Similarly, the rth moment J.1 r about the mean may be obtained from the formula J.1r = E«X - E(X»)') = coefficient of (-aYJr! in the power series expansion of ecxE(X) ix(a). Sometimes we are interested in the L.T. of the derivative of a function. If h(t) = dnfx(t)/dt n = ficn)(t), for n ;;;, 0, it can easily be shown that

The relation between d.f.'s and c.f.'s is one to one. There are several books deal with the properties of the c.f.'s and their applications. For the properti( the c.f.'s one may see Lukacs (1960); for applications of c.f.'s to probat theory see Neuts (1973), to stochastic processes see Parzen (1962), to stati (and probability) see Lukacs and Laha (1964), among others. The transfonn equation (I.4.2) is called the Fourier-Stieltjes transform, e tion (104.3) is called the Fourier transform, and equation (I 0404) is called Dirichlet series. Moment Generating Function and Cumulant Generating Function

Sometimes the moment generating function (m.g.f.) and the cumulant genen function are also used. The m.g.f. of a d.f. Fx(x) is given by Mx(w)

=

E(e Wx )

=J

e Wx dFx(x),

w real.

The two cases when X is absolutely continuous or discrete can be separated;

PRELIMINARIES

111

the case of the c.L However, one disadvantage of the m.g.f. is that it may not exist, whereas the c.L always exists. The cumulant generating function (c.g.f.) is the logarithm to the base e of the m.g.[ or the c.L, and in certain situations it is easier to calculate moments through the cumulant generating function. For relations between moments and cumulants, one may refer to Kendall and Stuart (1969). The p.g.f. Px(z), the L.T.fxCex), the c.f. Xx(w), the m.g.L Mx(ex), and the c.g.f. In XxCw) or In Mx(a) are related one to another, as may be seen easily from their definitions. For example, Px(e iW ) = E(e iWX ) = Xx(w) = MxUw) if Xis a discrete LV. If ex = iw, where w is a real number, thenix(ex) = Xx(-w). The Problem of inversion of Transforms

Moments of low order can conveniently be obtained from the p.g.f. or the L.T. of the LV. under study. We often find, however, that we need to obtain the distribution of our r.v. from its transform, that is, to invert the transform. Transform inversion is often very difficult, but sometimes it is simple, possibly with the aid of the relations between different types of transforms, as discussed above. Since we are mostly concerned with the p.g.f., we discuss briefly the inversion problem connected with the p.g.f. While an inverse transform can always be expressed as an integral over a contour in the complex plane, it may be difficult or impossible to obtain the inverse in closed form. For cases in which the inverse is known, the reader is referred to some standard text book in L.T. or tables ofL.T. Extensive tables of L.T.'s and their inverses are available; see for example, ErdelYi et al. (1954), Roberts and Kaufman (1966), or Oberhettinger and Badii (1973). A nice but elementary discussion relating to some queuing problems may be found in Kleinrock (1975). In the case of the p.g.f., one method of recovering the distribution is 1 px(n) = nl

d(n) Px(z)

dz

I

n

z=o

p§p) (0) = --. n!

Sometimes Px(z) may be expressed as the ratio of two power series in z. Abramowitz and Stegun (1964) show how to express the ratio of two power series as a power series, namely, 00

Px(O)----

I

;=0

where px(n) can be obtained recursively from _ ~ b.px(n-j) an L.. J px(n) = j=l Px(O) ( Px(O) 1,

PXCi)Zi

n = 1,2,3, ... n=O.

1.4

15

LAPLACE TRANSFORM

This recursive relation is good for numerical evaluation. In practice, when the function Px(z) is complicated, calculations may become cumbersome or even impossible. In that case some sort of approximation is desirable, such as the following method based on partial fractions. Let us suppose that Px(z) is a rational function (in application, many of the transform functions do take this form) of the form P (z) = Q(z) x R(z)

where Q(z) and R(z) are polynomials without common factors. Without going into the algebra of decomposing the function into partial fractions, we give below several results which hold under different conditions. For details, see Feller (1968). 1 If R(z) is a polynomial of degree m and Q(z) is a polYIJomial of degree n <m, andR(z) = 0 has m distinct rootszj,i = 1,2, ... m, then, for Izl <minlzd, Px(z) = ::£;;'=0 px(n)zn, where 1< i <m m

'" A·z:-(n+l) L..

i= I

"

(1.4.5)

(The case of repeated roots is discussed later.) Equation (1.4.5) is exact. It may be used with great accuracy when all the roots can be found exactly, as they can when m';;; 4. For m > 4, the zeros, in general, can be found only by algori~ic techniques. Several efficient algorithms, including Bairstow's and Muller's, are available for finding zeros (real or complex) of polynomials of fairly high degree. Although some of the methods fail to give repeated roots, Muller's method is successful in finding even the repeated roots. On the limitations of solving polynomials for zeros, the reader is referred to books on numerical analysis such as those by Conte and de Boor (1972) and by Henrici (1964). Bairstow's method has been successfully used by several authors, including Russell and Taylor (1977), in computing results that occur in the theory of bulk queues. When the calculation of m roots (for m large) becomes infeasible, an approximate expression for px(n) is desirable and is given in equation (1.4.6). 2 If one root, say z 1> is smaller in absolute value than all other roots, the first term of equation (l.4.5) is dominant as n increases, and thus we may write px(n)~Alzl(n+l),

asn""OO.

(1.4.6)

Once again it may be said that algorithms are available for finding the root of smallest magnitude of a polynomial; for example, see Bernoulli's method in Henrid (1964). Equation (1.4.6) often gives surprisingly good results even for relatively small values of n. The sign ~ between two sides of a relation indicates that the two sides are asymptotically of the same order, that is, their ratio tends to 1

asn-+oo. In the derivation of the result (1.4.6) we supposed that the degree of Q(z) is less

PRELIMINARIES

16

than that of R(z). If we now relax this ~estriction and say that Q(z) is of degree n = m + r, division by R(z) leads to the form already discussed, added to an rthdegree polynomial which affects only the first r + 1 terms of the p.g.f. under consideration, thus making equation (1.4.6) still true. If zi' i> 1, is a double zero of R(z), then the partial fraction expansion involves one more term, which will add an extra term to the exact form, equation (1.4.5). However, this will not make any difference to the limiting case presented in equation (1.4.6), provided the smallest root z 1 is a simple root. Thus under the relaxed conditions equation (1.4.6) still holds. 3 When z 1 is repeated, if R(z) = 0 has a root z! that is smaller in absolute value than all other roots, and if Z 1 is of multiplicity r, then px(n)-A!

(n+r-l)

zj(n+r)

r-1

where

and - means that the ratio of the two sides tends to 1 as n -+ co. 4 Douglas (1963) gives still another approximation. If Px(z) = ~;;' =0 px(n)zn, the method of steepest descent gives an expansion with leading term (1.4.7) where Px(Z») g(z) = In ( ---;;;-

and zn satisfies g(l)(zn) = 0, provided that g(2)(zn) > 0. Douglas gives a number of examples which show that the accuracy of approximation can vary widely. However, used with some care, approximation (1.4.7) can give useful information when it is difficult to get general explicit results for px(n). The approximation of Feller (1968) has been used by Cromie et al. (I979) to analyze the MX/M/c queuing system, and the results were good. We have not had enough opportunity to use the approximation of Douglas (1963), nor did we find his approximation used in the literature on bulk queues. It is planned, however, to test the accuracy of the approximation of Douglas at some later date.

1.5

ON CERTAIN FUNCTIONS USED IN QUEUING THEORY

1.5

ON CERTAIN FUNCTIONS USED IN QUEUING THEORY

17

stant. The gamma function has numerous properties, but we discuss only those that we shall use; other details and properties may be found in Abramowitz and Stegun (1964). The gamma function, denoted by rea), is defined by the integral

fo= za-Ie-zdz.

rea) =

The integral converges for all a with real part Re a> 0, but does not converge for Re a « O. However, it is possible to define rca) by analytic continuation for complex values of a, such that Re a =1= -n, n = 0, 1, 2, .... We do not go into further detail for the case Re a < 0, as we shall be concerned with rea) for real and positive values of a only. The value of the gamma function is obtained by series expansion or some type of numerical integration, unless a is a positive integer. Some of its properties are:

1

rea)

(a -l)r(a -1),

ifa>O

2

rCa)

(a-I)l,

if a is a positive integer

3

r(!)

4

I r=O

k

=.,fi rea + r) r1

1 rea + k = k! ex

+ 1)

(1.5.1)

ifa>O.

Properties 1, 2, and 3 are given in text books on calculus, and property 4 is proved below. By definition k tr Ik rCa . r) = ta-1e- t I - dt. (1.5.2)

t

f=

r!

r=O

r=O

0

r!

Integration of the right-hand side of equation (1.5.2) by parts leads to ta

k -e- t

a

trl

oo

L-

r=or!

f ex 1

+0

00

0

tae-tt k k!

1 r(a+k+ 1) dt = ---'-----'ex k!

since

ta k tr lim-e-tI =0. t-+= a r=O r! The vanishing of the limit may be seen by applying L'H6pital's rule to each term of the sum or otherwise. Equation (1.5.1) could also have been obtained more simply by applying induction on k and using only property 1. For more similar properties see Problems and Complements.

The Gamma Function

The Bessel Functions and Their Modifications

In Section 1.2, Case 2, while discussing the Erlangian distribution, we talked about the gamma distribution which involves the gamma function as a normalizing con-

Bessel functions, which occur in solutions to several queuing problems, are particular solution of the Bessel equation

PRELIMINARIES

18

(11)2] y=O

2

d- + y -1 -dy+ [ 1- 2

dt

t dt

t

1.5

19

ON CERTAIN FUNCTIONS USED IN QUEUING THEORY

(1.5.3)

The number n is called the order of the equation, and in our applications to queues n is a nonnegative integer. It is easy to verify by term-by-term differentiation that y = In(t) is a solution of equation (1.5.3), where

(1.5.5)

I

(O:I)k . (0:2)k ... (O:p)k

k=O (Pi)k • (PZ)k

... (Pq)k

Zk k!

where

- 26 31(n

+ l)(n~ + 2)(n + 3) + ....

is called the generalized hypergeometric function. The O:h'S are the numerator parameters, the Ph'S are the denominator parameters, and z is the variable. One notation prevalent for the function is

)

The function In(t) is called the Bessel function of the first kind of order n. The function In(t) is called the modified Bessel function of the first kind of order n, and is related to the function J net) through the equation In(t)

J (it) =~ , where i = .,;=:y. I

00

(t/2t+(k+l)m

L -'-'--1-<---m=O m .(n + mk)!

z)

or simply pFq when there is no confusion about the arguments of the function. As an example note that 0:1 '0:2

2Fi(0:1,0:2;Pl;Z)=1+~z+

+ 1)0:2(0:2 + 1)

0:1(0:1

1.2'Pl(Pl+ 1)

+ 0:1(0:1 + 1)(0:1 + 2)0:2(O:Z + 1)(0:2 + 2) 1 ·2 ·3 . PI (PI + l)(PI + 2)

In the theory of queues, the following modified form of In(t) has been used: I~ (t) =

Pl,P2, ... ,Pq;

pFq (0:1,0:2,""O:p;

(1.5.4)

It may be noted that I~ (t) is the same as In{t) defined above and thatI~(t) is not a solution of the Bessel equation (1.5.3) when k > 1. For the properties of I~(t) see Luchak(l956),and for the properties of In(t) and J n (t) see Abramowitz and Stegun (1964). It may be noted for future reference that the L.T. of I net) is given by

Z3

2 z

+ ....

This is called an ordinary hypergeometric function. Most of the very extensive literature on hypergeome-trirtunctions considers the cases when p and q take values between 0 and 2, both inclusive, such as 2FO' IFI, and so on. The following examples may further clarify the notation. The exponential function: (1.5 .6) and the negative binomial function:

(1 +z)-a =

~

(aM-zt

k=O

k!

L.

2Fl(a,Pl ;Pl ;-z) The Generalized Hypergeometric Functions Quite a number of important functions can be expressed by means of hypergeometric functions. The series

IFo(a;-z),

Izl

<

1.

(1.5.7)

It may be observed from equations (1.5.6) and (1.5.7) that when the parameter value occurs in both the numerator and the denominator, both parameters can be

20

PRELIMINARIES

omitted from the argument of the function, and the subscripts of F are each reduced by 1. Under certain conditions it can be proved by using the ratio test, that the series (1.5.5) (or its particular cases) converges in a certain circle Izi < r in the general case of p and q. For example, note that the series (1.5.6) and (1.5.7) are convergent for Iz 1< 00 and Izl < 1, respectively. For a more detailed study of these and several other functions see Luke (1969), Copson (1948), or Abramowitz and Stegun (1964).

21

PROBLEMS AND COMPLEMENTS

PROBLEMS AND COMPLEMENTS Section 1.1

Prove the following results, which are useful in many queuing problems:

=

1.6

(b)

PARTIAL DIFFERENTIAL EQUATIONS

npx(n)zn = zp(!)(z).

n=O

For some problems of multichannel queues in Chapter 5 we need to consider the solutions of linear partial differential equations in two independent variables, which may be solved by a general procedure known as Lagrange's method, which we now discuss briefly. Consider an equation of the form

oz oX

L

(c)

= L n(n -

+ l)px(n)zn

1)(n - 2)· .. (n - k

>

(d) If qx{n) = P(X

=

oz oy

L= hx(n)z"

(e) If hx(n) = P(X .;;; n), then

dy Q

P

dz R

<

(f) If gx(n) = P(X

=

n=O

2

In many problems ,euing theory we deal with an r.v. X which is continuous and whose density function vanishes for negative values. The mean of such r.v.'s may be obtained heuristically as follows:

S:

[1- Fx(x)] dx

Let two independent solutions of the first and second equations be if fx(x) exists and fxex) = 0 for x Hint:

vex, y, z) = constant

F{u(x,y, z),v(x,y, z)}

=

0

or

u

=

Sincefx(x) = 0 for x E(X) =

The general solution, which is a function of u and v may be written in the form

f(v)

or

v

= g(u).

fo=

<

xdFxex)

O.

0, lim yFx(Y)-

AsFx(oo) = 1, E(X) =

<

y->-=

lim y-+oo

The form of the function may be determined by the initial conditions. One may gain experience with Lagrange's method by attempting the problems given at the end of this chapter.

zP(z) l-z

L gx(n)zn

n), then

E(X) =

u(X, y, z) = constant

O.

P(z) . l-z

n=O

dx

>

1-P(z) . l-z

L qx(n)zn

n), then

n=O

P-+Q-=R where P, Q and R are functions of x, y, and z. We give here only an outline of a procedure for solving this standard-type equation. For the general theory of such equations, one may refer to standard text books, such as Forsyth (1929) and Piaggio (1958). For an application of these equations to generalized birth and death processes, see Bailey (1964). The associated Lagrange (also called subsidiary or characteristic) equations are

= zkp(k)(Z), k

n=O

fo

=

r 0

dx -

f=

fo= Fx{x)dx.

Fx(x)dx

0

[1- Fxex)] dx.

22

PRELIMINARIES

This problem shows that [l-Fx(;;)]IE(X) may be treated as a p.d.f. For a rigorous proof of this, see Chapter 2. In case of a positive discrete integer-valued LV., the analogous result is

5

It may be seen that if P(fV = n) = 1, then we recover the Erlangiai distribution (1.2.6). (a) If N in problem 4 is distributed according to the geometric law

=

E(X)

L

x=o

2

PROBLEMS AND COMPLEMENTS

P(X>x).

en = peN = n)

J-b

-b- b

n

,

n = 1,2, ... ,

°<

b

<

1,

then show that

Section 1.2

!WN(t) = )1(1 - b) exp [- )1(1 - b )t] . 3

Let Xi be independently distributed

LV.'S

with Wn =

:r;[!"1

Xi. (b)

(a)

If {Xi} are identically distributed as X, such that P(X = x) = pXql-X, X = 0, 1, with p + q = 1, then the LV. Wn is binomial with parameters n andp. Hint: The p.g.f. of X is (q + pz).

(b)

If {Xd are binomially distributed with parameters ni and p (p remaining the same), then Wn is binomially distributed with parameters :r;r=l nj andp. If {Xd are geometrically i.i.d. as X, such that P(X = x) = qXp, P > 0, q > 0, x = 0, 1,2, ... , with p + q = 1, then the LV. Wn has the negative binomial distribution.

(c)

Hint:

then show that !WN(t) = 6

(a)

The p.g.f. of X is pl(! - qz). N

4

If N in WN =

.:r; 1=1

Xi> instead of being a constant, as assumed in the

of the Erlangian distribution, is an r.v., we have a generalization of the Erlangian distribution, which may be called the modified Erlangian distribution and may be denoted by EN. Consider the sum (1.2.1), where N is now an r.v. and the Xi'S are independently exponentially distributed with a parameter )1. Prove that the density function of WN is given by

°

!F -

)1

t

[

exp -

(b)

a + )1t ] ( ) . II (2.;a;;t). 1 -exp -a

Continuation of Problem 4. If N follows a Poisson distribution witl mean A and P(X = 1) = p, P(X = 0) = q, with < p < 1 and p + q = 1 then using equation (1.2.4), show that

°

PWN(z) =

derivat~

where ~ Cn = peN = n) ~ 1 and :r;;=1 Cn = 1. The EN distribution, when considered for service (or interarrival) times, consists of an indefinite number of phases. In practice, however, the values of en and )1 are to be determined by equating a few moments of the experimentally determined service times to the theoretical moments which can be determined from this distribution. A large class of distributions that are of practical interest can be handled by this distribution.

If N has the zero-truncated Poisson distribution

e"P(Z-I)

which is the p.gJ. of a Poisson distribution with mean Ap. This show that the Poisson distribution is preserved under random selection. Thi property of the Poisson distribution is very useful in many applicatiom some of which may be found in Parzen (1962) or Feller (1968). We giv· two simple applications. In transportation and traffic problems N could be the number of car passing an intersection on a certain highway during a period of obser vation and p the probability of a car turning at the intersection. The] WN is the total number of cars that turned at the intersection. In biologi cal sciences, N may represent the number of female insects in a certail region and X the number of eggs laid by an insect. Then clearly W./I represents the total number of eggs laid by all the insects. In general, irrespective of the distribution of N and X, WN is the sum of; random number N of Li.d. LV.'S {X;}. From equation (1.2.4) or otherwise, show that: (i) E(WN ) = E(N)E(X). (ii) E(Wt,,) = E(N)Var(X)+E(N2)E2(X). If N is a geometric r.v. with peN = n) = qpn-l, n = 1,2,3, ... show that

24

PRELIMINARIES

PROBLEMS AND COMPLEMENTS

fo~ Fx ' (t -ca-bY) Ix ' (y)dy,

E(X) Fz(t) =

q

25

with a

>

O.

2

2 {,,(_1_+-,-p-,-)E_-"CX_),"-,q.!-.-+_V_ar....::(X'--'-!..)} E (WN ) = -

As an application of the above formula, let Xl and X 2 be exponential r.v.'s with the distribution functions

q

7

(a)

Prove that the density function of W2 = Xl + X 2 when XI and X 2 are independently exponentially distributed with parameters 111 > 0 and f.12 > 0, respectively, is

Fx , (t)

11

Fx,(t)

A.

> >

0,

t

~

0

0,

t

~

O.

Show that the dJ. of Z = Xl - X 2 is given by f.11 01= 112 l __ A._ exp(-f.1t)

A. +11

f.11 = f.12 .

(

(b)

'

t

~

Fz(t) =

The problem considered in (a) can easily be generalized. Consider the sum Wn = '2;[; I Xi where Xi are independently distributed exponential r.v.'s with parameters l1i > 0 (all l1i being different). Show that the p.d.f. of Wn is given by

~ exp(A.t),

0 "

t <; O.

1\+f.1

If X 2 is exponentially distributed as before and XI has an Erlangian distribution Ek with the d.f.

n

I

Bil1ie-Jlit

Fx , (t)

i=1

t

~

0

where then show that

1-e- Jlt [kII {I - (_P )k-r }(I1tY 1·' A.+ P r! r=O

Take the L.T. of Wn , break the result into partial fractions, and then invert. The distribution considered in this problem when all l1;'s are different is called the generalized Erlangian distribution En. In view of this, the Original En (when all Pi'S are equal) is called the ordinary Erlangian distribution. We simply use the term Erlangian En when there is no confusion with the generalized one, and refer to the ordinary Erlangian and the generalized Erlangian when both are being used in the same problem.

Hint:

8

(a)

If XI and X 2 are independently distributed and if Z = XI - X 2 , then show that

Fz(t) =

f~ Fx (t + y)fx (y)dy 0' ,

where XI and X 2 are continuous and restricted to nonnegative values. Limits are to be so taken that the inequalities y ~ 0 and t + y ~ 0 are satisfied. A more general case is the following. Let Z = aX! + bX2 + c. If X I and X 2 are independent, then

t

~

0

Fz(t) eAt (_11

A.+11

(b)

)k,

t <; O.

These results have applications in the theory of queues, dams inventories, and so on. In queuing theory Xl represents the service tim< of a customer and Xz an interarrival time. In several queuing problems we are interested in the distribution of th, LV. Y/ defined as

eX -

(X-Y)+ = max(O,X-Y)

=

a, {X-Y,

X-Y <; 0 X-Y> 0

where X and Yare nonnegative continuous r.v.'s Let us consider som cases where the nature of one of the r.v.'s, say Y, is known. The simple,

PRELIMINARIES

26

PROBLEMS AND COMPLEMENTS

case perhaps occurs when Y is' assumed to be independent of X and exponentially distributed with parameter, say A. Show that in this case, the L.T. of is given by

By using Fk(O) = 1, it can be shown that

eX - Yt

Fx(x) = 1 - exp [- (J.L(t)dt] Afx(a) - afx(A) A-a '

= \ - ANllO\)

(c)

and hence

+ fX(A)

A = a

where fx(a) = E(e- ax ) .. Hint: E(e-a(X-Y)+ IX = x) = g e-a(x-Y)Ae- AY dy + e- AX , and so on. Continuation of (b). Consider the problem in (b) and show that if Y has anEk distribution with p.dJ.

A(Ay)k-1

fy(y)

27

The function J.L(x) is also called the intensity function or conditional rate of failure function, and is widely used in renewal theory, reliability, actuarial science, and so on. In particular, it may be seen that if J.L(x) = A, a constant, then X is exponential with parameter A, namely, Fx(x) = 1 - e- AX . Conversely, if X is an exponential r.v., then

e-AY

k

(k -1)1

1,2,3, ... ,

F (x + tlX ~ t)

x

=

Fx(x + t)-Fx(t) I -Fx(t)

=

I _e- AX ,

then

k-

A fx(a)-

t

[k-l L

(-I)mf.r)(A)] 1

m=O

k.

•

~

[Ak(A-a)m-Am(A-atJ}/(A-a)k,

mt{ (9

which shows that, given that the service did not end until time t, the remaining service time has the original exponential distribution. It may be remarked that J.L(x) is not a p.dJ., conditional or uncon· ditional. The requirements that it must satisfy are

A *- a

J.L(x)

~

0,

the first of which is easy to see and the second can be seen to hold since

A)m::tm)(A)} ,

Fx(oo) = I. As an example of the use of the hazard function J.L(x) let Xl and X 2 be ii.d. exponential r.v.'s with parameter A. Then W2 = XI + X 2 will have an E2 distribution with mean 2/A. In this case

A = ex

where fl.°)(a) = fx(a). Hint: Proceed as in (b) and use the identity discussed in Problem 17 of Chapter 2. Another distribution that has been used in queuing theory may be discussed in relation to the following example. Let X be an LV. representing the service time of a customer with p.dJ. fx(x). It is sometimes useful to express the p.d.f. of a continuous nonnegative r.v. such as X in terms of the hazard function J.L(x) , where

J.L(x)dx = P{x

<

X';:; x + dxlX

fx(x)dx Fk(x)

_dFk(x) Fk(x)

and

Fk(x) = I-Fx(x).

>

x}

FWz(t) = l-e-At-Ate- At and consequently

J.L(x) Hint:

See Problem 7.

Section 1.3

10 (a)

Discuss the mathematical derivation of the hyperexponential distribution forn=2.

Hint: If Xl and X 2 are the r.v.'s representing the times a customer spends in channels 1 and 2, respectively, then

28

PRELIMINARIES

peS

(b)

>

> t Icustomer selects first channel) U 1 + P(Xz > t Icustomer selects second channel) Uz

t) = P(X 1

PROBLEMS AND COMPLEMENTS

(c)

Suppose that the arrivals to a given single-server queuing system follow the Poisson process with rate A, and the service times are exponential with mean 1/11_ Suppose further that it is known that the distribution of waiting time (in queue) consists of two parts, the probability mass concentration at the origin given by (i) P(Vq = 0) = 1 - P together with the p.dL (ii) !vq(t) = A(1 - p) e-/l(l-P)t, t> 0 where Vq is the LV. representing the time a customer will spend in the queue before getting served, and p = AIIl. Show that the truncated p.dJ. of waiting time (in queue) for those who have to wait is the exponential distribution with parameter (11 - A).

(d)

In the queuing problem discussed above, suppose that it.is known that the distribution of the LV. N, which represents the number in the system (queue + service), is given by

The derivative of this gives the required p.dJ. The case of a general value of n may be discussed similarly _ Show that the mean and the variance of the hyperexponential are E(S) =

f

U

r

r= 1 Ilr

Var(S) = 2

f u; - (I ur)2 .

r= 1 Ilr

29

r= 1 f.1r

n = 0,1,2,3, ....

The variability of the distribution of a nonnegative r.v. X can be measured through the coefficient of variation, defined by

Show that the truncated distribution of N truncated to the right of N=Mis

ax

C.V.(X) = E(X) .

Find C.v. (X) for the following distributions~ (i) Exponential with parameter 11. (ii) Erlangian given by equation (1.2.6). (iii) Deterministic. (iv) Hyperexponential with n = 2. Show further that for n> 1, the Erlangian distribution En has less variability (as measured by the coefficient of variation) than the exponential, whereas the hyperexponential HEn has greater variability than the exponential. These distributions are found to adequately represent many real-world problems in the application of queuing theory. In any application the values of It and 11 for the Erlangian distribution, and n, a r , and Ilr for the hyperexponential must be determined empirically. The hyperexponential distribution is very useful in approximating statistical distributions for which the coefficient of variation lies between 1 and 00, whereas the Erlangian distributions are useful when the coefficient of variation lies between 0 and 1. In practice the coefficient of variation can be used gainfully in identifying the distribution of r.v.'s involved in queuing theory. F or example, if the sample standard deviation is nearly equal to the sample mean, then one might suspect that the underlying distribution is exponential. If the sample standard deviation is significantly less than the mean, then the underlying distribution may well be Erlangian. Finally, if the sample standard deviation is greater than the sample mean, then it may be appropriate to test whether the hyperexponential fits the observed data.

0, peN = i) =

i

>M

(l-p)pn

(

i = 0,1,2, ... ,M.

I-pM+!

In the theory of queues, the truncated distribution of N is useful where the queuing system has a limited waiting room, say of size M. (e)

Find the p.dJ. of the truncated hyperexponential T.V. T (truncated on the right at T = 8) discussed in Section 1.3, (when n = 2).

Section 1.4

11 Show that the L.T. of En, namely, n is

(11 Prove from this that

n

= -, J1

+ at

~

1

PRELiMINARIES

30

PROBLEMS AND COMPLEMENTS

31

In particular, discuss the corresponding results for the exponential distribution. 12 Deterministic distribution, D, continuation of Problem 11. tl, n ->- co in such a way that lim fJ.->- = n .... =

Show that when

(~) is finite and equal to J.- , tl

15 If a> 0, prove that:

lim fJ.->-OO n->-oo

[1 + ~J-n

This is an L.T. ofa discrete LV.,X=

P(X

*-

This result can be extended to n independent nonnegative LV.'S. Section 1.5

tll

say, then

lw=(a) =

lx, (a)[x, (a)

1/111) = 0,

tlln

Woo,

=

e-CX1fJ., •

defined by

(a)

kf

(b)

kII rI'(a + r)

r=O

(k-r)I'(a+r) = I'(k+a+ 1) I'(a)·r! I'(a+2)·(k-l)!

r=1 (r - I)!

P(X = 1/l1d = 1. Hint:

X is called a regular, deterministic, or constant r.v., and is usually denoted by D. 13 Continuation of Problem 11. Show that the L.T. of EN (here N is an LV.) is

(a) Use equation (1.5.1). (b) Use equation (1.5.1).

Section 1.6

16 Let Q =Q(x, t). Consider the partial differential equation

aQ

-

at

14 If lx(a) =

k 1 ~ (k + n - 1)1 = E(WN ) = I i i... ( _ 1)1 /J. n=! n .

Fx(a) =

ax

Cn ·

where f( . ) is an arbitrary function to be determined by the initial conditions and X(y) is defmed below. Using the initial conditions, show further that the solution may be put in the form

f;;' e -cxx fx(x)dx, then show that:

(a)

aQ

-tl(l-x)- = -A{I-7T(x)}Q

with the initial conditions Q = Xi when t = O. Show that a general solution may be written as

wherec n =P(N=n). Prove by differentiation or otherwise that , Ilk

(ka + k -a)I'(a + k) (a + 2)(a + 1){(k - 2)!}

lxCa) , a

1 -lx(a)

Q = (q + pX)i exp

a Use (a) to prove the following: If XI and X 2 are independent nonnegative f.v.'s with dJ.'s F x , (x) and F x , (x) and respective p.dJ.'s tx, (x) andtx, (x), then if W2 = Xl + X 2 , show that

[~JX 11

q+px

X(Y)dY ]

where I-p,

and

X(y) = 1 - 7T(Y) . l-y

PRELIMINARIES

REFERENCES

33

32

Further show that if 1T(Y) = y, then

Q=

(q+px)iexpl~q(x-1)1·

An interpretation of 1T(Y) = Y is given in Chapter 5 on mUltichannebl qu;u:;~ion . . f P b1 16 Let A instead of being a constant, e a u 17 ContmuatlOn 0 ro e m . , h b e of t, say A(t). The partial differential equation of Problem 16 t en ecom s

oQ -/l(l-x)oQ = -A(t){l-1T(x)}Q at ax d·t· Q Xi when t = 0 Show that the solution in this . ..1 . with the IDltla con I Ion case is

. rl;1 Jr

Q = (q + px)' exp

x

q+px

1 -1T(Y) I\. (t

1- Y

+ l.-ln 1 /l

Y 1- x

)dY 1.

. h . t· of this with different initial conditions is discussed in Chapter A S119 t vana Ion 5 on multichannel queues. . h d If 18 The following equations arose in a study of the growth of anrrnal er s. oQ -x(Ax + /l)(1 -x) oQ = x at ox with the initial conditions Q = Xi when t =

Q

-(I\.~ /l)(1-x) Q

= 0, then show that

x [/l- /l exp [- (I\. + /l) t] + x {A + /l exp [- (A + /l)t])] i-l (/l + A)n ! . . [/l + A exp [- (A + /l)tJ + x {A -I\. exp [- (I\. + /l)t])] n-,.

For further details, see Bunday (1970).

REFERENCES ·tz M. and I. A. Stegun (Eds.) (1964). See Chapter 2. Wi! New York Ab ramoW!, , ey, . d N T J (1964) The elements of stochastic processes. Bailey, . . . . h (1961) A short table of z-transfonns an Beightler, C. S., L. G. Mitten, and G. L. Nem auser . . generating functions. Oper. Res. 9,574-578. .. . troduction to applied probabIlIty. Wiley, New York. J (1948) The life and works of A. K. Erlang. Blake, I. F. (1979). An m Brockmeyer, E., H. L. Halstr¢m, and A. ensen . The Copenhagen Telephone Co., Copenhagen. 1970) The growth of elephant herds. Math. GaZ. 54, 38-40. D ( Bunday, B.. . . il E lewood Cliffs NJ. 1973). Discrete-time systems. Prentlce-Ha, ng '. Y k Ca d zow, J . A . ( . I A I · McGraw-Hill New or. nd C de Boor (1972). Elementary Numenca na yszs. , D a Cont e, S ."' .

Copson, E. T. (1948). An introduction to the theory offunctions of a complex variable. Oxford University Press, London. Cromie, M. V., M. L. Chaudhry, and W. K. Grassmann (1979). Further results for the queueing system MX/M/c. J. Oper. Res. Soc. 30,755-763. Douglas, J. B. (1963). Asymptotic expansion for some contagious distributions. Proceedings of the International Symposium on Discrete Distributions, Montreal, 291-302. ErdeJ.yi, A., W. Magnus, F. Oberhettinger, and F. G. Tricomi (1954). Tables of integral trans· forms. McGraw·HiIl, New York. Feiler, W. (1968). An introduction to probability theory and its applications, vol. 1, 3rd ed. Wiley, New York. Forsyth, A. R. (1929).A treatise on differential equations. Macmillan, London. Gnedenko, B. V., and A. N. Kolmogorov (1954). Limit distributions for sums of independent random variables. Addison-Wesley, Cambridge, MA . Henrici, P. (1964). Elements of numerical analysis. Wiley, New York. Jury, E. 1. (1964). Theory and application of the z-transform method. Wiley, New Yv•.~L Kendall, M. G., and A. Stuart (1969). The advanced theory of statistics, Vo1.l, 3rd ed. Griffin, London. Kingman, J. F. C., and S. J. Taylor (1966).Introduction to measure and probability. Cambridge, London. Kleinrock, L. (1975). Queueing systems, Vol. 1, Wiley, New York. Loeve, M. (1977). Probability theory I, 4th ed. Springer-Verlag, New York. - - - - (1978). Probability theory II, 4th ed. Springer-Verlag, New York. Luchak, G. (1956). The solution of the single-channel queueing equations characterized by a time-dependent Poisson-distributed arrival rate and a general class of holding times. Oper. Res. 4,711-732. Lukacs, E. (1960). Characteristic functions. Hafner, New York. - - - - and R. G. Laha (1964). Applications of characteristic functions. Griffin, London. Luke, Y. L. (1969). The special functions and their approximations, Vol. 1. Academic Press, New York. Neuts, M. F. (1973). Probability. Allyn and Bacon, Boston. Neveu, J. (1965). Mathematical foundations of the calculus of probability. Holden-Day, San Francisco. Oberhettinger, F., and L. Badii (1973). Tables of Laplace transforms. Springer-Verlag, New York. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Piaggio, H. T. H. (1958). An elementary treatise on differential equations and their applications, rev. ed. G. Beil & Sons, London. Polya, G. (1954). Mathematics and plausible reasoning, Vol. 1: Induction and analogy in mathematics. Princeton University Press, Princeton, N.J. Roberts, G. E., and H. Kaufman (1966). Table of Laplace transforms. Saunders, Philadelphia. Russell, A., and R. G. Taylor (1977), Numerical analysis and relationships for the queueing models MIE~/1, EVM(l. Unpublished thesis, Royal Military College of Canada, Kingston, Ontario .

2.1

Chapter 2

SOME TECHNIQUES OF QUEUING THEORY The history of queuing theory goes back to the work of Erlang in the early years of the twentieth century, and important work was done in the 1930s and 1940s by Khintchine, Palm, Pollaczek, and others. About 1950 the theory of queues entered a period of intensive investigation by workers in a variety of fields, particularly in mathematics and operati~s research. Research in the theory of queues has developed new general methOds, including methods which can be used to give explicit solutions to many particular problems. At the same time, research has taken place in the development and application of new and old queuing models for particular applications. Queuing systems which have been studied include such everyday activities as waiting for service by a teller in a bank, as well as a wide range of systems in manufactUring, communications, transportation, and other fields. A systematic study of the theory of queues provides a base of knowledge which can be applied to improve the efficiency of many queuing systems in the real world.

2.1

BASIC MATERIAL IN QUEUES

A queue, in the sense in which we use the word, is a waiting line which usually is formed in front of some service facility as a result of an irregularity in the pattern of arrivals or departures, or both. Thus it appears that a waiting line has three elements a queue, a server who may be active (such as a bank teller) or passive (such as a parking lot), and a service facility. In the mathematical analysis, a queuing system is normally characterized by means of the following terminology due to Kendall (1951). The Arrival Process

This is the process that governs the arrival of some given service-seeking entities, which we shall call customers, be they human or otherwise, to the system. The customers normally receive service from a server or service channel and then leave the system. If a customer arrives when all servers are busy, the customer is usually assumed to wait for service. Here, as elsewhere in queuing theory, our technical 34

BASIC MATERIAL IN QUEUES

35

terms such as "customer" and "server" are often words taken from nontechnical English, but they are used here in an extended sense. For example, a customer might be a computer program waiting to be computed, and the server might be the computer itself which performs calculations on the program. The simplest hypothesis about the arrival process assumes that arrivals follow a Poisson process (random arrivals). The arrivals may occur singly or in bulk (groups or batches). If all the arriving customers are allowed to join the queue, we have an infinite waiting space queuing or delay system. We also consider systems, called finite waiting space systems, in which customers do not wait, but leave the system immediately if they arrive when a fixed number K of customers are already waiting. A loss system is the extreme case of a fmite waiting space system, in which K = 0 so that customers never wait, but leave unless they receive immediate service on arrival. Infmite waiting space queuing systems, which are si.,Inpler to handle mathematically, serve as a reasonable approximation to the (more realistic) fmite waiting space queuing systems, particularly when the traffic intensity is low (p «; 1), that is, when the traffic is light. In case the traffic intensity is large but less than 1, the finite waiting space queuing system converges very slowly to the infinite waiting space queuing system as the waiting space tends to infinity. When p> 1, it will be essential to consider finite waiting space queuing systems. ~ile queues with infinite waiting space may not exist in the real world, they provld~ excellent models for real systems in which the (large but finite) waiting space IS rarely fIlled. We may therefore say that in practice both infinite and finite waiting space systems are useful and worthy of study. ApprOximate (or exact) results obtained from the infinite waiting space systems (or finite waiting space sys~ems) may be good under the conditions stated above. There are also queues in which customers may not join the system if, on arrival, they get some information which deters them from joining the queue. Such customer behavior (not joining the. queue) is know.n asb.~~ing. Moreover, it is possible that a customer haVing waIted for some tIme becomes impatient and thus leaves the system without getting service; this is known a~.leneging. Connected with the arrivals are the interarrival times. The points in time at which groups of customers or individual customers arrive to join the queue are called arrival instants of time (or epochs), and the intervals between consecutive arrival instants (or epochs) are called interarrival times. Let the sequence of groups cU,stomers or .individ~al c~stomers arrive at the instants (or epochs) 0 = a~ , aI, a2, ... ; the mterarnval tImes are then defined by the r.v's Ur = a'r - a'r-l, . r = 1, 2, 3, .... It will be assumed that the U's are positive LLd. r.v.'s with common d.f. A (u) = P(Ur < u), 0 < u < 00. The corresponding p.d.f. will be denoted by a(u). When customers arrive singly and A (u) = 1 - exp (- Au), it is easy to show (see Section 2.2) that the number NCt) of customers who arrive in any time interval Js, s + t] of duration t follows a Poisson distribution given by

0;

P(N(t) = n) =

e-?-.t (At)n

n!

n ;;;. 0,

A > 0,

t>

O.

SOME TECHNIQUES OF QUEUING THEORY

36

U's, instead of being independent, may be correlated. For example, if the arrivals are scheduled for predetermined times and if a customer arrives early, there is likely to be a short interval before one arrival, followed by a long interval before the next arrival. However, we shall not consider such correlated interarrival times in this book, unless stated otherwise. The Queue Discipline

This is the rule that determines the formation of the queue and the take-up of customers for service. For example, customers may be picked up for service randomly or on some set basis. In the latter case, the discipline, which is the simplest in concept and also from an analytic point of view, is first-come,. first-served (FCFS), that is, co~encement of service is in the order of arrivaL Besides these disciplines, several others are possible, such as last-come, first-served CLCFS). One example, among others, of LCFS is last hired, first fired because of low seniority. Unless otherwise stated explicitly, we shall assume throughout our study that the queue discipline is FCFS. Service, like arrivals, might be in batches of fixed or varying size. Usually it is assumed that intake for service is instantaneous. However, this may not be true in certain situations. For example, if the service facility is manually controlled, the server may be available intermittently, in the sense that after a customer leaves the service facility, the server may not take up the next customer immediately, but may spend some time on a job left over by the departing customer before taking up the next customer. As another example, consider a one-way transportation system such as a ski lift in which the gondolas or chairs are regarded as servers. Then the servers are not instantaneously available for service after customers' departures, since each server must return to the lower station of the ski lift before picking up new passengers. Service Mechanism

The time interval from the instant of initiation of service on a customer to the instant when this service ends is called the service time (or holding time in telephone networks). Let the customers depart at the instants ai, a2, ... , a", ... , k = I, 2, 3, . . . . If, however, there is a departure at time t = 0, we may write' ao = 0, and call ao an initial departure instant. If V", k = 1, 2,3, ... , is the duration of the kth service interval, we assume that V's are independent of U's and also mutually independent, with the common distribution function B(v) = P(Vk

<

v),

o< v <

00,

k = 1,2, ....

Its p.d.f. will be denoted by b(v). As we did with interarrival time distributions, we consider various service time distributions. For example, consider the E" distribution for service times, in which service takes place in k exponential "phases". In general, phases may have no real physical significance. Recall from Chapter 1 that the Ek distribution may be obtained from a sum of k Li.d. exponential LV.'S. This artifice, introduced by Erlang and generalized by later writers, permits us to use

2.1

BASIC MATERfAllN QUEUES

37

the "memoryless" property of the exponential in studying a large, though not completely general, class of service time (or interarrival time) distributions. Generally speaking, the service facility can be thought of as being composed of a given combination of servicing stations arranged in series, in parallel, or in some more general configuration. In both applied and theoretical studies the most common configuration of service channels consists of c servers in parallel, where 1 < c < 00. The single'server case c = 1 has been thoroughly studied and much information is available about it. In the infinite-server case c = 00 all customers can be served without Waiting, and our interest is in the numb~r of busy servers. The infinite-server queue can be a useful and easily solved approximation for multiserver queues in which c is fairly large but fmite, and the system is lightly loaded, so that almost all customers can be served without waiting. Denote E(Uj) and E(Vj), i = 1, 2, 3, ... , by a and b, respectively, so that A = lla and Il = lib, and assume that 0 < a < 00, 0 < b < 00. The traffic intensity p plays a fundamental role in the study of queuing systems and is defined by _ mean arrival rate of customers p -. . maXImum serVIce rate when operating at maximum capacity In many, but not all, cases p is equal to the utilization factor, defmed as the proportion of time the system is busy. The cases in which p differs from the utilization factor include finite waiting space systems and single-server bulk-service systems (excluding the case when service is in batches of fixed size k), as well as other systems to be discussed later. For bulk arrival Single-server queues with mean arrival group size a, we have p = Aaill. Similarly, for the bulk-service queue MiCElI, p = AlB Il, where B is the capacity of the server. For multichannel queues with customers served singly and c( < 00) identical servers, p = A/ell. In the case when c = 00 or waiting space is finite, p does not play such a significant role as in the other systems. Two types of solutions of queuing systems are of interest - the limiting solution and the transient solution. There are other eqUivalent terms for both of these. For the former, we have time-independent, in transient, homogeneous in time, or steady-state, whereas for the latter we have time-dependent, nonlimiting, and so on. In the limiting case, the effect of the initial conditions is worn out. Let us explain these concepts by the following example. Suppose that one is interested in the distribution of the number of customers in a given queuing system. In the limiting case, although the probabilities are independent of time, the distribution t dete~inistic. The queue fluctuates following a distribution independent is of tIme. ThIS may be explained as follows. If the probability of 20 vehicles waiting at a toll ga~e on a highway between 2.30 P.M. and 2.31 P.M. (on a given day) is 0.15, we will have the same probability between the same times on other days, that is, the probability depends on the length of the time interval and not on the initial conditions. When the limiting solutions are inapplicable or do not exist, it is desirable to consider transient solutions. It may be mentioned here that the limiting solution, provided it exists, may be obtained by letting t -+ 00, or by setting the derivative with respect to t equal to zero and then solving the resulting limiting

n.a

38

SOME TECHNIQUES OF QUEUING THEORY

equations. A limiting solution will exist irO ~ p ~ 1. If p > 1, there is, in general, no limiting solution. However, there are exceptions to this. For example, when c = co or waiting space is finite, limiting solutions hold for any value of p, though in the finite waiting space case the limiting solution may be reached very slowly when p> 1. The transient solutions, of course, hold for any value of p. Transient solutions, however, are in general difficult to obtain in explicit form. Even if they can be obtained, numerical analysis seems difficult to carry out, and this is especially true when customers arrive or are served in batches. In view of this, we shall be discussing mostly the limiting solutions. Transient solutions are discussed when they have some other interesting feature. The most important problems connected with the probabilistic study of queuing systems include the determination of probability distributions for the system length, waiting tim?-{-actual and virtual), busy period, and idle period. We now discuss these problems. The Distribution of System Length. Let N(t) be the number of customers in the queuing system, including those in service, if any, at any epoch t. We will sometimes refer to N(t) informally as the system length, but the reader should be aware that the term "queue length or size" has been used in the literature in at least two different senses. Some authors use queue length to refer to the number of customers in the system, whether waiting or in service. Other authors use queue length to refer to the number of waiting customers only, excluding those being served. When we need to discuss the number of customers waiting, the number of customers in the system just before the nth arrival, or the number of customers in the system just after the nth departure, appropriate notation will be introduced to distinguish these quantities from one another and from N(t). For convenience, we have used the term queue size in Chapter 6. 2 The Waiting-Time Distribution. The time spent by a customer from the instant of joining the queue to the instant of entering service is called the customer's waiting time in the queue, and is generally denoted by V q • The customer's time in the system, or sojourn time, is the sum of the customer's waiting time in queue and service time, and is generally denoted by V. We must distinguish between the actual waiting time V~n) of the nth arriving customer and the virtual waiting time Vq (t), which is the waiting time experienced by a (fictitious) customer arriving at epoch t. The actual and virtual waiting times are connected by the relation V~n) = V q (o~ - 0), where a~ is the arrival epoch of the arriving customer. Similarly V(n) = V(a~ - 0), where V(t) is the virtual sojourn time at epoch t and V(n) is the sojourn time of the nth arriving customer. It should be clear that in the same queuing system, the two processes, though related, are qUite different. In fact, Vq(t) [or V(t)] is considered in continuous time, whereas V~n) is considered at a discrete set of points in time. In order to distinguish between the two processes, Vq (t) is called the virtual waiting time at epoch t, whereas v~n) is called the actual waiting time of the nth arrival. However, in the sequel, the adjectives virtual or actual will usually be omitted since the meaning should be clear from the processes under consideration. In the case of statistical equilibrium, Vq (t) or V~n) will be replaced by Vq in the continuous as well as in the discrete case,

2.1

BASIC MATERIAL IN QUEUES

39

although the two processes are in general different. This is done for notational convenience. The same argument applies to the time spent in the system. The d.f. 's of the actual waiting (in queue) time and the virtual waiting (in queue) time are denoted by W; (t) and Wq (t), respectively. If the two coincide, as they do in the steady state for Poisson arrivals, we prefer to use the notation Wq (t). Connected with the two waiting times (virtual and actual) is the occupation time of the server. The occupation time of the server at epoch t is denoted by ~(t), and is defined as the residual time at t during which the server will be continuously busy serving customers already in the system at t. Thus if no customers arrive after t, the server will be busy from t to t + ~(t) and will then become idle. Note that ~(t) = 0 if and only if (iff) the server is idle at t. We shall consider occasionally the limiting behavior of the process {t(t), t;" O}. The reader will see that in some situations the limiting behavior of the process {tCt) t;" O} coincides with the limiting behaVior of {Vq (t), t;" O}. One example in ~hich the limiting behaviors do not coincide is that of the queuing system MIME/I. For details, see Miscellaneous Problems in Chapter 4. It may be remarked here that some authors refer to the virtual waiting time at time t as the unfinished work (or backlog) in the system at time t [see, for example, Kleinrock (1975)]. The unfinished work at time t is defined as the time interval required to empty the system of all customers present in the system at time t. In view of the definition of unfinished work it is clear that it applies to all queue disciplines, whereas the virtual wait applies to FCFS disciplines. Since in this book we shall not be dealing with queue disciplines other than FCFS, we prefer the use of virtual waiting time to unfinished work. 3 The Busy-Period Distribution. For a single-server queue in which arrivals are by singlets, a busy period may be defined as the interval of time from the instant of arrival of a customer at an idle channel to the instant when the channel next becomes free for the first time. For this type of queue, the initial busy period may start with i> 1 customers and hence may be different from other busy periods which start with the arrival of a single customer. Results for busy periods of singleserver, single-arrival, Poisson-input queues can rather easily be generalized to the corresponding bulk-arrival queues. For multi-server queues, anyone of several different definitions of busy period may be appropriate. Some of the known results on busy period distributions and related problems, such as idle periods or number served during a busy period, will be discussed in later chapters. 4 The Idle-Period Distribution. For single-server queuing systems the duration of the period of time from the instant when the server becomes idle after serving a customer to the instant when the server restarts service on the arrival of a customer, is called the actual idle period and will be denoted by I. An actual idle period separates two successive distinct busy periods, and it therefore follows from the definition that I> O. One may also defme a virtual idle period, with duration Iv(n);"O, where Iv(n) is the duration of the actual idle period (if there is one) immediately following the departure of the nth customer, and Iv (n) = 0 if there is no idle period immediately following the departure of the nth customer. In steady state as n -+ 00, IvCn) converges in distribution to Iv, and

40

SOME TECHNIQUES OF QUEUING THEORY

the conditional distribution of (Iv Ilv > 0) 'will be the same as the distribution of I. For details, see the discussion of the queuing system GI/M/l in Section 2.2 and of Glk/M/I in later chapters. We will also make some use of the busy cycle, defined as the sum of a busy period and the following idle period. S Utility of These Distributions. At this point a comment on the utility of the system-length, waiting-time, busy-period, and idle-period distributions may be worthwhile. The distribution of system length is useful from the design point of view as an aid in estimating the cost of operation if a waiting room must be built. It is also useful:-from the customer's point of view. The customer on arrival is interested not only in the system length as measured by the number of customers in the system, but also in the interval of time he has to wait before service on him can be started or in the total time he has to spend in the system. The busy-period distribution is important from the server's point of view. In recent years, analysis of busy periods has found increasing use in the design and study of real-time computer systems, as in Kleinrock (1976). It may be mentioned here that the theory of queues is quite similar in its mathematical structure to seemingly unrelated mathematical theories of inventories, insurance, and so on. See, for example, Prabhu (1965 a), Takacs (1967), or Seal (1969). The theory of dams and reservoirs has been developed by Moran (1959), Gani (1957, 1969), Kendall (1957), Prabhu (1964, 1965 a), and others. The mathematical models of dam theory are, perhaps surprisingly at first sight, closely related to queuing models. For example, the wet period of an infinite reservoir corresponds to the busy period ofa queue. The study of idle periods also helps in the design of queuing systems. An analyst might probably like to strike a balance between idle time of the servers and customers' waiting time according to some cost structure. These distributions are related to each other and can give useful information to the management of an organization where there are congestion problems.

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

In Section 2.1 we gently introduced some of the ideas and terminology of queuing theory, with some mathematical symbols, but without mathematical manipulation of those symbols. Now we introduce some of the mathematical techniques used to solve queuing models, and some simple queuing models to illustrate the use of these techniques. The mathematical theory of queues is a small part of the more general mathematical theory of stochastic processes, which we now consider briefly. The word "stochastic" has a long and interesting history, but has now come to be used in the sense "depending on probability distributions," more or less synonymously with such words as "random," "chance," or "probabilistic." Stochastic Processes

For a formal treatment of stochastic processes, consult Bartlett (1978), Cox and Miller (1965), Parzen (1962), Karlin and Taylor (1975), <;inlar (1975), or more

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

41

advanced texts by Doob (1953), Levy (1965), and others. We give here a brief informal introduction to a few of the ideas contained in these books. A stochastic process N(t) [or TI(t), t(t), etc.] has three elements: 0) the state space S, (2) the parameter (or index) set T and (3) the dependence relations among the r.v.'s N(t) when t varies in T. The state space S, which is the set of values which may be taken by N(t) for typical values of t E T, could be discrete or continuous, scalar or vector. The parameter set T could also be discrete or continuous, scalar or .vecto!'. While in queuing theory t is normally a time parameter, in other applicatIOns It need not be so. [An example in which T is neither one-dimensional nor a time parameter set is that of waves in oceans. If one regards the longitudinal and latitudinal coordinates as the components of a 2-vector t, N(t) then is the height of the wave at the location t.] The arrival process, the distribution of system length, and so on, are further examples of stochastic processes th;}t arise in the study of queuing systems (see Section 2.1). Among the stochastic processes, the Markov processes occupy a prominent place. A stochastic process is called a Markov process if the future development of the process depends on its past only through its present and is otherwise independent of the past history of the process. More precisely, if for any n > 0 and any set of n points tl < t2 < ... < tn in T, the conditional distribution of NUn) given N(td, ... ,NCtn-l) is the same as the conditional distribution of N(tn) given N(tn-d only, then {N(t)} is a Markov process. In a Markov process, the dependence relatlOns (3) mentioned earlier between the r.v.'s N(t) and N(t + s) suffice to determine the probability structure of the process. For a discrete-state Markov process, often called a Markov chain, the dependence relations take the form of transition probabilitiesPu(t, s), defined by Pu(t,s)

=

P{N(t+s)

=

jIN(s)

= i},

i,j E S,

s;;.

o.

We will usually consider the important special case of a homogeneous Markov chain, in which P jj (t, s) does not depend on the initial epoch sET. The transition probability Pij(t, s) then may be written asPu(t), where Pjj(t)

=

P{N(t

+ s) =

jIN(s)

= i},

i,jES,

s;;'O.

Pu(t) satisfies the follOWing conditions:

t> t

O.

>

2

'fPjj(t) = 1,

3

fPik(t)Pkj(h) = Pjj(t

O.

+ h),

t,h

>

O.

(2.2.1)

Furthermore from (2.2.1) it is easy to establish the continuity of Pu(t) for every t> O. For let h > 0; then as h..,. 0, Pu(t + h)"" Pjj(t). Also Pj/t - h) ->PijCt). Thus Pjj(t) is continuous at each t> O. Similarly, using the continuity property of Pij(t), it can be shown that the derivative of poet) exists for every t > O. However, at t = 0, continuity and differentiability hold only from the

SOME TECHNIQUES OF QUEUING THEORY

42

right, if we define

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

43

and not, for example, on ll' As a consequence one may use the notation N(lz to represent bothN(t2) -N(td andN(t z + h) -N(tl + h). lim p;).(t) t .... o

=

3 .. 'J

= { 0, 1,

-

t 1)

j

j =t= i.

Orders of Magnitude o(h) and O{h)

One states informally that the function get) is o(lI. t) as lI. t It is supposed that the probability Pij(t) depends on the length of the time interval t and not on the position of the starting epoch s on the time axis, as indicated through the notation that we have adopted. Such processes are called time-homogeneous Markov processes or Markov processes with stationary transition probabilities. An excellent detailed mathematical study of these processes may be found in Dynkin (1965). In discussing time-homogeneous Markov processes, we shall make frequent use of equations (2.2.1), which are known as the Chapman-Kolmogorov equations. To begin with, the use of equations (2.2.1) is illustrated through one simple stochastic process, the bulk-arrival Poisson process, which will be frequently used in this book. The Time-Homogeneous Poisson Process wherein Events May Occur in Bulk

Since this process (also called the compound Poisson process) occurs frequently in the subsequent chapters, it is appropriate to discuss it here briefly. First let us consider informally certain other things needed for its discussion (and also needed later). Let N(t) denote the number of customers arriving at a queuing system during the time interval] 0, fl. For each fixed t> 0, NCt) is an LV. but if t varies, N(t) represents a single-parameter stochastic process to be denoted by {N(t), f ~ O}, or {N(t)} when the range of t is evident, or simply by N(t) for notational ease. The object here is to study the process {N(t), t;;' O} under certain assumptions, which are stated in terms of technical statements and notations. We proceed to explain these before taking up the process itself. independence of Increments

lim At .... o

-+

0 if

get) = 0 lI. t

that is,g(t) tends to zero faster than lI. t. Formally one may define o(g(x» as follows. We write [(x) = o(g(x» asx -+xo whenever lim X-+Xo

([(X») g(x)

= 0

.

In addition, it may be appropriate here, though it is not used until later, to introduce the big 0 as well. Let [(x) and g(x) be two real-valued functions defined on a set S of real numbers and assume that g(x) is nonnegative. We now write [(x) = O(g(x» for x in S if there exists a positive constant K such that 1[(x)1 ~Kg(x) for every x in S. Now we are ready to discuss the homogeneous Poisson process with bulk arrival. A stochastic process may be called a homogeneous Poisson process with bulk arrival if:

2 3

The process has stationary increments. The process has independent increments. Pik (h) = t..qk -i h + o(h), k - i ~ 0, where t.. > 0 is the mean rate of occurrence of the homogeneous Poisson process. If X is an LV. representing the size of batches arriving at each of the points of the homogeneous Poisson process, then qk = P(X = k)

1. k~l

Suppose that to < tJ < t2 < ... < tn < "". For every n> 1, if the increments N(td-N(to), ... , N(tn)-N(tn-d are mutually independent r.v.'s, the process {N(t), t ~ O} having this property is called a process with independent increments. This expresses the fact that increments of N(t) over nonoverlapping time periods are independent LV.'S. Stationarity of Increments

If the distribution of N(tz + h) - N(tl + h), h > 0, is the same as that of N(t z ) N(td, the process is said to have stationary increments, that is, the distribution of N(t2 + h) - N(t l + h) depends only on the length of the time interval t2 - tl

Here the existence of t.. and qk is assumed. We may also assume without loss of generality that to = 0 and let N(O) = O. DefinePon (t) to be the probability that n events occur by time t, that is, POn(t) = P(N(t) = nIN(O) = 0),

n

~

O.

It may be noted that POn (0) = 3 0n , where OOn is a Kronecker symbol. Moreover, we shall use the convention that POn (t) = 0 whenever n is negative. Now from the above defmitions,

PooCh) = 1 - t..

L Pan (h) n~l

I-t..h+o(h).

(2.2.2)

2.2

SOME TECHNIQUES OF QUEUING THEORY

44

Thus

Poo(t

+ h)

= Poo(t)(l- Ah

(2.2.4)

+ o(h))

which reduces, as h -+ 0, to the differential equation

pM)(t) = - APoo(t) where

p(1)(t) = dPoo(t) 00 dt, a notation that will be followed henceforth. Equation (2.2.3) has been derived after using conditions (1) and (2). On the same lines, we may write:

+ r..qIPoo(t) r..P02 (t) + AqIP01(t) + r..q2 P oo(t)

Pap(t) = - r..P01(t) pW(t) = -

45

In the subsequent chapters we shall frequently meet with the bulk-arrival as well as the unit· arrival time·homogeneous Poisson process. The above results hold when A, the rate of occurrence, is independent of the parameter t and the state of the process. However, if A is time dependent, for example, A= A(t), say, the process is known as nonhomogeneous in time. We shall meet with this case in multichannel queuing systems. For a detailed discussion of this process and various other processes connected with the basic unit-arrival time-homogeneous Poisson process, the reader is referred to Parzen (1962), Gross and Harris (1974), and Problems and Complements. It may be remarked here that for notational convenience, the first subscript 0 inPon (t), which represents the initial state of the process, will normally be omitted, except when we are dealing with the transition probabilities of the process, in • which case we shall continue to use both subscripts. A process that does not possess the Markov property is termed non·Markovian. In queuing processes discussed in this book, we shall consider both Markovian and non-Markovian processes. In general, the single-server system-length process {N(t)} is non-Markovian and difficult to handle directly. N(t) can often be represented by a vector Markov process of the form {N(t) , X(t), yet)}, where

(2.2.3)

= Poo(t)Poo(h)

TECHNIQUES AND HISTORY OF ANALYSIS

N(t) = number in system at epoch t X(t) = elapsed service time of a customer under service at epoch t n

-APon(t)+r..

L

Y (t) = time elapsed since last arrival.

qkPo,n-k(t).

k=1

Defining the generating functions Q(z) and Po(z; t) by

PO(z;t) = E(zN(t)IN(O) = 0)

L POn(t)zn,

n=O

multiplying the above equations successively by 1, z, Z 2,

Pal)(z; t)

=

... ,

and adding, we get

A{Q(z) -l}Po(z; t).

(2.2.4a)

This is an ordinary differential equation whose solution is easy t~ ?btain a~~ is given by Po(z; t) = exp [r..(Q(z) -1)tl, where we have used the imtial condition

Po(z; 0) = 1. P . = 1 = 0 k =1= 1 then we get the usual time-homogeneous Olsson If ql , qk , , . f h' h process (with unit arrivals) with the p.g.f. exp [r.. (z -1) tl , the inversion 0 w gives POn (t) = e-lI.t(r.. t)njn!, r.. > 0, n = 0, 1, 2, .... Also, if instead of taking to = 0, we start at to = s > 0, then

:c

Po(z;t-s) = exp[r..{Q(z)-l}(t -s)l· This is due to the assumption of stationary increments of the process.

The vector Markov process is often more tractable than the original non-Markovian process {N(t)}. The variables XCt), Y(t) are called supplementary variables (for details see below). In principle, it is possible to study multiserver queuing processes by introducing a sufficient number of supplementary variables; in practice the theory becomes more complicated. However, particular cases of both the singleserver and the multi-server queuing processes can be studied more elaborately by several techniques, some of which we proceed to discuss before illustrating their use in the chapters that follow. It should also be mentioned here that for singleserver queues with bulk input, in which the service times or the interarrival times, or both, have Erlangian distributions, some results have been obtained by studying the vector Markov process {N(t), XCt), yet)} (see Chapter 3)_ Phase Technique

This technique is essentially due to Erlang, although modifications have been made by various authors. We assume that service on a customer consists of k imaginary phases, which are mutually independent and exponentially distributed with the common expected sojourn time lip in any of the phases. A customer on arrival passes in sequence through all the k phases before it is discharged. After a customer leaves the server, a new customer is taken up instantaneously if one is waiting in the queue; otherwise the server remains idle. As pointed out in the first chapter,

46

SOME TECHNIQUES OF QUEUING THEORY

these distributions are known as k-Erlang distributions. For more details on the meaning of phases in particular cases, see Chapter 1. Gaver (1954) modifies this technique. He assumes that instead of a finite number k of phases there are (potentially) an infinite number of them, and further that an arrival demands j phases of service with probability Cj. It has been demonstrated by Gaver (1954) and later by Luchak (1956) that it is possible to obtain (or approximate) a wide class of service time distributions of practical interest by varying {Cj}. In this connection, see also Wishart (1959). In these papers the state of the system is defined by the number of phases in the system - either waiting or being processed. laiswal (1960) assumed the same modified Erlangian service time as Gaver (1954) and Luchak (1956), but in place of their phase process used a vector process {N(t), R(t)} as a system-length process. In Jaiswal's method, which we call a modified Erlangian method, N(t) is the number of customers (not phases), and R (t) is the number of phases of service that remain to be completed by the customer currently in service at epoch t. Since R (t) is the number of phases of service that remain to be completed, it is convenient to number the phases "backward." A customer, having entered phase j (l <.j < (0) with probability Cj, then moves after an exponential service time to phase (j - 1), and so on down to phase 1, and thence out of the system. The modified Erlangian method may be divided into two submethods, with N(t) being defined as the number of customers in the system in the first submethod, and as the number of customers in the queue in the second submethod. The difference between the two submethods will be important in some bulk-service queues.

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

t is normally replaced by nand N(t) by N n . Such a chain may be explained briefly as follows. Consider a physical system which is observed at a discrete set of points 0 1 2, .... Let the successive observations be No, N l , ••• , N n , ... , where N n is' a~ LV. Further, assume that each of these variables is capable of taking the values 0, 1, 2, .... The sequence {Nn , n;;;' O} is said to form a Markov chain if for all n, n = 1, 2, ... , the conditional distribution of N n , for given values of No, N I ,··· ,Nn- I , depends only on N n - I , the most recent known value. In symbols this becomes

P{Nn = kiNo == io,N l == il, ... ,Nn - I

== P{Nn = klNn _ 1 = in-I} where i o , iI, i2 , · · . , in-I, k take the values 0,1,2, .... Th~ system i~ said to be in state k at the nth time point (or step) if N n == k. Also If N n - I = I and N n = k, the system is said to have made a transition i .... k at the nth time point. The conditional probabilities P{Nn = klNn _1 = i} are called one-step transitional probabilities. These probabilities may depend on i, k, n, and n - 1. However, if they are independent of n, the discrete-time variable then the Markov chain is said to be a homogeneous Markov chain, and in thi~ case the one-step transitional probabilities may be denoted by

Fik = P{Nn = kiN n _ 1 = i},

i, k ;;;, O.

In the case of a homogeneous Markov chain, the n-step transitional probabilities are denoted by

Imbedded Markov Chain Technique

Before we explain this technique, it is essential to know about Markov chains which are a subclass of Markov processes in which the state space is discrete (finite or countably infinite), but the time parameter may be discrete or continuous. Essentially a Markov process is a Markov chain iff the state space is discrete. A discrete-state Markov process with discrete time parameter is known as a discretetime Markov chain, or simply a discrete Markov chain. Similarly ,a discrete-state Markov process with continuous time parameter is called a continuous-time Markov chain, or simply a continuous Markov chain. For example, the bulk-arrival Poisson process discussed earlier is a continuous Markov chain. We give here a condensed account of Markov chains. For further details see Feller (1968), Kemeny and Snell (I 960), Bharucha-Reid (1960), or for a deeper study Chung (1967). The choice of state space of a Markov chain or process (that is, the set of values that N(t) may take corresponding to the values that t takes in the set T) is not uniquely specified by a physical phenomenon under consideration. However, usually one particular choice stands out as the most suitable. The same remarks apply to the time parameter. A case of immediate interest to us is the discrete-time Markov chain in which

47

Pik(n) == P{Nm+n == klN m = i}

a~~ are indepe.n.d~nt of m. It, may be noted that Pik (1) =- Fik . The one-step transItIOnal probabilities are sometimes written by means of a matrix:

P

wherePik

;;;'

[P ik ]

/k

0

0

Poo

POI

Pm

PIO

Pu

P I2

2

P20

P 21

P 22

2

O,i, k = 0,1,2, ... , and

I

k=O

Pik = 1,

0,1,2, ....

SOME TECHNIQUES OF QUEUING THEORY

48

We define for later use Pjh(O)

0ih

k

= { 1,

i =1= k.

0,

Matrices whose elements satisfy the above requirements are known as stochastic matrices. In fact, these are infinite stochastic matrices (the number of rows and columns is co). In general it is more difficult to handle such infmite stochastic matrices than their counterpart finite square stochastic matrices. A systematic study of finite stochastic matrices is given by Romanovsky (1970) whose book deals with, among other things, discrete Markov chains. See also Seneta (1973). Homogeneous Markov Chains

In what follows we consider only homogeneous Markov chains, and the adjective homogeneous will therefore be omitted. The Markov chain {Nn , n;;;;' O} is completely determined once the value of No (or more generally the distribution of No) and the one-step transition probabilities are known. A state k of the Markov chain is said to be accessible from state i, i -+ k, if there exists a number n ;;;;, 0 such that P ih (n) > O. If two states i and k are accessible from each other, then they are said to communicate, i +l- k. A Markov chain is called irreducible if all its pairs of states communicate. Consider an arbitrary, but fixed state i. Let fu(n), n ;;;;, 1, be the probability that the system starting from state i returns to i for the first time in n steps. Notationally, fu(n) = P{Nn = i,Nr =1= i,r

=

1,2, ... ,n-lINo

=

i}

with fu(O) = 1. Note also that fu(l) = Pjj. The probability of the system eventually returning to i is denoted by f/i =

l: fu(n)

n=l

so that

fii

<

1.

2.2

49

chain is defined to be the greatest common divisor (GCD) of all integers n;;" 1 such that Pjj(n) >0. When the period is unity, that is, the GCD is 1, the state is called aperiodic. Also if Pu(n) = 0, for every n;;" 1, define the period as O. Two states of the Markov chain are said to be of the same type if both have the same period or both are aperiodic; if both are transient or if both are positive recurrent, or if both are null recurrent. All states of an irreducible Markov chain are of the same type. As an example consider a Markovian queue in which the process {N(t)} is a Markov chain. In Markovian queuing problems with infinite waiting space, it is possible to show that the classification criterion depends on the traffic intensity p. If p < 1, the states are positive recurrent, if p = 1, they are null recurrent, and if p > 1, they are transient. Intuitively, this concept may be explained by considering an%example MX/G/l in which p = il.if/f.1.

where A

mean arrival rate

if

mean of arrival group size

1/f.1. = mean service time

Then p represents the expected number of customers arriving during a mean service time. Consequently, if p> 1, then on the average more customers arrive per mean service time than the server can handle, and hence we could expect the system length to grow without limit. On the other hand, if p < 1, on the average there is less than one customer arriving per service time unit, and hence the server can cope with the load of customers. In this case the process reaches a steady state. The crucial case is when p = 1. In this case the mathematical system breaks down or degenerates, in the sense that all state probabilities become zero, or equivalently the system length becomes infinite. An irreducible aperiodic Markov chain possesses a limiting distribution limn->~' peNn = k) =Ph, k;;;;' 0, which is independent of the initial distribution P(No = i), i;;" O. Two cases arise:

Classification of States of Markov Chain

A state i is said to be recurrent iff fir = 1; otherwise it is transient (also called nonrecurrent). The number of steps required for the first return to i is called the recurrence time, its expectation being the mean recurrence time f.1.u, where f.1.u = I:;;'=1 nfu(n). A recurrent state can be further classified either as positive recurrent or null recurrent, according to whether f.1.ii < 00 or f.1.u = 00. In order to study the limiting distribution of the Markov chain, we need to define periodicity of the states of the chain. The period of a state i of the Markov

TECHNIQUES AND HISTORY OF ANAL YSIS

All states are null recurrent or transient. In this case Ph = 0 for every O. As n -+ oo,Nn -+ 00 with probability l. All states are positive recurrent. In this case Ph > 0 for every k;;;;' 0, and {Ph} is a probability distribution such that Ph = (f.1.hh)-l. Thus to determine Ph we need to determine f.1./,h which may not be easily determined. However, P's may also be determined uniquely as the solution of the following system of linear equations:

k;;"

2

Ph =

l: PiPih , i=O

1.

SOME TECHNIQUES OF QUEUING THEORY

50

Equivalently any nonnull solution of the system of linear equations with

LXjPjh = Xh i=O

when normalized gives {Ph }, that is,

Clearly the case of interest is case 2. Since the Markov chain possesses a limiting distribution {Pj }, it possesses a stationary distribution. A Markov chain is said to possess a stationary distribution {Ph} if =

Ph =

L PjPjh ,

Vk> O.

(2.2.5)

j=O

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

Thus to study the process {N(t)} of the queuing system MIG 11, let 01, 02 , . . . , an, ... be the epochs of departure of successive customers. If there is a departure at time t = 0, we may put 00 = 0 and call it an initial departure epoch. Let V"' n = 1, 2, ... , be the successive service times which are Li.d. r.v.'s with dJ. B(v) = P(V" ~ v). The sequence {Vn} is independent of the arrival process, which is Poisson with rate A. The service initiation is instantaneous as long as there are customers to be served. After an idle period during which there are no customers remaining to be served, neither in the queue nor with the server, let us suppose that the next count (increase in the index n) on the process occurs at the end of the service interval that begins with the first subsequent arrival. Define N:; =N(on + 0), that is, N:; is the number in the system immediately after the nth departure or (if N~ > 0) just before the service on the (n + I)th customer starts. If there is a departure at 00 = 0, write N; = !i(oo + 0). The stochastic process {N:;} is said to be imbedded in the continuous-time process {N(t)}. The process {N:;} forms a homogeneous Markov chain, as can be seen from the relation

If equation (2.2.5) holds, then

N:;+I = (N:; =

Ph = L PiPj/,(n),

"In.

i=O

It + Xn+1

where (xt = max (O,x) and Xn+1 is the number of customers arnvmg during a service period ending at 0"+1' The probability distribution of Xn+1 is given by

Consequently, if the initial unconditional distribution is P{No = i}=P;, then, for every n, the unconditional distribution

In other words, a Markov chain that starts in steady state will remain in steady state. In closing it may be mentioned that positive recurrent aperiodic states are called ergodic states. We have stated several important results of the theory of Markov chains, for which proofs are given in some of the references cited earlier. We are now ready to explain the concept of imbedded Markov chain (IMC). This can be done by means of two examples. As a first example, let us consider the non-Markovian system-length process {N(t)} in the queuing system MIGII. This process may be studied by considering the Markovian vector process {N(t), X(t)}, where N(t) represents the number in the system and X(t) the unexpended (or alternatively the expended) service time of the customer currently undergoing service. In the relatively simple M/G/l case we can give a full solution in continuous time for the vector process {NCt), X(t)}, as we do later (in steady state) by means of the supplementary variable technique. Alternatively, we may obtain useful information about the MIG/I queue with less effort by studying the process N(t) at a suitably chosen discrete set of time points. The discrete-time solution can indeed be used as an approximation to the continuous-time solution. For a more complex model, such as the GI/M/c queue, an exact continuous-time solution may be excessively difficult, and a discrete-time solution may therefore provide the most practical alternative.

51

t=P(Xn+1 = jlservice period

v) dB(v)

j> O.

Here we have used the independence of the arrival process and the service times, and the fact that in a Poisson process the number of arrivals during an interval of time of length v, the service period, depends only on AV. Also because of the independent increments of the Poisson process, the X's are i.i.d. r.v.'s. Note also that Xn is independent of N;,Nt, ... ,N:;. The one-step transition probabilities i,j

>

0,

n

>

I

are given by

Pij =

r;" kj -

0,

i+ l ,

0,

j > 0

i> 0,

j> i - I

j
i> 0

As a second example, consider the queuing system GIIMII, a single-server queuing system in which the interarrival times of successive customers are LLd. LV.'S and the service times are independently distributed exponential r.v.'s

SOME TECHNIQUES OF QUEUING THEORY

52

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

53

with mean 1//.1. If N(t) is the number 'in the system at any time, then let N;; = N( a~ - 0), where a~ = 0, and a;, a;, a~, . , . are the epochs of arrivals of su~­

Integral Equation Technique

cessive customers, The notation implies that N;; is the number of customers the system just before the arrival of the nth customer, and it can be seen that

This technique has been used for finding the waiting time (in queue) distribution for queues with general input and general service time distributions and may best be explained by first conSidering the queuing system GIIG/l and then its modified form. Later this technique will be used for some bulk queuing systems. Suppose that customers arrive at the instants 0 = a~, a~, a;, ... so that the interarrival times Un = a~+ 1 - a~, n = 0, 1,2, ... are i.i.d. LV.'S with common dJ, A(u) and finite mean. The customers are served individually by a single server on an FCFS basis. If Vn is the service time of the nth customer to be served, then it is assumed that {Vn; n = 1, 2, ... } is a sequence of positive LV,'S with common d.f. B(v). The two sequences {Vn } and {Un} need not be independent, but are often taken to be so for practical applications, and we shall assume that this latter condition holds. Define V~n) = V(a~ - 0) so that V~n) is the waiting time (in queue) of the (n + 1)th arrival (since first arrival is at a~ = 0). It can be seen that the following recurrence relation between the r. v.'s v~n + 1) and v~n) for the queuing system GIIGI! holds:

N;;+l = (N;;+ I-D n

t

where Dn represents the number of potential departures during the (n interarrival period. If {Dn} are Li.d. r.v.'s, their distribution is given by kj

=

fo= P(Dn

=

III

jlinterarrival time

=

+ l)th

u) dA(u)

j ;;;. 0

where A(u) is the dJ, of inter arrival times. Now the process {N;;}, which is imbedded in the continuous-time process {N(t)}, is a homogeneous Markov chain which has the following one-step transition probabilities: k i- j + 1 , Pij =

<

i ;;;. 0,

0

j = 0,

i;;;'

v(n) q

V(n+l) q

j ,s;; i+ 1

{

+S n,

+ Sn if v~n) + Sn if v~n)

0,

;;;. 0

,s;; 0

.

(

!

1- j=O L k·J,

0,

where Sn = V n + 1

The queuing systems M/G/! and GIIM/1, with bulk arrival or bulk service, will be studied in Chapters 3 and 4. One might think that in statistical equilibrium the limiting behavior of N(t) when t -+ 00 or of N;; (or N;';) when n -+ 00 should be the same, However, except in certain special cases, this is not so, as will be seen in the chapters that follow. The imbedded Markov chain technique is due to Kendall (1951, 1953), who gives a more precise definition based on the concept of regeneration points due to Palm (I 943). Later this technique was widely accepted and applied to various queuing systems. We may explain intuitively the concept of regeneration process, which is more general than that of a process having an imbedded Markov chain. Points on a time scale at which a stochastic process restarts itself probabilistically are called regeneration points, In other words, if the continuation of a stochastic process {X(t)} beyond a point Tl , which exists with probability 1, is a probabilistic replica of the process starting at 0, so that {X(t + T 1 )} has the same stochastic structure as {X(t)}, then the process is called a regenerative process. The points such as T 1 , T2 , T 3 , ••• are called regeneration points (the existence of the points T2 , T 3 , ••• is implied by the definition). As an example, note that a GIIG/l queue is a regenerative process with regeneration points at the start of each busy cy~le. A consequence of this is that if we can solve a queuing model between succeSSIve regeneration points, we can consider that we have a full solution.

-

Un + l

•

It may be remarked that when v~n) + Sn < 0, the (n + 2)th customer will arrive to find the server idle after service on the (n + 1)th customer has been completed. The above relation may be rewritten in a compact form: v~n+ I) = (v~n)

+ snt

where (xt = max (O,x),

-

00

<x <

00.

We note that the stochastic process {v~n)} is a discrete-time Markov process, since the behavior of v~n+ 1) is dependent on v~n) but is otherwise independent of the past history of waiting time. By virtue of the assumptions we have made about the interarrival time and service time distribUtions, the S n'3 are Li.d. LV.'S such that E(Sn) is finite. Also Sn and v~n) are independent. Now using probability reasonings, we can see that w;(n+I)(t) = p(V~n+l) ,s;; t) = p(v~n+!) =

0) + P(O

= p(v~n)

,s;; 0) + P(O

=

,s;; t).

+ Sn p(v~n) + Sn

<

v~n+!) ,s;; t)

<

v~n)

+ Sn

,s;; t)

SOME TECHNIQUES OF QUEUING THEORY

54

By use of the convolution formula' and the independence of v~n) and Sn we

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

equation (2.2.7) so that it has values for all t. Let

5!

W; (t) be defined by the equation'

get

w~(n+ l)(t) =

tOO w~(n)(t -

where Sn(x)

x)dSn(x),

t;;;'O

(2.2.6)

= P(S" ';;;x).

Undley (1952) has shown that lim n -+ oo W;;-(n)(t) = W~(t) exists and is a unique proper distribution iff p < 1, as one should expect intuitively. For the proof of existence of the limit, one may altematively see Prabhu (1965a) or Karlin and Taylor (1975). Using W;Ct) as the common limit of w~(n)(t) and w~(n+ 1)Ct) as n -+ "", we get W;(t) =

r~ W;(t -

x)dS(x),

-oo
(2.2.10:

Note that equation (2.2.10) is defined for the whole real line, whereas equation (2.2.7) is defined only for nonnegative values of t. We discuss equation (2.2.10; further as follows. First (when t < 0) we see that W;(t) = A

Substituting t - x = u and x

(2.2.7)

f~=_ocf:oWq(t-x)e-I\ZdB(X+Z)dZ. +z =

W;(t) = r...el\t

v, we get

f=v=o f~u=o W (u)e-l\(u+V) dB(v)"du q

or equivalently

W~(t)

= -

fo~W~(X)dS(t-X),

(2.2.8)

where

where Sex) is the common distribution of {SnCx)} and is given by Sex)

= {'''B(X+Z)dA(Z).

= a constant> 0,

(2.2.9)

It should be noted that equation (2.2.9) can be written in many forms. In practice, the integral is to be evaluated so that the inequalities Z ;;;, 0 and x + Z ;;;, 0

are satisfied. The integral equation (2.2.7) [or (2.2.8)] is of the Wiener-Hopf type, which requires sophisticated mathematical tools for a solution in its most general form. For methods of obtaining particular solutions in certain special cases, see for example Feller (1971), Prabhu (I965a), Gross and Harris (1974), and Kleinrock (1975). We give below Undley's solution for the M/G/1 queue. It may, however, be remarked here that this approach does not yield practical solutions to all GI/G/1 queuing systems. [In this connection, see also Smith (1953)J. The System M/G/1

We now employ Undley's approach to obtain the waiting-time distribution of the queuing system M/G/I. Let the d.f. 's of the service time V and the interarrival time U be B(x) andA(x), respectively. Both B(x) andA(x) vanish for x < O. Now using equation (2.2.9), Sex) = A fB(x

+ z)e-I\z dz.

Our interest now is to solve the integral equation (2.2.7) for t;;;' O. But the integral on the right-hand side of equation (2.2.7) does not have a value for t';;; O. In order to get around this difficulty, we redeflne the integral on the right-hand side of

and we have used B(t) = 0 for t < O. Second, t;;;'

O.

Finally, we can rewrite equation (2.2.10) as follows: t
(2.2.11 ) At t = 0 the limiting value of W~(t) is Wq(O +) when approached from the right (through positive values of t) and C when approached from the left (through negative values of t). It can be shown that C = Wq(O+), as shown by, for example, Kleinrock (1975). The solution is now obtained by taking the two-sided L.-S.T. of equations (2.2.10) and (2.2.11). Using equation (2.2.11), we have

L:

e-CY.t dW;(t) =

r~e-CY.tAcel\t dt -

Wq(O+)

+

fo:e-CY.t dWq(t).

(2.2.12)

The two-sided L.-S.T. of the right-hand side of equation (2.2.10) is wq(a)s(a), which is the product of the individual transforms of Wq(t) and Set), namely, and

* When the arrivals follow a Poisson process, Wq(t) is the same as Wq(t). Hence we shall not distinguish between them and instead write Wq(t). For details on this remark, see Section 2.1, Miscellaneous Problem 4 of Chapter 3, and so on.

54

SOME TECHNIQUES OF QUEUING THEORY

By use of the convolution formula and the independence of v~n) and Sn we

2.2

TECHNIQUES AND HISTORY OF ANAL YSIS

55

equation (2.2.7) so that it has values for all t. Let W;(t) be defined by the equation'"

get w,;-(n+ OCt) = where

fo= w,;-(n)(t -

x)dSnex),

t;;;;'

0

(2.2.6)

Sn(x) = P(Sn <;;;x).

Undley (1952) has shown that lim n .... = w;(n>(t) = W';-(t) exists and is a unique proper distribution iff p < I, as one should expect intuitively. For the proof of existence of the limit, one may alternatively see Prabhu (l965a) or Karlin and Taylor (1975). Using W;(t) as the common limit of W;
W;(t) =

(2.2.8)

Substituting t - x

= u and x + z = v, we get

W;(t) = Ae At L:o Ce At ,

where

where Sex) is the common distribution of {Sn(x)} and is given by Sex) = Jo=B(X+Z)dA(Z).

= (2.2.9)

It should be noted that equation (2.2.9) can be written in many forms. In practice, the integral is to be evaluated so that the inequalities z ;;;;. 0 and x + z ;;;;. 0 are sa tisfi ed. The integral equation (2.2.7) [or (2.2.8)1 is of the Wiener-Hopf type, which requires sophisticated mathematical tools for a solution in its most general form. For methods of obtaining particular solutions in certain special cases, see for example Feller (1971), Prabhu (1965a), Gross and Harris (1974), and Kleinrock (1975). We give below Undley's solution for the MIGII queue. It may, however, be remarked here that this approach does not yield practical solutions to all GIIGII queuing systems. [In this connection, see also Smith (1953)] .

The System MIG/1

We now employ Undley's approach to obtain the waiting-time distribution of the queuing system MIG/I. Let the d.f.'s of the service time V and the interarrival time U be B(x) and A(x), respectively. Both B(x) and A(x) vanish for x < O. Now using equation (2.2.9),

(2.2.10)

W;(t) = A D=-= f=owq(t-x)e- AZ dB(x +z)dz.

or equivalently

fo~W;(X)dS(t-X),

-00< t
Note that equation (2.2.10) is defined for the whole real line, whereas equation (2.2.7) is defined only for nonnegative values of t. We discuss equation (2.2.1 0) further as follows. First (when t < 0) we see that

(2.2.7)

W;(t) = -

f=Wq(t-X)dS(X),

[=0 Wq(u)e-A(u+V) dB(v)'tiu t
a constant> 0,

and we have used B(t) = 0 for t < O. Second, Finally, we can rewrite equation (2.2.10) as follows:

t
L:

e-O'.t dW;(t) =

r~e-O'.tAceAt dt -

Wq(O+)

+

50:e-O'.t dWq{t).

(2.2.12)

The two-sided L.-S.T. of the right-hand side of equation (2.2.10) is wq(a)s(a), which is the product of the individual transforms of Wq(t) and Set), namely, and

Our interest now is to solve the integral equation (2.2.7) for t;;;;' O. But the integral on the right-hand side of equation (2.2.7) does not have a value for t <;;; O. In order to get around this difficulty, we redefme the integral on the right-hand side of

* When the arrivals follow a Poisson process, Wq(t) is the same as Wq(t). Hence we shall not distinguish between them and instead write Wq(t). For details on this remark, see Section 2.1, Miscellaneous Problem 4 of Chapter 3, and so on.

SOME TECHNIQUES OF QUEUING THEORY

56

C(_A_)_ C+ wq(a) A-a

W;(t) =

A/(A - a),

wq(a) = C [1

TECHNIQUES AND HISTORY OF ANAL YSIS

(2.2.13)

+ A 1 =:(a)r

!:ooW;(t-X)dS(X)- [I-D(t)]

W;(t)

so that

In this case the following recurrence relations between the r.v.'s

V~n+l) and v~n)

and Tn can be seen to hold: V(n) +S {

q

n'

Tn+l ,

V~n) +Sn

>0

v~n) +Sn

';;;;0.

The sequence {Sn} is independent of the sequence {Tn}· Since Van) > 0, for every n, p(v~n) ,;;;; t) = 0 for t < 0. For t > 0, we find from the above relation W;(n+l)(t) = p(v~n+l),;;;; t) = p(v~n)

+ Sn';;;; t, Sn';;;; t) -

p(v~n)

+ Sn';;;; 0, Tn+1 > t),

Since Tn+ 1 is independent of Sr and V~r) for r = 1, 2, ... ,n, we obtain

t
t>O.

t>o.

W;(-x)dS(x),

t>o t
Sex) =

The result (2.2.13) has been obtained by several methods by many authors, and is now popularly known as the Pollaczek-Khintchine transform formula. Let us now consider a generalization of lindley's results to the queuing system GJ/G/l with the following modifications. In GJ/G/l it is assumed that if the nth customer arrives to find the server idle, service begins instantaneously. Now relax this restriction and assume that when the nth customer arrives to find the server idle, his service commences after a random time Tn' where the {Tn}'s are Li.d. with mean d and d.f.

foo

( 0,

where C = 1 - E(V) = 1 - p. E(U)

v~n+1) =

lim p(v~n),;;;; t) oo n-+

exists, is a proper d.f. (ifp < 1) , and is the unique solution of the integral equation

The constant C is found by using the normalizing condition E(U) = A-I and - E(V) = lim"' .... o [- 1 + b(a)] /a, we have

D(x) = P(Tn ';;;;x)

57

Note that this is a generalization of the result (2.2.6). Following lindley (1952), Finch (1959) has shown that for the queuing system under consideration the limiting waiting-time (in queue) d.f., namely,

Equation (2.2.12) then gives

and thus, since sCa) = E(a)

2.2

r

B(x

+ y) dA(y).

Supplementary Variable Technique

This technique was used in 1942 by Kosten [see Kosten (1973)]. The name appears to be due to Cox (1955), who used a supplementary variable to study the queuing system M/G/I. Kendall (1953) considered this technique, which he labeled "augmentation", but preferred the use of an imbedded Markov chain as leading to simpler calculations. In spite of this, supplementary variables have been used by many authors to solve a good number of queuing problems. Jaiswal's (1968) book on priority queues makes heavy use of supplementary variables. While Kendall's imbedded Markov chain technique is very powerful and elegant, it gives only approximate solutions to queuing problems considered in continuous time. However, in the steady state the apprOximation seems to be pretty good, and the mathematics involved become less cumbersome. Even in the steady-state case, many problems are more readily treated by the supplementary variable technique than by the imbedded Markov chain technique. See, for example, Regis (1973). Waiting-time distributions in queues with Poisson input are easily obtained by the supplementary variable technique. For details, see the discussion of waiting-time distribution for the system MiCE/I in Chapter 4, for the system MX/G/I in Chapter 3, and so on. In the supplementary variable technique a nonMarkovian process in continuous time is made Markovian by the inclusion of one or more supplementary variables. To illustrate this point, first consider the queuing system M/G/! studied by Cox (1955). In M/G/1, since the service-time distribution is general, let the state of the system be defined by a pair of variables, the number N(t) in the system [or Nq(t) in the queue] at epoch t, and the elapsed service time X(t) of the customer who is undergoing service. Then we may study the bivariate Markovian process {N(t), X(t)} in order to obtain results for the non-Markovian process {N(t)}. Thus Cox defines Pn(v, t) to be the joint probability and p.d.f. of n, the number of customers in the system, induding the one being served, and v the elapsed service time of the customer in service. The inclusion of a single supplementary variable makes the process Markovian in continuous time. The equations of the process can be written down by considering, in the usual way, the transitions occurring in time I::..t. In this

SOME TECHNIQUES OF QUEUING THEORY

58

case fI(V) dv is taken as the probability that the service is completed in the interval ] v, v + dV] , conditional upon its remaining incomplete up to time v. We illustrate the use of this technique by discussing the limiting behavior of the queuing system MIG/I. In Chapter 3 we use this technique to solve a typical bulk-arrival priority queuing system and in Chapter 4 the system MIGBII and its generalizations are considered using the same technique. In MIG/I customers arrive following a Poisson process with mean rate A and are served one at a time on an FCFS basis. Service times are Li.d. r.v.'s whose p.d.f. (see Chapter 1, Problems and Complements) is

2.2

TECHNIQUES AND HISTORY OF ANAL YSIS

59

Proceeding to the limit when !::"t -+ 0, gives

An equilibrium solution exists when p = AlIl < 1. In this case we can put the limiting derivatives with respect to t equal to zero; and by letting limt->= Pn(x, t) == Pn(x) and lim t -> = poet) ==Po, we get

o=

-APo

+

r

PICX)r!{X) dx

(2.2.15)

b(v) = fI(V) exp [- ( fI(X) dxl n;;;'l.

fo

with mean 1/11,0 < 1/11 = vb (v) dv< =. Also note that we have shown that the corresponding d.f. is given by exp

r-

f:

These equations are to be solved under the so-called boundary conditrons

n>l

(2.2.14)

flex) dX] = I -B(v)

a result we shall use in the sequel. Let us define:

2

Since {N(t), X(t)} is Markovian in continuous time, one can write the equations of the process in the usual Erlangian procedure by considering the transitions occurring in !::"t. First we may note thatPo(x, t)==O or in the equilibrium case Po (x) ==0. Then considering the empty state,

+ !::"t)

= Po(t)(l - A!::"t)

+ (l

- t-..!::"t)

(2.2.17) (2.2.18)

Pn(X, t) dx + o(dx), n > 0 as the probability that at time t there are n customers in the system with the elapsed service time of the customer undergoing service lying between x and x + dx. poet) as the probability that the system is empty at time t.

Po(t

(2.2.16)

r

and the normalizing condition Po

+

1: r Pn(x) dx = l.

The solution now can be obtained in closed form with the help of a p.g.f., Po(z; x), which we define as = Po(z;x) = I Pn(x)zn. (2.2.20) n=l

Then

PI (x, t)f/(x) dx!::"t + o (!::"t),

apo(z;x)

ax

so that

(2.2.19)

n=! 0

f n=l

apn(x) zn

(2.2.21)

ax

and Po(Z; 0) =

2: PnCO)zn.

(2.2.22)

n=l

Similarly, we get We multiply equation (2.2.16) by zn, add from n = 1 to =, use equations (2.2.20) and (2.2.21), and get

Pn(x+!::"t,t+!::"t) = (l-t-..!::"t)Pn{x,t)[l-fl(X)!::"t] + t-..!::,.tPn - 1 (x, tHI -fl(X)!::"t]

or Pn(X

+ !::"t, t + !::"t) -

Pn{x

+ o (!::"t)

+ !::"t, t) + Pn(x + !::"t, t) -Pnex, t)

!::"t

= - [A + fI{X)] Pn(x, t)

!::"t

+ APn -

oC!::"t) 1 (x, t) + - - . !::"t

apo(z;X)

ax

=

[AZ - A- fI(X)] Po(z;x).

(2.2.23)

Similarly from equations (2.2.17), (2.2.18), (2.2.22), and (2.2.15) we get zPo(z; 0) = fo= PoCZ;X)fI(X) dx

+ AZ(Z -

I)Po.

(2.2.24)

SOME TECHNIQUES OF QUEUING THEORY

60

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

k n = Pen arrive during V)

The differential equation (2.2.23) on solution gives

Po(z;x) = A exp

l

= foOO Pen arrive during

S:[A(Z -1) -1{(Y)] dY ]

where we have used equation (2.2.14) and A is a constant to be determined from the boundary conditions (not from the initial conditions, since the effect of the initial conditions (at t = 0) in ergodic queuing systems in equilibrium (in the limit as t -+ 00) is worn out]. The boundary condition, when x = 0, gives A = Po(z; 0). Consequently, (2.2.24a) Po(Z;x) = Po(z; 0)[1 - B(x)] exp [A(Z -l)x]. Substituting this in equation (2.2.24) and simplifying, we get

AZ(Z - l)P o Po(z; 0) = Z - b(A - AZ)

e-C>lV

Reu>O.

00

I knz n n=O

K(z) ==

The reader will observe in other chapters that the inversion of p.g.f.'s for the more complex queuing systems to be discussed later is not easy. However, it is possible to invert P(z) or P\z) given by equation (2.2.25) or (2.2.27), respectively. We now give a brief discussion of a rather subtle technique for the inversion of equation (2.2.25). Write equation (2.2.25) as I-b(A-Az) 1_ Z

i=1

= Po(z, 0) foo e- A( l - Z)X [1- B(x)] dx

AZ).

i=1

Using the theory of the L.T. to express b(t-. - t-.z) as an integral, we get

0

.IPiZ i = A .IPjzj fooe-A(I-Z)VP_B(V)] dv

b(A -AZ) -1

= zPo Z -b(A -AZ)

,=1

J=1

+ APoZ

where we have used oo

_

+ POb(A -

I PiZ i = A( f: PiZ i) 1-A(1-Z) b(t-. - t-.z) + t-.Poz 1- b(t-. - t-.z). A(l -z)

Po(Z) = foo Po(z;x) dx o

fo

(2.2.26)

= b(A - AZ).

This can be rearranged, after separating the term Po, to give

Thus

,

b(x) dx

n!

P(z) = P(z) b(v) dv,

= x)b(x) dx

we get

where b(u) is the L.T. of b(v), that is,

r

VIV

oo e-AX(Ax)n

= fo

= A exp [A(Z -l)x] [l-B(x)]

b(u) =

61

e- cw [l - B(v)] dv =

1 - b(u)

u

f:

0

e- A(1-Z)V [l - B(V)] dv.

Expanding e- A (1 -z)v and picking out the coefficient of Zi on each side of the equation, we finally get

.

The complete p.g.f. of Pn' n = 0, 1, 2, ... , is P(z), given by

P(z) == Po(z) + Po

(z -l)b(A - AZ)PO Z -b(A - AZ)

(2.2.25)

The value of Po, which may be determined by pel -) = 1 or by considering the idle time of the server, is given by Po = I - p. This completely determines the p.g.f. P(z). It is interesting to note that b(A - AZ) can be interpreted as the p.g.f. of the number of arrivals during a service time V of a customer. For, letting

+ APo

f

oo e-AV(Av)i-l (i -I)! [1 - B(v)] dv.

0

Relation between P~ and Pn

Now we show that if P~ represents the probability of having n customers in the system (in equilibrium) immediately after the departure of a customer, then P~ = Pn • To show this, one can write

SOME TECHNIQUES OF QUEUING THEORY

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

63

62

on such relations when the arrivals or services are in bulk, see Chapters 3, 4, and 6.

n;;;"O

where D is a normalizing constant to be determined later. The p.g.f. of P~ is then given by 00

P+(z) =

I

n=O

p~zn =

(1 - p)(z -nb(i\ - i\z) z-b("A-"Az)

(2.2.27)

The result (2.2.27) is the same as the result (2.2.25) and ~stablishes the assertion p~ = p n' To prove equation (2.2.27), we proceed as follows. P+(z) =

~ S:n~/n+l(X)1J(x)zn+l

= -D

z

=

dx

i"" Po(z;x)1J(x) dx 0

!!. Po(Z; 0)5"" exp [i\(z -l)x] b(x) dx z

0

The Busy-Period and the Idle-Period Distributions for the MIG/1 Queue

The busy-period distribution for the queuing system MIG/! could be derived by continuing the discussion of the system MIGII through the supplementary variable technique [see, for example,Jaiswai (1968) or Problems and Complements] _However, the following excellent method due to Takacs (1962) is more elegant. Since the busy period is independent of the order of service (the waiting-time distribution is not), we may assume LCFS queue discipline. Let T be an LV. representing the busy period of an MIGI1 queuing system, which starts with the arrival of a customer. Let G(t) be the distribution function of T, and let V denote the service time of the arrival with its dJ. B(v). Then sinoo each arrival during the initial service time of the busy period will generate its own busy period, we may write T = V + T j + T2 + ... + TN, where T i , i = 1, 2, .. _, N, is the r.v. representing the busy period generated by the ith arrival that arrives during V, and N is a LV. representing the number of arrivals that arrive during V. Now as T/s may be supposed to be independent of N and V and also to be independent LV.'S identically distributed as T, we have by conditioning on V and N

i\D(z - 1) b["A(1 - z)] Po z-b("A-"Az)

Get) =

The constant "ADP0 is obtained by the normalizing condi:ion P+(~ -) = 1, gi)~g "ADP = 1 - p. Thus P+(z) is completely known as state.d m equatlOn (2.2.2Jn· u~ o may be noted that the constant D is really 1/"A, the recIprocal of the mean P

rf

o n=O

t ""

I P(T Jo n=O

I ft Since we are considering a queue with FCFS queue disci~line, t~e c~~~o~;: ~e g behind by a departing customer are just those that ar~ve dur:n d customer was delayed in the system (queue + service). Smce arnvals are ran om,

Waiting-Time (in Queue) Distribution for the MIG/1 Queue

P+(z) = wq("A - "Az)b("A - "Az)

where W (ex) is the L.T. of the distribution of queuing time. We have use~ t~e 'ndepend~nce of the service time and the queuing time of the customer , an t~ e . . (2 2 26) Using the trans f orma lOn argument used in the derivatlOn of equatlOn : . . "A - "Az = ex and equation (2.2.27), we finally obtam

l

_ (l-P)ex Wq(ex) = ex-"A[1-b(ex)]

=

I

+ T2 + ... + TN

= v)'P(N = nlV = v) dB(v)

n, V

<; tiN =

rate "A.

it follows that

P(V + Tl

+ T2 + ... + Tn <;t-v)

f: nt e-"V~~v)n

e-A"("Av)n

dB (v)

n!

{G(t-vW' dB(v)

where {G(t)}n* is the n-fold convolution of G(t) with itself in which {G(t - v)}o* = v) corresponds to no arrival during the initial service time of the busy period. If i(a) and b(ex) are the L.-S.T.s of G(t) and B(v), respectively, then by taking the transform of the above equation, we find that

o(t -

i(ex) =

f

00

t=O

5 t

,,=0

L e-OI.t e-A"(i\v)n ! {G(t n. 00

vw* dB(v) dt

n=O

(2.2.28)

which is the L.T. of the distribution of the actual waitingyme (in .que~e). Because of Poisson input, the virtual waiting time distribution (dIscussed m Miscellaneous Problems and Complements) would be the same. The result p+ = P n was first proved for MIG/! by Khintchine (1932) and lat~r by several othe; authors. The derivation given here appears to be new. For details

where we have interchanged limits. Now making the transformation t then using the convolution property, we get

V =

Y and

SOME TECHNIQUES OF QUEUING THEORY

64

S; exp [- (A + ac -

Ai(ac)) v] dB(v)

b[ac + A- Ai(ac)]. This is a functional equation, inversion of which is difficult except in the simple case G -= M. For more details regarding· its inversion, one may refer to Takacs (1962). However, one can easily obtain moments from it. For example, the mean length of the busy period is

2.2

65

TECHNIQUES AND HISTORY OF ANALYSIS

d.f. 1 - e-"Ax, x;;' 0, with mean E(1) = I/A, where / is the length of the idle period. In view of this the mean length of the busy-period distribution for the queuing system MIG/1 can also be easily obtained without the use of the transform. For details, see Section 2.3. Next consider the system G/IM/l. Instead of service time being general as in MIG/l, we now have the assumption of general independent input. In this case the supplementary variable yet), which measures the time since the last arrival, removes the non-Markovian aspect of the system, and the system is solved by means of the backward Kolmogorov equations. Readers not familiar with renewal theory will understand the discussion below better if they first read the section on renewal theory.

E(T) = - g(l)(o:)lo:=o = -

=

Jj(l)

[0: + A - AiCo:)] • [1 - Ai(l)(o:)] 10:=0

~ [l + AE(T)] Jl

which gives

1 Jl-A'

E(T) = (

00,

ifp =

A

-< 1 /l

if p ;;. 1.

From this one can infer that (1) if p < 1, the busy period terminates with probability 1 and has a finite mean; (2) if p = 1, the busy period terminates with probability 1, but has infinite mean; (3) if p > 1, the busy period may not terminate at all. This corresponds to positive recurrent, null recurrent, and transient cases in

The System GI/M/1

AIl the assumptions in the discussion of the queuing system G/IMll are the same as those in the syst,em MIG/1, except that interarrival times are now arbitrarily and indepen~entl~ distribu~ed with d_f. A(t), p.d.f. aCt), t;;' 0, and I/A = f'; taCt) dt, and semce times are mdependently exponentially distributed with p.d.f. pe- Ilt , p >0, t;;;' O. Since the limiting behavior of the system does not depend on initial conditions, ~e .may formulate the model with any convenient initial conditions before taking lirmts as t -+ 00. We choose the simple initial conditions N(O -) 0 and N(O) = 1. Thus t = ~ is an epoch at which a customer joins an empty system and immediately enters serVIce. Throughout our discussion, it will be assumed that p = AlP < lLet us now define the following probabilities and probability densities:

=

Markov chain analysis. Although we have considered the busy period starting with the arrival of a single customer, the busy period may also start with the arrival of a group of size r> 1 (r fixed) of customers. Then one may conceive each customer of the group generating its own busy period in an independent way, and consequently one may write the L.-S.T. of the distribution of the busy period &(0:), generated by a group of

Pn(t,y) dy

= P{N(t) = n and last customer arrived in

[t-y-dy,t-y]IN(O-)

size r, as with il (0:) -=i(ac). We may also interpret ir(O:) as the transform of the duration of the busy period initiated by the existing r customers in the queue, which is called the initial busy period. If arrivals are single, the other busy periods will have transforms il (ac) or

2

i(ac), as discussed earlier. It is an easy matter to find the idle-period distribution for the queuing system

MIG/l, for when the busy period terminates, an idle period must start, which in turn terminates as soon as the next customer arrives. In the language of renewal theory (see later), an idle period may be thought of as the residual interarrival time. Thus using the characteristic property of the exponential, the idle period has

+ o (dy)

3

= O,N(O) =

I},

n ;;. O.

In words, Pn(t,y) is the joint probability and the probability density for the system state in which there are n customers in the system [one in service and (n -1) in the queue], the last customer having arrived at (t - y). Related to the above probability is the probability Pn(t, 0) dt + o (dt), which is associated with the arrival of a customer in the interval] t - dt, t 1. The joint probability and probability density Pn(t, 0), which is denoted by P~(t), has meaning only for n ;;. 1, for it refers to the case when there are n in the system after an arrival. Oearly then Pg(t) == O. P~(t) is the rate at which customers join the system in state (n -1).

Pn(t)

= P{N(t) = nIN(O -) = O,N(O) =

1}.

In words, Pn(t) is the probability that there are n in the system, whatever may be the time of arrival of the last customer.

SOME TECHNIQUES OF QUEUING THEORY

66

The inclusion of the single supplementary variable Y = yet), which is measured backward from t to the epoch of arrival of the last customer, makes the queuing process {N(t), yet)} for the GI/M/l queue Markovian in continuous time. Thus one can write the equations of the process by using the backward Kolmogorov equations rather than the forward ones, which were used while discussing the process M/G/I. First we investigate P~(t), n > 1, and its limiting behavior before discussing the other probabilities. Let =

pact) =

L

= aCt) + S>aU-y)a(y) dy,

t>O.

(2.2.32)

=

L p~ n=1

Equation (2.2.32) is called the characteristic or operator equation. Consequently, if 1';, i = 1, 2, 3, ... , are the roots of equation (2.2.32), then the elementary theory of difference equations (see Appendix A.9) gives the solution ~ p~ =

(2.2.29)

Note that equation (2.2.29) is simply the ordinary renewal equation given later as equation (2.3.9a) with met) = pact). We may therefore appeal to renewal theory for a rigorous proof of the intuitively reasonable result

t-+ oo

67

interval after the arrival in (1) is of length y; and (3) m customers depart during the interval y, with a customer in service at the end of the interval y. This result summed over all m = 0, 1, 2, ... and any y, 0 < y < t, gives the reqUired pro babili ty and probability density P~(t) of the left-hand side. We now proceed to solve the set of equations (2.2.31a). Replacing p~ by zn in equation (2.2.31a), we get z = fo= a(y)e-/l(l -z)y dy

P~(t).

pOet) then represents the renewal density related to the event of an arrival of a customer in the interval] t - df, fl. paCt) is related to the interarrival density aCt) through the integral equation

lim paCt) = pa =

TECHNIQUES AND HISTORY OF ANAL YSIS

= a[fJ.(l -z)].

n=1

pact)

2.2

(2.2.30)

where I/A = f';u dA(u). We now proceed to discuss the probabilities p~ = limt-> = P~(t) for n > 2. The integro.difference equations for these probabilities are given (for any t) by

L; Aar,

(2.2.33)

n>l.

i

Equation (2.2.33) is the solution of equation (2.2.3la) which holds for n > 2, but we have assumed that equation (2.2.33) holds for n = 1 as well. The reason for this assumption will become apparent later. Since by equation (2.2.30) 2:;;'=1 P~ must be a convergent series, only those 1'; for which Ir;l < 1 can be included on the right-hand side of equation (2.2.34). Now using Rouche's theorem (Appendix A.5), it can be seen that equation (2.2.32) has only one root inside the unit circle Izl = 1 when p = A/J.I. < 1. Let this root be 1'1' Consequently we may write equation (2.2.33) as

Using equation (2.2.30), we finally write P~ as p~ = A(l-1'lhr- l ,

n>l,

l1'd<1.

(2.2.34)

P~

(2.2.31)

or in the limit as t

-+

00,

n>2.

(2.2.31 a)

given in equation (2.2.34) gives us the limiting probabilities just after an arrival. It may be stated that p~ is an arrival rate and not a true distribution, as 2: P~ is A. From equation (2.2.34) one can, of course, get the true distribution of the number just before an arrival. Number in System just Before Arrival, P;:' We now proceed to find the distribution of the number in the system GI/M/l just before an arrival epoch (imbedded Markov chain). Since P;:'

Since the first and the second equations of (2.2.31) may be explained similarly; we choose to explain the second. Its left-hand side is the probability that the state of the system has just increased to n9" 3) by an arrival at epoch t. The term P~_I +m(t - y)a(y)(fJ.y)me-WY/m ! on the right-hand side is the joint probability and probability density which is obtained by considering three cases: (1) the state of the system increases to (n -1 + m) at (t - y) by an arrival: (2) the interarrival

= DP~+I'

where D is a normaliZing constant, we have, using 2:;;'=0 P;:' = 1, n ;;;'0.

(2.2.35)

Equation (2.2.35) gives the distribution of the number in the system just before an arrival epoch, and this can easily be used to get the waiting-time distribution.

SOME TECHNIQUES OF QUEUING THEORY

68

2.2

69

TECHNIQUES AND HISTORY OF ANAL YSIS

=

Waiting-Time (in Queue) Distribution for'the GUM/1 Queue

.

Let W-(T) dT be the probability that the waiting time Vq of a customer h.es betwee~ ] T, T + dT]. Then using the characteristic property of the e.xp?nentlal distribution and the FCFS queue discipline, the L.T. w~(a) of the p.dJ, IS gIven by =

w;(a)

_(

= j~O Pj

J.1)i J.1 + a

= (1-1'1)

=

(l - 1'1 )(J.1 + a) J.1 + a - J.1rl

+ J.1rl (1

-1'1) J.1+a-J.1rl

(2.2.36)

where (l - I'd is the concentration of probability mass at the origin. Equation (2.2.36) on inversion gives P[Vq = OJ = (I-I'd and (2.2.37) T>O. Number in System at Random Epoch, Pn Let {P n} represent the limiting distribution of the number in th~ s~s~em. ~IIM~l a.t

a random epoch. To get the distribution {Pn }, we first get the hmltmgjOmt dlstnbution (2.2.38) and then (2.2.39) We now derive a probability expression for Pn(y) in terms of P~. Recollect the definition of P n(Y) in equation (2.2.38), which represents the limiting probability and probability density that the number in the system is n and the last customer joined the system y time units before the current epoch. Define AC(y) = P{interarrival time> y} = dA(t). Since during the period y only departures can occur, adapting the argument used to get equation (2.2.31a), we have

P n = p.

n=l

Thus Po

=

(2.2.43)

l-p.

For an alternative derivation of Po using Po(Y), see Problems and Complements. Equations (2.2.42) and (2.2.43) completely determine the distribution of the number in the system at a random instant of time. Distribution of Virtual Idle Period and Actual Idle Period for the GI/M/1 Queue

First we obtain the virtual idle-period distribution for the Gi/MII queue, which can be found by using some of the analysis discussed earlier. Then the actual idle-period distribution can be deduced from the virtual one. For an alternative and elegant derivation of these distributions, see Miscellaneous Problems and Complements. The procedure which is used here can be extended easily to get the idle-period distributions for the bulk-arrival queue GriM/I. For details of this, see Miscellaneous Problems in Chapter 6. Recall that for the M/G/I queue the idle-period distribution was simple and easily obtained, while the waiting-time distribution was more complicated in form and more difficult to derive. It is therefore not surprising to find that for the GIIM/I queue, the dual of the M/G/I, it is the waiting-time distribution which is simple and easily found. The idle-period distribution can be found by a somewhat more complicated argument given below. Recall that I is an LV. which represents the actual idle period, and that Lv is an LV. which represents the virtual idle period. The distribution of Iv consists of two parts given by P(Iv = 0) 1 -P (arrival finds system empty)

I-PO' dP(Jv';;; T) dT

r;

P ( ) = A C( ) ~ P~+m(J.1y)me-J.lY, n y y m'-:.o m!

L

1 -Po =

T>O.

The distribution of I is related to that of Iv by the following relation:

n> 1.

(2.2.40)

F1(T) = P(I';;;T) = P(Jv ';;;TII" >0) _ F1v(T)-(1 -PO')

Using equation (2.2.34), we get Pn(y) = M C(y)(1-rlh?-l e-J.lY(I-'Yl>,

n>l.

(2.2.41)

(2.2.42)

n>l

where we have used equation (2.2.32) to obtain

fo AC(y)e =

-J.l(l-'Yl)Y

d = l-a[J.1(I-rd] y (1 _ ) J.1 1'1

The term Po is obtained by the normalizing condition

J.1

or equivalently,

Ff(T) = P(J>T)

Putting equation (2.2.41) into equation (2.2.39) gives, on integration,

Pn=p(l-rlhr- 1 ,

PO'

= P(J~~T) = F~i)

(2.2.44)

(2.2.45)

In some situations it is convenient to use equation (2.2.45) rather than the equivalent form (2.2.44). We now derive the distribution of Iv, and thus encounter the same difficulty as in the derivation of the waiting-time distribution for MIG/I. As the waiting-time distribution for MIGll (see Miscellaneous Problem 4) depends on the residual service time of the customer undergoing service, so the idle-period

SOME TECHNIQUES OF QUEUING THEORY

70

distribution for GIIMll depends on how long an interarrival period has already elapsed since the start of the idle period. Now aCr + y)IA C(y) represents the probability density that an interarrival interval lasts for time r + y, given that it did not end before or at time y; Po(Y) represents the limiting joint probability and probability density that the system is idle at some instant of time (all customers having departed during y). Hence we have, if we define FfJr) to be the probability that the system is idle at some instant of time and that the idle time thereafter is grea ter than r,

=

Ff,,(r) =

f

0

a(y + r) PoCy) AC(y) dy.

(2.2.46)

To evaluate the integral in some compact form, we need to know first the value of Po(y). This may be evaluated from 1:;;'=1 Pn(y) +Po(Y) = t..

AC(y)

(2.2.47)

AC(y)

where Pn (y) is given by equation (2.2.41). Note that since 1:;;'=0 Pn (y) represents the p.d.f. of the past life of an arrival (see Section 2.3), it must be t.. [1 - A (y)]. Using equation (2.2.41) in equation (2.2.47), we get

Po(Y) = t.. [1 - e- Il (I-,)" ):1']. AC(y)

(2.2.48)

For an independent proof of equation (2.2.48), see Problems and Complements. Ignoring the norming constant t.. and using equation (2.2.48) in equation (2.2.46), gives Flv(r) = a(y + r) dy e- ll (I -')',»)/ a(y + r) dy

s:

s:

Considering the complement, we get FI (r) = ACr) + v

f= exp [- /l(I - 1'1 )(u T

r)] a(u) du,

r ~0

(2.2.49)

which is the d.f. of Iv' Note that equation (2.2.49) is true for r = O. For when r = 0, F1v(0) =P(Iv = 0) = 1'1, as it should be. See the definition of P(Iv = 0) given earlier. An alternative derivation of the dJ. of Iv is discussed in Problems and Complements. The moments of Iv may be obtained by taking the L.T. of Iv, as discussed in Problems and Complements. The distribution of I may be obtained by using equation (2.2.45) or equation (2.2.44). In particular, note that for the system

2.2

TECHNIQUES AND HISTORY OF ANALYSIS

71

case and the distribution {P n} given in equations (2.2.42) and (2.2.43) are contained in Conolly (1958). Conolly (1958) derives P n by first considering the transient solution and then takes the L.T. and the limit. Our procedure here has been based on several results due to Conolly (1960, 1975). We have attempted to unify many of the results for the queuing system GIIM/l. Some of the results discussed here are obtained later in this chapter and also in other chapters using other techniques. In this connection see Cohen (1980), Prabhu (l965a) and the references therein, among others. Wishart (1961) has solved the system MIG/! by defining the supplementary variable as the time to service completion instead of the time since the beginning of the last service. Henderson (1972) extends Wishart's work by giving a transient solution for M/G/! using the time to service completion as supplementary variable, and also discusses GI/M/! using the time to next arrival as supplementary variable. In this connection, see also Cooper (19&1). Discrete Time Analysis

Most studies of queuing systems either consider time as a continuous parameter or define a convenient set of discrete points in time (imbedded Markov chain case). However, in practice, situations do arise when events occur at discrete points in time which may not coincide with the time points of the imbedded Markov chain. Such systems may be found in electronic installations whose operations are governed by internal clocks, or missile bases which fire at oncoming airplanes at somewhat regularly spaced intervals of time. By a discrete-time queuing system we mean a system in which customers arrive or are processed· only at fixed time points separated by uniformly spaced fixed intervals of time., Thus arrival of customers or their processing is initiated and completed in an integral number of periods of time. Mathematical formalism required for the study of discrete-time queuing systems is analogous to that required for the continuous-time case. However, from the practical point of view it may be worthwhile to study discrete-time queuing systems in order to show the modifications needed in their study, and the ease with which computer calculations can be handled. For more details of the advantages of discrete-time queuing models, see, for example, Dafermos and Neuts (1971). We shall study some discrete-time queuing systems with bulk service or bulk arrival, which have been discussed in the literature. The study of queuing systems in discrete time was begun by Meisling (1958), who studied the system Geom/G/I and in a way used a hidden Markov chain. We shall consider Meisling's (1958) approach while discussing the system Geom/G B /1. The geometric distribution is used here as the discrete analog of the negative exponential, whose memoryless property it shares. Similarly the binomial distribution is the discrete analog of the Poisson distribution.

MIM/1,

Conservation Principle

as it should be. The distribution of the number in the queuing system GIIMII in the transient

Conservation principles play a fundamental role in severai fields, such as physical sciences, engineering, and economics. The ideas of conservation of mass, energy, momentum, and charge often provide us with a system of equations which can, in

72

SOME TECHNIQUES OF QUEUING THEORY

many cases, be easily solved to give the desired information. The principle of conservation states that in equilibrium the flow must be conserved in the sense that the input flow must equal the output flow. This principle can be used to ascertain some limiting characteristics of queuing systems. We use this principle to discuss a few of the simple queuing results in the limiting case. For applications of the principle of conservation to some other queuing models we refer the reader to Kleinrock (1976) or Krakowski (1973, 1974). However, the reader is warned here that the result (S.23) in Krakowski (1974) appears to have been incorrectly reported. The correct version of tlus may either be obtained by taking correctly the probability of the position of a customer within a group [see, for example, Problem 15(e)] or else it may be seen in Chapter 3, where it has been derived using other techniques. For the bulk-arrival queuing systems M X /G/l or Glx/M/I with mean group size a, the mean arrival rate is Aa, and the mean service (or departure) rate when the system is occupied is J1. Since the mean departure rate J1 is conditional upon the server being busy, we have, in the limiting case, using the fact that the average input is equal to the average output,

BASIC RENEWAL THEORY

73

which are being used by a certain machine or equipment. Also suppose that as soon as the item under use fails, it is replaced by a similar item from the stock. If the life lengths Un, n = 1, 2, 3, ... , of the items are LLd. r.v.'s, then N(t), which represents the number of items renewed (or replaced) during the time interval] 0, t], is a renewal process. This section is largely self-contained. It discusses that portion of renewal theory which is needed for the study of bulk queuing theory, and some other basic properties of renewal theory. These properties are best discussed through the L.-S.T. (or the L.T. when the r.v.'s are continuous, as we shall assume here). For the discrete case we refer the reader to Feller (1968), Prabhu (1965b), or Neuts (1973). For a more detailed study of renewal theory in the case of continuous r.v.'s see Cox (1962), Feller (1971), or Karlin and Taylor (1975). It may also be remarked here that a renewal process is called by some authors a rCl.current process [see Feller (1968) and Takacs (1962)]. and suppose that Un = Let the renewals occur at instants of time a~ - a~-l' n = 2, 3, ... , are i.i.d. r.v.'s with common distribution

a;, a;, ...

A(u)=P(Un";;;;'u),

n:>2

and common mean

that is,

Aa

Po = 1 - - = I-p J1

which is the probability of the server being idle for both M X /G/l and Gl x /M/1. In subsequent chapters many of the results of this section will be derived using other techniques.

2.3

2.3

BASIC RENEWAL THEORY

As we have remarked earlier, queuing theory originated with the probabilistic study of problems connected with telephony. Similarly, renewal theory was first applied to problems connected with the failure and replacement of equipment. Later, however, in renewal theory, as in queuing theory, models developed for one type of application turned out to be useful for other applications. Thus renewal theory can be applied to nuclear particle counters and to queues. Renewal theory deals with the study of renewal processes. A process {N(t), t:> O} whose state space belongs to a denumerable set {O, 1, 2, ... } and for which the interarrival (or interoccurrence) times Un = a~ - a~ -1, n = 1, 2, 3, ... ; a~ = o between successive arrivals (or occurrences) are positive LLd. r.v.'s, is called a renewal process. Clearly, as N(t) counts the number of arrivals during a time period ] 0, t], it may also be called a counting process. The interarrival times are called the renewal periods, and the arrival instants (or epochs) the renewal instants (or points in time). As an example, consider airplanes arriving singly at an airport. If interarrival times are Li.d. and N(t) is the number of arrivals during] 0, t], then {N(t)} is a renewal process. As a further example, suppose that we have a stock of items

n:> 2.

Further, let U1 = a~ - a~ be independent of other U's and

[To signify the positiveness of the LV.'S Un' n :> 1, one may define A (0+) = 0.] We have taken the distribution of the first renewal period to be possibly different from that of other renewal periods. In practice, however, two possibilities arise: 1 Al (u) = A (u). In this case the renewal process is said to be an ordinary renewal process or simply a renewal process. This situation arises when the renewal instant occurs at a~ = 0, thus making U1 identical with the other U's. We may, however, caution the reader here that some authors do not count the renewal at a~ = 0 while studying an ordinary renewal process, whereas others do. We have preferred to use the former approach and as such may call the renewal at a~ = 0 an initial renewal. The essential effect of counting the initial renewal at a~ = 0 is that the total number of renewals in [0, t] is N(t) + 1 instead of N(t) in J 0, tl. The real distinction between the two versions, though trivial, needs to be made. 2 Practical cases do arise when Al (u) *A(u). This situation arises when the renewal instant does not occur at a~ = 0, thus making U 1 not identical with the other U's. To illustrate this point, consider the second example discussed before. Suppose now that we started observing the process when the first item was already in use. Thus in this case it is natural to suppose that U 1 need not be distributed as the other U's. However, U1 will be taken to be independent of the remaining U's. In this case the renewal process is said to be a modified (delayed or general) renewal process. One particular type of modified renewal process is extremely important, in which

SOME TECHNIQUES OF QUEUING THEORY

74

2.3

BASIC RENEWAL THEORY

75

1

-f a

Al(U) =

u

0

•

Thus

[1-A(x)] dx

P(N(t);;;;'n) = P(Wn
where a =

L= U dA(u).

In this case the process is said to be an equilibrium (or stationary) renewal process. As we show later [see equation (2.3.14)], the equilibrium renewal process can be obtained as a limiting case of a renewal process which has been in operation for a sufficiently long time. Since Un is nonnegative for all n, we see that a> 0, except in the trivial case in whichP(Un = 0) = 1. We assume a > henceforth. Consider the partial sum

°

Wo = 0.

(2.3.1)

We say that a renewal occurs at time t if Wn = t for some n. Clearly, Wn gives the waiting time until the nth renewal. It is common practice to refer to either the counting process {N(t), t ;;;;, o} or the partial sum process {Wn, n ;;;;, o} as the renewal process, for as we shall see later, one is related to the other. Using independence and identity of the r.v.'s, we have from equation (2.3.1), (2.3.2) where * indicates convolution, and A(n-1)(u) is the (n - I)-fold convolution of A(u) with itself. Note that A(o/u) = for u < 0, A(o)(u) = 1 for u > 0, and

°

A(1)(u) =A(u),

= A1(t)*A(n_l)(t)

Now

Introduce the transforms

Pn(t)

==

P(N(t) = n) = P(N(t);;;;'n)-P(N(t);;;;'n

+ 1)

= P(Wn
= Al (t)* A(n-l)(t) -

n;;;;' 1.

At (t)*A(n)(t),

(2.3.5)

(t) This gives the distribution of N(t). We could also write P (t) = Fw (t) - F n n Wn+ 1 ' n ;;;;, 0,1'f we de fime F Wo (t) = 1 for t;;;;' 0, or zero otherwise. The Renewal Function

The mean value of the renewal process N(t) is known as the renewal function. Thus the function M(t) = E(N(t» is the mean value or renewal function of the process {N(t), t;;;;' OJ. Much of renewal theory is concerned with the properties of M(t). It can be seen easily from equation (2.3.5) that =

M(t) = E(N(t»

=

I A! (t)* A(n-nCt ), n=!

t;;;;'

°

(2.3.6)

where we define M(O) = 0. For the ordinary renewal process, since A! (t) = A(t), we can write equation (2.3.5) as Pn(t) = A(n)(t)-A Cn +1)(t), n;;;;'O (2.3.7) and consequently for this case

Re 0:>0

=

M(t) =

au(O:) =

eJf= o

Re 0: > 0.

cw dA(u),

.

lw n (o:) = fo= e- au dFwn(u)

P(N(t) = n)

(2.3.3)

Distribution of the Number of Renewals N(t) during the Time Period J 0, tl

iff

Wn

< t.

=

[1 - ~t! e-A~;At)j] - [1 - f e-At.~At)jl J-O

J.

j=O

J.

_ e-A1(AtY

In order to obtain the distribution of N(t), we need the basic relation that exists between N(t) and the sequence {W n }. By definition of N(t) and Wn and for t> 0, N(t);;;;' n

Acnlt).

A simple case of the ordinary renewal process arises when the renewal periods are positive LLd. r.v.'s. If the renewal periods are exponentially distributed with c?~on. mean, a = 1/1.., then using the convolution formula for the exponential distnbutlOn, we have from equation (2.3.7),

From equation (2.3.2) we have

= au, (a)[au(a)] n-l.

I n=!

(2.3.4)

n!

n;;;'O

which is a Poisson distribution and has been discussed earlier. The process {N(t)}, when t varies in T, is called a Poisson process, and is a particular case of the ordinary renewal process for whichM(t) = At.

SOME TECHNIQUES OF QUEUING THEORY

76

The Renewal Density

The derivative of the renewal function M(t) is called the renewal density met). Thus from equatiOn (2.3 .6) we have =

I

al(t)* a(n-ll(t)

(2.3.8)

n=1

where al(t) and a(n-I)(t) are the p.dJ.'s corresponding to the dJ.'s AI(t) and A (n -1)(t), respectively, and we have assumed that the dJ.'s of the U's are absolutely continuous. Note also that a 1 (t)*a(n _ 1)(t) is a notational way of writing the derivative of A 1 (t)*A(n-l)(t). The probabilistic interpretation of met) is that it gives the probability density of a renewal at time t. It can be seen that for the Poisson

2.3

BASIC RENEWAL THEORY

77

which is the required density. Since the inverse transform uniquely determines the function, m (t) is unique. If al(t) and a(t) are given, the interest lies in finding the unknown function met) as a solution of equation (2.3.9). We must caution the reader here that though met) is called a renewal density, it is not a p.d.f., for its integral diverges even in the typical case of a Poisson process for which limt-+= f~ m(x) dx = limt-+= At .... 00. m (t), in fact, gives the renewal rate at time t. In view of this equation (2.3 .9) may be given a probabilistic interpretation. Roughly speaking, u~ to first order in !::.t, the left- and right-hand sides of equation (2.3.9) are: LHS of eq. (2.3.9) = P (a renewal occurs in Jt, t

+ !::.t])

= m(t)!::.t

RHS of eq. (2.3.9) = P (first renewal in] t, t + !::.t]) + sum over u of P (last renewal before t in 1t - u, t - u + M]) x P (renewal component introduced at epoch t - u fails in interval ]u, u + !::.u] of renewal life)

process, m (t) = A. The Renewal Equation for m (t)

The equation met) = al (t)

+ (m(t -

u)a(u) du

(2.3.9)

where a(t) is the common density of U's and al (t) the density of U I , is known as the renewal equation for met). In the special case al (t) == aCt), equation (2.3.9) reduces to the ordinary renewal equation met) = a(t)

+

I:

met -u)a(u) duo

G(t) = H(t)';"

Then as we got equation (2.3.3) from equation (2.3.2), we have, by using equation (2.3 .8),

2:=

+ L:;m(t -u)t::.toa(u)!::.u

and the result follows. A similar probabilistic interpretation may be given to the renewal function (2.3 .12). The renewal equation (2.3.9) for met), which is an integral equation, is a particular case of the more general Volterra integral equation of the second kind:

(2.3.9a)

There are several methods of finding the renewal equation (2.3.9) for the renewal density m (t), and one of them, which we adopt here, is through the transforms. Let

mea) =

= al(t)t::.t

where

J:

G(t - x) dA(x),

(2.3.11)

G~t),

H(t) and A(t) are defined for t;;;' O. In equation (2.3.11) as in equation If H(t) and ACt) are given, then the interest lies in finding the unknown functIOn .G(t) as a solution of the integral equation. In renewal theory, equation (2.3.11) IS usually called the renewal equation because many interesting results of renewal theory satisfy particular cases of equation (2.3.11). The solution of the renewal equation (2.3.11) in the case when A 1 (x) = A (x) is easily obtainable and is given below. In fact, from equation (2.3.11) it can easily be shown that (2.3.~),

au, (a)[au(a)] n-l

G(t) = H(t)

n=l

+ (H(t-U)dM(u)

(2.3.11a)

where

au, (a)

(2.3.10)

1 -au(a) .

=

M(t)

= 2:

An(t).

n=l

Rearranging, we get mea) = au, (a)

By inversion, we get met) =

aj

(t)

+

+ m(a)au(a).

r o

met -u)a(u) du

We wish to prove equation (2.3.11a). Taking the L-S.T. of equation (2.3.11) and solving it for g(a), we get g(a) =

lie::)

1 -a(a)

SOME TECHNIQUES OF QUEUING THEORY

78

2.3

BASIC RENEWAL THEORY

79

Key Renewal Theorem [Smith (1958)J

where

h(O'.)

r o

If the function H(t) is such that: 1 It is nonnegative for t > 0, 2 It is nonincreasing, 3 fO' H(u) du < 00 and A(u) is nonlattice, then

e-e>t dH(t)

il(O'.) = fooe-e>t dA(t). o

lim [I H(t-u) dM(u) =

t-4-ooJ o

Now rewriting g(O'.) and using equation (2.3.8), we have

g(O'.) = h(O'.) [ 1 + 1

Another result is the asymptotic normality of the r.v. N(t). For a proof of asymptotic normality of N(t) the reader is referred to Karlin and Taylor (1975). 2 Thus if a < 00 and a < 00 are the mean and the variance of Un, it can be shown that

~~lO'.)]

= h(O'.) + h(O'.)iii(O'.). The inversion of this gives the desired solution (2.3.11a). It may be noted that we have used A 1 (x) =A(x) while using equation (2.3.3).

Iim t-+oo

2

lim M(t) t

t~oo

One of the limiting cases is easy to derive by using the result limt-+oof(t) = lime>-+o+ 0'.1(0'.). Thus . _ 1 lim met) = hm O'.m(O'.) = -.

p[N(t)-t/a Vta /a 3

';;;y

]

1

= --

v'2ii

fY e -x'l2 dx " -00

that is, N(t) is ~symptotically normally distributed with mean and variance given by

Limiting (or Asymptotic) Behavior of Renewal Functions

and

a

. VaT (NCt» 11m t-'l>-OO

Residual life

We approach the topic of residual life by reconsidering the renewal equation for

Analogous to this, we also have

met) given in equation (2.3.9). Using the convolution property, we have from

~.

equation (2.3.9),

lim (M(t») = t-+ oo t a

m(x) = aj (x)

TIus result is known as an elementary renewal theorem. An intuitive interpretation of this theorem is that the average number of renewals in ] 0, t], for large t, is approximately the product of the reciprocal of the mean renewal p.eri.o~ l/a and the time t. For nonrigorous non probabilistic derivations of the above lirrutmg results and some other limiting results, which are stated below without proof, the reader is referred to Problems and Complements. For rigorous probabilistic proofs the reader is referred to several of the existing sources, for example Feller (1971), Smith (1958), Takacs (1962), Neuts (1973), or references contained therein. Blackwell's Theorem

+ foX a(x -

u)m(u) duo

Integrating this from 0 to t gives

M(t) = Aj(t)+ f: [f:a(x-u) m(u) dU}dX = A 1 (t)

+

fat {(a (x -u) dx}m(U) du

= Aj(t)+ (A(t-u)m(u)dU = Aj(t)+ (A(t-U)dM(u).

If A(u) is a nonlattice distribution function with mean a < 00, then

M(t)-M(t-c) for all real numbers c? o.

t

respectively.

a

a-"'Q+

t....:,..oo

~fooH(u) duo a 0

c

-+-

a

as

t -+ 00

(2.3.12)

Consider the modified renewal process {N(t)} and let r(t) be the time measured from t, the instant at which we start observing the process to the next renewal instant after time t. Then since WN(t)+ 1 is the time when the [NCt) + 1] th renewal occurs, we have

SOME TECHNIQUES OF QUEUING THEORY

80

ret) =

WN(t)+!

t>O.

-t,

ret) is called the residual (excess or remaining) life at time t. The distribution of ret) is given by x>O, t>O. P{r(t)';;x} = A 1 (t+x)- S:P-A(t+X-U)] dM(u), (2.3.13)

We proceed to prove equation (2.3.13). First note that

{ret) .;; x}

*

{at least one renewal in ] t, t

+ x] }

* U {Wn
2.3

first answer, which is normally given, is a12. This is based on the following argu· ments. Since the arrival epoch of the person is chosen at random (so that it has a uniform distribution over the interval), it is argued that the expected waiting time should be half the expected waiting time a between the two buses. Thus we may say that limt-+= E(r(t») = a12. However, this argument is fallacious, since it ignores the fact that the renewal period is an LV. The answer is right if the distri· bution of renewal periods is deterministic, that is, if the buses always arrive after a constant period of length a. Now if we base our arguments on the forgetfulness property of the exponential, then we obtain the second answer limt-+= E(r(t») = a. Clearly, the two answers are contradictory, but the contradiction can be resolved by more sophisticated arguments, which take into account that the renewal periods are r.v.'s. To this end, by looking at equation (2.3.14), one can easily see that the required expected value (see Problem 12) is given by

'This implies (since Wo = 0) that

a lim E(r(t)) = -

=

P{r(t) ';;x} = P{t';; WI

81

BASIC RENEWAL THEORY

< t + x} + L P{Wn < t, t';; Wn + Un + 1 < t + x}

2

t-+~

0

2

+ -2a

n=l

= Al(t+X)-AI(t)+

= A1(t+x)-A1(t)+

=

t

n=1

0

=

• t

L S P{t-u';;Un+1
L J0 {A(t+x-u)-A(t-u)}dFwn(u).

where 0 2 is the variance of the renewal periods. Since for the exponential distri· bution 0 2 = a2 , this limit becomes a, which is the answer obtained by using the forgetfulness property of the exponential. Thus the correct answer is a and not a12. Note that if (and only if) 0 2 = 0, then the answer is a12. Past Ufe

n=1

Hence, usingM(u) = 2:;;'=1 Fw (u), we finally get equation (2.3.13). Now we discuss the limiti~g behavior of the distribution of ret) when A(u) is nonlattice. Using the key renewal theorem with H(t) = 1 - A(t + x), we have from equation (2.3.13), = l-A(u +x) lim P{r(t)';;x} = 1du

f

t-+ oo

a

1 =-

a

let) = t -

A1Ct)- f:-xp-A(t-U)] dM(u),

f= [l-A(u)] du

P{l(t) ';;x} =

x

fX [l-A(u)]du

WN(t).

let) is called the past life (or age or current life) at time t. The distribution oft(t) is given by

a

0

1 1--

If let) is the time measured backward from t to the last renewal instant, then

(2.3.14)

0

where we have used the fact that f;;'([1 - A (u)] la)du = 1. The Paradox of Residual life Let us explain this paradox by means of an example. Suppose that buses arrive at a certain bus stop following a Poisson process, so that the renewal periods are exponentially distributed with mean a. Suppose further that a person arrives at the stop at an epoch t. We discuss here the case when t is large, that is, we assume that the buses have been operating for a sufficiently long time. The question usually posed is this: What is the expected waiting time of the person for the next bus? The

( 1,

O<x
x;;;. t.

To prove this, first note that let) > x iff no renewals occur in ] t - x, t] ,x.;; t, or equivalently iffr(t - x) > x. Thus P{l(t)~x}

= P{r(t-x).;;x}.

The result now follows from equation (2.3.13). The case of unity follows by considering the complement of the impossible event I (t) > x, for x > t, noting that P{l(t) = t} = 1 - A I (t). It can be shown, by using the key renewal theorem, that limP{I(t)';;x} =

t-+=

x

l-A(u)

0

a

I

du

provided A(u) is nonlattice. Thus the limiting distribution when t -'> 00 of the residual life ret) and the past life let), both tend to the common distribution whose

82

SOME TECHNIQUES OF QUEUING THEORY

p.d.f. is [1 - A(u)] la. It is seen later that this common distribution and its discrete version play important roles in queuing problems. We now discuss another renewal process called an alternating renewal process. Alternating Renewal Process

As an example, consider an equipment which is subject to breakdowns. Suppose that initially the equipment is running and that it remains in this state for a random amount of time To. As soon as it breaks down, repairs are started. Suppose that the repair time is T~. If the cycle of running times and repair times continues, it generates two sequences of r.v.'s To, T J , ••• and T~, T~, .... Suppose that the two sequences Tn and T~, n;;' 0, are LLd. as T and T' with c.d.f.'s F and G, respectively. If the sequences {Tn} and {T~} are mutually independent, the two-state process defined by the state of the equipment is an alternating renewal process. While this assumption of mutual independence of running and repair times is often made, it is not needed [see Franken (1978)] and is not used in the derivation of equation (2.3.15) below. Also suppose that H is the common distribution of the r.v.'s Tn + T~, that is, H is the convolution of the c.d.fs F and G. The process defined by the state of the equipment referred to above may be considered as an alternating renewal process. The interest in this case might be to find the limiting probabilities such as lim R (t) = limoo P {equipment is running at time t}. t-+oo

t-+

The evaluation of this probability is discussed later. As another example, consider the queuing system M/G/I. Suppose that initially the server is idle and that he remains so for a random time T~. As soon as a customer arrives, he gets busy and remains busy for a random amount of time To. As in the last example, the cycle of idle periods and busy periods will generate two mutually independent sequences which can be studied as an alternating renewal process. Similar applications of alternating renewal processes arise in the theories of dams, inventories, and so on. We now return to finding the limit R (t) as t -+ 00. We wish to show that if E(T + T') < 00 and H is not a lattice distribution, then lim R(t) =

t .... ~

E(T) E(T) + E(T')

(2.3.15)

where E(X) is the expected value of an r.v. X. In words it may be stated thus: The limiting probability that the equipment is running is equal to the ratio of the expected time the eqUipment is running to the sum of the expected running and repair times. To prove equation (2.3.15), we condition on To + T~ and get R(t) =

Jo~ P{runningat tl(To + T~)

Since the process regenerates itself at time To

= y} dH(y).

+ T~, the conditional probability

2.3

BASIC RENEWAL THEORY

83

within the sign of integration breaks up into two parts according to whether y < t ory >t. Thus R(t- y ), ifytl(To + To) = y}, ify > t. Consequently, R(t)

=

JotR(t-y) dH(y)

+ Jt""P{To>tl(To +T~) = y}dH(y).

But as T~ is nonnegative, we must have

Jt~ P{To > tl(To + T~)

>t P{To > t}

= y} dH(y) = P{To =

and

To

+ T~ > t}

= 1 - FT(t)."

Therefore R (t) satisfies the following renewal equation: , R(t) = I-FT (t)+

S: R(t-y) dH(y)

which is of the form (2.3.11). Consequently, by using equation (2.3.11a) as the solution of equation (2.3.11), we have R(t) = 1 - FT(t)

+

S: [I - FT(t - y)] dM(y)

(2.3.16)

where ~

M(y)

= 2:

Hn(Y)·

n=1

Equation (2.3.16), on applying the key renewal theorem (since H satisfies its properties), leads to IimR(t) = fO'[1-F T (y)] dy = E(T) fO' x dH(x) E(T) + E(T')"

t .... ~

If we suppose that Set) = P{ eqUipment under repair at time t}, then Set) = 1 R(t), and therefore E(T') lim Set) = E(T) + E(T') . (2317) t .... ~ . . It may be noted that the limiting probability remains unaffected whether the equipment was initially running or under repror. From equations (2.3.15) and (2.3.17) we may observe that

E(T) = lim R(t). E(T') t ..... = Set)

(2.3.18)

Now we apply result (2.3.18) to find the mean busy period E(T) in the queuing system M/G/I, in which E(T)/E(I)=p/(l-p), where E(T) and E(I) are the expected busy and idle periods, respectively. Now because of the forgetfulness property of the exponential, the calculation of the idle-period distribution for the

SOME TECHNIQUES OF QUEUING THEORY

84

PROBLEMS AND COMPLEMENTS

85

system M/G/l is trivial. It is, in fact, the saine as the interarrival time distribution, and therefore E(J) = l/A. Consequently, 1

E(T) = - .

Using Ll.t = t/k and passing to the limit as k .... 00, show that

P.-A

Formulas for the mean busy period E(T) in systems more general than M/G/l, including bulk queues, are obtained in Section 6.7.

2.4

P(NCt) = n) = Jim (k)CALl.t)n(1- ALl.t)k-n k~oo

n

OVERVIEW

While we have now considered several techniques which have been used to analyze queuing problems, there are other techniques which we have not discussed in detail, but which are mentioned here. The first one is the method of collective marks. Though this method gives an interesting probabilistic interpretation of p.g.f.'s, no problems seem to have been solved by this technique which cannot be solved by other techniques. For some details of this method, the interested reader is referred to Kleinrock (1975) and the references therein. Four other methods are the combinatorial technique for which the reader is referred to Takacs (1967) or some other books on queuing theory; the spectral theory approach for which the reader is referred to Prabhu (1965a), Cohen (1980), and Bhat (1968); the random walk method for which the reader is referred to Conolly's (1975) lecture notes and the references contained therein; and finally an algebraic method for which the reader is referred to Kingman (1966). For the use of fluid approximations, diffusion approximations, bounds, and so on, one may consult chapter 2 of Kleinrock (1976) and the references contained therein, and also Newell (1971). For the use of a semiMarkov approach to nonbulk queues, we refer the reader to a book by <;inlar (1975), and for applications to bulk queues to the works of Fabens and Neuts, cited in other chapters. Recently Neuts has been advocating the use of a new class of matrix methods in queuing theory. References to Neuts's book and papers using matrix methods for the study of bulk queues are made in other chapters. An excellent and comprehensive work connected with the mathematical derivation of results leading to the limiting results when t..,. 00, n .... 00, or p .... 1, and many other interesting queuing results is that of Cohen (1980).

PROBLEMS AND COMPLEMENTS Sections 2.1 and 2.2

Unit arrival time-homogeneous Poisson process. Here we consider its alternative derivation. Divide the time interval 0 to t into a large number, say k, of subintervals of length Ll.t such that kLl.t = t. Then let the probability of the occurrence of an event in Ll.t be ALl.t + o (Ll.t). We also assume that the probability of more than one event occurring in Ll.t is o(Ll.t). Then N(t) will have a binomial distribution, and

2

(a)

Continuation of Problem 1. One important property of the Poisson process is that the time between occurrences of events is exponentially distributed. Thus if Ur = tr - t r - 1 , r = 1, 2, 3, ... are interVals of time between successive events, then show that {Ur } are Li.d. LV.'S with P(Ur>x)=e-il.x,

(b)

X;;;.O,

r=1,2, ....

If N(t) is the Poisson process with unit arrivals, then show that for u < t, k
(c)

A>O,

=

kIN(t)

=

n} =

(n)k Uk(t -t u)n-k n

Let {Ur , r ;;;. 1} be a sequence of Li.d. r.v.'s such that fur(u)=p.e- IlU ,

U;;;.O,

fJ.>O.

Define N(t) such that N(t)

N(t)+1

r=1

r=1

L Ur';;;t< L

Ur

where N(t) is a nonnegative integer and t > O. Show that {NCt), t ;;;. O} is a Poisson process with mean p.t. Hint: Use N(t)';;; n '* I:.:.'=? Ur > t and the convolutions of U's discussed in Chapter 1. 3

An r.v.X. is said to have a "memoryless" (or forgetfulness) property iff, for any x > 0, y > 0, P(X>x + yIX>x) = P(X> y). (1) That the exponential LV. satisfies equation (1) is easy to see. To show the converse, define FHx)=P(X>x); then FHx+Y)=FHx)FHy). The proof is now based on a well-known theorem in analYSis [Hille (1964)] which states: If Fi-(x), x> 0, is a real-valued function satisfying the functional relation FHx + y) = FHx)FHy) and is bounded in every finite interval, then either FHx) vanishes identically or there exists a constant A> 0 such that F3.:(x) = e -I\x ,

X>O.

SOME TECHNIQUES OF QUEUING THEORY

86

4

This shows that the only continuous LV:'S having the property (1) are exponential r.v.'s. The property (1) is also called a characteristic property of the exponential distribution. For a different approach to this problem, see Problem 9 of Chapter 1. Continuation of Problem 1. Let {N(t), t E T} be a stochastic process with finite second-order moments. The autocorrelation function RN(tl' t z ) of the process N(t) is defined for all t 1, t z ETas the joint moment of the r.v. 's N(t I) and N(t z). Thus notation ally , RN(tI,tz)

==

E[N(tdN(t z )]·

Prove that for the Poisson process with unit arrivals

5

Hint: Consider t2 ;;;;'t 1 and use the property of independence and stationarity of the Poisson process. It may be noted that RN(tl' t z ) = RN(tz, tl)' Continuation of Problems 1 and 4. The correlation coefficient function of a stochastic process N(t) is defined for all t I, t z E T by p ( tl,t z )

=

where oh(t) = E [N 2 (t)] - E2 [N(t)] . For t2 > t1> show that for the Poisson process, the correlation coefficient function is given by

Order statistics. Let Xi' i = 1, 2, 3, ... , n, be a sample of n elements from a population having continuous p.d.f. [(x). Further, let X(I) be the smallest, X(z) the second smallest, ... , X(r) the rth smallest, X(n) the largest of the sample of values (XI, X 2 ,· •• , Xn)· Clearly, -oo<X(l) ~X(2) ~X(3) ~ ... ~ X(n) < 00. The LV.'S X(i)' i = 1, 2, ... , n, are called the order statistics of size n (n-order statistics) of the sample (XI' X z , ... ,Xn ). Assume that X;'s are Li.d. r.v.'s and that their joint p.d.f. exists. Show that the joint p.d.f. of the order statistics X(i) is given by

The order statistics are used extensively in certain branches of statistics. For more details, see Kendall and Stuart (1969). We illustrate their uses in the next two problems. 7

Continuation of Problems 1 and 6. We have already given one important property of the Poisson process in Problem 2. Another one is its relation to order statistics. Given that n events of the Poisson type occur at epochs tl < t2 < ... < tn in the interval] 0, tJ, prove that the LV.'S t l , t z , ... , tn have the same joint distribution as the n-order statistics corresponding to n independent r.v.'s Xi uniformly distributed over the interval] 0, t] , that is, prove that

Proof: In view "of Problem 6, it suffices to show that the probability that an epoch of occurrence of an event lies in ] x, x + dx] , ~ x ~ t, given that it lies in] 0, t], is dx/t, that is,

°

P[x

< epoch ~x + dx I epoch lies in 10, tJ]

n [(Xi),

i=1

1

= t'

°

~x ~ t.

In other words, the epoch of occurrence has a uniform distribution over the interval. To prove this, we first fmd the joint probability that n events occur in the given time interval, ] 0, tJ ,and one out of the n epochs lies in lx, x + dx]. Let the interval be divided into three parts, x, dx, and t -x -dx, as shown below: Events I x x+dx Time

°

Let the number of events occurring in the first part be i, i = 0, 1, 2, ... , n - 1. Then the number occurring in the last part must be (n - 1 - i), so as to have one in lx, x + dx] and maintain the total of n. Since the events occur in accord with the Poisson process, the probability that they occur in this fashion given i events occur in ] 0, xl, is the product of the two appropriate Poisso~ probabilities and thus is given by

"A dxe-A.dx

In particular, if Xi are uniformly distributed with the common p.d.f.,

[(x)

dx

=-,

pen events occur in ]0, t] out of which one in ]x,x +dxJ liin ]O,xj)

n

g(XI,X2,··' ,x n ) = n!

then

87

E[N(tl)N(tz)] -E[N(tr)l E[N(tz)] ON(t,)ON(t,)

6

PROBLEMS AND COMPLEMENTS

xE]O,t],

-I -i {A(t - x - dx)}n . e-A.(t-x-dx) (n-I-I)!

t - -d )"-I-i = An - i ( x X

(n-I-i)!

-A(t-X)

e

dx.

Consequently the unconditional probability is

SOME TECHNIQUES OF QUEUING THEORY

88

Pen even ts occur in ] 0, t] out of which one in ] x, x

PROBLEMS AND COMPLEMENTS

+ dx])

'line-At n-I (n -1) --L i xi(t-x-dx)n-I-i dx (n -I)! ;=0

1=1,2,3, ....

(b)

[A(t-dx)]n-I e-l\tAdx (n - I)! = (At)n-l e-l\tAdx

(n-l)!

'

where we have ignored the powers of dx higher than 1. If this joint probability is divided by the probability that n events occur in ] 0, t], we get a conditional probability:

Density function of the least r.v. As in (a), suppose that customers come to a certain counter which has k par alleY channels. Now the k service channels, instead of serving only one customer at a time, serve independently k customers if they are available; otherwise the channels serve the lesser number of available customers so that, for example, if 5 < k customers are available, only 5 channels will be busy. If k independent exponential service channels, each with parameter Il, are busy at any given time, then let Xl, X 2 , ••• , X k be their remaining service times. If Z = min (Xl' X 2 , ••• , X h ), then show that fz(z)

P(one out of n epochs of occurrence lies in ]x, x

8

+ dx lin

n lie in ] 0, t]) = - dx. t

It is equally probable that anyone of the n events is responsible for causing an epoch of occurrence to all in the dx interval. Thus the conditional probability that an event occurs in dx, given that it is one of the n occurring in ] 0, t], O";;;x";;; t, is dx/t. Since this conditional probability is independent of 11, it is the unconditional probability sought. Because of this property, the events occurring in accord with the Poisson process are often called random events. (a) Density function of the largest LV. Suppose that customers come to a service counter which has k parallel service channels, such that only one customer can be served at a time. Each job is distributed among the k channels, which are assumed to act independently. Let the channels have service times Xl' X 2 , •.• , X k which are independently exponentially distributed with a common parameter Il. Show that Z = max (XI, X 2 , ... , X k) has a p.d.f. fz(z) and a c.f. tPz(w) given by fz(z) = kll{(l-e-/J.Z)k-l -(1-e-/J. Z)k} and

where

(~) is a combination sign and i = V=T From this c.f. or other-

wise, show that

=

klle-h/J.z.

Note that we have used the forgetfulness property of the exponential. In words, if Xi, i = 1, 2, ... , k, represents the remaining service time of the ith customer, then Z represents the remaining service time of the first of the k customers who are going to leave the counters, and its p.d.f. is fz(z). This property is used extensively in multiserver queuing systems. Problems (a) and (b) can both be generalized to the case when the LV.'S connected with the service times are exponentially distributed, but with different parameters III , 1l2, ... , Ilk' 9

(a)

Show that the solution of the differential-difference equations for the bulk-arrival Poisson process (2.2.4a) can be put in the follOwing recurrence relation form, from which it is easy to compute numerically the probabilities Pn(t) of the process:

poet)

= e-I\t n = 0,1,2, ....

(b)

If the group-size distribution qk follows the geometric distribution qk = qk -I (I - q), k;;;;' 1, 0 < q < 1, then it is easily seen that the p.g.f. Po(z; t) takes the form Po(z; t) = e-I\t(l - qz)

I: Ln ( n=O

'11(1 - q) t) (qz)n q

from which we get (3(1 ) £(Z) = Il 2(3(2) - «(3(1»)2

Var (Z) where

89

PoCt) = e-I\t Pn(t) = e-Atqn [Ln( -A(lq-q)t) -L - (-A(lq-q)t )] n I

n = 1,2,3, ...

SOME TECHNIQUES OF QUEUING THEORY

90

where Ln(x) is a special case (a =! 0) of the more general form of the Laguerre polynomial L~"')(x). This polynomial is related to the general. ized hypergeometric function discussed in Chapter 1 by the relation L("')(x) = n

( 1 +a) n IFI (- n; 1 + a;x). n!

Hint: Case 1: M(t) = It m(x) dx, and so on. Case 2: M(a) = I';; e-"'tM(t) dt, and so on. 14 Continuation of Problem 13. Show that the result of Problem 13 may also be obtained by conditioning on U 1 • Hint: M(t) = E(N(t» = I'O E(N(t)1 U I = u) dA I (u). But 0, E(N(t)IUI = u) = { 1 +M(t-u),

The generating function of Ln(x) is found by setting a = 0 in formula (22.9.15) of Abramowitz and Stegun (1964), and is -xz (l_Zfl exp [1- z

1=

I=

Ln(x)zn.

n=O

For more details on the Laguerre polynomials, the reader is referred to Rainville (1960) or Abramowitz and Stegun (1964). (c) If groups arrive in batches of fixed size, say m, then one can see that Po(z; t) = exp [A(Zm - l)t] and consequently Pkm(t) = (At)k exp (- At)/k!, k = 0,1,2, .... to Show that for the bulk·arrival Poisson process {N(t), t;;" O}, E(N(t)IN(O) = 0) = Aqt

Var (N(t)IN(O) = 0) = [a 2 where q = 1:;;'=1 kqk and arrival instant.

a2

+ cr J At

is the variance of the number arriving at each

Section 2.3 11 Show that the d.f. of N(t) is given by

91

PROBLEMS AND COMPLEMENTS

Hint: show

o

udA(u)
This can be proved by several methods. Perhaps the easiest one is to lim f=e-"'Uh(u)dU = l. ","""0

0

Show that the mean of the r.v. having a p.d.f. h(u) is !J.2/2a, where !J.2 = I';; u 2 dA(u). 13 By integrating equation (2.3.9) or otherwise [by taking the L.T. of M(t) given in equation (2.3.6)], show that M(t) = A 1 (t)

+ rM(t-U)dA(U)

which is a renewal equation for the mean value function M(t).

and so on

1,

12 Prove that h(u) = [1- A(u)] fa, u;;,. 0, is a p.d.f. where

r

u ~ t,

This may be stated in words as follows: If the first item did not fail up to time t, then there are no renewals in [0, t] . On the other hand, if U = u < t, then there is a renewal at u and, on the average, M(t - u) more renewals will occur in time t - u. In the terminology of Section 2.2, we use the fact that u is a point of regeneration for the regenerative process {N(t)}. 15 (a) If the renewal periods of the ordinary renewal process are exponentially distributed ~th means l/A, A> 0, then from the renewal equation for met) show that met) = A, and hence P[r(t) ~xl = 1 - e- AX , x;;,. 0, t > O. This proves the forgetfulness property of the exponential distri· bution, and also that in a Poisson process the residual life has the same distribution as every other life. Hint: For finding met), use equation (2.3.10). For finding the d.f. of ret), either (1) take the L.T. of equation (2.3.13) or (2) use the fact that a Poisson process has stationary independent increments. (b) (i) Show that in a Poisson process with rate A, the d.f. of the past life I (t) is given by O~x
P[N(t) ~ n] = l-AI(t)*A(nlt).

O
u>t

(c)

(li) Show further that E(l(t» = [l-e-At]/A. Hint: (i) Proceed as in (a). (li)Firstmethod: I';; x dFI(tix)=I& Axe-AX dx + te-At, and so on. The contribution te-At comes from the concen· tration of probability mass e- At at x = t. Second method: IS e- AX dx, and so on. Total life. If we define T(t) as the total life of the component in service at time t, then clearly T(t) = let) + ret) and consequently one may show by using (a) and (b) that for a Poisson process, 2 _e- At

E(T(t)) =

A.

which shows that the total expected life is significantly larger than the expected life l/A = E(Un ) of any particular renewal period. The signifi· cance becomes sharper if we let t -+ 00, in which case E(T(t)) = 2/A = 2E(Un ). These facts seem to be paradoxical. However, in reality there is

92

SOME TECHNIQUES OF QUEUING THEORY

(d)

no paradox involved, and this seeming discrepancy may be explained as follows. Let us see how we define T(t). First we fix t, an arbitrary point on the time axis. Then T(t) is defined to be the duration of that renewal period which contains t. Since some renewal periods are definitely going to be of longer duration than others, the procedure adopted indicates that there is a higher probability of a long renewal period containing the point t than of a period of short duration. In the language of statistical sampling theory, we may say that T(t) is determined by length-biased sampling of a renewal period from a sequence of (long and short) renewal periods. In (d) below, we proceed to make a precise statistical interpretation of the intuitive idea oflengthbiased sampling, and hence give an alternative derivation of the limiting distribution of residual life. Let Up denote the renewal period which contains an arbitrary epoch P fixed in advance in a certain arrival process, say 10 A.M. in a bus arrival process. Now renewal periods of length u occur with a probability density a(u) = dA(u)ldu. The length-biased sampling phenomenon states that the probability the renewal period ends in Ju, u + du] should be proportional to u. Hence one can say that the probability of the renewal period thus selected is Ku dA(u). Applying fO' Ku dA(u) = 1 immediately leads to K = lla, the common mean of the renewal periods. As a consequence, the p.d.f. ap(u) of Up is given by

(e)

of the residual life Y is more general (in a sense that here the renewal periods need not be independent) than the earlier discussion leading to the distribution of ret) as t ..... 00, though it is interesting to observe that the two are identical, provided the latter one exists. As a consequence, the above discussion applies to a wider class of stochastic processes than the renewal processes. Note also the fact that the discussion leading to the distribution of the residual life Y is based on the assumption that the sampling point is randomly distributed over the renewal period in which the sampling point falls. The earlier discussion leading to a derivation of the limiting distribution of ret) was based on the assumption that the sampling point was fixed at an epoch t after the process had been in operation for a sufficiently long time. Suppose we are interested in the distribution of the position qf a random customer in an arrival group of size X. Intuitively, the arguments used in (d) may be adapted to the case of a discrete distribution. For mathematical details, see Burke (1975). The distribution of the size of the group in which the random arrival falls should be proportional to ia;, where P(X = i) = a;, i;;;' 1, and is thus given by ia;!li, where a = Li= 1 ia;. Consequently, if the position of the random customer within the arrival group is J, then rj

I

or

ada

(f)

limP[r(t)~x,l(t)~y] =!a = -1 a

Y Uo

=

1 -A(y)

a

a dy

that is, the p.d.f. of Yis [1 - A(y)] la, which is the required density. It should be noted that the present discussion leading to the distribution

ai

In particular, if (i) ai = 0ib k = 1,2,3, ... , then rj = 11k, as it should. (ii) ai = (I - a) ai-I, 0 < a < 1, i;;;' 1, is a geometric distribution, then rj = (1 -a)a i - 1 , the original distribution. This, in a way, proves the forgetfulness property of the geometric distribution. Show that the limiting joint distribution of ret) and let) is given by

O
P[y~Y~y+dy] =rdYUodA(Uo)

1-

i =j

t~QO

and consequently the unconditional probability is given by

i~1 a. [ ;=1]/

=

P(J = j) ==

ap(u)du = udA(u). a

Knowing the distribution of the duration of the renewal period Up that contains the fixed point P, we can find the distribution of Y, the residual life measured from the fixed point. In fact, conditioning on Up = uo, we note that, by the random property, Y is rectangular (uniform) over the interval] 0, uo] . Thus the conditional probability of Y is

93

PROBLEMS AND COMPLEMENTS

+ y)-A(u)]

du

fY [A(u +x)-A(u)]

du

fX[A(U 0

0

Hint: Noting that the event {ret) > u, let) > v} ~ no renewals in ] t u, t + v] .... {ret - v) > u + v}, lim P{r(t»u, let) >v} =

t-+ oo

lim P{r(t -v»u

t-+ oo

= -1 a

J=

u+v

This implies that the jOint p.d.f. is a(u

+ v}

[l-A(x)] dx.

+ v)la. Thus

SOME TECHNIQUES OF QUEUING THEORY

94

lim P{r(t) ';;;x, let) ';;;y}

t-+ oo

(g) (h)

PROBLEMS AND COMPLEMENTS

= 'f.X.y J a(u a+ V) du dv 0 0

~oof: _Let F(t) =-¥(t) -M(t -c). Then taking the L.T., we get F(a) =M(a) - e-C"'M(a). Hence

and so on. Show from (t) that the limiting marginal distributions of ret) and let) as t -? 00 are the same as obtained earlier. Continuation of (t). Show that for a Poisson process with rate A, lim P{r(t) > u, let) > v} =

lim aF(a) =

Q;-+O+

=

(iv) Key renewal theorem.

Pro~f:

. M(t) 1 lim-- =-. t a

t .... ""

and

[(il(a»)am(a)] ex

a-+O+

= f&H(t -

=H(a) m (a). Hence, lim aF(a) =

0:-+0+

0

v) dM(v). Then taking the L.T., we

lim H(a)am(a)

a-+o+

t-+oo

f"" H( v) dv -a

0

provided f'; H(v) dv < 00.

(b)

The r.v. X, representing the life of a certain item has a p.d.f. given by

f x (x)=4xe- zx ,

Thus lim aM(a) = 1 + lim

a

· F() 11m t=

Proof: (i) Since limt .... ""f(t) = lim"''''''o+o:1(a), and M(t) = fb m(x) dx, t > 0, by definition, we have from equation (2.3.12) on taking its L.T.

a-O+

Show that

0

Le.!. F(0.

get F(a)

a

a

t 1 fO<> lim f H(t - v) dM(v) = H(v) dv.

t-+co

-? 00.

a

1 -e- c ", lim am(a) - - -

c = -'c = a a

t .... ""

M(a) = ill (a) + il(a) m(a).

lim m(a)(l-e- C "')

Q--}oO+

"' .... 0+

e-;>"(u+V)

This implies that the limiting distributions of ret) and let) as t -? 00 are independent, as they should be. 16. As stated in the text, in (a) below we give nonprobabilistic nonrigorous deri· vations of some of the limit theorems, using the L.T. and based on the assumption that the L.T. is applicable to the functions under discussion. (a) Show that:

(ii)

lim aM(a)(1 - e- C"')

a-+o+

=

t->""

(i) lim M(t)

95

x>O.

Supposing that the renewal process starts with a new item, show that the renewal function M(t) then is given by 1 e- 4t M(t) = t - - + -

4

This implies

lim M(t) = limoo t-+

t-+- co

=

(ii) Now lim M(t) t-+- co

(iii)

t

=

·t

j

0

m(x) dx

lim_ aM(a) -? 00.

a-+O+

lim f&m(x) dx t-+co t

=

Hint: Use the L.T. of the renewal equation for M(t) and then invert. If N(t) is a Poisson process with mean At = y, then show that

17 An important identity. Y

e-xx n

n

e-Yyk

Io -n!- d x = l - I -k! - . k=O

lim met) t-"'"co

where in getting the last step, we have used L'Hopital's rule since by (i)M(t)-+ 00 as t -? 00. The last limit, we know, tends to l/a. Blackwell's theorem. Show that c lim [M(t) - M(t - c)] =-.

t-+fXI

4

a

Hint: This can be proved by several methods. (i) Integration by parts of the left-hand side gives the right-hand side. (ii) Let

Differentiation gives

SOME TECHNIQUES OF QUEUING THEORY

97

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

96

Mz(t) = M(t)

+ 2 S:M(t-r)dM(r),

t>O.

or

MISCELLANEOUS PROBLEMS AND COMPLEMENTS Integration by parts gives g(y) = g(y) - 1 + c

=?

C

= 1

which proves the result. and the convolution of exponential (iii) Using Wn ";;; t * N(t);;;' n we get Fwn(t) =

S: f(n)(x)dt

'A('Ax)n-le-AX

t

-

Hence,

LV.'S,

S0

(n -1)1

e-At('At)k

=

dx

= k L= nk'·

"-1 -At('t)k ~ e __ .o...I\-,-

.Ate-zz n - I

--dz=l-L.. k=O Jo (n - 1)1

I·

k.

. d tfng 'At = y gives the desired result. Various Changmg n - 1 to n an pu 1 , St . ··ans · t: ns of this identity have been discussed. For details, see eml] generallza 10 (1971) and references therein. 18 For the equilibrium renewal process, prove that: (a) (b) (c)

1

met) = -. a t

M(t) = -. a M(t

+ c) -

(d)

P(r(t)";;;x) ==

(e)

P(lCt) ";;;x) ==

Hint:

-

M(t)

r 0

c

a

,

l-A(u) a du,

x1-A(U)

fo

c;;;.o.

a

du,

x>o. x>O.

(a) show that mea) = l/aa. (b) and (c) follow from (a). (d) follows by

using (b).

. Sh w , h Let H(t) = l/c for 0 < t";;; c and zero otherWIse. 0 19 Blackwell s t eorem. al theorem that Blackwell's theorem follows by using the key renew . ()_ 20 Show that for the ordinary renewal process, the second moment M2 t E(N 2 (t)) satisfies the renewal equation

2

In many physical situations we are interested to know the nature of the sum or difference of two Poisson processes. As an example, if {N1 (t), t;;;' O} and {N2 (t), t;;;' O} are two independent Poisson processes with mean rates 'AI and 'A 2 , respectively, then show that: (a) The sum N(t) = Nl (t) + N2 (t), t;;;' 0, of the two Poisson processes is a Poisson process with mean rate 'AI + 'A 2. (b) The difference N(t) = Nl (t) - N2 (t), t;;;' 0, of the two processes is not a Poisson process. The process {N(t), t;;;' O} has as state space the set {O, ± 1, ± 2, ... }. (c) In both (a) and (b) {N(t), t;;;'O}has stationary independent increments. [The result stated in (a) can easily be generalized to n independent Poisson processes.] Hint: Use the p.g.f. Continuation of Problem 1. As opposed to pooling the two independent Poisson processes to obtain a new Poisson process, we can also get two independent Poisson processes by branching a given Poisson process, provided the branches are selected independently. Example: consider a Poisson stream of vehicles approaching a fork in the road. Assume that each vehicle, independently of all other vehicles, takes the left-hand road with probability PI and takes the right-hand road with probability P2, where PI + P2 = 1. If the rate of the given Poisson process of vehicles entering the fork is A, then the units going into the ith branch follow a Poisson process with rate APi' i = 1, 2. Hint: Let N(t) be the number arriving in time t in the original Poisson process and Ni(t) the number going into the ith branch over the same time interval. Then show that

where nl + n2 = n. Remove the conditioning on the value of N(t) by multiplying by the probability that N(t) = n to get

98

3.

SOME TECHNIQUES OF QUEUING THEORY

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

:u:

Since the joint distribution of NI (t) and 'N2 (t) factors into two Poisson distributions, the units going into the two branches are following independent Poisson processes, This result can easily be extended to the case when the number of branches is k instead of 2, using the multinomial distribution instead of the binomial one for the conditional joint probability. Consider a stochastic process N(t) = :Zf;!-f) Z;, where the Zi'S are U.d. r.vo's distributed as Y and X(t) is a stochastic process independent of y, The process {N(t), t;;:;;' O} is called a compound process, to which various names may be given, depending on the nature of the process (X(t), t;;:;;' O}. Thus, for example, if {X(t) , t;;:;;' O} is a Poisson process, then {NCt), t;;:;;' O} is called a compound Poisson process, Suppose that {X(t), t ;;:;;. O} is a Poisson process with mean rate A, then: (a) Find the c.f., XN(t)(W), of {N(t), t;;:;;' O}. (b) (c)

(d)

(d)

= Po + J=

x=o

wq(a)

exp [At(XY(W) - 1)],

(f

+

= e-CiUb(u x) u=o 1 _ B(x)

)" du dx

eA[b(O<)-lJx

{f=u=o b(u+X)e-CiUdU} dx

J=

eA[b(Ci)-lJ x

(f=

C(a)

=

= Po +C(a) f~ x-o = Po

AtE(Y), AtE(y2),

+ C(a)

x=o

y=x

b(y)e-Ci(Y-X) d ) dx y

where

AtIE(y2),

if tl "'- t2

{ At2E(y2),

ift2 "'-t l

.

P(N(t)

=

n

n)

= h~O

A _ wq(a) = (l-p) ( l-;[l-b(a)]

)-1

P yl1

~here = and Po, has been determined by using Wq(O) = 1. Clearly wq(a) obtaIned here IS identical with equation (2,2.28). This confirms our assertion made earlier that the virtual waiting time of a fictitious customer is the same as the actual waiting time of a real customer.

e-At(At)h k! {qn}h*

where {qn }h* is the n-fold convolution of {qn} with itself, and {qn}o* = onO' This shows that the bulk-arrival Poisson process may be seen as a Poisson stream with randomly varying batch size. Waiting time (in queue) distribution for an MIG!! queue, (a) In this problem we discuss an alternative derivation of equation (2,2,28) and, in fact, discuss the virtual waiting time (in queue) distribution. It will be seen that for MIG!l the distribution of the actual waiting time (in queue) of an arrival is the same as that of the virtual waiting time (in queue) of a fictitious arrival.

ArhCa) - 1] Po b(a) - b [A - Ab(a)] ,

~ow in. the double integral, first changing the order of integration, then IntegratIng W.r.t. x, we get, on simplification,

We may remark here that a compound Poisson process has a c.f. of the form exp [A.t(Xy(w) -1)], The bulk-arrival Poisson process whose p,g,f. is exp [At(Q(Z) - 1)] (discussed in Section 2,2) is therefore a compound Poisson process, Furthermore, the distribution of N(t) can be written in the following explicit form without the use of either the p.g.f. or the c,f.;

4

Po(lj(a);x) b(a)

where PoCz; x) is_defined in Section 2.2. Using equation (2,2.24a) to find the values of Po(b(a);x) and Po (h(a); 0), we get

O}.

Answers: (a) XN(tlw) (b)

y random instant of time in steady state, Pn(x) is the joint At probabIllty and probability density of the number n of customers in the system, including the one in service, and the elapsed service time x of the customer undergoing service, The time required from the random instant to ~om~lete service on the n customers is the convolution of complete semce tImes of the (n - 1) waiting customers and the residual service of the customer undergoing service. Hence the LT. of the distribution of the virtual waiting time (in queue), including zero waits, is

Find the mean and the variance of {N(t), t;;:;;' O}, Show that the process {N(t), t;;:;;' O} has stationary independent increments. Find the covariance, cov (N(tl)' N(t2»' of the stochastic process {N(t), t;;:;;'

99

(b)

The limiting waiting time (in queue) distribution for the queuing system

MIGII may be derived by still another method. If vn denotes the Waitin.g time of the (n + 1)th arrival occurring at o~ (with o~ = 0 corresponding to the zeroth arrival), then V;+l = (V;

+

Vn + 1

-

Un+1f,

n = 0,1, .. ,

where Vn + 1 is the service time of the (n + l)th arrival and U is the interarrival time between the (n + l)th and the (n + 2)th arri;~s. The

SOME TECHNIQUES OF QUEUING THEORY

100

r.vo's Vn + 1 and Un + 1 are indepen'dently distributed with p.d.f.'s b(x) and A exp (- AX), x;;;;. 0, respectively. Show that in the limiting case when n -+ 00, the L.T. of Vq is the same as that given in (a) above. Hint: w q (a,n+1) = E(exp [-aV;+l)) = E(exp [- a(V;

+ Vn+ 1 -

A A-a

-

_ _a_p(V;

+ Vn + 1

Un + 1

(c)

(d)

S

-

(1 )

where the integral is the d.f. of an Em r.v. and represents the probability that all the m customers complete service before the time period y expires. The right-hand of equation (1) may be evaluated by several methods. Show that it is equal to

Un + 1 <0)

by Problem 8 (b) of Chapter 1. Now using the fact that peV; + Vn+l -Un +!
MC(y)[l-e-).LO-'Y,)y]. Hint: Different authors have evaluated the expression on the righthand side of equation (1) by various methods. The simplest way is to use first the value of P~ and then evaluate the integral (note that Pg == 0). Another way is to replace first the integral by the summation

~

wq(a) = (l-p)

L

Find the L.T. pa(a) of paCt) given in equation (2.2.29) and show that lima->o apa(a) = A, as it should. Note that this corresponds to the sum of probabilities discussed in equation (2.2.30). Show that the probability Po(y) can be written as

rn

= - - wq(a, n) b(a)

A-a

101

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

[pR(a)]n

n=O

which on inversion gives wq(t) = (l - p) L;=o p"{R(t)r* ~here R(t) is the residual service time (see Section 2.3). R(t) has the L.-S.T. R(a), and is given by R(t) = J.!

6

f:

and then sum over m. This is done as follows:

[1 - B(x)] dx.

MC(y)

The above formula for wq(a) has an interesting interpretation. It shows that the waiting time (in queue) has a p.d.f. similar to that of MIMIl, if time is considered in such a way that the remaining service time is taken as the fundamental unit for all the n potential customers which a customer finds on arrival. Consider an MIMll queuing system. Show that the mean waiting time (in the system) and the mean busy period for this system are the same, namely, 1/(J.! - A), but the corresponding variances are different, that is, 1/ [J.!(1 - p)] 2 and (1 +p)/[J.!2(1-p)3J respectively.

= MC(y)

(a) (b)

Show that a[J.!(l- z)] of equation (2.2.32) represents the p.g.f. of the number of services completed during an interarrival time. Show that the root used in equation (2.2.34) is, in fact, real and lies in] 0, 1 [. Hint: Use (a) and the Appendix A.2.

1

[1- ~~1 (J.!y);;-ILY]

[1 _1 -11 f 11

r=o

(J.!yY,er.

ILY

I 1r] m=r+1

and so on. Still another method, using complex variables, is given by

Problems on GI/M/l

7

mt (l-'Ylhr-

11

(e)

where r is a suitable inversion contour and b(a) = J.!/(J.! + a). Now first perform the summation over m and then invert by using partial fractions and inversion tables or by using the residue calculus (see Appendix A.3). Note that once equation (l) is evaluated, one can see that equation (2.2.47), which we have used in the text, is satisfied. In addition, note also that if we use P;" instead of p~ in equation (1), then the norrning constant is automatically ignored if the new value of Po(Y )/A C(y) is used in equation (2.2.46). UsingPo(y),show!hatPo = I-p.

SOME TECHNIQUES OF QUEUING THEORY

102

Hint: Po =

=

fo P (y) dy

[1'

/I. - -

=

l-a[p(l-/I)]]

A.

0

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

The moments of I may be obtained through the relation [implied by equation (2.2.44)]

1 -II

P(jE(r) = E(1;),

= l-p

8

where in getting the last step we have used equation (2.2.32). Continuation. In this problem we discuss an alternative derivation of FrJr). This approach is based on the discussion of waiting-time ~istribution f~r GI/G/I by the integral equation technique. Therefore the notatIOn used here IS the same as that in Section 2.2. Define the r.v.Iv(n) as

(b)

_ W = q

P(Vq

+ V < (u -7»

-7»

9

-7 -

E(12) --2E(/)

r

dA(u).

But as P(Vq + V < (u = J~-T P(Vq < u v)pe-}J.Y dv and P(~q ~ t) = 1 - I I exp [- p(1 - 'YI )tJ, substituting appropriately, we get the deSIred Fr (7). Compare this result with the result obtained in equation (2.2.49). C;ntinuation: moments of Iy in GI/M/l and GI/G/!, and some other relations in GI/G/I. (a) Show that the first three moments about the origin of Iv in a GI/M/l queue are given by

2E(S)

Hint: By Section 2.2 and Miscellaneous Problem 8, v~n+J) = (v~n) + Snr and Iy(n) = - (V~n) + Sn)- imply v~n+ J) - Iy(n) = v~n) + Sn' Squaring both sides, and noting that v~n+J) IyCn) = 0, gives [v~n+J)J 2 + [2v (n) = v(n)] 2 + S2n + 2 v(n) V(n) q q Sn' Taking expected values'since q and Sn are independent, we have, in the limit, on using P(jE(12) = E(Ji,) from (a), the desired result.

Then Iy represents the virtual idle time, and its distribution function is defined as follows: Fly(7) = P(1y>7)

T

£(S2) -

where S = V - U is an LV. representing the difference between the service time and the interarrival time, and I is an LV. represel'lting the actual idle period.

where (x)- = min (0, x). The corresponding relation in the limiting case as n ~ is

00

r

r = 1,2, .....

The moments of the idle-time distribution and the waiting-time (in queue) distribution are related and may be obtained even for the more general queuing system GI/GII. For the queuing system GIIGII in steady state (p < 1), show that

Iv(n) = _(v~n)+ Vn+ 1 -Un +d-

=

103

(c)

Continuation of (b).

wq

Show that the result in (b) can be written as 2

= A. (a; +a;)+(1-p)2 _£(12) 2(1-p) 2£(1)

a;

where and a~ are the variances of interarrival times and service times, respectively. (d)

Continuation of (c). Specialize the results obtained in (c) for the queuing systems MIG/l and GIIMII and compare with the results obtained elsewhere. Hint: For MIG/I, E(1) = I/A. and £(12) = 2/A.2, and for GI/MII use £(1;) = P oE(I2) and (a).

(e) s _ _ E(1y) - as

Hint:

}l

3E(1;) (1 ).

inCa) = hy(a)iv(a)

-II

where D, Iv, and V are interdeparture, virtual idle period, and service times, respectively.

Take the L.-S.T. of equation (2.2.49), which is given by _ fry (a) = a(a)

a['Y1 -a(a)] + a - p(1 - 'Yd

where lrvCa) = J'(;' e- at dF1v(t), and so on, and a r = J~ ura(u) du with al == I/A., r = 1,2, .... Expand the right-hand side of fr/a) as a power series in a and pick the coefficients appropriately.

Show that the L.T. of the distribution of interdeparture interval for the queuing system GIIGll, when p < I, is given by

(t)

Hint: Let Dn be the time between the nth and (n + I)th departures. Thus Dn is the service time of the (n + l)th customer plus the virtual idle time: Dn = Vn+1 + Iy(n). The proof now follows on using the independence of the LV.'S V n + 1 and Iy(n) and proceeding to the limit as n -+ 00. Continuation of (e). Show that the mean and the variance of Dare

SOME TECHNIQUES OF QUEUING THEORY

104

£

CD)

1

"+ 2

av

= a~

Cox, D. R. (1955). The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables. Proc. Cambridge Phi/os. Soc. 51, 433-441. - - - - (1962). Renewal theory. Methuen, London.

£([2) _ (l - P)2 V

+ 2a~

,,2

2(1-p) Wq

where the various quantities have been defined earlier. Hint: Consider the derivatives ofJv(ex) at ex = 0 and use (a) and (c). Continuation of (e). Show that for the special case MIG/1 ex)f- ( ) f-D (ex) -- p(J.L+ ,,+ex vex

Hint:

For M/G/I,

Po = 1 -

1;/ex) (h)

105

Cooper, R. B. (1981). Introduction to queueing theory, 2nd ed. Elsevier North Holland, New York.

"

(g)

REFERENCES

p,

and by equation (2.2.45),

p(/1 + ex) = '--'.:.."-+-ex-"-

Continuation of (e). JvCex) in (e) indicates that if Vis exponential with parameter /1, then D is exponential with parameter ". Conversely, if Dis exponential with parameter A, V is exponential with parameter /1, implying thereby that the exponential distribution with parameter /1 is the only service time distribution for the system MIG/1 for which the output process is Poisson with rate A.

REFERENCES Abramowitz, M., and L A. Stegun (Eds.) (1964). Handbook of mathematical functions. Applied mathematics series 55, National Bureau of Standards, Washington, DC.; also reprinted 1965, Dover, New York. Bartlett, M. S. (978). An introduction to stochastic processes, 3rd ed. Cambridge University Press, London. Bharucha-Reid, A. T. (1960). Elements of the theory of Markov processes and their applications. McGraw-Hill, New York. Bhat, U. N. (1968). A study of the queueing systems M/G/l and GI/M/l. Lecture Notes in Operations Research and Mathematical Economics, Vol. 2. Springer-Verlag, New York. Burke, P. 1. (1975). Delays in single-server queues with batch input. Oper. Res. 23,830-832. Chung, K. L. (1967). Markov chains with stationary transition probabilities, 2nd ed. SpringerVerlag, Berlin. <;:inlar, E. (1975). Introduction to stochastic processes. Prentice-Hall, Englewood Cliffs, NJ. Cohen, J. W. (1980). The single server queue, 2nd ed. North-Holland, Amsterdam. Conolly, B. W. (1958). A difference equation technique applied to the simple queue with arbitrary arrival interval distribution. J. R. Stat. Soc., Ser. B 20, 168-175. - - - - (1960). Queueing at a single serving point with group arrival. J. R. Stat. Soc., Ser. B 22,285-298. - - - - (1975). Lecture notes on queueing systems. Ellis Horwood Ltd., Chichester, Sussex, England.

- - - - and H. D. Miller (1965). The theory of stochastic processes. Methuen, London. Dafermos, S. C., and M. F. Neuts (1971). A single server queue in discrete time. Cah. Cent. Etud. Rech. Oper. 13, 23-40. Doob, J. L. (1953). Stochastic processes. Wiley, New York. Dynkin, E. B. (1965). Markov processes. Springer-Verlag, Berlin. Feller, W. (1968). An introduction to probability theory and its applications, Vol. 1, 3rd ed. Wiley, New York. - - - - (1971). An introduction to probability theory and its applications, Vol. 2, 2nd ed. Wiley, New York. Finch, P. D. (1959). A probability limit theorem with application to a generalization of queueing theory. Acta Math. Acad. Sci. Hung. 10,317-325. Franken, P. (1978). A remark on the stationary availability. Math. Oper. Ser. Optimization 9 143-144. ' Gani, J. (1957). Problems in the probability theory of storage systems. J. R. Stat. Soc. Ser. B 19,181-206. ----(1969). Recent advances in storage and flooding theory. Adv. Appl. Probab. 1,90-

110.

Gaver, D. P. (1954). The influence of service times in queueing processes. Oper. Res. 2, 139149. Gross, D., and C. M. Harris (1974). Fundamentals of queueing theory. Wiley, New York. Henderson, W. (1972). Alternative approaches to the analysis of the MIG/1 and G/M/1 queues. J. Oper. Res. Soc. Jpn. 15,92-101. Hille, E. (1964). Analysis, Vol. 1. Blaisdell, New York. Jaiswal, N. K. (1960). Bulk-service queuing problem. Oper. Res. 8, /39-143. - - - - (1968). Priority queues. Academic Press, New York. Karlin, S., and H. M. Taylor (1975). A first course in stochastic processes, 2nd ed. Academic Press, New York. Kemeny, J. G., and J. L. Snell (1960). Finite Markov chains. Van Nostrand, Princeton, NJ. Kendall, D. G. (1951). Some problems in the theory of queues. J. R. Stat. Soc., Ser. B 13, 151-185. - - - - (1953). Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24, 338-354. - - - - (1957). Some problems in the theory of dams_ J. R. Stat. Soc. Ser. B 19, 207212. Kendall, M. G., and A. Stuart (1969). The advanced theory of statistics, Vol. I, 3rd ed. Griffm, London. Khintchine, A. J. (1932). Mathematical theory of a stationary queue. Mat. Sb. 39,73-84. Kingman, J. F. C. (1966). On the algebra of queues. J. Appl. Probab. 3, 285-326; also Methuen's monographs on applied probability and statistics, supplementary ser., Vol. 6, 1-44. Kleinrock, L. (1975). Queueing systems, Vol. 1. Wiley, New York. - - ' - - - (1976). Queueing systems, Vol. 2. Wiley, New York. Kosten, L. (1973). Stochastic theory of service systems. Pergamon, Oxford.

106

SOME TECHNIQUES OF QUEUING THEORY

Krakowski, M. (1973). Conservation methods in queueing theory. Rev. Fr. Autom. In! Rech. Oper. 7, 63-84. . . ----(1974). Arrival and departure processes in queues. Pollaczek-Khmtchme formulas for bulk arrivals and bounded systems. Rev. Fr. Autom. In! Rech. Oper. 8,45-:-56 .. Levy P. (1965). Processus stochastiques et mouvement brownien, 2nd ed. Gauthier-Villars, Paris. Ph 7 S Lindley, D. V. (1952). The theory of queues with a single server. Proc. Cambridge lOS. oc. 48,277-289. . Luchak, G. (1956). The solution of the single channel queueing equations characte~lzed .by a time dependent Poisson distributed arrival rate and a general class of holdmg times. Oper. Res. 4, 711-732. Meisling, T. (1958). Discrete time queueing theory. Oper. Res. 6,96-105. Moran, P. A. P. (1959). The theory of storage. Methuen, London. Neuts, M. F. (1973). Probability. Allyn and Bacon, Boston. Newell, G. F. (1971) Applications of queueing theory. Chapman and Hall, London. . Palm, C. (1943). IntensiHit sschwankungen im Fernsprechverkehr. Ericsson TechnIcs no. 44, 1-189. Parzen, E. (1962). Stochastic processes. Holde~-Day, San Francisco. -46 Prabhu, N. U. (1964). Time-dependent results m storage theory. J. Appl. Prob. ba 1, 1 '. - - - - (1965a). Queues and inventories - A study of their basic stochastic processes. Wiley, New York. - - - - (1965b). Stochastic processes. Macmillan, New York. Rainville, E. D. (1960). Special functions. Macmillan, New York. Regis, R. C. (1973). Multiserver queueing models of multiprocessing systems. IEEE Trans. Comput. 22, 736-745. Romanovsky, V. I. (1970). Discrete Markov chains. Wolters Noordhoff, Groningen, The Netherlands. Seal, H. L. (1969). Stochastic theory of a risk business. Wiley, New York. .. Seneta, E. (1973). Non·negative matrices An introduction to theory and appizcatlons. George Allen and Unwin, London. Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge Phi/os. Soc. 49, 449-461. . 43 302 - - - - (1958). Renewal theory and its ramificatIOns. J. R. Stat. Soc .• Ser. B 20, 2 . Steinijans, V. W. (1971). Crossing times in cumulative processes. South Afr. Stat. J. 5,63-66. Takacs, L. (1962). Introduction to the theory of queues. Oxford University Press. Ne,,: York. ----(1967). Combinatorial methods in the theory of stochastic processes. WIley, New York. . d h' Wishart, D. M. G. (1959). A queueing system with service-time distribution of mlXe c 1squared type. Oper. Res. 7, 174-179. . - - - - (1961). An application of ergodic theorems in the theory of queues. Proceedmgs of the Fourth Berkeley Symposium on Mathematical Statistics and ProbabIlitY, Vol. 2, University of California Press, Berkeley and Los Angeles, 581-592.

Chapter 3 BULK-ARRIVAL QUEUES The study of bulk-arrival queues may be said to have begun with Erlang's solution of the MIEk/l queue [see Brockmeyer et a1. (1948)] since this gives, implicitly, the solution of the Mk IMI! queue (see Section 3.1). Explicit consideration of bulk-arrival queues [see Gaver (1959)] seems to have begun several years after the work of Bailey (1954) on bulk service. We have arranged the chapters in the reverse order, which appears more natural. In this chapter we discuss single-server queuing systems with bulk arrival, and in Chapter 4 queues with bulk service. For interpretation of some bulk-arrival queues as single-arrival queues, or bulk-service queues as single-service (When the server services one customer at a time) queues, the reader is referred to Chapter 6 and the references therein. In "ordinary" queuing problems it is assumed that customers arrive singly at a service facility. However, this assumption is violated in many real-world queuing situations. Letters arriving at a post office, ships arriving at a port in convoy, people going to a theatre, restaurant, and so on, are some of the examples of queuing situations in which customers do not arrive singly, but in bulk or groups, where the size of an arriving group may be an r.v. or a fixed number. Mathematically and also from the practical point of view, the cases when the size of an arriving group is an LV. are more general, and also more difficult to handle. Many authors have contributed to the theory of queues with bulk arrival, and in this chapter we discuss some of their works.

3.1

THE SYSTEM MX/E.,/1

Let us start with the discussion of a queuing system in which customers arrive in batches according to a time-homogeneous Poisson process with mean rate ft.. The batch size X is an LV. and m

1,2,3, ...

with

/z/ <: 1. We shall assume that O
108

BULK·ARRIVAL QUEUES

small interval of duration ~ t, the probability of no arrivals at the queue is 1 A~t+O(~t) and that of arrival of m customers is Aam~t+O(~t). Batches are served in the order of their arrival. The order of service of customers within batches is not considered in this model. If a batch arrives to find the server idle, a customer of the batch receives immediate service. The processing times or service times of customers who are served singly by a single server are Li.d. r.v.'s, having a modified Erlangian dJ. B(v) with

3.1

THE SYSTEM MXjE J ll

at time t

109

+ ~ t with that at time t, it can be seen that

Pn,r(t+~t)

=

n

Pn,r(t)(1-A~t-P~t)+Pn,r+l(t)P~t+A

L ajPn-j,r(t)~t j=l

+ P n +1, 1(t) PCr ~ t n ;;",

1,

+ AanPo(t)Cr~ t + O(i:l t),

r;;'" 1.

From this we get the following differential· difference equation: s

dB(v) =

I

j= 1

()j-I Cj p .pv I e-1LVdv (J - 1) .

O
c

~

fo

vdB(v) = p

<

poet) = PCN(t)

n, R

r),

f ajPn-j,r(t)+PCrPn+l,l(t)

n

+ AanCrPoCt),

n ;;", 1,

r;;'" 1

(3.1.1)

where

P~~~(t) = dP n, r(t)/dt. Following the procedure used to get equation (3.1.1), we have corresponding to the empty state of the system, the equation

p~l\t) = - APo(t) + PPl.l (t).

(3.1.2)

The differential· difference equations (3.1.1) and (3.1.2) are known as the forward KOlmogorov equations of the process. These are to be solved under some initial conditions which we SUppose to be

00

where c = 1:;f jCj. As pointed out in Chapter 2, the phases are labeled in reverse order, and a new customer is taken up for service only when service on all phases of the customer under service is over. The system length process {N(t)}, in this case, is clearly non·Markovian. However, it can be made Markovian by considering the vector (N(t), R) where N(t) represents the total number of customers in the system at time t and where, for N(t) > 0, R denotes the number of service phases to be completed by the customer who is being served. Thus the states of the system are (0), (l, r), (2, r), ... ,(l <, r <, s), and consequently we may define for any time t the following probabilities: * P n, ret) = P(NCt)

= -(A+p)Pn,r(t)+pPn,r+l(t)+A

j=l

with b(a) = J;' e- cxv dB (v) = 1:;f Cj {pf(p + O'.)}i. The service times are also indepen· dent of the interarrival times and the batch sizes. In other words, service consists of] Li.d. exponential phases which need not necessarily have a physical meaning. Here] is an r.v. such that P(] = j) = Cj, 1 <,j <, s, with 1:;1=1 Cj = 1. The modified Erlangian distribution (random number of phases) is very useful in modeling real queuing systems, as a generalization of the Erlangian distribution (fixed number of phases). The method of phases (Erlangian procedure or its modified form) helps one to replace a non·Markovian process by a Markovian one, as will be seen shortly, and thus to study the queuing system through the use of Kolmogorov equations. For more details, see Chapter 2. The mean service time is easily obtained as b =

P~~~(t)

>

0,

1 <, r <, s

0).

Pn,r(O) = crol n ,

I

> 0,

1 <, r <, s

(3.1.3)

that is, at time zero there are I units in the system, and the unit under service is in the rth service phase with probability Cr. In the case 1= 0, we suppose Po (0) = 1, but it will be seen that the final result automatically includes the case 1= O. To solve the set of equations (3.1.1) and (3.1.2) subject to the initial conditions (3.1.3), we first repeat the definition of the L.TJCa) of f(t) by [(a) =

Jo= e-cxtf(t) dt,

Rea> O.

(3.1.4)

Integration by parts of the L. T. of dPn, r(t)/dt gives the relation dt lo=e-cxtdPn,r(t) dt

e-cxtPnr(t)I;+O'.J=e-cxtPn,r(t)dt ' 0

Obviously Po, r(t) = 0, 1 <, r <, s. By comparing the state of the queuing system = -olnCr

* In order to simplify the notation used here, the dependence of P n ret) and Po (t) on the value of N(O) is not indicated. This convention will be followed henceforth.

+ aPn,r(a)

(3.1.5)

which has been obtained using equations (3.1.3) and (3.1.4). Applying equation (3.1.4) to the set (3.1.1) and (3.1.2) and using equation

BULK·ARRIVAl QUEUES

110

It may be noted that as x -+ 1, z -+ 1, P(l, 1; 0:)

(3.1.5), we obtain, after transposing tenus, . - (0: + A + f.l)Pn,r(O:) + f.lPn,r+1 (ae) + A

.t ajPn_j,r(O:)

+ f.l C r Pn+l, 1 (0:)

J= I

n~l,

+ AanCrPo(a) + crain = 0,

r~l

(3.1.6) (3.1.7)

(0: + A)Po(O:) + f.lPI, 1 (0:) = O.

Next let us define the p .gJ.'s

s

P(z,x;o:) =

I

00

I

r=1 n=!

Pn ,r(OI.)zn x r.

(3.1.8)

Multiplying equation (3.1.6) by znxr, summing over r (l ~r~s) and n (l ~ n ~ 00), and using equations (3.1.5), (3.1.7), and (3.1.8) and A(z), we obtain, after exercising some patience,

AC(x)A(z)Po(a) + C(x)zl - C(x)(01 + A)Po(OI.) -f.l{l-C(x)z-l}

x

I

Pn,I(O:)Zn.

(3.1.9)

1

The solution of equation (3.1.9) for P(z,X;OI.) can be greatly simplified by considering equation (3.1.9) first when x takes the value /1/ {a + A + /1 - AA (z)} == y. Defining H(z; 01.) == 1:;;:=IPn, I (OI.)zn ,and setting x = y, equation (3.1.9) becomes P o (o:)["A(A(z)-l)-OI.] +zl f.l [(C(y»-1 - Z-I]

(3.1.10)

It can be shown, using RoucM's theorem (see Problems and Complements), that the denominator of the right-hand side of equation (3.1 .1 0) has exactly one zero inside the unit circle Izl = 1. Let this zero be denoted by Zo = zo(OI.)· Since H(z; 0:) is a proper p.g.f., it must be analytic within the unit circle and in particular at z =zo(O:). It follows that the numerator of H(z;OI.) must vanish at z =zo(O:), and hence that

Po(OI.) = 01.

+ A[1 -A(zo)]'

(3.1.11)

Also substituting H(z; 01.) in equation (3.1.9) yields the relation

_

111

I

C(x) ]P(z,x;OI.)[a+A+/1(1-x- I )-AA(z)] = /1lC(y) -1 H(z;a). (3.1.12)

k:

+ Po (0:) -+ 0:- 1 , implying thereby

= 1 Pn, ret) + poet) = 1 so that the distribution of the number in the that k;;'= 1 system remains fmite in fmite time. The distribution of the number of phases N 1 (t) in the system M /EJ/l has been discussed by Luchak (1958). However, from the results (3.1.11) and (3.1.12) one can get the distribution of N(t) directly, even for the system M/EJ/l, which is more useful than finding the number of phases in the system. We next show that the distribution of NI (t) in M/EJ/l, obtained by Luchak (1958), is identical with that of N(t) in M X /E Ill.

The system M X /E 1 /1. In this system the service times are exponentially distributed. This system may be obtained from equation (3.1.12) by simply letting Cr = Orl. Thus the distribution of N(t), in this case, is given by

_ _ P(z,l;a)+Po(OI.)

=

f.l Po(o:)(z-1)+zl+1 Z{o:+A+f.l-AA(Z)}-p

(3.1.13)

The result (3.1.13) is a particular case of Gupta's (1964) result, which is given here in a slightly modified form. However, we observe that Luchak (1958) has obtained the same result for the system M/Ex/l. It is thus interesting to see that the systems M X /E1/1 and M/Ex /l are equivalent. But it must be remembered that N(t) represents the number of units in the system MX/EI/l, whereas it represents the number of phases in Luchak's model. Intuitively, this may be interpreted as follows: In Luchak's model, a customer on arrival demands a random number X of exponential phases through which he must be served before another customer is taken into service, whereas in MX/E1/l, a group contains a random number X of customers, each of which is to go through a single exponential service phase. Therefore a single customer in Luchak's model (M/Ex/I) has the same service time distribution as the group in M X IE 1/1. The mean service time in both systems being lilf.l (for a customer in M/Ex/l and for a group in M X /El /1), the traffic intensity p (= AIi/f.l) remains unchanged. Thus the process representing the number of customers at any time t in the system M X lEI /1 is identical with the process representing the number of uncompleted service phases at the time tin the system M/Ex /l, provided that the initial states (customers in the former case and phases in the latter case) are the same. We shall see later that the waiting-time distribution is also the same for M X /EI/1 andMIEx/l. If we could invert equations (3.1.12) (that is, find a p.g.f. P(z, x; t) which has P(z,x;a) as its L.T.), we would have a p.g.f. for {Pn,r(t)}, the time-dependent state probabilities. Time-dependent (transient, nonequilibrium) solutions for state probabilities are known for some queuing models [see, for example, Ledenuann and Reuter (1954), Sack (1963) and Takacs (1962)], but even in these cases the transient solutions are complicated and intractable. For most queuing models we have at most an L.T. or other integral transform of the transient state probabilities. In such cases we must rely on asymptotic solutions, valid as t -+ 00, or use numerical methods to study the time-dependent solution.

BULK-ARRIVAL QUEUES

112

3.1

THE SYSTEM

MXIEJI1

113

(3.1.14), (3.1.15), and (3.1.16) yield the relations

The Limiting Distribution of N (t)

H(z) = We consider first the traffic intensity p (defined in Section 2.1), which plays a fundamental role in the study oflimiting distributions in queues. For the M X /EJ/l queue, the mean total service time for a group of customers is (ie/p. Since the mean interarrival time between groups of customers is 1I "A., we have p = "A.(ielp. It can be shown (see Problems and Complements) that if p < 1, there exist limiting state probabilities for the number in the system, which are independent of initial conditions, so that

p

P(z,x) P(z)

t .... =

L

z [A(z) -1] Po zy k - 1

(3.1.17)

P [(Xly)k -1] H(z) "A.+p(1-x- I )-"A.A(z)

(3.1.18)

Po(z -1) zy-k - 1

(3.1.19)

where in all these results

s

lim

.?:

lim Pn(t) = P n t .... =

Pn,r(t)

r=1

y

and

J1. p + "A. - "A.A(z)

k

and

=

L L r=1 n=l

P(z,x)

Pn rzn xr. '

~

=

L

n=O

Pn

l.

2

We define the steady-state r.v. N with peN = n) = Pn' If p ;;;. 1, no proper limits exist, as Pn (t) -+ 0 for any finite n. These results are intuitively clear except in the case p = 1, since if p > 1, customers arrive faster than they can be served, and the queue must then grow without limit. Applying the Abelian theorem given in Appendix A.8 to equations (3.1.1 0) and (3.1.12), we get

The systems MX/E1/l and MX/D/L To obtain the distribution of N for the system MX/El/I, simply put k= 1 in equation (3.1.19). For the system MX/D/l, in which service time is constant and equal to b, letting k,p-+ 00 in equation (3.1.19) so that k/p = b, results in

P z _ where we have used limd .... = (1

=

H(z) = LPn,lZn 1

"A. [A(z) -1] Po p [(C(y»-l - Z-I]

(3.1.14)

3

and

P(z x) = p [{C(x)/C(y)}-l] H(z) , "A.+p(l-x- 1 )-"A.A(z) wherey = p ["A. + p- "A.A(z)] -1. It is easy to see that at a = 0, C(y) length distribution has the p.g.f.

P(z) = P(z, l)+P o

= fi ("A. -

4 (3.1.15)

"A. A (z» and hence that the system

Po(z - 1) {z/fi("A. -"A.A(z»}-l

(3.1.16)

where fiCa) is the L.-S.T. of B(v). From the normalizing condition P(l -) = 1 we obtain Po = 1 - p, from which we again infer that the limiting distribution exists only when p < 1. For p;;;' 1, all probabilities tend to zero, and hence we say the system degenerates. Next we consider certain special cases of the system M X /EJll. The system M X IEk/l. In this case the service time distribution is Erlangian with k phases, each with rate p. Letting cr = Ork, equations

Po(z -1)

() - ze A1,{I-A(z)}_1

+ x/d)d

(3.1.20)

= eX.

The system MkIEJ/l. For this system, in which the size of the arriving batch is constant and equal to k, the distribution of N may be obtained by putting aj = Ojk in equations (3.1.16). Various other cases corresponding to variations in C r may be discussed similarly. The system MIEI/1. This is the Simplest system, in which arrivals are single and follow a Poisson distribution with mean rate "A. and service is exponential with mean lip. In this case, the distribution of N may be obtained by either putting am = Oml and C r = Orl in equation (3.1.16) or am = om! and k = 1 in equation (3.1.19), and is given by P(z) = Po(I - pztl, where p = A./p. Since p < 1, it follows that plzl < 1 for Iz I ~ 1. Thus the binomial expansion is valid, and hence

P n = pnpo ,

n;;;'

0

which is a geometric distribution. It may be noted that Po = 1 - p. Although the present derivation of equation (3.1.16) uses the Erlangian method, the result was first obtained by Gaver (1959) by using the imbedded Markov chain technique and renewal-theoretic arguments. In fact, Gaver considers the system MX /G/I in which the distribution of service time is unrestricted. The present discussion, though restricted to M X /EJ/l, covers most of the cases of practical interest. For a discussion of the system M X /G/I using the supplementary variable technique, the reader is referred to Miscellaneous Problems and Complements.

BULK-ARRIVAL QUEUES

114

115

The results (3_1.17), (3_1.18), and (3'.1,19) have been obtained independently by Restrepo (1965). In the case am = Om! they have been discussed by Morse (1958). But while comparing the two results, one must make allowance for differences of notation. The service rate in each phase is kp in Restrepo (1965) or Morse (1958), whereas here it is simply p. A class of queues of the type presented here, but where the distribution of the number of customers in each arrival group is restricted, has been considered by Cohen (1963) using his method of derived Markov chains. The Waiting-Time (in Queue) Distribution - The limiting Behavior of Vq'(tl

Since the waiting-time distribution in the transient case is cumbersome, we consider this distribution only in the steady state. Let Vq (t) be the virtual waiting time of a customer arriving at time t, and let Vq (t) converge to Vq in distribution as t -+ 00. By the characteristic property of the exponential distribution, we can suppose that the remaining time of completion of a unit in a given phase recommences at the instant of arrival of a batch. This property will be used whenever the service time is exponential, or is composed of exponential phases, without repeating this argument in the future. Now the arriving batch must find the system in one of the states 0, (1, r), (2, r), .... If w qI (T) dT is the probability that the waiting time of the first customer of an arriving group ends during the interval [T, T + dT], then

wq,(T)dT = Poo(T)dT+

L:

PI

r=I'

r

(r-1)1

W (0:) qi

pdT

-

Po

1 - (A./O:){l - A(C(e»}"

(3.1.23)

It can easily be seen from wq , (0) = 1 that Po = 1 - p, a result obtained earlier. In the particular case when Cr = Orl, equation (3.1.23) gives wq , (0:) = (1 - p)1 [1 - (Ala)(l - A(e»] , which is eqUivalent to a result of Luchak (1958) for MIEJ/l in the limiting case. It may be remembered here that p will be the same X in both systems M IE 111 and MIEJ/l if in MIEJ/l the mean service time is taken as alJ1. so that p for this system is Aalp - the same as in system M X IE l /l. These equilibrium waiting time results are valid for p < 1. If p;;' 1, the distribution of waiting time does not exist. As an example, consider the case of the simplest system MIE 1/1..,It can readily be seen from equation (3.1.23) that

(1- p)(p

+ 0:)

O:+J1.-A (l-p)+ (l-p)A

O:+P-A

(3.1.24)

Inversion of equation (3.1.24) gives and

(pTy-l e-wr

S

when substituted in equation (3.1.22), leads to

P(Vq

=

0)

=

1- P

Wq(T) = A(1-p)e-
T>O

a result which is found in elementary textbooks on queues and on operations research.

s

+L: r=1 s

Measures of Efficiency

&-1

+L: L: r=1 ;=0 + ... ,

(3.1.21)

where OCT) is the Dirac delta function whose L.T. is unity. Applying the L.T. to equation (3_1.21) gives the result

WqI (0:) = P o

+ p(c(e), e) C(e)

(3.1.22)

where e = p/(p + 0:) and P(z, x) is the generating function given in equation (3.1.15). Substituting x = e,z = C(e) in equation (3.1.15) givesP(C(e), e), which

The expected value of an r.v. connected with a process may not be an adequate measure of efficiency, since the expected value can be the same for several systems with very different input or service rates. Nevertheless, in many physical situations one is interested in the means or other moments of the distributions rather than in the distributions themselves, at least when it is difficult to get sufficient information about the latter. As an example consider mean system length and mean X queue length in an M IEk/l queue, and mean waiting time and mean sojourn time for the first customer in an arriving group for the same queue. First we evaluate the mean number L = E(N) in the system. To do this, we find the derivative of f(z) given in. equation (3.1.19) at z = 1 by expanding the denominator of P(z) about the pomt z = 1 and canceling the factor (z - 1) in numerator and denominator. To obtain the expansion of the denominator of P(z), let

x(z) = y-k = [ 1

+;

(1 -A(Z))f

BULK-ARRIVAL QUEUES

116

Then

111

THE SYSTEM GI'/M/l

times a~+1 - o~ > 0, n = 0,1,2, ... , are Li.d. r.v.'s with common distribution functionA(u). Let the L.-S.T_ ofthis distribution be denoted by

x(!)(1) = -p

X(2)(l) =

3.2

_p[- (k~1)P + 0; +~2 -a]

a(a) =

r~

.0

e- CtU dA(u)

and let the mean interarrival time be denoted by l/'A, where where p = Akalv. Expanding the denominator of P(z) in a Taylor series about the point z = 1 and canceling the factor (z - 1), we get

Po P(z) = 1+ZX(1)(l)+!z(z-iTl2)(l)+ ... ·

Thus (I)

P

_

(1) -

which, after using the values of mean number in the system as

Po [X(!)(1) +! X(2)(l)] (1 + X(I)(1))2 X(l) (1),

~

L = E(N) =

/2) (1),

and simplification, gives the

s

I

I

n=1

r=1

nPn,r = pO)(1)

k+ 1 p2 pro; +a2 -al =p+----+ . 2k 1 - p 2a(1 - p)

(3.1.25)

If N q is the number in the queue, then mean number in the queue is given by

I

I

n=1

(n-l)Pn,r = L-p.

(3.1 .26)

r=1

The mean waiting time of the first customer of an arnvmg group may be obtained from wq , (a) given in equation (3.1.23), and is given by wql = E(Vql ) = -w(I)(O) q) p

keo; a

[/7 + + 2 )] 2v(l-p)a

(3.1.27)

and consequently the mean time spent in the system by the first member of an arriving group is given by k (3.1.28) W = W +-. q,

V

Since cce) = b(a), one can see that equation (3.1.23) agrees with Gaver (1959), and for the special case MkIE1/l, equations (3.1.25) and (3.1.27) agree with Ross (1970).

3.2

<

l/'A = -a(l)(O)

<

00

The size X of the arriving batches is such that P(X = r) = 1. The customers are served individually by a single server. If V n is the service time of the nth customer to be served, then it is assumed that {Vn ; n = 1, 2, 3, ... } is a sequence of positive Li.d. r.v.'s with the common exponential p.d.f. Ve-J1.V, v> O. It is further assumed that the two sequences {Vn} and {o~ + I - a~} are mutually independenf~ Let N(t) be the number of customers in the system at time t and put N;; = N( a~ - 0), n = 0, 1, 2, . . . . In words, N;; represents the number in the system, including the one, if any, in service, just before the arrival instant o~. By using the characteristic property of the exponential distribution, it is easy to see that the chain {N';-} is a homogeneous. Markov chain imbedded in the continuous time process {N(t)} and has a countable infinity of states. Now if Dn represents the number of departures (real or potential - potential when there are no customers in the system before an interarrival period ends, otherwise real) during an interarrival period of arbitrary duration, and k j = P(Dn = j), then

s

~

Lq = E(Nq ) =

o

kj

=fo~ P(Dn =jjinterarrivaltime

~ e-J.LU.~u)j dA(u), Jo 1·

= f

We next consider a special case of the system Glx /M/1 wherein batches of customers of exact size r arrive at epochs 0 = a~, a'! , a; , ... , a~, .... The interarrival

0,1,2, ....

j

The p.g.f. K(z) of the sequence {Kj } is given by

fo

K(z) =

kjz i =

J~~

e-J1.0-Z)udA(u)

= a(v(1 - z)),

which is the p.gJ. of a Poisson LV. It should be clear from the context that we have supposed {Dn' n = 0, 1, 2, ... } to be a sequence of Li.d. r.v.'s with the distribution stated above. Define j-I {3j-1

1-

I

kn'

j

>-

1

n=O

THE SYSTEM GI"IM/1

The Limiting Distribution of N;;

= u)dA(u)

Now one can easily see that the following relation holds between the

LV.'S

N;;,N;;+! ,Dn and the fixed number r:

N;;+I

(N;;+r-Dnt

(3.2.1)

BULK-ARRIVAL QUEUES

118

3.2

where

and then take limits as n

(xt = max (x, 0)_

1

( kr+i-i> ~r+i-l

<j<

r+ i,

i ?o 0

j = 0

,

0

(3r-l

I

P

2

kr- 1

3

r

r+ 1

kr- 2

k r- 3

ko

0

<

>

<

+P(N;+r-Dn

>

>

0)

>

0)

O)P(N;+r-D n

O)P(N;+r-Dn

= zrE(zN;'-DnIN;+r-Dn

i ?o O.

2

to find P-(z). Thus

+E(zOIN;+r-Dn

<

0)

O)P(N;+r-Dn

0).

(3.2.5)

Now

E(ZN;'-D n ) 0

00

= E(zN;'+r-DnIN;+r-D n

(3.2.2)

The transition probability matrix P of {N;} is therefore given by

./

-+

119

E(zN;'+I) = E(z(N;' +r -D n)+)

If Pjj(n)=P[N;=jiNo=i], i, j?oO, n?o 1, are the n-step transition probabilities of {N;}, the one-step transition probabilities Pij == Pij(1) may be obtained from equation (3.2.1) and are given by

Pij =

THE SYSTEM Gl rlM/1

r+ 2

r+ 3

0

0

(3,

kr

kr - 1

kr- 2

kl

ko

0

0

(3r+ 1

kr+l

kr

kr- 1

k2

kl

ko

0

E(zN;'-DnlN;+r-D n

=

>

+E(zN;'-DnIN;+r-Dn

O)P(N;+r-D n

<

>

O)P(N;+r"D n

0)

<

0). (3.2.6)

From equations (3.2.5) and (3.2.6) we have

E(zN;'+I) = zr[E(zN;'-D n ] -E(zN;'-DnIN;+r-Dn

<

P(N;+r-D n Proceeding to the limit as n

-+

00,

<

0)

0)] +P(N;+r-Dn
we have ~

Since k j

P-(z) = zrr(z)K(z-I)_zr

> 0, it can be seen that the matrix Pis:

L

z-m-rp(N-+r-D = -m)

m=o

Irreducible, since every state can be reached from every other state in a finite number of steps with positive probability. 2

Aperiodic, since the diagonal elements are positive.

3

Ergodic if and only if p = rT < 1, where interarrival time) = "A/p.

T

+

L

m=o

P(N-+r-D = -m)

= zrp-(z)K(z-I)+

= (mean service time )/(mean

=

L m=O

where

bm (1-z-m) =

bm = P(N-+r-D = -m) =

In the ergodic case we denote the limiting distribution which exists independent of the initial state of the process by

Pj-

=

= N(a~ -

lim peN;

n->~

0)

~

Lo p j-z

where "11 , "12 , •.. ,"1 r are the

r

. 1

n 11 -"II r

=

1

-

Z"ll

'

Izi

o

K(z) = zr.

i +m ~

IimP(N;+1 =O)=P(N-+r-D
n->-oo

m=O

<

1

(3.2.3)

~bm(1-z-m) P- (z) = _o-:-----:~:__;:-:-­ l-zrK(z I) .

~

L

+

The relation (3.2.7) could have been obtained, as is normally done, by using the one-step transition probabilities (3.2.2), but the method used above is more elegant. Now equation (3.2.7) finally gives

roots inside Iz I = 1 of the equation

kjz i =

L Pi-k r

i=o

and therefore

Po=

= n.

The limiting probabilities are then given by the p.g.f.

r(z) =

(3.2.7)

(3.2.8)

(3.2.4)

To prove equation (3.2.3), we use equation (3.2.1) to find the p.g.f. of N;+l'

To evaluate the constants b m in equation (3.2.8), we first show that equation (3.2.4) has exactly r roots (distinct or coincident) inside the unit circle Izl = 1. Consider absolute values of f(Z) = z' and g(z) = - K(z) on the circle Iz I = 1 - 8,

BULK-ARRIVAL QUEUES

120

or

where 0 is positive and sufficiently smalL Then If(z)1 = (1 = 1 - or + 0(0), and since K(z) is a power series with nonnegativ~coefficients, Ig(z)1 ~ K(l - 0) = 1 - oK(I)(1) + 0(0) = 1 - 0 Il/A + 0(0) = 1 - (r/p) 0 + 0(0), where p = rA/Il_ Hence for p < 1 and 0 sufficiently small, If(z)l> Ig(z)1 on Izi = 1 -0. Sincef(z) and g(z) satisfy the conditions of Rouche's theorem, it follows by applying that theorem and letting 0 -+ 0 that equation (3.2.4), which is equivalent to fez) + g(z) = 0, has exactly r roots in the interior of the unit circle. Let these roots be 'YI> 1= 1,2, ... ,r, and now consider the zeros of the denominator of equation (3.2.8). Since equation (3.2.4) has r roots 'YI inside the unit circle, the denominator of equation (3.2.8) has r zeros l/'YI outside the unit circle. AsP-(z) is. an analytic function of z for Izl ~ 1, the function

3.2

THE SYSTEM Gl rlMI1

and r roots z = 'Y/ ,I = 1,2, ... , r, for which Izi > 1. Thus we have an identity

O-z)

r

r(z) = UsingP-(1 -) = 1, we finally obtain

• (3.2.1 0)

Thus in the special case of exponential interarrival times, the generating function P-(z) can be written in a form not explicitly involving the roots 'Y/. The Waiting-Time (in Queue) Distribution

bm(l-z-m)]

1 -ZrK(Z-l)

Let V q , (t) be the virtual waiting time in the queue for the first customer of a group = V q , (a~ - 0). Then is the waiting which arrives at time t. Denote time of the first customer in the nth arriving batch. We consider the limiting distribution W~ (T) lim p(V~n) ~ T)

vt)

Since in this expression all the zeros of the denominator outside the unit circle are also zeros of the numerator, it follows that B(z) is not only analytic for Iz I > 1, but is analytic on the whole complex plane. But it is a well-known result of complex analysis that a function which is analytic on the whole complex pJane must be a constant, and therefore B(z) = constant = B, say. As a consequence,

vt)

n-+oo

1

1

If we define

w-(a) ~

r

D

+ (p/r)(1 _zr)}

(l-p)(1-z) r(z) = 1 -z{l + (p/r)(l zrn

must be analytic for Iz I ~ 1. Now let Bez) be defined for Izi > 1 by

r(z) = B/

I

1 -z{l

I

B(z) =

=

(I - z) n (1 - 'Yz)

r

[Jo

(I (1-'YIZ) 1-z [1 +;- o-zr)].

It follows that, in this case, we have

B(z) == r(z)D O-'YIZ)

[/~1 (I-'Y/ Z)1

121

1

=f"" 0

e-OI.T dW- (7) ~

then

(l -'YIZ)

/~l

r

W';; (a)

and usingP-(I -) = 1, we obtain equation (3.2.3). As a simple exam pIe, let us consider the system Mr/M /1, in which A(u) = 1 _e- AU ,

u ;;;. O.

Then ilea) = A/(A + a), K(z) = A/[A + 11(1 -z)], and therefore the denominator of equation (3.2.8) equated to zero gives 1-AZ r [A+Il(1-z-1)r 1 = 0 or

1-z

[1 +;-0 -zr)) 0 =

(3.2.9)

which is an equation of degree r + 1 with exactly r + 1 roots, one root at z = 1

=

I}

1 - 'YI 1 -'Yill/(ll + a) .

(3.2.11)

We now proceed to prove equation (3.2.11). If an arriving batch finds j customers in the system, the waiting time of the first customer in the batch will have the L.T. {ll/(P. + anj. As the limiting probability of j customers in the system is Pj-, we have

Wq~ (a) = ~ Pj- (ll ~ a

r

= p- (11

~ a)

which gives the required result by using equation (3.2.3). The relations (3.2.3) and (3.2.11) are due to Foster (1961). Using the supplementary variable technique, Conolly (1960) has considered the limiting behavior of Gr/M/l in continuous time. Some numerical work on the parameters discussed by Conolly (1960) has also been carried out by Barber

BULK·ARRIVAL

122

THE SYSTEM M rlM/1

(1964). In this connection, see also Pike (1963). Some properties of the Gl"/M/l/Finite have been discussed by Shanbhag (1966). In the limiting derived from continuous time and just before arrival instants, we establish in. Chapter 6 a relation between their p.gJ.'s of the queue size. Suzuki (1963) (see Problems and Complements) has extended the (3.2.3) and (3.2.11) to the case when the size of an arriving group is an r.v. transient behavior of N;; for CrlMll has already been discussed by Takacs ( but the results are given in implicit form. However, one special case in explicit results are available is MriMll, which is discussed in the next section.

123

=

Po(n) =

L

PI-(n -1)13r+I-1

= L

PI-(n -I)k z+ r -j ,

1=0

Pj-(n) =

O<j
(3.3.5)

1=0

=

Pj-(n) =

L

PI-(n -l)kZ+r-i>

r

I=j-r

< j.

solution of the set of equations (3.3.5) is given by the joint p.g.f.P-(z, w),

3.3

THE SYSTEM M r/M/1

The Transient Behavior of N;; qPo{w}(1 -z) -PZOi,O + zi(q -z)(1-o~o) q -z + pwz r + 1

All the assumptions in the present case are the same as in the previous section the following exceptions: The interarrival times are exponentially distributed with dJ.

A > 0, 2

u;;;'

(3.3.6)

obtain equation (3.3 .6), we proceed as follows. First obtain recursive (3.3.7) for the Pj-(n), n;;;' 1, by substituting 13r+l-1 and k l + r - j from (3.3.4) into equation (3.3.5):

O.

The initial number in the system is i. This assumption was not needed the previous section where we considered only the limiting behavior n ~ 00, in which case the effect of the initial distribution wears out.

qPj(n) = pPo(n)

(3.3.7)

Define with

Pi-CO) = Oi,i

=

pnw} =

that is, we shall assume that the initial state i is arbitrary but fixed. Now

k j = P(j potential departures during an arbitrary interarrival period) e-(A+/.I)U

du ,

and let

j = 0,1,2, ...

j-l {3j-l

= 1-

Also let p = A/(A + /J.), q (3.3 .3) that

= 1-

Lkn o·

j ;;;. 1.

p. Then it follows from

The definition of {3 j _I holds for j = 0 if we define {3 -1 = 1. It is easy to see that the probabilities Pj-(n), n;;;' 1, satisfy the

Iwl

<

1

Izl

<

1.

=

r(z, w) = = Af= (/J.u)i O'j J.

Lo P;(n)w n , L

i=o

pnw}zj,

equations (3.3.7) by w n, adding, using the initial conditions given equation (3.3.1), and Pi {w } gives a new set of equations. Multiplying the new by zn, adding, and using the second p.gJ. P-(z, w) leads finally to equation .6), which is completely known except for the constant Po {w}. An explicit . for Po {w} may be obtained by Lagrange's theorem (see Appendix A.6) is given by .

x [q(Xn,i-Xn,i-l)

+ Xn,i -Xn,i+l]}

(3.3 .8)

3.3

125

THE SYSTEM MT{M/1

BULK·ARRIVAL QUEUES

124

which states

where Xn,i

I

(nr

+J~ +

i)

pi qnr+n+i-j

;nn (UV)

with

U

j=O n-l (nr+n+i-l)

-p(r+ 1 +in- 1)

q - pr - ipn- 1

L:

.

j=O

J

the result (3.3.8) is obtained by substituting in the series (3.3.10) from equation (3.3.11). The terms Xn,i appearing in equation (3.3.8) can easily be determined from published tables of the binomial distribution for values of nr + n + i up to 150. For large values of n, binomial probabilities can be calculated by using the recur· renee relation, where /3 + r = 1,

pjqnr+n+H-I

.. . + i). pI qnr+n+!-1

nr+n+i (nr+ n

-

.

L

l=n+l

+p(r+ 1 + in-I)

J

nT+n+i-l (nr

.L:

+n +

i-I)

n+l

L:

r=c

n+ r

1) /3T r n+I-T

pjqnT+n+i-i-l.

j

J=n

(

As demonstrated previously, using Rouche's theorem, it can be shown that the denominator of equation (3.3.6) has a single simple zero within the unit circle, which we denote by ~. The numerator must vanish at to since P-(z, w) is convergent for Iwl < 1, Iz I ~ 1. Now we indicate a procedure for determining Po {w} given by equation (3.3.8). Since ~ is a zero of the numerator of equation (3.3.6), we have, say, (3.3.9)

or by using the method of continued fractions described by Bahadur (1960). Alternatively the binomial probabilities can be approximated with the aid of Stirling's formula. The limiting Behavior of N;;

The limn .... .., Pj-(n) =Pj- always exists and is independent of the initial distribution. But Pj > 0 for every j = 0, 1,2, ... , if p = r"A/J.l < 1, whence ~;;' Pj - = 1, and Pj - = 0 if p;;;' 1. To get the p.g.f. of Pj-, we apply Abel's theorem (see Appendix A.8.c) to equation (3.3.6). Thus r(z) =

Po{w} =

0;,0

+

n

L --,pnw

n=1

n.

{D n - 1 [f(l)(n~nr+n]}

l"=q

As

~(n---l-)-!

I

nx . n 1 ~nT+n+i) l"=q = pn:\! { D - (l_D2

-+ 1-

(1 -w)r(z, w)

1 -z{l

(3.3.12)

+ (p/r)(1-zT)}'

UsingP-O -) = 1, we get the constant Po = 1 - p. It is interesting to note that

I (3.3.10)

and

lim

Po(1-z)

From Lagrange's theorem (see Appendix A.6), we have the expansion of Po {w} given by ..,

w

(3.3.11)

where the latter derivative has been obtained by using Leibnitz's rule from calculus,

0, lim X n,1.=. { q -pr, n .... ..,

if

(nr+n)p>n

if

p ~ 1

or

p>1

and this leads once again to Po = (1 - pt. It may be observed that the deviation of Xn, i from its limiting value involves in the first case (p > 1) two lower tail binomial probabilities and in the second case (p ~ 1) two upper tail binomial probabilities. When r= 1, the system length distribution for theM/Mil queue in continuous time was first published by Ledermann and Reuter (1954) using spectral theory of simple birth and death processes. Subsequently Bailey (1954), Morse (1955), Clarke (1956), Champernowne (1956), Sack (1963), and Pegden and Rosenshine (1982), among others, gave alternative methods of solving the same problem.

127

BULK-ARRIVAL QUEUES

126

Once again it may be observed that the limiting distribution ~iven in e~uation (3.3.12) is the same as the one given by equation (3.2.10). In thIS connectJ~n, see also Problem 11. It can be shown that in some but not all queues the contmuous time process has the same limiting distribution as the corresponding imbedded Markov chain. For further discussion, see Chapter 6.

~ Po(n)w n

n=1

~ Po(n)w n

n=O

Po

The Waiting-Time (in Queue) Distribution

Let V (t) be the virtual waiting time in the queue at time t and define V~n) = Vq (a~q - 0). Then V~n) is the time that elapses between the arrival of the (n + I).th batch and commencement of service on its first member to be served, IrrespectIve of his position in the queue. If w~n (1') is the p.dJ. of V~n), then

T> and

and use the convolution property to get

(3.3.13)

0

0) = Po(n).

Using the value of {w} from equation (3.3.9) with i = 0, we finally get equation (3.3.14). To prove equation (3.3.15), use Lagrange's theorem as discussed earlier with = (t - q)/pt and get

fcn

from which we recover f oo(n) as given in equation (3.3.15). The results of this section are given by Brockwell (1963), and for the special case r = 1 are contained in Takacs (1962), though at several steps they have been incorrectly reported.

The Distribution of the Number of Batches Served in a Busy Period

Let foo(n) be the probability that starting from the empty state (0), the s~stem returns to this state for the first time at the nth step. This means thatfoo(n) IS the probability that n batches complete their service during a busy period. In mathematical notation we may write, for n ~ 1, foo(n)

=

peN;;

=

0, N;;' =1= 0, m

=

1,2,3, ... ,n-lIN

o = 0).

The limiting Behavior of N;'; when Service Time Depends on Number in System

In the system M X /G n /l, the input is through the homogeneous Poisson process with rate "1\, and the p.g.f. of batch size is A(z) = L ~ am z m , as described in Section 3.1. The arrivals, in fact, form a compound Poisson process so that if a total of e(t) customers arrive during 10, tJ , we have

Po

Furthermore, since by definition (0) = 1, we can write Po(n)=P(N;;= OINo = 0). Now the p.gJ. of foo(n) is given by ~

foo(w) =

I

foo(n)w

n

(3.3.14)

where {ait* is the n-fold convolution of {a;} with itself, with {aj}o* = {ai}l * = {ai}' Also the p.gJ. of ai(t) is

Iwl
I aj(t)zi = e-At(l-A(Z». (3.4.1) ° The customers are served by a single server on a FCFS basis. When a group arrives

(3.3.15)

to find the server idle, service on anyone member of the group starts instantaneously, otherwise the group joins the queue. The service time V of a departing customer is conditioned on the number of customers at the instant of service initiation. Service times Vn of customers beginning service with the same number n in the system are independent identically distributed r.v.'s with dJ.

n=1

where t=t(w) is the simple root of the equation z=q+pwz inside the unit circle Izi = 1. Further from equation (3.3.14) we get

1(nr + n- 2) pn-l qnr.

foo(n) = n

n-l

To prove equation (3.3 .14), form the generating function of Po(n) =

I. k=l

foo(k)Po(n -k),

n

~

1

r

i = 0,1,2, ...

(jiO

and

~

+!,

BULK.ARRIVAL QUEUES

129

128

From the form of the matrix it is clear that the chain is irreducible, and since the diagonal elements are positive, it is aperiodic. Crabill (1968) has shown that sufficient conditions for ergodicity are

whose L.-S.T. is

Denote i;;' 0

ma m

!

-b~l)(O)

= E(V,,) =

(3.4.4)

lim sup {Pi

=

a = A(I)(1) = L

It can be shown (see Problems and Complements) that equation (3.4.4) can be put in an equivalent form

L~ vdBvn(v).

lim sup {Pi = i\aE(Vj)} Let N(t) be the number of customers in the system at time t, and defi~eN; ':' N(o" + 0) so that N; represents the number of customers in the system Im:ed.l' at ely after the nth departure. We are interested in the behavior of the s:oc ~stlC process {pI;:-} when n .... 00. The process {N:'} is an imbedded Markov cham wIth. a countable infinity of states. The one·step transition probabilities Pij of the cham for i;;;' 1 are given by ( kj

- i

j ;;;. i - I

+ I ,;,

j

\ 0, and for i = 0,

i>O

<

1

in which Pi appears to be in the usual form of traffic intensity. Define

P/ =

lim peN; = N(on

n .....

~

+ 0) =

j).

In the ergodic case, the probabilities P/ are given by the system oflinear equations (3.4.5)

(3.4.2)

< i-I

with

j ;;;. 0

where

Now using equation (3.4.2), we can change equation (3.4.5) to the form

P/ = kj,oP;

+

j+l

L

kj

- i + 1 ,i

Pt·

(3.4.6)

i~1

i> =

L

i = 0

j";;3 0,

ankj-n+I,n,

1

that is , kJ,l .. = pc]' arrive during a service period Ii were in the system when service began). The transition matrix P = CPij) has the special form j

j/

0

2

3

koo

klO

k 20

k30

kOI

kll

kZl

k31

2

0

k02

kl2

k22

3

0

0

k03

k l3

4

0

0

0

k04

0

P -

The p.g.f. of the set of equations (3.4.6) is given by r(z) = P; Ko(z)

j+!

kj,o

0

+I

Pt Kj(z)zi-l

(3.4.7)

1

where we define r(z) =

Io P/ zj

and Kj(z) =

f:o

kj,i Zi ,

i = 0,1,2, ....

(3.4.8)

To obtain equation (3.4.7), multiply equation (3.4.6) by zi, sum over all j, and use the convolution property. It has not been possible to put equation (3.4.7) in closed form as a function of the {Kj(z)} and a few unknown probabilities. Thus the direct application of this result is limited to some special cases that follow.

BULK-ARRIVAL QUEUES

131

The System MX/G/l. In this case the service time distributions are independent of the number in the system at the time of initiation of service. Thus we may assume, for C;> 1,

This would amount to a situation in which servers work at a slower (or faster) rate .when they s~ an empty. queue. We now have what appears to be the simplest s~eclal. ca~e o~ M /G n /l, wInch preserves arrival groups of random size and service tIme dlstnbutlOn dependent on system size. Thus noting that {ajf* is the coefficient of zi in the expansion of (alz + (l-ad z2 )n, we have from equation (3.4.3)

130

kj,i == k j = P(j arrivals during a service period) j;;;' O.

kj,

Then it follows from equation (3.4.8) that Ki(z)

=f

kjz i

=

I

=

= i Jo I

i;;;' 1

K(z),

o

and

=

=

KoCz) = A(z) K(z) z

.

(3.4.9)

+ _

fo

(

[il2]

'C"

L..

e- AV

1=0

i

-/1 V

1\2

I

dv,

j;;;' 0

-Z) I

---

( ]. -Z) .I

,i-2hZ "I

1\2 V

j-I

Ji.1 e

-/1 1

v dv

k j = P(j arrivals during a service time Imore than one in the system

when service started)

The constantPt may be obtained fromP+(l-) = 1 and is given by Po -

1\1

(j-n)! (2n _j)! Ji.1 e

where [nJ is the greatest integer contained in n, and

whence the p.g.f. P+(z), given in equation (3.4.7), gives the solution P+(z) = Pt [A(z) -11 {z/K(z)}-l

vn,zn-j,i- n

e- AV

[(i+l)f2]

= kj,l with /-t! replaced by /-t, j ;;;. O. It can easily be shown by integration that

1- p

_

a

where p = Aiib < 1, b being the mean service time. Note that we can replace K(z) given in equation (3.4.9) by b(A - AA(z»). 2

The System MX/Gn/l with Some Modifications. Let us modify this system by simplifying the input disfribution and the service time distribution. Suppose first that each arrival group contains either one or two customers, so that

P(X

where Al

+ A2

= j) ==

ai

=

j =

f::

kj = k j ,1

with Ji.1 replaced by Ji.

and from equation (3.4.3)

(3.4.10) j

0

j>

o.

j = 2

lo,

j> 2

= A.

The p.gJ_ Ki(Z) given in equation (3.4.8) takes the form

This means arrivals are by singlets and doublets with rates A.I and A. 2, respectively. Next let the service time distribution of those alone in the system be a different exponential from that governing all other customers, so that

n n

= > 1.

i = 0,1 i ;;;. 2.

BULK-ARRIVAL QUEUES

3.5

132

Thus, from equation (3.4.7) we have ' P+(z) = HzKo(z)-K(Z)}P~+z{Kl(Z)-K(Z)}pn![Z-K(Z)], + d p+ One equation h' h is completely known except for the constants P 0 an 1· . W IC . ' f P+(l _) nd the other from equatIOn involving these constants IS obtamed rom a (3.4.6). Noting that K;(l-)

~

"~ J·k·},I. o

=

i = 0,1

f Pj,

l P,

+ Po -p)+Pt(Pl

(1

-p) = I-p

The bulk queuing systems which we have considered so far have had either exponential interarrival time distribution or exponential service time distribution. Studying more general queuing systems such as Glx/G/ I involves great difficulties if one wishes to go beyond mathematical generalities to a computable solution. Many Glx /G/1 queues can, however, be approximated with sufficient accuracy for practical purposes by Erlangian queues, such as the Elf/Er/1 queue discussed below. As before, customers arrive in groups of size X such that P(X = l) = aI, 1= 1, 2,3, ... , where the mean a and the variance of the group size are such that 0< a< 00 and 0 < < "". The service times of the customers are i.i.d. r.v.'s with

+ Pll)P~

=

P;(Pu -PI2) +P+I PlI

with Al + A2

PI2

JlI

A2 JlI

PI = PH + P12,

Jl dB(v) =

and Po =

a;;

a;;

Al + 2A2 P

THE SYSTEM Er/Er/1 WITH GENERALIZED ERLANG INPUT ~

The Limiting Behavior of N(t)

and after using equation (3.4.10), the second equation as

Pll

133

the inputs by singlets and doublets. For an independent proof of equation (3.4.9) and its generalization, see Problems and Complements. Ivnitskiy (1975) discussed the transient and the limiting distribution of N(t), the number of customers in the system M;{n/Gn/l, using the supplementary variable technique. In this system, the interarrival rates, the group size probabilities, and the service rates all depend on the number in the system. Sufficient conditions for the existence of the limiting distribution are discussed, and recursion fonnulas for the L.-S.T. of the distribution are obtained. However, as one can expect from the discussion of the system MX/Gn/l, the results are cumbersome. Consequently they are not presented here.

3.5

i ;;;;, 2

we get the first equation as P;(l

THE SYSTEM Ef"tEr /l WITH GENERALIZED ERlANG INPUT

Io

PI2(1-Pl +p) +Pl jkj,o

(3.4.11)

Pu

(See Problems and Complements for the calculation of Po·) Solving the above set of equations for P; and Pt, we get Pu(l-p)

Jlr v r - 1 e-}J.fJ dv ( ) , r-l !

o
which is an Erlangian distribution with mean b satisfying the inequalities 0 < b = Analogously, the interarrival times of the groups are ii.d. r.v.'s with

fo~vdB(v)=r/Jl<"'"

h

dA(u)

I

'YiA.j e- AiU du

i=l

where

Pll(PI -p)(l + Pu)+ PI and

.. gP+ J'2 2 may be obtained iteratively from equation (3.4.6). Th e remamm j , "'" , h h .d d Most of the results of this section are due to Harris (197.0),.w ~ as conSI e~e certain other particular cases by varying the service time dlstnbutlOn but keepmg

This is a generalized k-Erlang distribution which may be thought to be composed of a sequence of k exponential phases or, equivalently, the sum of k independently distributed exponential LV.'S Xi with

Ai> 0,

x;;;;' O.

BULK·ARRIVAL QUEUES

134

3.5

THE SYSTEM EttEr!1 WITH GENERALIZED ERlANG INPUT

135

where d = [(i - l)/r], [x] being the greatest integer contained in x. For j = 2 3, ... , k, ,

The mean arrival rate for groups is

(3.5.2) This distribution Ek is useful in approximating statistical distributions having coefficients of variation between 1 and k-1!2, the first being the coefficient of variation for the exponential and the second for the ordinary k.Erlang distribution. The stochastic process to be studied now is a vector process {N(t), X(t), yet)}, where N(t) is the number of customers in the system (including the one in service, if any) at time t, XU) is the number of service phases yet to be completed at time t, and Yet) is the phase in which the arrival is found at time t. The arrival phases are marked in their natural order, whereas the service phases are in the reverse order. Clearly, using the phase technique, the process can be made Markovian. First we define X(t) == 0 when N(t) = 0, and then P~,j(t)

PO,j(t)

= P(N(t) =

= s, Yet) = j), n ;;;, 1, 1 < s < r, 1 < j < k = P(N(t) = 0, X(t) = 0, yet) = j), s = 0, n = 0, 1 < j < k. n,X(t)

(

/1

+ Aj) p. /1

._

'-I,J

(b) JJ.

2,3, ....

Pi-I,j-l,

(3.5.3)

Also k

Po =

I

PO,j

(3.5.3 a)

j=l

the .pro)bability that the whole system is empty (no customers in the serVlce .

q~ueue or in

T~de di~ference equations (3.5.1) to (3.5.3) completely define the system under consl eratlOn But these e t' th h '. qua IOns are not completely tractable analytically oug numencal wo~~ on the computer is possible. Furthermore, after com utin' P~j, to get the probabilIty of n customers in the system it is essential to calcul~te g nr

Pn =

The notation and the further analysis can be simplified if we make the transform· ation 1 < i = (n -1)r + s. Such a transformation has been used by Morse et al. (1954), Jackson and Nickols (1956), and others. With this transformation we can modify the notation and write P~,j(t) = Pi,j(t) and PO,j(t) = PO,j(t). Obviously, since n represents the number of customers in the system (queue + service), i represents the number of phases in the system. Thus the probabilities Pi,j(t) are the joint probabilities representing the number of phases in the system together with the arrival being in the jth phase at time t. Assume that the probabilities P~j = Iimt .... = Pi,j(t) exist, which they do if p = '?..7ib < 1, as we expect intuitively. As we are trying to study the limiting behavior of the system, it will be assumed that p < 1. One can now write the basic differential· difference equations, and by proceeding to the limit as t -7 co get the following recursive relations. For j = 1, /1 ( ~)p

P~j

lim P(N(t) = n) =

t-+oo

= 2,3, ... ,r

n ;;;, 1.

'-

i= (n-l)r+ 1

~oweve.r, one can ~iscuss certain things in more detail and thus come across certain mterestIng .propertles of the system. Later on, of course, we shall show that closed. f~rm solutlOn~ of the above equations are possible for certain particular cases of t e syste.m EkIEr/I. We shall also discuss measures of efficiency such as mean f . . ' number In the system or I'n th . e queue, or the ongInal system EX/E /1 'th generalized k-Erlang input. k r WI To proceed further, consider first the p.g.f. Pj(z) =

=

I

n;O

Pn,jZn,

Izi

<

1,

1

<j <

k.

(3.5.4)

Multiplying the set of equations (3.5.1) by appropriate powers of z using e uation (3.5.4) and the convolution property, and Simplifying, we get, for j ~ 1, q

{l-Z(l+

0,1

'\'

~l)}PI(Z)+ ~kZA(Zr)p,,(Z)

=

(l-Z)PO,l'

Analogously, from equations (3.5.2) and (3.5.3), for j = 2, 3, ... , k,

i = r + 1,

r + 2, . " (l-z)Po,j'

(3.5.5)

BULK-ARRIVAL QUEUES 136

Before going further, we can find Pj(l); which helps us in evaluating Po == Po,j == 1 - p, since from equation (3.5.5), Pl(l)

P 2 (1)

P,,(l)

~

M/t...!

M/t...2

fJ./t..."

fJ.

Consequently,

Pj(l) = ~/t...j,

(3.5.6)

1,2, ... ,k.

j

'L}<= 1

The equation (3.5.6) has an interesting probabilistic interpretation. Si~ce 1/~ and l/t.... are the average times the arrival group spends in the whole arnval chan~el i and its jth phase, respectively, (l/t...~/(l/~) is the probability of.the group bemg in the jth phase. Next we derive Lj=l Po,i' Observe from equation (3.5.5) that, since p == ra~Pk(l)/fJ.,

PO,j ==

_(t...~I)Pf~\(l)+(~)PJ!)(l)+Pj(l)'

j = 2,3, ... ,k.

Adding these equations and using the normalizing condition

Lj"=l

P j (1) = 1

3.5

THE SYSTEM

D(z) = A(l - z)

"-I

k

=

L. j=!

Po,i

=

(3.5.7)

1 - p.

Since the set of equations (3.5.5) is linear in generating functions, .one c.an easily obtain the determinant D(z) of the matrix of the set by expand~ng with respect to the column k. Note that as the coefficients of the elements mvolv.ed in the expansion are triangular determinants, they are easy to expand. The result IS D(z) =

Ih

(

/I..) 1+(-l} ,,/1.

1 -z ( 1 + J

j=l

1

+1

/1. 2

fJ.

. • . /l.k

h

J1

~ am z rm+h .

'm=l

X

n

I=j

i\

As usual, by Rouche's theorem it can be shown that the expression D(z) has \z\ = 1, and a simple zero, z = 1, on the given circle (see Problems and Complements). Let us now look at the p.gJ.'s Pj(z). The second equation of the set (3.5.5) expresses the dependence of Pj -1 (z) on Pi(z), j = 2,3, ... ,k. We therefore need to find P" (z). By Cramer's rule one can find from equation (3.s .5) that

n m=1

i-I (

P,,(z) k;;;' 2

l-z

(

/l. m )] "-1 t...IZ 1+nJ1

l=j

J1

(3.5.9)

(z - zoc)

iJ

(_1)2h+i Po,i

Iii [1 -

z(

~

1+

/l.J1m )

1

(3.5.10)

J.l

where A is a constant. When equation (3.5.10) is substituted in equation (3.5.9), we get 1

F,,(z) = [A

c~ (z -Zoc)r

where A can be calculated by using equation (3.5 .6). Finally we have (3.5.11)

(3.5.8) k _ 1 complex zeros within the unit circle,

137

where the product nJ=i>(') is to be taken as unity whenever ~>'Y, ~ and 'Yare nonnegative integers, and 'Y may be zero. Since the p.g.f.'s must converge within the unit circle Iz 1= 1, the k - 1 zeros z 1 , . . . , Zh -1 of the denominator of equation (3.5.9) lying within the unit circle must coincide with the zeros of the numerator of equation (3.5.9), which is a polynomial of degree k - 1 [excluding the factor (1-z)]. Substitution of the zeros Zl, . .. ,zk-l into this polynomial leads to a system of k - 1 equations linear in Po, 1 , P O,2, ••• ,FO,h which, along with equation (3.5.7), can be solved for all the unknown probabilities PO,j, j = 1, 2, ... ,k. When the P O•i are known, equation (3.5.9) can be solved for Ph (z), and then equation (3.5.5) can be solved recursively for Pj(z), j = k - 1, k - 2, ... , 3,2, 1. While all these calculations can be carried out in principle, they are complicated unless k, the number of interarrival phases, is small. Fortunately if we can assume that the arrival group size does not exceed a finite lnteger q, it is possible to express the generating function in a simpler and more explicit form. Now as D(z) is a polynomial of degree qr + k [since the upper limit of the sum in equation (3.5.8) is q], it has qr roots which lie outside the unit circle. Let us denote them by ZOI, zoo, ... , ZOqr' Since all the zeros of D(z) lying inside the unit circle coincide with the zeros of the numerator of equation (3.5.9), we have

immediately leads to Po

E1!tE,.I1 WITH GENERALIZED ERLANG INPUT

The case q == 1 corresponds to the generalized k-Erlang input with single arrivals.

Measures of Efficiency Once again it may be pointed out that the remarks made earlier while discussing measures of efficiency for the queuing system M X jE"jl apply here as well. L, and hence L q , may be found by differentiating equation (3.5.5), but this approach requires lengthy and tedious calculations. The following derivation of a

BULK-ARRIVAL QUEUES

3.5

THE SYSTEM

ElfIEr!1 WITH GENERALiZED ERLANG INPUT

139

138

It is clear that the calculation of L (or Lq), in the general case, requires evaluation of the complex zeros of a polynomial. It is possible to use approximations by considering bounds on L.

formula for L is simpler and more elegant. First, noting that k

~

k

L pf1le l )

L L

=

j=l

iP~j = E(J)

i=O

j=l

gives the expected number of phases in the system (queue + service) and then taking the expected value of the transformation I = eN - 1)r + X, where I, N, and X are the r.v.'s whose values are i, n, and s, we frod that

Lq

= E(N-lt =

r-

1

Ltl

Pjl)(l)-E(X)l

The System Elf/Erl1 with Ordinary k-Erlang input

As a special case of the Elf1Er/1 queuing system with heterogeneous phases discussed above, let us consider interarrival times following the ordinary Erlangian distribution, that is, Aj = A, j = 1,2,3, .. , . In this case one can easily see from equation (3.5.8) that D(z) turns out to be

where E(X) is the expected number of unserviced phases of a customer. E(X) may be found as follows: k

P(X

=

Ybeing an

0)

LV.

= peN =

j~l

0)

k

= j)

peN = 0, Y

=

j~l

PO,i

I-p

whose value isj, andP(X = siX> 0) = l/rimply

E(X) = E(XIX

>

O)P(X

>

per + 1)

j

r

0) =

L -r

If the size of the arrival group is bounded above by q, then one gets the corresponding p.gJ. for P" (z) from equation (3.5.11). The mean number of customers in the system, in this case, may be found from equation (3.5.13) and is given by

P

2

(3.5.15)

j=l

Consequently, Also L is given by equation (3.5.12). But to evaluate L we first need to know pI!) (l), which can be obtained by differentiating the second equation of the set of equations (3.5.5) with Aj = A,j = 2, 3, ... , k. Now k-jk= 1

and therefore using L = Lq

Pj

+ p,

(I) - 1

_

ra

(1) - - k p PO,j

+

ra + P (1) (1),

k2P

j

and using this equation to express pP)(l), (3.5.12)

Fainberg (1974) obtains, equation (3.5.12) by a method similar to that given above and also states without proof an alternative formula for L, L

=

A -

L(a 2 +a 2 +a)+ ~ Pk1) (1). 2a

a

p~l)(1), we obtain

L =

If the size of the arrival group is bounded by q, then L may be ~xpressed in terms of the qr zeros of D(z) lying outside the unit circle. From equatIOn (3.5.13) by using equation (3.5.11), we obtain (3.5.14)

pi l )(1), ... , p//2 1 (1)

in terms of

Finally, using equations (3.5.12) and (3.5.15), we get

(3.5.13)

Jl

.

J = 2, 3, ... , k

Ii

".

k(l - p)

'" /:1

(J-l)P o

P .+--

.J

1- p

x [a~ + liZ + a _ (k - l)a _ pCr 2a

2kp

1)].

2r.

(3.5.16)

If r-+=, Jl-+= so that rlJl-+ 1, then equation (3.5.16) gives L for the system

ElflDIl which has constant service time.

140

BULK-ARRIVAL QUEUES

The Transient Behavior of the Joint Distribution of the Customers in the System In previous sections we have considered queuing systems in which customers, all of the same type, arrive to get service at a given service facility_ However, situations do arise when customers of more than one type, say j(> 1) types, arrive at the service facility and are served according to some kind of priority discipline. While several kinds of priority discipline have been studied, we shall be concerned only with head-of-the-line priority discipline, which is defined later. Normally, a priority index i, 1 .;;; i .;;; j, is associated with each type of customer where 1 indicates the type with top priority and j the lowest, that is, as the index increases, the priority rank decreases. Since our discussion will be confined to the case j = 2, we can without ambiguity refer to customers of priority type 1 simply as priority customers, and to customers of priority type 2 as nonpriority or ordinary customers. Our discussion of priority queues will make frequent use of busy-period processes, discussed in Chapter 2, and of completion-time processes, which we now introduce. The interval of time starting from the instant the service of a customer begins, and ending when the server is free to take up the next customer of the same type, is defined as the completion time of the customer under consideration. Thus for queuing systems with two priority types, the completion time for a priority customer is simply his service time, and is of no special interest. The completion time for each non priority customer may be defined as the sum of his service time and the time it takes to clear the system of any priority queue that may have formed during his service time. In other words, this is the interval of time that elapses after the start of service of a nonpriority customer until the next nonpriority customer can begin service. These ideas will be used when we discuss the waitingtime distribution of a nonpriority customer. Introducing bulk arrival or service into priority queues, particularly when the number of priorities is greater than 2 and the interarrival-time or the service-time distributions are general, creates tremendous analytical difficulties and thus makes mathematical modeling of such queuing systems impractical. This is the major reason that not much seems to have been done in the direction of priority bulk queues. The difficulty might be better realized if we consider the queuing system Mf' , Mf' /G I , G 2 /1_ This is a queuing system in which two types of customers, to be called priority and nonpriority customers, arrive according to two homogeneous independent Poisson processes with mean rates Al and A2' Both types of customers arrive in batches such that the batch sizes Xl and X 2 are r.v.'s with distributions

r

= 1,2,3, ....

3_6

THE SYSTEM Mf' • Mf'/G " G, 11

Customers are served one at a time, and thel'r serVl'ce tl'mes are independent an d general with the p.d.fo's

b 2(v) =

f

dt ) 172(V)exp {-rT/2(t)dt}

b l (v) = 'Til (v) exp ( -

'Til (t)

and the mea~s. b ~' b~ for the priority and nonpriority customers, respectively. The queue dISCIpline IS FCFS within each type, but a priority customer is always served. b~fore a nonpriority one. However, if a priority customer arrives to find a nonpnonty customer in service he cannot preempt the no "t h' d' . ' npnon y customer w 0 IS un ~rgomg ser,?ce. The service on the priority customer begins. only on th.e c~mplet!On of service on the nonpriority customer. This type of queue discipline IS known as nonpreemptive or head-of-the-line priority diScipline and was ill~ro~uce~ ~y .Cobh~ (1954). For nonbulk priority queuing systems and other (;~o;~Z dlsclplmes whIch we do not discuss here, the reader is referred to Jaiswal . To discuss this s~stem, we use the supplementary variable technique discussed ill Chapter 2, and defme the follOWing state probabilities. Pm, n (v, t) dv + .o(~v), m ;;;. 1, n;;;' 0, is the joint probability that at time t

th~re. are m pnor~ty a~d n nonpriority customers in the system, and a pnonty customer IS bemg served with elapsed service time between v and V+dv.

2

Qm. n (v, t) dv:- ~(dv), m ;;;. 0, n;;;' 1, is the joint probability that at time t th~re. are m pnont~ an~ n nonpriority customers in the system, and a nonpnonty customer IS bemg served with elapsed service time between v and

v+ dv.

3

poet) is the probability that the system is empty at time t.

With. the inclusion of a single supplementary variable v, the process becomes Marko",:an and thus can be studied by Writing the differential-difference equations. ~ollowlllg the usual procedure (discussed in Chapter 2), we get the set of equations gIVen below:

(3_6.1) m-l

The corresponding p.gJ.'s are denoted by AI(z) and A 2 (z), and the means by 0< al < "", < ii2 < 00. The numbers in different batches are independent.

°

141

n-l

Al r~1 Pr,n(V,t)al(m-r)+A2 m ;;;. 1,

n;;;' 0

];0 P

m

,r(V,t)a2(n-r), (3.6.2)

BULK-ARRIVAL QUEUES

142

+

oQm,n(V, t) ot

OQm,n(V, t) 0V

+ [r.. + r.. + 1

2

m-l

=

772

(V)] Q

m, n

143 Mu~tiplying equations (3.6.2) and (3.6.3) by zn w m , summing over m and n, and USlllg the transforms, we get after some algebraic acrobatics , .

(V t) ,

oP ov + {r..l(l-A1(w»+ r..2(1-A 2(z»+ a+

n-l

r..l L

Qr,n(V,t)al(n-r)+r..2

r=O

L

Qm,r(V,t)a2(n-r),

r=l

m ;;;. 0,

n;;;' 1

oQ

3v + {r..! (l -

(3.6.3)

Al (w»

+ A2(1 -

A2 (z)

771 (v)}P

= 0

(3.6.10)

O.

C3.6.11)

+ a + 772 (V)} Q

Using equation (3.6.4) and the initial condition (3.6.7), the transforms of the boundary conditions (3.6.5) and (3.6.6) become (3.6.4) P(z,w;O,a) = -1

It follows from the above definitions that Pm n (V, t) = 0 for m < 1 or n < 0, and Q m n (v, t) = 0 for m < 0 or n < 1. Equatio~s (3.6.2)-(3.6.4) are to be solved

w

f""-P(z,w;v,a)77I(v)dv 0

under the following boundary conditions:

Pm,n(O,t) =

f;

P m +l ,n(V,t)771(V)dv+

f:

Qm,n+I(V,t)772(V)dv 1 -

+ OOn r..1 Po(t)al (m) Qo,n(O,t)

=.r:

Qo,n+I(V,t)772(V)dv+ (

00 _

j0

(3.6.5)

+-

(3.6.6)

--z1 f ""_QoCz;v,a)1?2(v)dv

z

Q(z,w;v,a)1?2(v)dv

P1,n(V,t)771(V)dv

+ r..2 PO(t)a2(n)

0

and an initial condition which we suppose to be (3.6.12)

(3.6.7)

PoCO) = 1

that is, we start with an empty system. To solve the system of equations (3.6.2), (3.6.3), and (3.6.4) subject to the boundary conditions (3.6.5), (3.6.6), and the initial condition (3.6.7), we define the following transforms:

Pm(Z;v,t)

""

= L

P(Z,w;v,t) =

QoCz;O,a) = -1

z

+

Pm,n(v,t)z"

n=O

L

m=l

Pm(z;v,t)w m

e-O'.t P(z,

+ A2 (1

- A2 (z»

+ a} Po (a) + 1.

(3.6.13)

L V

<

1

(3.6.8)

w; v, t) dt, . ..

Re a

>

O.

y

(3.6.9)

771(Y)dY

a})

(3.6.14)

-V{r..10-AI(W»+A2(1-A2(Z»+a}1.

(3.6.15)

-V{Al(1-Al(W))+vA20-A2(Z))+

Q(z,w;v,a) = C2 (z,w;a)exp [-

Qm(z;v,t)w m

t""

1

P(z,w;v,a) = C 1 (z,w;a)exp [-

m=O

P == P(z, w; v, a) =

L"" P (z;v,a)77I(v)dv

- {A!

_

I

0

Equations (3.7.1 0) and (3.7.11) give the solution

Izi .;;; 1, Iwl

Q(Z,w;v,t) =

f""-Qo(z;v,a)772(v)dv

fo 772 (Y)dY

.remains only. to find the functions C 1 (z, w; a) and C2 (z, w; a). Since a nonprIOrIty customer IS taken for service only in the absence of any priority customers,

145

BULK·ARRIVAL QUEUES

144

°°

°

we must have Qm n (0, t) = 0, for m > and all t, and Qo n (0, t) *- for t> 0. It then follows that 12m (z; 0, a) = for m> 0, and Qo(z; 0, a) *- 0. Hence Q(z, w; 0, a) = QoCz; 0, a). Now from equation (3.6.15) it follows that Cz(Z,w;a) = Q(z,w;O,a)= Qo(z;O,a) does not contain w, and we therefore

Now if P(z, w; t) is the joint p.gJ. for the number in the system at time t, its transform is given by P(z,w;a) =

denote Cz(z,w;a) by Cz(z;a). Substitute equations (3.6.14) and (3.6.15) into equations (3.6.12) and (3.6.13), noting that Qo(z; v, a) = Q(z, 0; v, a), to obtain

Jooo P(z,w;v,a)dv+

_ C (

-

1

C1(z,w;a)[1-{bdi''l(1-A1Cw»+Az(I-Az(z»+a}/w}j = C z (z;a)[bz {Aj(l-A I (w)+Az(1-A z (w»)+a}

00

I

(3.6.16)

(w)Po(a)

Cz(z;a)[l-{bz{AI +Az(l-A 2 (z»+a}/z)] =

S:

Pj(z;v,a)rl!(v)dv-{Aj +A2(1-A z (z»+a} Po (a)

Applying Rouche's theorem (see Appendix A.5) one can see that w - bdAI (1 - Al (w» + Az(l -A z (z» + a} has one and only one zero, w(z; a), inside the circle Iwl = 1 for Re a> 0, Izl';;; 1. Equation (3.6.16) then gives

. ) 1 -bl [AI(1-A I (w» + A2(l-A 2(z» + a] z,w,a /..1 [1 -AI(w)] + AZ [1-A 2 (z)] + a

+ P-() 0 a.

Inversion of P(z, w; a), even in the limiting case t -+ "", is difficult, but it is possible to find moments. We first discuss the joint limiting distribution which is later used in finding the waiting·time (in queue) distribution for priority and nonpriority customers.

The limiting Behavior of the Joint Distribution

The limiting behavior is detemlined by applying Abel's theorem (see Appendix A.8.a) and is given by the joint p.g.f. P(z,w) =

oo

lim aP(z,w;a)

",-+0

Po K I (z,w)(l-l/w)

fo Pj(z; v, a)1'/j (v) dv

1-(1/w)K J (z,w)

= AjAj(w(z;a»Po(a) + [Cz(z;a)!z] [bz{Aj(l-AI(w(z;a»))

bz {AI

+ Az (1 - A z (z» + a)].

This when substituted into equations (3.6.16) and (3.6.17) gives CI(z;w;a) and Cz (z; a), which contain Po (a). To obtain Po(a), Rouche's theorem is applied again to the new denominator of Cz (z; a), which has a root z'" inside Iz I = 1 for Re a> O. Finally, C j (z; w; a), Cz (z; a), and PoCa) are given by Cz(z;a) =

Q(z,w;v,a)dv+Po(a)

+ 1. (3.6.17)

+ Az (l -A z (z» + a} -

0

. )l-bz[Al(1-Al(W»+Az(1-Az(Z»+O:] +C( z z,a Al [1 -AI(w)] + A2 [l-A 2 (z)} + a

-bdA! +Az(l-Az(z»+a}]/z

- 50 PI (z; v, a) 111 (v) dv + Al A

r

1 -Po(a)[Aj(l-Aj(w(z;a») + Az(l-A 2(z» + a] 1 - (l/z)bz(AI (l - Aj (w(z; a») + Az(l - Al (z» + aJ

+Cz () z

l-K2(z,w)- [1 Kz(z,w)/zj [l-K l (z,w)]![1-K l (z,w)/w] /..1(1-A 1 (w»+Az(1-A 2 (z»

(3.6.18) where Kl(Z, w) = b l [AI(l -A1(w»

+ Az(1-A z (z»]

Kz(z,w) = b z [Al(l-A I (w»

+ Az(l-A 2(z»]

C2 (z) =

lim aCz(z;a)

0<-+0

C (z· W' a) = 1 -Po(a)[Aj(1-A I(w» + Az(1-A z (z» + al !" 1-(1/w)b l [AI(1-A!(w»+A2(l-A 2 (z»+a]

-C ( 2

z, a

) 1-(1/z)bz[AI(l-A j (w»+A2(l-A z (z»+a] ) 1 -(l/w)b t [AI (1 -Al(W» + A2(l-A z (z) + aJ

Po(a) = [AI (1 - A I (w(z",; a») + Az (l - Az (z",»

+ ar l .

(3.6.19) and w(z) is the smallest root of the equation w =

b1 [/..1(1 -AI(w» + /..2(1 -Az(z»].

BULK-ARRIVAL QUEUES

146

THE SYSTEM Mfl, Mf'/G" G,/1

3.6

147

Since by the normalizing condition w'e must have P( 1, 1) = 1, it follows that Po = 1 - PI - P2, where Pi = A; hi aj, i = 1, 2. The limiting distribution exists if PI + P2 < L From equation (3.6.18) one can obtain the expected values of the number of priority and non priority customers as Al al [AI al EI(V 2) + A2 a2 Ez(v Z )] + (~lladpl LI = PI + 2(1 - PI)

Now to proceed any further, we first need to where Cz(z) is given by equation (3.6.19) and CI(z,w) =

respectively,

where Ei(V 2 ) =

f;; vZbj(v)dv

and

J3i = [dZA;(z)/dzzlz=1

Consequently,

the values of C (1) 2

,

C l (1, w),

lim aCI(z,w'a)

0-+0

'

(1/w)b 1 P'-1 (l - AI (w»

L2 = P2

+ A2 a2 [AI ifl EI (v 2) + 1\2 a2 E Z (v 2 ) + PI bl(~I/ifl)] + P2 (J3z/a2)(1 - pd 2(1-pd(l-Pl -pz)

~now

+ AZ (1

- A2 (z»]

-C (z) l-(l/z)bz [Al(l-AI(w))+ Az(1-A z (z»J 2 1 (l/w)bd AI(1-A I (w»+A2(1-A 2 (z»]

Cz (1) = A2 az

for

where we have used Po = 1 - PI - pz and

i = 1,2.

Next we discuss the limiting waiting-time (in queue) distribution of priority and nonpriority customers.

C1 (1,w) = _PO[A\-A\A1(w)J +Aziiz[l-bz{Al 1

The Waiting-Time (in Queue) Distribution

AIA\(w)}J

AIAI(W)]

Also from equation (3.6.14)

Our discussion of waiting time in the Mf', Mf'IG 1 , Gz /1 queue makes use of (and the reader is therefore referred to) an outline given as Miscellaneous Problem 3 of this chapter, and the discussion in Chapter 2 of the busy-period distribution for the simpleMIG/l queue.

P(1, w; v) = a lim aP(l ,W'V a) -4 0 ),

,

.

=

C1(l,w)exp [- (77I(Y)dY -V(AI- /lJ A I(W»].

Fmally,replacmgAI(w)bYwinP(J , w'v)P'(1 .). , " , w, V IS given by

Priority Customer First we discuss the distribution of waiting time (in queue) for a priority customer, which is easier to discuss than the one for a nonpriority customer. If we identify a priority group with a single priority customer, then as its (the group's) service time is the total service time of the members constituting the group, the L.-S.T. of this service time is A I (iii (a» = b{ (a), say, The mean arrival rate of groups is AI, and its (the group's) "group size" p.g.L is simply w. To facilitate further discussion, it is convenient to use primes (or some other notation) on the functions that are altered in the preceding discussion, and the notation + for probabilities immediately after a departure of a group. Thus the probability that a departing priority customer (group) leaves behind him m priority customers (groups) is, in the new notation,

where

P'(l,w;v) = C;O,w)exp [-VeAl -AIWJf(V)

C{(l, w) = - Po (AI(1-W)] + Az if2 (I-bdAI -AIWU 1 (l/W)b{(AI -AIW)

I(v) = exp [ - (

r'(w) = C{(1,w)b{(Al -AIW)!AIW.

==

I

m=O

p:;,'w m =

J= P'(l,w;v)rMv)dv/Alw, 0

(3.6.20)

(3.6.21)

The customers (groups) left behind by a depa t' . r mg customer (group) are J'u t . , ,s those that arnve during the time it spends in the queue and in are random, it follows that semce. Smce arnvals W

where Al is the normalizing constant, t is omitted in the above notation since we are dealing with the limiting case, and

77~ (Y) dY ]

and substituting in equation (3.6.20),

P+'()

r'(w)

(l/w)b 1 [AI

where

:~~:p

-, = wq , (AI -AIW)b{(AI -AIW)

(3,6.22)

~~, (a) is the L.-S.T, of the distribution of queuing time for a riorit (I.e" for the first customer of the group). Equations (3.6.21) and (~.6.22~ W~, (a)

=

Poct + Az ilz [1 - bz(a)] Al b{(a)-Al +a

(3.6,23)

149

BULK-ARRIVAL QUEUES

148

If the customers within an arrival group are selected randomly, then the probability of a priority customer beingjth in its arrival group is 1-

Xi

(3.6.24) Now as the waiting time (in queue) of this jth customer in a group is the sum of the queuing time of the first customer of the group together with the service times of the first j - 1 customers of the group, it follows (since we are dealing with priority customers) that the L.-S.T. of the distribution of waiting time (in queue) for a random priority customer is given by

I

I

=

L: Xdbj(a)]i-l j=1

Foa+/I,2a2[1-b2(a)]

l-A 1 (b1 (a»

A1A1(bl(a))-/I,l+a

al[1-b1(a)]

(3.6.25)

where we have used equations (3.6.23) and (3.6.24). Next we consider the equivalent distribution for nonpriority customers. Nonpriority Customer

To find the equivalent distribution for nonpriority customers, we need to use the concept of completion time for each nonpriority customer, which has been defined earlier. Given that the service time of a nonpriority customer is v, the probability of n priority groups arriving during this time is, since the arrival process is a Poisson process, e-A., v 0\1 v)n /n!. The conditional L.-S.T. of the distribution of the time taken to clear the queue which has built up during the time v is [g(a)] n, where g(a) is the L.-S.T. of the busy-period distribution for a queue of priority customers, considered in isolation. Thus the L.-S.T. of the distribution of completion time c(a) is given by

To consider the waiting-time (in queue) distribution for a nonpriority customer, we identify, as in the priority case, each nonpriority group with a single nonpriority customer, so that the service time of the group is the total service time of its mem..?ers. Th~ ,~.-S_T. of the group's service-time distribution will then be, say, A 2 (b 2 (a» = b 2 Ca). We shall use double primes to identify the corresponding functions which we use for a group of non priority customers in the present case. As we did for priority customers, we first consider the waiting-time (in queue) distribution of the first served customer of a nonpriority arrival group. If the first customer of a non priority group is served only in the absence of priority customers, the order of service of priority customers and nonpriority customers, other than the first, does not affect the waiting time (in queue) of any first customer _ Hence we can use the previous discussion by taking A 2 (z) == z and assume tha.t the nonpriority customers in a group are served without interruption, ~, as to gIve a service-time distribution (for the entire group) whose L.-S.T. is b2 (a). Consequently, the number of nonpriority groups present in the system when the service on the first customer of such a group is about to start has the p.gJ. F"(z) == Q"(z, w; 0) = C;'(z) (3.6.27) A2 /1,2 where we have used the limiting form of equation (3.6.15). The multiplicative constant 1//1,2 is the normalizing constant, as in the priority case. The groups present at the start of service are the ones whose service is about to start and those that have arrived during its queuing time_ If the queuing-time distribution of the initial customer (group) whose service is about to start has L.-S_T. w~' (a), then the p.gJ. of the number of nonpriority groups arriving during its (the gro~p's) queuing time is zw~: (A2 - A2Z), where the factor z corresponds to the initial customer (group). But this p.gJ. is also given by equation (3.6.27). Thus we have C;'(z)

/1,2

exp [-av-Alv{l-g(a)}] dB 2 (v)

- /l,d(a)].

PO[/I,I-/I,lAl(w"(a»+aj a-A2 [l-A 2 (b 2{Al -/I,lAI(w"(a)

+ a})]

(3.6.28)

where"we ha~e .used equation (3.6.19) with A;(z)=z and b;(a)=A2(bz (a», and w (a) satisfIes the equation

C

+ AI

= ZW~2(A2 -/I,2 Z)

which gives

w~:(a) =

b 2 [a

b2 (a) is replaced by

g(a) = Adbda + AI -/I,d(a)}].

i-I L al(m)

m:;::l

and the p.gJ. of {Xi} is

Wq (a) = w~ (a)

priority group. Consequently, if in the above equation A I (b l (a»), we have

(3.6.26)

Arguing as above (or independently as in Chapter 2), one can also obtain g(a). In this case the service time of the first customer will be the service time of a

Now as the queuing time of the jth customer of a nonpriority group is the queuing time of the first customer of the group together with the completion

BULK-ARRIVAL QUEUES

150

times of the first (j - 1) customers, the L_-S_T. of the distribution of a randomly selected non priority customer is (3.6.29)

w;,

where (a) is given by equation (3.6.28). While exp~cit results do not seem to .be accessible, moments can be obtained as usuaL In partlcular, the expected que~mg times for the priority and nonpriority customers may be found from equatIOns (3.6.25) and (3.6.29), and are given by - w(1)(O) q,

It may be noted that once again Little's formula is satisfied.

. . The results of this section are due to Hawkes (1965). For single arnvals, that IS, when Al (z) = z and A 2 (z) = z, the joint distribution of the number in t~e system has been discussed by Jaiswal (1968)_ In the case of single arrivals, Miller [see Jaiswal (1968) for details] has also discussed this problem using the imbedded Markov chain technique. By some minor modifications in Hawkes's work, one can obtain similar results for the case of preemptive priorities.

3.7

OVERVIEW

The first six sections of this chapter discuss a number of bulk-arrival single-server queuing systems, chosen from the considerably larger. number. of such systems which have been studied. Some other systems are discussed ill Problems and Complements. In this final section we give references and some brief co~ments on some of the research we have not discussed in detail for one or a vanety of reasons. Some queuing systems are omitted because they are fairly readily available in books such as Takacs (1962, 1967) or in published lecture notes such as Bhat (1968). Some systems appear to be solvable only ~y mathemati~al techniques which we consider to be inaccessible at the mathemahcal level of this book. Some worthy papers have, no doubt, eluded our search, and others will ~pp:ar in .the interval between writing and publication of this book. The publicatlons Cited below are listed in Additional References at the end of this chapter. In his lecture notes, Bhat (1968) discusses the time-dependent and the limiting behavior of the queuing system MIG/1 with group arrivals. Using spectral theo~, he discusses the queue length and the waiting-time processes for the above que~mg system. Although analytically explicit results are available in the form. of multJple integrals, they require sophisticated computer programs for co~vement. com~u­ tation. Neuts (1977) uses the technique of semi~arkov .proce~ses ill t~e dl~cusslon of several properties of the queuing system M ISM/l ill whlch semce times are

PROBLEMS AND COMPLEMENTS

151

semi-Markovian. A more general Single-server queuing system (with bulk arrivals) than that of Neuts (1977) has recently been examined by Ramaswami (1980), using the matrix-geometric method [see Neuts (1981)] . In this paper Ramaswami discusses, among other things, the busy period, the busy cycle, the steady-state queue lengths (at departure and random epochs), the virtual waiting-time distribution, and the moments of some of these distributions. He also discusses the computational aspect of several formulas which he investigates. Other contributors whose work we describe briefly are Chaudhry (1974) who analyzes a bulk-arrival queuing system with intermittently available server; Foster (1964) who discusses some bulk-arrival or bulk-service queuing systems using renewal theory; Gupta and Goyal (1965) who discuss the queue length process for the bulk-arrival queuing system MX/HEnll with hyperexponential service times; Keilson (1963) who considers, among other things, the asymptotic behavior of the queue MX/M/l using Bessel-type functions; Kerridge (1966) who gives a numerical method for a typical simple bulk-arrival queue; Kleinrock et al. (1971) who discuss a processor-sharing queue with bulk arrivals; Moore and Bhat (1972) who give a computational approach to some mean value functions in the bulkarrival queuing system MX/Ey/l [see also Moore (1975)]; Rosenlund (1973) who considers the L.-S.T. of the joint distribution of the length of time and the number of customers served during a busy period of the bulk-arrival queuing system with finite waiting space. Several authors, including Fainberg (1974) and Yao (1981), have given bounds and approximations for various bulk-arrival queuing systems. We now mention, for completeness' sake and without any description, the work of a few other researchers whose investigations are related to single-server bulkarrival queues. They are Kambo and Bhalaik (1982), Lippman and Ross (1971), Mohanty and Jain (1970, 1971), Murari (1969, 1972), Nakamura (1968), Narasimham (1968), Neuts and Chakravarthy (1981), Shanbhag (1969), Sharda (1973), Sharma (1975), Soriano (1962) and Van Hoorn (1981).

PROBLEMS AND COMPLEMENTS Section 3.1

(a)

Show that the denominator of equation (3.1.10) has one zero inside the unit circle B. Recall that a point z on B satisfies z = e iO (i = '\1'=1), for some e, 0 .;;; e < 21T. Proof: By Rouche's theorem (see Appendix A.5), the functions fez) = z and g(z) = z - C(y) have the same number of zeros inside the unit circle, if (i) f(z) and g(z) are analytic on and inside B. (ii) If(z)-g(z)I 1,

BULK.ARRIVAL QUEUES

153

PROBLEMS AND COMPLEMENTS

152

Consequently, and thus g(z) has no singularities inside B. Also on B,

o;ZI+I(o;)

Po =

~l cr~o;+Il+t-.Il-t-.A(ei8»)

f(z)-g(z) =

mdu

0, { == I-p,

i8

100+1l+t-.-t-.A(eie)l;;;' 1l0;+1l+t-.I-t-.lA(e )ll ;;;. Re

11

if!

lo;+t-.+Il-t-.A(e )1

(b)

~

0;

Re

+ 11

~ + 11 <

if Re

1,

0;

>

0. 2

_ (z)l < 1 = If(z)l. Consequently, as all the con· Thus on B, If(z) g . f d (z) has one zero inside the ditions of Rouche's theorem are sabs Ie ,g unit circle, since fez) has one. For the system MX/E 1/1 in equilibrium, show that

=

II _p,

p

11

Il+t-.-t-.A(z)

<

=

(z-I)Po

z - b(t-. - t-.A(z» .

C;

C;

where and time, respectively.

3

+ 1) + a(C; + 1) -

1]

are the coefficients of variation of group size and service

When the system MXjEJ/l is in statistical equilibrium, show that the L.T. of the distribution of waiting time for the first customer of an arriving group when the group has to wait is given by

b(t-. - t-.A(z)).

[obtained by setting the denominator of equation (3.1.13) equal to

lim

Moreover, for p > 1, it is clear that Po = lim", -+ 0 + o;Po (0;) = 0, since then z (0;) ...... z such that 0 < z < 1. This proves the existence of the limiting distribution for M X /E 1 /l whenp < 1. If N q denotes the number in the queue (excluding those, if~any, in service) when the system MX/EJ/l is in statistical equilibrium, then show that the p.g.f. of N q is given by

2(I-p)

z(o;)[o;+t-.+Jl-t-.A(z(o;»] == Jl

a-+O+

lim

a ......... 0+

ifp
p [p(C~

°

0;

Il [1 - z(o;)J

1.

. f hi eq uation is the p.g.f. of the number Clearly the right·hand side 0 t s f t Let us call this K(z). d . .ce time 0 a cus orner. of arrivals unng a serVl hi ess theory (see Appendix Now applying a result from branc ng proc h that < z < 1 that z = K(z) has a real root z suc ' ~.2), lwe c~ s~y == 1 If such a root exists, then as 0; ...... 0, Z\o;) ...... z, Iff K( ) (1) - t-.alll P>.· h & 1) z(o;) ...... I. Besides, dlfferen· where 0 < z < 1, otherwise (wen p '" , tiating the equation

zero] , with respect to

0+

if P

p ;;;. 1

. th t for p::;;' 1 all the probabilities Pi' j;;;' 0, are zero ThiS means a 9" , 19) when k = 1. . (3 1 19) and for p < 1, the p/s are given by the p.gJ. (3 J . Proof Consider the zeros of the denominator. of equatIOn .. 'k - 1 that is consider the roots of the equation wen -, ' h

z

-+

Show further that

f 0,

Po =

a:

lim

_ 'APo [I - A(C(e»] Wqc(O:) = Qo [o;-''A{I -A(C(e))}] where Qo = 1 - Po is the probability that the arriving group finds the system busy. In particular, show from this or otherwise that for the systems M X /Ek /l and M X /D/l, respectively, k(o; +a 2 )+a 2Ila(1- p)

and E(V

qj

1

vqj >

0) =

b( 2 +-2) Oa a 2a(1 - p)

(1)

and letting 0; ...... 0, we get

zO)(o;) ==

~

l/(H- Jl), 00,

p

<

1

p == 1.

where b is the constant mean service time of a customer, and p = t-.ab in equation (1). The result (1) in its unconditional form and when ar = Or1 , may be compared with M/D/c (when c = 1), which has been considered by many authors, including Erlang (1920) and Crommelin (1932). An extension

154

BULK-ARRIVAL QUEUES

PROBLEMS AND COMPLEMENTS

155

4

MXIDlc of the system MIDlc is discussed in the chapter on multichannel queues. For the system MX IE,,!1 in statistical equilibrium, the mean waiting time spent in the queue may also be obtained as follows. If a group arrives and finds the system in state (n, r), the mean waiting time in the queue for the first member of the group is [en - 1) k + r] Ill, and consequently this quantity for any group is given by

f

L...

f .L...

n=l

r=1

q

Pn = -A [1 -

11

6

dP(1,x) 11 dx

'A] [ a +!::.]

pO - a)

n-l

p

n

,

>

O.

For the system MXIEJII in statistical equilibrium, show that P+i-I = P i,1 I.L...Pn, f I,

i = 1,2,3, ...

I

(n-l)k+r -'-----'---- Pn r 11 •

=!5...L + ~ 11

for the number in the system can be shown to be given by

is the probability that there are i - I customers in the system immediately . b after departure of a customer. Hence show that the p .g . f • of {P+}' n IS gIVen y

I

r(z) =

x=1

'A Po (A(z) - 1) pp[{zlb('A -AA(z)}-lj

Obtaining the constant Po by P+(l) = 1, show finally that where P(l, x) may be obtained from equation (3.1.18)_ Consequently it can be shown that dP(I,X)1 dx

r(z) =

p(k + 1) 2

x=1

q,

= kL q 11

+ p(k + 1) 211'

After substituting for Lq from equation (3.1.26) into the above equation, one gets equation (3.1.27). It may be observed that if S is the number of unserviced phases of the unit under service at the instant of arrival of a group, then E(S) = p(k + 1)/2. For an independent elegant proof of this, see Section 3.5. 5

Section 3.2

7

(a)

Th~. system GIIMll. Show that when r = 1 in GrIM/!, the distribu ~on of th~ .num?er m the system is geometric, and that the distributIOn of waltmg time (in queue) for those who have to wait is exponentIal.

(b)

The system Er,IE/l. distribution, namely,

In the system MXIEI/I in statistical equilibrium, if the size of an arrival j group follows a geometric distribution, that is, if aj = (1 - a) a -I, j = 1,2, 3, ... , then show that

wq,(a)=

1l(1-a)-A 1l(1-a)

11(1 -a) + a Il(l-a)-A+a

11(1 -a) - A Wqc (a) = ---'---'---'---

Il(l-a)-A l-az P(z) = (1 -a) . Il-Z(A + Ila)

Wqc(a) is defined in Problem 3. It may be noted from the expression for wqc(a) that conditional waiting time in the queue for the first member of the arriving group is exponential with parameter 11(1 - a) - A. The distribution

Here the interarrival times have an Erla

A(u) = 1 _

h~1

(t..ku)j

j=O

1·

.L...

'!

. ng13n

-"-Iou

e

with mean l/'A. _I~ t~is case a(a) = [Ak/(Ak + a)] k, and so K(z) = [TIC T + (1 - z) k )] , and therefore the denominator of equation (3.2.8) equated to zero gives

l-z r [T(T+(1-z-1)k- 1

a+ 1l(1-a)-A

and

.

ThiS. result is obtained independently through the imbedded Markov chain in SectIOn 3.4. For another proof, see Miscellaneous Problem 1.

Thus W

(1 - p)[A(z) -1] a [zlb('A - 'AA(z) -lJ

r

l

l"

= 0

or

(kT)h zr+k - {(l

+ Tk)z -

l}k = O.

This equation of degree r + k has exactly r + k roots, 1, 1hl, Ih2, . " ,

Ih" 1/"(10 ,

1h20 ,

•. -,

I/-v'(10-00

BULK-ARRIVAL QUEUES

156

the roots 1hz, 1=1,2, ___ ,r, being inside and 1hjo,j=1,2,_ .. , k - 1, being outside Iz 1= 1. By virtue of this we may write

PROBLEMS AND COMPLEMENTS

157

are used in the discussion of the waiting-time problem: I'ln

r

P(x)

Therefore we have

I ( _ ) i=1 X I'i

xr

p'

(I) (

)'

O~m~r-I

I'!

r

P(x)

I'i

I ( _ )P' (1)( I'i ) ;=1 x I'i

J+

where r

The normalization finally gives

r(z) =

(c)

(kT)k-l k(1- p)(J -z)

(kT)kzr+k - {(l

+ kr)z _l}k

P(x) = k-l 1 -"/'joz

n

j=l

1 -I'jO

For the system GrIMII, show that the mean waiting time (in queue) of the first member of an arrival group (including zero waits) is given by W;, = [S - rJ IJ-!, where r

S =

I

8

(x - 1'1),

'Yz,

i=I, ...

,r,

,. are the r roots inside Izi = 1 of the equation (3.2.4), and p(1)(x) indicates the derivative of P(x). Hint: For details of the proof of the first identity, see Miscellaneous Problem 1 of Chapter 6. On the same lines one can prove the second identity. In the limiting case of GriMI 1, show the following: (a) The p.dJ. of the waiting time (in queue) of the first member of a group of size r is

{l/(l-I'I)}.

1=1

Cd)

n i=l

P(waiting time in queue = 0) = Po = P(l)

Let p=n[=j (I-I'zf l . Show that the probability that an arnvmg batch finds the system empty is liP. Use this to show that the mean waiting time (in queue) for the first member of an arriving group (excluding zero waits) is given by E(Vq , iVq , > 0) = (S - r)PI(P - 1) J-!. These results have also been derived by Conolly (1960), using the supplementary variable technique. Barber (1964) has calculated numerical values of P and S for the system DrlMll. For more discussion on the system GrIMIl, see Chapter 6 and problems therein.

t

>

O.

The moments of the wait (in queue) may be obtained either directly or by using the transform of the above density. In particular, show that

Wq_,

Continuation of Problem 7 (b). Show that when k is odd, all the I'jO'S are complex, and when k is even, exactly one I'jO is real and positive. In particular, show that for k = 2,

Hint:

= -1 [ Ir

w;, (x)

jJ.

1=1

(

-=1

1

I'i

) -r ]

1'. = -1 Ir --'-.

jJ. i=1 1 -I'i

00

I

Pen exponential services completed in time

n=1

4r(l-p)(l-z) l-I'z r(z) = 4r2 zr+2 _ {(I + 2r)z _1}2 1-1'

x IN- = n)P;'

where 1'-1 is the unique real zero of 4r 2 z r + 2 -{(I +2r)z-1}2 lying within ]0, 1 [. Hint: Consider the intersection of the pair of curves y = (kr)kzr+k and y = {(I + rk)z _l}k for realz. 9

The system GrIMIl. This problem deals with the waiting-time distribution for the system GrIMII. First consider the following identities, some of which

(b)

The transform of the p.dJ. of the waiting time (in queue) of the second member of a group of size r, since its wait (in queue) is the convolution of the wait (in queue) of the first member and its (the first member's) service time, is iii;, (a) jJ.I(J-! + a). This procedure can be easily extended

BULK-ARRIVAL QUEUES

159

PROBLEMS AND COMPLEMENTS

158

(c)

(ii)

. ) of any member of a newly arrived to deal with the wait (in queue group. [Conolly (1960).J .. In the limiting case of GnM/l, show that the ~.~.f. of t:e ;a~tmg time (in queue) of a random customer of an arnvmg bate 0 SIze r

waiting time (in queue) of a random customer of an arriving batch has L.-S.T. I

w~(a) =

fI I

has the L.T. given by

wO(a)

~ ~ ~~ W'~' ("~a)i1 [

=

I

11

given by

q

10

[

I

-. (eue) W- in this case is Show further that the mean waItmg tIme m qu q

W- =

fI 1

1-,m

I-Jl,m/(Jl+a)

Jl a [/1 J1 +- . J Jl+a aa

-J [{I-A - -

Hint: Use arguments used in Miscellaneous Problem 3. (a) Continuation of Problem 10. Show that the mean waiting time (in queue) of a random customer of an arriving batch is given by

W.;

rll=11-71 i _ 1 -~1//1· 2

W.;

=

-w;(l)(O)

=[t

O-,mf 1 - l +hm2/ ii -l))!Jl

wherem2 =Llj2 ajanda=Lljaj.

. f N-' Gl x /Mjl wherein the size X of Consider the limiting distnbutlOn 0 n m .. _ _ . _. h' t fixed but rather follows a probabilIty dlstnbutlOn arnvmg bate es IS no _ _ .) = a- . = 1, 2, ... , I. Show that correspondmg to equat-Ion gIven by P(X -, h ]> '1 t' N- = (N- + size of a batch arrived at the n . if (3 2 1) we have t e re a Ion n + 1 _ . . t' I _ D )+ Use this relation to show that the transItIon probab Ity n' . b mstan an matrix (Lp.m.) for the chain N;;, for i;;;' 0, IS gIVen y

(b)

In the particular case of Problem 10 when the interarrival times are exponentially distributed, that is, for the system MX/M/l, show that the p .gJ. of N - can be expressed in the form (1 -z)(l-p)a

_

P (z) = a -z{if+ pel -A(z»} where

_ ({3I+i-j, Pij I+i-l 1 - L, {3n' o

I

if =

j = 0

(31+i-j -

a1kl+i_j+al-l

If p ==

k

I+i-j-l

+ .. . +a1ki-j+l, + .. - + a-1-'-ko ,

)...ii//1 =)... Ll jaj//1 < 1,

jai'

j=l

where _ tazkl+i-i+al-lkl+i-i-l

L

Oi=j':;;i+l i+l<j':;;l+l.

it is possible to prove the existence of the

limiting distribution for the process {N;;) v.:hen_n ~~. Show that, in the limiting case, the dlstnbutlOn of. (i) N;; has a p.gJ.

a e the reciprocals of the roots outside the unit where 71,72,'" ,71 r - I ' hAC)- Ll a-z l circle, Izl = 1, of the equation A(z) K.(z ) = 1, Wlt z - I I , ~ k J and K(z) =a(/1(l-z» = Lo j Z '

It is interesting to see that for the system M X /M/l, P- (z) = P(z), where the latter p.gJ. may be obtained from equation (3.1.19). This is true because the input process is Poisson. [In view of the input process being Poisson, the relation r(z) = P(z) is true even for the more general queuing system MX/G/l.] This means that the limiting distributions of the number in the system considered at an arbitrary instant and at an instant just before an arrival are the same, when the arrivals are in groups of random sizes and the interarrival times and the service times are exponentially distributed. In fact, when the arrival batch sizes are fixed, relations between generating functions of the queue size considered at an arbitrary instant and at an instant just before the arrival of a group and just after a departure have been considered in several papers of Foster, Foster and Perera, Chaudhry, and Chaudhry and Templeton. Though some of the results of these authors have been discussed at appropriate places in various chapters, most of them and other related material are the subject matter of Chapter 6.

12

Show, by eliminating

Po {w}

from equation (3.3.6), that P-(z, w) is

BULK-ARRiVAL QUEUES

161

PROBLEMS AND COMPLEMENTS

160

given by

r

(z, w) =

Section 3.4

1 _--:-~f-)+~~i..:;.(~::.....----,-,q):-(:.....1_-_z"-,,)] . O~i,.QIOP~(:l.~~z!...)+~(l:..--_o~i~,o~)~[Z~i(:!-q-;:-.;:Zp) (0:. r 1 (1-n(pwz

-z+q)

+

15

K;(z) = b;(A -AA(z»

Consider the transient behavior of N;; in the system Mr/M/l. Show that the first two moments of N;; about the origin are given by

13

r

00

E(N;;) =

L

n

iPj-(n) = i+ P

j=O

_.

16

_

For the system MXIGnl 1, show that for i;;;' 1,

that is, Ki(Z) is related to the L.-S.T. of the service-time distribution. For the system M X /G n /1 ,show that for i> 0,

L (Po(J)-P o )

Pi =

j=1

~ L- jk·1,1. = AiiE(V·) , i=1

E«N;;)2) =

f

j2 Pj-(n) = i

2

and

+ (2r + 1)(i -E(N;;»

j=O

Po =

I.

jkj,o = ([-1

+

where Po and E(N -) refer to the limiting case when n

Hint:

-+

0:>

and are given by

Po = I-p

17

E(N-) = p(1 + r) 2(1-p)

18

n

19

E(N;;) = coefficient of w

in

dr(Z,W)\ dz

aiPi'

Since Po < 00, the condition of the existence of a limiting distribution can be changed to lim sUPi > 0 {p;} < 1, as stated in Section 3.4. Hint: To get Pi, i > 0, use either equation (3.4.1) or problem 15. Continuation of Problem 16. By using Problem 16, show that Po has the value stated in equation (3.4.11). For MX/Gn/I of Problem 15, show that limsuPi>o{pj}<1 reduces to (AI + 2A 2 )/J.( == P < 1, regardless of the value of PI' For special case (b) of M X /G n /l as given in equation (3.4.10), show that the expected number in the system is given by

, ...

}{1-p+p 2 +Pod

(1-p)2L+ =Pt[{P120-Pl+P) +PI

z=1

f

i=1

j=l

PlI

14

where P-(z, w) is given in equation (3.3.6). Continuation of Problem 13. In the same system Mr/M/l, show that the m th moment about the origin of the waiting-time distribution is given by 1

=

_

_(p2

.

E(N;;) =--J1 _

Wqn

=

E(N;;)2

+ P02)(1-Pl)]

Section 3.5

+ E(N;;)

/12

.

Hint: Take the L.T. of W~n(T) given in equation (3.3.13), etc. H may be seen that the moments discussed in Problems 13 and .14 can be calculated numerically once we know Po (n), which can be determmed from equation (3.3.9).

+ PI2)(1- p)

where L + = E(N+), P02 = A2/J.(, and other quantities have been defined earlier. Hint: Differentiate the requisiteP+(z)and use Kj(z) as given in Problem 15.

and in particular,

2

+ P12) + P12(p 2 + P + P02)}]

+ pt[(Pl - p)(l - p) + (p2

(J'-I)I

] =1

(1- p) {(Pu - P12)(P!

Pll

U+m-l)!

-m.LPj(n) /1

+

20

Show that the right-hand side of equation (3.5.8) has k -1 zeros within the unit circle, Izl = 1, and the simple zero, z = 1, on the unit circle. Hint: Take the circle Izi = 1 + €(€ > 0, € -+ 0), consider fez) and g(z) as the first and second terms of equation (3.5.8), and apply Rouche's theorem_ To show that z = 1 is a simple zero, note first that D(z) has no zeros on

BULK-ARRIVAL QUEUES

162

PROBLEMS AND COMPLEMENTS

163

the unit circle, Izi = 1, except z = 1, since If(z)1 =1= Ig(z)1 at all other points on it. Besides, since

DCz) 1 -z

I

Hint· .'

~~)

(~~»)

= (_l)k-l Al'A2 ... Ak (1- )=1=0 ~k-!

z=!

p

1\J1.

= 1 is the simple zero on Izl = 1 of DCz). For k = 2, verify equation (3.5.13) by using equation (3.5_12). Hint: Solve the set (3_5.5) for Pl(z), P2(Z). Find pP)(l), pil)(1). Use Po 1 + Po 2 = 1 - p and p = faA! A2/J1.0'1 + A2), wherever necessary. One ca~ also ~erify by direct differentiation of equation (3.5.5). For k > 2, the method is straightforward though laborious. (a) The system Ef/Er/1 with generalized 2-Erlang input. Using equations (3.5.9) and (3.5.7), or the set (3.5.5) with k = 2, show that Z

21

22

P2(z) = [J1. P02(1-Z)-Al(1-P)z]fJ1.D(Z)]-1

,

ll-z

Show further that, in this case, D(z)/{l - z) given in equation (3.5.8) has a single root within Izi = 1. If this root is Zo, then show that 0 < Zo < 1, and also that Po, 2 = A!Zo(1 - P)/J1.0 - zo). (b) Show that in (a)

-P_[(2aFl(G~

L =

+a2 +a)_(P)2 a A1A2P

l-p

+

23

_per-I)] 2r

{Al(~~P) }P

L =

2a(I -p)

24

2'

The system E k /M/1. Let the arrivais . . d h be b~ slll~ets, the Illterarrivai times follow the ordinary Erlangian E distribution. Define the followi k, an t ~ serVice hmes follow an el'ponentiai ng generating function: k

k

QO

P(z,y) = \' PJ.(z)yj = \' \' P . L.. L.. L.. n,jZn y J , j=l n=o i=l

lyl";;;;l,lzl";;;;1.

Show that

P(z,y)

"

(I -z) /d/o,jyi

+ (A/J1.)zy(yk -z)Pk(z)

1 z{l+A!J1.}+(Azy/fJ.) This equation is true for all Izi ~ 1 and I,;::: . . Consequently we can write y I "'" 1, and III partIcular for z = yk.

1

A

p, p

pCGa2 +-2) a +E.

L =

k

P(z) = Po 1 - [ J1.

=1-

:y us~ng PCz) ~d the transformation used in Section 3 5 III Y usmg equatIOn (3.5.13) if k = 1 and A = A .. Keepi . . d 1 • . ng III nun the notation used in Section 3 2 'f obtallled here a .h h . , ven y that L as grees Wlt teL as given in equation (3 1 25) Show that for the system MX/D/l, " .

P(y)=P(y\y) =

O ,2'

Further, if Al = AZ = A, then show that this value of L agrees with the one which can be obtained from equation (3.5.16). Discuss L for the system Er/D/l with generalized 2-Erlanginput and constant service time. (a) The system MX/Er/l. Take k = 1 and let Al = A, PI (z) =P(z) and Po =Po, l ' Show that the first equation of the set (3.5.5) (other equations being redundant in this case) gives uniquely the p.g.f. of the number of phases in the system (queue + service) as

where Po

(b)

Lb' c.an e obtallled by one of the following methods; By usmg equation (3.5.16).

L QO

m=1

am

rm]-I L zn

n=1

= Arii/J1.. In this case L is given by

p (2 +-2 Ga a + r _l-::;\+p(r-l) a, . 2al-p 2r ( )

Here L is given completely in terms of parameters of the input and the service distributions.

(I_yk)y 2: P . j - I i=l o,JY yk{l + (AIJ1.)} + (AifJ.)yk+l·

Using Rouche's theorem, show that the deno . say Yo, such that IYol> 1. Using this fact ;mator of P(y) has one zero, '~. ~ convergence of P(y) within and on the unit circle, Iyl = 1 and th , e norm lZlllg condition, show that P(y) can be written as k

P(y) =

(Yo -1) .2: yi ]=1

k(yo -y)

-_

(1 -Yo!) k

- Ik I

QO

i=1

v=o

(

y)V

yX_

Yo

BULK-ARRIVAL QUEUES

164

PROBLEMS AND COMPLEMENTS 165

25

Now as Pn,j represents the joint probability that there are n customers in the system (queue + service) together with the arrival being in the jth phase, Pn = Lf=l Pn,i gives the probability of n customers in the system. Show that the distribution of the number in the system is given by Po = 1 - p, Pn = p(y3 1)Yonk, n;;' 1, p = ,,/k/l, and therefore L = [k(yo - 1)] -1 • Lcan also be found from equation (3.5.16), which gives an expression not involving the root Yo. Hint: Po = sum of the coefficients of y, y2, . .. , yk in P(y), Pn = coefficient of ynk+j in P(y), and use l-y~{l + ("//l)} + ("//l)y~+1 = 0, wherever necessary. Here we have found directly the distribution of the number of customers in the system. Jackson and Nickols (1956) obtained the same results for Pnand L by first considering the number of phases in the system and hence obtaining the distribution of the number of customers in the system. Accounts of Jackson's work may be found in textbooks by Saaty (1961), Prabhu (1965), and Gross and Harris (1974). (a) Continuation of Problem 24. Find the waiting-time distribution for the system E,,/MII with ordinary k-Erlang input. To consider the waitingtime (in queue) distribution for the system E k/M/l, we first need to know the probability that an arrival when it enters the queue finds n customers in the system (queue + service). Let this probability be P;;. In other words,

(b) Show that the conditional waiting-time density w;c(1') for those who have to wait is w - (1') = dP( Vq .;;; l' I Vq qc d1'

k-

= Yowq(1'),

1'>0

whic~

is an expo~ential distribution. This assertion holds even for qu~umg systems Wlth general input and exponential service times [see SmIth (1953) or Problem 7(a)J _ For numerical work on particular cases such as M/M/l, D/M/! of the model Ek/M/I, the reader is referred to Jackson (1956).

26

Continuation of Problems 24 and 25. Verify Little's formula L = A W- f

th~ system E,,/M/!. discussed in Problems 24 and 25, where "e is the :ffecti~:

arnval rate and W IS the average time spent by an . al' th . amv ill e system. Hint: Note I\, = A.jk.

27

!he system Ez /M/l. Consider the system E 2 /M/I with generalized 2-Erlang mput and exponential service time. Show that for such a system, Lq = _ P I - P

where O ,2

P

P;; = peN = nlarrival about to join the queue)

P

= peN = nlarrival in phase k of the arrival channel)

Hint:

= P(N=n, Y=k) = Pn,k = kP P(Y = k) Pk(l) n, "

where we have used equation (3.5.6). Thus by using the expression for Pn,k - the coefficient of yn"+k in P(y) - we find that

> 0)

[p{l_L} +~P "1 "2

"1

]

0,2

=~(~:-;)+(~(~:-;r +~;p(1-p)r2 =

"1 "2

------:-......0.......:0._

/1("1

+ "2)'

Use the root Zo of the quadratic equation Z2 -

(..!..+L)z "1 "2 +L "I "2 --

0

p

which lies in ] 0, 1 [_

28

n ;;. O. Consequently, if w;(1') is the p.d.f. of the waiting time (in queue) Vq of an arrival, then W;(1') = P(Vq .;;; 1'), and therefore

Continuation of Problem 27. ordinary 2-Erlang input, Po, 2

W;(O) = Po = (1 - Yok)

L

Show that for the system

Ez/M/l

with the

HI + v'I+8P] - p (4p -1) + v'I+8P 8(l-p)

29 = /l(1_Yok)Yok exp {_/l(1- yok)1'} = "(Yo-l)Yo"exp{-"(Yo-l)1'},

where we have used the expression giving the root Yo.

1'>0

F~nd the w~ting-time (in queue) distribution for the queuing system EX/E /1 WIth generalIzed k-Erlang input. " r (a) Show tha~ ~he L.T_ of the waiting time (in queue) of the first customer Of(an) .am.VIng group is given by (a) = (A,,(i.V,,(/l/(/l + a» where P z IS !71ven b .k . . D" Y equa rIOn (3 .5_9), and the corresponding mean 'waiting tIme IS gIven by

w;,

BULK-ARRIVAL QUEUES

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

167

166

, W-

q,

q,

pew)

A

= -w-O)(O) =

':.P~!)(l).

~A

where Po = 1 - PI and the corresponding expected number in the system is L _ + (Alad 2E I(V2) + (f3dadpl I PI 2(I-PI)

W~,(T)dT = ~ lpO,kO(T)dT ( T~ )

r

(n-l)r+S-le-/lT

+];1 ~IP(n-!)r+s,k {(n-l)r+ s - 1}!

(b)

1

~T •

.

Hint: For getting pew), use A2 == 0, Z == 0, and b 2 (v) == 0 in equation (3.6.18), and for getting L I , use A2 == 0 in the value of L 1. (b) Show that the L.-S.T. of the waiting time (in queue) of priority customers alone is given by _

Now use the transformation i = (n -1)r + s and take the L.T. Show that the L.T. of the waiting time (in queue) of a random customer

Wq

,

() _ 0: _

Po 0: l-A I (b 1 (a)) • --...:...:",.:.-"-''"AdA 1 (b 1 (0:))-lj +0: adl-bl(o:)]

where Po is given in (a), and

of an arriving group is given by

W

l-A(b(o:))

w~(o:) = W~,(o:) a(1-b(o:))

q,

where b(O:) = [~/(~ + 0:)] r, A(z) = L;= I amzm, = A (1)(1),. and w~, (0:) is defined in (a). ShoW further that the mean waiting time (m queue) of

a

this random customer is given by

+ C;) -1]

Ak b[a(l W- = --=- p~1)(l) + ~A

q

-AIAI(w)](w -1)

w - b l P\I - AlAI (w)]

Note that W-q, can also be obtained without using W~, (o) Hint: Using Problem 25, we get

=

= Pob l [AI

2

where is the coefficient of variation of the arrival grot~'p size. _ -It is now easy to see that Little's formula Lq = A<: Wq' where Ae - A.a and L can be obtained from equation (3.5.13), smce L = Lq + p, IS

= Al a I E I (V2) + b 1 f3diii 2(1 - PI) .

Note that the results in Problem 31 are the same (except for notation) as obtained in Miscellaneous Problems Sea), 4, and 3, as they should be. lt may also be noted that the same results could be obtained in the case of nonpriority customers alone in the system, by taking the corresponding arrival parameters and the service density equal to zero and using the appropriate formula.

C;

30

satisfied for the system E:/Er/1Hint: See Miscellaneous Problems and Complements. x Continuation of Problem 29. Show that for the system M /Er/1 with poisson input, W q,

=.!..p(1)(l)= ~

P

))[r(a~+a2)+a]

(2a(1 - P

~

where P(z) is given in Problem 23 (a). The expression for Wq, obtained here is the same as the one given in equation (3.1.27). Section 3.6. It is interesting to see that many particular cases can be deduced from th~ qu.euing MX /G G II We state below one of them, whose denvatlOn 18 system M xI 1 , 2 ' 1 , 2 • simple and is left to the reader. . . 31 (a) Show that the p.gJ., in the limiting case, of p~o~ty customers alone, that is, when there are no nonpriority customers, IS given by

MISCELLANEOUS PROBLEMS AND COMPLEMENTS The system MX/Gjl with modifications. In the limiting case as n .... 00 and =- AW < 1, consider the following modification of the system MX/G/l. In the system MX/G/l (see Problem 5 below), it is assumed that if a group arrives to find the server idle, service begins instantaneously. Now we relax this restriction and assume that when customers arrive to find the server idle, then the service begins after a random time T whose dJ. is D(t) = P(T< t) so that d = J'; t dD(t) and dCo:) = J'; e-O!t dD(t). In reliability theory, the time T is called the "warm-up" time of the service apparatus. Show that the transition matrix is given by the relations

P

+

N n+l

_ { -

N~ + X n + I

-

1,

X+T/+X n + 1 -1,

ifN~>O ifN~ = 0,

n

=

0,1,2, ...

where X n + I is the number of customers arriving dUring the time of service of the (n + l)th customer; X is the size of the group arriving at the instant of

BULK-ARRIVAL QUEUES

168

beginning of the time period T; and 7)' is the number of customers arriving during the delay of beginning of service for this group. The process {N:} is an imbedded Markov chain, and when p = Aab < 1, the limiting probabilities P/ exist and have the p.gJ.

P+(z) = P~[A(Z)J(A-AA(z))-1][z{1/b(A-M(z))}_l]-1,

zE(zN~+l) = E(zN~+Xn+lIN:> O)P(N~>O) = O)P(N: = 0).

NotingE(zN~IN~ >O)P(N~> 0) = E(ZN~) -peN:; = 0) =P+(z) -pt, where p~zn, Pt =P(N~ = 0), using independence of the r.v.'s involved and letting n -+ 00, we get the required p.g.f. As a particular case, show that if peT = 0) = 1, then P+(z) reduces to equation (3.4.9). Further show that if peT = 0) = 1 and ar = Sri' i = 1, 2, 3, ... , then

P+(z) =

}:;;'=o

+

P (z) -

__(-,--l-:-~p..!..)-,--(z_r---,1)~

- r[z(l/b(A-Azr»)-l}

2

where p = Arb, (a) Continuation of Problem 1, It is possible to discuss the limiting waitingtime (in queue) distribution of the queuing system of Problem 1. If p < 1, the limiting waiting-time (in queue) distribution of the first member of an arriving group, namely, W';-, (r) = limn"''''' P(Vq, (n)
169

00

L P(service time of the nth arrival group

B 1 (x) =

r=1

<xlnth arrival group is of size r)a r

=

L ar{B(xW'

lzl < 1,

with P~ = (1 - p )(ii(1 + 'A.1i), Finch (1959) has shown the existence of the limiting distribution for a more general queuing system GI/G/l when p < 1. His arguments can be used to show the existence of the limiting distribution for Gl x /G/l when p < 1. The case under discussion is a particular case of Glx/G/l. Hint: Taking the p.g.f., we have

+ E(zx+1)+xn+1\N~

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

r=1

{Bex)}"

= ({B(X _y)}(r-l)* dB(y),

r = 2,3,.,.

{B(xW' =B(x), and A(x) and B(x) are the interarrival-time and the service-time distributions, respectively, We now explain the use of the above integral equation by discussing the waiting·time distribution Wq , (r) for the queuing system Nlx/G/l. N~te here that since the arrivals are by POisson process, the notation Wq , (7) can be replaced by Wq , (r). To discuss this problem, we need to define the following additional transforms:

sea) Further, define the function W;,(r) =

=

L:

e- ax dS(x).

W;, (r) by the following equation:

roo Wq,(r-x)dS(x),

(Ml)

where, since arrivals are random,

Since Bl (r) = 0, for r < 0, so for r < 0, we have, following the procedure discussed in Chapter 2: W;.(r)

=

AL:_ooJ:=o

Wq,(r-x)dB1(x+y)e-AYdy

CeAT 0,

W.;-,

r
d) foo w.;-, =

r""

where C>O. (r - x) dS(x) - (1 - D(r) x

Thus we can rewrite equation (Ml) as

Wq, (-x) dS(x), r;;'

where Sex) = fooBt(X o

+ y)dA(y)

But using equation (Ml) one can see that C= which we assume to hold. Consequently,

O.

r:: Wq, (-x)dS(x),

BULK-ARRIVAL QUEUES

170

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

171

W;, (r)

ceil:r

=

{

'

Wq , (r)

j -1

r':;:;O

+ (1 -

D(r)) C,

r~O.

Vq = Vq 1

(M2)

5

e-O!T dW*q, (r)

= -=

AC =-A-a

-(

-

1,

(MS)

I·

Now as the sequence of group b k the size of th 1. scan e ta en to form a renewal process with e group p aytng the role of the waif t· discrete form of the distribution of past life (ChaPte~~) l.;e, ~;e ~anhuse the ability that the customer, though selected randomly t~S 1 X,j IS t. e probith in its group, then WI n e group, IS served

(M3)

Cd a) + w q, (a).

j~

where Wq (x)=P(V ":::x) B( )_P( L~:l S ' q, "" , x Si ':;:;x), and Vq is independent of 1-1

e-O!'T dW;, (r) = s(a)wq, (a)

p

i=l

From equations (Ml) and (M2) we get, respectively,

L:

+~ " s.

.thl

(M4)

From equations (M3) and (M4) and using sea) = A(fi(a))A/(A - a), show that

p(it! Si':;:;X)

=

'-1

wq,(a) = C[(A-a)d(a)-A][A-a-f...A(b(a))]-l

f

Xj{B(x)}(i-l)*

i=l

But since Xj is the equilibrium (discrete) density at j - 1 of the past Chapter 2), we h a v e .

where

}-l

Xj =

The constant Cis evaluated by using the normalizing condition wq , (0) = I and hence I-p C=l+Ad· (b)

(c)

wq(a) = E(e-O!vq)

-- E(e -O!v) - ~j-I q' E(e O! "'1=18;) = Wq , (a)

f Xj(b(a)]

= Wqj(a) {

(C~ + 1 + C~ Iii)

a

1

_i~lai(b(aw)

I-b(a)

l-b(a)

= Wq (a) 1 - A (b(a» 1 - b(a) .

where C~ and C~ are the coefficients of variation of group size and service time, respectively. Continuation of Problem 2: waiting-time (in queue) distribution of a randomly selected customer of an arrival group. Let Vq be an r.v. representing the limiting waiting time (in queue) of a customer selected randomly from a group so that its dJ. is Wq(r)=P(Vq ':;:;r). If j~ 1 is the position of the customer selected within the arrival group, then

i-I

j=l

t )

3

e see

1- La· _i=1 I

Continuation of (b). Show that the mean waiting time (in queue) of the first member of an arrival group is 2 I-p

1

a Consequently, taking the L.T. of equation (MS), we get

If peT = 0) = 1, then show that the L.-S.T. of the waiting-time (in queue) distribution of th~ first member of an arrival group is wq , (a) = -(l-p)a/[A-a-AA(b(a))] . It is interesting to observe that if we identify a single customer with a group whose total service-time distribution function has L.-S.T. equal to A (b(a)), then Wq (a) could be obtained from an ordinary MIG!l queue with b(a) replaced by A(b(a)). For example, see Miscellaneous Problem 6. This is due to the input being by the Poisson process. Consequently, these observations apply equally well to the problem discussed in (a).

-b

n (

J

a

It may be noted that this agrees with Problem 31(b) (Section 3.6).

4

Continuation of Problems 2 and 3 Sh queue) of a random customer of an· arriva~:ro~;\:he expected waiting time (in

Wq

[a(C~ + 1)+ p(cg + 1)-1] b 2(1 - p)

BULK-ARRIVAL QUEUES

172

MISCELLANEOUS PROBLEMS AND COMPLEMENTS 173

which may be obtained by adding b [a(l + C;) -1] /2 to the expression given in Problem 2 (c)_ It may be observed [in view of this relation, which could be obtained from Miscellaneous Problem 5 (a), that is, from equation (M6)] that Little's formula Lq = i\eW~, where i\e = Aa, holds for the system MX/G/l. In the slightly less general case MX/Er/l, the formula can easily be verified by looking at the value of Lq found in Problem 2_ Quite a number of authors have given derivations of Little's formula, including Jewell (1967), Eilon (1969), Maxwell (1970), and Stidham (1974) among others_ If a! = 0lr> 1= 1,2, 3 _.. , in addition to peT = 0) = 1, then show that

equation (2.2.25), or by followin the . vation of equation (2.2.25). g procedure discussed in the deri-

(b)

Continuation of (a). Show that the lim'r 1 illg p.g.f. of the number in the system conSidered immediat 1 f e yater a departure instant (imbedded Markov chain) is given by

_ () _ () 1 - (b(a)Y a = w q , a . r(l- b(a))

=

I

p;zn = l=p A(z)-l n=O a {z/b(i\ - M(z»)}-l' (M7) Hint: Use the pro d . . . ce ure used to get equation (2 1 27) I tiS mterestmg to see th l' .. . e re atlOn between P(z) and P+(z), which is r(z)

Wq

s

Most of the results discussed in Problems 1 to 4 are due to Sahbazov (1962). Following Burke (1975), we have corrected the error in the original derivation of wq(a) by Sahbazov. The question discussed in Problem 3 has also been discussed by Soriano (1966), though under a different notational garb. The system MX/G/l. In this problem we discuss the system MX/G/l using the supplementary variable technique. The present solution generalizes the solution in Section 3_1 and derives some of the above problems as particular cases. A comparison between the solution to this problem and the Erlangian solution given in Section 3.1 thus shows both the usefulness and the limitations of the Erlangian technique (or its modified form). Section 3.1 also shows the formal identity between the MX/E dl and M/E x/I systems. In view of this, one might question the need for Section 3.2. Although for some purposes the Erlangian technique has been superseded by more general methods, we consider it to be still worthy of study for several reasons, in particular its relative simplicity for some problems requiring explicit solutions, its wide use in the literature, and its continuing usefulness in some models which yield most easily to a combination of two or more techniques. Since the procedure and steps used are quite similar to those discussed in Section 2.2 and also the notation is the same, we do not give the details here, but rather give the results with brief descriptions. Chaudhry (1979), while discussing the advantages of the supplementary variable technique over some other techniques, unifies several results for the queuing system M x /G/I and gives the correct expression for L q which has been incorrectly reported by Krakowski (1974), using the conservation principle. (a) Show that the limiting p.g.f. of the number of customers in the system M X /G/l is Po(z -1) PCz) - --::---'''-'----''--:(M6) - [Z/b(A - i\A(z))-1] where Po = 1 - p = 1 - ("ajM), A(z) = 2:;"=1 arz , and b(a) is the L.-S.T. of the service time distribution. Hint: We could obtaln this by replacing z by A(z) in b(/\ - i\z) of r

P+(z) = A(z) -1 a(z - 1) P(z). (c)

(M8)

In ~apt~r 6 we discuss many such relations. Wrutmg-tIme (in queue) distributi group. nus is essentially Mi IIon for the first customer of an arrival ous ~oblem 2~b) whose discussion depends on the use of the inst~e r:: alternative deriVation which d g d qUatlOn technIque. The follOwing n chain t~chnique, is m~re elega:r s on the use of the imbedded Markov FOflfwe identify a group with . . a sl.ngle customer, then its (the group's) service time is just the total fVlc e tune of the members constituting the group. Consequently the se p.g.f. A(z) = z The m' ean gro.uP Ccustomer) will have as its group size . amval rate will be " d h group's service time distribution ill b ' an t _e L.-S.T. of the w e Sg(a) = A(b(O')). Using this information, we have from (M7)

P;(z) = (1-P)(Z.=-l)Sg(A-Az) (M9) z - Sii\ - i\z) where we have used = 1 + . Pg (z) may now be interpreted as the f p.g.. of the number of customers (groups) left behind by a d ' customer (g ) 'T'l.-! also be obtained as follows epartmg g th roup. uus p.g.f. may departing customer (group)' Ithnce e customers (groups) left behind a . are ose that arrive d ' h . . ill the queue and in service and . . unng t e tIme It spends , smce arnvals are random, it follows that

a

w

P;(z) = q, (i\ - i\Z)Sg(A - AZ) _ where w q , (a) is the L.-S.T. of the di ' .

(M

10

. stnbutlOn of queumg time for a customer (that is for the fi (MI0) and (M9) f;llows: ust member of a group). From equations

Wq , (a)

(l-p)O' a - A(l - Sg(a)' which is essentially th 1 . e resu t contruned in Problem 2{b).

175

REFERENCES

BULK.ARRIVAl QUEUES

174

6

(a)

Using in the above equation the normalizing condition PN'(l -) = 1, we get peN' + \)I = 1) = [1 - E(1/1)] fa. One can also see easily that P.p(z) = b('AM(z)), where b(O!.) = fO' e-Wb(v) dv and E(1/1) = 'AalJ1 = p.

'From formula (MS) show that for an Continuation of Problem 5. MXjGjl queue, Po = aP~.

Since the groups are arriving by a homogeneous Poisson process, the steadystate distribution of N- = (number in the system just before an arrival epoch) must be the same as the distribution of N = (number in the system at a random epoch). Suppose now that a customer arrives at a busy server in the interval Ct, t + dt] , t time units after the beginning of a service period whose length is Vb' The probability of this event [by Problem 15(d), Chapter 2J is

This somewhat paradoxical result states that the s~stem ~s more likely - > 1) at a random time than Immediately after a to be empty (for a departure. . (b) Obtain the result of (a) directly, without the use of generating fun~tlOns. Hint: (PriT I mean number of customers served in a bus~ penod .. In this problem we consider an alternative solution XjGjl The system M . . [ procedure for Problem 5 (a) using the length·biased samp~ng phen.omenon see Problem 15(d), Chapter 2] and the imbedded Markov cham technique. . . Let N~ be the number of customers present in the system at the begmmng

=

7

vb(v) dt p E(V) -;. Consequently, the p.g.f. of N(or N-) is

of the service of the nth customer. Then as

+ 1/1n - 1 ;;. 1 ifN~ + 1/1n = 1

if N~

Pw(z)

(MU)

= m) = am,

PNb(tiz)

m;;' 1

d- - A(l)(l) an a - · ..' The interarrival times of the groups are exponentially dlstn.buted mdependent r v's with mean rate 'A. The service times are i.i.d. r.v.'s With p.d.f. b(V? suc~ th~t 1/J1 =E(V)= vb(v)dv. Define p = 'A~/J1. ?early , the process {Nn+d 1S a Markov chain whose limiting distribution eXists if? < 1. . Now the p.g.f. of the LV. N~+ 1 defined by equatIOn (MIl) IS

= m withA(z)=~m=lamz

,

+ 1/1n -1 ;;'1] PeNn + \)In -

+ E[zxIN~ + \)In

=

l)P(N~

+ 1/1n =

1 21)

1).

E(zN~+.pn) = E(zN~+.pnIN~ + 1/1n;;' 2)P(N~ + \)In;;' 2) l)P(N;,

+ \)In =

and X is independent of N~

1)

+ 1/1n, we have on proceeding to the limit as n -+ PN'(z) = Z-l [PN'(z)P.p(z)-zP(N' + 1/1 = 1)] +A(z)P(N' + 1/1 = 1).

Hence

where the

00,

z(A(z) - OpeN' + \)I = 1) z -P.p(z)

N~ and \)In are distributed as N' and 1/1, respectively, with PN'(z) =

E(zN') andP.p(z) =E(z1/J).

f f =

v=o

v t=O

PNb(t)(Z) b(V)) dt + (1 - p) E(V

= PN'(z)exp

[-At(l-A(z))].

PN(Z)

=

~1 - p)(z -1) {zlb(A - 'AA(z))} - 1

which is the same as equation (M6). [Adiri (1973)] .

REFERENCES

"'"

But as

= + E(zN~+.pnIN~ + \)I =

=p

Making various substitutions, one can finally get

ro

P' (z) = E[zN'n+t] Nn+l = E[zN~+.pn-IIN~

PN(z)

where peN = 0) = 1 - p and Nb(t) denotes the number of customers in the system t time units after the beginning of a service on the first member of an arrival group. But as Nb(t) =N' + \)I,

n = 1, 2, ... , where 1/1 n is the number of custom:rs that arrive during the service of the nth customer and X is the size of an arnval group such that

P(X

=

Adiri, 1. (1973). Cyclic queues with bulk arrivals.J. Assoc. Comput. Mach. 20,416-428. Bahadur, R. R. (1960). Some approximations to the binomial distribution function. Ann. Math. Stat. 31,43-54. Bailey, N. T. J. (1954). A continuous time treatment of a simple queue using generating functions. 1. R. Stat. Soc. Ser. B 16, 288-291. Barber, B. (1964). Some numerical data for single server queues involving deterministic input arrangements. Oper. Res. Quart. 15, 107-115. Brockmeyer, E., H. L. Halstrs"m, and A. Jensen (1948). The life and works of A. K. Erlang. The Copenhagen Telephone Company, Copenhagen, Denmark. Brockwell, P. J. (1963). The transient behaviour of a single server queue with batch arrivals. J. Aust. Math. Soc. 3,241-248. Burke, P. J. (1975). Delays in single-server queues with batch input. Oper. Res. 23,830-833. Champernowne, D. G. (1956). An elementary method of solution of the queueing problem with single server and constant parameters. 1. R. Stat. Soc. Ser. B 18, 125-128.

BULK-ARRIVAL QUEUES

176

Chaudhry, M. L. (1979). The queueing system M X /G/1 and its ramifications. Naval Res. Logist. Quart. 26,667 -674. Clarke, A. B. (1956). A waiting line process of Markov type. Ann. Math. Stat. 27,452-459. Cobham, A. (1954). Priority assignment in waiting line problems. Oper. Res. 2, 70-76. Cohen, J. W. (1963). Applications of derived Markov chains in queueing theory. Appl. Sci. Res. lOB, 269-303. Conolly, B. W. (1960). Queueing at a single servin!e !,oint with ~rou!, arrival. J. R. Stat. Soc.

Ser. B 22, 285-298. Crabill, T. B. (1968) Sufficient conditions for positive recurrence and recurrence of specially structured Markov chains. Oper. Res. 16, 858-867. Crommelin, C. D. (1932). Delay probability formulae when the holding times are constant. P.O. Elect. Eng. J. 25,41-50. Eilon, S. (1969). A simple proof of L == Oper. Res. 17,915-917. Erlang, A. K. (1920). See Brockmeyer, E., et al. (1948), chap. 1. Fainberg, M. A. (1974). Servicing a nonordinary flow by a one-channel system with waiting. Eng. Cybern. 12,82-89. Finch, P. D. (1959). See Chap. 2. Foster, F. G. (1961). Queues with batch arrivals LActa Math. Acad. Sci. Hung. XII, 1-10. Gaver, D. P. (1959). Imbedded Markov chain analysis of a waiting-line process in continuous time. Ann. Math. Stat. 30,698-720. Gross, D., and C. M. Harris (1974). Fundamentals of queueing theory. Wiley, New York. Gupta, S. K. (1964). Queues with batch Poisson arrivals and a general class of service time distributions. J. Ind. Eng. 15,319-320. Harris, C. M. (1970). Some results for bulk-arrival queues with state-dependent service times. Manage. Sci. 16, 313-326. Hawkes, A. G. (1965). Time dependent solution of a priority queue with bulk arrival. Oper.

"W_

Res. 13, 586-595. lvnitskiy, V. A. (1975). A stationary regime of a queueing system with parameters dependent on the queue length and with nonordinary flow. Eng. Cybem. 13, 85-90. Jackson, R. R. P., and D. G. Nickols (1956). Some equilibrium results for the queueing process Eh/M/l. J. R. Stat. Soc. Ser. B 18, 275-279. Jaiswal, N. K. (1968). Priority queues. Academic Press, New York. Jewell, W. S. (1967). A simpler proof of L == "W. Oper. Res. 15, 1109-1116. Krakowski, M. (1974). Arrival and departure processes in queues. Pollaczek-Khintchine formulas for bulk arrivals and bounded systems. Rev. Fr. Auto. Inf Rech. Oper. 8,45-56. Ledermann, W., and G. E. H. Reuter (1954). Spectral theory for the differential equations of simple birth and death processes. Phi/os. Trans. R. Soc. London. Ser. A 246, 321-369. Lindley, D. V. (1952). See Chapter 2. Little, J. D. c. (1961). A prooffor the queueing formula L := "W. Oper. Res. 9, 383-:-387. Luchak, G. (1958). The continuous time solution of the equations of the single ~hannel q~eue with a general class of service-time distribution by the method of generatmg functIOns. J. R. Stat. Soc. Ser. B 20,176-181. Maxwell, W. L. (1970). On the generality ofthe equation L == "W. Oper. Res. 18, 172-173. Morse, P. M. (1955). Stochastic properties of waiting lines. Oper. Res. 3,255-261. _ _ _ _ (1958). Queues, inventories and maintenance. Wiley, New York. Morse, P. M., H. N. Garber, and M. Ernst (1954). A family of queueing problems.J. Oper. Res. Soc. Am. 2,444-445.

REFERENCES

177

Pegden, C. D. and M. Rosenshine (1982). Some new results for the MIM/1 queue. Manage. Sei. 28, 821-828. Pike, M. C. (1963). Some numerical results for the queueing system DIEhll. J. R. Stat. Soc. Ser. B 25,477-488. Prabhu, N. U. (1965). Queues and inventories, a study of their basic stochastic processes. Wiley, New York. Restrepo, R. A. (1965). A queue with simultaneous arrivals and Erlang service distributions. Oper. Res. 13,375-381. Ross, S. M. (1970). Introduction to probability models. Academic Press, New York. Saaty, T. L. (1961). Elements of queueing theory with applications. McGraw-Hill, New York. Sack, R_ A. (1963). Treatment of the non-equilibrium theory of simple queues by means of cumulative probabilities. J. R. Stat. Soc. Ser. B 25, 457 -463_ Sahbazov, A. A. (1962). A problem of service with nonordinary demand flow Sov Maih 3 1000-1003. . . ., Shanbhag, D. N. (1966). On a duality principle in the theory of queues. Oper. Res~ 14, 947949. Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge Philos. Soc. 49 449-461. ' Soriano, A. (1966). On the problem of batch arrivals and its application to a scheduling system. Oper. Res. 14, 398-408. Stidham, S. (1974). A last word on L == "W. Oper. Res. 22,417-421. Suzuki, T. (1963). Batch arrival queueing problem.J. Oper. Res. Soc. Jpn. 5, 137-148. Takacs, L. (1962). Introduction to the theory of queues. Oxford Univ. Press, New York.

Additional References Bhat, U. N. (1968). A study of the queueing systems MIG/] and GI/M/l. Lecture Notes in Operation Research and Mathematical Economics, vol. 2. Springer-Verlag, New York. Chaudhry, M. L. (1974). Transient/steady-state solution of a single channel queue with bulk arrivals and intermittently available server. Math. Operationsforsch. Stat. 5, 307 -315. Fainberg, M. A. (1974). See references. Foster, F. G. (1964). Batched queueing processes. Oper. Res. 12,441-449. Gupta, S. K., and J. K. Goyal (1965). Queues with batch Poisson arrivals and hyper-exponential service. Naval Res. Logist. Quart. 12, 323-329. Kambo, N. S., and H. S. Bhalaik (1982). Bulk arrival heterogeneous queueing systems. Opsearch 19,97-105. Keilson, J. On the asymptotic behaviour of queues. J. R. Stat. Soc. Ser. B 25,464-467. Kerridge, D. (1966). A numerical method for the solution of queueing problems. New J. Stat. Oper. Res. 2,3-13. Kleinrock, L., R. R. Muntz, and E. Rodemich (1971). The processor sharing queueing model for time-shared systems with bulk arrivals. Networks 1, 1-13. Lippman, J. A., and S. M. Ross (1971). The streetwalker's dilemma - a job shop model. SIAM J. Appl. Math. 20, 336-342. Mohanty, S. G., and J. L. Jain (1970). On two types of queueing processes involving batches. Can. Oper. Res. Soc. J. 8, 38-43. ____ (1971). The distribution of the maximum queue length, the number of customers served and the duration of the busy period for the queueing system M/M/l involving batches.INFOR 9, 161-166.

BUI.K.ARRIVAI. QUEUES

178

Moore, S. C. (1975). Approximating the

beha~or

of non'stationary single server queues. Oper.

Chapter 4

Res. 23, 1011-1032.

al e functions in dUN Bhat (1972). A computational approach to some mean v u ;)!E il·queue. Tech. Rep. CP72015 CS/OR Centre, SMU (U.s.A.). . Y . . al . b t h s of variable size and service rate an _ Murari, K. (1969). A queueing problem with arnv s rn ace depending on queue length. Z. Angew. Math. Mech. ~9, 157 162~th correlated batch _ _- - (1972). Time dependent so~uti?n ?f a que~emg prob~m . al and general service time dlstnbution.Memka 19,201 208. . amv s . ' t with bulk arnval. Electron. Nakamura, G. (1968). Analysis of a discrete·tlIDe queuerng sys em

Commun.lpn. 51·A, 27-32. h to tic distribution of the traffic·time Narasimham, G. V. L. (1968). A note on t e asymp average in Gl!G!"" with bUI~.arrivals.l. Appl. Prob. 5, ~76~4:~~avior of the queue with . M F (1977) Some expl1clt formulas for the steady sta Neuts, . . ' I Pr b 9 141 157 semi·Markov service times. Adv. App . 0 . , -:-. I orithmic approach. _ _ _ _ (1981). Matrix.geometric soluti~ns to stochastic models - an a g The Johns Hopkins Univ. Press, Baltimore. . hase _ _ _ _ and S. Chakravarthy (1981). A single server queue with platooned arnvals and p . 2 222-261 type services. Eur. I. Oper. Res. 8, 379-389. . V (1980). TheN!G!1 queue and its detailed analysIs. I. APPI. Prob. 1 , . .

~:::::'I'S. ~. (1973). On the length and number of served customers of the3~~~3~~lOd of a

liz d M!G!1 queue with finite waiting room. Adv. Appl. Prob. 5, . genera e . tern with several types of customers. Ann. Inst. Stat. Shanbhag, D. N. (1969). A queuerng sys Math 21,367-371. M t 'k 20 . . blem with batch arrivals and correlated departures. e rz a , Sharda (1973). A queuerng pro

81-97. . ' t with arrivals in batches of D (1975) On continuous/discrete tlme queuerng sys em S. . ' h Sarma, J A I Pr b 12 115-129 variable size and correlated departures.. pp . 0 . , .'. h d lin . 1962 On the problem of batch arrivals and its applica:lOns to a sc e u g sonan~r~ie~. D1~toral dissertation, The Johns Hopkins University, Baltlffiore. L (1962) See references. Wil N T ak'acs,. . . . h d . the theory of stochastic processes. ey, ew _ _ _ _ (1967). Combmatorzal met 0 s m York. babilities in a general class of single Van Hoorn, M. H .. Sci. 27,1178-1187. server queuerng sy al . f b lk arrival queueing systems". M.A.Sc. W (1981) "Contribution to the an YSls 0 u Yao, D~h~~iS, Depart~ent of Industrial Engineering, University of Toronto.

(198!I~:S1~~~~~:u~o~:i~:I:.t~:n~~~.

BULK-SERVICE QUEUES In Chapter 3 we have considered single·server queuing problems in which arrivals occur in bulk. We now turn to queuing situations in which arrivals occur singly, but service is in bulk. Many transportation processes involving buses, airplanes, trains, ships, elevators, and so on, all have a common feature of bulk service. It.may happen that the server (or carrier) has a fixed maximum capacity, say B, or else the server may take units depending upon the number already present with the server. Such systems may serve as a model for a shuttle or automatic elevator. At times it may also be necessary to change the capacity at the time of initiation of service. For example, consider a train to which additional cars may be added at the starting station when the queue length at that station reaches or exceeds a certain number E, or may be removed if the queue length drops below A B, then the first B customers will be taken into service. This description may well fit, at least approximately, the operation of an unscheduled car ferry or a single ground.floor station of an elevator. Another application of this could be to traffic flow where a minor road merges into a major road. The traffic flow on the major road is interrupted by a traffic light after a certain period of time if at least k cars have activated a trip plate on the minor road. Otherwise the light stays green until k cars have arrived. The duration of the red signal is such as to allow at most B cars per cycle to merge into the traffic on the major road. If the successive service times are taken as the times required for successive batches of cars to merge, together with the fixed length of the green signal on the major road, then the study of the queue on the minor road may well be described by the above model. Such a bulk·service system has been called a system with general bulk·service rule as opposed to other types of bulk·service rules. Another illustration arises if we consider places like an amusement park, a museum, or an art gallery where there are guided tours. Suppose that a guided tour is not started until there are at least k prospective customers (visitors). This introduces what we normally call a "quorum," which occurs in other service systems as well. In another type of bulk service, the server always takes a fixed number of customers. Such bulk queuing systems have some intrinsic interest, as well as some practical applications, and can be considered as a special case of the general bulk·service rule. 179

180

BULK-SERVICE QUEUES

There is a close analogy between the' theory of queues and the theory of reservoirs and dams, a still closer one between single-server bulk queues and models of storage and inventories, We pause briefly to point out this analogy by giving a few references and examples, For the analytical probabilistic study of models of storage and dams, the interested reader is referred to books such as Moran (1959) and Prabhu (1965)_ In models of storage of inventories, which may be considered to consist of discrete units (such as storage of newspapers by a newsboy or cigarette boxes by a tobacconist), or to be measurab1.e on a continuous scale (such as electricity, water, or even wheat), the output is random, but the inpat may be somewhat controlled, whereas in models for dams the input is random, but the output may be controlled, In the theory of queues, both the input as well as the output are, in general, random, Clearly, the element of randomness appears in all three subjects - theories of queues, storage, and dams, In addition, a large number of problems arise in the three studies depending on whether the queue (dam or store) is of finite or infinite capacity, and on whether the queue length (commodity stored or quantity in dam) and the time variable are considered as discrete or continuous, The mathematical theory of dams of "infinite capacity" is closely related to the theory of infinite waiting-space bulk-service queues of the type GeomlG B 11 (see Section 4.4), if we identify the content of the dam at each instant of release with the number B being served at the corresponding service instants. For details on the discussion of models of this kind for dams, see Moran (1959). A different analogy exists between the queue DIG/I (a speCial case of the GI/G/I queue discussed in Chapter 2) and dams of infinite capacity, For details, we again refer the reader to Moran (1959) or Smith (1953). Stidham (1974) considers a queue (or stock in inventory) which builds up until it reaches a certain level, say 1, and then is instantaneously cleared and the situation allowed to repeat itself. Assuming that the epochs of clearance are regeneration points, Stidham studies the properties of the net quantity in the system at a given instant of time. Ghosal (1970) approaches some bulk-service queuing and inventory problems from a unified viewpoint. The introduction of some kind of dependency, queue disciplines other than FCFS, correlation, and so on, make bulk-service queuing systems more cumbersome and less tractable analytically, If such a model is tractable at all, computational work is likely to become more tedious and time consuming even on large compu ters, In view of this, we do not consider such systems here although the lists. of references and additional references include papers on some such systems, on which analytical work appears to have been done, In this chapter we limit ourselves to an analysis of bulk-service queues with independent arrivals and FCFS queue discipline,

4.1

THE SYSTEM MIG B /1

4.1

THE SYSTEM M/GB /1

181

customers, instead of being served singly, are now served in batches of maximum size B. Entry into service is instantaneous so long as customers are waiting in the queue, otherwise the server waits for a customer to come and starts service as soon as the customer arrives, In the study of MIG/I in Section 2.2 we obtained the distribution of the number in the system, For the present system it is more convenient to obtain the distribution of the number in the queue, which immediately leads to the imbedded Markov chain results and the waiting-time distribution, Let Nq(t) be an LV, representing the number in the queue (waiting for service) at time t. The process {Nq(t), t;;;' O}, though non-Markovian, can be made Markovian by the introduction of a single supplementary variable X(t), which may be taken as the elapsed service time of the group under service at time t. We study the steady state of the vector process {Nq(t), X(t)} as t ~ 00. Since the analysis runs parallel to what has been done in Section 2.2, details are omitted. Let 4$ now define the following probabilities:

1

l1(X) is the conditional service rate, so that the service-time density b(x) and the d.f. B(x) are given by

b(x)

l1(x)exp [-

B(x)

1 - exp [-

and

with

o<

1//1

=

r

r r

T/(t)dt]

l1(t)dt ]

xb(x)dx

< X(t)

< 00. + dx],

2

Pn,l (x, t)dx = P[Nq(t) = n, x

3

Thus Pn , 1 (x, t)dx is the probability that there are n in queue [not in system (queue + service) J and the elapsed service time lies in ] x, x + dx] . Poo{t) = P[Nq(t) = 0 and the server is idle].

.;;; x

n ;;;. O.

Let customers arrive at the arrival epochs 0 = o~, o~, , .. , a;, , . , and let 00, 01 , an, ... be the departure epochs. In addition, we shall use the following notation: 02, , . , ,

lim Poo(t) = P oo

t-+=

The limiting Behavior of Nq(t}

Pj = lim P[Nq(t) = i], t-+=

B

In the system MIG 11 the customers arrive singly by a Poisson process with rate A and wait in the queue until served, The service-time distribution is arbitrary. The

Note that~, is the limiting probability (as t ~ 00) of j customers in the queue at a random epoch of time.

BUlK-5ERVICE QUEUES

182

Assuming that the various limiting probabilities exist, as they do when p = t..,!CB/l) < 1, we find the p.g.f.

f

P(Z) =

4.1

THE SYSTEM MIG B /1

18

where

Consequently

Pj'}

j=O

Pq(z) =

of {Pj } from the following partial differential· difference equations:

o=

apOa:(x) aPnax1 (x)

+ J[= Po ' 1 (x}ry(x)dx

(4.1.1)

= -(t.., + 1)(x))PO,l(X)

(4.1.2)

- APoo

o

= -(t.., + 1) (x))Pn , 1() x

,

D

+A.rn-l,l

(

r

Pq(z;x)dx

= P (z'O) I-b(A-"Az) q, A(l-z)

(4.U

and finally

P(z) = Poo

n

) X,

>

(4.1.3)

0

+ Pq(z).

(4.1.5

Equation (4.1.9) gives the p.gJ. of the number in the queue, but involves B + unknown probabilities fo= Pn ,l(x)7)(x)dx, n=O,l" ... ,B, where we have use equation (4.1.1). Evaluation of these unknown probabilities is discussed later.

which are to be solved under the boundary conditions

r= P + 1(x)7)(x)dx, Jo n B'

Pn 1(0) =

,

n

>0

(4.1.4)

(4.1.5) and the normalizing condition

P oo

+

fo

r

Pn,l(x)dx

If N(t) is the number in the system at time t, then

PI

=

lim P[N(un

n->-=

+ 0)

= j].

This means that {P/} is the limiting distribution (as n ->- =) of the number ( customers in the system immediately after a departure epoch. We have, since there will be j customers in the system just after a departure and only if there were j customers in the queue before the departUre,

(4.1.6)

1.

j

~

0

(4.1.1 1

0

where D is a normalizing constant. The p.gJ. P+Cz) = Lj=o P/z! is then given by

Define the p.g.f. Pq(z;x) by

Pq(Z;x) =

f

(4.1.7)

Pn,l(X)Zn.

n=O

Multiplying equations (4.1.1) to (4.1.3) by appropriate powers of equation (4.1.7), we get

Z

and using

Pq(z;x) = Pq(z;O)(l-B(x))exp [-t..,(l-z)xJ.

= DPq(z; O)b(A. -"Az) beA -"Az)

.~

J=O

{4.1.1

p/(zB - z!) (4.1.1

Similarly, from equations (4.1.4) and (4.1.5) and the above equation which gives

one finds

By first putting the value of Pq(z; 0) from equation (4.1.8) in equation (4.1.11) al then using the normalizing conditions Lj=o Pi = 1 and P oo + Pq(l) = 1, one c; find that D = [(1 - Poo )llr 1 . In view of this, we can put equation (4.1.10) in the form (4.1.1

BULK.SERVICE QUEUES

184

Before proceeding further, it would be appropriate to discuss t~e evaluation of the unknown probabilities in P+(z). It can be shown by RoucM s theorem that the denominator of equation (4.1.12) has B-1 zeros inside and one on the umt circle, Iz 1= 1. As P+(z) is analytic within and on the unit circle, the ~u~erator. must vanish at these zeros, giving rise to B equations in B unknowns PO,P 1 , · · · , p+ _ . In order that P+(z) be uniquely determined, it is necessary to show that t~es~ B equations are linearly independent. This condition is satisfied for ~he system MIE~ 11 in which the service·time distribution is k·Erlang, the convolutIOn of k independent exponential distributions with common mean 11M. The System MIEf /1

THE SYSTEM M/G B /1

185

Since IZil < 1 and Zj -:/= zj, 1 ~ i <j ~ B-1, b" -:/= 0, and hence P+(z) is uniquely determined. For details of the evaluation of b" see Problems and Complements. Relation between P oo and

P;

Now we return to MIG B 11 and the evaluation of the unknown probabilities Jo~ Po, 1 (x)'T/(x)dx, JO~Pl,l(X)'T/(x)dx, and so on. Once P;, ... ,PB- 1 are known, we can find these probabilities by using equation (4.1.13), since P oo may be expressed in terms of P;, which is known. From equations (4.1.1) and (4.1.13) we can easily get (for M/C B /1, in which M- 1 is the mean service time and Bp = "AIM) P

In this case we can show that P+(z) is unique. For MIE~ 11, the denominator of equation (4.1.12) becomes, since now P = (k"A)/(BM),

ZB_(

\k =zB_f 1+ BP (1-zT

M

l

"A + 11 - "Az J

k

'j

k

which has B-1 simple zeros zi, i = 1,2, ... , B-1, inside \z I ~ 1. together. with one simple zero zB = 1 on lz \ = 1. That the zeros in or pn the umt ClIC\e are Simple is easy to see. For if Z =z is a multiple zero with \zl ~ 1, =

ZB

[1 +B:O-Z)f

and, differentiating both sides, ZB-I

=

P[l+B:(l_·Z)r

h

l -

Dividing these two equations gives rise to p;;' 1 for Iz\ ~ 1, a condition which is excluded by hypothesiS. Now the B equations containing the B unknowns are B-1

I

4.1

p/(zf -z{) = 0,

i = 1,2, ... , B-1

j=O

together with j=O

fl}::\(zj-l)21l P2lJl(Zj- Zi)1(-1) B- 1

0

Bp +P;

(4.1.14)

" This completes the discussion of the number in the queue for the system M/ G B /1. For more details and references to numerical work, see Problems and Complements. It may, however, be remarked here that since arrivals are by a Poisson process, the number in the queue as seen by an arrival and by a random customer must be the same. The above results and derivations are due to Chaudhry and Templeton (1981). In the special case when C = E R, that is, when the service is through a random number R of exponential phases, similar work has been done by Jaiswal (1960a). For correction to equation (29) of laiswal (1960b) see Problems and Complements. The result (4.1.1 2) was first presented by Bailey (1954) through the imbedded Markov chain technique (see Problems and Complements). In Bailey's case the queue length is measured at epochs just before service is to start, or equivalently just after service ends. In his analysis, Bailey assumed that the server is "inter· mittently available," and continues to serve even when there are no customers waiting. Thus in such cases service is virtual; otherwise it is real. However, it can be seen that the result (4.1.12) still holds even if the server interrupts service whenever he finds no one waiting in the queue, and resumes service as soon as the subsequent arrival comes, as assumed by Jaiswal. Bloemena (1960) also made this additional assumption, and using the method of collective marks due to Van Dantzig (1947, 1948, 1955), showed that equation (4.1.12) holds. The system M/G B /l has also been discussed by Novaes and Frankel (1966) as a particular case of their system MIG B /1 with balking. They reproduce the analysis due to Bailey, but give a different approach for the numerical analysis of the problem.

(B-j)P/ = B(l-p).

These equations are linearly independent if their determinant b" -:/= 0, where b,,=

-

The Waiting-Time (in Queue) Distribution

B-1

I

p+

_ 00

[B-1 j - I

B

1

-.

Let Vq(t) be the virtual waiting time in queue for a customer who may occupy any position from 1. to B in his group, and consider the limiting behavior of Vq(t) as t -+ 00. In the hmiting case, Vq is the time spent in the queue by an arriving customer. Let wq(1') be the p.d.f. of Vq so that wq(1')d1' is the probability that the waiting time of a customer, who may occupy any position from 1 to B in his group, lies in [1', l' + dt]. Since we are considering a queue with FCFS queue

BULK-SERVICE QUEUES

186

discipline, the number of arrivals during the waiting time (in queue) of the customer under consideration must equal the number in the queue immediately after the start of service on his group. Since arrivals are random, it follows that (4.1.15) P + Pq(z) = Wq(A - Az)

oo

where wq(a) is the L.-S.T. of the waiting-time dJ. Wq(t) , defined by wq(a) =

C

e-

at

dWq(t).

From equations (4.1.8), (4.1.11), and (4.1.15), we get, using also a = A - Az, wq(a) = Poo

+ [.l(l -

P+[1 - (a/A)]

_

Poo)(l - b(a))

ab(a)

(4.1.16)

It is now an easy matter to find the distribution of waiting time for those who have to wait. For ifw c (1'11'>0)d1'is the conditional probability that a customer waits in the queue between 7 and r + dr, given that Vq > 0, then from equation (4.1.16) on removing the additive constant Poo (corresponding to the zero wait) and the multiplicative constant (1 - Poo ), we get wc(al Vq

>

-

0) = J.l.(l - b(a))

P+[1 - (a/A)]

ab(a)

.

(4.1.17)

It may be observed that once Pq(z) is known, it is easy to find wq(a) given by equation (4.1.16) or wc(a\Vq >0) given by equation (4.1.17). For the system M/E~/1, laiswal (1960a) derived equation (4.1.17) by a cumbersome though straightforward method, which has already been explained in Chapter 3 while discussing the system M X/ER/l. Downton (1955) derived the same equation through the imbedded Markov chain technique, using Bailey's (1954) results. In the above approach, which gives immediate results, we use neither Downton's technique nor laiswal's method. Little's formula is satisfied (see Problems and Complements) as it should be since his formula is independent of the distributions involved, though it requires careful interpretation in some situations.

THE SYSTEM M/G B /1/M WITH CUSTOMERS SERVED IN BATCHES OF VARIABLE CAPACITY

4.2

The Limiting Behavior of

N;

2

Consider the following modifications of the system M/G /1 discussed in the last section (more specifically, see Problems on that section). Everything remains the same as in M/G B /1, save the following: The waiting room has capacity M, so that not more than M + B customers can be present in the system at any time, M customers waiting for service and B customers being served. If a customer arrives to find M - 1 customers

187

waiting in .the queue, the input process stops after accepting the arriving customer; It does not restart until M customers only are present in the system, that is, until the current service period is completed. The ~ustomers are served in batches of variable capacity, the maximum capacIty of the server still remaining B.

Althou~ the. infinite waiting-room queues serve as a good approximation to queues ,:Ith finIte waiting room in light traffic (p ~ 1), the finite waiting-room queu~s gIve better results in heavy traffic (p -+ I) or where p> 1. In practice, all queuIng systems have finite waiting room, and often this waiting room is small so that for practical applications there is a real need for a theory of finite waiting: room que~es. Let now 00 = 0, 01, 02, . . . , an, ... be the instants of departures of successIve batches, 00 corresponding to the initial departure. '[he service times {Vn} are Li.d. r.v.'s with dJ. B(v). The sequence {Vn} is independent of the arrival process, which is Poisson with rate A. If Y n customers are already present with the server at epoch an, then the server takes min (B - Yn , whole queue length) customers at an. We are assuming that the service batches are not accessible in the sense of Bhat (1964). T~at is, if a customer arrives to find m, < m < B, in service, he ~annot be accepted Into the batch already undergoing service, but has to wait untIl the next service instant. Suppose that {Yn } are Li.d. r.v.'s with distribution given by bm , P(Yn = m) = ( 0, m>B

°

so that L;g bm = 1. Also s~ppose that {Yn } are independent of the arrival process. Let NCt) be the ~umber In ~he .system at any time t. Define N~ = N(on + 0) and let P/ = hmn-+~ peNn = I). Denote
B

1

THE SYSTEM MIG B111M

4.2

= fo~P(if;n+! = jjserviceperiod

v)dB(v)

_ J_ oo e-AV(Av'j 0 ., dB(v). ] . Let us assume k j > for all j;;' 0, and b m > for all permissible m. The finite chain is then irreducible and aperiodic, so the limiting probabilities p:<- exist independent of the initial distribution and are the unique solution Jof the equations

°

°

BULK-SERViCE QUEUES

188 M

L piPij, i=O

P/

4.2

THE SYSTEM M/GB/1/M

189

j = O,1,2, ... ,M-l

+ Pt(¢B-1k j + bBkj - 1 ) + P;(¢B-Zkj + bB-1kj - 1 + bBkj - z ) +. ,,+PB_l(¢lkj+bZkj-l +. ··+bBkj - B + l ) + PB(¢okj + b 1 k j - l + .. , + bBkj - B) + .. ,+PM-l(bOkj-M+B+l +b1kj-M+B + .. ·+bBkj-M+l) + PM(bOkj-M+B + btkj-M+B-l + .,. + bBkj - M ),

= P;kj

j = 0,1,2, ... ,M-l

M

C;

=

B i+r-B

L L L F7b i=B r=o j=o ,rk M-jZ i+r-B-j .

. _To get equation (4.2.4), multiplying equation (4.2.3) by zi and adding for

(4.2.1)

and PM = IMP;+PWM¢B-l +IM-1bB)

J - 0,1,2, ... ,M - 1, we get the following:

left-hand side:

p\Z)-PMZ M

right-hand side:

C1

+ Cz + C3

where Cl , Cz , and C3 are calculated as follows:

+ ... + PB(boIM + b 1 l M- l + ... + bBIM-B) + ., . + PM- 1 (boI B + 1 + bilB + ... + bBld + PM(boIB + ... + bBlo)

B-1

(4.2.2)

.L

C1 = (k(z)-kMzM)

Pi¢B-i

'=0

where Ir = kr + kr+l + .... It can be verified that the equation (4.2.2) is implied by the first M equations (4.2.1) and the normality condition L~ p/ = 1. In view of this, equation (4.2.2) is redundant and will not be considered henceforth. Equation (4.2.l) may be rewritten to give B-1

p/

= kj

+

Pi¢B-i+

M

B

;=0

B-1

B-1

L

L

i=1

Pi

L

(if we put j - i

r=B-i+l

L Pi L kj-i+B-rbr> ;=B

0,1,2, . .. ,M-1.

j

(4.2.3)

r=O

The p.g.f. of these probabilities is given by

=

i B r Piz - r=Ji+l brz

k(z)

B-1

7

JI

I-I

M

j=O

kjz

j

.

L.

M

L

Ptb k Zm+i-B+r r

m

.

After a certain number of algebraic manipulations it triple sum in the last expression for C simplifies to ' can be shown that the z

L

L

[(Jo -m=M~i+B_Jkmzm]

r=B-,+l m=M-i+B-r

B-1 zM ._

k(z) =

< 0)

Ptz' [B(Z)-B-i(Z)] B

B-1

where

r = m and use k m = 0, m

XII

- ,=1 ,L (4.2.4)

+B -

B

"t...

r=B-i+l

F7b I r

i+r-B "t... k M-}Z . i+r-B-i i=O

BULK.SERVICE QUEUES

4.3

THE SYSTEM MIG h /1

191

190 B-2

L

B-r-I

f

r=O

t=l

j=O

= ZM

4.3

L L Pi+rbB_rkM-jzt-i

TBHE SYSTEM M/G k /1 WITH CUSTOMERS SERVED IN ATCHES Of EXACTLY SIZE k

The limiting Behavior of N~

In the present case the input is by Poisson ro e . . : c ss ':Ith rate X, .but customers are served in batches of exactly size k b sm than k customers are present in the Y:e e s~rv~r m the order of arrival. If fewer q ue .at t. e mstant when the server becomes free to serve a new group the , server remams Idle until th k then restarts Let V be th . . ere are customers and {Vn} is a seq~ence ;fij.d. ~.~~;s~~:~I~:~fthe nth batch to be served. Suppose that

Similarly,

L

L

i=O

i=O

1

= k(Z)[M ---zj ptZi - B-1 Piz i
P(Vn ,;;;;; v) = B(v) :,ith b = f;;vdB(v) and g = lIb. Define p = A/kg. 'The mdependent of the sequence of interarrival times. sequence {Vn} is also

After some simplifications, left·hand side = right-hand side gives

1

P +(z) [k(Z) l----zj
z

1

L

;et N(t! be ~e number of customers in the system at time t, including the batch un er servIce. urther, let an be the instant at which the nth batch (of' k departs from the system on completing service. Put N; = N( on + 0) and let sIze )

. = k(z) ---zj [B-1 Pi+ {zB
z

i=O

P/ This implies equation (4.2.4). If expression (4.2.4) for P+(z) can be expanded in a power series in some suitable region of convergence, then the second term will be a power series in z with powers of z at least M + B, which implies that pt, j = 0, 1,2, ... , M, is the coefficient of zi in the first term. Let us denote the first term of equation (4.2.4) by Q(z).

=

nlim -+00

peN;

~e d~~t~:~e:t:~ eap;~ sect~ons, o

nSumt~er 40f2 cDustf~mers

" = 0 , 1, 2 ,

...•

.te

distribut.ion {p/} exists and is independent of e sys em 1 and only If p < 1. The probability k· that the arriving during a service period is j remains the as in

~ame

ec Ion . . e me ~

P+(z)

Thus

= L p/zi. o

B-1

B

.

:k Pj{z
Q(z)

i=O

= ...:...--[z----B-Ik-(z-)---
(4.2.5)

When the constants pri ,pi, ... ,PB are evaluated, as usual, by Rouche's theorem and the analyticity of Q(z), the function Q(z) becomes completely known. The results of this section are due to Singh (1971-72) but the arguments used are different at least for calculating the last probability PM· An imbedded Markov chain analysis of a more general bulk-service queuing system than the one discussed above has been carried out by Dick (1970). In his system, besides the assumption in the above system, Dick (1970) assumes that (1) the service does not start until there are at least k in the queue, (2) capacity of the server is B, and (3) when the waiting room is full (that is, when there are M in queue), the customers' arrival rate changes from X to Ap. The rate Ap has an interesting interpretation in that if O,;;;;;p';;;;; 1, the rate reduces, that is, 100(1 - p)% of all arrivals balk, whereas if p > 1, the faster rate Xp could be interpreted to mean that a crowd of M in queue attracts more customers. The case when the waiting room is infinite is discussed later.

= j) '

We show that P+(z) is given by (4.3.1) where z!,zz, ... ,Zk-l are the k-l roots i f th h = k(z) inside the unit circle, Iz 1= 1, and n z o e c aracteristic equation

Zk

"V!e ~roceed :~ prove equation (4.3.1). Clearly, the process {N+} is a Markov cham WIth transItion probability matrix described by the relations: n N;+l =

(N;-kt+ l/!n,

n = 1,2, ...

:~e~eat~h. is the number of customers who arrive during the service time of the

Now the p.gJ. of the

LV.

(N; - kt is

T

1 BULK-SERVICE QUEUES

192

= E(ZN';,-k \N~ =

> k)P(N~ > k) + E(zO \N~ .;;; k)P(N~ z-kE(ZN;'\N~ > k)P(N~ > k)+P(N~ .;;; k)_

.;;; k) (4.3.2)

THE SYSTEM GEOM/G B /1

4.4

193

distribution. Since the arrivals are random, the number of new arrivals during the waiting time in the queue and the service time of a customer must be equal to the queue size at a departure epoch in the limiting case. Thus as in Section 2.2, (4.3.6)

E(z

Wn\N+ n

> k)P(N+ >

+

k)

n

t

m=O

E(zWn\N~ = m)P(N~ =

m). (4.3.3)

. . (4 3 2) we get a new form of the p.gJ. of Using equation (4.3.3) m. equ.atlOn f' ~ g\ves us the limiting p.gJ. of (N'" - kt (N~ - kt. Letting n --l> 00 m this new or as

P+(z)z- k -

k-l 'I;'

L... o

Equation (4.3.5) follows immediately from equation (4.3.6). It may be remarked here that Wqk(ex) given in equation (4.3.5) has an interesting interpretation in that it gives the waiting-time (in queue) distribution for the system EkIG/I. For details, see Chapter 6.

4.4

Pi+ (z i -k - 1)

The limiting Behavior of N;

. d (N+ - k)+ are independent, we have the relation which, on whence, smce IjJ an simplification, reduces to k-I p:'"{Zk -zi) (4.3 .4) P+(z) = ~ z~ /k(z) - 1 .

. f (4 3 1) by considering the zeros of the Equation (4.3.4) can be put m the orm . ' denominator within the circle, \z \ = 1 + 0.. (4 1 12) (B is noW k). Although the The result (4.3.4) is th~ same as eq~~::~~nc~ does not show up at the instants I ron of the MIG k 11 system, the underlying processes are different, the . h h M k chain For the transient so u I examined in tear o v , ' U. th theory of semi-Markov processes, t e reader is referred to Takacs (196~). sm; ; bens (1961). See also corrections to same system has also been conSidered Y a Fabens's results by Fabens and Perera (1963).

This section deals with the same problem as discussed in Section 4.2, the only difference being that time is considered as a discrete LV. rather than a continuous one. In the discrete time queuing system the events occur only at definite time points called "time marks" by Meisling (1958). Thus let the arrivals occur at these time marks, which we suppose are regularly spaced with an interval of magnitude /:;.t. The service is initiated at one of these time marks and completed at another time mark. A service period may consist of several of the intervals 1::..t with a given probability distribution. The system is discussed under the following assumptions:

2

3 4

The Waiting-Time (in Queue) Distribution - The Limiting Behavior of V(n)asn--l>OO

q

.' f h customer who arrives last (and hence Let V~k) be the waiting tIme m queue 0 ht e tomers served in the nth batch. t' ) among t ose cus . , has the shortest waltmg Ime Further, let lim P(V~'!1 .;;; r).

n-=

Consequently if we define

we have

THE SYSTEM GEOM/G B 11

(4.3_5)

-( ). he L -S T of the service-time where P+(z) is given by equation (4.3.1) and b ex IS t . .,

5 6

The service is in batches of maximum size B. The server is continuously busy, so that the service is virtual in case no customers are available at the time of initiation of a service period, otherwise it is real. The service times are Li.d. f.v.'s. The arrival of a customer at any time mark is independent of the arrival of a customer at any other time mark, and the probability of an arrival is p and of no arrival it is q = 1 - p. Not more than one customer can arrive at a particular time mark . The queue discipline is FCFS.

From the above description of the system it follows that the probability Ai. k of j arrivals during a time period tk consisting of k time marks is the binomial probability given by

otherwise.

As the expected value of j, E(j) = kp, the mean arrival rate A is given by A= kp/k/:;.t = pi/:;.t. If Sk is the probability that a service interval V contains k time

194

BULK·SERVICE QUEUES

THE SYSTEM GEOM/G B 11

4.4

195

marks, then S,. = P(V = kilt), k = 0, 1; 2, ... , with ~;;' Sk = 1. Note that here the service times have a discrete distribution as opposed to a continuous one, and the interval under consideration is open on the left and closed on the right. One can easily see that E(V) = ilt ~;;' kSk and that E(V(V - ilt)) = (ilt)2 ~;;' k(k - 1)Sk' Also p, the traffic intensity, is defined by p = AE(V)!B = p ~;;' kS,.!B. We assume p to be less than unity, so that the system has a limiting distribution. Let now N(t) be the number in the system at time t. Define N; = N( an + 0). The process {N;} is an imbedded Markov chain and thus can be analyzed as in the previous sections. If P/ = limn --+= P(JV; = j), then the p.gJ. of P/ is given by B-1

t

=

L p/zj

r(z) =

fo

Sk(PZ

+ q/

(4.4.2)

j ;;;, 0

j

< Oor>k.

We get equation (4.4.2) by taking the p.gJ. of k j • Also as kj,

0 .;;;; i .;;;; B

kj - i + B ,

B .;;;; i .;;;; j+B i

0,

B-1

00

PtPij

=

kj

L0

+L

> j+ B

B

kH+BPi,

p

p.

or 1

It

= (pz

Voilt

+ q) Vo/!:,.f

+ q)-Vo/!:"t -

1

.

(4.4.3)

within the unit circle are simple were not so, we should have, for some z, zB = (pz

and

(

+ q)Vo/!:"t

pz+q) V o/<.>t-1Vopjilt. A

Dividing these two equations gives z = (1 - p)j(p - p)

h' h

.

I I< 1 b .. , W IC reqUIres p > 1 for ~ , ut ~hI~ IS excluded by hypothesis. Thus expression (4.4.3) has B-1 sImple zeros InsIde the unit circle and one simple zero z - 1 Now' p+ . t' I I -. SInce (z) IS convergen In z .;;;; 1, the numerator of equation (4.4.1) must vanI'sh at _

where {z.}

1";:''';: B

•

J '.

~J ~

_

z - zj,

1, are. the ~eros of expression (4.4.3) within the unit circle. ThIs gIves nse to B-1 equatIOns Involving the B unknowns One e t' from P+(l-) - 1 h ' . qua JOn COmes + . T e B equatIons to determine the B unknowns P+ P+ PB - 1 are 0, I,· •

Lo NCB-I)

and ~

L.

o

j = 0,1,2, ....

As in earlier sections, one equation involving the B constants P;, Pt, ... , P~_I is determined by the normalizing condition P+(l -) = 1, and the others can be determined by applying Rouche's theorem (see Appendix A.S) to the denominator of equation (4.4.1) by taking the circle Iz I = 1 + 0, where 0 > 0 is sufficiently small. Rouche's theorem shows in this case that there are B-1 roots zj, j = 1, 2, ... , B-1, other than z = 1 within the unit circle. This gives rise to B-1 more equations involving the constants. More details regarding the zeros of the denominator are given in the examples that follow. Constant Service Time. a constant. Thus

+ q)ko

VJ -

~~w>we;ho~ th~; .the zeros of expression (4.4.3)

B-1

00-

Pi

otherwise

B-1

equation (4.4.1) is obtained by taking the p.g.f.of

= Lo

k = ko

whence the denominator of equation (4.4.1) becomes

BZ B-l =

is the p.gJ. of the number of arrivals during a service period. Clearly,

p/

k(z) = (pz

1

k(z) =

if

E(V(V - ilt» =

zB(pz

o

(

E(V) = Vo,

and

Pt(ZB - z') (4.4.1 )

I, [ 0,

Sk =

and consequently

.

where

Po =

P(V = kilt)

Let the service time V be equal to koilt = V o ,

p+

B i (Zj -

.

zj) = 0

'

= B-Bp

'

j = 1,2, ... ,B-1.

It ~an be .se~n t~at the determinant of the matrix of coefficients of P+ does not ~msh. ~hlS Imphe~ that the set of equations involving P+ is linearly independent ence P (z), for thIS case, is completely determined and is given by

.

(4.4.4) Mean. and varianc~ of system length may be more easily calculated if we ex ress equat~on (4.4.4) In terms of roots inside the unit circle. As the numerat:r of

equatIOn (4.4.4) can be written as B-1 ~

L.

o

P+ B . i (z - z')

B-1

Io

B-1

Pi(z - 1)

.n

J =0

(z -

Zj)

BULK-SERVICE QUEUES

4.4

197

THE SYSTEM GEOM/GB 11

196

P+(z) given in equation (4.4.1) becomes

P+(l-) = 1 gives the constant B-1

~

B-Bp

B-1

IN o

P+(z) =

.

Pi(zB -z')

i=O

ZB {I -d(pz

+ g)}/(l-d)-1

.

(4.4.9)

The denominator of equation (4.4_9),

and finally B-1

(B-Bp)(z-l) j~O [(z-zj)/(l-Zj) ZB(pZ

P+(z)

+ q)

]

zB {I -d(pz

(4.4.5)

Yo/At - 1

'on (4 4 5) one can find the mean and variance of the system From equa t 1 . . f 11 L t tl enumerator len th distribution. To get the mean, we proceed as 0 ows. e 1 an: the denominator of equation (4.4.5) be denoted by (z) and 1jJ(Zh)' respec.h 1 b t sP+(l-)= 1 wemust ave tively. Clearly (z) and Hz) vams at z = , u a , (1)(1) = 1jJ(1)(1). (4.4.6) .h t to z and taking the limit as Also differentiating equation (4.4.5 ) Wlt res~ec . L'W ital's we get a result which becomes indetermmate even after usm~ op h ;ule on~e because of equation (4.4.6). Using L'H6pital's rule a second Hme, we ave

+ q)}/(1

(4.4.10)

-d)-1

has B-1 simple zeros inside the unit circle, one simple zero z = 1, and one simple zero zB say, outside the unit circle. Then we can write equation (4.4.9) as B-1

C(z - 1) (z - 1)

n

j=l B

.n

J=1

(z -

(z -

Zj)

C

Zj)

where C is a constant to be determined. Since P+(1-) = 1, C = 1 -

Z B.

Therefore,

.... 1-

(2)(1) _1jJ(2)(1)

P+(z) = ZB - 1 . ZB -z

(4.4.11)

From equation (4.4.11) we can get

21jJ (l) (1 )

+ I E(N) = - - , ZB -1

Similarly, one can determine the variance of system length as var (N

+) = __ B~I Zj +Bp(l-Bp)+t.?(VJ-L:l.t·Vo) j';;l (l - Zj)2

{)l.2(VJ - L:l.t.

Vo)-B(B-l)}{)l.2~_V5 -L:l.t· Vo)-B(B-l) + 2B(1-P)}

+

4B2(1-P)2

)l.3(VJ - L:l.t . Vo)(Vo - 2L:l.t) - B(B -n(B - 2) . (4.4.8) 3B(1 - p) Note that ~ is taken to be zero when B = 1, and that we have ass.umed that the roots are simple. If the roots are not simple (repeated), a slIght modlficatlOn to t~e above procedure as indicated by Wishart (1956) is necessary. In the next examp e ots are simple without any conno such modification seems necessary, f or th e ro

+

straint on P except that p < 1. 2

Geometric Distribution of Service Time. k = 0, 1, 2, 3, . . . . Since k(z) =

fo d k(1-d)(pz + q)k

Let S" = d k (1-d), 0
By the transformation z = l/y we can change expression (4.4.10) to an expression having zeros which are reciprocals of those of expression (4.4.10). Then the zero YB whose modulus is less than 1 will be the only zero of the transformed expression within the unit circle, which can be determined either numerically or by Lagrange's expansion (see Appendix A.6). It is interesting to observe that the imbedded Markov chain results for the system MIDB /1 (Problem 5) can easily be obtained from the discrete time results for the system Geom/D B /1 given by equation (4.4.5) by assuming p .... 0 and L:l.t .... 0 so that the limit ofthe ratio p/ L:l.t .... a constant = A. Similarly Geom/Geom B I 1 becomes M/MB I 1 if in addition we suppose that 1/f.1 = E(V) = L:l.td/(l - d). In this connection, see equation (4.4.9) and Problem 3. For in the first case, we can see that (pz+q)-vo/At-*exp[Bp(l-z)] and in the second case [l-d(pz+q)]/ (1 - d) -* [(1 + Bp) - Bpz] . This equivalence, in fact, is based on the fact that the binomial distribution tends to the Poisson distribution and the geometric distribusion tends to the exponential distribution. The results for B = 1 are due to Meisling (1958), and for the general value of B they are due to Natarajan (1962). However, the arguments given here are more complete, at least for the examples considered above, than those given by Natarajan (1962) who assumed throughout that the roots are Simple.

198

BULK-$ERVICE QUEUES

For the discrete case the waiting-time distribution has been obtained by using the results from the continuous-time case. To get it, consider the waiting-time distribution discussed in equation (4.1.17). Transform equation (4.1.16) so as to write

= ~[:J--1]P+(1-t::) a b(a) ;>.

P+(z)

where is given in equation (4.1.12) and the condition W(O) = 1. Thus

_

wq(a)

THE SYSTEM MIMB 111MB

199

The Waiting-time (in Queue) Distribution

wq (a)

4.5

customers. The service tim f . 1 - e-"u, < v< 00. The w::ti~g :~~c~s~~v~f ~atc:es are ~.i.d. LV.'S with d.f. B(v) = Let NCt) be the number of t . 1m e capacIty ME. N:; = N( On + 0) and r(n) = P(cN,us.. ~m.e)rs.1ll the system at any time t and define J n -I If k m is the proba bil 1't y th at there are m potential . I d' J =. 0 ' 1, 2 ' .... , MD D. arnva SUrIng a servICe period, then

°

= P(m potential arrivals during an arbitrary service . . penod)

km

_ foo

Iis a constant to be determined by

-

(i\v)m e- AU m!

0

l1e- llu dv

= pmq

[1] =

P+(l-a/;>..) b(a) - 1 E(V)a

(4.4.12)

where E(V) is the expected service time. Now we transform this result to the discrete case, in which b(a) = E(e-O: v ) = Z;;' Sk exp (-a:k~t), and hence k(z) = (pz + q)k = b [-In (pz + q)/ ~tJ . This means that we can use the transformation a = -In (pz + q)/ 6.t to get the value of z in terms of a, whence equation (4.4.12) becomes

(4.5.1)

wherep = V(;>.. + 11) andq = I-p Write I = Z"" k 1 . . m J=m j. t IS easy to show that

1m = pm. It can aho be easily verified (4.5.2) following recurrence relations: that the probabilities P/(n), n;;;" 1, satisfy the

P/(n

j

+ 1)

S~l

(4.4.13)

Pl3+s(n)kj-s

+ (PB(n) + PB - 1 (n) + ... + P;(n»)kj,

j = O,1,2, ... ,MB-B

From equation (4.4.13) one can get moments of the waiting time. In particular, for constant service time we get

BP]

+ Wq = ~1 [ E(N)-"2

P/(n

+ 1)

MB-B

L:

-

P+ ()k

B+sn .j-s+(P13(n)+PE - 1(n- 1 )+"'+P;(n»kj,

S=I

= MB-B, ... ,MB-l MB-B PMB(n+ 1) = L: P+ ()/ + s=1 B+sn MB-s+(PB(n)+P13-t(n)+"'+P;(n»IMB' j

and for the geometric service time case,

Wq =

.!.(_l_ + 6.t/l..). ;>..

4.5

ZB

-1

2

Using equations (4.5.1) and (4 5 2) . (4.5.3) " , equatIOns (4.5.3) lead after Simplification to PMn + 1) = (PE(n) + ... + P;(n»q

THE SYSTEM MIMB 111MB B

The Transient Behavior of

+

P/(n

P PMB-B+s(n

N;

+ 1) + 1)

= qP/+B(n)

= pSqPMB(n

+ pP,7_ I (n + I) J , + 1),

j = 1,2, ... ,MB-B

s = 0,1,2, ... ,B-1. system M/G B /1/M

We now consider a particular case of the which has been discussed in Section 4.2, for which we present a transient solution through the imbedded Markov chain and hence obtain more explicit results. The limiting behavior is also considered. In M/MB 11/MB the inter arrival times are Ll.d. LV.'S with common distribution functionA(u) = 1- e-"u,
°

When Nt
peN:;

n>

= j

1,

INt

= i) =

j;;;" 0,

for

peN:;

= j

[Nt

= 0),

i = 1,2, ... ,B-1.

Thus we shall determine the distribution {P+ j

.

(n)} subject to the initial condition

BULK-SERVICE QUEUES

200

Nt

4.5

THE SYSTEM

M/MB/1IMB

< B. Equivalently, we can write

XMB(W) = YMB(W)XO(W) + 1.

B-1

L.

201

j ;p B.

P/(l) = 0,

P/(I) = 1,

j=O

Because of the above remark, it is sufficient to determine the distribution P/(n) subject to the initial condition PMl) = 1, and it will be supposed that this is so throughout our discussion. In order to solve the set of equations (4.5.4) we define =

pt{w} =

L.

P/(n)w

(4.5.9)

The normality condition LXj(W) = 1/(1 - w) gives Xo(w) = w(I - W)-l

[%: Y;(W)r

and c~nseq~ently the set (4.5.9) is completely known if the Y's are known. To find the Y s, wnte

f

Yew, z) =

n 1 - •

Yj(w)zj.

j=O

n=!

Then from equations (4.5.8) we obtain, for a suitable domain of z and w for example, Izl
Then from equations (4.5.4) with Pt(!) = 1 we obtain Pt{w} = 1 + qw{Pt{w} + .,

. + P;{w}} 1 ,;;;; j

Pt{w} = qwp/+B{w} + pP/- 1 {w}- P01,j,

~

Y(w,z) = (l-z)[I-Zp-I(1_qwz BW l

(4.5.10)

f

(4.5.11)

MB-B = (1 - z)

o ~ s < B.

pBP.itIB-B+s{W} = pSqPMB{W}

(4.5.5) where Fk(w) is given by

Write Xj(n) = PMB-in). Then it follows that and hence

(4.5.6)

0 ,;;;; j ~ MB

Xj(w) = PMB-j{W},

Fk(w)zk

k=o

Fo(w) == ]

B

XMB(W) = 1 + qw

L.

k=O

k ;p 1,

XMB-k(W) B ~ j

0<

s

~

< ME

(4.5.7)

B.

It follows from the third equation of the set (4.5.7) that 2:~ XiCw) = XB(w)/q, and

hence that the first equation can be derived from the second and third equations, together with the normality condition MB MB 1 Xj(w) = P/(w) = - - . o 0 j-w

L.

[x J being the integral part of x. From equatio~ (4.5.11), Yj(w) = Fj(w) - F j - 1 (w),j;P 1, and as Lf;~ Y{w) = FMB(W), we obtam, from equations (4.5.6) and (4.5.9), J ~+{w} =

OO,j

+ w(l-W)-l

(4.5.8)

piYj(w) = qYo(w), pYj+l(w) = Yj(w)-qwYj-B(w),

Then it is easy to show that X's and Y's are related as follows: O~j<MB

(4.5.13)

PMB{W} = w(l-w)-1[FMB(W)r

1

•

Equat.io~ .(4.5 .13) ~ive~ the ge~erating function of the transient probabilities p/{n). The lImiting beha~lOr IS consIdered below. Various special transient and limiting cases are discussed m Problems and Complements. The Limiting Behavior of

Yo(w) = 1

[FMB-j(w)-FMB-j-I(W)] [FMB(W)rl,

j = 0,J,2, ... ,MB-l

L.

Thus we consider the second and third equations of the set (4.5.7) for further discussion. Introduce an infinite sequence {lJ( w)} defined by the equations

(4.5.12)

N;

The limiting behavior m.a y be obtained by rewriting equations (4.5.3) with n supp~essed an~ then solVing the resulting equations along the lines of the above tran~le~t solutIOn, or else by multiplying equations (4.5.13) by (1 - w) and takin

the lImit as w

-+

1. The required probabilities, in this case, are

P/ = (FMB-j-FMB-j-d[FMBr

1

,

PMB = [FMBrl

j

g

O,I,2, ... ,ME-l

(4.5.14)

202

BULK-SERVICE QUEUES

PROBLEMS AND COMPLEMENTS

203

where Fo=Fo(l)=l, Fk=Fk(l), k;;l, and Fk(w) is given by equations (4.5.12). The results of this section are due to Finch (1962).

4.6

OVERVIEW

This section, like Section 3.7, gives brief descriptions of some work we have not discussed in detail. Publications cited here are listed in Additional References at the end of this chapter. The first and foremost reference here is the work of Bhat (1968), who provides a combinatorial analysis of the transient and limiting solution of the queue GI!M!1 with bulk service, and gives references to work on bulk-service queues before 1968. Chang (1970) discusses many examples of the application of bulk-service queues to computing systems. Another application to computing systems is given by Frank (1969). Chaudhry and Templeton (1968) discuss a different kind of singleserver bulk-service system in which time is discrete and arrivals are correlated. A bulk-service queue with single intermittently available server has been investigated by Chaudhry and Lee (1972). A paper by Crane (1974) deals with a linear network of N + 1 terminals served by S vehicles of fixed capacity B. Murao and Nakamura (1973) investigate some relations between the queuing systems GI!D!c and GI!D e 11. Jain (1971) investigates, among other things, the joint distribution of the length of a busy period and the number served during the busy period for the bulk-service queuing system GI!Mk !1. A queuing system with "concentrated service," loosely connected with a bulk-service queue, is studied by Fukuta (1969). In a concentrated service system, an idle server will delay the start of service until 1> 1 customers are waiting, and will then serve customers one by one until the system is empty. A service discipline of this type may be useful in reducing start-up costs. Another prolific researcher on the theory of queues is Marcel Neuts. In the sixties, besides other studies of queuing systems, he and his collaborators published several analytical results on bulk-service queues using the semi-Markov approach. Some of his bulk queuing system papers have been quoted earlier, and he has many others along the same lines. In particular Neuts (1965) obtains the transform of the busy-period distribution for the queue M!G~!1 with service-time distribution dependent on queue size. Some aspects of this queue have also been studied by Runnenburg (1965). The distribution of occupation time and virtual waiting time of the more general queuing system M!G~,B/l have been investigated by Nair and Neuts (1972). No computational results corresponding to the above analytical results appear to be available. Recently, however, Neuts has published and advocated a computational method of solution to queuing problems using matrix methods as an alternative to closed-form analytical methods, in particular methods using Rouche's theorem. For the application of this technique to some bulk-service problems, see Neuts (1981), an excellent research monograph which gives a full account of recent work by Neuts and his collaborators. We have referred to the queuing system concerning the unscheduled car ferry before Section 4.1 and have discussed some properties of this in Miscellaneous

Problems. Some properties of a bulk-service u . t' q ~U!ng system wherein a car ferry operates at regular intervals have bee' This latter system could also haven mvet Ig~te by Stuart and McMahon (1966). pp processing work that can arrive only a IcatllOn~ to a plant of fixed capacity, a t regu ar mtervals Kash (1966 · dlscusses h' h h . yap , 1969) a bulk-service queuing syste which taxis and customers wait for eac: ~ IC . e calls a double-ended queue, in bulk. The model is interesting but the 0 1 ~~' ~Ith the former serving customers in Dave (1971) develops an analy'tical mOd:~~ y lca ~es~lts appear to be very complex. resume rule, with bulk service for ord' oCr a pn.ofl.ty queue under the preemptive A ~ . mary nonpnonty) customers ew papers conSider optimal control of b l k ' . Serfozo (1973), Frank (1969) Handa (1971) u -servtce queues. See Deb and As in Chapter 3 we now'll'st so th ' and a recent paper by Weiss (1979). , me 0 er contributio 1t d . ns re a e to smgle-server bulk-service queues' Abramov and T . k (1969), Bahary and Kolesar (1972) Bor:~~k yn(l~;:)8~ Alagaraja (1976), Bahary rav~n (l963a, b), Georganas (1976), Goyal (1967), Harris and' Markin (1969), Mohan and Murari (1972) M h ' IVllltsklY (1971), Kotiah et al Murao (1968), and Sharda (1981).' 0 anty (I 972), Murari (1969), Nakamura and

(t972) ,

PROBLEMS AND COMPLEMENTS Section 4.1

(a)

. The system M/GB/I' I t · (1954) problem dis~u:see~n:lv;e~~i~~lo: rro~:t~:· b Cohnsider Bailey's . . . n e t e number of customers waiting i th Show that n e queue Just before service interval n + I begins.

+ N,n+1

- [N;;-B+l/.tn+l, l/.tn+!,

-

if

N; < B

where l/.tn+l is the number of customers h . . interval ending at a w 0 arnve dunng the service n+1·

Define p.+ = lim P(N,+') h' } n"'''' n = I , w Ich exists if = Ab!B . mtensity, is less than 1. Show that h + ~ -. ' the traffiC (4_1.12). t e p.g.£. of Pj IS given by equation .

Hint: .use the procedure discussed in Section 4.3. In some applIcations one may be interested in . . bulk service scheduled to take I fi a queUIng system m which ' P ace at Ixed intervals of t' t k . h' elt er m those in tervals or not at all Th" Ime, a es place (I 968). . IS varIant has been discussed by Mercer (b)

Show that the results of (a) can also be ob . t" tamed from the matrix given in Section 4 2 l'f that . ma nx IS extended t . f . capacity of the server is fixed ra ther than var7a~e:mty and the maximum

BULK-SERVICE QUEUES

204

PROBLEMS AND COMPLEMENTS

205

Continuation of Problem 1_ For the system MIEf! II of Problem 1, that is, when the service-time distribution is k-Erlang, show that b(A -"Az) = [1 + (t..II1)(1 - z)]-k and consequently

2

B-1

hHint·. h Use equa rIOns (4.L17) and (2). Here, following Downton (1956) we . ave c anged the notation of Problem 2 so that the service rate in each phase ~ nowl'd~/l· The me~n service time of a customer is therefore 1/J.I. as required ' or va 1 lty of equatIOn (4.1.17). If we define 1j;(0I) = In () th d' . . we 01, e can lhonal mean and the conditional vanance may be found from equation (3) and are given by

.

L p7(zB -zJ)

o J P+(z) - -=--~-----,.-;-­ - zB {I + (A/I1)(1 - Z)}k - 1

(I)

Show also that P+(z) can be put in the form -1

B+"-1

nB

P+(z) =

where zj(B"';;'j"';;'B

+k

(2)

E[

Zj-Z

VqjVq>O]

=

l

l k = -IB+kI --_---==2

-1j;(!)(0)

t..

- 1) are the roots of

j=B

1

Zj -

2kJ.l.

(I +~(1-Z)r

zB =

outside the unit circle. Through the use of the cumulant generating function or otherwise, show that the mean and the variance of queue size are given by

5

The system MIDB /1.

B-1

(B - Bp)(z - 1) i~1 [(z - zi)/(l - za] P+(z) =

B+k-l

I

L+ = E(j'r) =

where p = AbIB and b = lim k Hint:

B+"-!

I

(4)

ZB exp [Bp(1 - z)] - 1

(Zj -1)-1

B

var (N+) =

In this case show that

kl J.l.

J..L -+00

-;.CO,

Write the numerator of equation (1) as

Zj(Zj - 1)-2.

B-1

I

B

o

Show that the mean can also be calculated from equation (1) and is given by

B-1

I

P/(ZB -zi)

o

B-1

P/(z-l)

n

i=1

(Z-Zi)

and use P\I-) = 1. Using equation (4), prove that

2(B - Bp)

3

+ _

A-(t..+/l)z+/lZB+l = 0

4

L

Hint: Use P+(1-) = 1. Continuation of Problem 1. In particular, for the system M/Ef /1, show that the queue size distribution forms a simple geometric series whose common ratio is the unique root 'Y of the equation

such that 0 < 'Y < 1. Continuation of Problem 1.

01)"

+ kll

IJ

Zj

] B+k-l

-1

BO -

p)2

t

B-1

+

2(1-p)

(I-Z i )-1

Also show that wc(o:lVq

For the system M/E: /1 show that [(1

1-

-

Zj

0) =(1- AbJB1--;:;;::-;----eCY._b_---:.,1::--_ iii eCY.b(1 O:/Af - 1

-1

+ OI/t.. -

>

1

where {Zj} are the roots with modulus greater than unity of the equation

(3)

where {Yi} are the B roots of yB eCY.b(1 -y)

E(Vq IVq

> 0)

=

1

[l-o:/t..-Yi] 1 - Yi

1 = 0 wI'thl'n an d on IY I -- 1 , whence

-

B-1

I

1

;=1

A(l - Yi)

+ B(Ab + 1 - B) 2ACB - Ab)

PROBLEMS AND COMPLEMENTS

BULK-5ERVICE QUEUES

207

206

/1/(/1 + a), and

>

var (Vq ! V q Hint: 6

0)

From this show that Pn , l = (1 - Poo)P~ = BpPoor- n

Use equation (3).

Show that

where r is the unique real root outside the unit circle of the charac· teristic equation with k = 1, defined in Problem 2, and

zf - zf- 1

zf -1

z~ _Z~-l

z~ -1

r-I P oo = - - ' - - - Bpr+r- l

A Z~-l -1 B

Z~-l

-ZB-l

B-1

zi Z2

z~

8

(a)

B-1

n

(zi -1).

j=l

(b)

Z1-1 1

1

P(z) = P+(z).

*"

Hint:

where C may be found t~ be ((a) The system MIG 11. Show p.gJ.'s Pq(z) and P+(z) holds:

~at

lZ

"=.!. ~L..

Z n=O

Pn + 1 Z"+1

1 -[P(z)-Pool

z

the following relation between the

Pq(z) _ 1 - bCA - Xz) ~ P+(z) - (1 - z)b(A -"Az) Bp

~ L.. Pn

Pq(z)

n=Q'

i == 1,2, ... ,B-l I)B-l

laiswal (1960b) obtains relation (6) and states that it~holds for the system M/E~/l. However, it should be pointed out here that his relation is true only for the system M/Ef /1 and not for the system M/E~/1. Equation (5), in fact, gives the correct relation even for the more general system M/G B /1 of which laiswal's system M/E~/l is a special case. Continuation of Problem 7: the system M/G/l. For this system find independently Pq(z) and r'"(z) [without using equation (4.1.12)J and then show that the result given in Problem 7(b) is true. The system M/G/l. If Pn is the probability that there are n customers in the system M/G/l (as opposed to n customers in queue) and P(z) = 2:;;'=0 Pnz n is the corresponding p.g.f., then show that for the system

MIG/I,

. f the first proceed as follows. d determmant rom , Hint: To get th e secon determinant starting with the second from Subtract each column of the first determl'nant After taking common . I ft Th's leads to a new . the column to Its e. 1 . T et the final answer, note that h ond determmant. 0 g factors out, we get t e sec. 'f _ 1 or z. == z. i j. Therefore the second determinant vamshes 1 Zi • J,

7

(6)

I-Poo .

(5)

_ 1and consequently it follows If B == 1, then first show that Poo p, from equation (5) that _ P(z)_ I-b(A-"Az). P~(:l.) - (l - z)b(A -"Az) h llll, then as b(et)= (c) If the service times are exponential wit mean

9

and so on. The result P(z) =P+(z) was first obtained by Khintchine (1932) and later by other scholars. The above derivation is due to the authors. For references and more details on such types of relations refer to Chapter 6. The system MIEf 11. In case the service times are exponential with mean 1Ill, one can get explicit results for the unconditional waiting-time distribution. Show that in this case,

(b)

inversion of which gives P(waiting time in queue

0) = Poo

BULK.SERVICE QUEUES

PROBLEMS AND COMPLEMENTS

209

208

w (7) =

dW (7)

-q-

d7

q

=

1\:1'(1 - 1'-1»

Bp'Y+'Y- 1

exp [-11(1-'1

-B

7>

)7),

°

(:~~;~

where '1 is defined in Problem 3. For another independent derivation of this, see Miscellaneous Problems and Complements.

10

th~ t~anSient

Jaisw~l

2 (1 -Poo)L+ + ~ I-I(l-Poo ) (0 - :2)

12

whereL =P~l)(l),L+ =P+(I)(l), and 0 2 is the variance of the service time. q Observe that if the service is exponential, then Lq = (1 - Poo)L +. Further· more, if B = 1, then L is the average number in the system, and equals L +, as it should. Besides, it can be seen from equation (4.1.15) that

I, br = { 0,

"AWq = Lq

r = r =1=

° °

and hence ¢Bto -i = 1 =

B

which shows that Little's formula is satisfied for the system MIC 11. Some numerical work on the system MIE~ /l has been done by Downton (1955,

B-1

.2: pi(zB _Zi)

1956). Q(z)

=

_'_=-;0,,--_ _ __

(7)

zB/k(z) - 1

Section 4.2

11

Bail~;

sy:tCe:~is~u~S~:~o P970 di~cusses

For the system MIC B 11 show that

Lq =

btThe lnalYSiS for the infinite case was carried out by Jaiswal (1961) He o ame .the same results through the phase technique which can als~ gIve solution, if desired. Both (1961) and custom:::~~e t at the server is continuously busy, even when there are no th I d be sen:ed . However, the limiting solution holds good even under e re axe assumptIOns as stated in S l' 4 2 limiting results for the same queuing :he 1) time dist ib r d ' . . y alswa, wIth serVIcer u Ion etermmlstlc and batch size dependent on I h The system M/CB /l /M .. queue engt . . ThiS IS a bulk-service system with the "service . " capaClty fixed. Thus we have

Thc system MIC B 11 with variable capacity. Considerable simplification occurs in the theory of queues, particularly in bulk queues, if we assume the capacity of the waiting room to be M = 00, although in practice this is unlikely to be true. The transition probability matrix (t.p.m.), in this case, is infinite and may be obtained from MIC B 111M with columns and rows corresponding to M _ 1, M omitted. The Markov chain corresponding to this new matrix is 13

irreducible and aperiodic, and is ergodic if

It may be remarked that equation (7) h Id . k(z) is extended to infinity f thO o. s even If the power series for M + 1 probabilities P+ + ,or + IS ~xtenslOn would not affect the first Lwin and Gho I 1~7Pl" .. '~M' ThIS problem has also been discussed by (1958) wh d' sa (d 1) followmg a procedure similar to that of Finch o Iscusse the same problem with B = 1. For a detailed account of th~ latt~r case, see also Gross and Harris (1974). Con.tm~atIon .of Problem 12: the system M/D/I/M. equillbnum, smce For this system in

v ;;. 1/1-1

I,

B(v) = { 0,

where E(V) is the mean service time. This restriction on P is a necessary condition for the limiting distribution to exist; for A£(V) is the average

k j = e-Ppj/j!, where p = t-../I-I, show that

input per service period and

B-
LB mb o

m

= Bb o + (B-l)bt

cep_1)iP;,

+ ... + (B-B)b B

r -I -

is the mean service capacity. For queuing systems in the steady state, the average input must be less than the mean service capacity. Show that the p.gJ. P+(z) = 2:j'~o Pt zi in this case is B

=

.

P;

_p+

i~l eprC-pr)i-r-1

0 r~1 i....

=

[J'~l

rtl

epr((J~;)): •

;=0 -:...-~------­

zB ik(z) -
(J - r)'.

(.J-r-l)!

'

where

2: Pilz
P+(z)

j=O,l

t epr(~pr)j-r

p/ = P;

(

B-1

otherwise

14

-r -

f f j

j~Zr=1

~

-IJ-l

eP(r - pr)j -r J--r-l)!

Continuation of Pro bl em 12 : the system M/M/l/M.

For this system in

BULK-SERVICE QUEUES

210

PROBLEMS AND COMPLEMENTS

equilibrium, b(v) = p[lJv', V ~ 0, and k j = p)q, where p = A/(A + Jl), q = 1 - p, Using k(z) = (1 + p - pZ)-I, we have, from equation (7)

Q(z) = P;[l - pZ]-l

21

p.a~tia1 fractions, and consequently we can pick up the desired coef.

flclents, Thus

(8)

h

I

P/ = P;

where p = 'AIJl, Two cases arise: (i) If A=1= Jl, expand the above expression within Iz 1< € = min (1, JlIA), This gives i ... A ... j 0,1,2,.",M Pj P. ( ) Po,

A,-z,:-(j+!),

j

i=l

0,1,2" .. ,M,

where we note that

=

and

and normalization gives

P; (ii)

= [l_p][1-pM+lr 1 •

(9)

lim qk(l-Z)(Z;-z) Z-Zi

If A = Jl, expand equation (8) within the region Iz I < 1, Hence

P/

N,

=

qk(z;-l)

j = 0,1,2, ... ,M

(p(k

whence

+ l)z; -l)(l-pzi)h-I'

i = 1,2, ... ,k.

The normalization gives

P,+ = -1- , ) M+ 1

15

qh -z(l-pz)h

j = 0,1,2, ... ,M.

Note that we could have obtained this distribution from equation (9) when p .... L (a) The system M/Eh/lIM. For this system B = 1 in MIE~ 111M and the service time distribution is Erlangian with dB (v) = JlhVh-l e- lJv I(k - I)!, Jl> 0, and k ~ 1 being an integer. In this case kj is a negative binomial probability given by j

0,1,2, ...

A problem of a similar nature occurs in the theory of dams. For this we refer the reader to Chapter 6 of Prabhu (J 965). Section 4.3

Since the system discussed here is related to those studied in Chapter 6, for problems related to this sectIOn see Problems and Co mp1ements In . Chapter 6.

and consequently

k(z) =

(_q)h I-pz

where p,q ,k(z) are defined in Problem 14. Thus we have from equation (7) of Problem 12,

Q(z)

P;(1-z)q" qh _ z(l - pz)h

°

Clearly q" - z(l - pz)" = has one root equal to unity = zo, say, Let the other roots be ZI,Z2,'" ,Zh_ We consider further the case when the roots z 1, Z2, , .. ,Zh are all simple, and none are equal to unity. However, the case of repeated roots can be treated similarly. If Zo = I and Zl ,Z2" ,. ,Zh are all simple, then Q(z) can be broken up into

Section 4.4

15

(b)

Derive equation (4,1,12) from equation (4A.1), Since bee:) = E(e- av ) = 2:k'= e-akt::. t S k() = b-(-l ( )/ t::..t) If t::.. 0 h, Z n pz + q . _ t .... 0, p .... 0, such that pi t::..t .... A, a constant then show that k(z!:+ b(A - 'Az). It is thus interesting to see that ~nder the above Hint:

l~mltIng process, as binomial input tends to Poisson input, the servicetIme dlstnbutlOn becomes continuous, and thus we get equation (4J.l?) as a hmlt from equation (4.4.1). All the results deduced from equatIOn (4.4.1) hold good under the above limiting process. SolutIOn:

k(z) =

lim t::.t

-> 0

b (_ In (pz + t::..t

q))

BULK-SERVICE QUEUES

212

.

r=

1 exp

hm

Ll.t~O

-0

PROBLEMS AND COMPLEMENTS

17

l;lnCr-.Atz+l-r-.At)]f()d v v At

213

If P~( 1) = 1, then the generating functions Ph" {w} 0 < w are given by "

<1

0 ~ k ~ MB "" "" ,

where we replace the probability mass function (p.m.f.) {Sk} with the continuous p.d.f. of service time, thus replacing summation by integral. The integrand being indeterminate when At -+ 0, we employ L'H6pital's rule and get

O~k<MB

k(z) = [

eV(Az-r..)

r

f(v)dv

e-(r..-Az)V

N(fB {w} = w(l- W)-tt Ai (W)]

f(v)dv = b(r-. - f-.z).

where ),·(w) I , j -- 0 , 1 , =O,md

... ,

[Jo

Ai (WhTMBr

B ,areth eB+l distinctrootsofn-z+ B+l • ~

Section 4.5

16

(a)

Show that the equation, for

a< w < 1,

p - z + qwZ B + 1 = 0

has distinct roots, and further that it has only one root z = )'(w) within the unit circle, \z \ = 1. This root is given explicitly by {'Y(w)}i=pi+j

I= -1 m=lm

(mB+j+m-l)

m qmpmB+iw ,

18

(I + mt

(m

+ 1)-1

e=)

where {7(w)VJ;;;;' 1, is given by equation (10) of Problem 16. Hmt: Let M-+ 00 in Problem 17.

1

19

Hint:

CO~i~atiOn of.~r~blen; 18. J=O

k,n"..,

Ag

A~

(pqw)m ]

(11)

=

= p,

..!.. (mB + m) qmpmB+l

m

m-l

= ..!..qmpmB+k [k(mB m

Use the binomial expansion

(1 -

XY12

= 1- 2

( .)m+ I= em + 1)-1 (2m) :: m=O m 4

Show that the explicit results for the limiting L~_-2 Ai :::;" 2 J-O o,n".. ,and . ere

(lI! ) probab~ltJes Pk(n) are given by Prien) = 1 P~(n)=L~-2AJ :::;"2 k>a H "

= (2qW)-1 [1 - (l - 4pqw)I!21·

-1

P':{w} = oo,k+w(l-w)-l(I-)'(w))(Y(W))k,

A.6). If B = 1, show that equation (10) becomes

)'(w) = p

m ;;;;. 1,

+k

+m m-l

-1)

m

'

-(k

>a

+ l)p (mB +

k + m-j

m)] ,

k;;;;' 1.

1

Note that when B = 1, )'(w) could have been more easily obtained by solving the quadratic equation

Hint: Expand the funet'IOn gIven . . Pr oblem 18 and use equation (10) of Problem 16. m

20

If B = I,M < 00, andP;(l) = 1, then show that M

p - z

+ qwz 2

o.

w

.l~). ( )).

Contm~ation of Problem 17. If Pri(l) = 1, M=oo, the limitin (M-+oo) generating functions P':{w}' k;;;;' a,are g

j ;;;;. 1. (10)

r

~etp- ~ +qwZ B + 1 =p(I -Z)'ol (w))(1 -z)'1 1 (w)) ... (l-z fractIons for the function Yew, z) given in equation (4.5

~artJal

m-l

Hint: A repeated root implies w = (Bp-l)B{q(B + l)B+l ;;;;. 1, which is a contradiction. Hence roots are distinct. That only one root is within the unit circle follows from Rouche's theorem, and the explicit expression for this root follows from Lagrange's theorem (see Appendix (b)

Hint: Use

k~i P;;{w} = w(1 - W)-l(),(W))i[l- (p-lqW)M+l-i{'Y(W)}2M+2-Zjj •

PROBLEMS AND COMPLEMENTS

BULK-SERVICE QUEUES

207

214

J.l./(J.I. + a),

(12) '}2M+Z]-1 1 < j < M [l_(p-lqW)M + 1 {'y(w) , . . . 'on (11) of Problem 16 and ('y(w))',r~ 1, is where 1'(w) IS gIVen by .equatl) f P bl 16 with B = 1. The probabilities given bythese riesexpanslo n .(10 0 ro em P/(n) are given by the equatIOns

IM

p/;(n) =

T? + Tl

prien)

1-

+ ' .. +

j

>

n

Tn-2 ,

2,

1

From this show that

Pn ,1 = (1 -

<j <M

M

n

P/;(n),

>

(13)

2

-y-1 Bp-y + -y - 1

where [kl(M+1)] . "" (-1 L.. p q )MS +sr2s(M+l) k-s-Ms +] T1.'] = s=O

""

(

L..

. 1)-] P -1 q )MS +s -j r2s(!l1+ k+J-s-Ms'

k

>

0,

j>

1.

S

(a)

8=1

10

(

The second sum in Tj is zero for k

r].

_(2m + j - 1) p = jm 1

m

+.) < (M + 1) and the J

'

m

m+j m

q,

>

rim are given by (b)

0,

j

>

1.

Jaiswal (1960b) obtains relation (6) and states that it 1T0ids for the system MIER/l. However, it should be pointed out here that his relation is true only for the system MIEf/l and not for the system M/ER/l. Equation (5), in fact, gives the correct relation even for the more general system MIG E /1 of which Jaiswal's system M/ER/l is a special case. Continuation of Problem 7: the system MiG/I. For this system find independently Pq(z) and P\z) [without using equation (4.1.12)J and then show that the result given in Problem 7(b) is true. The system M/G/I. If Pn is the probability that there are n customers in the system M/G/l (as opposed to n customers in queue) and P(z) = 2:;;'=0 Pnz n is the corresponding p,gJ., then show that for the system

MIG/I, P(z)

From equations (4.5.13), withB = 1, we have

Proof:

IM

m-1

Bppoo-y-n

P oo = - - ' - - - -

k=1

[(1<+j)/(M+l)]

Poo)P~ =

where -y is the unique real root outside the unit circle of the characteristic equation with k = 1, defined in Problem 2, and

k=j

I

(6)

J-P oo ·

)r1 ,

P;{w} = w(I - w)- 1 FM-i(W) [FM( W

1

< j < M.

Hint:

= P+(z).

Pq(z)

k=j · (12) it is sufficient to prove the following: , . Therefore to prove equa t IOn FM_i(W)[FM(W)]-l = (1'(w)ill_(P-lqw)M+H(,y(w»2M+2-2J] •

[l_(p-lqw~+1(1'(w))2M+2]-1

I - [P(z) - P oo J

z (14)

. 5 12) 'th B = 1 By the definition of where Fk(w) is given by equatIOn (4.. .Wl _ 1 ~nd x = (p-l qWYI2 Z , Fk(w), we have from equation (4.5.12) With B - ,

f k=O

Flo (W)(p-l qW)-k/2 X k = (1 - 2x cosh 8

+ x 2)-1 ,

2)-1 = 2:=- Xk -112 h 8 B t x sinh 8(1 - 2x cosh 8 + X 10-1 where (pqw). = 2 cos . u+ 2)-1 = 2:=- Xk-1 (sinh 8)-1 sinh k8 and sinhk8, implIes (1-2xcosh8 x k-l hence

9

and so on. The result P(z) = r(z) was first obtained by Khintchine (1932) and later by other scholars. The above derivation is due to the authors. For references and more details on such types of relations refer to Chapter 6. The system M/Ef 11. In case the service times are exponential with mean I/J.I., one can get explicit results for the unconditional waiting-time distribution. Show that in this case, iii (a) = P oo q

+

ABp1'Poo

A-Y- A + a

inversion of which gives P(waiting time in queue

Thus

0) = P oo

BULK·SERVICE QUEUES

208

T>

dWq(T) = A.-yel _:.y-B) exp (-I1(l-1'-B)T], Wq(T) = ----;;;Bp'Y + l' - 1

PROBLEMS AND COMPLEMENTS

0

where 'Y is defined in Problem 3. For another independent derivation of this, see Miscellaneous Problems and Complements. For the system MIG B 11 show that

Lq = (1 -Poo)L+

+

~ /1(1 -Poo) (a

2 -

:2)

12

- P(l)(l) L + = P+(l)(I) and a2 is the variance of the service time. ' . L+ F h where L q - q , Observe that if the service is exponentIal, then Lq = (1 - Poo) . urt+ er· more, if B = 1, then L is the average number in the system, and equals L , as it should. Besides, it can be seen from equation (4.l.15) that

The analysis for the infinite case was carried out by J aiswal (1961). He obtained the same results through the phase technique, which can also be used to give the transient solution, if desired. Both Jaiswal (1961) and Bailey (1954) assumed that the server is continuously busy, even when there are no customers to be served. However, the limiting solution holds good even under the relaxed assumptions as stated in Section 4.2. Murao (1970) discusses the limiting results for the same queuing system discussed by Jaiswal, with service· time distribution deterministic and batch size dependent on queue length. The system M/GB/IIM. This is a bulk·service system with the "service capacity" fixed. Thus we have br =

AWq = Lq

and hence B-i Q(z) reduces to

B

which shows that Little's formula is satisfied for the system MIG II. Some numerical work on the system MIEf! 11 has been done by Downton (1955,

209

= 1=

I, { 0,

r = 0 r =f= 0


B-1

1956).

.

L Pt(zB -z')

i=O

11

(7)

Q(z) = -zB--I-k-(z-)--1-

Section 4.2

It may be remarked that equation (7) holds even if the power series for k(z) is extended to infinity, for this extension would not affect the first M + 1 probabilities P;, Pt, ... , PM' This problem has also been discussed by Lwin and Ghosal (1971) following a procedure similar to that of Finch (! 958) who discussed the same problem with B = 1. For a detailed account of the latter case, see also Gross and Harris (1974).

The system MIG B 11 with variable capacity. Considerable simplification occurs in the theory of queues, particularly in bulk queues, If we a.ssum~ the capacity of the waiting room to be.M = 00, a~though in p:actice thIS I~ u.nh~ely to be true. The transition probablhty matnx (t.p.m.), In thIS case, IS infInite and may be obtained from MIG B 111M with columns and rowh~ correspotndm.g to M - 1, M omitted. The Markov chain corresponding to t IS new rna fiX IS 13

irreducible and aperiodic, and is ergodic if = AE(V) p - B -
<

1

Continuation of Problem 12: the system MID/l/M. equilibrium, since I, B(v) = { 0,

where E(V) is the mean service time. This restriction on P ~s a necessary condition for the limiting distribution to exist; for AE(V) IS the average

For this system in

v ;:;. 1//1 otherwise

input per service period and B -
B

I

o

mb m = Bb o + (B -1)b 1

+ ., . + (B -

B)b B 2";;' j";;' M

is the mean service capacity. For queuing systems in the steady state, the average input must be less than the mean ~ervice capacity. Show that the p.g.f. P+(z) = 2:;}',.0 p/z i in this case is B-1

2:;

;=0

where

P; _ [M

( )] Pi+ [z B B-i- Zi", ""B-i Z

i epr(-pr)i- r

- .I I

]=1 r=1

ZB Ik(z) -
14

M i-I epr(_pr)i-r-I]-I

c)1 -.2:: I c--1)' Jr. Jr . ]=2 r=1

Continuation of Problem 12: the system MIMII/M.

For this system in

BULK-5ERVICE QUEUES

210

PROBLEMS AND COMPLEMENTS

211

equilibrium, b(v) = pe-}Jv, v~ 0, 'and kj = plq, where p = A/(A + p), q = 1 - p. Using k(z) = (1 + P - pZ)-l , we have, from equation (7)

Q(z) = PHI - pzl- 1

Pf.a~tial fractions, and consequently we can pick up the desired coef lClents. Thus -

(8)

"

P/ = Pi; "~

where p = Alp. Two cases arise: (i) If A -=1= p, expand the above expression within Iz I';;; € = min (1, pIA). This gives j

;=1

0,1,2, ... ,M,

0,1,2, ... ,M and

Pi; = [1- pHl- pM+I)-l.

(9)

Ai =

lim qk(1 - z)(z; - z) q" - z(I - pz)"

z -+Z;

If A = p, expand equation (8) within the region Iz I .;;; 1. Hence j

= 0,1,2, ... ,M

(p(k

whence

q"(Zi - 1) 1)(1 - pZit -1

+ l)z; -

'

1,2, ... , k.

The normalization gives

p.+ = -1- , ) M+ 1

15

j

where we note that

and normalization gives

(ii)

A·z:-(i+t) l I ,

j = 0,1,2, ... ,M.

Note that we could have obtained this distribution from equation (9) when p .... 1. (a) The system MIEkl 11M. For this system B = 1 in MIEg 111M and the k 1 service time distribution is Erlangian with dB (v) = pkV - e-}Jv I(k - 1)\, p> 0, and k ~ 1 being an integer. In this case k j is a negative binomial probability given by j

0,1,2, ...

Pi;=

[

-M]-I k A 1+L:-f1-z~ . l i=lZ;

l-Z i

A problem of a similar nature Occurs in the theory of dams F thO refer the reader to Chapter 6 of Prabhu (1965). . or IS we Section 4.3

Since the system discussed here is related to those studied in Cha t 6 problems related to this section see Problems d C I ' P er , for an omp ements III Chapter 6,

and consequently k(z) =

(-q)" 1 -pz

where p,q ,k(z) are defined in Problem 14. Thus we have from equation (7) of Problem 12,

Q(z)

Pi;(1-z)q"

q" -z(l_pz)k

Clearly qk - z(1 - pz/ = 0 has one root equal to unity = zo, say. Let the other roots be z 1, zz, ... , Zk. We consider further the case when the roots z 1, zz, ... , Zk are all simple, and none are equal to unity. However, the case of repeated roots can be treated similarly. If Zo = 1 and Zl,ZZ,'" ,zk are an simple, then Q(z) can be broken up into

Section 4.4

15

(b)

D~rive equatio,:: (4.1.12) from equation (4.4,1). Hlllt: Since b(o:) =E(e- av ) = ~= -akt:..ts k() At) If At .... O k=Oe ", z=b(-In(pz+q)/ k z· .... E _' p .... 0: such that pI At .... A, a constant, then show that (A "!I.z). It ,IS thus Illteresting to see that under the above l~mltlng, process, as blllomial input tends to POisson input the s ' b ' erVlcetIme dl t 'b t' ' (4 I 12) s n u "IOn ecomes continuous , and th us we get equatIOn a(s a hmit from equation (4.4.1). All the results deduced from . " equat,lOn 4.4.1) hold good under the above limiting process Solutlon: '

,( ?

k(z) =

lim L>.t-+o

E(_ln (pz + q)) At

BULK-SERViCE QUEUES

212

.

roo

1 exp

hm

lV In ()..L).tz + 1 -

At~O -0

)"At)l f( ) d V

At

PROBLEMS AND COMPLEMENTS

17

v

where we replace the probability mass function (p.m.f.) {Sk} with the continuous p.d.f. of service time, thus replacing summation by integral. The integrand being indeterminate when At -+ 0, we employ L'H6pital's rule and get k(z) =

r

e V (1v<-A)

If Pt(1) = 1, then the generating functions Ph {w}, 0 < w are given by

P;{w} = OO,k

[Jo

+ w(1 - W)-1 Aj(WhTMB

f(v)dv

roo e-(A-1v<)V f(v)dV

Pt1B {w} = w(1 - W)-1

be" - Az).

=

213

[.t

[Jo

Aj(w)(1 -

(W)f, Aj(W)]

J-O

')'j (w)hT

0 .;;; k

[.f

where ')'j(w), j = 0, 1, ... ,E, are the E = 0, and

,

0';;; k .;;; ME

,

MB +k(w) ] •

< ME l

A j (whj-MBr

J

1=0

-0

<1

+ 1 distinct

roots of p - z.+ pZB+ 1

Section 4.5

16

(a)

Show that the equation, for 0 < w

P - z

+ qwZ

B

+

< 1, 1

= 0

has distinct roots, and further that it has only one root z = ,),(w) within the unit circle, Iz I = 1. This root is given explicitly by .

.

{'Y(wW = pi

+j

L.= -1 m=l

(mB+j+m-l)

m

. m qmpmB+I w ,

j

:>

Letp -z +qwZ B+1 =p(l-z')'oi (w»)(l-z')'li (w) ... (l-z,),-I(w ) fractions for the function Y( w, z) given in equation ).

~artJal

18

(4.5.l~.

Contm~ation of Problem 17.

If Pt(l) = 1, M = 00, the Iimitin (M -+ 00) generatmg fUnctions Ph {w}, k:> 0, are g

1. (10)

+

mt

(m

+ 1)-1

e:)

19

Con~i~ation of.~r~ble~ 18. Show that the explicit results for the limiting (M ) probab~ltJes PkCn) are given by Pt(n) = 1 - ~ry_-z Aj :;:;, 2 d P:;'(n) = '£j=-oz A~, n:> 2, k > O. Here }-o 0, n"" , an

Ag

Hint:

A~

(pqw)m ]

= (2qW)-1 [1 -(1-4pqW)lIZ].

(11)

=

m :> 1,

A2

= pkq

m-1

A km = ..!. m mB+k [ (mE mq p k

.)m+1 L.= (m + 1)-1 (2m)( ~ m=O m 4

Note that when B = 1, ')'(w) could have been more easily obtained by solving the quadratic equation

= p,

..!. (mE + m) qmpmB+l,

m

Use the binomial expansion

(l-x)lIZ = 1 -2

- ')'(w»(')'(W»)k,

where {')'(w) VJ:> 1, is given by equation (10) of Problem 16. Hint: Let M -+ 00 in Problem 17.

A.6). If B = I, show that equation (10) becomes

')'(w) = P [1

+ w(1 - W)-1 (1

Ph {w} = OO,k

m-l

Hint: A repeated root implies W=CBp-l)B{q(B+l)B+lfl:>l, which is a contradiction. Hence roots are distinct. That only one root is within the unit circle follows from Rouche's theorem, and the explicit expression for this root follows from Lagrange's theorem (see Appendix (b)

Hint: Use

+k +m m-l

1)

- (k

m

>

0

+ l)p (mE + k + m)] m-l

'

k:> 1.

Hint: Expand the function given in Problem 18 and use equation (10) of Problem 16. 20

If E = I,M < 00, and PtCl) = l, then show that M

p - z +qwz Z

o.

k~/k{w} = w(I -

Wfl (,),(w)i[l

- (p-l qW )M+l-j{')'(W)}2M+2 -Zj] •

214

BULK·SERVICE QUEUES

PROBLEMS AND COMPLEMENTS

(12)

FM_j(w)[FM(w)]-l

M

Pi,(n) =

TP + T/ + ... + Tp-2 ,

n

~

k=j

M

P;(n) = 1 -

I P~(n), k=1

n

~

(13)

2

21

> BM t.. < BM t..

if if where l' is a root of the equation p - z

k

~

+ j) < (M + 1), and the Pm

°

=

inside the unit circle.

°

s=1

The second sum in Ti" is zero for (k

+ qzB + I

(16)

Hint: + To do t~is, consider equations (4.5.4) with n suppressed. Define Xj = PMB _j, ";;;j ,,;;; MB, and the infinite sequence {Yj} satisfying

j ~ 1.

0,

(15)

Discuss independently the limiting (n -+00) behavior of the system MIMB/l and show that

where

[(k+j)/(M+l)] " (p-l q)MS +s -j r2s(!\1+ 1)-j L... "+I-s-Ms,

.

Substituting cosh e = (pqw)-1! 2 /2 in the identity e-e = cosh e __ 2 (cosh e - 1)1/2 and using equation (11) of Problem 16 we obtain e-e = -1 )112 ' . ' (p ~w 'Yew). Puttmg thiS expression for e-e in equation (15) gives equatIOn ~14! and hence equation (12). Expansion of equation (12) as a power senes m 'Y(w) and using equation (l0) of Problem 16 with B = 1 gives equation (13).

1";;; j ,,;;; M

2,

(p-l qw )-iI2 e -i{)[1_e-2(M+I-j){)] •

[1-e-2 (M+I)Or l

where 'Y(w) is given by equation (11) of Problem 16 and ('Y( w)i, j ~ ~ ,is given by the series expansion (10) of Problem 16 with B = L The probablhtles P/(n) are given by the equations

I

215

Yo

are given by

==

1

j = 1,2, ... ,B

m

~

0,

j ~

1.

j

~

B.

Then From equations (4.5.13), with B = 1, we have

Proof:

I

Pk'{w} = w(l-w)-IFM _i (W)[FM (w)]-I,

l";;;j";;; M.

and

"=j

Therefore to prove equation (12), it is sufficient to prove the following: F - (w) [F (w)r 1 = (,y(w))i[l_(p-l qW )M+I-i('Y(W))2M+2-2j] • M M i [1_(p-l qw )M+I('Y(w))2M+2r1

(14)

where F,,(w) is given by equation (4.5.12) .with B = 1. By the de!~nitioe2of F,,(w), we have from equation (4.5.12) with B = 1, and x = (p qw) z,

2 -I

~~

k

where (pqW)-1I2 = 2 cosh e. But x sinh e(l - 2x cosh e + x) = "=1 x sinh ke. implies (1 - 2x cosh e + X 2 )-1 = ~k=l X"-1 (sinh eFl sinh ke and hence

Now write p - z + qzB+l = p(l - z)(1 - 'Yil z) ... (1 - 1'1:1 z). One can see that the roots 1,1'1, ... ,'YB, of P - z + qzB+ 1 = are distinct, except when t.. = BM, in which case z = 1 is a double root. When t.. = BM, we suppose that 1'1 = 1. From equation (17) Y(z) = ~$=I A·(1 - z'Y:- 1 )-1 where A. = B _lk 1 J, J ni *i (1 - 'Yj'Yi-1)-1 . Th us Y k = ~i=1 Aj'Yi ,and

°

p~

=

[.f

1=1

Aj'YT

MB k +

]

[.f

A;(! -'Yi MB - 1)O-'Yi1)-I]-1

1=1

k = 0,1,2, ... ,MB,

t..

'*

BM.

(18)

If t.. = BM, the expression for P~ is valid if we replace the indeterminate ratio (1 -- 'YiMB -I )(1 - 'Yil )-1 by its limit at 'YJ = 1, namely, MB + 1. Thus

216

BULK-SERVICE QUEUES MISCELLANEOUS PROBLEMS AND COMPLEMENTS

The limiting case when M ...,. 00 may now be derived from equation (18). By means of Rouche's theorem, one can easily see that the equation P - z + pzB + 1 = has only one root inside the unit circle if A< BIl, and no root inside the unit circle if A;;'" BIl. Thus from equation (18) one obtains equation (16), a result which has been derived by many other authors. For example, see Foster and Nyunt (1961) or Problem 3 of Section 4.1. In the limiting (n"'" 00) case of the system M/M/l/M, that is, when B = 1 in M/MB /1/MB and n"'" 00, show that

kj,

°

22

k

Fk =

I p-l 1=0

p .. =

1 _p-l

'

pkj - i + 1 +qkj

IJ

(

-

i +2 ,

2~i~j+1

qko,

i = j+ 2

0,

i>j+2

w~~re k j = P(j arr~e during a service period). Show further that the probabIlIty sequence {~ } has the p.gJ. given by

I_p-k-l

=

217

k;;"'O

P+(z) = (qPt

where p = p/q = A/p. Using this value of Fk and equation (4.5.14), show further that

+ Po)(z -l)[z + {qP
The normalizing condition P(1-) = 1 leads to qPt+Pt(1 +q) = 2-p-A.b I

j = 0,1,2, ... ,M

which gives one equation involving the unknown constants P
which is a classical result and has been derived by many authors. See, for example, Morse (1958) and Finch (1958).

MISCELLANEOUS PROBLEMS AND COMPLEMENTS Consider the following oversimplified analysis of a computer core storage system. Suppose that the input to the computer core storage system comes from a large number of sources and thus can be reasonably approximated by a Poisson process with rate A. Suppose further that during each service period, one or two storage requests can be serviced, depending on the size of the queue. If the queue size at the completion of service is 1, the next batch taken for service is 1. However, if the queue size is ;;", 2, the next batch taken for service is either 1 or 2. The selection of the batch size 1 or 2 is assumed to be independent of the previous selection. The probability that a batch will be of size i is Pi(n),i = 1,2, where PI (1) = 1 ,P2(l) = 0, and PI (n) = P, pz (n) = q if n ;;", 2, with P + q = 1, n being the size of the queue at the instant of initiation of service. Any new arrivals during the service period will be assumed to join the queue. The service periods are assumed to be Li.d. r.vo's with common dJ. B(v). Let b(v) = f; e-O'v dB (v), with b i = fo= vdB(v), and bz = v 2 dB(v). If N(t) is the number in the system at any time t, then let N~ = N(on + 0), where On is the nth departure instant. Define P/ = lim n .... = P(N~ = j), j = 0, 1,2, ... , which limit exists under certain can· ditions discussed later in Miscellaneous Problem 2. The process {N~}forms a homogeneous Markov chain. Show that the transition probability matrix P = (p/ j ) of the chain is given by

f;

°

2

~iscuss the ergodicity condition for Problem I, that is, find the traffic mtensity p and show that the condition reduces to the one for M/G1./I if P = O,q = 1 and to the one forM/G/l ifp = l,q =0.

3

Continuation of Problem 1. If B(v) = l-e-J.l.v, Il>O, v;;", 0, then show that the denominator of P+Cz) has three zeros, one of which lies within the unit circle, Izi = 1. Show further that this root is - [{(p/A)2 + 4q(Il/A)}1/2 _ (Il/A)] /2.

4

Continuation of Problem 1. If in Problem 1 the service times are constant 6 and equal to p-l = 10- per second, find the characteristic equation for this c:se and solve it using a digital computer or otherwise. Draw the curves for L + for various values of A/Il and p and show that the optimum (minimum) for L occurs at p = 0, q = 1. Observe the effect on L + for light and heavy traffic conditions, that is, for small or large values of Alp.

5

~ontinuat~on of Problem 1. If in Problem 1 we suppose that the service time reqUired to handle two requests simultaneously is longer than that for one request, two different service-time distributions must be used. Let therefore the service time dJ. in the former case be B1 (v) and in the latter case B2(v). If P+(z) is the p.gJ. of the number in system in this case show that '

P+Cz) = (qN

+ Pt)z 2 b 2 (A -'Az) -pPtzb2 (A- 'Az) -q(p[z + Pt)bI(A Z2 -

pzb 2 (A - 'Az) - qb 1 (A - 'Az)

'Az)

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

6

Since P+(z) is analytic for Iz I ~ 1, the constants are determined by using Rouche's theorem. Continuation of Problem 1. Show that the waiting time in the queue for the bulk-service system of Problem 1 has the L.-S.T. _ wq(a)

=

_

w (a) -

q

Station one

Station two

Station three

Units

Units

Units

A series of stations on a single loop.

P+(1 - a/A) b(a)

where P+Cz) is defined in Problem 1 and bea) is the L.-S.T. of the servicetime distribution. In particular, show that for the system MIG/l (that is, when p = I and q = 0 in the original system of Problem I),

Loaded carrier Terminal

Figure 1

219

7

P;a

----'~:--

- a-A+Ab(a)

where P; = 1 - p. Consider a grossly oversimplified or a hypothetical model of a real-world transit system wherein carriers with common fixed capacity provide service at a series of stations on a single loop, as shown in Figure I. In this system, fixed capacity carriers leave the terminal area empty. Each carrier visits every service station while serving all customers waiting, up to the capacity of the carrier. Carriers are equally spaced in time with no early turnoffs or passing permitted. The capacity of each station is infinite. All customers remain on board the carrier until it reaches the terminal where all customers are unloaded. The empty carrier is then free to run the route again. Customers who are not served by the first carrier passing their station after they have entered the queue are lost to the system. Such customers may be thought to seek an alternative mode of transportation in order to get to work on time. A simplified model of the type considered here might be used as a first approximation to a problem faced by suburban transit facilities during certain peak hours of the day. Let now the parameters of the system under consideration be specified as: t = time spacing between carriers, in minutes n = number of stations to be served B = capacity of each carrier Ai = mean arrival rate of customers seeking service at station i, per minute

Suppose customers arrive at station i following a Poisson distribution with mean rate Ai such that p(x, Ait) = probability that x customers arrive in a t-minute interval at station i (Ait)X e-A-it, x!

i = 1,2, ... ,no

With these assumptions it is possible to derive probability distributions for a 218

BULK-SERVICE QUEUES

220

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

Note that for n = 1, we have

variety of LV.'S of interest. First suppose that we are interested in finding the probability distribution for the number of customers on board a carrier at any point along the route. To do this, we need to consider the sum of LV.'S Xi> representing customers present for service at each station thus far served. Then the variable Yn + 1 which represents the total number of customers on board as a carrier approaches the terminal is easily seen to be given by

Yn + 1

=

min

a~y
y = B.

Show that the mean and the variance of Y n + I are E(Yn + l

{.f X;,B}.

)

tl1n [1 - PCB - 1, tl1n)]

var(Yn+d

tfJ.n [1 - PCB - 1, tl1n)]

(Ml)

z=l

The unused carrier capacity Un + 1 may be represented by

As customers are assumed to leave the system if not served by the oncoming carrier, each Xi may be taken to be identically distributed for every carrier that passes through the system. The sum l:i'=1 Xi, being a sum of Li.d. Poisson LV.'S, is therefore a Poisson LV. with mean tl1n where I1n = l:i'=l Ai' The following notation associated with the Poisson distribution is used for further. results: m p(k,m) = e(M3)

I

Hint: (b)

Use equations (M6) and (M?).

Units are lost to the system whenever an approaching carrier is unable to provide service. This occurs every time the total number of customer arrivals during the passage of a carrier around the loop exceeds carrier capacity. If L is the number of customers lost per carrier, then show that the distribution of L is given by

gn+! (I)

(M4)

p(j,m)

j=k

where m represents the mean. Now it is easy to verify the following identities:

I

jp(j,m)

= mP(k-l,m)

2, tfJ.n)]

+ BP(B, tl1n)P.

(~~)

P(k,m) =

+ BP(B, tl1n) + (tfJ.n)2 [1 - PCB -

+ B2 PCB, tfJ.n) - [tl1n {I - PCB - 1, tl1n)} (M2)

Un +! =B-Yn +!.

221

=

a,

I

1 - PCB + l,tl1n),

I

( p(l+B,tl1n)'


1= 1,2,3, ....

Show further that the mean and the variance of L are E(L)

(MS)

j=k

var(L)

tl1n P (B - 1, tiJ.n) - BP(B, tl1n) tl1n P (B, tl1n)(I - 2B) + (tfJ.nl PCB - 1, tl1n)

k

I

(M6)

m[l-P(k,m)]

jp(j, m)

+ B2 PCB

+ 1, tl1n) -

[tl1nP(B - 1, tl1n)

j=O

- BP(B, tl1n)F. k

I

j2p(j,m)

m[l-P(k,m)] +m2[1-P(k-l,m)]

(M7)

L should not be confused with the usual meaning of mean system length.

j=O

I

j2p(j,m) = mP(k-l,m)+m 2P(k-2,m).

Hint: To. derive E(L), use a modified form of equation (MS). It may also be denved from the following necessary condition for eqUilibrium: On the average, the average number arrived during an interval t must equal the sum of the average number lost and the average number served. Thus

(M8)

j=O

(a)

Show that each carrier will arrive at the terminal of an n-station system with the probability distribution of the. number of passengers given by

°

0,

y

p(y, tl1n),

y = O,1,2, ... ,B-l

PCB, tl1n),

y=B

0,

y

<

> B.

tl1n = E(L) - E(Yn+l ). (c)

The unused c~pacity u on a carrier is represented by equation (M2). Show that thlS r.v. has the distribution given by the probability mass function (p.m.f.) hn+l (u), where

BULK.SERVICE QUEUES

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

223

222

<

0,

u

P(B, tlln ),

u = 0

pCB - u, tll n ),

O
0,

u

0 Pq(z)

==

I

n=O

Pn

lZn

ZB -

'

b(/-... - r...z)

1 -b(A - r...z) A-r...z

> B.

(M12)

n:)

For numerical calculations one may use the tables for p(k, and P(k m) which are reproduced in standard handbooks, such as Bunn~t~n and'May (1958} The above problems are based on the work of GrIffm

8

(1966). p . The system M/Gk,B /1. In this system arrivals are b~ the Olsson proc:ss 'th ean rate A. The service times are Li.d. r.v.'s wlth p.dJ.. b(v), fimte WI m . .' ps With the rule ean 1/11 and finite variance 0 2 • The service IS ill grou . ~at the server does not start service until there is (a: least) a flXed nu:n~\k of customers in the queue, and the maximum capacity of the server IS n other words, service is in groups of sizes m such that k <; m <; B. We say t at

h

the server has quorum k and capacity B. f S r 4 I shoW that the steady. Using the notation and procedure 0 ec IOn ., k B 1 state partial differential-difference equations for the system M/G ' / are

(M9)

o=

-APr,o

+ APr-t,O +

Jo=P

r , 1 (x)1)(x)dx,

1 <; r <; k - 1 (MIO)

ClPO,I(x) = -(A + 1) (x))P o, 1 (X)

ax

n

aPn,I(X) = -(A+1)(X))Pn,l(X)+APn - 1,l(X),

>

9

0

ax

and p = AIBIl < 1. There are several pOints to note about the expression in equation (M12). The most important one is that there is no explicit dependence on k, the minimum number in a batch. In view of this, the p.gJ. of Pn 1 'takes the same form for the systems MIG k II and MIG B11, where the service batches are of fixed size k (=B) and variable up to B, respectively. This implies that the imbedded Markov chain considered at departure epochs (see Problem 11) would be the same for the three systemsMIG k /l,MIG B /I, andMIGk,BIl. Since equation (M12) is the same as equation (4.1.8) for MIG B11, evaluation of the constants involved is as discussed in Section 4.1. Explicit results are available only in the special case when G == E, which is discussed later. In other cases one has to do numerical work [see Holman (1977)]. The distribution of the number in the system, among other things, for the more general queuing system MIGk,B 11 in which service times may depend on the batch size has been discussed by Neuts (1967) using the theory of semi-Markov processes. The results, however, are given in forms that are computationally inconvenient. It may be noted that the special case when k = 1 is discussed in Section 4.1 and the one whenB = k is discussed in Problem 9. (a) Continuation of Problem 8. The system MIG k 11. If B = k, then show that equation (MIl) becomes

. d the server is idle). Other probabilities have where Pn 0 = Pen ill queue an .. been defi~ed in Section 4.1, and the boundary conditions are

P(z)

(b)

Pn

,

1 (0)

=

J' = Pn+B, 1 (x)1)(x)dx,

n

>

O.

0

Zk

(M13)

-b(A -r...z)

The distribution of the number in the system MIG k 11. Once we know the p.gJ. of the number in the queue, it is easy to find the p.gJ. of the number in the system. Show that the p.gJ. of the number N in the system is k-l

(zk -1) r=O :l: Pr,O z' Show that the p.gJ. of the number in the queue is k-l

P(z) =

L

r=O

P r , ozr

+ Pq(z)

(M14) (Mil)

Hint:

N = q

where

N-k, { N,

if

N). k

if

N

<

k.

BULK-SERVICE QUEUES

224

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

Take the p.gJ. of N q , etc:, and use the result of Problem 9(a). For one alternative proof of equation (M14) see Chapter 6; for still another one, see Takacs (1962). By taking a look at equations (MI3) and (MI4), one may observe that p.(z) = P(z)b (A - Az). This has the interpretation that t.he p.gJ. of N equals the p.g.f. of N q times the p.g.f. of t~e. number that arnv.e during the service of a group. Furthermore, the probablhty that the server IS idle may be obtained from equation (MI3) or (MI4) by using the normalizing condition. Show that it is given by k -1

L Pr,o

(d)

Now substituting for Pq(z) and using A- Az = a gives the desired result. Takacs (1962) obtains the transform w(a) by following a rather complex procedure. The procedure adopted here is unclassical and the result for w(a) agrees with the one obtained by Takacs, except that his re.sult has a sign error. In the case k = 1, the result may be compared wIth the now familiar result for the waiting time for M/G/I. Show that in the special case MIMk /1 the distribution of ~ is given in explicit form by

A p =-. kp.

= 1 -p,

r=O

225

=

P(~

0)

=

1- P

WeT) = Ap(y-l)exp [-A(,y-I)t]

(c)

The procedure discussed in Problem 8 has another advantage. Itk quickly leads to the occupation time distribution for the system MIG 11. For if Ht) is the occupation time at instant t, then we are interested in its limiting distribution. Let Ht) converge to ~ in distribution as t -+ 00. Show that the L.T. w(a) of the p.d.f. of ~ for the queuing system MIG k /1 is given by

_

_

w(a) - (1

_

+ Ak(1 p)

p)(l-

b(a»)

k-l

Akb(a)-(A-a)k,[I1

where,), is the unique real root outside the unit circle of the equation

A

~int:. In thi~ case set b(a) = fJ./(p. + a) in the formula for w(a) in (c), sImplIfy, and Invert to get the desired result. However, the following method would be simpler. FindPq(z) = p('{- 1)/(,{- z) from Problem IO(a) and follow the procedure used in (c) and invert. It may be noted here that this distribution can also be obtained as a special case of Problem 13(a)whenB=k.

[1 -_a_1' A-Az r

where zr> r = 1,2, ... , k - 1, are the k - 1 roots, inside the unit circle, \z I = 1, of the characteristic equation zb = b (A - Az). Hint: As for the queuing system M/G B /1 (see Section 4.1), we have in this case

l-,),-k

- = ---. iJ. ,{-I

10

(a)

Continuation of Problem 8: the system M/Mk,B/l. one can obtain explicit results. Show that, in this case,

c

k-l

L

Pr,o+Pq(z)

When G=M,

W(A - Az)

r=O

where

and hence k -1

L

Pr,o

n ;;;, 0

I-p

r=O

which is discussed in (b). Further analysis can now be done either by using Problem II(c) and k equation (4.3.1) or as follows. In MIG /1,

T

k,e

I-b(A-Az) r I... Pr oZ zk - b(A - Az) r=O '

W here Z r,

'1+ 1

k-lZ-Z r

n-1

r=O

-zr

which are inside the unit circle and are also zeros of Z

-1

Po=Poo r, y(,{-I) '

r= 1 2 ... ,k-I, are the zeros of L~;J Pr,ozr, all of "

= A/iJ..

(M16)

Solving equation (MIO) recursively after using equation (MIS), one can show that

and (l-p)

(MIS)

where '{ is t~ root outsi.de the unit circle, \z 1= 1, of the equation (A + p. - Az)Z = fJ. and C IS a constant From equations (M9) and (M1 0) on~ can show that

k

-

-

b (A - liz).

l';;;r';;;k-l

(M17)

Poo is obtained by the normalizing conditionP00 + ,L=P where ",k -1 . . n - 0 n, 1 "'-'r=l Pr,o = 1 and IS gIven by P 00 = '{k -I (,{-I )2 [(k+T)(,{k+l_Y')_(,{k- 1 1 .

)r

+

BULK·SERVICE QUEUES

226

(b)

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

Show that Lq = [k-l+ 2 (k-1)+2r k + I (1"-k 2 +k) + rk(k 2 -k - 21" - 2) + 2kr - 2(k -1)] /[2(r - 1)·

(c)

Show that for the system M/Gk,B /l

{~+l (k + 1") - rk(k + 1" + 1) + I}].

(c)

The results of this problem are due to Borthakur (1971), but our procedure in deriving the results is slightly different. Medhi and Borthakur (1972) have also found the p.gJ. of the number in the queue for the two·server queuing system M/Mk, B /2. The system M/Mk,~/l. The results obtained in (a) are simplified if the capacity B of the server is assumed to be infinite. Show that in this case Pr,o = (1- r-r-l)(k + (t·JMh-k)-l, 0 ~r ~ k - 1, and Pn , I = (}.JM)r-npo,o, n;;;'O, where r is given by r=(A+M)/t-... Since the distribution is known completely, one can easily get numerical values of probabilities and expressions for moments. In particular, k-l Lq =

(1- ~~: Pr,o)

Pq(z) = I-b(A-Az). Jl A-Az b(t-..-Az)

r(z)

(d)

(a)

In the special case G = M, this relation takes the simple form P (r(z) = 1 - ~~;J Pr o. q " Use (c) to show that k-I

Lq =

I

rPr,o

+ p~l)(l)

7=0

~

I rPr , + n=O I nPn,1 r=o 0

where L + = P+(!)(l). In the special case when B = k, show that tl becomes Lq - L + = (k - 1)/2 - kp.

k(k - 1)/2 + (t-../M)(k + t-..IM) k + (t-../Mh k

11

Continuation of Problem S: the imbedded Markov chain of the system M/Gk,B /1. Show that for the system M/Gk,B /1, _

Hint (speci.al case): Di~f:;entiate equation (!'113) once to get ~~;J Pr , a second tIme to get ~r=o rPr 0, and substItute thelf values in the found earlier. '

1

(e)

Rewrite equations (M9) and (Ml 0) in terms of

P; as follows:

B-1

b(t-.. - Az) ~ (zB - zr)p; r=Q

P\z) = --z=B----=b=-(-t-..---Az-)--

Show that D, defined in (a), is given by

(Ml

(MIS)

This system is discussed by Holman and Chaudhry (1979) and Holman et a1. (1981) and, in more detail, by Holman (1977). Some of Holman and Chaudhry's findings concerning this system are reported in Chapter 6. For the special case when k = 1, see Chaudhry and Templeton (1981), and when B = k see Section 4.3. Observe again that equation (MI8) has no dependence on k. It may be further observed that though the underlying processes for the three systems M/G k /1 (if k = B), M/G B/1, and M/Gk,B /1 differ, the difference does not show up at the instants examined in the Markov chain. In his paper concerning the M X /G Y /1 queuing system, Bhat (1964) has made this same obser· vation. However, he has not considered the presentM/Gk,B/l queuing system. Hint: Define P~ = D fo~ Pn , I (x) 77 (x) dx, where D is a normalizing constant, and proceed as in Section 4.1. (b)

-.

APi, 0 -APi-I,o =

k-I MPt (1 - r=o I Pr,o ) ,

1 ~ i ~ k - 1. (M2'

This gives a set of k equations expressing k unknowns P00, P 1 0, .. Pk -I, 0 in terms of Pt, 0 ~ i ~ k - l, the evaluation of which has bel discussed in Section 4.1. Solving equations (M19) and (M20) t Cramer's rule, one finds rather complicated results even for small valu of k such as k = 3 Of 4. However, the following procedure is mo elegant. Multiply equation (M19) by k and equation (M20) by (k - i Add and Simplify to get I -

k-l

I

r=o

A. Pr,o = -

[A-+ k-I I (k-l)Pt ]-1

P. M

(M2

;=0

Now add equations (Ml9) and {(M20) for i == I to j} and use equatic (M21) to get

BULK-SERVICE QUEUES

228

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

j

APoo'Y Wq (7) = I' _ 1

~ P/ r=O

[AP. (I' -

229

1) exp {- A('Y - 1)7}

k-l

AI/1 + ~ (k - i)pt i=O

+ exp(-Ar)

In the special case B = k, the expression for Pj,o can be simplified. Show that when B = k, then j

Pj,Q

=

L r=o

..2:..,

k

P;

Continuation of Problem 10: the waiting-time (in queue) distribution. Nair and Neuts (1972) have discussed the virtual waiting time when service times may depend on the size of the batches to be served. The expressions are unwieldy, and a look at their paper shows that even if the service times do not depend on the size of the batches to be served, numerically manageable results do not appear to be possible. Analytically explicit results are available only for the actual waiting-time (in queue) distribution for the simpler queuing system MIMk,B /1, which we proceed to discuss in this problem. The service discipline is FCFS, and the other assumptions are the same as in Problem 8, except that now G =M. Further, it is assumed that the servicetime distribution is independent of the batch size. The discussion given below is similar to that of Medhi (1975). The distribution of the number in queue for the two-server queuing system MIMk,B /2 has been investigated by Medhi and BOl·thakur (1972). We are interested here in the limiting distribution of Vq(t) as t -->- 00. Let Vq(t) converge to Vq in distribution as t -->- 00. Let wq(r) be the p.dJ. of Vq . Further, Jet us define the following functions: vItI-I e- vt (1- I)! '

t

>

0,

Z = 1,2,3, ....

This is the p.dJ. of an Erlang distribution with L.T. [v/(v bution function corresponding to f(v; Z; t) is

eX

F(v, l;x) = I, f(v,!; t) dt = 1 .0

r~o

7(ArY

] {1-exp(-A(I'-I)r)},

0

where I' (introduced in Problem 10) is the root outside the unit circle Iz 1= 1 of the equation AI/1 = (1 -z-B)/(z -1), an equation which is ofte~ used i~ changing the results from one form to another.

p+

An alternative proof of this for the system MIG k 11 is given in Chapter 6. Furthermore note that when k = 1, the relation between Poo and turns out to be the same as the one discussed in Section 4.1. Hint (B = k case): Use equation (MI8) withP+(l) = l.

f(v, Z;t) =

7>

k -2

+ a)11. The distri-

I-I e-VX(vxy

L

r=O

r!

with L.T. [vl(v + a)] l/a. Show that the distribution of Vq is given by P(Vq = 0) = Ph-I,O and

Hint: An arriving customer will find the system in one of the following classes of states:

(i)

(k - 1,0)

(ii)

(s,O),

0~s~k-2

(iii)

(n,l),

(iv)

(n,l),

iB+k-l ~ n ~ iB+B-l} iB ~ n ~ iB + k _ 2

i

=

0, 1, 2, ....

In case (i) the arriving customer immediately goes into service and thus P(Vq =0)=Ph-1,O' In case (ii) the arriving customer has to wait for the a.rrival of ~ - 1 - s customers before service on his group can be started. The time reqUIred for k - 1 - s arrivals has an Erlang distribution with p.dJ. f(A, k - 1 - s; r). In case (iii) the arriving customer has to wait for the

completion o~ service on i + 1 groups before service on his group can be started. tIme :e.quired for this has an Erlang distribution f(p., i + 1; 7). In case (lV) the arnvmg customer has to wait until k - 1 - s customers arrive -:here n iB + s, 0 ~ s ~ k - 2, and the service on i + 1 groups is over. The tIm,e r~qUlred for thiS to happen is the maximum of two independent Erlang r.v. s With parameters (A, k - 1 - s) and C/1, i + 1). Let us define this maximum by y = max {Erlang r.v. with parameters A, k - 1 - sand Erlang r.v. with parameters /1, i + I}.

The =:

Sin~e the t~o Erlang r.v.'s under consideration are independent, the distributIOn functIOn Fy(y) and the corresponding p.dJ. fy(y) of Yare given by Fy(y) =P{Y ~y}

= F(A, k - 1 -s;y)F(p, i + 1 ;y)

fy(y) = Fi!\y) = f(A,k-l-s;y)F(/1,i+ l;y)

+ F(A, k -

1 - s;Y)f(p., i

+ 1 ;y).

From the above discussion one gets k-2

Wq(r) =

L

8=0

PS,QJ(A,k-l-s;r)

230

BULK-SERVICE QUEUES

00

B-1

+ I. I.

Ps + iB , d(ll, i

;=0 s=k-l

00

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

k-Z

I. Ps + iB , dy(7), +I. ;=0 s=o

Hint:

+ 1; 7)

7>

231

Consider three mutually exclusive classes of states:

(i)

(r,O),

r = 0,1,2, ... , k - 1

Oi)

(n,l),

n = iB+r

O.

r=O,l, ... ,k-l, (iii)

Note that the integral of Wq (7) from 0 to 00 is I-Pk-l,o, as is expected. Now there are two ways to proceed: (a) Use the probabilities given in Problem 10(a), simplify, and get the desired result. (b) Use the probabilities, take L.T., simplify, and then invert. However, in this case we need to use the following expression for the L.T.:

0,1,2, ...

n = iB + r,

(n,l),

r = k, k

+ 1, ... , B-1,

i = 0,1,2, ....

Then use P(~ = 0) = Z;~;J Pr,o and 00

W(7)

k-1

=.L L Pr + iB , d(ll, i + 1; 7) + I z=o r=o

B-1

L Pr + iB , d(/-I,j + 2; 7)

i=O

CT}

7n e

L -{

n!

-

1

- (a-c)n+l

,

n = 0, 1,2, ....

and simplify to get the required result. (b)

(e)

r=k

Show that the first and the second moments of Vq are given by

If B = k = 1, then show that the distribution of ~ is the same as that of Vq , which is given by P(Vq = 0) = I-p,

p

= -A 11

Z 2 _ (2(A//-I)"{ A E(Vq) - Poo ( )3 r-I

+ (k -

l)k(k + 1)"{ () 3 r - 1

Wq (7) = :\(l- p) exp (-/-1(1 - p)7),

14

(a)

+ (k-kl)+(k z +k-2),,{-2(-yl +r3 + .. '+r k )] ~(r -1)3

(d) (e)

13

(a)

from which one can obtain the variance of V q • Using Problem 10, show that Little's formula Lq = AWq is satisfied. The system ErIMk,B /1. This system differs from the one discussed above and in Problem 10 in that the inter arrival times are now r-Erlang distributed. The distribution of the number in the system, the waitingtime distribution, and the computational aspects of some measures of efficiency have recently been discussed by Easton and Chaudhry (1982). Continuation of Problem 12: the distribution of occupation time ~(t). Let ~(t) converge to ~ in distribution as t ..,. 00 and let W(7) be the p.d.f. of t. Show that the distribution of ~ for the queuing system MIMk,B 11 is given by

P(~ =

r-l

0) = P oo

(:2

1-

rk~:;~l)Z]

W(7) = ArPoo [(l-r-k)e-A(-r-l)T + (-yB-k -1)(e-A<-r- 1 )T -e-IJ.T)1.

(b)

7>

0.

A typical bulk-service problemMIE!Ez ... Ekll. In this problem we diSCUSS the steady-state waiting-time (in queue) distribution of the queumg system in which arrivals are by the Poisson process with rate :\. The service is performed in a manner discussed in Problem 8(a) of Chapter 2, and thus the p.d.f. of service times is the one given in that problem. Assuming that the queue discipline is FCFS show that W = E[Vq] =A,6(2)/[/-Il(l-p)] and var[Vq ] =2A,6(3)![~\1-p)J +W2 where p = (A,6(1) Ill) < 1, Vq is the steady-state waiting-time (in queue)' and (3(1), I = 1, 2, ... , is defined in Problem 8(a) of Chapter 2. ' Hint: In this case the waiting-time (in queue) distribution is the same as equation (2.2.13). Consequently to get moments, use equation (2.2.13), and for notations use Problem 8(a) of Chapter 2.

If in Problem 8(a) of Chapter 2, we now assume that service times Xl, Xl, ... ,Xk are independently but not identically distributed so that EeXi ) = I//-1i, i = 1,2, ... , k, then show that k

fz(z) =

.L

/-Ij exp [-,ujz] -

+

L

(/-Ij

+ /-II) exp

[-(tJ.j

+ /-Iz)Z]

i
]=1

L

(Ill +,um

l<m
+ (-l)k-l(1l1

exp [-(tJ.1

+ Iln) exp [-(Ill + Il m + /-In)Z]

+ /-Iz + ... + ,uk)

+ /-Iz + ... + ,uk)Z]

- ...

BULK-SERVICE QUEUES

232

and the characteristic function of Z is

t

I

-.!!l.- _

<J>z(W) = j";:l fJ.j - iw

+ I
j

+ III _ + 111 -IW

I1j

< Il1j

+ 11m + I1n + fJ.m + fJ.n -

111 /11

iw

k-l fJ.l+112+···+fJ.k +(-1) Ill+1l2+"'+ l1k - iw

(c)

wherel,m,n=1,2,3, ... ,k. . h e Find E(Z), var (Z), and verify that when fJ.i = fl, for all I , t ey agre with the results found in Problem Sea) of Chapter 2. If the service-time distribution for Problem 14(a) is the one given in Problem 14(b), then show that, for k = 3, A.

Wq

C

{3

#!

3 I-I

1

I I }

1~2

1:1

(Ilj

2A. ~

3

1

fJ.J -

+ Ill? + (111 + fJ.2 + fJ.3)2 3

1

1-1

-C j=lflj I 3"- 1=2 I?: .+ }=l (11) fJ.z) +

where

1 (Ill +fJ.2 +fJ.3)

3

3} + W~

C = 1 - A.E(Z)

I-A.

l

3

1

3

I-I

1

1)

I - - 1=2i=1Ilj+fJ.l I I - - + 111 + 112 + /l3 '=lfJ.j

with A.E(Z) < 1. Discussion of Wq and var [V q] for an arbitrary value straightforward. [Fujisawa (1962).]

fk' 0

IS

now

REFERENCES . . h bu lk s ervice J R Stat. Soc. Ser. B 16, .· . Bal'I ey, N . T . J . (1954). On queuing processes Wit 80-87. . . b Ik J Aust Bhat, U. N. (1964). Imbedded Markov chain analysIs of a single server u queue.. .

Math. Soc. 4, 244-263. '. k' Bull Inst 0) On queuing processes with a certain type of bul serVIce. . . BIoemena, A.R, (196 . Int. Stat. 37,219-226. . S . Soc . 71) A Poisson queue with a general bulk serVIce rule. J. Assam CI. A (19 Borth ak ur,. . XiV, 162-167. .. 'th t bles . d D C M (1958) Handbook of probability and statistics WI a . Burmgton, R. S., an . . ay .

REFERENCES

233

Handbook Publishers, Sandusky, Ohio. Chaudhry, M. L., and J. G. C. Templeton, (1981). The queueing system MIG B II and its ramifications. Eur. J. Oper. Res. 6,57-61. Dick, R. S. (1970). Some theorems ofa single server queue with balking. Oper. Res. 18,11931205. Downton, F. (1955). Waiting time in bulk service queues. J. R. Stat. Soc. Ser. B 17,256-261. _ _ _ _ (1956). On limiting distributions arising in bulk service queues.J. R. Stat. Soc. Ser. B 18,265-274. Easton, G., and M. 1. Chaudhry (1982). The queueing system EklM'" b II and its numerical analysis. Comput. Oper. Res. 9,197-205. Fabens, A. J. (1961). The solution of queueing and inventory models by semi-Markov processes. J. R. Stat. Soc. Ser. B 23, 113 -11 7 . _ _ _ _ and A. G. A. D. Perera (1963). A correction to the above. J. R. Stat. Soc. Ser. B 25,455-456. " Finch, P. D. (1958). The effect of the size of the waiting room on a simple queue. J. R. Stat. Soc. Ser. B 20, 182-186. _ _ _ _ (1962). On the transient behavior of a queueing system with bulk service and finite capacity. Ann. Math. Stat. 33, 973 -985. Foster, F. G., and K. M. Nyunt (1961). Queues with batch departures. Ann. Math. Stat. 32, 1324-1332. Fujisawa, T. (1962). On a waiting-line process with a particular bulk service. Yokohamo Math. J. X, 45-51. Ghosal, A. (1970). Some aspects of queueing and storage systems. Lecture Notes in Economics and Mathematical Systems 23. Springer-Verlag, New York. Griffin, W. C. (1966). A simplified model for bulk service at a series of stations. J. Ind. Eng. 17, 430-436. Gross, D., and C. M. Harris (1974). Fundamentals of queueing theory. Wiley, New York. Hoiman, D. F. (1977). Some problems in the theory of bulk queues. Thesis, Royal Military College of Canada, Kingston, Ontario. _ _ _ _ and M. L. Chaudhry (1979). A unified approach to some results for the queueing system MIG A , B II. Presented at the XXXIV International Meeting of the IMS, Honolulu, June 18-22. _ _ _ _ and A. Ghosal (1981). Some results for the general bulk service queueing system. Bull. Aust. Math. Soc. 25,161-179. Jaiswal, N. K. (1960a). Bulk-service queueing problem. Oper. Res. 8,139-143. (1960b). Time-dependent solution of the bulk-service queueing problem. Oper. Res. 8,773-781. _ _ _ _ (1961). A bulk-service queuing problem with variable capacity, J. R. Stat. Soc. Ser. B 23, 143-148. Khintchine, A. Y. (1932). Mathematical theory of a stationary queue (in Russian). Mat. Sb. 39, 73-84. Lwin, T., and A. Ghosal (1971). On the finite capacity treatment of Bailey's queueing model. Calcutta Stat. Assoc. Bull. 20,67 -76. Medhi, J. (1975). Waiting time distribution in a Poisson queue with general bulk service rule. Manage. Sci. 21,777-782. _ _ _ _ and A. Borthakur (1972). On a two-server Markovian queue with a general bulk service rule. Cah. Cent. Etud. Rech. Oper. 14,151-158. Meisling, T. (1958). Discrete-time queuing theory. Oper. Res. 6,96-105. Mercer, A. (1968). A queue with random arrivals and scheduled bulk departures. J. R. Stat. Soc. Ser. B 30, 185-189.

234

BULK-SERViCE QUEUES

REFERENCES

235

Moran, P. A. P. (1959). The theory of storage. Methuen, London. Morse, P. M. (1958). Queues, inventories and maintenance. Wiley, New York. Murao, Y. (1970). An analysis of the bulk service queue with stochastic variable batch size (depending on queue length) (in Japanese). Keiei Kagaku (Jpn.) 14,34-51. Nair, S. S., and M. F. Neuts (1972). Distribution of occupation time and virtual waiting time of a general class of bulk queues. Sankhyii Ser. A, 17 -22. Natarajan, R. (1962). Discrete-time bulk service queueing process. Def. Sci. J. 12, 318-326. Neuts, M. F. (1967). A general class of bulk queues with Poisson input. Ann. Math. Stat. 38, 759-770. Novaes, A., and E. Frankel (1966). A queueing model for unitized cargo generation. Oper. Res. 14,100-132. Prabhu, N. U. (1965). Queues and inventories - a study of their basic stochastic processes. Wiley, New York. Singh, V. P. (1971). Finite waiting space bulk service system. J. Eng. Math. 5,241-248. _ _ _ _ (1972). Addendum to the above papeLJ. Eng. Math. 6,85-88. Smith, W. L. (1953). On the distribution of queueing times. Proc. Cambridge Phi/os. Soc., 449-46l. Stidham, S. (1974). Stochastic clearing systems. Stochastic Processes Appl. 2,85-113. Takacs, L. (1962). Introduction to the theory of queues. Oxford University Press, New York. Van Dantzig, D. (1947). Kadercursus statistiek. Stencilled notes of lectures given at the University of Amsterdam. (1948). Sur la methode des fonctions genera trices. Colloques Internationaux du Centre National de la Recherche Scientifique, 13,29-45. - - - - (1955). Chaines de Markof dans les ensembles abstraits et applications aux processus, avec regions absorbantes et au probleme des boucles. Ann. Inst. H. Poincare, Sec. B 14, 145-149. Wishart, D. M. G. (1956). A queuing system with x' service-time distribution. Ann. Math. Stat. 27,768-779.

Additional References Abramov, A. Kh., and A. D. Tsvirkyn (1968). On the optimal designation of speed service. Autom. i Telemekh. (USSR) 2,76-80. Alagaraja, K. (1970). A queue with particular bulk service having dependent service times. Yokohama Math. J. 18,15-22. Bahary, E. S. (1969). MultHevel bulk service queues. Thesis, Columbia University, New York. _ _ _ _ and P. Kolesar (1972). Multi-level bulk service queues. Oper. Res. 20,406-420. Bhat, U. N. (1968). See Chapter 3. Borthakur, A. (1971). An additional special channel Poisson queueing system with a general rule for bulk service. Inst. Math. Stat. Bull., 72t-8. Chang, W. (1970). Single server queueing processes in computing systems. Int. Bus. Mach. Syst. J. 9,36-71. Chaudhry, M. L., and A. M. Lee (1972). Single channel constant capacity bulk service queueing process with an intermittently available server. INFOR 10,284-291. _ _ _ _ and J. G. C. Templeton (1968). On the discrete-time queue length distribution in a bulk service system considering correlated arrivals. Can. Oper. Res. Soc. J. 6,79-88. Crane, M. A. (1974). Queues in transportation II - an independently despatched system. J. Appl. Prob. 11,145-158.

Craven, B. D. (1963a). Some results for the bulk service queue. Aust. J. Stat. 5 49-56 - - - - (1963b). Asymptotic transient beha . f ." Soc. 3, 503 -512. VIour 0 the bulk serVIce queue. J. Aust. Math. Dave, H. B. (1971). A bulk queue with priorities M A Sc . Engineering, University of Toronto. . . . . theSIS, Department of Industrial Deb, R. K., and R. F Serfozo (1973) Prob. 5, 343-361. .

°

f 1 P lma control of batch service queues. Adv. Appl.

Frank, H. (1969). Analysis and optimization of disk stora e ' . J. Assoc. Comput. Mach. 4,602-620. g deVIces for time sharing system. Fukuta, J. (1969). Concentrated service queue with Ii 't d . (Jpn.} 12,21-35. ml e umt Source. J. Oper. Res. Soc. Georganas, N. D. (1976). Buffer behavior with Poisson arriva IEEE Trans. Commun. 24, 938-940. Is and bulk geometric service. . ~ Goyal, J. K. (1967). Queues with hy er-P . 11,157 -167. POisson arnvals and bulk exponential service. Metrika Handa, J. M. (1971). Optimal control of service rate and s . '. stochastic service systems MAS th . D erVJce batch sIZe m single channel ' . . c. eSIS, epartment of Industrial Engineering, University of Toronto. Harris, C. M., and P. G. Markin (1972). A note on fe d . Comput. Mach. 19, 727 -733. e back queues With bulk service. J. Assoc. Ivnitskiy, V. A. (1971) A . I ha . . smg e c nnel bulk service system with 1066 -1077. queue. Eng. Cybern. 6, Jain, J. L. (1971). Some contributions to the theory of ueue . . slty of Delhi. Abstracted in Opsearch 8, 293-294. q s and dams. Ph.D. theSIS, UmverKashyap, B. R. K. (1966). The double-ended ueue w' h . . . Oper. Res. 14, 822-834. q It bulk serVIce and lImited waiting space. (1969). The double-ended queue w·th b Operational Research _ Proc ItS . I atch dep~rtures. Advancing Frontiers in (India), New Delhi, 1969, 139:"1~:i. emmar, New Delhi, 1967. Hindustan Pub!. Corp. Kotiah, T. C. T., J. W. Thompson and W A O'N W gh for single-server queueing sy'stems ,j A' 1 Pr' a u (1969). Use of Erlangian distribution b 6 ,584-593. ' . pp. o. Mohan, ~., and K. Murad (1972). Time dependent solution of a . With variable capacity. Metrika 19,209-215. correlated queuemg problem Mohanty, S. G. (1972). On queues involving batches. J. Appi. Prob 9 430 43 9 ., - 5. Murao Y and G N k '~h' a ~mura (1 73). On the relation between C/D/s and C/D/1 (bulk serVI'ce WI size s).Kelel Kagaku (Jpn.) 17,26-31. Murari, K. (1969) A queueing bl .h b t' Z A' pro em Wit correlated arrivals and general service time distriu IOn. . ngew. Math. Mech. 49, 151-156.

,

.

Nair, S. S., and M. F. Neuts (1972). Distribution of occupation time and virtual w .1' . f a general class of bulk queues. Sankhyii Ser. A 34, 17 -22. al mg tIme 0 Nakamura, G., and H. Murao (1968). A solution on bulk service waitin . Denkitsushin Kenkyujo Kenkyu Jitsuyoka Hokoku 17, 1599_161~:ueues (m Japanese). Neuts, M. F. (1965). The busy period of a queue with batch service. Oper. Res. 13,815-819. - - - - (1981). Matrzx-geometric solutions to stochastic models - an aigorith . h The Johns Hopkins University Press, Baltimore. mlC approac . Runne';!';;g'SyJ'mTph ' ~1965). con the use of the method of collective marks in queueing theory o SlUm on ongestlon Theory W L S . h d . . . . sity of North Carolina Press, Chapel Hill, 339'_438~lt an W. E. WJlkmson, Eds. UniverSharda

(1.9~ll). A limhited space correlated queueing problem with departures in batches of vana e SIze. Ca . Cen. Etud. Rech. Oper. 23,87-96.

BULK SERVICE QUEUES

236 M M hon (1966). A queuemg system with bulk service. Oper. Res. 14, Stuart, 1. M., an d G .B' c a 728-731. 'hb h . . H J (1979) The computation of optimal control limits for a queue WIt atc servIces. W elSS,

.

.

.

Manage. Sci. 25,320-328.

Chapter 5

MULTICHANNEL BULK QUEUES In Chapter 3 we discussed single-server queuing systems with bulk arrivals, and in Chapter 4 a similar analysis was carried out for queuing systems with bulk service. In this chapter we discuss a different class of queuing systems knowt'! as multichannel (or multiserver) queues having more than one server in parallel, and either bulk arrival or bulk service. Multichannel queues constitute an important class of queuing processes and have broad practical meaning. Examples of such systems may be found in banks, telephone exchanges, hospital admission systems, seaports, toll booths, military tactics, courts of law wherein servers are judges and customers are cases, and information transmission systems wherein messages containing a (small) random number of characters (a batch of characters) arrive according to a Poisson process and must be transmitted to some destination. It will be seen that multichannel queues are more complex to deal with than single-channel ones, and this complexity increases when customers either arrive or are served in bulk. Some cases arise in which the number of channels is so large (infinite for all practical purposes) that waiting times are negligible and only the level of server occupancy is of analytical or practical interest. An intensive care unit in a hospital, where delay in service may result in loss of a life, may provide an example of such a situation. Other examples are self-service stations such as large parking lots, theaters, or auditoriums. In addition, a large telephone exchange, the number of claimants drawing unemployment insurance or workmen's compensation, may be considered to fall under the category of infinite-server queues. Analytically, infinite-server queues are easier to handle than their counterpart finite-server queues. Both the finite- and the infinite-server queues having either arrivals or service in bulk, are discussed in the present chapter. The reader will find that the algebra, at times, becomes rather involved. We wish we were able to find some guiding thread, but we are sorry to say that we were not as successful in unifying the various models or simplifying the algebra as we were in the other chapters. Whenever we did succeed in simplifying some algebra, we have indicated this by our remarks at various points.

5.1

THE SYSTEM MX (d/M/co WITH THE INPUT RATE A(t)

The Transient Behavior of N(t)

In the system under consideration the customers arrive in groups following a 237

MULTICHANNEL BULK QUEUES

238

Poisson process with a variable rate A(t), wl1ere A(t) is an arbitrary dif~erentiable function. The group size X has a distribution P(X = m) = a,:,' m.~ I, w:th A (z) = ~;;'=1 amz m . Furthermore, the group sizes are independently Identically. dlstnbuted. The system has an infinite number of homogeneous servers whose service t.lmes are independently exponentially distributed with parameter Jl. The group Sizes, the service times, and the arrivals are independent. Let N(t) be the number of busy servers at time t and define

Pn(t)

= P(NCt) =

17

17),

= 0,1,2, ....

5.1

THE SYSTEM MX{t)IM/~ WITH THE INPUT RATE A(t)

239

with the initial condition F(z; 0) = fez). To solve equation (5.1.5), we consider the associated ordinary differential equations

dt

-

=

1

dz

dF

- /1(1 - z)

- A(t)[l - A (z)] .

(5.1.6)

The set (5.1.6) is solved on the lines discussed in Chapter 1. The general solution is >/;2 = C (>/; d, where C(·) is an arbitrary function and >/; 1, >/;2 are two independent solutions of the first and second equations of (5.1.6). The first equation of(5.1.6) gives

Then the system of differential-difference equations describing the system under consideration has the form:

1 >/; 1 = t - - In (I - z) J..t

and the second equation of (5.1.6) gives n

P<~)(t) = - [A(t)

+ 17J..t]Pn(t) + A.Ct)

+ (n + 1)J..tPn+ 1 (t),

17

I

m=!

amPn-m(t)

(5.1.1)

r

>/;2 = F(z;t)-..!. J..t

X(U)t..(>/;l +..!.In(l-u)'dU

'J

/1

0

where X(z) is the p.g.f. of {Xr }, with Xr = ~;;'=ram+1 = P(X > r), that is,

= 1,2, ....

~

We solve this set by introducing the p.g.f.

X(z) =

I r=O

Xrzr = (l-A(z»(1-Z)-l.

~

I

P(z; t) =

(5.1.2)

Pn(t)zn

n=O

which is continuously differentiable in the domain D = {O < z < 1; 0 < t since for Iz 1< 1 ,equation (5.1.2) converges for any t. Let g(z) be the p.gJ. of the initial distribution Pn (0), so that

g(z) == P(z; 0) =

I

< T},

oP

oP

ot

oz

--J..t(l-z) -

and summing, we obtain

= -A.(t)[I-A(z)]P

zn

I

amPn-m(t) = P(z;t)A(z).

X(U)A(-..!. In(1 - z) +..!. In (1 - u)ldu = c(_..!. In (1 - z») J..t

0

whence

J

J..t

_ 1. Jl

F(Z;t)-..!. Jl

of oz

J (5.1.7)

J..t

I-e -J.LZ

(I

J..t

~

X(u)t.. -In{(I-u)eJ.LZ} duo J..t

0

(5.1.8)

Eliminating the arbitrary function C(·) from equations (5.1.7) and (5.1.8), we obtain after simplifica tion

Thus the set (5.1.1) has been reduced to a single partial differential equation which we proceed to solve. Consider the function F(z; t) = In P(z; t) and let fez) == F(z: 0) = In g(z), which is possible since pez; t) > 0 for t, zED. As a result, equatIOn (5.1.4) reduces to an equation for F == F(z; t),

of at

r

(5.1.4)

m=l

--J..t(i-z)- = -A(t)[l-A(z)]

'J

J..t

n

~

n=l

X(U)A(>/;l +..!.In(I-U)\dU = C(t-..!.ln(l-Z)\.

0

C(z) = f(l-e J.LZ) __ J

where we have used the relation

I

fez) -..!. J..t

Z2, . . .

r

Putting t = 0 in equation (5.1.7) and using the initial condition F(z; 0) = fez), we get

n=O

Multiplying equation (5.1.1) successively by 1, z, an equation for P == pez; t),

F(z;t)-..!. Jl

(5.1.3)

PnCO)zn.

Thus >/;2 = C(>/;d gives

(5.1.5 )

r v

X(U)A(t+..!.ln l-U)dU = f(v) J..t

I-z

wherev= l-(l-z)e-J.'t From this we have

P(z;t) = g(v)ex p [;

I:

X(U)t..(t+;lnC=:))dU]

as the final solution of our equation (5.1.4).

(5.1.9)

MULTICHANNEL BULK QUEUES

240

5.2

THE SYSTEM MX/M/c

241

The result (5.1.9) is due to Abol'nikov (1968). For some results which are given in implicit form for the more general system M X (t)/G/=, see Shanbhag (1966).

5.2

00

n

I zn m=l I amPn-m(a) n=l and transposing we get

The Transient Behavior of N(t)

i+l

c-l

-P.(l-z)n~o(c-n)Pn(a)zn

Z

In contrast to the previous section we now have constant input rate. The input process is a Poisson process with constant rate A for all values of t. For simplicity we now assume N(O) = i, that is, initially there are i customers in the system. All other assumptions are unchanged. The group size X is an LV. with distribution given by am =P(X=m), m??-l. X has mean (i, O';;;(i= :E;=lma m <00, variance o< a~ < 00, and p .g.f. A (z) = :E;;'=1 am zm. It will be assumed that A (z) can be expanded as a power series about the point z = I. The service-time distribution is exponential for each busy server with parameter p. so that the service rate of the system is np., for 1 ,;;; n ,;;; c, and c p., for n ??- c _In short, the present system may be considered to be a c-channel analog of the corresponding infinite-channel system treated in the previous section. Let Pn(t)=.P(N(t)=n) be the probability that there are n customers in the system at time t. It is not difficult to see that the process {N(t)} is a Markov process with an enumerable number of states. The differential-difference equations for the present system are

a;,

11/ \t)

- APo(t) + p.P1 (t)

J"nl )(t)

- (A + np.)Pn(t) + A

J"nl) (t)

n

I m=1

+ l)p.Pn+ 1 (t),

amPn-m(t) l';;;n
(5.2.1)

n

== - (A +

+

Fn_m(a)zn-m

n-m=O

= A(z)P(z;a)

THE SYSTEM M X IMlc

+ (n

00

I

cp.)Pn(t) + A

cp.Pn+ 1 (t),

I

Define the generating function P(z; a) == :E';;=oPn(a)zn, where P n(a) == J;;'e-O!t Taking the L.T. of equation (5.2.1), multiplying by appropriate powers of z and summing over all n, we obtain, after some algebra,

c-l

I

(c - n)Pn(a)z~ =

n=o

i+l

Zo

L

(c-n)Pn(a)zn

n=1

(5.2.3)

The relation (5.2.3) together with the first c - 1 equations of the system (5 2 1) offers the possibility of finding all the P (a) n = 0 1 2 - 1 d'l . 'th th em a1so t h e f unction P(z; n" , "'" c WI a). ' an a ong Some further analysis is carried out by expressing :E C - 1 (c _ )F ( ) n . f P ( ) S' h' n=o n n a z In terms d o 0 a. Ince t IS cannot be done by directly using the system (5 2 1) on the following lines. . . ,we procee Let us consider ~e system MX/M/oo of Section 5.1, which differs from the pr~~e~t system only. In the p:esence of an infinite number of servers. Let the probabilItIes and generatIng functIOns of Section 5 1 now be denoted b I . th th . 1 ..' y p aCIng a caret over. em, at IS, et them be Pn(t), P(z; t), and so on. Taking the L.T. of the equatton (5.1.4) wIth respect to t with ACt) = A, we have

+ {A[A(z)-lj

-a}P(z;a) p.(z - 1)

Zi

=

I

amPn_m(a)zn

(5.2.4)

where we have used the initial condition

P (z; 0)

(5.2.5) Comparing the systems (5.2.1) and (5_1.1) with ACt) = A, it is easy to observe that

I

n=o

(c - n)Pn(a)zn

is expressed by exactly the same form in terms of Fo(a) as is

n=l m=l

c-I

where we have used Pn(O) == lS in . Using

•

p.(I - Zo)

c-l

c-I

=

By using Rouche's theor~m. it is simple to see that the denominator of equation (5.2.2) has a umque root wIthIn the unit circle, Iz 1 == 1, which we den"ote by Zo = z.(a). ~he n.umerator must vanish at z 0, for otherwise P(z; a) would not exist. PuttIng z - Z 0 In the numerator of equation (5.2.2), we obtain

az

P n (t)dt, Re a??- O.

I

(5.2.2)

where X (z) is defined in Section 5.1.

ap(z;a) =

n ??-c.

zi+ A

az-(l-z)[cp.-AX(z)z]

amPn-m(t)

m=l

[a+A+cp.(l-z-I)]P(z;a)-p.(l-z-l)

F(z; a)

I

n=o

_

(c-n)Pn(a)zn

MULTICHANNEL BULK QUEUES

242

in terms of

Po (a). Also,

5.2

THE SYSTEM MXIMlc

whence, from equations (5.2.2) and (5.2.3), we obtain c-l

I

-;::-

(c-n)Pn(O:)Z

n

-(. ) _ }J.(z-l)R(z;o:)Po(O:)-}J.(z-J)T(z;O:)+Zi+l h(z;o:) P z,o: -

n=o

is the coefficient of yC in the expansion of the function {y/(l - y)2 }P(yz; 0:) in = powers of y, for

y i'(yz, 0:) = (1-y)2 '

I

I

(c -n)pn(o:)znyc.

I (c n=O

aC- 1

P(yz; 0:)\ n)P n(o:)zn = (c _ 1)' ayC-l (l - y? 1

y=o

(5.2.10)

= z(o:) is a zero (for which Izo I < J) of the function h(z; 0:) = a:z - (J z)[c}J. - r...zX(z)]. Equation (5.2.9) shows that we have expressed P(z; 0:) in terms of one unknown quantity Po (0:) which is given by equation (5.2.10).

where Zo

n=O

c=1

(5.2.9)

- r) _ T(zo;O:) + __-'--'-'-c (zo)i+1_ __ P0\0: R(zo;O:) R(zo;O:)}J.(l-zo)

c-I

Consequently. c-l

243

(5.2.6)

The limiting Behavior of N(t) ~

Thus if instead of Jhe initial condition (5.2.5) for the solution of equation (5.2.4), one first uses P(O; 0:) =:oP 0(0:) = ~o(O:) and then equation (5.2.6), one can express 2:~:~ (c - n)pn(o:)zn in terms of Po (0:). This is shown next. . To solve equation (5.2.4), let us multiply both sldes by the mtegratmg factor ex

1- du ] [- fo}J.-r... X (u) du + -0:}J. S0 -l-u z

p

and write it as -G(Z)(l-ZY"II'Zi

A -d [P(z; o:)G(z)(l - z)Cl.11'] dz

}J.(1 - z)

(5.2.7)

where G(z) = exp[- It (r.../}J.)X(u)duj. Integrating this, we obtain

P(z;o:)G(z)(l-Z)Cl.I}J. = where A is a constant. Puttingz

-~

S:

G(u)(l-U)Cl.II'-luidu+A,

(5.2.8)

= 0 in equation (5.2.8) gives A = Po (0:). Thus

~f

) = (1 - Z)-Cl.11' fP (0:) _ P-;:"(. z,o: G(z) 0 }J.

l

Z

0

G(u)ui(l _U)Cl.II'-l dul.

Now consider the zeros of the function h(z; 0); these are the roots of the equation

z = K(z)

- G (z)(1 - z )Cl.11' =

Z

Let us assume that the limiting probability distribution {P n} of NCt) exists, where n» O.

where K(z) = [cp + r...zA(z)J j(cp + r...). It is not difficult to see that K(z) is a p.g.f. of an LV. with mathematical expectationK(l)(J) = (r... + M)/("A + c}J.). Consequently (see Appendix A.2) equation (5.2.11) has a root z such that 0 < z < 1, iff K(1)(J) > 1, or what is the same thing, iff p = r...a/c}J. > 1. In this case, Iimo:->o+z(o:) = z where 0 < z < 1, since assuming the contrary, we would obviously arrive at a contradiction to the fact of uniqueness of the zero z(o:) of the function h(z; 0:) inside the unit circle. If, however, p';;; J, then h(z; 0) does not have a zero in the interval J0, 1 [, and consequently IimCl."o+z(o:) = 1. Differentiating the relation (0: + C}J. + "A)z(o:) - C}J. - r...z(o:)A (z (0:» = 0 with respect to 0:, we obtain

(0:

+ C}J. + r...)z(l)(o:) + z(o:) -

r...z(o:)A(I)(z(o:»z(1)(o:) - iIz(l)(o:)A(z(o:» = O.

Letting 0: -+ 0 +, we get since z(O +) = J p
Consequently, _

~l

I

(c-n)pn(o:)zn = R(z;o:)Po(o:)-T(z;o:)

p = 1.

n=O

where 1

c

a-

r

1

1 =

ac - I

1

l(I -yY

R(z;o:) = (c-l)! ayC-l T(z;o:)

(5.2.1 1)

[

1

(c - 1)' ayC-l (1 - y)2

x

f

zy

o

(l-ZYfCl.II') G(zy)

(l - zyfCl.II' G(zy)}J.

G(u)ui(l _U)O:II'-l dU]

I y=o

I

It can be shown that Iimo: .. o+R (z; 0:) and Iimo:.. o+ T(z; 0:) exist. In particular, .

=

1

C

a

1

11.!!R(z;0:) =r(z)

y=o

-

(c-I)!ayC-1

[ex P{(r... M /(Zy)}ll (J_y)2 y=o

where

fez) =

=

L k=l

and consequently,

/k

-zk. k

'

k-I

1-

L

j=l

aj

MULTICHANNEL BULK QUEUES

244

Po

5.2

p

limPo(t)

THE SYSTEM MXIMlc

being Aii/M and (T~ = A(2)(1) - (a)2

t-+=

lim "-+0+

245

+ a. For the systemMx IMI2,

P _ a(l - p) [

a[z(a)]i+l R (z(a); a)M(l - z(a))

p~l

0,

a+p'

0-

(5.2.12)

Hence

~ cel ~p),

_ P «(T~

-

L

+ aZ + ii)

· ~I\a-/2M. 2a( 1 - p) p b emg For the system M X /M/3, p = Aii/(3M),

where r = lim,,-+o+ R (z(a); a). Furthermore,

MCI - p)

Po =

2a L

Hence it is clear from the analysis carried out above that the limiting distribution of N(t) exists iff p < 1. For certain purposes, however, it is better to derive P(z) from equation (5.2.2), that is,

2p

+-a+p

+ (5 -al) P + (3/a) p2 (T~ + a2 + a 2 a(1- p) .

_6p {3(l + (p/a)) - ad 2ii + (5 -a!) P + (3/ii) p2

+ 1:

The Waiting-Time (in Queue) Distribution

P(z) =

lim aP(z;a) = M "-+0+

Using P(l) = 1, we get

c-l

(c -n)Pnz n

n=O

cM-"AzX(Z)

L

c-\

L nPn

(5.2.13)

(5.2.14)

= c(Qo - p)

n=O

where Qo = P (all servers are busy). The results (5.2.13) and (5.2.14) are used in calculating expected values, which are discussed below. If Land Lq represent the expected number in the system and in the queue, then in the case of c channels, it is easy to show that

Let the queue disc~plin~ be FCFS by groups, and random within the groups. Let. Vq be the waltmg tIme of a random customer in an arriving group. In this sectIOn we find P(Vq > 0) and P(Vq > t), t > 0, by a method which is instructive though a bit laborious. There is, however, a much simpler method, discussed i~ Problems and Complements, for finding these two probabilities. To discuss P(Vq > 0), we reason as follows. If a group containing m customers arrIves .t? find .the system in state n, 0 < n < c - 1, then for given m and n, the probablhty of Immediate service for the customer under consideration is 1 if 1 < m < c - nand (c - n )/m if m ~ c - n + 1. Consequently, for all m and given n,

c

L-Lq = c+

L

(n-c)P n ·

P(Vq

=

Oln)

c-n

= L

f

m=l

n=O

m=c-n+l

c -n ma m m 11'

O
C-n

This relation, though intuitively obvious and easy to prove otherwise, is true irrespective of the nature of the distribution of interarrival times, service times, and X size of arrival groups. To particularize this for the model M /M/c we get, using equation (5.2.14), L - Lq = 8 a, 8 = "A/M, which represents the average number of busy servers. From equation (5.2.13) one gets, by using equation (5.2.14) c-l

8{ii+A(2)(1)/2} + ~on(c-n)Pn L = p(1)(z) 1z--l

-

c-8ii

Once Po, PI, ... , P C- l are determined,L and consequently Lq are determined. In particular, for the system MX/M/l, (T~

1

L =

"2

p

+ (aY + a a(l - p)

c-n- L: (c-n-m)a m=l

(5.2.15)

m

where mam/a is the probability that the customer under consideration arrives in a group of size m. For details, see Chapter 3 (Miscellaneous Problems and Comple. ments). Since th~ probability of Vq > 0 is the complement of the probability of Vq = 0, the unconditIOnal probability of Vq > 0 is P(Vq

> 0)

= 1 -P(Vq = 0) c-l

L

1-

n=Q

P(Vq = Oln)Pn

1

c-I

a

n=o

1--::-

L

{c-n c-n L

m=1

}

(c-n-m)a m Pn .

(5.2.16)

MULTICHANNEL BULK QUEUES

246

5.2

247

THE SYSTEM MXIMlc

The result (5.2.16) can be simplified. For d'etails, see Problems and Complements. Let us now tum our attention to P(Vq > t), t> O. We show that P(Vq

1

> t)

=

L d=l

== - e- cilt P

d-l

Pc+d

(C/ltY

t ;;. 0,

(5.2.17)

L -.,-. ;=0 l.

1,2,3, ....

d

This gives finally P(Vq

> t)

To prove equation (5.2.17), first we need to discuss the relations between the limiting probabilities which may be obtained from equation (5.2.1). One can easily show that the limiting probabilities Pn , in fact, satisfy the following relations: n-l = -e L Pi L am' n i=O m=n-i

Pn

e

n-l

c

i=O

L

Pi

f

m=n-i

am'

l
=

ma m li

2.. == m a

d

L

m=c-n+d

== 1,2, ... ,

O
°

is 1

d+c-l

a

n=o

=

L m=c-n+d

Pc+d p

d

=

d

d=1

Cfl

L -

[It may, however, be remarked here that the probability corresponding to d == is P(Vq == Oln), 0';;; n';;; c - 1, which has already been discussed and which will not be used further while discussing P(Vq > t).] Consequently, the probability that a random customer in an arrival group waits for d departures before beginning service

L. Pn -_ '"'

where we have used the important relation between the Erlang distribution and the Poisson process. . If Wq is the mean waiting time (in queue), then

n;;'c

To get equation (5.2.17), we proceed as follows. If a group containing m customers arrives to find the system in state n, c - m < n < c - 1, then c - n are served immediately and m - c + n are delayed. Of the m - c + n who are delayed, the customer under consideration must wait for d departures,d == 1,2, ... , m - c + n. On the other hand, if the group arrives to find the system in state n, n;;' c, all the m customers are delayed. Thus, of these m customers who arrive when n == c + k, k == 0, 1, 2, ... , the customer under consideration must wait for d departures, d == k + 1, k + 2, ... , k + m. The conditional probability, given the state of the system n just before the arrival of the customer's group, of the customer waiting for d departures before his service commences is given by

L m=c-n+d

(5.2.19)

1,2, ... ,

where we have used equation (5.2.18). Since the service time is exponential with the same mean 1/ Jl for each of the c servers, the time until the first of the c servers becomes empty is exponentially distributed with mean equal to l/cJl. Thus the time the customer under consideration must wait for d departures from the system is the d-fold convolution of this and is the Erlang distribution with d degrees of freedom, yielding

1 -Pc + d P

~ dPc+d == Lq

J:l

Ali

Ali

which shows that Little's formula holds for the system MX /M/c. One can easily find the higher moments of V q . In particular, the variance of Vq is var(Vq )

_ -

~ d(l L.

d=1

+ d) Pc+ d p(Cfl)2

-

2

W

q'

. Now consider the distribution of two different components of the waiting hme V q . (m. que~e) of a randomly selected customer of an arrival group, namely, (l) .waltmg-tim: (m queue) distribution of the first member of the group, (2) distributlOn of waltmg time (in queue) due to the service times of the members of the group served before the selected customer. One first needs to know the distribution of the number of departures from the queue the typical customer has to wait for ?efo~e his service commences. The consideration of these two components of waitmg tIme leads to the generalization of the results discussed in Chapter 3 for the single-server case (see Miscellaneous Problems and Complements). We now proceed to discuss the two components of V q ' In case (1) the distribution of the number of departures is Poisson or the waiting-time distribution is Erlang, as before. Thus if V represents the' waiting time (in queue) of the first member of an arrival group, ~~e can easily see that if a group arrives to find d;;' 1 customers already in the queue, and the system in state n == c + d - 1, d == 1, 2, 3, ... , d

This gives the unconditional distribution of Vq, as

1,2, ....

MULTICHANNEL BULK QUEUES

248

P(Vql

> t)

=

. (

I.

=

P d + c - 1 e-

cllt

(Cllt

d-I

L ;=0

d=1

i)

-'-1-

(5.2.20)

.

l.

From this, or otherwise, one can easily see that E(Vq1 ) = :Ed=1 (d/CIl)Pd+c-l' The higher order moments may be obtained similarly. This completes the discussion of the distribution of V q , • The discussion of case (2), however, is a bit more involved. For that we need to use the fact that the probability that a random customer has the jth position in his arrival group is =

L

r·J =

(5.2.21)

a

(see Miscellaneous Problems and Complements, Chapter 3). Let D; be the event that a customer, whose waiting-time distribution is required, experiences a delay due to the commencement of service of i> 0 members from within his arrival group before his service commences. Consequently, if a group arrives to find the system in state n(>c), thenP(D;ln) = ri+! ,i> O. In other words, the probability that the customer whose waiting-time distribution (in queue) is required is 1st, 2nd, 3rd, ... , in his group is rl =P(Doln), r2 =P(Dlln), r3 = P(D2In), .... On the other hand, if a group arrives to find the system in state n, 0';;; n .;;; C - 1, then there is no delay for the first C - n customers in the group, and P(D o In) = :E Y;f rj. However, if his position is c - n + i, i> I, then P(D; In) = rc-n+ i, i> I. Hence we obtain the unconditional probabilities c-1

00

L n=c

Uo = P(D o ) =

P(Doln)Pn

n=c

+

rlPn c-2

rl

I

+

Pn

n=O

U; =

P(D;) =

+ L

I: Pnr;+l +

n=c

P(Doln)Pn

n=O

c-1

00

L

c-n

L

Pn

I

rj

n=O c:'n

L rj

j=l

(5.2.22)

i=2

c-I

L

n=O

Pnrc+;-n

c-2 r;+1

+

L

THE SYSTEM

MXIMlc

249

If V q , is the component of the waiting time (in queue) of the customer under consideration due to the queuing times of the members of his arrival group, then P(Vq, >t) = e- cilt

(rc+;-n-ri+dPn,

i>l.

n=O

Now define the "mean service position" or Ec(SP) as the expected number of customers from within a random customer's arrival group that enter service before the random customer. Then

f

Ud 'I1(cllt

Y

d=1

i=O

i!

'

t>O.

The expected value of V q , is therefore E(Vq ) = ,

a~

m=j

5.2

f= P(Vq 0

'

> t) dt

cll

Numerical calculation is possible in all the above cases, but analytically explicit results can be achieved only for geometric distribution of the arrival group size. For more details, see Problems and Complements. ~ Many of the results of this section are due to Abol'nikov(1967) or Kabak (1970). Hawkes (1965) obtained the distribution of waiting times of ordinary and priority customers in the priority queuing system MX"x, /G I , G2 /1, from which one can obtain the waiting time distribution for the system M X /G/l. Later Burke (1975) derived the waiting·time distribution for M X /G/I, and cited both correct and incorrect formulas for waiting time in the work of earlier authors. It is easy to show using Burke's results, that Little's formula holds for M X /G/l. In this connection, see Chapter 3 (Miscellaneous Problems and Complements). Those results of Kabak which were erroneously reported have been corrected by Cromie and Chaudhry (1975), a revised version of which is presented in this section. Abol'nikov and Yasnogoridskiy (1972) have also discussed the distribution for the number in the systems MX /M/c /c and MX /M/c/Finite, but give the results in impliCit form. For explicit results for certain particular cases of MX/M/c/c, see Problems and Complements. Abol'nikov (1970) has transient solutions of MX/M/c. Cromie (1974) has discussed more elaborately the numerical aspect of the system M X /M/c (limiting case) for the three cases where the arrival group size has (1) constant input, (2) geometric input, and (3) positive Poisson distribution (left truncated). He has provided numerical values 'and curves for certain measures of efficiency such as Lq and PC Vq > 0) for certain combinations of values of the average group size a, the traffic intensity p, and the number of channels c for the three cases mentioned above. In this connection, see also Cromie et al. (1979) who have simplified further some of the results of Cromie, and have given independent proofs for others. For numerical evaluations of the cumulative distributions of Vq and V q1 , Grassmann (1974) has shown that the following expression, which may be obtained either directly or by interchanging the summation in equation (5.2.19), for P(Vq > t) converges faster than equation (5.2.19):

=

Ec(SP) =

L i=O

iUi

A(2)(l)

c-2

=-L 2a n=O

c-n-l

c-n-l- ~

t

• =1

)

(c-i-n)r; Pn · (5.2.23)

The above results for the system M X /M/c have been extended to the system Elf/M/c. For details, see Holman et al. (1980) and Holman (1977) .

MULTICHANNEL BULK QUEUES

250

5.2

THE SYSTEM MXIMlc

The Busy-Period Distribution

The distribution of a busy period for a multi-server queue may be defined in X many ways. Here we initiate the discussion for the M /M/c queue when c = 2. The procedure, of course, runs on the same lines for the case of more than two servers. Two cases need to be distinguished, depending on whether a busy period is considered to be a period during which both servers are continuously busy, or a period during which the two servers are not simultaneously idle. Two Servers (at Least One Busy). The length of a busy period in this case may be defined as the interval of time from the instant of arrival of a unit that makes at least one of the servers busy (that is, the initial number, i = 1) to the subsequent instant when both servers become free for the first time. The p.d.f. for the distribution of a busy period is given by dPo(t)/dt. To compute dPo(t)/dt, we have the equations p.P I (t)

- (A + M)P 1 (t)

then zg' =

251

(1=1f: Cjai-1)n pn ~ U + 2n + m - 1)! . am + m f: b ·a 2n m n! U+n+m)! nl -

n=l

l +

L..

+

j=O

The inverse of zg' is L-1(zg') =

'!!. [C 2 /lt)m + t

m!

L

(p2Mtt = (2/lt)i+n+m ] nj (j+n+m)! exp[- (l -n-!j~O b n=!

+ p)2MtJ. (5.2.26)

Taking the inverse of equation (5.2.25) with the help of equation (5.2.17), we fmally get the p.d.f. for the busy period starting with i customers in the form dPo(t) = /lP (t) l dt

+ 2MP2 (t) n-I

- (A + 2p.)Pn (t)

+A

Define P(z;~) =

L

amPn-m(t)

+ 2MPn+1 (t),

n;;"

2.

m=l

=

L

x exp[-(l +p)2/lt). For ~he MkjM/2 queue with ar = 0 rk' since bn,n(h-l) = 1, b nj = 0, j =F n(k - 1) and I = 1, we have the p.d.L

Pn(~)zn.

n=l

Taking the L.T. of the above equations, usingPiCO) = 1 and the generating funcHon, we get _ zi+1 + p.z(z - 2)1'1 (~) (5.2.24) p~.~ = . , (A + 2M + ~)z - 2M- AzA(z) The denominator of equation (5.2.24) has one zero Zo inside the unit circle, 1. As the numerator of equation (5.2 .24) must vanish at zo, we have

Iz 1=

(5.2.25)

To invert PI (~), we proceed as follows. Let p = A/2M, a = 2M/(A + 2/l + 0:), W = zA~), and put z m = f(z). Here m is an arbitrary integer, and fez) is an analytic function whose Lagrange expansion (see Appendix A.6) is given by

11./(11. + 2/l + ~), g(z) =

fez) z=a

If we let

dPo (t) dt

This p.d.f. may be expressed in terms of a function 1). Thus

I/;, (x)

(defined in Chapter

dPo(t) = 1k ( ) -d- = m 2 2 - m /lp-{(m-O/(k+!)} ~ exp[-(l t m=l r

L

+ p)2Mt)

where r/2 = 2p 1/(1 +k)Mt. Furthermore, if k = I, that is, if arrivals follow a Poisson process, dPo(t) _ exp {- (A. + 2/l)t} = (2A.)-mI2 dt t m~l m -;; 1m (2 ~t) where 1m(r) == 1!n (r) is a modified Bessel function of the first kind of index m.

2 Two Servers (Both Busy). The busy period in this case may be defined as the mterval of time from the instant of arrival of a unit that makes both servers busy (that is, i = 2) to the subsequent instant when at least one of the servers becomes free for the first time. The p.d.f. for the duration of a busy period is now gIVen by dP l (t)/dt. We compute dP l (t)/dt from the following equations:

MULTICHANNEL BULK QUEUES

252

5.3

THE SYSTEM MX/D/c

At

PP )(t) = 2J1P2 (t)

1Tn+l (t)

pI'i\t) = -(A+2J1)P 2(t)+2J1P3(t)

p~l)(t) = - (A + 2J1)Pn(t) + A

n-2

L

amPn-m(t) + 2J1Pn+l (t),

Pij(t)

(A + 2J1 + a)z - 2J1- AzA(z):

(5.2.27)

give below the result.

2J1P2(t)

= rl(z~-l)

r

= i - I (2J1t)H

t

+

lO-I)!

x exp[- (l

I

(p2J1tt

n=1

n!

~

(2J1t)i+n+H I)!

L bnj (.J + n +.I j=O

(n - j

+ l)a n-i+l1Tj(t),

n = 0,1,2, ...

j=O

Each customer receives service for exactly b units of time. The queue discipline is FCFS for the batches, and the system is in statistical equilibrium. Let NCt) be the number of customers in the system at time t. Then the transition probability matrix (t.p.m.) of the process {N(t), t? o} is given by

m=l

The denominator of equation (5.2.27) is the same as that .of equation (5.2.15) Therefore the zero inside Iz 1= 1 and the inversion can be discussed as before. We

=

n+1

1To(t) = e- At .

As in case (l), we now have

dPl(t) dt

n

L

n?3.

Define

P(z; a)

253

As before, the results for the k-arrival and single-arrival cases when i = 2 are, respectively, and

P{N(t)

=

j IN(O)

= i},

j?O.

i?O,

Although {N(t) } is non-Markovian, if we examine the sequence N~ =

n = 0,1,2, ...

N(nb),

(5.3.1)

we see that it is Markovian. In fact {N~, n = 0, I, 2, ... ,} is a homogeneous given by Markov chain with one-step t.p.m. (Pij ) = (P(N~+l = jIN~+=

i»

1

+ P)2J1tJ .

=

Pij =

(

1Tj(b)

O<;i<;c

1Tj-i+c(b),

c
0,

j +c < i.

Since the input process is a compound Poisson process, the limiting distribution of the number in the system encountered by an arbitrary arriving batch is the same as the limiting distribution Pj = 1imt .. ~P(i(t), j = 0, 1, 2, ... , of the process {N(t), t? O}. If Pj exists, so does P/ = Iimn .. ~P {N~ = j IN~ = i}, j = 0,1,2, ... , and they are equal. We therefore do not distinguish between probabilities such as P/ and Pi> and will use Pj instead of P/ in the sequel, although our interest is in the limiting distribution of the imbedded Markov chain (5.3.1). The Chapman-Ko!mogorov equations for this distribution are

Pi

where r has been defined earlier.

n+c P n = 1Tn(b)dc

5.3

THE SYSTEM M X IDle

The limiting Distribution of N~

In the system MXjDjc, the input scheme is the same as discusse~ in Se.ctions 5.1 and 5.2. Once more, let X(t) be the number of customers that arnve dunng J0, t]. Then if 1Tn(t) = P(X(t) = n), it can be shown (see Chapter 2) that the p.g.f. o~ 1Tn(t) is II(z,t) = k:=O 1Tn(t) zn = exp [tJ3(z)] , where J3(:? = A(A(z) - ~) an~ A~z)­ k;;-I amz m . Further, we shall write J1l =/3(1)(1) = Akjrnam and J12 - Akl. rn am, tha~ is J11 and J12 are the mean and the variance of the total number of arnval~ per unit ti:ne. The probabilities 1Tn(t) can be computed through the recurrence relatIOn

+

L

m=c+l

Pm 1Tn-m+cCb),

n = 1,2, ...

(5.3.2)

where d n is the probability that not more than n customers are in the system and is thus given by (5.3 .3)

Equations (5.3.2) may be paraphrased by writing pen in system at end of an interval of length b) = PCqueue empty at beginning of interval) pen customers arrive during interval) + k~~C+l P(rn customers in system at beginning of interval) Pen m + c customers arrive during interval). Here we use the fact that during any interval of length equal to the constant service time b, all customers in service at the beginning of the interval, and only those customers, complete service and leave the system. As

MULTICHANNEL BULK QUEUES

254

usual, we assume that p = J..l.l ble < I so that the limiting distribution exists. Con· sequently, we need to solve the set of equations (5.3.2) for the limiting probabilities Pn' To do this, we introduce the p.g.f. P(z) = 1:;;=0 Pn zn . Multiplying both sides of equation (5.3.2) by zn and summing over n, we have, after simplification,

P(z) = d c exp(b;3(z»

+ [P(z) - g(z)]

exp (b;3 (z» c

Z

e-f..b exp

= exp[bplO

-(Plb-c)

D(z) =

(l-zd(l-z 2 )

Since {Pn} are probabilities, the function P(z) is regular in z ~ I and, therefore, the zeros of the denominator within and on the unit circle must coincide with those of the numerator. Now for Iz I = 1 + 0, where 0 is sufficiently small and positive, we have

~

255

The proba?ilit~ that a batch on arrival does not have to wait for getting service started on Its fIrSt member is equivalent to the probability dc-I that the batch finds at mos~ c - 1 s~rvers busy. In computing d C - 1 it is convenient to introduce the generatmg functIOn D (z) == 1:;;=0 d" z". Since d - d _ = P n == 1 2 3 b equation (5.3.3) we get (I -z)D(z) =P(z), a~d co~s~que;;IY by ~aki~g' ~~; o~ equatIOn (5.3.5) we have

(5.3.4)

I

THE SYSTEM MXIDlc

Probability of No Service Delay

where g(z) = 1:~=o Pn z". Finally, the p.gJ. of the sequence {P", n = 0, 1, 2, ... } is given by

\ eb!l(z)

5.3

lb tA ~ak\Zlk} 1

.•.

The coefficient ofz c - I inD(z) is the desired probability d C-I --

c - /lIb (1 - z d

equation

c-l

J;: has exactly c roots within the region Iz I ~ 1 + O. Let these roots be denoted by Zl, Z2, ... , ZC-I, zc(= 1). It can be shown that these roots are all distinct. For note that Zc = I is simple because 1 - ZC e- b{3(zc) lim C =blll-ci=O. zc"'l Zc - 1 To show that the remaining roots z;, i = 1,2, ... , c - 1, are distinct, first show that these roots are within the unit circle, Izi = 1. Suppose that Iz; 1= 1 for some i = 1, 2, ... , c - 1. Then from the equation giving the roots, Iexp [b;3(z;)] I = 1, which in turn implies that the real part of b!3(zj) must be zero, that is, Re [!3(zi)] = O. Thus one should have Re [13 (z;)] = - Re [A1:~ ak(1 - zf)] = O. Since all terms within the sum are nonnegative, one should have Re [1 - z~ 1 = 0, for all k, and therefore Z; = 1. This is contrary to our assumption that the root z = 1 is simple. It then follows that Iz d < 1, i = 1,2, ... , c - 1. Since the numerator of equation (5.3.4) is a polynomial of degree c, it can be replaced by A(z - l)(z - Zl) ... (z - ZC-I)' where the constant A is determined by P(1 -) = 1. Thus

P() z

(Plb-c) .(z-l)(z-zd.·.(z-zc_d (1 - Z I ) ... (1 -zc-I ) 1 -z c e-b!l(z)

... (1

- ZC-I )

whence In d c- I ==In(c-J..l. I b)-1: cI - I 1n (1 - Zi. ) U' '. smg t Ile generalized argument pnnclple (see Appendix A.4), one can get an expression for In dC_I independent of t~e r~ots zl' To do this, let ¢(z) = e b{J (z) - zC, and note that ¢(z) has simple zeros a zZ2, . . . , ZC-I, and no pole inside the circle D, Izl = 1 - E. Here E> 0 is chosen III such . a way that all the roots z·" i = 1" 2 . • • , C - 1, I'Ie IllS! . 'd e D an d . zc(= I,> IS ?utslde D. Choose x(z) as the principal branch of In (1 - z) which is analytic mSlde and on D. The generalized argument principle yields

z:'

+o(o)J

< (l+or = Izlc where we have used the fact that bPI < c. Consequently, by Rouche's theorem, the

=

.(Z-ZC-I) l-z c e- b{3(z)

(Z-ZI)(Z-Z2)"

(1-zc_d

(

)

5.3.5

.L

In(1-z;) =

t=1

~ rh In(1-z)d[ln¢(z)]. 2m 'Yv

It will now be shown that

J = In (c - bpI)

+I n

n=1

and hence In d C- 1 =

I

1Tj(nb)

j=nc

=

I

=

n=l

n

j =nc

I - I

1Tj(nb).

It i~ea~ to see that the prin~ip~l branch of (1 - z) In [ZC-I (1 - z)] is analytic in Izi "" 1 E, and consequently Its Illtegral on D is zero by Cauchy's integral theorem Hence we get . 1

J = 21Ti

•

p In(1-z)d[lnz

<11 1n(1-z)d[ln ¢(z)] -

D

21Ti

rf... [ 1 'f. In( 1 - z) d In D

Integration by parts gives

Z

-c

b{J(Z)J

e_

l-z

1

.

C-l

(1 -z)]

MULTICHANNEL BULK QUEUES

256

J=

21fi

1;

D

'1 - z-c e b(3(z)

In

1 - Z-I

dz --. Z- 1

z -c e b(3(z)]

As a simple example, consider the case of single arrivals, in which case a 1 == 1, ar ==

d~l

I - cP,D, z"(zdz-

In(1-z-1) = 1 ---'------"- d z - Z - 1 n=1 n

1)

which is seen to be zero by calculating the residues at the poles z = I (order 1) and z = 0 (order n) and invoking the residue theorem (see Appendix A.3). Consequently, the value of J is as stated above. Once again as Iz- c e b (3(z) I < I on D 1 , by using the argument that the power series expansion forln [1 - z-c e b !3(Z)j converges uniformly on DI . and termwise integration is permitted. we find 1 J = In(e-bp.l)+-.

=

21fl

D,

(~~ )i e->-.nb. J.

The Waiting-Time (in Queue) Distribution

gm (t) == P (among the customers present at some time to, at most m of them will remain in the system at time t + to, 0 < t ~ b, m;;;' 0)

and the p.g.f. G(z; t)= L:;;',=ogm(t)zm. Considering the state corresponding to gmc+c-l (t), it can be seen that, at most, me + e - 1 of the customers preceding the given batch will be in the system at time t later, and thus, at most e - 1 of them at time mb + t later. This is the condition for the waiting time (time until a first customer from the batch enters service) to be less than mb + t, that is, P(Vq

< mb + t) n

I

1f n - m

(t)gm (t)

m=O

which gives

z-

== gmc+c-l (t).

Now in the limiting case we have the relation dn =

1 ,f., e bn!3(z)z-nc 'Yr 1 dz

I n=1 n

I

..!.

n=1 n i=nc

To obtain the limiting distribution of Vq(t), define the probabilities

z

so that D,

I

In d C- 1 =

where we have used the fact that the integral of [lnC1 - Z-I)] /(z - 1) on Dl vanishes. This fact is easy to establish. For as 1z 1-1 < 1 on D 1 , the power series for In [1 - Z-I] converges uniformly on D 1, and termwise integration is allowed,

1>

257

THE SYSTEM MXIDlc

O,r* 1. Then

Now we want to change the contour D to D 1 , Izl = 1 + 8, where 8> 0 is chosen in such a way that the only zeros of (z) in Izi ~ 1 + 8 are 1,ZI ,Z2"" ,Zc-l' The integrand above has a pole z = 1 which is inside DI but outside D. Since the residue of the integrand above at the pole z = 1 is In(e - bP.l), we have J = In(e - bP.l) - _1_. rh In [1 21f1 ~D,

5.3

==

=

I

dnz n == e t(3(Z)G(z; t).

Now the integrand has poles at z = 0 (order ne) and at z = 1 (order 1). Clearly the residue at z = 1 is 1. The residue at z == 0 is the coefficient of Z-1 in the expansion of _e bn(3(z)z-nc (1 - Zfl as a power series in z, that is, !t is the coefficient of Z-I in _ L:;'oebn(3(Z)z-
By using equations (5.3 .3), we have, as before,

theorem, '

Substituting the values of D(z) andP(z) in equation (5.3.6), we obtain

D(z)

(l-z)D(z) = P(z). (" b - e ) e- t (3(z) ( G(z; t) = _ 1.1-'1 z -ZI) ••• (Z - Zc-I ) [1 - ZC e- b (3(Z)J(1 - z d ... (1 - ZC-l)

which is the required result. Since the probability of no delay =

dc-I'

P(delay> 0) == 1 - d c - 1 1 - exp -

t

I-I - nc-l I 1fj(nb)

= 1 [ n=1 n

)=0

1} .

(5.3.6)

n=O

(5.3.7)

.

(5.3.8)

One can get the explicit probabilities, P(Vq < bm + t), by first obtaining a Laurent series expansion of G(z, t) in the annulus 1 < Izl< 11/1, where 1/ is that root of ZC exp [- bj3(z)] = 1 which has the smallest modulus exceeding unity. It is easy to see that such an 1/ exists. For let us consider the function XC exp bj3(x)]1 == f0c) (x = real part of z). Then as f(1 + 8) == 1 + e(l - p)8 + 0(8) > 0 (since p < 1) and f( 00) = - 1, the function f(x) which is continuous must cross the x axis between 1 + 0 and "". This proves that there exists at least one real root of f(x) == 0, and consequently at least one root of ZC exp [- bj3(z)] == 1, outside the circle Iz I == 1 + 8, where 8 is sufficiently small and> O. Select 1/ such that 1/ has the minimum modulus, that is, 1/ is the next pole (in order of magnitude) of the expression

5.4

THE SYSTEM OS/M/c

259

MULTICHANNEL BULK QUEUES

258

for G(z; t), the first pole being at z = 1. Here we have used the fact that the numerator of G(z; t) does not vanish at 1) for p < 1. For 1 < Iz I < 11) \, the absolute value of z-c exp [b/3(z)] is less than 1. (Consider Iz\ = 1 + 0 and choose 0 in such a way that 1 + 0 < 11)\.) Expanding the denominator of equation (5.3.8) in powers of z-c exp [b/3(z)J, the exponential function in powers of z, and collecting like powers of z, we get for the annulus 1 < \z \ < \1) \ (5.3.9) where gn is the coefficient of zn in the polynomial (;;'1 b - c)nf;l [(z - z;)/O Zi)] . Since z = 1 is the only singularity of G(z; t) in the circular region \z \ < \1) \ (a simple pole with residue -1), G(z; t) + (z _1)-1 is analytic in Iz 1< 11) \, and hence for Izl

< 11)\'

2.' ... , where P(O~+l - o~ = I) = 1, for every n and a~ = O. Since the interarrival and equal to ' I we have 0'n = n' The servI'ce t'm . dependtimesI are constant . . . I es are m enty exponentially dlstnbuted with mean I/p. While in other models we have considered the number of customers in the system just before arrival instants here we follow Cox and Hinkley (1970) in an analysis of the imbedded Markov ~hain {Nn } formed by considering the numbers of customers present immediately after (potential) arrival instants. Let the .number of customers arriving at time n be An, having a binomial distributIOn ~f mdex s and parameter p. Further, let N(t) be the number of customers present m the system at time t and let N n = N(o~ + 0). Also let Y n be the number of possible serv~ce completions during a unit interarrival interval from n to n + I. Clearly the dlstnbutlOn of arrival groups is given by

ah=p(Ai=h)=(~)phqs-h,

0,1,2, ... ,s

(5.4.1)

where E(A;) = sp, var(AJ = spq, and

Therefore, for \z \ < 1, we must have

1 G(z; t) = expansion (5.3 .9) +-I -z ~

=

expansion (5.3.9)

+

L zn. n=O

From this by picking the coefficient of zmc+c-I , we get c-l

L

P(Vq<mb+t) = 1-

n=O

00

qn

L 1T(m+i+2)c-n-dU+l)b-t],

ifNn - Yn;;;'O if N n

-

(5.4.2)

Y n ';;;;0.

Now ~ecause the distribution of service times is exponential, the distribution of YNn , }given N n = l, IS independent of the values of N n-l , N n-2,"" and h ence f { n . o.rms a Ma:ko~ chain. Let N n converge to N in distribution as n -+ =. To find the lImltmg dlstr.lbutlOn of N which exists subject to p = sp/)J.C < 1, we need first to find the tranSitIOn probability matrix of N n . To get this, consider first the distributIOn of Y n , gIvenNn = l. For this, we have

i=O

m=0,1,2, ... ,

e-CP()J.C )l-m

O';;;;t';;;;b.

The results of this section are due to Kuczura (1973), although we have corrected some minor errors in the original paper. The corresponding results for the single poisson arrival case M/D/c have been discussed by Erlang (1920), Pollaczek (1930a, b) and more elegantly by Crommelin (1932, 1934), among others.

5.4.

h

G(Z;t)+(Z-l)-1 = expansion (5.3.9).

THE SYSTEM DS/M/c

The limiting Distribution of N n

_ qlm = peNn - Yn = m INn = I) =

(I)

m e- mp (1 - e-Pl- m ,

I,m ;;;'C

(5.4.3) I, m ,;;;; c

because in the first case c servers are working throughout the unit time period, wherea~ m the second case each of the I customers has independently a chance 'Y of sefV1ce c~mpletlOn where 'Y = J6 pe - P t dt = 1 - e - p. This leaves the case m < c < I. For thiS It may be noted that the time x measured from n until the number of customers first falls from I to c has the probability [(x)dx, where [(x) is the Erlang p.d.f. given by [(x) -

In this section we consider a multiserver queuing system in which customers are scheduled to arrive in groups of exact size s at unit time intervals. There is an independent constant probability, q = 1 - p, that a customer scheduled to arrive in fact does not do so. In application, servers may be judges, customers cases. All cases are scheduled to start at the beginning of a day, and nonarrival corresponds to a case being settled out of court shortly before the time set for its hearing. Suppose that s customers are scheduled to arrive at each instant o~, n = 0, I,

I

(l-m)! '

cp(c;;'XY-C-l e- cpx (l-c-I)!

Also, i~ x < 1, the probability that in the time 1 - x, measured from x to the end of the mterval, further c - m services are completed is

(:) {1_e-P(I-x)}c-me-cPO-x).

260

MULTICHANNEL BULK QUEUES 5.5

THE SYSTEM

M/MB/c

261

Thus mlNn = I)

=

f

CII(Clly)I-C-1 e- clJX (C) e-mP.(l-X) ,...,.-o (/- C - I)! m 1

x {I - e-P.(!-x)y-m dx,

m

of. this section are due to Cox and Hinkley (1970) wh 0 h ave con.d Thed results ' ere . approXlma~lOns. to the present model, numerical calculations, and further generahzatlOns to sltuatlOns wherein not all customers are scheduled. Sl

< C < I.

(5.4.4) Besides, it may be noted that qlm = 0 if 1< m. Now the entries Pij of the t.p.m. P may be obtained by using equations (5.4.1) and are given by

5.5

THE SYSTEM MIMB Ie

j

Pij

=

P(Nn+1

= j INn =

i) =

I

qihaj-h

(5.4.5)

The Transient Behavior

h=O

where we have used the independence of An+! and (Nn - Yn). With P determined from equations (5.4.1), (5.4.3), (5.4.4) (or its equivalent form - see Problems and Complements) and (5.4.5), the limiting distribution 1fj = limn ... ~ PeNn = j) can, in principle, be found from the relation''TT = 'TTP, where 'TT is the row vector {ITo, 'TT 1, 'TT2, ... }.

T~is sec~ion deals with a queuing system having C homogeneous servers, each With maXImum capacity B. When one of the servers is available, then: ~ If number in queue> B:;;:' 1, then the first B customers enter service immediately

2

If 0.< number in queue <'B, then all the waiting customers are taken up for service

3

If number in queue = 0, service stops until a new customer arrives after which service resumes immediately. '

Measures of Delay

As an index of delay to customers, we may define the relative measure ,f,

_

'Yo -

~ j1f c +i L. - -

j=l

sp

(5.4.6)

which is the ratio of the expected number delayed to the expected number of arrivals. As in many practical situations, a customer delayed at time n stands a good chance of starting (and, possibly, even of finishing) service before n + 1, whereas a delay beyond n +! may be regarded as serious. An analogue of equation (5.4.6), calculated at n +! instead of n, is a useful index and may be evaluated thus. Given N n = l > c, the probability that at time n +! there are m :;;:. C customers present, as in equation (5.4.3), is 1 (~ C/J)l-m exp(-'2 C/l) (l-m)!

Arrivals follow a Poisson distribution with mean rate A, batch service time is exponentially distributed with rate /J, and FCFS queue discipline is observed. We first discuss the transient probabilities of the number of customers in the queue and the number of busy channels. Let Nq(t) and Lb(t) be the number of customers in the queue and the number of busy servers at any time t, respectively. Let us define Pm,l(t) = P(Nq(t) = m,Lb(t) = I). It can be easily seen that these probabilities are nonzero for 0 <,1< c, m = 0 and 1= c, 0 <, m < 00. The state equations for the system are:

P<';'~c(t)

=

-(A

+ C/J)Pm,c(t) + APm-1,C(t) + C/lPm+B,c(t),

Accordingly, the half-period analogue of equation (5.4.6) is

(5.5.2)

p&:lct) sp

where bh =

m>O (5.5.1)

= - (A + 1/1) PO,I(t) +APo,I-1 (t)

(5.4.7)

(! C/J)h exp(-! c/l) h !

Given p and /J, feasible combinations of C and s may be found to satisfy tolerance limits on CPo, CP112, and p, that is, the feasible number of appointments per unit time for a given number of servers may be determined.

Prl,& (t)

= - APo,o (t)

+ /JPO,I (t).

+ (l + l)/JPO,I+! (t),

O
(5.5.4)

We solve this set of equations under the initial condition P 0, c (0) = 1,th . II at'IS, at time zero a servers are busy and no customer is waiting to be served. Define

P(z; t)

(5.5.5)

MULTICHANNEL BULK QUEUES

262

5.5

THE SYSTEM

MIMBlc

263

and let the L.T. of PO,I (t) be denoted by q;(OI.) == qz. Then from equations (5.5.4)

}.I.lh = (A+OI.)ilo

and (5.5.3) wehave (l + l)}.I.ilZ+l

and

(A.) /-j r(OI.!f.J. +.) I •I _. I j=o}.l. r (OI./}.I.)]. (l ]).

I '\' ql = qo £...

(5.5.6)

(A + a + l}.l.)liz - i\(jZ-l,

1 ~ I < c.

(5.5.7)

= qo

where To obtain a general solution for equations (5.5 .6) and (5.5 .7), we first rewrite equation (5.5.7) usingf(l) = ill, 0 ~ I ~ C - 1, and get

±

(;r*

(5.5.8) The generating function of f(x) is

Gf(x) == u(z) =

I:

The following two well-known results concerning generating functions can be easily verified. [For details see Problems in Section 1.1 of Chapter 1 or Jordan (1965).]

+ r)

Gxf(x+r) =

= (:r) [u(z) - f(O) - ... - fer - l)zr-I],

z(~)

lFo

(-l,;;-~)

where 2F O is a generalized hypergeometric function discussed in Chapi,er 1. Using the definition of
f(x)zx.

x=O

Gf(x

(5.5.11)


(x + 2)}.I.f(x + 2) = [A + a + (x + 1)}.I.] f(x + 1) - Af(x),

=

O~l";;;c-l

r> 1

[zlr {U(Z)-f(O)- ... -f(r-J)zr-J}],

(5.5.9)

r> 1.

ilC-l = ilo
(5.5.12)

Now we deal with P m,c (t), m > O. Multiplying equations (5.5.1) and (5.5.2) by appropriate powers of z and summing over all m leads to an equation whose L.T. after simplification is

F(z; 01.) =

[

1 + C}.I.

B-1

I:

k=o

(5.5.10)

(1 - zk-B)Fk,c(OI.) - c}.I.J5o,c(OI.) +

/[01. + A. - A.Z

Using equations (5.5.9) and (5.5.10) in equation (5.5.8), we have

+ C}.I. - {C}.I./zB}].

APO,c-l (01.)

]

(5.5.13)

The zeros of the denominator of equation (5.5.13) are the roots of

!!:.)[dU(Z)_f(I)]_}.I.dU(Z)_OI.+A. [u(z)-f(O)] Hu(z) = 0 (z dz dz z or

}.I.(l-Z) du(z) -lOl.+ A. -A.} u(z)-..!.. [}.I.f(l)-(OI.+ A.)[(O)J = z dz z z From equation (5.5.6), }.I.f(!) = (01.

+ A.)f(O), and consequently,

o.

01.

+ (A. - AZ + C}.I. - {c}.l./zB})

(5.5.14)

Applying Rouche's theorem to equation (5.5.14), we find that there are B roots inside and one root outside the unit circle, Iz 1= 1. Let the outside root be denoted by Zo· Also since the degree of the numerator in equation (5.5 .13) is one less than that of the denominator, the ratio P(z; 01.) can be expanded in terms of the external root Zo as

-

A

A.Z-(A+OI.) u(z)-----;;;-= }.I.(z-l)

P(z;OI.) = - Zo -z

) = A exp [A.z/}.I.]

where A is an unknown function of a to be determined later. From the expanded form of equation (5.5.15)

1 du(z)

which gives

u (z

(5.5.15)

(l-Z)CXi/L = f(O) exp [i\z/}.I.} . (l-z)CiI/L

(5.5.16) and in particular

A Po c(OI.) = , Zo

Thus x

(A )X-i

f(x)=f(O)j~O}.l. and

= O.

r

(OI.!}.I. + j) r(OI./}.I.)j!(x-j)!

(5.5.17)

Rewriting equation (5.5.13) at z == 1 and substituting from equations (5.5.15), (5.5.17) and (5.5.12),

-

A

P(1;OI.) == - Zo -1

a

5.5

THE SYSTEM MIMB/c

265

MULTICHANNEL BULK QUEUES

264

This gives

m:;;'O

1 + Moc-I (a)

A = zo(zo - 1) ruo

+ cJl(zo - 1)'

where Po,o is given by

. th e L.T. of the sum of all probabilities at any time t is l/ex, A1so, smce

A

--+

Po,o

1

c-I

L

ql = loa

Zo -

. _ lIm aqo "'->0+

=

[(AIJlr (l-I/k)c!

+ c-I L 0

. (A/tJ.) I Po, I = hm aPo lea) = Po 0 - - - , 'l!

"'''0+'

cJl(zo - 1)

(AIJly]-1 r!

(5.5.25)

O
(5.5.26)

(5.5.19)

+ {ruo + CJl(zo -

Equations (5.5 .24), (5.5 .25), and (5.5 .26) completely determine the limiting state ~. probabilities.

I)} cf z(ex)]

From equations (5.5.16), (5.5.18), and (5.5.19),

zorn (zo - 1) [Ac-I (a)

=

From equations (5.5.11), (5.5.20), and (5.5.21) we get

Substituting for A and ql and simplifying, we get

qo = a[Azo c-I (a)

(5.5.24)

(5.5.18)

The Waiting-Time (in Queue) Distribution

+ ex cf z(ex)]

If Vq is the time spent in the queue by an arriving customer, then we find P(Vq > t) as follows. Given that all servers are busy, an arriving customer jOins the nth batch awaiting service, according to the following scheme:

From equations (5.5.11), (5.5.12), and (5.5.19) we get

I (ex)CJl(zo - 1)

Po lea) .

= ql =

[

a AZo C-I (ex) + {exzo + CJl(Zo - I)}

CI

t

]'

The Limiting Behavior

To obtain the limiting state pro babilities which exist for p == A/CpB < 1, first observe that (5.5.20) O~l~c-l lim z(a) 0;-+0+

lim cxA = ",->0+

where k ( -1' - llTI", .. O+Zo )

I'S

rlc! (i-l/k) (A/Jlr +I

0

k(~)C (~) lim aqo Jl c!

(~)r ~1-1 Jl

(5.5.21)

if B ~ i ~ 2B - 1

customer joins the

if (n - I)B ~ i ~ .nB - 1

nth batch,

where i is the number of customers in the queue when a customer arrives. Now a customer joining the nth batch must await n service completions before entering service with his batch. With all servers busy, the output from C independent, homogeneous servers, each serving with an exponential rate Jl, follows a Poisson process with mean ctJ.t, where t is the time measured from the customer arrival instant. Thus if we define do == 0 and d n to be the probability that less than n service completions occur in time t, then

dn =

nf r=O

(5.5.22)

. cuc . 1e, Iz I - 1, of the equation the root outside the umt

From equations (5.5 .16) and (5.5.22) we have

customer joins the second batch,

r!

",->0+

A kf3 - 1 CJl = kf3 (k - 1) .

i ~ B-1

if 0

l(ex)

. d exp1··t! T 's of the state probabilities. ICI y the L ., Thus we have obtame

l~~exqo =

~

customer joins the first batch, O~l
(cJ11Y exp

(~CtJ.t) r.

Also, let an = P (arriving customer jOins the nth service batch), then ao = P (arriving customer receives immediate service) =

(5.5.23)

and

c-I

L

1=0

P 0,1

MULTiCHANNEL BULK QUEUES

5,6

267

THE SYSTEM GllMBlc

266

i=~f-:)Ji"

Po,c

Hence combining the two cases, we see that the probability that the time spent by an arriving customer in the system is greater than t, is given by

n = 1,2, ...

[(~rr-ll(k~1~~-11

= Po,c

and n = Po,c

l(k

)B

1 rlCk-1)0-1\1 n=l~o

e-p.t

(GlltY e-cp.t r! .

kB - 1

n-i

. '11 be delayed in the queue Thus The probability that a customer arriving at a~y time WI for a time greater than t, is obtained by summmg and n over n. P(Vq>t) =

'" 1;;

1n=l{;o

(GIlt)" e-cp.t r!

I

{(~) 1

r 0 -1 Po,ck-B(n-l)lck_l)0-1

n-l

_

0 -

- Po,c

I

(k-l)0 k

1

'" (cpt)' e-cp.t r!

{;o

-c}J.Bp(k-l)t

Po,c - e k- 1

n-l=r

B n-l

k

I

(5.5.27)

,

d - A/CpS An alternative derivation of where we have used equation (5.5 .23 ) an d P d in'Section 5 6 is much simpler. _j f of Vq' following the proce d ure ISCusse . , tle p. d .. For details, see that section. The Waiting-Time (in System) Distribution Since the p.d.f. of the time spent in the queue can be seen to be -A(k-l)t

get) = Po,c kAe enters service with his, her or its batch at the proba?ilit y that a random c~s~omae~er its arrival is given by g(y) dy. Further, Y h the probability that the batch will coma time lymg between ~ and ->.' if the batch starts serVIce at tIme y, t en f d m customer in the batch f b tween x and x + dx a ter a ran 0 plete service at a l.me _eP.(X_Y) dx x> y. Thus the probability that a customer who ~ntere~ the queu~ It~~: t = will' complete service (with his or her batch) at a time Joms t e queue a l ' IX ( )dy pe-P.(X-Y) dx. The probability that . I d lying between x and x + dx ater IS 0 g y 'd ration will not yet have completed serVIce at an e apse d the customer un er conSI e . x ( ) e-P.(x-y) dy But this does not time t after entering the queue IS It dx I o g Y fJ ... h t T include the probability of the event that a customer enters service WIt ou wal mg,

°

00

which is

k

c-j

L.

PO,I =

I-Po,c k- f

1=0

The probability that such a customer has not completed service at a time t is just e- p.t.

P

_k_lJ e-p.t

o,c k - I

+ Jt~ dx foX PO,c kAe-A(k-I)Y fJe-p.(x-y)

. for the probability that a customer joins the nth batch Now the expresslO n . awaiting service and is still awaiting service at time t later IS

l

f1 L

P (system time> t) =

+

dy

kP [e-J.Lt -e-A(k-I)tl o,c . (k-l){(AlfJ)(k-l)-I}

The jOint distribution of the number of customers in the queue and the number of busy servers was found by Ghare (1968), but the various other distributions and results are due to Cromie (1974). An elaborate discussion on the computational performance of the results presented in this section is available in Cromie (1974) and Cromie and Chaudhry (1976). Cromie has drawn curves for Lq or A. Wq , tables of a root k of the characteristic equation (5.5.23) for various values of the parameters P and B, tables of state probabilities Po,c for various combinations of B, p and c, and so on.

5.6

THE SYSTEM GI/MB Ie

The limiting Distribution of N~ The MIMB Ie queue in continuous time, a particular case of the system CliMB Ie, was discussed in the last section using the Erlangian procedure. In the present section we discuss CliMB Ic in full generality, but in discrete time, through the imbedded Markov chain technique. In CliMB Ic, customers arrive at the sequence of epochs o= a~, a~, ... , a~ such that the interarrival times a~+! - a~ > 0, n = 0, 1,2, ... , are Li.d. r.v.'s with the common distribution function A(u) and mean 1lA., 0< A < co. The service times and the mode of servicing are as discussed in the last section, with the mean service time 11fJ. It is convenient to define the state of the system at epoch t asN(t), which now is taken to mean the sum of the number of customers waiting in the queue and the number of groups of customers being served. Then N;. = N( a~ - 0) is the state of the system just before the arrival of a customer at epoch a ~. N;. = i may be taken t6 indicate (1) if i > e, that e servers are busy and i - c customers are waiting in the queue; and (2) if i ~ e, that i of the c servers are rendering service to i groups of customers, with no customers waiting in the queue. To get the one-step transition probability matrix P = [Pij 1 = [P(N;.+l = j IN;. = i)], i, j > 0, we use the notation introduced by Kendall (1953) in his study of the CI/Mle queue, which Simplifies many things. We have the following notation: [II m; u] is defined as the probability that I servers complete service during an interarrival period of duration u, given that m servers were busy at the beginning of the period, where I ~ m ~ c. The probability that I independent exponential

5.6

MULTICHANNEL BULK QUEUES

268

THE SYSTEM GlfMBfc

269

servers will complete service in time u is giv~n by (1 - e-P.u)l, and the probability that m -I servers will not complete service in time u is given by (e-P.u)m-I. Thus, we have

If we let [lIm] =

J:

=5:

=

I lc. - J Ie;

[i - + BJ" } C B

'

j
where [x] is the greatest integer contained in x. [llm;ul dA(u),

Fin~~y, we h~ve the case when j:;;' c and i:;;'c. If (llc; u) is defined to be the probabIlIty that In tune u, Z groups will complete service from a P . whose mean rate is cll, then Olsson process

(7)(1-e-p.U)I(e~p.U(m-!»)dA(U)

then Pij = [i + 1 - iI i + 1] when j";; i + 1, i";; c - 1. Clearly, this situation corresponds to the case when i";; c - 1, so that the newly arriving customer will find c - I or fewer servers busy. The situation when i:;;' c is a bit more complicated. In this case define {lie; m; u} to be the probability that I servers become free after an interarrival period u, given that m groups were waiting in addition to the c servers busy at the beginning of the interarrival period. Assume that a time V has elapsed when the last of the m groups has been received into service. Until time V has elapsed, the c servers have been processing at a rate IlC. When the time V has elapsed, m groups have been processed since the beginning of the period. While all c servers are busy, departures from the system follow a Poisson process with rate c 11, the convolution of c independent Poisson processes, each with rate Jl. The time V, then, is Erlang-distributed, being the sum of m exponential r.v.'s with mean IlC. Thus the p_d.f. of V is given by g(v) =

so if we write

(lie) = fo= (l1e;u)dA(u),

(~) (1 -

e-(u-V)P.)! e-p.(u-V)(c- I).

The probability that the system ends a period u with I servers free given that it started period u with m groups waiting to enter service is therefore given by

z:;;.o

then

and Pij=(tle),

i:;;'c+tB,

j=i-tB+l

where t :;;. 0 is a nonnegative integer. . It :nay also be noted that [0 Ie J = (0 Ie). The complete form of the matrix is gIven In Table 1.

Let bl == (lie); then the p.gJ. of the sequence {b l } is given by

(llc)(!1Cv)m-1 e-P.cv (m -I)!

When the time required for all the waiting groups of customers to enter into service is less than the interarrival time, that is when V < u, then there is time u - V remaining to have I groups complete their service. The probability that exactly I groups will be processed before time u - V expires is given by [lie; u - V], where

[lie; u - V] =

Pij

K(z)

=

f

b!zl

=

!=o

0

=

~fte:

J-= e-p.c(l-z)u dA(u) a(!1C(1 - z».

kno,",:ing the form of the t.p.m., it can be seen that the Markov chain

~Nn }lhS ure~ucI~le. That the chain is aperiodic may be seen as follows. For any state

I, we ave / -r / + 1 -r 0 -r 1 -r 2 -r -r" h' h II . . . . . . " 1, In W IC a the transitions occur with posItIve probabilItIes. Therefore p .. (i + 2) > 0 On the oth h d I i -r i + I -r 1 -r 2 -r ..." . er an, we a so have H ... -r /, In whIch all the transitions are also with positive prob'I't' ab I lIes. ence we have P (i + 1) > 0 B . + '" ii . ut / 1 and i + 2 are relatively prime and ~h~~ I.mphes that the chain is aperiodic. As a consequence of irreducibility and ap~no lClty of the Markov chain, limn-+=P(N':; = J) =P~ ==A . . = 0 eXists. When P="A./(B )< .] 11),/ ,1,2, ... , . .b h Cll 1, all stages are ergodIc. The limiting distribution is then gIVen y t e vector

{Allo,AIlI, ... ,Allc-I ,A,Aw,Aw2 ,

... }

(5.6.1)

where Ifwe now let {llc;m} =

then

s:

A =

C-I /;0

{11e;m;u}dA(u),

1 -

rL lli+--w 1

]-1 (5.6.2)

w is the unique root within the unit circle , Iz I - 1, of the equa.lOn t' and

a(llc(1 - wE» = w

(5.6.3)

MULTICHANNEL BULK QUEUES

271

5.6

THE SYSTEM G/IMBle

e

e+1

e+2

e+ 3

e+B

e+B+I

e+B+2

e + 2B

e+2B+I

0

0 0

0

0 0

0 0

0

0 0 0

0

0 0

0 0 0

0

0

0 0 0

0

0

0

0

0

0

0

0

0 0

0

0

0 0

0 0

0 0 0

0

0 0 0

270 Matrix P of transition probabilities for GI/MB ie

Table 1:

4

2

0

0

0

0

0

0

I 2

{1\1l [2\2J [3\3J

{O\I] [1\2J [2\3J

0 [012J [1\3 J

[0\3J

0 0

0 0

0 0

[e-2IeJ

[e - 31eJ

{e-4IeJ

(O\e)

[cleJ

[e- l\eJ

[ I\e]

e-I

{e- 3\e; l} {e- 31e; I} {e - 31e; I)

{e 4\e; I} {e-4ie; I) {e-4ie; I}

0

e e+ I e+ 2

{el e; l} {cle; I} {e\e; I}

{c - 1\ e; !} {e - I ic; I} {e- I\e; I}

{e 2\e; l} {e- 2\e; I} {e-21e; I)

{I\e; l} (l Ie) {I Ie; I} (I\e) {I Ie; I) (lie)

0 0

0 (O\e)

0

0

0

0

(Ole) 0

0 (O/e)

0

0 0

0

0

0 0 0

0

(O/e)

0 0

0

0 (0 Ie)

0 0

0

0 0 (lIe)

0

0

(Ole)

0 (21 c)

0 0

0

(2 Ie)

0 0 0

0

0

e + 2B

+2

0 0

............. I' =

e+B-I e+B e+B+I

e + 2B I e + 2B c + 2B + I

{elc; l} (e\e; 2) {ele; 2}

{ele; 2} {ele; 3} {cle', 3}

{e- I\e; I} {e - lie; 2} {e-I\c;2}

{e- lie; 2} {e- lie; 3} {e - lie: 3}

(e - 2\e; 1) {e - 2\e; 2} {e- 2\e; 2)

{e-2\e;2} {e-2Ie;3) {e - 2\ e; 3}

{e-3\e; I) {e - 3\e; 2} {e-3I e;2}

{e-4\e; l} {e-4\e;2) {e-4\e;2}

{e-3I e;2} {e - 31e; 3} {e-3Ie;3}

(e-4\e;2) {e-4\e;3} (e- 4 Ic;3)

r

.,.

{lie; 1 } (l\e) {I\e; 2) (2ie) {lie; 2) (2\e)

...

{II e; 2} (2ie) {I Ie; 3} (3 Ie) {lie; 3} (3 Ie)

0,1,2, ... ,c-l

0 (lIe)

(lIe)

0

0 0

(lie) 0

x

(5.6.4)

0 0

(0

( lie)

= {XO,Xl,X2,"'},

f

;=0

0 0

\c)

0

0 0

(Ole)

0 0

0

(0 Ie)

Ixd<=

such that

where

x = xP.

(5.6.8)

It can be seen that the coefficients of the variables in the kth, k;;' c + 1 equation

c

are the same. Thus we may write

L r=j+!

k;;'c+1. j = 0,1,2, ... ,c-l,

Uc

(5.6.5)

(5.6.9)

In order to solve the difference equation (5.6.9), let (5.6.10)

with Substituting equation (5.6.10) reduces equation (5.6.9) to k

1,2, ... , c

(5.6.6)

w = bo +b1wB +b 2 w 2B B

and

aj = iiUJl) =

f'

= K(w )

e-JJ.i u

dA(u),

j = 1,2, ... , c.

(5.6.7)

In order to prove the ergodic property, it is sufficient to prove that there exists a nonzero vector

+ ... (5.6.11)

which is equation (5.6.3). It can now be shown, by branching process theory (see Appendix A.2) or otherwise, that equation (5.6.11) has a real root in J 0, 1 [, iff BJlc/A. > 1. It follows that this real root is the unique root of equation (5.6.11)

MULTICHANNEL BULK QUEUES

272

inside the unit circle, and hence that the Markov chain is ergodic iff p == A/JlCB < 1. It can also be shown by substituting equation (5.6.10) in the (e + 2)th equation (5.6.8) that Xc == 1 so that equation (5.6.10) holds even for k == e. In order that equation (5.6.1 0) hold for all values of k > 0, Xh == w h - c must satisfy the remaining first (c + I) equations of the set of equations (5.6.8). Substitu ting equation (5.6.10) into the equations from the second to the (e + 1)th we obtain e equations with e unknowns, Xo, X I, X2, . . ., XC-I' Since the determinant of the coefficients is [0 I I]. [0 I 2] ... [0 Ie - 1]. [0 Ie] =F 0,' these equations are linearly independent, and consequently we find Xi == Ili, 0 .:;; i < e. Since P is the t.p.m., the row sum is unity, so that the x == {Ilo, 111, . . . , Ilc-l, 1, W, w 2 , ... } obtained in this way will obviously satisfy the first equation of (5.6.8) where not only X=FO, but also ~Ix;\
Finally, we have to prove equation (5.6.4) for Iln 0 .:;; r < e, which may be done as follows. Define

5.6

THE SYSTEM GI/W/c

where we have used x· i-c ';;" Diff . ~rentlating equation (5.6.12) letting z == 1, we have fo~:V= I' ~~ e. , , ... , e 1 that

Ur = a r U,+a r Ur_1+

so that

U]· ==

.!.. djU~z) ] j!

dz

I

,

Ur =

1 ~a

Ur-I r

i=O

I

i W )

i=O

L

_~ (e +r I)" I-a,

:~sbi: :0;::~f(~:s~oprderdlineAar9d)ifferenc~ equations with variable coefficients, which pen

IX

Co

== 1

by lettIng

•

c

XjZi

jt (t-I

XiPij)

zj

Ui ==

Cj{~c -rt+lUr~r-=-a~(kld)(~)C(l_:B)_r-C~1 (C;l)J}, j == 0, 1,2, ... , e - 1.

By use of the identity

Ct j

r=j+l

i=i-l

~)_+(C)= +f Cj

Cr (1-a r )

Cc

f'

C (e+l) ' j

4- -C r-l

r=J+l

r

j = 0,1,2, ... ,e-l

which can easily be proved by using the relation

(c ; 1) = where k(i) == l(i - e + Bt IB] when i > e, and [xl is the greatest integer contained in x. This, after simplification, gives the following integral equation:

s:

C

e(1-w B )-r

Dividing both sides of equation (5.6.13) by C and t h " . to e, and using that U = Xc == 1, we have r, en summIng over r from J + 1

]=0

U(z) ==

r

r=1,2, ... ,e.

II + fo~ ell.± o

(t" (e)

j==0,1,2, ... ,e

Clearly, Uc == Xc == 1. This fact is used in the sequel. Now by using equation (5.6.8), we can see that

rf

+~ I-a,

(5.6.13)

i=O

==

aT

%=1

U(z) ==

XjZi ==

i=O

Uj(z-l)i

c

it

(t"1 Wi) (e)r e(l-wB)-r e(ar-w) _ (e+ 1) r

j=O

and let

U(z)

r times and

where we have used equation (5.6.3), or

c

L

U(z) ==

273

(n c~ J +

and after some Simplification, we obtain equation (5.6.5). Since

(l-e-f.L u +ze-f.L U) U(l-e-f.L U +ze-f.LU)dA(u) by inversion we have Xr

(5.6.12)

=

L (- l)j-r

c j=r

("f ' )

r

Uj ,

r

0,1,2, ... ,e-l

MULTICHANNEL BULK QUEUES

274

5.7

(see Appendix A.IO). This determines 110', 111, ... , Ilc-l' that is, equation (5.6.4). Thus we have completely determined the vector x, the normalization of which gives equation (5.6.1) with the normalization constant A given by equation (5.6.2). For an explicit form of A, see Problems and Complements. An alternative procedure for getting the values of x" r = 0, 1,2, ... ,c - 1, which is based on solving the linear equations containing x" is much simpler, and may be used for oomputational purposes. For details, see Problem 36.

5.7

P(Vq = 0) = A

to

B-1

w~(x) = A

(

L J1.i

;=0

)

J1.c exp [- c)J.X (l - w B )],

Wi

(5.6.14)

x>o.

It is easy to see the result of the first equation of(5.6.14). To prove the second equation, we first recall k(i) = [(i - c + Bt IB] , which may be interpreted as an LV. representing the number of groups awaiting service (including the arrival that has just joined). A customer joining the k(i)th batch must wait for k(i) service completions before his batch enters service. Thus under the condition that the system state is i ;;;. c, that is, k(i) > 0, the conditional density function for k(i) service completions is

(CJ1.X)k(i)-1 fk(i)(X) = CJ1. (k(i)-l)!

e-cp'x

and so the p.d.L of waiting time in the queue is

w~(x) =

OVERVIEW

The contribut~ons of some researchers in multiserver bulk queues, whose work COUld. not be Included In the main text, for reasons given in other chapters are mentIOned here. '

T1n)

c-l

275

. The limiting behaVior of {N,";"} for the system GI/MB /c when B = 1 has been dIscussed by Kendall (1953), of {N,";"} and {N(t)} by Takacs (1962). The transient behaVIOr of {N(t)} has been discussed by Wu Fang (1962) and of {N-} and {N(t)} by Xu Guang·Hui (1965). n

The Waiting·Time (in Queue) Distribution

Suppose that the service discipline is FCFS, and let v~n) be the waiting time in converge to Vq in distribution as n -l> 00. Then the queue of the nth arrivaL Let the distribution of Vq is given by

OVERVIEW

In recen~ years, as we have stated earlier, Marcel Neuts has been lldvocating the use of matnx methods in queues, In the system Gl x /M/c with bulk arrivals Neuts (1979) shows that the distribution of the number in the system is matrix-ge t . rrovlded ~he gro~p size cannot exceed a certain limit k, He also investig~:: ;~c: .Imltlng dlstnbutlOn of the number in the system, both at a random epoch and ImmedIately precedmg an arrival epoch, and the limiting waiting.time distribution MedhI and Borthakur (1972) discuss the steady·state distribution of the numbe; m th~ system f~r the ,two-server queueMIMk,B /2 using Tauberian arguments. Later MedhI (1979) investIgated the waiting.time distrl'butl'on for the MIMk,B I . c-server queue _ c. Numencal values of mean and variance are also discussed for the cases ~- \.~nd c = 2_ R~late? work has been done by Neuts and Nadarajan (1980) on 1M /c by algOrIthmIC methods, and by Cosmetatos (1983) and by Sim and Templeton (1983) onMIMk·~/c by analytic methods. X In the queui.ng system M IDlc with bulk arrivals, Polyak (1968) investigates ~mong o~her thmgs, the distributions of queue length and waiting time, and som: me~uahtles. Oth~r contributors who have done some work on bulk·arrival or bulkservIce queues WIth a finIte or infinite number of servers include Baily and Neuts (1981), Pearce (1965), Murari (1968), and Sharda (1969,1970,1981).

=

L Pi fk(i) (x) i=c

= AcJ1.

B-1 (

~o

)

Wi

exp[-CJ.!X (i-w B )],

O<X<=.

The results of this section are due to Shyu (1960). Shyu (1964) has also given waiting-time distributions for random order of service and for last·come first·served. This author's name may be found with at least three spellings. Xu Guang-Hui, Hsu Kuang-Hui, and Shyu Kwang-Huei all occur in the English.language literature, and all refer to the same author. Roes (1966) has also discussed a system of the GI/M Y /c type using the theory of derived Markov chains due to Cohen (1969). Furthermore, the limiting behavior of {N';-} in Ek/M r Ic has been discussed by Love (1970) using the theory of semi-Markov processes which had previously been applied to M/G'/1 by Fabens (1961) and Fabens and Perera (1963). In both cases exactly r customers are always served together.

PROBLEMS AND COMPLEMENTS Section 5.1 Using equation (5.1.9), show that poet) is given by

P,(tl 2

~ g(1 - '-"'l exp [-t

s:-'-"' X(")A~ + t !n(1-"l)

Using the transformation -..!.In 11

l~] l-z

=X

d"]

PROBLEMS AND COMPLEMENTS

MULTICHANNEL BULK QUEUES

276

The result (1) is due to Sahbazov (1962) and equation (2) is due to Reynolds (1968).

show that P(z; t) of equation (5.1.9) assumes the more convenient form

P(z;t) = g(v)exp [ -

fot (I-A(U))A(t-X) dX1

6

where Show further that in this case

poet) = g(1 - e-J.l.t) exp [3

{I - A (1 - e-J.l.X)} A (t - x) dx

I

=

[A(I )(1)

z=1

f:

A (y)eJ.l.Y dy

+ ll)(I)]

l 7

Show that P(z) of Problem 6 may be put in the form

~

r

X(u) dU) .

+ pZ)i exp [~fZ J.! q+pz X(u) dU]

f X(u) du Clearly rp( 1) = 1, and as rp(z) =

iJ

A

limL(t) = -X(I). t .. ~ J.!

.!. k

~ {l-rp(z)}]

(4)

= k [1 - rp(z)J.

r f: 0

(5)

1

Xru r du - .!. f Xrzr+ r=o - k ~o 7+1

it follows that rp(z) is the p.g.f. of the distribution p = X /kr r = 1 2 3 ... , whence equation (4) follows from equations (3) a~d (5).1, '"

In Problem 6, if X follows a geometric distribution with am = a m-I (1 - a) 1, ~ a < 1, then show thatP(z) = [(1 -az)/(1 -a)]-AIJ.l.a. '

m;;:'

°

Section 5.2

(1)

9

If am = 0 m I show that in the limiting case Po is given by

Po = { (2)

C I

iJ - exP(cpy)}-1 c! (I - p) iJf- 1 (I - y)2 1

I

y=o

c-I (c -k 1) (k + I)! (cp)C-k-l )-1 {c! (1 1- p) :fo = {

L

(3)

rp(z ) -- k- I JZ0 X(U ) duo Equation (4) shows that the 1tIng I:n .dlstnbution of the number of busy servers is a compound Poisson dlstnbutIOn. Solution: To prove equation (4), we can write

8

where p = e-J.l.t, q = 1 -po Then using equation (2), show that the mean number of busy channels is

Further, if t -- 00, then

{~ fX(U)dU} .

rwhere .. k = .JIa.X(u) du , and

Further, if the initial conditions are Pn(O) = On;, then show that

L(t) = E [N(t)IN(O)

= exp

pez) = exp [-

Letting A(t) = A = constant, show from equation (5.1.9) that

P(z; t) = (q

Continuation.

e-J.l.t

Consider the following numerical illustration of the system discussed in Section 5.1. In the time interval [0, TJ let all the hypotheses assumed in that section be satisfied. To further simplify the analysis, assume that each of the arriving groups contains two customers so that a2 = 1, am = 0, m =1= 2, and that the arrival rate of the groups is represented by X(t) = e at where a is some arbitrary parameter, and that at the initial instant the system was empty. Show that under such assumptions the probability that the system is empty at time t is given by 2(e-J.l.t _ eat) e- 2 J.1.t _ eat) , Po(t) = exp [ a + J.! a + 2J.!

P(z; t) = g(v) exp [-

J~l!lP(z; t)

This res~lt confirms the existence of an equilibrium distribution independent of the InitIal state of the queue.

and that the mean number of busy channels is given by 2(e at _ e-J.l.t) L(t) = + ' a J.! 5

Continuation: limiting distribution of N(t) in MX IMloo with ACt) = A and t -- 00. Show from equation (1) or (2) of Problem 5 that

P(z) =

Continuation. From Problem 2 derive the conditional mean number of busy servers for a given initial distribution {P n (0) } and show that it is given by

L (t) == iJP(z; t) iJz

4

f:

277

(cp)C c!(l-p)

+

cf (C )k}-l k=O

P k!

a well-known result for the system MIM/c. Hint: Use equations (5.2.12).

MULTiCHANNEL BULK QUEUES

278

10 If am =

Omk

PROBLEMS AND COMPLEMENTS

and c = 2, then show that 'in the limiting case,

e

(l + pz)Po P(z) = I-pz(1-zk)/(l-z)

(b)

where we have used PI = ("Ajp.)P o , a relation which can be obtained from the first equation of the basic equations (5.2.1) in the limiting ~ase. Show further that, in this case,Po = (l-kp)/(l + p),p being < 1. Also usmg

= "A111,

279

e

e(l _ a)' and Po may be obtained by normalization.

p =

In view of the definition of rn = Z::;"=nam!li, the limiting probability equations for the system MX IMle may be written as

lie

.(n + j-l) .

Pn --

(-1)1

{

J

n-I

-n j=O L Pjrn-i>

1 <;n<;e

n-I

P

.L Pjrn-i,

n~e,

leO

where

n! (n-m)!m!'

m <;n

!hi~ rel~ti~n is true irrespective of the group size distributioll. and helps m slmphfymg certain results. In particular, see the geometric input case discussed in (c) below or the discussion that follows immediately. Hint: Use equations (5.2.18),

and zero otherwise, show that the explicit probabilities are given by

The results of the above problem may be used to determine the probability that a customer chosen randomly from a group of random size is in the nth I;,osition in the system (including those being served). If this probability is P n , then where n

= j + ik + m. In particular verify for k = 2, n = P7 = POp 4 [2 p 3

+

llp2

Pn

7, that

npn lie '

+ 16p + 5].

Hint: Use equations (5.2.13). m 1 11 (a) If the arrivals follow a geometric distribution, that is, if am = (1 - a)a - , m = 1, 2, . , . , 0 <; a < 1, show that Pn satisfies, in the limiting case, the following difference equations:

"APo = p.P 1

{

n

L

am-1Pn_m+ep.Pn+l,

n~e.

m=l

Prove that the solution of this set, subject to p true when p < I, is given by

+ a (1 -

p) < I, which is

I<;n<;c

Pn

n~e.

p'

Furthermore, the second of these probabilities may be used to obtain a simpler derivation of the distribution of V"' since we have P(Vq >t) =

l<;n<e

("A+q()P n = "A(I-a)

= P(nth in system) =

f

P

d=1

)j

d-I (

L

c+d i=O

~e-c}J.t i!

S~bstitution of Pc+d leads to the desired result given in equation (5.2.19). The dlstnbutlOn of Vq , may also be obtained similarly. (c)

With the same assumptions as in (a), show the following: (i) rn = (1 -a)a n - 1 , n = 1,2,.,.

(ii)

(0:) P(Vq, = 0)

1-

Pc (I-a)(l - p)

(I-a)(! _ p) exp [-el1(1 -a)(1 - p)t], where

t>O

MULTiCHANNEL BULK QUEUES

PROBLEMS AND COMPLEMENTS

281

280

an~!~(~)n),

PO P (iii) (a) un = an p(l _ p)'

n;;;" 1.

where

= ( Pc {p+a(l-p)}n-c,

eta, and 1 c-I a n 1'(f3 + n) aC 1'(~ + c) - = 1+ + -----''---'-Po n=l n!1'cm (l-p)(1-a)c!1'(f3)

L

(~) P(Vq, = 0) = Uo = 1- p(l-p)(l-a) (r)

~=

n

l';;;n';;;c

If in Problem 12 we have c = 1, show that

13 Continuation.

P(V >t) = aPe exp[-cp(l-a)t], q, p(l-p)(l-a)

Po = (l-p) t>O

Pn = (l-a)p(1-p)(p+a(l-p)t- 1 ,

This shows that the distribution of Pn has a probability mass concentration at

(0) E(Vq,) = pel _ p)(l -a)2cp (iv)

(a) P(Vq

=

0)

=

1-

n = 0, and is otherwise a geometric distribution expect for a multiplicative constant p. Hence the truncated distribution peN = n IN> 0), n = 1,2, ... , is geometric with parameter (p + a (l - p)).

Pc{p + a(1 - p)} p(l-a)(l _ p)

p + a(l - p) exp [- cp(l-a)(l - p)t], (~) P(Vq > t) = Pc pel -a)(1- p)

t>O p+a(l-p) (r) E(Vq) = Pc p(1-a)2(1-p)2cP'

(d)

Observe that E(Vq) = E(Vq,) + E(Vq,), as it should. A~so note that the truncated distribution P( Vq , > t I Vq , > 0) is exponentIal, as are the truncated distributions of V q " V q • . Show that the distribution of waiting time in the system (queu~ +.semce) for the first customer of a group, where g(y) = p.d.f. of V q " IS glVen by P(system

time>

t) = [1 -P(Vq,

>

0)1 e-

1Lt

+

t:J:=o

The assumption of geometric group size is as restrictive as the exponential assumption for continuous time models. Nevertheless, both these distributions serve as useful first approximations, even in situations where the application of these distributions may seem unrealistic. Once these exact results are known, computations are easier. The computational results for other group size distributions may be compared against those for the geometric distribution. One may then see how the two differ for various parameters. For some work in this connection, see Cromie et al. (1979). 14 If am = Omk in MX/M/c, show that the limiting probabilities Pn for k;;;"c satisfy the following difference equations:

+ n)Pn

= (n

(e

+ c)Pn

= cPn +!,

where

e = A/fl. Solving this set recursively, show that

Hence show that if the group size distribution is geometric, thenP(system time> t) is given by

e- Ilt l1-P(Vq , >0)+

C(l-P)(~ -a)-l

x {cP - P(Vq, > 0) exp{- (cCI - p)(l- a) - 1 )/·.tt}}1 c Hint: Use the arguments of Section 5.5. 12 Continuation of problem 11. If in problem !lea) a =1= 0, show that the results may be expressed in terms of gamma functions as follows:

+ 1)Pn + 1 ,

(e + C)Pk+n = cPn + n +! + epn ,

g(y) dy

x pe- Il (X-Y) dx.

(e

P

n

=

l

1':~e;n~) Po, (0 +c)n-c r(e +c) P c

1'(e)c!

0,

where 1'( . ) is the gamma function. Show that the p.g.f. corresponding to equation (5.2.13) reduces to c-I

P(z) =

L Po n=O

(c - n) I'(e 1'(e)n!

+ n)

zn k-I -

c-ez(l+Z+ ... + Z )

MULTICHANNEL BULK QUEUES

282

from which, by using P(l) = 1 (and the results on gamma functions discussed in Chapter 1), find Po = c! r(8 + 2)(1 - p )/r(8 + c + 1), where p = "Ak/Cf-L < 1. Prove further that for the system iW' /M/e, where k ;;;. e, the mean number of customers L in the system is given by

L = B(e -1) + _kB...'-(k_+_l-"-) B+ 2 2(e - 8k)

[Abol'nikov (1967).]

15 Numerical question.

Let us consider the operation of receiving and checking luggage for passengers arriving at an airport by bus. Suppose that statistical analysis has demonstrated sufficient correspondence between the theoretical model and empirical data (exponential interarrival times and service times). Suppose further that two types of buses transport passengers to the airport type 1 carrying 10 customers each and type 2 carrying 35 each. On the average, 8 buses arrive per hour so that A = 8 per hour, where the probability of an arrival of a bus of type 1 is 0.75 and that of type 2 is 0.25. Thus A(z) = 0.75z 10 + 0.25z 35 • Suppose that e = 3 with Jl = 60 per hour for each server. Under these conditions it is easy to verify that p = J 3/18 < 1. Show that P", n = 0,1,2, ... ,satisfy

-8Po

°

+ 2P1 + P2

= 5/6.

where we have used approximations in computing the coefficients. Show that Lq = 39 customers. If e = 4, then show that Lq = 12, but Po increases to 0.37. 16 This problem is concerned with an alternative derivation of the probabilities Pn for the system MX /M/e without the use of a generating function. Solving recursively the set (5.2.1) for the first few P" in terms of Po and noting that A I = 1, show that the probabilities Pn are connected by the relations "-I

L PI
n = 1,2, ... ,

(6)

12=0

where yen) = A./Jl(n), Jl(n) and defining G(n) = yen)

= Jl min (n, e) andA j =

L;'=iam' Letting C(O) = I

n-\

L A n - k C(k),

n = 1,2, ... ,

(7)

12=0

show that

and after normalization

Po =

[l

G(n)]-1

[Kabak (I970).J

Note that we could have obtained equation (6) by using equation (5.2.18). In applications of queuing theory to telephone systems, two concepts _ loss system and delay system - are frequently used. Systems which have been considered so far are delay systems. In a loss system, no queue is permitted, so that In a loss system with c channels, no more than e customers can join the system. If e channels are already busy when a customer arrives, he leaves with. out service and is said to be lost to the system. Some probabilities concerning these two systems are discussed in the problems that follow. For various other concepts in telephone congestion theory, we refer the reader to Riordan (1962). 17 Verify that equations (6), (7) and (8) of Problem 16 hold for n = 1,2,3, ... , u, where u = c refers to the loss system defined above (which may be denoted X by M /M/e/e) and u = 00 refers to the delay system discussed in Problem 16. Note that in the loss system Po is given by [Kabak (1970).]

Prove that, on the average, the servicing apparatus will be free one·fourth of the operating time. Prove also that the p.g.f. of P", n = 0, 1,2, ... ,is (approxi. mately) P - 6z 3 - 14z2 - 205z + 225 (z) = lO(z 36 + 3z II - 94z + 90)

Pn = yen)

283

o

+ 120P2

- 68P 1 3Po

+ 60P1

PROBLEMS AND COMPLEMENTS

n = 0,1,2, ...

(8)

18 In the case of the system M X /M/c (limiting case), when am = 0mk> there are two interesting cases for which explicit results can be obtained. (a) Show that if am = Omk, then P(Vq>O) = 1- c-I Lmin ~1,e- n ) P n . n=o k (b)

lfa m =omk> where k;;;'c, then

Hint:

(a)

This may be proved on lines similar to the ones which lead to the derivation of equation (5.2.16).

(b)

For this case the following method is more elegant. If a group of size k(;;;' e) arrives to find the state of the system n, ~ n ~ c, then k - e + n arrivals are blocked. However, if it arrives to find all the servers busy, then all the k arrivals are blocked. Consequently the conditional prob. ability of Vq > 0, given that the state of the system is n, is

°

P(V. >01,) ~ (k-;

h n ;;;'e.

MULTICHANNEL BULK QUEUES

284

PROBLEMS AND COMPLEMENTS

Thus the unconditional probability is given by

P(Vq

> 0)

=

c-I

k-c + n

n=O

k

I

k = ~ k

Pn +

+ k- 1 [C-l I

Pn

+ I=

n=c

X

CPn

P(V >0) = q

(c)

k

n-!

-nlJ. j=n-k L Pi'

1 .

X

n-!

- L

CIJ. j=o

Since the last expression within brackets is easily seen to be the average number of busy servers, it is equal to Xk/ IJ. and hence

k-c

n-l

-nlJ. j=o L Pj,.

=

I

n=c

nPn

n=O

X

285

Pj,

clJ. j=n-k

In general show that the result (5.2.16) can be simplified to

Pj,

P

Show that the p.gJ. of the sequence {un' n ;;:. O}, defined by equation (5.2.22), is given by

U(z) =

I=

n=o

+

(R

(z) ) ( c-2 ) unz n = Uo+ - - - r l 1- I Pn Z

%: {R(Z) - ~ riz

i

}

cn Pn/z -

By using U(z), verify equation (5.2.23), and in particular show that E 1 (SP) =

A (2)(1)

--:u;--

= R(l)(I)-I,

[Burke (1975)]

A(2)(l) i i - I E 2 CSP) = - 2 - - - - - Po· a a

20 Continuation of Problem 16.

I

1 '

where Po = c! Jnf=l [CX/IJ.) + iJ. In particular, show that the probability that a group on arrival finds the system full is X/(X + CIJ.) = e/ce + c).

r

21 Continuation of Problem 20: the loss system Mk /M/c/c with k;;:' c. th"

in th" "''',m

P( V,

> 0)

~

1_

i (1 + ;;

Show

[K,b'" (1968).1

Hint: Use arguments of Problem 18 and 'Z~=onPn = Xc/(X + IJ.), which can be obtained by using the first expression for Pn given in Problem 20. For k = 1, the loss system M/M/c/c is known as Erlang's loss system, and the delay system M/M/c/"'" is known as Erlang's delay system. These systems have been completely analyzed; see, for example, Parzen (1962).

n=O

where R (z), the p.gJ. of the sequence {r n, n;;:' I}, is given by R (z) = z(l-A(z))/(l -z)o. (b)

n;;:'k.

= P n-l X/IJ.+i non .+ ,=0

19 (a)

n ;;:'c,

Typically for the loss system Mk /M/c/c, one can see that the jirst two expressions together with 'Z~=oPn = 1 must hold. In the case of the loss systemM k /M/c/c with k;;:' c, show that the solution is given by

P(Vq>O) = (l-ntPn)/P which is easy to evaluate numerically. Hint: This may be done recursively by taking c = 1, 2, ... and using equations (5.2.14) and (5.2.1) in the limiting case. In view of this value of P(Vq > 0), one may conclude that equation (5.2.19) holds even for t = O. For an independent proof of this and some other interesting results, numerical evaluation of certain measures of efficiency, and so on, see Cromie et al. (1979).

n~k

X n-l

- I

X

+-. IJ.

n;;:'c,

If am = 0 mk' then show that

22 The distribution of N(t), the number in the system for MX /M/2(p.1 ,1J.2)' Assume that an arriving customer who finds the system empty goes to the first server with rate IJ.! and probability p and the second server with rate 1J.2 and probability q, where p + q = 1. Define PI (1, 0, t), PI (0, 1, t) as the probability that one unit is in the system and that it is with the first server or second server, respectively. Let PI (t) = PI (1,0, t) + PI (0, 1, t), and let PnCt) be the probability that n units are in the system at time t, n;;:' O. Show that the L.T. of the p.gJ. for P n (t) is given by P(z a) = ,

Zi+l -

(l - Z)[(IJ.I + 1J.2 )Po(a) + z {1J.2 P l (1,0, a) + 1J.1 PI (0, 1, a)}] az-(l-z){(111 +1l2)-AZA(z)}

where PiCO) = 1. Discuss the evaluation of the constants involved and also the cases (a) ar = Orl and (b) IJ.I = 1J.2·

MULTICHANNEL BULK QUEUES

286

of

Section 5.3

287

PROBLEMS AND COMPLEMENTS dc-I

suggests the application of the generalized argument principle with

X(z) as (1 - zr l and ¢(z) as before. Then we have, say,

23 Show that the constants Po, PI , ... , PC-I in the numerator of P(z) given in equation (5.3.4) are uniquely determined. Hint: Write equation (5.3.4) as

11 -

ZC

==

1 L1-z

c-I ;=1

I 2m

= -.

i

P. -

e- b{3(z)

-+'YD

I

d [In ¢(z)]

D1-z

whereD is the contour of the circle, Izl = I

Pi(Zi - ZC)

c-i

P(z)

K

-e, e> O. Now as

_1- d [ln{zC-I(1-z)}] = 2ni(e-1) l-z

since the integrand has a simple pole at z = 0, where

K=

j

Qj =

I

Pi,

r~o.

e -1

+ 2~i

~D 1 ~z d[ln {1-I~c1el:{3(Z)}]-

;=0

Integration by parts yields

The condition P(l -) = 1 gives c-I

I ;=0

In addition,

(9)

Qi

1 P. In [1 -

K = e-l-2ni

I

Q;zj = 0,

(10)

j = 1,2, ... , e - 1.

i=O

Consequently, Equations (9) and (10) determine the constants Qo, ... , QC-I uniquely if the determinant of the coefficients of Qo, ... , QC-I is nonzero, and that it is so can be easily seen. Finally Q's determine P's. 24 If the group size distribution ak, follows a geometric distribution qk-I (1 - q), k ~ 1, 0 < q < 1, then show that

_

-

-

b{3(Z)]

I-liz

D

dz I-z '

-2

The integrand has a pole at z = 1 (order 2). Its residue there is c-I

In de-I

[C e

e-"Abn = . [ ( - >--(l I'" - L q' L j n j=nc q

n=1

q)

nb)

-

Lj_1

(- >--(l - q) nb)] q

where Ln(x) is the Laguerre polynonial, some details of which are given in Problem 9, Chapter 2. 2S For M X IDle in limiting equilibrium, show that the mean numbers in the system and in the queue are given, respectively, by

e(e - 1) - bill (bill - 1) - bll2 2(e-bJJ.l)

_-+-

K = e(e-l)-bJJ.l(bll l -1)-bJJ.2 __1 2(e-bJJ.d 21Ti

One can eliminate the roots Zi, i = I, 2, ... , e - 1, from the expressions for L and consequently for L q . The method used while discussing the evaulation

In [l-z-Ceb{3(Z)]~ I-liz (l-Z)2

'fv,

where DI is the contour of the circle, Iz 1= 1 + 0,0

> O. Now since

In (1 - liz) (1 _ Z)2 dz = 0 by considering the expansion of In (1 - liz) [since Il/zi grating term by term, we can evaluate the integral

< 1 on DI

and inte·

1 ~. In [1 _z-c b{3(z)] - dz -e 2ni D, (I - Z)2'

As discussed in Section 5.3, Iz- c e b{3(z) 1< 1 on D 1 , In [I - z-c e b {3(z)] has a power series expansion in Z-c e b{3(z) which is uniformly convergent onD t , and hence term·by·term integration is permitted. Thus the integral becomes I

=

I

2ni

n=l

n

+- L and

(e - 1).

~ D,

e nb {3(z) z-nc

(1 - z)

2

dz.

The integrand now has poles at z = 1 (order 2) and z = 0 (order ne) with residues n (bJJ.t - e) and L.;~;ol (nc - k) nk(nb), respectively. Finally combining all the results and using the residue theorem (see Appendix A.3), we have

MULTICHANNEL BULK QUEUES

288 =

L = bJ.1.1

+

1

Ln=l n

PROBLEMS AND COMPLEMENTS

nc-l

.

[n(bJ.1.J -e)+

L

an average customer, he must (on the average) be in the middle of the group. As service times are constant, e customers are departing from the system at a time. (We assume that the first batch of m = kc customers finds an empty system.) Consequently the average queuing time of the customer under consideration arising from the delay due to the members of his group is ke-e.J?. = (k-l)b 2 e 2

(ne-k)tTkCnb)J.

k=O

This is easy for computation, but it may be written in a compact form as L

bJ.1.1

+

=

289

1 =

L - I ktTk+nc (nb) n=l n k=l

where we have used

Finally his total average queuing time is the sum of this expression and the expression calculated earlier, that is, 26 (a)

Continuation of Problem 24. The value of Lq can be used to give Wq , the average delay in queue. If the queue discipline is FCFS for the groups but random within the groups, then using little's formula, we have Wq =

L

1

J.1.1

J.1.1

_.9. = -

L= -1 L= ktTk+ncCnb). n=l

n

k=l

When groups arrive in batches of fixed size m, then Wq may be put in a form which is easy for computation. Thus Wq = -

1

I= -1 {nCbJ.1.1- e + ee -r..nb)+ nc-l L

J.1.1 n=1

(b)

n

Wq=(k-l)b+ kbp 2 2(1- p)

If b is added to the above expression, we get the average time the custome.r will spend in the system. Also if k = 1, then Wq = bp/2(1 - p), that IS, the average queuing time of an average customer in M C /D/e is the same as the corresponding quantity in M/D/ 1, as it should be on the basis of intuition, but the average number in the queue in M /D/e will be c times the corresponding number inM/D/l. Section 5.4

} (ne-k)tTk(nb).

k=l

This form with b = 1 was used by Kuczura (1973) to do some numerical work, but he points out that the convergence of the series involved is slow in heavy traffic (when p is near 1). Therefore in the interest of speedy computations it may be necessary to solve for the roots of the denominator of equation (5.3.4). But as, in general, finding the many roots accurately is difficult, it might be better to find the root with minimum modulus and then apply some approximate procedure, such as one of those suggested in Chapter 1. The same remarks apply to the cal· culation of delay probability. Since the service times are fixed (= b) and if groups arrive in batches of fixed size k ;:;. 1, then an interesting phenomenon occurs. One can observe that the average queuing time of the first customer of a group in lIf' /D (with mean b)/l is the same as the average queuing time in a one-server queuing system with single Poisson arrivals and service time kb, M/D (with mean kb )/1, both being equal to kbp/2(1 - p), p = A.kb. The statement remains true if the queuing systemMk /D (with mean b)/1 is replaced by Mkc /D (with mean b)/e, that is if the batch size is an integer mUltiple of the number of channels e, that is, if m = ke. In this case we can get a simple expression for Wq' the mean queuing time of the average customer of the arrival group. For as the customer under consideration is

27 Show that equation (5.4.4) can be put in the form peNn - Y n = m INn = l) =

e-mJ.LCl-cc!

c-m-l

e-J.Li (-l)i

m!

j=O

j!(e-m-j)!(c-m-j)l-C

+ e-cJ.L (CJ.1.)I-C (i-c)!

L

(e ) m

e-cJ.Lcl-C e ! c-m-l

m! /1 (c - m -

k!

n

(-l)j

l-c-l

L k=O L j=O

k

X

(_l)c-m

j!(c-m-j)!

k - l+ c

,m<e
Hint: Use the identity discussed in Chapter 1 to transform the integral into a summation. 28 Prove that ifp = 1 in DS/M/c, then tTo = tTl = tTz = ... =

tT -

s 1

= O.

Section 5.5

29 Show that the root k of equation (5.5.23) which lies outside the unit circle is real and bounded above by min(p-l , 1 + (B P)-1). 30 In the caseB=l inM/MB/e, we may define Pm ,c =Pm+c, m::;"O c ..... o. In P, :;::;::-

MULTICHANNEL BULK QUEUES

290

the limiting case, show that the state probabilities for M/M/c may be deduced fromM/MB /c and are given by

Section 5.6

35 Show that c

A =

Hint:

where

(cp)C ]-1 I -n!- + ----'---'-'--c! (1 - p)

291

PROBLEMS AND COMPLEMENTS

(c) (1-a

L __r __ • ( r=IC r

q

Show further that

1=0

PO,1

P(N~ = l) =

1=0

=

q

a customer

Note from Problem 31 that Little's formula Wq = Lq/A stands verified.

00

A

l-w

bow

B'

= P(H = n).

00

where Lq is defined in Problem 31. It may be noted that when B = 1, E(H) = L q , as it should be.

(

(

Ic-l

)

Ilj + 1 ,

I = 0

i~O AW'

[>0.

Show also that E(N~) =Awl(l- W)2. 38 The sytem M/MB /1. In G/M BIe, if A(u) = 1 - e- u , so that the arrival rate is unity and e = 1, show that the vector x is {Ilo, 1, w, w Z , ... }, where w is a root of the equation IlWB+1 - (11 + 1) w + 1 = 0 and 110 = 11. The normalization of the vector gives the probability distribution

Po

Po,c k A(k-l)2'

34 Define H to be the number of batches awaiting service and let b n Show that nB L E(H) = I nb n = I n I Pk,c = - q o 1 k=(n-I )B+I Bkp

=

A

32 In the limiting case of MIMB Ie, show that the expected number of busy servers is c-l kBp-1 c- I (c-l)Po,1 = Bpc-cPo,c-k_l .

W

1 1 1+----.

L Ili

;=0

r,

[l-E(Nq)+k~

33 From equation (5.5 .27) show that the expectation of the time spends waiting in the queue is given by

1-w

37 Let N~ = (N- - c where N- is the number in the queue plus the number of busy servers in the limiting case. Then N~ represents the number of customers in the queue. Show that the limiting distribution of N~ is

Po,c k Lq = E(Nq ) = (k -1)2

var(Nq ) = E(Nq )

)-1

36 There is an alternative procedure for finding the values of XO, Xl, . . . , X c -l . This is done by solving the second to (e + l)th equations of the set ofeej,uations (5.6.8) recursively, starting with the value of Xc = 1. In particular, show that

X rl

f (~k)r

+ Po,c 7=1

1

+ __

Use equation (5.6.5) and c-l

31 In the limiting case of MIMB Ie, if N q is an LV. representing the number of customers in the queue (excluding those in service, if any), then show that the p.gJ. of N q is given by

r )-r

c(l-wB)-r

r)

Uo = U(1) = 1 +

and p = "/ell. For independent proofs, see Saaty (1961) and Jackson and Henderson (1966).

±

I

1=0

c-\ (cp)n n=o

PN (z) =

(1-a

C(Bi:,lW i )

=

[1

Pn_ = -1 11

+; (1 W

n-l

P-

0,

W)-lr n;;;'

1.

Remark: It is interesting to see that the same result is obtained for the system MIMB II as in Section 5.5, although the processes considered in the two sections are different. It may, however, be noted that the state of the system in the present section and in Section 5.5 is defined in the same manner, although the notation used is different. 39 The system M/MI2. If in a GIIMB Ie system, C = 2, B = 1, and A(u) = 1 e- u , then show that the vector x is {Ilo, Ill, 1, W, w 2 , .. , }, where w = p == 1/211

REFERENCES

MULTICHANNEL BULK QUEUES

293

292

and 110

= 1/2p2, 111 = lip. n

The normalization of x gives

Po = (1 -

p)/(l

+ p)

o

andr=2p p n;;;'1. 1 f fth' Thne same re~ark applies here as in problem 3~. For a genera proo 0 IS for the system MIMic the reader is referred to Takacs (1962).

40 (a)

(b)

Show that

Show that W~ may also be obtained from

-

wq

~

= L. CJl i=c

(o-c + Bt1 B

r;I

where [x] is the largest integer less than or equal to x. 41 (a) For the single-server system CliMB /1, show that

Po _

Pn =

(b)

1 - w - (1 - wB)(l -ii(I1)) = 1 _ _ _W_B_ii-"i:(Jl~)_ B ii(Jl) + w B - w 11(11) + w - W 11(Jl)(1 ii(Jl)

w) w

n+B-1

n ;;;;. 1

' + wB - w

where ii(l1) = Je-I'u dA(u) and a(l1(l - wB)) = W. - 1 - e- u the results in (a) reduce to the results of Show t h at f or A() uProblem 38.

REFERENCES Abol'nikov, L.M. (1967). A multichannel queuing system with group arrival of demands. Eng.

Cybern. 4 , 3 9 - 4 8 . . .h infinite number of _ _ _ _ (1968). A non stationary queumg problem for a sys~em Wit channels for a group arrival of requests. Problemy Peredaci Informacll 4,99-102. English . translation in Probl. Inform. Transm. (1968),4,82-85. _ _ _ _ (1970). Transient regime in the system MX/M/n with nonordinary entenng flow.

m:.

.

Eng. Cybern. 5,881-885. . 'd k' (1972) Investigation of many channel nonstatlOnary _ _ _ _ and R.M. Yasnogo n s I~ . • ern 10 636-642. Markov systems with non-ordmary mput flow. Eng. Cyb . , Burke, P.J. (1975). See Chapter 3. . . C h J W (1969). The single server queue. North-Holland, Amsterdam. o en, . . d D V H' kJe (1970). Some properties of multi-server queues With appomt.. m y Cox, D.R., an ments.J. R. Stat. Soc. Ser. A 133, 1-13.

Cromie, M.V.(1974). Measures of efficiency for the bulk-service queue M/MX/c/= and the bulk-arrival queue MX/M/c/= with computing algorithms, tables and charts, etc. M.Sc. thesis, Royal Military College of Canada, Kingston, Ontario. - - - - a n d M. L. Chaudhry (1975). Further discussion of the results for the queuing system MX/M/c. Presented at the joint national meeting of ORSA and TIMS, Las Vegas, November 17-19. - - - - (1976). Analytically explicit results for the queuing system M/MX/c with charts and tables for certain measures of efficiency. Oper. Res. Quart. 27, 733-745. - - - - a n d W.K. Grassmann (1979). Further results for the queuing system MX/M/c. 1. Oper. Res. Soc. 30,755-763. Crommelin, C.D. (1932). Delay probability formulae when the holding times are constant. P.O. Elect. Eng. J. 25,41-50. - - - - (1934) Delay probability formulae. P.O. Elect. Eng. J. 26,266-274. Erlang, AX. (1920). See The life and works of A.K. Eriang, (1948) by Brocfmeyer, E., H.L. Halstr¢m and A. Jensen, Copenhagen. Fabens, A.J. (1961). The solution of queuing and inventorY models by semi-Markov processes. J. R. Stat. Soc. Ser. B 23, 113-127. -~--and A.G.A.D. Perera (1963). A correction to Fabens (1961).1. R. Stat. Soc. Ser. B 25,455-456. Ghare, P.M. (1968). Multichannel queuing system with bulk service. Oper. Res. 16, 189-192. Grassmann, W.K. (1974). The steady state behaviour of theM/Ek/l queue with state dependent arrival rates. INFOR 12,163-173. Hawkes, A.G. (1965). Time-dependent solution of a priority queue with bulk arrival. Oper. Res. 3,586-595. Holman, D.F. (1977). Some problems in the theory of bulk queues. M.sc. thesis, Royal Military College of Canada, Kingston, Ontario. - - - - " M.L. Chaudhry, and W.K. Grassmann (1980). Some results of the queuing system Elf/M/c. Naval Res. Logist. Quart. 27,217-222. Jackson, R.R.P., and J.C. Henderson (1966). The time-dependent solution to the many server Poisson queue. Oper. Res. 14,720-722. Jordan, C. (1965). Calculus offinite differences, 3rd edition. Chelsea, Ne\\ York. Kabak, LW. (1968). Blocking and delays in M(n)/M/c bulk queuing systems. Oper. Res. 16, 830-840. ----(1970) Blocking and delays in Alx)/M/c bulk arrival queuing systems. Manage. Sci. 17,112-115. Kendall, D.G. (1953). Stochastic processes occurring in the theory of queues and their analysis by the method of the imbedded Markov chain. Ann. Math. Stat. 24, 338-354. Kuczura, A. (197 3). Batch input to a multi-server queue with constant service times. Bell Syst. Techn. J. 52,83-99. Love, R.F. (1970). Steady-state solution of the queuing system EwiM/s with batch service. Oper. Res. 18,160-171. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Pollaczek, F. (1930a). Uber eine Aufgabe der Wahrscheinlichkeitstheorie 1. Math. Z. 32,64100. ----(1930b). Uber eine Aufgabe der Wahrscheinlichkeitstheorie II. Math. Z. 32, 729750. Reynolds, J.F. (1968). Some results for the bulk-arrival inrmite-server Poisson queue. Oper. Res. 16,186-189.

294

MULTICHANNEL BULK QUEUES

Riordan, J. (1962). Stochastic service systems. Wiley, New York. Roes, P.B.M. (1966). A many server bulk queue. Oper. Res. 14,1037-1044. Saaty, T.L. (1961). Elements of queuing theory with applications. McGraw-Hill, New York. Sahbazov, A.A. (1962). Some problems of queuing theory (in Russian). Author's abstract of dissertation submitted for the degree of Candidate of Phys. Math. Sciences. Shanbhag, D.N. (1966). On infmite server queues with batch arrivals. J. Appl. Frob. 3,274279. Shyu Kwang-Heui (1960). On the queuing processes in the system GI/M/n with bulk-service. Acta Math. Sinica 10, 182-189. English translation in Chinese Math. (1960) 1, 196204. - - - - (1964). See Xu Guang-Hui (1964). Takacs, L. (1962). Introduction to the theory of queues. Oxford Univ. Press. New York. Wu Fang (1962). On the queueing process GI/M/n. Sci. Sinica ll, 1169-1182. Xu Guang-Hui (1964). The waiting time distribution for the queuing processes in the system GI/M/n with bulk-service. Acta Math. Sinica 14,796-808. English translation in Chinese Math. (1965) 6,195-207. - - - - (1965). The transient behaviour of the queuing processes GI/M/n. Acta Math. Sinica. 15,91-120. English translation in Chinese Math. (1965) 6, 393-421.

Chapter 6

RELATIONS AMONG QUEUING SYSTEMS This chapter seeks to give a unified discussion of several queuing system~ by considering some common features of those systems. In particular, it deals with:

Additional References Baily, D.E. and M.F. Neuts (1981). Alorithnric methods for multi-server queues with group arrivals and exponential services. Eur. J. Oper. Res. 8,184-196. Cosmetatos, G.P. (983). Closed-form equilibrium results for theM/MCY.,=/n queue. To appear in Eur. J. Oper. Res. Medhi, J. (1979). Further results in a Poisson queue under a general bulk service rule. Cah. Cen. Etud. Rech. Oper. 21,183-189. - - - - a n d A. Borthakur (1972). On a two server Markovian queue with a general bulk service rule. Cah. Cent. Etud. Rec. Oper. 14,151-158. Murad, K. (1968). An additional special channel,limited space queuing problem with service in batches of variable size. Oper. Res. 16,83-90. Neuts, M.F. (1979). An algorithmic solution to the GI/M/c queue with group arrivals. Cah. Cen. Etud. Rech. Oper. 21,109-119. - - - - and R. Nadarajan (1982). A multiserver queue with thresholds for the acceptance of customers into service. Oper. Res. 30, 948-960. Pearce, C. (1965). A queuing system with non-recurrent input and batch servicing. J. Appl. Prob. 2,442-448. Polyak, D.G. (1968). Some problems in the theory of multi-channel queuing systems with constant service time. Eng. Cybern. 2,86-95. Sharda (1969). A many server queuing problem with variable departures. Z. Angew. Math. Mech. 49, 163-166. - - - - (1970). A discrete time limited space queuing problem with batch arrivals forming a Markov chain and service in M parallel channels in batches of variable size. Cah. Cen. Etud. Rech. Oper. 12,178-194. ----(1981). A limited space correlated queuing problem with departures in batches of variable size. Cah. Cen. Etud. Rech. Oper. 23,87-96. Sim, S.H. and J.G.c. Templeton (1983). Computational procedures for steady-state characteristics of unscheduled multi-carrier shuttle systems. To appear in Eur. J. Oper. Res.

2

Relations among three different steady-state p.g.f.'s for the number in the system considered at different epochs: P-(z) is the p.gJ. just before an arrival epoch, P(z) the p.g.f. at a randomly chosen epoch, and P+(z) the p.g.f. just after a departure epoch. Formulas connecting these three p.g.f.'s are found for the queuing systems GrIM/! and MIGr/l with arrival or service in groups of exact size r. For systems in which all three p.gJ.'s P-(z), P(z), andP\z) are known, we may wish to use any or all of them for a particular purpose. For example, an arriving customer may be interested mainly in the p.g.f. P-(z) at an arrival epoch and in the corresponding waiting-time distribution. In other systems we may know exact relations among P-(z), P(z), andP+(z) without knOWing a formula giving any of the three p.gJ.'s in terms of the interarrival and service-time distributions. Either P-(z) or P+Cz) may serve as an approximation to P(z) in queuing systems in which P(z) is difficult to obtain. For convenience, where the context makes it clear which p.gJ. is intended, we can use the term "queue size p.gJ." for anyone of P-(z), P(z), and P+(z). Relations between bulk-arrival and unit-arrival systems, or between bulkservice and unit-service systems.

Several authors have considered some of the above properties. Cox and Smith (1967, ex. 2, 3, p. 166) state that the limiting distribUtion of queue size considered at an arbitrary instant for M r IE/I is eqUivalent to the limiting distribution of the number of phases considered at the corresponding instant for M/Er/l. [For explicit analytical details of this, see Gross and Harris (I974).J A more general statement in the transient case (continuous time), the equivalence of the distribution of the number in the system M X /EI /1 and phases in the system M/Ex/l, where X is an LV., has been discussed in Chapter 3. Brockwell (I963) discusses the relation of r M /E 1/1 to M/Er/l in the transient case (imbedded Markov chain). Some numerical work on a particular case of MY/EI/I, namely, for M/E2/1 and consequently for M2/EI/l, has been carried out by Kerridge (1966). Finch (1962) discusses the 295

RELATIONS AMONG QUEUING SYSTEMS

296

relation of M/Mr/l/finite to E r /M/l/finite in: the transient case (imbedded Markov chain). Khintchine (1932) proved that the limiting distributions of queue sizes considered at an arbitrary instant and just after a departure for M/G/l are the same. For some more general cases, the reader is referred to Takacs (1962). A unified approach in the limiting case for single-server bulk queues was made in Foster (1961,1964), Foster and Nyunt (1961), and Foster and Perera (1964, 1965). These authors and, more recently, Chaudhry (1978,1979), Chaudhry and Templeton (1981), and Holman et al. (1981) give results showing relations between different queuing systems and between different limiting solutions (random times, before arrivals, after departures) for the same queuing system. Since the results of the last four papers have already been discussed in Chapters 3 and 4, we consider here the results of the five papers cited above by Foster and his collaborators. We also try to derive some of the results of the earlier authors from the work of Foster, Nyunt, and Perera. It is shown, for example, that the limiting distribution of the number considered just before an arrival instant for the system GI"/E I !1 is equivalent to the limiting distribution of the number of phases considered just before the corresponding arrival instant for the system GI/Er/l. In passing, the reader is reminded of the notational equivalence of E1 ,E, and M. In what follows, we use El == E == M indiscriminately.

6.1

THE SYSTEM Gl r /E/1

Queue Size just before an Arrival instant

Define the probability lim peN;; = j)

Pj -

n-=

lim P(N(o~ - 0) = j).

n-=

It has been shown in Chapter 3 that the p.gJ. P-(z) of the limiting probabilities Pj - is given by equation (3.2.3). Queue Size just after a Departure Instant ~

Let

01,02, . . .

be the instants of departure of customers from the system. Define

P/(n)

= P(N~ = j)

= P(N(on

+ 0) = j).

Since the customers arrive in groups of size r, P/(n) is nonzero only when j + n, the number of arrivals up to the instant of the nth departure, is a multiple of r, that is,

P/(n)

6.1

297

THE SYSTEM Gl r /E/1

{

- 0 - , 0,

*

if

j+n

* sr,

if

j

+n

= sr,

s=I,2,3, ....

(6.1.1)

Equation (6.1.1) shows that the ordinary limit limn -+= P/(n) does not exist. Let us then consider the Cesaro limit, The limiting Distribution of Queue Size

We start out by considering the system GI" /E/l. In Chapter 3 we discussed the p.gJ. P-(z) of the limiting probabilities of queue size considered just before an arrival instant. As in Chapter 3, batches of customers of fixed size r arrive at the I I , h h h' . I t' / sequence of instants 0 = 00,01, . . . , on, ... , suc t at t e mterarnva Imes 0n+ I o~ > 0, n = 0,1,2, ... , are Li.d. LV.'S with common dJ. A(u). The customers are served individually by a singJe server with an exponential service time distribution of mean 1/11. It will be convenient to take the mean of A(u) as klA so that when the interarriva! times have the Ek distribution, I/A would represent the mean of each of the k phases. Further, define 7 as the ratio of mean service time to mean interarrival time so that 7 = A/kll, and p = rr = AT/kll. Assume that p < 1; this is a necessary condition for the existence of the limits to be considered next. In this section we use techniques used in renewal theory to derive steady-state distributions of N (the number of customers in the system at a random time), N(the number of customers just before an arrival), and N+ (the number of customers just after a departure), all for the system GF /E/l. A supplementary-variable derivation of the steady-state distribution of N, and related results not accessible by the techniques of this section, are given in Miscellaneous Problem 1.

P/

=

lim

n -+00

[.!.n

±p/(m)].

m=l

(6.1.2)

Again by virtue of equation (6.1.1), one can see that this Cesaro limit, if it exists, is also the Cesaro limit of the sequence 1

+

-; Pj (sr - j),

s=1,2,3, ....

It is shown later that the ordinary limit of this new sequence exists. It then follows that the Cesaro limit (6.1.2) exists. Let us now write 1

P/ = - lim P/(sr - j) r S-+!XI

(6.1.3)

and define P+(z) =

f

p/zi.

j=O

For details of the existence of the Cesaro limit when an ordinary limit exists, we refer the reader to Parzen (1962).

RELATIONS AMONG QUEUING SYSTEMS

298

Queue Size at an Arbitrary Instant

Let

(6.1.4)

Pj(t) = P(N(t) = j)

and Pj =

(6.1.5)

lim Pi(t).

6.1

THE SYSTEM G/ r IEI1

299

to finite limits as n -+ 00. Consequently, the limit of the left-hand side of equation (6.1.10) exists when n -+ 00, and by taking s = 1,2, ... recursively in equation (6.1.10), one can see that the limits of the individual probabilities in the summation on the left·hand side of equation (6.1.10) also exist. Thus we see that the limit given in equation (6.1.3) exists for all j. From equations (6.1.9) and (6.1.10) after taking limits when n -H'O and using equation (6.1.3), we have, respectively,

t--=

Define

P/ =

=

I

P(z) =

PjZi.

Lto /r,

j


(6.1.11)

[Jo -ll

j ;;;, r.

(6.1.12)

P i-]

and

j=O

j~O P/- ir

[For existence of the limits (6.1.5), see Problems and Complements.] Relationships among the Functions P-(z), P+(zl. and P(z)

=

Pi

r,

Replacing j by j - r in equation (6.1.12) and subtracting, we get (since P/-(s + I)T = 0 in view of the relationj - sr = m < r)

We establish that for GI"/E/I rcz) P(z)

r(l---z) P +() z 1 _zr

(6.1.6)

1 - p + pzP+(z)

(6.1.7)

1 _zr

I-p

+ pz - - - r(z).

(6.1.8)

r(1- z)

Thus if one of P-(z), P(z), and P+(z) is known, the others can easily be determined. To prove equation (6.l.6), we consider separately the two cases j ~ r - 1 and j ;;. r. If j ~ r - I , then N(anr-j

+ 0)

= j

¢>

anr-j

<

a~+l

¢>

N(a~+l - 0) ~ j.

Thus j

p/(nr-j) = P(anr-j

<

I

a~+d =

Pi(n

+ 1).

(6.1.9)

i=O

However, if j ;;. r, say, j = sr + m, where s is a positive integer and m is an integer


+ 0)

= ir

+ m,

j ;;. r.

The result (6.1.6) follows from equations (6.1.11) and (6.1.13) on taking generating functions. Foster also discusses equation (6.1.6) using heuristic arguments which make no assumptions about the input or service distributions, and hence concludes that equation (6.1.6) should hold even for GI" /G/l. To prove equation (6.1.7), we first give heuristic reasonings based on the forgetfulness property of the exponential distribution before giving more sophisticated arguments. Since the service times are exponentially distributed, we can consider the server as generating a sequence of potential departure points constituting a Poisson process, a potential departure being a real departure if there is at least one customer in the system; otherwise it is virtual. In other words, services (real or virtual) end at random. Thus an instant chosen at random may be considered as a service end point (real or virtual). With the traffic intensity p, the probability that no customers are present at a random instant is 1 - p. The service end point in this case is virtual and

Po = I-p.

Pj = pP/- 1 , s

i=O

Ftr+m(nr-j) = P(anr-j

< a~+d

j

=

I pi-en + 1). ;=0

(6.1.14)

On the other hand, if the system is not empty (with probability p), the random instant chosen is a real service end point, and the queue size is about to be reduced by unity. Consequently,

i = 0, 1,2, ... , s

which gives

I

(6.l.13)

(6.1.10)

--=

If p < 1, it can be shown that the limiting probabilities Pj- = limn Pj-en) exist. It follows that the right-hand sides of equations (6.1.9) and (6.1.10) converge

(6.1.15)

Taking the p.gJ. of equations (6.1.14) and (6.1.15) gives equation (6.1.7), and then equation (6.1.8) follows by using equation (6.1.6). An independent rigorous proof of equation (6.1.8) is now given. Pj, which is the limiting probability that the queue size isj at an arbitrary instant of time, can now be related to the limiting queue size probability just before the

RELATIONS AMONG QUEUING SYSTEMS

300

S.1

THE SYSTEM Gl r /E/1

301

preceding instant of arrival, if the distribution of the time interval Y bet:veen th.ese two instants of time is known. The limiting distribution of the elapsed mterarnval time Y then follows from renewal theory (see Chapter 2) and is given by P(y

<

Y ,;;; y

r=

-0

1 -z

r(z).

k

provided A(y) is a nonlattice distribution. Let D* be the number of departures (real or virtual) during the time interval Y and define gj = P(D* = j) so that the p.gJ. of the sequence {gj} is G(z) = L::j=o gjzj. It then follows that gj =

1 _zr TZ - -

P(l-) = 1 gives C and consequently the result (6.1.8).

A - (l - A(y)) dy

+ dy)

P(z) = C+

As an example, let us consider the system E'kIEI/1 [Problem 7(b) of Section 3.2]. Here the interarrival time distribution is k.Erlang with

P(D* = jlY = y)0.0- A (Y»)dy

k

and mean kiA. In this case we have proved (see Section 3.2) that j = 0,1,2, ... r(z) =

and therefore

0 I -'Yzz -'Yz 1

(6.1.17)

1

G(z)

1 -K(z)

(6.1.16)

=T~-~

l-z

where

T

= A/(kJ.t) and

'Yl, 'Yz,

where ZT

K(z) = a(J.t(l - z»)

... ,'Yr are the roots inside Iz 1= 1 of the equation

=

K(z)

and

Now Nand N- being the r.v.'s representing the limiting queue sizes at the arbitrary chosen instant and just before the preceding arrival instant of time, respectively, we have N = (N- + r - D*t in distribution. Since Y and D* are independent of N-, we have, on taking the generating functions and proceeding as in Chapter 3, P(z) = z" r(z)G(z-l)

+

=

I

d;'O -

z-m)

=

1 -z )-k (1. +Tr

(6.1.18)

[In comparing results in this section with the corresponding results in Section 3.2, note that the mean interarrival time has been taken as l/A there and as klA here. Thus T is the ratio of mean service time to mean interarrival time in each case, and equation (6.1.18) is unambiguous, since it is given in terms of T and not A.J It can be shown (see Problems and Complements) that the roots of equation (6.1.18), and in particular the 'Y1'S, are all distinct. Consequently we may write equation (6.1.17), using the method of partial fractions, as

m=O

where (6.1.19)

d;;' =P(N-+r-D* = -m).

Use equation (6.1.16) and then eliminate K(Z-l) by using equation (3.2.7) to get, on simplification, P(z) =

TZ

l-z" - - r(z) l-z

+ T:Eb m

l-z-

1-

m

*

+ L::dm(l-z

-m

where

~ l-'Yz

).

[Recall from Section 3.2 that b m = P(N- + r - D = -m), where D is the number of real or potential departures during a complete interarrival period.] Now since the left·hand side of this equation is a power series with nonnegative powers, so must be the right·hand side. Consequently all terms on the right.hand side with negative powers must cancel out, leaving at the most a constant C, say. Thus

=

n

l-'Yi , i*zl-'Yi/'YI

1,2, ... ,r.

i=1

Thus comparing the coefficients of zi on both sides of equation (6.1.19), we have T

Pj- =

I

cI'YL

j;;;' 0.

Z=1

From equations (6.1.11), (6.1.13), and (6.1.20) we obtain

(6.1.20)

RELA nONS AMONG QUEUING SYSTEMS

302

j

<

r (6.1.21)

p:
,

"

l'

- L... r 1=1

Czrlj-r+l --rz -, 1 -rz

j ;;;.

r

and from equations (6.1.14), (6.1.15), and (6.1.21) we obtain j = 0

p,

T

,

1=1

It m~y ~e ~marked here that Wishart's equation (in our notation) giving the roots bz! IS z =a [r(1- z)/b I , w~ere b is the ~ean service time of a customer in GIIE,/!. ThIs confo:ms to Our equatIOn (3.2.4) If and only if b = rlJ.(, the mean service time of a group III GI'/Ell (see discussion later). We. ne.xt s~ow that the distribution of Nt given in equation (6.2.1) is the same as the dls~nbutl~n of N- in GI'/E1 /l given in equation (3.2.3) by means of some algeb~aIc mampulation of the roots {rz} of the equation K(z) = zr. First consider equatIOn (3.2.3), which is repeated here as equation (6.2.4):

j_,

C1rl

1-

ri

---, I-rl

r(z) =

r

pi = peNt = i) = K

=( i

1=1

I on!,

j ;;;.

0

(6.2.1)

1=1 al

£~ 1 -zrz

r

1-rd

1 = 1,2, ... , r.

Now using equation (6.2.3), we can write, say, C

'

az

1

exp [-J.lU(I-z)] dA(u)

(6.2.2)

= A.

Consequently from equation (6.2.5), r

Pi- =

I cat

r

= A

I aat,

i ;;;. O.

1=1

1

I

1,

Cl'.z

1=1 r

K

I 1=1

Po

r

I Cz = 1=1

aZrl = I-po l-rz

I~=l

[see Section 3.2, in particular, equation (3.2.4)] , and I = 1,2, ... , r.

rz)

r

service) just before an arrival instant, rl, 1= 1,2, ... ,r, are the r roots (distinct or coincident) of the equation

r

(6.2.5)

where

(6.2.6)

~omparing equations (6..:2.6) and (6.2.1) and applying the normalizing condition gIves A/K = 1. Hence Pi for the system GrIE/I is equivalent to pi for the system GIIE,/!. We also have the following relations, which are often useful in problems:

pi is the probability that there are i phases in the system (including those, if any, in

z' = il[J.(l -z)} =

(6.2.4)

1=1

- z = n (I -

The results for the system GI/E,/l can be derived from those for the system Glr/E/1. Wishart (1956) has shown that for the system GI/Er/l

K

1 -rz . l-zrz

For convenience we suppose that the roots rl, 1= 1, 2, ... , r, are all 'Ciistinct. If one o.r more of the roots are repeated, the analysis proceeds on similar lines. Now equatIon (6.2.4) can be rewritten, using partial fractions, as

j ;;;. r.

THE SYSTEM GI/E,/1

where

n I

(6.1.22)

Once we have discussed the relationships among P-(z), P+Cz), and P(z) for the system GI'/E/l, several other interesting properties follow. First we show that the number N- in the system GI'/E/1 has the same distribution as the number Nt of phases in the system GI/E,/ 1. This use of the notation Nt in this section and in Section 6.2 is not to be confused with the earlier use of N;; = N«(J~ - 0) to denote the number in the system just before the arrival epoch (J~. Further properties of the system GI'/E/1 are considered in Section 6.2 in connection with the relations established there between the systems GI'/E/l and GI/E,/l.

6.2

303

r(z) =

C1 - - - ' l -rz 1=1

I

THE SYSTEM GI/£,/1

l-iz

I

,

T

6.2

1=1 1 - r l (6.2.3)

[use rei) = 1 in equation (6.1.19) or otherwise J .

K

304

RELATIONS AMONG QUEUING SYSTEMS

For an independent derivation of pi one may see Wishart (1956). Once the limiting distribution {pi} of the number ofphases in the system GI/Er /1 is known, one can derive the limiting distributions of the number of customers in the system or in the queue, the limiting waiting-time distribution, measures of efficiency, and so on, (see Problems and Complements). The equivalence between customers in Gr/E!/! and phases in GI/Er/l holds even in the case of transient distributions of queue sizes, provided that the two systems start with the same initial conditions (number of customers in Gr/E!/! and phases in GI/Er/ I). The transient distributions of queue sizes (number in the system Gr/E l /1 and phases in the system GI/Er/l) have been independently discussed by Jain and Rani (1971) and by Wu Fang (1960). Since both used the supplementary variable technique, their equations representing the two systems in continuous time can easily be seen to be equivalent if the initial states (number in Gr/E l /] and phases in GI/Er/I) are the same. The above remarks hold also for the distributions of waiting times, busy periods, and so on, for the two systems. (For some explicit results on waiting times, see Problems and Complements.) To see the equivalence in the case of busy periods, the reader is referred to Takacs (1962) or Conolly (1960) for the queuing system Gr /M/I, and to Prabhu (1965) for the queuing system GI/Er /!. In closing this section it may be remarked that the results considering instants just before arrival for the system GI/E,/I (or for Gr /E/I) can also be derived from those for the system ErIG/! (or for E/Gr/I). For such derivations the reader is referred to Prabhu (1965). In this case each system is called the dual of the other, for the reason that the distributions of interarrival and service times are interchanged.

6.3

RELATiONSHIP BETWEEN GI/Er/1 AND GV/E!1

We showed in Section 6.2 that the number of customers in the system GriE/l can be identified with the number of phases in a GI/Er/l system such that the service-time distribution of a group in GI' /E/1 is the same as the service-time distribution of a customer in GIIEr/l. In other words, suppose that in the group arrival system Gr/E/! the groups retain their identity, in that a group is regarded as being present in the system until all its members have departed. The total service-time distribution of a group will have an Erlangian distribution E r • The resulting system, in which the groups are now to be treated as the individual customers, is a GI/Er/l queue. Thus if 01,02, ... are the departure instants in' GI'IE/!, then Or, 02r, ... are the departure instants in GIIEr/l, the instants of arrivals being the same in both the systems. Now if Ht) represents the number of customers (not groups) in the system GIIEr/l (including the customer under service, if any) at time t, we have w) =

[N(t) : r -1]

6.3

RELATIONSHIP BETWEEN GIIE,/1 AND GI'IEI1

wher.e [~] denotes the integral part of x. We shall consider the system length dlstnbutlOns of GIIEr/l at the three instants of time and their relations to the corresponding distributions of GrIE/I. Let us now define for GI/E,/l:

2

qj-(n) = P[Ho~ - 0) = j] qj (t) = P[W) = j].

3

q/(n)

= P[Honr + 0) =

jj

=

P(~;;

=

P(~~r

=

j).

=

j).

Case 1

From equation (6.3.1) we get at the instant t =

0' n

0, j = 0 j

>

O.

Since the probabilities on the right-hand side of the above equations converge as .n ...,. 00, so do the ones on the left-hand side. Consequently q.- = lim q:-( ) eXIsts and J n ->-= J n j = 0

j

>

O.

(6.3.2)

Let us define q -(z) = L~=O q·-zi and put R·- ="'<'f p.- d T:- _ ",i h J J J "'-"=0, an J - "'"'i=O qi so t at Ri and Tj are the d.f.'s corresponding to the two distributions involved. It is easy to show that To- = Po- , T j- = R r- ) . . . , T·and J = R·Jr

r(z)

(6.3.3)

1 -z

~ T.-

j

_

q-(z)

L..,JZ---· j=O I -

z

But ~n vie~ of equation (6.3.3), the left-hand side of which is a power series, we may wnte, uSing complex variable theory (see Appendix A.3), R-:- = _1 rf,

(6.3.1)

305

1

rev) dv

21Ti'Yc (I-v)vi+l'

Here C is a contour, traversed in the positive (counterclockwise) sense around the

306

RELATIONS AMONG QUEUING SYSTEMS

origin, excluding the poles of P-(z)/(l - z) artd the point i = q-(z) l-z

=

f

=

RTrzi

i=O

or q-(z) = I - z 2ni

_1_ 2ni

RELATIONSHIP BETWEEN GIIE,./1 AND Gl r lEI1

A. Thus

r

L j=!

I


6.3

r(v)cv-rzY dv c i=Q (l-v)v

Xj

307

r

l-z[1

+ p(l-z)/rj"

Therefore

¢,

(6.3.6)

rev)-dv '--c(l-v)v(1-v- rz)

Since the poles of the integrand within C are at wi zl/r, j = 1, 2, ... , r, where wi is an rth root of unity, the residue at Wiz llr is .!. p-(Wiz llr ) r 1 - w i z1!r

It is interesti~g t~ see that the p.gJ. q-(z) of the limiting queue size considered just befo.re an arnval I~stant for the system M/Er/l agrees with the corresponding p.g.f. .[WhICh can be denved from equation (3.1.19)J which is considered at an arbitrary mstant. Case 2

Therefore by the residue theorem (see Appendix A.3) we have I -z

_

q (z) = - r -

r

2:

j=l

Again from equation (6.3.1) we get, at the instant t,

p-(wiz llr ) 1 izl/r

(6.3.4)

-w

j = 0

where P-(') is given by equation (3.2.3). Equation (6.3.4) expresses the p.g.f. of queue size for the system GI/Er/l in terms of the corresponding p.gJ. for the system Gr/E/l. As an example, let us consider the case of the system M/Er/l in which the customers arrive singly. In this case, the mean arrival rate being A and the mean service time riM, the traffic intensity p = },r/M remains the same as in the case of M r /E/l. Using equation (3.2.1 0), we have

j>

o.

On the ba.sis of convergence of the probabilities [which do converge when p < I a.nd A(u) IS nonlatticeJ on the right·hand side, the probabilities on the left.hand SIde converge under the same restrictions, and consequently

qi = t-+oo lim q·(t) J ,

j = 0, 1,2, ...

1/r

. (1 - p)(l - WiZ ) p-(W1Zllr) = _--"-.....,.-;-'c...c::-=-_=----'=--_ I - wlz llr [l + (p/r)(l - z)]

exists, and we can write j = 0

where we have used (wi)' = 1. Therefore q-(z) = I - p I- z r

i i=t

.

1

1 - w 1z llr [l

+ (p/r)(l

It is possible to obtain a simpler expression for q-(z) by expressing the sum on the right·hand side of equation (6.3.5) as the sum of roots of a certain equation. Define Yi = wi zllr[1 + p(1 - z)/r] so that the sum on the right.hand side of equation (6.3.5) becomes Lj=l (l-Yj)-l = Lj=l Xi, where we set Xj=(I-Yj)-l. Now {Yi}, j = 1, ... , r, are the roots of the equation yr = Z [1 + p(1 - z)/r 1r. Setting x=(l-y)-l or Y=(X-l)/x transforms this equation to (x-1Y=X"z[1+ p(l - z)/rJ r, which can be written as Xr

[I -z {I + pCI r-Z)rJ +

it

(;)(-I)iX-i

=

O.

For this last equation the sum of roots is easily found by taking the ratio of co· efficients ofX r - 1 and Xr, and is

(6.3.7)

j> O.

(6.3.5)

- z)]

By the arguments used in Case 1, we can write the p.gJ.'s q(z) and P(z) as follows: q(z) = 1 - z r

± j=t

P(WiZ.lIT) . 1/T W 1Z

(6.3.8)

1-

Case 3

Once again from equation (6.3.1) we get , at the instant t =

(J

M

j = 0

j> O.

+ 0,

RELATIONS AMONG QUEUING SYSTEMS

6.4

THE SYSTEM E/Gkl1

309

308

where we have used By virtue of equation (6.1.1), this may be written as qj(n) = Pj~(nr),

j = 0, 1,2, ....

r(l) =

But we have shown in Section 6.1 that the limits Pt = (1/r) limn -~ Pt(nr - j)

J

=

~ 1 -'YI

= l.

Also by equations (6.3.7) and (6.1.22) we obtain

exist. Consequently the limits q"t

t

I=l

lim qj(n)

n

~CX)

.t eXls an d qj+ -- rP+jr' " -- 0 , 1, 2 , and from equation (6.1.13),

Pj~ =

L

T =

qo = 1 - p

r

r

= (l/r)Po

qj

t

;=0

r

j> O.

Finally, by using equation (6.3.2), we have qj = qi, that is,

(6.3.9)

q+(z) = q-(z).

k -I

L j=O

1-'YI

±

1=1

Equation (6.3.9) shows that the distribution of the limiting queu~ size just before an arrival instant is equivalent to the one just after a departure mstant for the system GI/Er/l. It may also be observed t~at q~, qi, and ~j for GI/Er/1 are identical with Pt, Pj-, and Pj, respectively, for GI / E/ 1 If r = 1, as IS expected. .. It is possible to elaborate more on the system E,,/Er/1 and get more explIcit expressions by using the results of the exampl~ cO.nsid.ered in S~ction 6.1. If GI = E" in GI/Er/l with E" having the dlstnbutlOn functIOn A(u) = 1 -

C U-2)r+l (1-'Yr)2 r ~ L..!"YI -- ,

q(z) = 1 - p(l - z) - rz(l - z)

r-l-

,

(6.3.12)

j

>

l.

The p.gJ., in this case, is given by

j = 0

~jr-i

-_

1=1

r

L

1 -iz" I )2 - 'Yl

q 1 =p-r"C-v L.. I J/ (1 1=1

j> O.

jr-i,

Po , q]

1 +( ) - Pjr nr

P-

i=O

Therefore

~oo

Now from equation (6.1.11), P~

..•.

r-l

. = r nhm

e-AU(A.u)j

.,'

J.

CZ'YI(l - iz") (1-'Yz)2(1-iz"z)

(6.3.13)

where {'Yz} are the roots within the unit circle, Iz 1= 1, of equation (6.1.18), provided Er is taken such that the mean service time per customer is r/J.!. This is because a customer in Gr/E/l is identical with a phase in GI/Er /l (or equivalently a group in Gr /E/l is identical with a customer in GI/Er/l) if the rate of service of the customer in GF /E/ 1 is the same as the rate of change of phase in GI/Er /1 (or equivalently the rate of service of a group in GFIE/1 is the same as the rate of service of the customer in GI/Er /l). However, if Er is taken in such a way that if its total mean is 1/11, that is, the mean of each service phase is l/rl1, then in equation (6.1.18) we must change J.! to rJ.! so as to get the desired equation giving the roots {'Y/} within the unit circle. It will be shown later that these distributions can also be derived by considering the system E/E~/l, a particular case of the system E/Gk/l which we proceed to discuss next.

and mean k/A., then by equations (6.3 .9), (6.3.2), and (6.1.20) we obtain

-

+

qj = qj =

t1~1

C/,

(6.3.10)

r

"

L..

6.4

j = 0

THE SYSTEM EIG k 11

The limiting Distribution of Queue Size

r

C

1=1

U-!)r+!

z'Yz

1-'YI 1 - 'YI

'

j

>

O.

This means that the corresponding generating functions are q-(z) = q+(z)

f

1 - (1 - z) L..

1=1

CZ'YI r ) (l-'YI)(1-'YZ Z

(6.3.11)

In Section 6.1 we have discussed the distributions of the r.v.'s N-, N+, and Nand the relationships among their p.g.f.'s for the system Gr/E/!. Since this section runs parallel to Section 6.1 and is also related to Section 4.3, some details will be left to the reader. Briefly, in EIG" /1 the customers arrive singly by a Poisson process with rate A and are served in batches of exact size k, the service-time dJ. B(v) for the batches being arbitrary with mean service time r/J.!=!;; vdB(v). We have chosen our units so that r/11 is the mean service time. This choice will be

RELATIONS AMONG QUEUING SYSTEMS

310

6.4

THE SYSTEM E/G k /1

311

convenient when we consider the special case EIE~/l of EICk/l [see equation (6.4.15)] . In EIE~ 11 with mean service time rill the mean of each service phase will be 111l. Define 71 = 'AT11l, the ratio of mean service time to mean interarrival time, and take p, the traffic intensity, as p = 71lk = 'ATlkll. The limiting distributions to be considered exist if p < 1, which we shall assume to hold.

Define

Queue Size just before an Arrival Instant

For the existence of the limits (6.4.4), see Takacs (1961).

Define the probability pi-en) = r(N;; = j) = P(N(o~ - 0) = j), j = 0,1,2, ... , where o~ - 0 is the instant just before the nth arrival, o~ being the instant of arrival of the nth customer. Since we are interested in the limiting distribution, the initial queue size is immaterial. However, for ease of analysis, one can take N(O) = O. Since the customers are served in groups of size k,Pj-(n) is nonzero only when n - j - 1, the number of customers departed from the system, is a multiple of k, that is,

Queue Size iust after a Departure

Pj

lim Pj(t).

(6.4.4)

t-+=

=

P(z) =

I

Pjzj.

j=O

Let P/ = P(N~ = j) =

lim P(N(on

n-+=

+ 0)

= j).

It has been shown in Chapter 4 that the p.gJ. of {PI} is given by equation (4.3.1).

Relationships among the Functions P-(z), p+(z), and P(z)

if

n - 1 - j =1= ks,

if

n - 1 - j = ks,

s = 1,2, ....

(6.4.1)

Here we want to establish that r(z)

Clearly, the ordinary limit of Pj-(n) does not exist for k> 1. Let us therefore consider the Cesaro limit

(6.4.2)

1 1 -Zk

= -k - P\z) = P(z). l-z

To prove the first equation of (6.4.5), we proceed as follows. Two cases arise according to whether j k. Ifj < k - l, then On

<

O~k+j+l

= peOn

<

O~k+j+l) =

N(O~k+i+l -0) = j

By means of the relations (6.4.1) one can see that this Cesaro limit, if it exists, is also the Cesaro limit of the sequence

k1 P;(sk + j + 1),

¢}

¢}

N(on +0)

< j.

This implies Pj-(nk

+ j + 1)

i

I

pt(n).

(6.4.6)

i=O

j=0,1,2, ....

However, if j > k, say j = sk + rn, where s > 0 and rn It will be shown later that the ordinary limit of this new sequence exists. It then follows that the Cesaro limit (6.4.2) exists. Let us then write

(6.4.3)

N(O~k+j+l - 0) = ik

+ rn,

< k, then

i = 0,1,2, ... ,s

which implies s

L

and define

(6.4.5)

;=0

Pik+m(nk + j

+ 1)

= peon

<

O~k+j+l)

r(z) j

L Let Pi(t)

and

Pt(n),

j> k.

(6.4.7)

i=O

Queue Size at an Arbitrary Instant

= P(N(t) = j)

The existence of the limits of the left-hand sides of equations (6.4.6) and (6.4.7) and consequently of equation (6.4.3) is to be proved on lines similar to the ones discussed in Section 6.1. Thus assuming that the limits exist, we have from equation (6.4.6), on proceeding to the limit when n -l- 00 and using equation (6.4.3),

312

RELATIONS AMONG QUEUING SYSTEMS

6.4

THE SYSTEM E/Gk/l

313

(6.4.8)

I-p

(6.4.11)

we have

Similarly, from equation (6.4.7) we have

C

j :;;;, k.

(6.4.9)

1<-1

j

-LLFt A j=O i~O

I-p

that is, k-l

~ I

Replacingj by j - k in equation (6.4.9) and subtracting, we get

A j~o

(k-j)P/ = 1 -po

k-l

_ 1 " P+ Pj = k- L... i-I,

(6.4.10)

Now since P+(l) = 1, we have from equation (4.3.4),

i~O

k -1

where we have used Pj-_ (8 +!)k = 0 in view of the relation j - sk = m < k. The first equation of (6.4.5) now follows from equations (6.4.9) and (6.4.10) on taking generating functions. Foster has also discussed the first equation of (6.4.5) using heuristic arguments which make no assumptions about the input or service distributions. He states that the first equation of (6.4.5) should hold even for the more general system CliCk 11. To prove the second equation of (6.4.5), again two cases arise according to whether the arbitrary instant considered falls in an idle period (j';;; k - 1) or in a busy period (j:;;;' k). For the case j';;; k - 1 we argue as follows. The expected numbers of arrivals and departures during an arbitrary period of length T are AT and AT/k, respectively. But as the expected length of each service time is rill, the expected time the server is busy is (ATjk)(rIIl) = pT. Thus the probability that the server is busy at an arbitrary instant of time is p, and consequently the probability that he is idle is 1 - p. The server is idle when i = 0, 1,2, ... , k - I customers are present in the system and gets busy as soon as k customers are present. The length of a period when the server is idle (idle period) is divided by the instants of arrivals into k - 1 intervals, each of expected length l/A, and consequently the expected length of an idle period starting with i customers is (k - i)/A. The probability that an idle period selected arbitrarily starts with i customers is jointly proportional to Pi and to the length of the idle period, namely, (k - i)/A. Now as P(there are j customers at an arbitrary instant of time selected in the idle period I an idle period starts with i customers) = I/(k - i), j = i, i + 1, ... , k - 1, we have j

I

i~O

k-i

(1)

C-,-Pi k-' 1\

~±r A i~O

!

L (k-j)p/

= k(1-p).

j~o

Thus CIA = 11k, and therefore (6.4.12) For the case j:;;;' k, the arbitrary instant selected falls in a busy period. Let us then take Y as an r.v. representing the time between the start of service on the customer undergoing service and the instant of time considered above. Then reasoning as in Section 6.1, we get

P(y

<

Y .;;; Y

+ dy)

= P(y

<

y .;;; y

+ dylserver is busY)P(server is busy) A

Il = P - [1 - B(y)] dy

k [I -

r

B(y)] dy

provided B(y) is a nonlattice distribution. Let A be an r.v. representing the number of arrivals during the time period Y, and define hj = peA = j) so that the p.gJ. of the sequence {hj } is H(z) = Lj~o hjz j . It then follows that

hj

f

~

0

A

peA = jlY = y) k [I-B(y)] dy

A f~ e- AY (i\y)i 0 j! [I-B(y)] dy

k and therefore

1 1 -k(z) ---::-~~ k l-z

Hz) ( = j = 0,1,2, ... , k - 1

"

where C is a normalizing constant. As the probability that the server is idle is

where k(z) = L7'~o kjz i = b[A(l - z)] , as defined in Section 4.3. Now Nand N+ being the limiting queue sizes at the arbitrarily chosen instant

314

RELATIONS AMONG QUEUING SYSTEMS

and just after a departure respectively, we have

THE SYSTEM E/Gkll

315

and mean rlJ.l. In this case we have (see Section 4.3)

+ A,

N = max (N+,k)

6.4

N??:k

+

where A is the number of arrivals during the period Y that is still going on at the arbitrary instant considered above. Making use of the convolution property and taking generating functions, we get E(zN) = E(zmax(N+'k»E(zA)

= [E(ZN+IN+ ??: k)P(N+??: k)+E(zkIN+

P (z) = k(1 - p)(z - 1)

k-I (z -zj)/(l-zj) (Zk Ik(z» _ 1

Q

(6.4.14)

where ii, Z2 , ... ,Zk -I are the k - 1 roots within the unit circle, Iz I = 1, of the equation zk = k(z) = [1 + 7\ (l - z)lrrr, where 71 = t.r1J.l. This may be written as

< k)P(N+ < k)JH(z).

(6.4.15)

But as In view of equation (6.4.15), the denominator of equation (6.4.14) has k + r zeros, I, Zt, ... ,zk-I, ZlO,Z2Q,· .. ,zrO, the last r zeros being outside the unit circle. " Consequently we can write the denominator of equation (6.4.14) as

r(z) = E(ZN+) k -1

= E(ZN+IN+ ??: k)P(N+ ??: k)

+

2:

E(zN+ IN+ = m)P(N+ = m)

m=O

k-\

r

C(z-I) nCZ-Zj)nCz-zjo)

we have

I

k-I P(z) -

2:

Pizi

(6.4.13)

j=O

By equation (4.3.4)

kr.

1

where C is a constant to be determined. Replacing the denominator of equation (6.4.14) by this expression and normalizing, we get

I -Z'o n __ r

P\z) =

[~-I} P+(z). k(z)

p/(zk -zi) =

j=O

I

1_.

I

Z "- ZjO

If we put ZjO = l/ej, we can write P+Cz) as

Using this and equation (6.4.12), k-I

1

1'=0

P1'Zi = k

2:

1 -e. n--l 1 - e1'Z r

k-l

P\z) =

j

2: 2: ptzi

(6.4.16)

1

1'=0 ;=0

where l/€j, j= 1,2, ... (6.4.15).

,r,

are the roots outside the unit circle of equation

It can be shown that all the roots of equation (6.4.15) are distinct and hence that the 1/€j are distinct. Consequently we can write equation (6.4.16), using the method of partial fractions, as

=

1 (z - I)k

f~-

[~ _ I]P+cZ ). k(z)

where

B l = (1 - €z)

Substituting this and H(z) in equation (6.4.13) finally gives the second equation of

0.4~.

k . .

As an example, let us consider the system EIE, distribution is r·Erlang with B(v) = 1-

e-lJ.v(pv/

j=O

,.

2:

.,

nr

1 -e.

'

;*zl-€;/ez ;=1

.

11. In thiS case the serVlce.t!me

r-I

(6.4.17)

/=ll-€/z

Note that since P\l) = 1,

2:

1=1

Bz

-= 1 -€z

(6.4.18)

RELATIONS AMONG QUEUING SYSTEMS

316

and equation (6.4.1 7) gives r

P/

=

I

(6.4.19)

BZE{.

l=!

Once P/ is known, we have, using equation (6.4.5),

! k

±Bz(l_e\+l), 1=1

1 - €l

O<j
(6.4.20) k

1

1 ±B1€t + (1-€f),

k

Z=1

j ;;;. k.

l-EZ

Thus for the system EIE~/l the three probabilities P/, Ph and Pj- are known in terms of the roots 1I Ex, I = 1, 2, ... , r, outside the unit circle of equation (6.4 .15).

6.5

RELATIONSHIP BETWEEN E,,/G/l AND E/G"/1

6.S

equations (4.3.5) and (4.3.1) for the system E/G"/l and show that the result is equivalent to the one for Ek/G/l given in Prabhu (1965). Furthermore, the joint distributions of duration of a busy period and the number served during a busy period, as well as the occupation-time distributions, will be the same. To see analytical equivalence in the former case, the reader is referred to Takacs (I 962) and to Prabhu (1965) who discuss independently the systems M/G k /1 and EkIG/I, respectively. For the distribution of occupation time for the queuing system M/G" /1, the reader is referred to Problems and Complements in Chapter 4. However, no independent derivation of an analytical expression for the occupation-time distribution for the system EkIG/I seems to be available for comparison. The relations between the limiting distributions of queue sizes considered at three instants of time just before an arrival instant, at an arbitrary ins1;.ant, and just after a departure instant for the system E,,/GII, and the corresponding distributions for the system E/G k /1, are considered next.

THE SYSTEM Ek IG/1 6.6

We have shown in Section 6.2 a relation between the two systems GIIEr/l and GI" /E/1. A similar relation exists between the two systems EIG k /l and E"IG/l if a group in the former system is identified with a customer in the latter, so that the interarrival time distribution for the group in EIG"/l is the same as the one for the customer in Ek/G/l. At each departure instant the queue for the system E/G k /1 is reduced by k, whereas the number of phases of Ek/G/l is reduced by k. If the interarrival-time distribution for E/G"/I is E with mean llA (or for a group Ek with mean k/A), the interphase distribution for E,,/G/l is E with mean l/A (or the interarrival-time distribution for a customer is Ek with mean k/A). This makes the traffic intensity p equal to A/kl1 for both systems, since 1/11 is the mean service time for both systems. In Ek/G/l a customer passes through k exponential phases (marked frontward as 1,2, ... , k), each of mean 11A, before actually entering the system. Consequently if Nl (t) represents the number of arrival phases completed by a customer at time t, then Nl (an + 0) converges in distribution to an r.v. Nt. Hence Nt represents the limiting number of arrival phases completed by the customer just after a departure. In view of this, the distribution of Nt in EklG/l and that of N+ in the corresponding E/G" /1 must be the same. As a check, one may easily see that the distribution of Nt relating to Ek/G/l given in Prabhu (1965) is equivalent to the one given in equation (4.3.1). This statement remains true even in the transient case, provided the initial states (customers for E/G k II and phases for E,,/G/I) are taken to be the same. It can be shown (see Problem 4) that the waiting-time (in queue) distribution for the system GI/Er/l is equivalent to the corresponding distribution for the first member of a group in GI" /E/l. Similarly for the systems E/G k /1 and E,,/GII, the waiting time (in queue) of the last customer of a group in E/G k /1 is identical with the corresponding distribution for the system Ek/G/l. For an analytical verification of this statement, one may use

317

RELATIONSHIP BETWEEN Ek/G/1 AND E/G k/1

As in Section 6.3, we consider the relations between the limiting probability distributions of queue size in the systems E,,/G/I and EIG" I 1. First recall from Section 6.5 that a group in EIG k II is identified with a customer in Ek/G/l. Thus if a~, a;, ... are the instants of arrival in E/G k 11, the instants of arrival in the corresponding E,,/G/I are a~, a;k, ... since a customer in E,,/G/I is considered to have arrived if and only if it has passed through all the arrival phases. A customer leaving the system E,,/G/I reduces the number of phases in the system by k, which is equivalent to a group of size k leaving the system E/G k II. The instants of departure for both systems E/G k /1 and E ,,/G/1 are the same. Now if Ht) represents the number of customers (including the customers, if any, undergoing service) at dme t for E hIG/1, then we have

w)

=

[NZ)]

(6.6.1)

where [x] is the greatest integer contained in x. We shall discuss the three distributions of Ek/G/l at the three instants of time and their relation to the corresponding distributions of EIG k I I. Let us define

q/(n)

= P(Ha n + 0) = j) = peW) = j)

2

qj (t)

3

qj-(n) = P(Ha~" - 0) = j).

318

RELATIONS AMONG QUEUING SYSTEMS

Case 1

6.6

RELATIONSHIP BETWEEN Ek/G/1 AND E/Gk/1

where

(vi is a kth root of unity, the residue at

+ 0,

From equation (6.6.1) we get at the instant t = un

L

1 p+(wiZllk)wizllk 1 - wiZ llk

m=O

P/k+m(n).

Therefore by the residue theorem (see Appendix A.3), we have +

Since the probabilities on the right-hand side of this equation converge as n -> 00, p < I, so do the ones on the left-hand side. Thus we may write k-l

q/

L

=

j

P/k+m,

~

O.

(6.6.2)

m=O

Let us define q+(z)

~i p+ = Lj'=o qi+ Zl. and put R+j = ""';=0 ; an d

T+ j

= Li;=0 q+i so

that.

{Rj} and {Tn are the dJ.'s corresponding to the two distributions involved. It 18 easy to show that

q (z) =

I-z Bl dv q +(z) = - ~ 2rri c (l - w)(l - v)(v k

= R;'-l,' ..

and =

Rjz

P+(z)

j

'"

L

RU+ l)k - I zi

j=O

(6.6.3)

l-z

i=o '"

L

T/z

j=O

j

=

L... j=O

2' m

c (1

z)

(6.6.6)

Case 2

Again from equation (6.6.1) we have

j=o

.:!... ~

-

EhIEr!I.

~ j L... RU+!)k-l Z

;,

z)

by evaluation of the residue at the pole v = I!E outside C (see Problems and Complements). Equation (6.6.6) can also be derived alternatively by using equations (6.6.2) and (6.4.19). This procedure is discussed later when we consider the system

where C is a contour around the origin excluding the poles of P\z)/(1 - z). Thus =

-

(6.6.7)

1 P+Cv) dv Rj = 2rri
1- z

q+(z) = 1 -.z ~ (l-E)dv 2m 'Yc (1 - EV)(l - v)(v k

q+(z). 1 -z

Now reasoning as in Section 6.3, we can write

q+(z)

(6.6.5)

where EI == E is the unique root inside Iz 1= 1 of the equation kZk+l - (k + p)z _ p = 0, and p = A./kll is the traffic intensity for the system Ek/E/I, I!/-.. being the mean for each arrival phase. But as by equation (6.4.18), Bl = 1 - E, we have

T/ = RU+1)k-l

L

1 -z k P+(WiZllk)wizllk 1 - WiZllk

~ i~l

where P\-) is given by equation (4.3.1). Equation (6.6.5) expresses the p.g.f. of queue size for the system Ek/GII in terms of the corresponding p.g.f. for the system EIG k 11. As an example, let us consider the special case EkiE/l of the system 4k/G / 1 in which G = E. Using equation (6.4.17), which gives P+(z) for Ek/EII, we have from equation (6.6.4) for the system Ek/E!l

T; = R k- 1

Ti

= WiZ llk is

zk

k-l q/(n) =

Vi

319

k-l P+(v) dv

qi(t) =

-V)VU+l)k

or l-z
(6.6.4)

Pik+m(t).

The probabilities on the right-hand side of this equation converge [Takacs (1961)] when t -> 00, provided that p < I and B(v) is nonlattice. It follows that qit),j = 0, 1,2, ... , exists, and under the same restrictions qi = limt

_=

Since the poles within C of the integrand are j = 1,2, ... ,k

L

m=o

k-l qj =

L

m=O

Pjk + m

,

j ~ O.

(6.6.8)

320

RELATIONS AMONG QUEUING SYSTEMS

If we define q(z) = 2:j=o %zj, then proceeding as in Case 1 we can show that

q(z) =

1 -z tf, 211i ~ (1

P(v)dv v)(v k -- z) .

6.6

RELATIONSHIP BETWEEN Ek/G/1 AND E/G k /1

321

From equations (6.4.20), (6.6.2), and (6.6.10) we get k-l

(6.6.9)

L P/k+ m

qT = qj =

m=O

In particular cases, however, it may be convenient to use equation (6.6.8) rather than equation (6.6.9). (See example following Case 3.)

t

'k

BI

k

€I ) ,

j

1 -€I

1=1

Case 3

C; (l -

~

0

(6.6.11)

which implies

Once again from equation (6.6.1) we have q;(n) = PJi,(nk) + PJi,+ 1 (nk) +

"

q-(z) = q+(z) =

,+PJIHk-l(nk)

= Pii,+k-l(nk)

kflk~

n

lim q;(n)] =

~CQ

e?)

(1- €l)(l - €7z)

(6.6.12)

Also from equations (6.4.20) and (6.6.8)

since other terms vanish in view of equation (6.4.1). Now it follows from Section 6.4 that the limits (i/k) Iimn""'~ Pj-(nk + j + 1) = Pi exist, provided that p < 1. Consequen tly, q; =

Bl(l -

Lr 1=1

k[~k

n

h -1

Pjk + m

,

j~l

(6.6.13a)

m=Q

1," r = - L...

lim PJk+k-l(nk)l

~CQ

L

qj =

k

]

1=1

(k)2

j"-k+l 1-€1

Bl€z

--1 - €I

But by equation (6.4.1 0), (6.6.13b) Therefore qT =

L7;J P/iHi, which gives, by equation (6.6.2), qj- =

qj,

that is

q-(z) = q+(z)

(6.6.10)

where q-(z) = Lj=O qj-zj. This is the result corresponding to equation (6.3.9) for the system GI" /E/l and shows that, for ilie system E/G k /1, the limiting distri· bution of queue sizes just before an arrival instant is equivalent to the one just after a departure instant. It may also be observed that qj, qj, and qT for E k iG/l are identical with P/, Pi, and Pj -, respectively, for EIG k 11 if k = 1, as is expefted. In this case, asP/,Pj,Pj- are all equal for k = 1, so are qj,qj,qT and, in fact,qj = qj = qj-

where we have used equation (6.4.18). The p.gJ., in this case, is q(z)

(6.6.14)

= P/ = Pj = Pi·

We have already discussed the system Ek/Er/I. Now we give an alternative derivation of the same system. In the problems we consider a proof that the results for the system EkiEr/l derived by the two different methods are, in fact, identical. The System Ek/G1/1

If in the system EkIG/l, G == Er (with mean rl/1), then it is possible to get and qj explicitly in terms of the roots 1/€j of equation (6.4.15).

qT, qj,

We have shown in Section 6.6 that the two systems E/G k /1 and Ek/Gll behave identically if we identify customers in E k /G/l with groups in E/G" /1. Similarly, the results for the bulk·queuing system Ek /G 1/l, in which arrivals follow a k-Erlang distribution and the service consists of batches of fixed size l, may be obtained from the bulk-service queuing system E/G K /l, where K = kl. In particular, if N is the number of customers in the queuing system Ek/GljI, ilien peN = n) = Z]~;i:f-l Pj, where th~ probabilities Pj have been analyzed in Section 6.4. The queuing system EIG K /1 discussed here may also be compared with an (s, S)

322

RELATIONS AMONG QUEUING SYSTEMS

inventory model which has attracted a great deal of attention in inventory controls. For details, see Fabens (1961).

6.7

EXPECTED BUSY AND IDLE PERIODS

In this section we obtain expected busy and idle periods for some bulk queuing systems by means of the alternating renewal process and its generalization, as discussed in Chapter 2. This method, in general, is more elegant for finding averages than that of finding these averages through their distributions. Of course finding the distributions gives more information than finding the averages only. Instead of giving these averages in the discussions of various bulk queuing systems in other chapters, we thought that it would be nice to keep all these results together. This chapter was considered to be the best choice to contain these results. As in Chapter 2, let To, T 1 , . • . and T~, T;, ... be two sequences of r.v.'s representing busy and idle periods, respectively. Suppose that {Tn} are Li.d. as T, and that {T~} are Li.d. as I, but that {Tn} and {T~ } are not necessarily mutually independent.

6.7

EXPECTED BUSY AND IDLE PERIODS

323

k = 1,2,3, ...

(6.7.3)

where Po is the probability that an arrival finds the system idle. Equation (6.7.3) is easy to prove otherwise; for E(I k I arriving group finds system idle) P o- + E(I k I arriving group finds system busy) (l - Po) = E(Ik I ... idle) Po + 0 . (1 - PO) = E(I;;). Although equation (6.7.3) is true for k= 1,2,3, ... , we need to use it only for k = 1. If, however, the higher moments of Iv are known, equation (6.7.3) gives higher moments of I. Higher moments of Iv for the system GFIM/l are given in Miscellaneous Problem 1. Now for the system GFIMII, since E(lv)=(l-p)/A by equation (6.7.2), E(l) is given by I-p

E(l) =

APo

(6.7.4)

where Po may be found either from Chapter 3 or from Miscellaneous Problem 1 and is given by r

Po =

n (1 ;=1

'Y;)

{'Yi} being the r roots within the unit circle of the equation

The System Gl r /M/1

Since for this system, in the limiting case, the proportions of times the server is busy and idle are p and 1 - p, respectively, we have, by equation (2.3.18), E(T) p -- =--. E(I) 1- p

or

A.r p =-, p

(6.7.2)

Recall that E(Iv) is called the mean virtual idle period (since it includes the zero idle period) as opposed to the actual idle period which excludes the zero value. If we define I as the actual idle period as observed by an arrival, then as we know (see Section 2.2) the moments of these r.v.'s are related through the equation

with

a(a) = ( " e-ata(t)dt.

Using equation (6.7.4) in equation (6.7.1), we get

(6.7.1)

To find E(l), we use the conservation principle that in the limiting case, mean input rate equals mean output rate. Thus in Gr IMII treat a group as a single supercustomer so that the group is in the system as long as its last customer has not been served. Let Iv be an LV. representing the duration of the idle period between two successive groups (Iv = 0 if service continues without interruption between two groups) and lip the mean service time. Then as E(Iv) + r/p is the mean time between completion of service of successive groups, we must have

I-p E(Iv) = -A-'

a(p(1 - z)) = zr

E(T) =

r

-r---

(6.7.5)

p ;~1 (l-'Y;)

which is the expected busy period for the system GF IMI 1. The latter expression for E(T) has been obtained by using equations (6.1.6) and (3.2.3). The System GI/Er/1

Recall the remarks made toward the end of Section 6.2 that if a group in Gf" IMI! is identified with a customer in GIIEr/! so that the total mean service time of a group in Gf" IMI! is the same as the mean service time of a customer in GIIEr/!, the distribu tion of a busy period for the system Gf" IM/! is identical with the distribution of a busy period for the system GIIEr/I. Consequently the mean busy period of the server for the system GIIEr/! is given by E(T)

r

where l/p is the mean duration of each phase of service in GIIEr/l. In terms of the

RELATIONS AMONG QUEUING SYSTEMS

324

OVERVIEW

325

service-time distributions whose coefficient of variation is < I, and has the advantage that numerical calculations are possible. The initial probabilitiesPt, i = 0,1,2, ... , k - 1, may be found either as in Section 4.1 or as in Section 6.1. The computational work done by Holman (1977) may be extended and used to evaluate Pt, i = 0, 1,2, ... ,k - 1, and hence E(T) for various values of the parameters k, r, B, and p = AIBfl.

notation of Section 6.3, E(T) could as well be written as r

E(T) =

6.8

fJJlo

The System M X /G/1

For the system MXIGI 1,since E(l) = l1A, we have by using equation (6.7.1) E(T) = a/(Il- Nf) = (!1Pt)-l , where P(X = x) = ax is the distribution of group size, a = 2:;~1 xa x , llA is the mean interarrival time and 1/11 is the mean service time. For more details on this, see Chapter 3. The second expression for E(T) has been obtained in the same way as equation (6.7.5) for GI" IMII.

The Systems M/G B /1 and M/G k /1

The systems MIGB/l and MIG h/1 are special cases of MIG",Bll, and results for them may be obtained by putting k = 1 and B = k, respectively. We have

The System MIGk,B 11

This discussion should be read in conjunction with Section 4.1, where the notation is explained, and Miscellaneous Problem 8 in Chapter 4. For the system MIGh,B 11 the relation between the mean busy and idle periods [analogous to equation (6.7.1)] is h-l

E(T) E(l)

1-

2: Pr

r=O

0

'

h-l

2: Pr

r=O

'

(6.7.6) The System Ek/G/1

0

We first discuss the mean idle period E(l), which is easy to evaluate. Let N(t) be the state of the system at time t. N(t) enters the set of idle states, S == {O, 1,2, ... , k - I}, at the termination of a service period. The conditional probability that N(t) enters state i, given that N(t) enters S, is therefore cPt, i = 0,1,2, ... , k - 1, where I/C= 2:7~-J Pt. Now N(t) leaves S when k-i customers arrive, which happens in expected time (k - i)/A. Thus k-I

2:

E(l)

i~O

The derivation of the mean busy period of EkIG/l from MIG k 11 is analogous to that of GIIEr/1 from GrlMII. Recall the remarks made in Section 6.5 that if a group in MIG h11 is identified with a customer in E"IG/I so that the mean interarrival time for a group inMIG h/1 is the same as the one for a customer in EkIGI1, the mean busy period for the system E"IG/I is the same as that for the system MIG k /1. Thus the mean busy period of the server for the system E"IG/1 is given by

Ft(k -i) k -1

j~O

E(T) =

(6.7.7)

A 2: p7 J

First using equation (M21) of Chapter 4, and then equation (6.7.6), we finally get E(T) =

For MIG k 11, E(T) may also be written as E(T) = plAPh _I , where we have used equation (6.4.12) and p = Alkll. In these special cases, some computational work has been done by Allan and Carignan (1978).

h-l ]-1 [ fl.2: Ft

(6.7.8)

.~o

For a generalization of this result, see Problem 16. The System M/E,k-B 11

If we take the r-Erlang distribution of service times, we have the system MIE:,B II. This system is a good approximation to the more general case discussed above, for

k

(

,=0

where p = Alkfl = A'IIl and IIA is the mean input rate is Alk = 'A'. In terms as well be written as E(T) = I/fJJlo. result obtained earlier for GIIEr/i. It EhIG/I, 1/fl is the mean service time system GIIEr/I.

6.8

-1 Pt)-1

fl.2:

= -p-

APh -

1

mean of each input phase so that the total of the notation of Section 6.6, E(T) could This tallies, as it should, exactly with the may, however, be noted that in the system whereas rip is the mean service time in the

OVERVIEW

Shanthikumar and Chandra (1980) discuss, among other things, relations between different limiting probabilities for general multiserver bulk-arrival queuing systems

326

RELATIONS AMONG QUEUING SYSTEMS

of which special cases are M X IGlc and GlxlMlc. For some other similar results relating to non bulk queues, the reader is referred to references contained in Shanthikumar and Chandra, and to the recent work of Franken et al. (1981) and Miyazawa (1977), among others. In reading these publications, one should keep in mind that some authors describe arrival-point (departure-point) and random-point probabilities as arising from customer and time processes, respectively.

PROBLEMS AND COMPLEMENTS

3

327

If V~n) is an r.v. representing the waiting time (in queue) of the nth arrival, then show that the limiting waiting-time distribution function for the system GIIEr/l is given by lim p(v~n) .:;; t)

W';-(t) =

n .... =

j-K

r ex''Y' exp [-~.u(l-'Y;)J, .=11 -'Yi

.I -'-'

t ;;;. 0

(2)

PROBLEMS AND COMPLEMENTS where K is defined in Section 6.2, and deduce that the mean waiting time W~ is given by

Sections 6.1, 6.2, and 6.3

2

Using renewal-theoretic arguments, Foster and Perera (1965) have proved, for the system GI' IE 111, that Pj = limt .... = P/t) exists, provided that p < 1 and A(u) is nonlattice, as is expected on the basis of our experience gained through the previous chapters. Show that the mean queue size L (mean number in the system considered at an arbitrary instant) for the system GI'IEll is given by L=p

1 r [ i=11-'Yi

r-l]

I - - - 2-

Hint:

W-(t) q

(1) 4

and deduce that

Lq =

p

[itl 1~

'Yi - r

~ 1)-

Po

·t =

+ Joj /'\'.... PiI =1

(f.1X)i-l e -J.LX d C'-I)! J1 X, I

t ;;;. O.

.

Use equation (6.2.1) and the other relations found while discussing the system GIIEr/1. Continuation of Problem 3. Show that the waiting-time (in queue) distribution obtained in Problem 3 for the system GIIEr/l is really equivalent to the corresponding distribution for the first member of a group in GriEf!. Hint: From equation (3.2.11) we have

w~(ex)

Hint: Usep-(I)(1), which may be obtained from Chapter 3 [equation (3.2.3)] and equation (6.1.8). The result (1) has also been independently obtained by Conolly (1960) for the system GI'IEI/I considered at an arbitrary instant by using the supplementary-variable technique. For details, see Miscellaneous Problem I. Observe that by using p-(I)(l) and equation (6.1.6) one can also obtain the mean queue size just after a departure instant. It may be observed by looking at Lq and Wq (Problem 9 of Section 3.2) that Little's formula Lq = A., Wq , where Ae = Ar, for the system GI'IM/l is satisfied. The formula is true even for the more general system Gl x IM/1 [for details see Easton (1981)], as it should be. It is true without any constraints on the input distribution, the service-time distribution, or the group-size distribution, if the customer whose mean waiting time is required is chosen at random from an arrival group. For the verification of Little's formula for MXIG/l, see Miscellaneous Problem 4(a) in Chapter 3. It should be pOinted out that in both GlxIM/1 and MX/G/l, the queue size Lq is calculated as a steadystate average over all instants of time, whereas W';- (or Wq in M X /G/1) is calculated at arrival instants.

-

=

t C(1 + 11(1 - l1'Yl'YI) + ex ]

j=1

z

where q is defined in Section 6.1 and ex is the parameter of the transform. Also recall that L[=I Cz = K, Cz = Kexz. Get the result by inverting or by taking the L.-S.T. of equation (2). It has been assumed that the roots {rz} are distinct. As a consequence of Problems 3 and 4, it follows that the waiting-time distributions for GI'IEl/1 and GIIErfl are identical. These remarks apply equally well to both systems if they are considered at arbitrary instants of time. 5

(a)

The system GI/Erll. Show that the limiting distribution of the number ~- in the system GI/Er/l considered just before an arrival instant is given by

Po = Po

r

1-

.I ~j

)=1

n>O

RELATIONS AMONG QUEUING SYSTEMS

328

PROBLEMS AND COMPLEMENTS

(c)

where c'l.jlj .

(3j = K

r

Po

r

I

=

K

r

I 0:1 = K = A = 1=1 n 1=1

zr_ 7(zr-l

and the p.gJ. of ~- by 1

r

I

r(z) = Po+Z

r

{3j~' 1 -zl}

j=1

(1 -Iz)

where II are the roots of the equation

0,1,2, .. ,

d

(3j1}d,

j=!

(b)

Show that for the system MIErl 1, Po = 1 - p. Hint: Equation (6.2.1) gives

1 -Ij

The cumulative distribution of ~- is given by P(~- ~ d) = 1 -

329

(3)

6

(a)

Show that equation (3) can also be obtained by using equation (6.3.11). Show that the mean queue size L - (the mean number in the system) is given by

+ .. '+Z)-7

= O.

Substitute z = 1 - Y and then 1 - I I are the roots of the equation in y, and so on. The system GF lEI 11: the waiting-time distribution. Let Vq(t) be the virtual waiting time (in queue) at the instant t. Show that the limiting distribution for the first member of a group Wq, (x) =

lim P(Vq(t)

t ..... =

~

x)

has the L.-S.T. given by L- = E(n =

r

I

j= 1

_{3j

1-

r 'Yj

Wq (0:)

=j.= e- ax dWq ' (x) =

'0

P (-p-) . p+ ex

and deduce that the mean number L~ in the queue is

t

L- =

(b) {3j1} •

j=1

q

1 -I}

Hint: To discuss this problem, first note that the relation between the number I of phases and the number ~- of customers in the system GIIEr/l is given by I =

W

q,

7

r(~--l)+S

r

I

P(I = r(n -1) + s)

s=1

r

=

L

8

P;(n-l)+s

s=1

and so on. Also note that equation (6.3.1) gives

Po = 1 -

f

i=!

pi =

1-

Ii

= P [ ft.. - ;=11- li

t

{3j.

(l-p)(1-z) . 1 - z {I + 7(1 - zr)}

Hint: Use equation (3.2.10) or equation (3.1.19). Intuitively, the equivalence of P-(z) andP(z) may be argued on the basis of Poisson (random) input. Such an equivalence has already been observed in Chapter 3 (see Problem 11), even in the more general case when the arrival group follows an arbitrary distribution. But here, of course, we have given the proof in the case when the arrival group size is fixed. Show that all the roots of the equation zr =

j=l

1] I

+ r-+2 - . p.

For the system MrlEt!l show that r(z) = P(z) =

where S is the phase of service of the customer undergoing service at the instant of the last arrival. In addition, the system is uniquely defined if it is supposed that whenever ~- = 0, I = 0. Now for n > 0, P(~- = n) =

Show that the mean waiting time for the first member of a group in (a) is given by

1 (

1 -z +-,;)

-k

(4)

are distinct. Proof: Differentiating equation (4), dividing one equation by the other, and

330

RELATIONS AMONG QUEUING SYSTEMS

PROBLEMS AND COMPLEMENTS

setting rT = p, leads to the equation 'z = (pk + r)/(k + r). Substituting this value of z in equation (4) gives the condition for the roots to be repeated, which is pk/(k +r)

9

=

pk + r k+r'

1

= k

Let us now see if this equation holds for the traffic intensity p < 1. Putting p = 1 - E, 0 < e < 1, in equation (5), we get (l - e)C = (1 - ce), where c = kiCk + r). But this is impossible, unless E = O. Thus the roots of equation (4) are distinct if p < 1. Since the roots can be computed only up to a certain degree of accuracy, they may become computationally inseparable as p -+ 1. It appears from our computational experience that when p ~ 1, the roots will be computationally distinct so long as the degree of the equation to be solved is kept smalL Muller's algorithm [see Conte and de Boor (1972)] may be used if the roots are repeated. IfGI=E1 inGIIEr/l,thenshowthat

1=1

zt

t=o

L ef-v v=O

One particular case of EIE~ /1 is EIE~ 11, when r = 1. Show that for this case j .,;; k - 1

j ;;;. k

where e is the unique root inside the unit circle of the equation Zh+! -(1

13

+ kp)z+kp

= 0,

p = AlkjJ..

(6)

Evaluate the integral given in equation (6.6.6). Proof: 1

rh

211i

'Yc

=

(l-€)(l-z)dv (1 - Ev)(1 - v)(v h - z)

2~i


-(l---:v-~-(-:---V-) (1 - v-v:-_---:)dV (l-e)(vh -l)dv (l-ev)(v-l)(v h -z)

Sections 6.4, 6.5, and 6.6

Show that the roots of equation (6.4.15),

since the neglected part has no poles inside C. Now transform the integral by making the substitution v = 1/ V. Then the integrand has the pole V = € inside C. The residue at V = e being (1 - e h )/(1 - €h z ), we have by the residue theorem (see Appendix A.3)

are all distinct. Hint: Proceed as in Problem 8. Show how to get the results of equation (6.4.20). Hint: P(z)

t

-I

and so on.

12

This shows that for the system El /Er /l, the distributions of C, ~+, and ~ are identical. This is true even for the more general system MIG/1; see Section 6.4 for k = 1. Hint: Use equations (6.3.8) and (6.3.9) and the procedure discussed in the example of Section 6.3. See also comments after equation (6.3.6). We can also discuss this problem by using the example of Section 6.3 and Problem 7.

11

Er B z [I
(5)

_ + (l-p)(l-z) q (z) = q (z) = q(z) = l-z{I+7(1-z)y

10

331

I

k-l

k

u=o

= r(z) = - L I = -

k

r

ZU

= E Bz [h-I EE r

1=1

v=o t=v

=

L B z m=o L (ezz)m

1=1

ef-vzt ]

1 _€k q+(z) = --h-' 1-

14

(a)

€ Z

Show that when r = 1 in Eh/Er/!, that is, for the system E h/E/1, qi- = qj = €ik(l - €h),

j ;;;. 0

qi = P€(i -1)hO

j;;;'

_ Eh),

qo = I-p,

and the corresponding p.g.f.'s are

j = 0

332

RELATIONS AMONG QUEUING SYSTEMS

PROBLEMS AND COMPLEMENTS

333

(12)

But by equations (l I) and (12)

q(z) = 1 -

(b)

pO 1_

z)

€h

Z

where t! == € is the root inside the unit circle of equation (6). Hint: Use equations (6.6.11), (6.6.l3a), (6.6.13b), and (6.4.18). Show that for the system EhIE/l

q(z) = 1 - p + pzq-(z).

IS

It may be observed that this relation between q(z) and q-(z) is true even for the more general system GIIEI1, as may be seen from equation (6.1.8). We have discussed queue· size distributions {qj}, {qj-}, and {qj} for EhlEr/1 by discussing two different systems E'klEI1 and EIE; /1. Now we show that equation (6.3.10) is, in fact, identical with equation (6.6.11) and equation (6.3.12) with equations (6.6.13a) and (6.6.13b). To do this, we first need to establish relations between the roots I and lit of equations (6.1.18) and (6.4.15). Since the traffic intensity for both systems is p = rv-Ikil, we rewrite equations (6.1.18) and (6.4.l5) in terms of p. Thus r

z =

[

z)r]-h 1+--(I -

kp

which shows that Ii is a root of equation (7) and also I/jl < I by equation (12). Relation (10) gives a one·to·one correspondence between the r roots 1/Ej outside the unit circle of equation (8) and the r roots Ij inside the unit circle of equation (7). Now from equation (10) '"

kp r

(.!.. €i

-1)

1,2, ... , r

1-

(13)

I = 1,2, ... , r.

II,

By subtracting we get

(7) i =1= l.

(14)

gives the roots Ii> j = 1,2, ... , r, within the unit circle, whereas By dividing equation (13) by equation (14) we get

Zh[l+k;(1-zf = 1

(8)

11-

gives the roots I/tj, j = 1,2, ... , r, outside the unit circle. Since l/Ej is a root of equation (8),

Ei Ed€1

=

1 - Ii. II -

Ii

Using the definitions of C and B (Sections 6.2 and 6.4) we have

(9) which implies

Now if we define

_B _ 1_

( 1) I,' = 1 + kp r 1 - Ej ,

j = 1,2, ... ,r

(10)

1-

(15)

Ell-II

Since both equations (6.4.11) and (6.6.13b) give the value of qo, we have on comparing

then it follows that

kp

(11)

p

1

r

Recall that

1

-Er

- L BIEI (l-EI) 2

k 1=1

and by equation (9),

C 1-r = --l1L.

•

(16)

RELATIONS AMONG QUEUING SYSTEMS

334

B

Ir

_1_ 1=1 1-€1

(a) (b)

= 1

and

t

1=1

~= 1-'YI

1.

Show by using equations (12), (15), and (17) that equation (6.3.10) is identical with equation (6.6.11). Show by using equations (12), (13), (15), (16), and (17) that equation (6.3.12) is identical with equations (6.6.13a) and (6.6.13b).

Equation (6.7.8), derived above for MIGk,B/l, is valid for GJxIGk,B/l, as indicated by the following heuristic argument: E(T) = E(service time)' E(number of groups served in a busy period) since service time and service group size are independent. Hence E(T) =

n ;;;. r

P~ =

0,

< r.

n

(M2b)

It may be noted, as indicated, that equation (M2b) is true for n < r, for the number in the system after an arrival cannot be less than r. Equation (Ml) corresponds to the case n = r. Write in words equations (Ml) and (M2), which completely describe the system GI" 1M/I. Show further that the solution of the set given by equations (Ml) and (M2) is given by

(M3)

.r

F(x)

n (x -'YD

(M4)

'=1

dF(x)

i=O

F(1)(x) = - - = F(x)

dx

and 'Yi, i

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

r

I

(x - 'Yi)-1

;=1

= 1,2, ... , r, are the r roots inside the unit circle of the equation a(ll(l - z)) = zr

In this problem we discuss the queuing system GI" IMII already considered in Section 6.1. The analysis used here runs parallel to Section 2.2. For notation and other terminology, see Sections 6.1, 2.2, and 3.2. In addition to rediscovering some of the results of Sections 6.1 and 3.1, we discover some new results which are difficult to obtain using other techniques. Briefly, GI" IMII is a single-server queuing system in which groups of size r arrive. Interarrival-time distribution is arbitrary with mean l/A., and servicetime distribution is exponential with rate 11.

Show that the equations for GI" IMII corresponding to equations (2.2.30) and (2.2.31a) of the system GIIM/I are (Ml)

= pen in system just after arrival in the limiting case)

(MS)

where

In writing the solution (M3), it has been assumed that the roots of equation (MS) inside the unit circle are simple, as they are at least when GI = Ek . For details, see Pro blem 8 of this chapter. Hint: Proceed, in general, as in Section 2.2 for GI/M/l. Replacing P~ by zn, we get equation (MS). Now corresponding to equation (2.2.33), we have

The limiting Distribution of the Number of Customers in the System

P~

(M2a)

together with

jh-I 1Pt.

where

+1

where

E(service time) P(the end of a service is the end of a busy period)

1 = 11

335

(17)

Section 6.7

16

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

p~ =

r

I i=1

Ci'Y?

(M6)

whe~e. the r co~stants {C;} which are independent of n are to be determined by requmng equatIOn (M6) to satisfy equations (M 1) and (M2b). Hence

I

~ i

CaW -

'Yi)-1 = A

I (M7)

C;A.f

0,

1 .;;; s .;;; r - 1 )

RELATiONS AMONG QUEUING SYSTEMS

336

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

337

where the summations are over i = 1 'to i = r, a convention which will be adopted in all problems in this section unless otherwise stated. Solving the set (M7), one can show that 1 .;;;; i .;;;; r.

(M8)

One way to get equation eM8) is to solve the set (M7) as si:nul~aneo.us equations. Another way is to first consider the identity (M9), which IS easily proved by considering partial fractions of XS [F(X)]-1 ,

x' F(x) =

t

~f

(x -

(M9)

~i)F(l)(~5'

Then use equation (M9) with x = 1 and s = r - 1 and show that equation (M8) satisfies the first equation of (M7). Similarly, use equation (M9) with x = 0, ] .;;;; s .;;;; r 1, to show that equation (M8) satisfies the second equation of (M7). This, in turn, shows that equation (MS) is the solution of equations

2

(M7). . . One more point which needs attention is that equatIOn (M3) IS true f~r n? 1, for its right-hand side automatically vanishes when 1 .;;;; n .;;;; r - I, as It should. This can easily be seen by using the identity (M9). Continuation: {P~}, the distribution of the number in the system just before an arrival epoch. (a) Show that P~ is given by ~!,,+r-I

P;; = F(!)

(b)

i= ;(1)(~i) ,

n ? O.

(MIO)

and thenPn = Jo'" Pn(y)dy, and so on. (iii) Po may be obtained either by using (1 - xr)/(! - x) = ~~ :,10 Xk, the identity (M9), and 1 - Po = ~:=l P n ; or independently by PoCy) dy, where Po{y) is given by equation (M13) (see later). (a) Continuation: the virtual idle-period and the actual idle-period distributions for the queuing system GI" /M/I. Adapting the arguments used in the discussion of the virtual idle-period distribution for the queuing system GI/M/l in Section 2.2, show that the distribution of Iv, the virtual idle period, for G1' /M/I is given by

J;

4

JI '"

FJ(7) = AC(7)-F(1) v

Hint:

e-}J.(l -'Yi)(U-T)

( Til

-~i

)F (1)(~i )a(u)du.

(MI2)

Use (M13) where Pn(y) is given in Hint (ii) to Problem 3. Note that equation (M13) for the bulk-arrival system G1' /M/! remains the same as for the singlearrival system GI/M/l.

(b)

Continuation: moments of Iv. are given by

Show that the first two moments of Iv

1 r = ---

'A

Hint: Let P;; = KP~+r(O), where K is a normalizing constant, and proceed as we did in getting equation (2.2.35). Show that the p.gJ. of P;; is the same as the one obtained in Chapter 3.

Hint: Use the identity (M9). The waiting-time distribution for the first customer and a random customer of an arrival group may be obtained by using P;;. For details, see Chapter 3 and problems therein. Continuation: {Pn }, the distribution of the number in the system at a random epoch. Show that the distribution {Pn } is given by

(c)

3

n ? !

E(I2) = a2 _ ~ [E(l ) F(l)(l )_ v Jl v F(I)

Hint: (i) Adapt the procedure used to get equations (2.2.42) and (2.2.43). (ii) First note that Pn(Y) here is the same as that given in equation (2.2.40),

+ r(r 2Jl

1)]

where, if need be, one can replace F(l\l)jF(I) in terms of the roots given by F(l)(1)/F(1) = ~i (1 - 1i)-1 , and az = u1a(u) du, I = 1,2,3, . . . , with at == 1/'A. The moments of I may be obtained through the relation

J;

1=1,2,3, .... Hint:

(MU)

Jl

Take the L.T. of equation (MI2), which is

f1v(a)

_

= a(a)

+aF(1) '" a(a)-~r -JlF(l)6i)(1 - ~i)[l -

T

~i -

a/Jll

where we have used equation (MS) and Trv(a) = fo'" e- cxt dFr/t), and so on. Now adapt the procedure used in problems on GI/M/l discussed in Chapter 2 and the identity (M9) and its derivatives.

338

5

RELATIONS AMONG QUEUING SYSTEMS

MISCELLANEOUS PROBLEMS AND COMPLEMENTS

The system Gl x /G/l. The distributiohs of waiting times and their moments for the queuing systems Glx /M/1 and MX/G/l have been discussed in Chapter 3. The discussion of these distributions for the more general case Gl x /G/1 poses problems. It is, however, possible to obtain some relations between expected values by completely elementary arguments, even for the most general case Glx /G/l. This problem, and the next one which deals with GIIG k 11, are devoted to these relations. (a) Show that for the queuing system Glx /G/l in the steady state, p = Ni/J1 < 1, the expected waiting time (in queue) of a random customer of an arrival group is given by

(mza )

W - = W -I+ - - - l q

q1

2J1

E(S) = E(V*)-E(U) =

and since Vn\

+ a~/J1z)

;:=

var(S) = a;

= P(X =

Ii = E(X), a~

+ a~

= aa;

a2

+ -f+ a~ p.

(M14)

1 LI (Ii=j L ~a ).~ J1

j=l

E(l)

which can be simplified to get the desired result. The sum of the two parts gives the result (M 14). (b)

x = 1,2, ... , m2 = E(X2)

k

The system GI/G /1. Everything else in this problem is the same as in (a), except that service is now in batches of fixed size k, and arrivals afe by Singlets. Show that in this case (when p = i\/kp. < 1) the mean wait (in queue) of an arrival is given by

variance of service time of a group = lia~

a~

x),

i\

where we have used equation (MIS) on the right-hand side. Now one can use Problem 9(b) of Chapter 2 to get W;, ' the first part of W;. To get the second part of Wq given in equation (M14), one may pr;ceed as follows. Since the probability that the customer selected randomly in his batch is positioned jth is L!=j ada, where I is the upper bound of the size of a group, the mean wait (in queue) of this customer during service of other customers in the same batch is given by

an r.v. representing the size of a group such that

ax

= p-I

i\

and Un + 1 are independent,

_ E(lz)

2i\(1 - p)

x

1

expected wait (in queue) of the first customer of a group i\2(a~ +aa~

~_.!. J1

where W~l

339

m2 -

W-=W-+ k - 1 q qk 2i\

+ a~/J1z

a2

= variance of group size.

Other quantities are as defined in Problem 9(b) of Chapter 2. Hint: Since the procedure used in this problem is similar to the one used for a special case in Problem 9 of Chapter 2, only a brief discussion is given here. Let Vr n be the service time of the rth customer of the (n + 1)th arrival gr~up (since the first arrival group is at a~ = 0; see definition in Section 2.2) and define x

L

Vr,n'

(MIS)

r=l

Now if V~~) is the waiting time (in queue) of the first customer served in the (n + l)th group, then (M16) where Sn = V~+l - U n+1 and U n+ 1 is the time between the arrival of the (n + 1)th and the (n + 2 )th groups. In steady state as n -+ 00,

(M!?)

where W~k is the mean wait (in queue) of the customer who is the last (kth) arrival in his group. Hint: Let Ur , n, 0 ~ r < k, be the inter arrival time between the (r + 1)th and the (r + 2)th arrivals of the nth group to be served and let b.e the wait (in *queuv_~f the last (kth) customer of the nth group. Then Ifwe defme Un = Lr=o Ur,n, we have

vW

n v'
+ Vn -

U*)+ n

(MiS)

where Vn is the service time of the entire nth group. Now since equation (~lS) is similar to equation (M16), one can find, in the steady state, Wqk = E(Vqk ). The second term on the right-hand side of equation (Ml?) is the expected wait of a random customer before his service group is completed and made available for service by the arrival of its kth customer. The rth customer in a service group must wait for k - r arrivals, and the mean interarrival time is 1/i\. Therefore the mean wait of a random customer for completion of his service group is k-l =--. 2i\

RELATIONS AMONG QUEUING SYSTEMS

340

(c)

Continuation of (b): the system M/G k /1. Show that if in (b) GI =M, then Little's formula for M/G k /1 is satisfied. Hint: To show that Little's formula Lq = i\Wq for the queuing system MIG k 11 is satisfied, we first need to evaluate L q . Let N q be an LV. representing the number in queue at a random epoch m the steady-state case. Then as if N;;;' k N-k, Nq = { N, if N < k where N is the number in the system at a random epoch, N P(z) Pq(z) = E(z q) = b(i\ - i\z)

(M19)

where P(z) is given in equation (6.4.5). Now from equations (6.5.5) and (M19), k-j i\ (M20) L = L+ + - - = Lq +2

11

and from wqk(a) given in equation (4.3.5),

Wqk

i\ = L + --.

(M21)

11

The use of equations (M17), (M20), and (M21) gives the desired result. Most of the results of this problem are due to Marshall (1968).

REFERENCES Allan, S. E. A., and P. L. B. Carignan (1978). Queueing theory - A derivation ~f the length of the queue for the model E k IE, 11 .. _ . Engineering Report, Royal Military College of Canada, Kingston, Onto Brockwell, P. J. (1963). The transient behaviour of a single server queue with bulk arrivals.J. Aust. Math. Soc. 3,241-248. Chaudhry, M. L. (1978). Marriage between the supplementary variable technique and the imbedded Markov chain technique - I. Transactions of the 8th Prague Conference on Information Theory, Statistical Decision Functions, Random Processes, vol. A, 133-141_ _ _ _ (1979). The queueing system MXIC/I and its ramifications. Naval Res. Logist. Quart.

26,667-674. _ _ _ _ and J. G. C. Templeton (1981). The queueing system MICE 11 and its ramifications. Eur. J. Oper. Res. 6,57 -61. Conolly, B. W. (1960). Queueing at a single serving point with group arrival. J. R. Stat. Soc., Ser. B 22, 285-298. Conte, S. D., and C. de Boor (1972). See Chapter 1. Cox, D. R., and W. L. Smith (1967). Queues. Methuen and Co., London. Easton, G. (1981). Some problems in th~ theory of single-server bulk queues. M.Sc. theSIS, Royal MilitarY College of Canada, Kmgston, Ont.

REFERENCES

341

Fabens, A. J. (1961). See Chapter 4. Fang, Wu (1960). See Wu Fang. Finch, P. D. (1962). On the transient behaviour of a queueing system with bulk service and finite capacity. Ann. Math. Stat. 33,973-985. Foster, F. G. (1961). Queues with batch arrivals I. Acta Math. Acad. Sci. Hung. 12, 1-10. _ _ _ _ (1964). Batched queueing processes. Oper. Res. 12,441-449. _ _ _ _ , and K. M. Nyunt (1961). Queues with batch departures I. Ann. Math. Stat. 32, 1324-1332. _ _ _ _ , and A. G. A. D. Perera (1964). Queues with batch departures II. Ann. Math. Stat. 35,1147-1156. _ _ _ _ , and _ _ _ _ (1965). Queues with batch arrivals II. Acta Math. A cad. Sci. Hung. 16,275-287. Gross, D., and C. M. Harris (1974). Fundamentals of queueing theory. Wiley, New York. Holman, D. F. (1977). Some problems in the theory of queues. M.Sc. thesis, Royal Military College of Canada, Kingston, Onto _ _ _ _ , M. L. Chaudhry, and A. Ghosal (1981). See Chapter 4. Jain, J. L., and R. G. Rani (1971). The transient behaviour of the queueing system CIIM/l: (=/FCFS) with arrivals in batches. Ekonom.-Mat. Obzor. 7,299-304. Kerridge, D. (1966). A numerical method for the solution of queueing problems. New J. Stat. Oper. Res. 2,3-13. Khintchine, A. Y. (1932). Mathematical theory of a stationary queue (in Russian). Mat. Sb. 39,73-84. Marshall, K. T. (1968). Bounds for some generalizations of the CI/C/l queue. Oper. Res. 16, 841-848. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Prabhu, N. U. (1965). Queues and inventories - A study of their basic stochastic processes. Wiley, New York. Takacs, L. (1961). Transient behavior of single-server queueing processes with Erlang input. Trans. Amer. Math. Soc. 100,1-28. _ _ _ _ (1962). Introduction to the theory of queues. Oxford Univ. Press, New York. Wishart, D. M. G. (1956). A queueing system with x' service-time distribution. Ann. Math. Stat. 27,768-779. Wu Fang (1960). Some results about the queueing system GI/Ekll. Chinese Math. 1,205-216. ADDITIONAL REFERENCES

Franken, P., D. Konig, U. Arndt, and V. Schmidt (1981). Queues and point processes. Akademie-Verlag, Berlin. Miyazawa, M. (1977). Time and customer processes in queues with stationary inputs. J. App/. Prob. 14,349-357. Shanthikumar, J. G., and M. J. Chandra (1980). Application of level crossing analysis to discrete state processes in queueing systems. Working paper #80-010, Dept. of Ind. Eng. and Oper. Res., Syracuse University, Syracuse, N.Y.

A.3

A.3 Proofs of the results and theorems stated here without proof may be found in many books; some references are given for the convenience of the reader.

b(U) a(u)

h(x,u)dx

where h (. , .) is a function of x, u and a ( .) and b (.) are functions of u only. Then the derivative of this integral with respect to u is given by

dg(u) -- = du

Sb(U) a(u)

There are several inversion methods, such as series expansion or the partial fractions method, for inverting the various transforms. One such method for the inversion ?f the z transf~rm F(z) = L;;'=o Inzn (which is a p.g.f. if the In are probabilities) ~ mvolves evaluatmg the following complex integral: 1Tl

In some situations the solution of a given problem is easily obtained if we differen· tiate, with respect to a parameter, a function which is given in the form of a definite integral. Of course, one way is to perform first the integration and then differentiate with respect to the requisite parameter. It is often easier and may be necessary, however, to differentiate without actually performing the required integration first. In such situations we may use the following elegant procedure due to Leibnitz. Let the definite integral g(u) be given by

S

TRANSFORM INVERSION AND RESIDUE CALCULUS

1 In = -2.

DIFFERENTIATION OF A DEFINiTE INTEGRAL WITH RESPECT TO A PARAMETER

g(u) =

3h db(u) da(u) -du+h(b(u),u)---h(a(u),u)-au du du

¢C Fn(~~ dz,

_

1

1Tl

If K(z) = 2::;;" kjz i is a p.gJ., and ko > 0, 0 I [Prabhu (1965), Feller (1968), and Parzen (1962)] . Proof: This lemma may be proved by several methods. The following geometrical proof seems to be elegant and simple and hence is given here. Since K(z) - z has a zero at z = 1, we define [(z) = (K (z) - z )/(1 - z). Note that we have defined z) in such a way that [(0) = ko > o.

[e

342

v=T

(1)

where C is a closed contour around the origin and is traversed in the positive (counterclockwise) sense. If F(z) is a rational function, one can also use a modified form of the above formula. Let F(z) = FI (z )/Fz (z), where FI (z) and F z (z) are polynomials. Suppose that the degree of FI (z) is less than that of F2 (z), or if not, consider the rational function R(z)/Fz(z) where R(z) is the remainder when FI(z) is divided by F2(Z), and therefore R(z) is of lower degree than F2(Z). There is another procedure if F(z) i~ of the form z k F I (z)/F2(z) where k is a positive integer and the degree of FI (z) IS less than that of F2 (z). In such cases one can either use the transformation z = l/wkin order to evaluate equation (1), or evaluate equation (1) by omitting the factor z and then use an elementary transform to get the final result. For details of the latter procedure, see Kleinrock (1975), and for details of the discussion on complex rational functions, see Green (I 939). Now if F(z) = FI (z )IF2(z) and the above assumptions are made, then clearly F(z) -+ 0 as z -+ 00, and by using this argument one can show that the inversion formula (1) modifies to In - --2.

FUNDAMENTAL lEMMA OF BRANCHING PROCESS THEORY

i =

Z

wherever the operations on the functions are defined.

A.2

343

To show that I(z) has a root 7/, 0 < 7/ < 1, we must show that [(0) and 1(1) have opposite signs. It is easy to see that this is so, since 1(1 - 0) = 1 - K(l)(J) + terms containing 0 and its higher powers. The converse may be seen to be true by retracing the steps backward. The uniqueness of the root follows by Rouche's theorem.

Appendix A

A.1

TRANSFORM INVERSION AND RESIDUE CALCULUS

cp c'

F(z) Z

n+l

dz

(2)

where C I is a ~losed contour large enough so as to include all the poles of F(z) but exclude the pomt z = O. It may be noted that the degree of the denominator in the integrand in equation (2) is at least 2 higher than that of the numerator. Furthermore equation (2) holds whenever F(z) -+ 0 as z -+ 00, but we use it only for the rational functions. The contour integration is easily performed by making use of the residue theorem, which may be stated as follows. Residue Theorem: Let C be a closed contour within and on which a function ~(z).is analytic except for a finite number of isolated singular pointsz l ,Zz, ... ,zn mtenor to C. If K 1 , K 2 , . . • , Kn denote the residues of F(z) at these points, then

344

APPENDIX A n

CP

F(z) dz = 2ni

C

I

Kj •

(3)

z=a

A.5

THE RIEMANN-STIELTJES INTEGRAL

{d

m

ROUCHE'S THEOREM

1 1)1 lim d m-t [(z -a)mF(z)] } . m. Z~a Z

z=a

If fez) and g(z) are functions of z, analytic inside and on a closed contour C, and if If(z) 1< Ig(z) I on C, then g(z) and g(z) + fez) have the same number of zeros inside C [Titchmarsh (1939)] .

A.6

LAGRANGE'S THEOREM

If f(z) and g(z) are functions of z, analytic on and inside a contour C surrounding a point a, and if w is such that

Iwg(z) I ~ =

(4)

fm = f(a)

FI(a) ResF(z) = F(I)( )' z=a

2

a

The method can be extended to poles of order greater than I. For. details, see Churchill (1960).

GENERALIZED ARGUMENT PRINCIPLE

Suppose that tJ;(z) is a function of a complex variable z which is analytic within and on the contour C. Suppose also that tjJ(z) is another function analytic within and on C except at a finite number of poles. If CXk, k = 1,2, ... , are the zeros of order (mUltiplicity) rk and B k , k = 1,2, ... , are the poles of order (multiplicity) Sk of the function tjJ(z) within C [tjJ(z) must not have poles or zeros on C], then the generalized argument principle states that _1_. th l/J(z)tjJ(l)(z) dz = 2m 't'c tjJ(z)

I

rkl/J(CXk)- ISkl/J(B k ).

The proof is based on the residue theorem [Whittaker and Watson (1927) and Ahlfors (1966)].

IZ-al

a + wg(~)

regarded as an equation in ~, has exactly one root in the interior of C. Further any function fCn of ~ analytic on and inside C can be expanded as a power series in w by the formula

z-+-o

If F(z) = F t (z)/F2(z), where Ft(z) and F 2(z) are both analytic at z =a, and F t (a) =1= 0, F2(a) = 0, and Fil)(a) =1= 0, then F(z) has a simple pole at z = a. Equation (5) then reduces to

<

is satisfied at all points z on the perimeter C, then the equation

1

A pole of order 1 is called a simple pole. For a simple pole equation (4) simplifies to (5) Res F(z) = lim {(z - a)F(z)}.

A.4

345

j=l

For our purpose, equation (3) may be taken as a definition of residue. A function F(z) of a complex variable z is said to be analytic (regular, holomorphic) in a region R of the complex z plane if it is single-valued and differentiable at every point of R. A point of the complex z plane at which a function ceases to be analytic is called a singular point. A function which has no singular point, finite or infinite, in the whole complex plane, must be a constant. Furthermore, a singular point of an analytic function F(z) is called an isolated singular point if it has a neighborhood which does not contain further singular points of F(z). An isolated singular point of f(z) at z = a =1= 00 is a pole of order m, m = 1, 2, ... , if and only if (z - a)mf(z) but not (z -a)m-tf(z) is analytic at z = a. If F(z) has a pole of order m ;;;. 1 at z = a, then the residue of F(z) at z = a is given by

ResF(z) = (

A.7

+ n~l

~~ :~:~ldf(1\~)[g(~)Jn}1 ~=a

[Whittaker and Watson (1927)] .

A.7

THE RIEMANN-STIEl TJES INTEGRAL AND THE LAPLACE-STI EL T JES TRANSFORM

The Riemann-Stieltjes Integral Integrals may be viewed essentially as sums, since they are limits of approximating sums. Readers may recall the definition and the usefulness of the Riemann integral which we study in elementary calculus. For some applications an extension of the definition of the Riemann integral, known as the Riemann-Stieltjes (or StieJtjes) integral, provides a substantial simplification in notation. It also affords a useful tool in fields such as physics in which mass distributions are partly continuous and partly discrete, or the theory of probability in which LV.'S are discrete, continuous, or mixed. Before defining the Riemann-Stieltjes integral, we discuss below certain concepts of elementary calculus. A function f(x) is said to be nonincreasing (either decreasing or stationary) in the closed interval [a, b 1 if and only if fCX2) Xl, where Xl and X2 is any pair E [a, b J , and is called strictly decreasing if and only if f(X2) XI. Similarly, f(x) is said to be nondecreasing (increasing or stationary) or strictly increasing according to whether f(X2) > f(x I) or f(X2) > f(x I) for X2 > Xl.

346

APPENDIX A

A function which is either nonincreasing ot nondecreasing is called monotonic. A function [(x) is called bounded on [a, b] if and only if m ~[(x) ~M for some finite values of m,M and all x E [a, b] .

By a partition of the interval [a, b 1 we mean a finite set {xo, x I, points such that

=

< ... < x n - I < For any partition of the closed interval [a, b 1 define a

Xo

<

Xl

<

X2

xn

=

... ,

x n } of

b.

i = 1,2, ... , n.

A function [(x) is said to be of bounded variation [see Apostol (1974) for this and some of the other statements given here 1 if and only if there exists a number M such that n

L IN;!

~ M

;=1

for all partitions

a = Xo

<

XI

<

X2

< ... <

Xn -

I

<

Xn

= b

A.7

THE RIEMANN-STIElTJES INTEGRAL

347

integral) is that either

1 2

g(x) is continuous and [(x) is of bounded variation, or [(x) is continuous andg(x) is of bounded variation.

Although we have defined the Riemann-Stieltjes integral for real-valued functions, the definition easily extends to complex-valued functions. For details, see Apostol (1974). Alternatively, one may define the Riemann-StieJtjes integral of g(x) with respect to a function [ex) by considering the least upper and the greatest lower bounds of g(x). For this we refer the reader to Rudin (1964) or Apostol (1974). One can easily see that if [(x) = bx, where b is a constant, the RiemannStieltjes integral reduces to the usual Riemann integral. Whereas the Riemann integral represents the area under g(x), the Riemann-Stieltjes integral does not do so, except when [(x) = x, in which case the two integrals coincide. To give such a geometrical interpretation to the Riemann-Stieltjes integral, the sum may be considered as a weighted sum, weighted by the values that [(x) takes in [a, b 1. As an application of the Riemann-Stieltjes integral to the theory of probability, let us consider [x(x) to be a density function, discrete, continuous, or mixed. Its (cumulative) distribution function or dJ. may now be written as

of the interval [a, b 1. Alternatively,r(x) is of bounded variation if and only if it can be written in the form [ex) = [1(X)-[2(X) where the functions [I (x) and [2(X) are nondecreasing and bounded on [a,bj. Also, if the function [(x) has a finite number of relative maxima and minima, a finite number of discontinuities, and is bounded in 1a, b [, then it is of bounded variation in 1a, b [. Now we are ready to define the Riemann-Stieltjes integral. Let g(x) and [(x) be real-valued functions on [a, b] _ Further let C; be a point in the ith interval of a partition of the interval [a, b J , that is, let

with

If Fxex) possesses a derivative, then Fx(x) = r=F§p(Y)dY

L:[x(X)dX

with

.C,,[x(x) dx = 1 Form the sum n

I

g(c;)!:"[;.

;=1

The limit of this sum if it exists as &m -+ 0 and n -+ 00 is called the RiemannStieltjes integral of g(x) with respect to [(x) and is denoted by

fb g(x)d[(x) .

•a

A sufficient condition for the existence of the limit of the sum (and hence of the

which is the usual way of writing the distribution function. The above discussion follows immediately from one of the properties of the Riemann-Stieltjes integral (see property 1 beloW). The Riemann-Stieltjes integral, when it exists, has a number of properties, some but not all of which are analogous to those of the Riemann integral, as may be seen from the properties stated below: If [(x) possesses a continuous derivative on [a, b 1, then

J:

g(x)d[(x) =

f:

g(x)[(!)(x)dx

APPENDIX A

348

where the integral on the right is in the Riemann sense. 2

A.8

SOME THEOREMS ON lAPLACE-STIEL T JES TRANSFORM

A.S

If at and a2 are constants, then

349

SOME THEOREMS ON lAPlACE-STIElTJES TRANSFORM AND GENERATING FUNCTIONS

laplace-Stieitjes Transform

If for some real number (3;;;;' 0,

An Abelian Theorem.

C

lim F(t) =

c E [a, bJ. 3

r«(3 + 1)

t(3

t->oo

and

If g is a constant in [a, b 1, then

f: e-

1(0:) =

o.t dF(t)

exists for Re 0:;;;;' 0, then

rgdf = g[f(b)-f(a)].

lim o:f31(0:) = 0:

4

-')00+

c.

Iffis a constant in [a, bJ, then 2

In this book we have used the special case of this theorem when (3 = O. A Tauberian Theorem. If F(t) is increasing or stationary and

exists for Re 0: > 0, and if there is some number (3;;;;' 0 such that 6

If g is continuous at a and b, and f is the constant r in Ja, b [, then

lim o:f31(a) = C 0.->0

then

f:gdf = g(a)[r-f(a)] +g(b)[f(b)-rJ.

.

PCt)

c

t(3

r«(3+ 1)

lIm - -

7

t->oo

From integration by parts, one can see that [Widder (1946)].

S:g(X)df(X) = g(b)f(b)-g(a)f(a)- I:f(X)dg(x). Generating Function

Abel's Theorem. then

The laplace-Stieltjes Transform

The definition of the Laplace-Stieltjes transform is analogous to the definition of the Laplace transform. Whereas the latter corresponds to the Riemann integral, the former corresponds to the Riemann-Stieltjes integral. Thus the function g(o:), defined by

lim

z~l-

2

provided the integral on the right exists for some range of values of 0:, is called the Laplace-Stieltjes transform of g(x).

If limn ->

[Titchmarsh (1939) J . Tauber's Theorem. If

and

00

an = a and ~;=o anz n converges for

{(l-Z) I

n=O

anzn} = a

Iz 1< 1,

APPENDIX A

350

lim n(a n -an-d

n .....

A.10

INVERSION OF A DISCRETE TRANSFORM

0,

n

Ix

~

=

lim an = a

n ..... ~

[Hardy (1949) and Whittaker and Watson (1927)J .

LINEAR DIFFERENCE EQUATIONS WITH CONSTANT AND VARiABLE COEFFICiENTS

We give below outlines of the solution procedure of two types of difference equations which have frequently been used in the theory of queues. They are:

2

I

Cj"lt·

j=l

then

A.9

Linear difference equations with constant coefficients. First-order linear difference equations with variable coefficients.

For more details on these and other difference equations we refer the reader to the publications mentioned below.

If a root, say "II, is repeated m + 1 times, then the term C1 "If in the solution is replaced by (CIO + Cux + C 12 x 2 + ... + C1,m_IX m - 1 + Clmx m ) "If. Since the complex roots occur in pairs, if need be, two terms corresponding to complex conjugate roots "/ = a ± ib may be combined into I'Y IX (c cos x8 + d sin x8) (8 = arc tan b/a). The constants Cij, C, d, ... must be chosen so that the solution satisfies the boundary or initial conditions on the Ix. So far we have discussed the solution procedure for the homogeneous difference equation. Several special methods are used to find a particular solution of the nonhomogeneous difference equation. One such method is the method of undetermined coefficients, another one is the z-transform method, still another one is by appropriately guessing from the form of the function Ix. Since the particular solutions have not been used in the text, we omit their discussion and refer the interested reader to the publications mentioned below. linear First-Order Difference Equation with Variable Coefficients

Consider the following first-order difference equation with variable coefficients:

linear Difference Equation with Constant Coefficients

x=I,2, ... ,n.

An equation of the type

The general solution of this set of equations is

x = 1,2, ... where the aj are known constants, Ii are unknown functions to be determined, and b x is a given function of x, is called a nonhomogeneous linear difference equation of order n. If b x = 0, for all x, then it is called the homogeneous linear difference equation with constant coefficients. A general solution to the above nonhomogeneous equation consists of two parts: 1 2

351

A linear combination of all solutions to the homogeneous equation, plus A particular solution to the nonhomogeneous equation.

The solution to the homogeneous part of the equation proceeds along the following lines. Lettinglx = C,,/x in the homogeneous equation leads to aoC,,/x+n +aIC,,/x+n-l

+ ... +an_IC,,/X+l

+anC,,/x =

°

and

This last equation in ,,/, being an nth-degree equation, gives n roots (real or complex, distinct or coincident). As a consequence, assuming that the roots are distinct, the general solution of the homogeneous part is written as

f

= C x

x

[InC _ ~

j=x+!

n

bj Cj

]

,

x

1,2, ... , n

where j

q

=

n

j=1,2, ... ,n.

ai>

i=1

The details of the solution are given in the text [Milne-Thomson (I 933), Jordan (1965), and Hildebrand (1965)J .

A.10

INVERSION OF A DISCRETE TRANSFORM

Let us suppose that an r.v. X assumes only nonnegative integer values. Let Ph =

P(X = k), k = 0, 1,2, ... ,N. The mth binomial moment of X is defined by

b = E(X) = m m

I (k )Pk,

h=m

m = 0,1,2, ... ,N.

m

If the moments bm determine {Ph} uniquely, then we have

APPENDIX A

352

Ph

=

f (_l)m·-h(m)b m. k

AppendixB

m=h

Proof: get

Putting the values of b m in the right-hand side of the second equation, we

I

m=k

(_l)m-h(m) k

~(k)Ph k-m

m

=Ph

BULK-ARRIVAL, BULK-SERVICE QUEUES

since 0,

k

<

m

(:) = 0,

k

>

m

(:)

(:) = (:) .= 1,

k

m.

REFERENCES Ahlfors, L. V. (1966). Complex Analysis, 2nd ed. McGraw-Hill, New York. Apostol, T. M. (1974). Mathematical analysis, 2nd ed. Addison-Wesley, Reading, MA. Churchill, R. V. (1960). Complex variables and applications, 2nd ed. McGraw-Hill, New York. Feller, W. (1968). An introduction to probability theory and its applications, vol. 1, 3rd ed. Wiley, New York. Green, S. L. (1939). The theory and use of the complex variable. Pitman, London. Hardy, G. H. (1949). Divergent series. Oxford Univ. Press, Oxford. Hildebrand, F. B. (1965). Methods of applied mathematics. 2nd ed. Prentice-Hall, Englewood Cliffs, NJ. Jordan, C. (1965). Calculus of finite differences, 3rd ed. Chelsea, New York. Kleinrock, L. (1975). Queueing systems, vol. 1. Wiley, New York. Milne-Thomson, L. M. (1933). The calculus of finite differences. Macmillan, London. Parzen, E. (1962). Stochastic processes. Holden-Day, San Francisco. Prabhu, N. u. (1965). Queues and inventories. Wiley, New York. Rudin, W. (1964). Principles of mathematical analysis, 2nd ed. McGraw-Hill, New York. Titchmarsh, E. C. (1939). Theory of functions, 2nd ed. Oxford Univ. Press, London. Whittaker, E. T., and G. N. Watson (1927). A course of modern analysis. 4th ed. Cambridge Univ. Press, London. Widder, D. V. (1946). The Laplace transform. Princeton University Press, Princeton, NJ.

As one might expect, the study of queues in which both arrival and service take place in batches is not easy. These bulk-arrival, bulk-service queues have all the complexity of the bulk queues discussed in Chapters 3 to 6. The'se queues also have some additional complications due to the necessity of breaking up arrival batches in order to construct service batches. Most of the literature we have found on these queues uses techniques which are now familiar from their use in Chapters 3 and 4 for queues with bulk arrival or bulk service. It is sometimes useful to use two or more techniques for a single model, for example, the Erlang method for the arrivals and the imbedded Markov chain method for the services. Dagsvik has presented analytical methods for bulk-arrival, bulk-service queues which are very general, but mathematically and computationally very difficult. Since we found no general method for bulk-arrival, bulk-service queues that we could present at the mathematical level of this book, we have omitted these queues from our discussion. We provide below, however, a list of publications for readers interested in such systems.

REFERENCES Alfa, A. S. (1982). Time-inhomogeneous bulk-server queue in discrete time: A transportation type problem. Oper. Res. 30,650-658. Bagchi, T. P. (1971). Contributions to the theory of bulk queues. Ph.D. Thesis, University of Toronto, Toronto, Ont. Bagchi, T. P., and J. G. C. Templeton (1972). Numerical methods in Markov chains and bulk queues. Lecture Notes in Economics and Mathematical Systems, 72, Springer-Verlag, New York. _ _ _ _ and _ _ _ _ (1973a). Finite waiting space bulk queueing systems. J. Eng. Math. 7,313-317. _ _ _ _ and (1973b). A note on the MX/G Y/l,K bulk queueing system.J. Appl. Prob. 10,901-906. and (1974). Some finite waiting space bulk queueing systems, in Mathematical methods in queueing theory. Lecture Notes in Economics and Mathematical Systems, 98, Springer-Verlag, New York, 133-137. Bhat, U. N. (1964a). Imbedded Markov chain analysis of single server bulk queues. J. Aust. Math Soc. 4,244-263. 353

354

APPENDIX B

_ _ _ _ (1964b). On the busy period of a single server bulk queue with a modified service mechanism. Calcutta Stat. Assoc. Bull. 13,163-17l. Borovkov, A. A. (1965). Some limit theorems in the theory of mass service. Theory Prob. Appl. 10,375-400. Borthakur, A. (1975). On busy period of a bulk queueing system with a general rule for bulk service.Opsearch 12,40-46. ______ and J. Medhi (1974). A queueing system with arrival and service in batches of variable size. Cah. Cen. bud. Rech. Oper. 16, 117 -126. Chaudhry, M. L.,and J. G. C. Templeton (1972). The theory of bulk-arrival bulk-service queues. Opsearch 9, 103-121. Chiamsiri, S., and M. S. Leonard (1981). A diffusion approximation for bulk queues. Manage. Sci. 27,1188-1199. Cohen, 1. W. (1980). The single server queue, 2nd rev. ed. North-Holland, Amsterdam. Dagsvik, J. 0975a). The general bulk queue as a matrix factorization problem of the WienerHopf type. Part I. Adv. Appl. Prob. 7,636-646. _ _ _ _ (1975b). The general bulk queue as a matrix factorization problem of the WienerI-Iopftype. Part II. Adv. Appl. Prob. 7,647-655. Delbrouck, L. E. N. (1970). A feedback queueing system with batch arrivals, bulk service and queue dependent service time.J. Assoc. Comput. Mach. 17,314-323. Gaur, R. S. (1972). A limited queueing problem with any number of arrivals and departures. Rev. Fr. Autom. Inf Rech. Oper. 6,87-93. (1973). An intermittent MI(X) IG(Y)/l system with multiphased capacity of the service channel. Rev. Fr. Autom. Inf Rech. Oper. 7,97-106. Goldman, A. J. (1968). Fractional container-loads and topological groups. Oper. Res. 16, 1218-1221. Hirasawa, K. (1971). Numerical solutions of bulk queues via imbedded Markov chain. Elec. En~Jpn. 91,127-136. Keilson, J. (1962). The general bulk queue as a Hilbert problem. J. R. Stat. Soc. Ser. B 24, 344-358. Kinney, J. R. (1962). A transient discrete time queue with finite storage. Ann. Math. Stat. 33, 130-136. Lambotte, J. P. (1968). Processus semi-markoviens et files d'attente, Cah. Cen. bud. Rech. Oper. 10, 21-3l. LeGal!, P. (1962). Les systemes avec ou sans attente et les processus stochastiques, vol. l. Dunod, Paris. Loris-Teghem, J. (1966a). Un systeme d'attente it arrivees et services en groupes d'effectif aleatoire. Cah. Cen. Etud. Rech. Oper. 8, 179-191. - - - - (1966b). Systemes d'attente it plusieurs guichets et a arrivees en groupes d'effectif aJeatoire. Cah. Cen. bud. Reel!. Oper. 8,98-111. _ _ _ _ (l966c). Condition necessaire d'ergodisme pour un processus stochastique lie 11 un systeme d'attente it arrivees et services en groupes d'effectif aleatoire. Bull. A cad. R. Belg. 52,382-389. Madan, K. C. (1976). Interrupted service queueing with arrivals and departures in batches of variable size and general repair time distribution. Math. Operationsforsch. Stat. 7, 139-149. Miller, R. G. (1959). A contribution to the theory of bulk queues. J. R. Stat. Soc. Ser. B 21, 320-337. Murari, K. (1972). A queueing problem with arrival and service in batches of variable size. Metrika 19, 27-35.

REFERENCES

355

Powell, W. B. (1981). Stochastic delays in transportation terminals - New results in theory and application of bulk queues. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA. Prabhu, N. U. (1965). Queues and inventories - A study of their basic stochastic processes. Wiley, New York. Rana, R. K. (1968). On a certain type of bulk queueing problem with general service time distribution. Z. Angew. Math. Mech. 48,495 -497. Rao, S. S, (1965). Bulk queues with arbitrary arrivals and exponential service time distributions. Def Sci. J. 15,1-9. Schellhaas, V. H. (1971). Ein Algorithmus fUr das zeitabhangige Verhalten eingebetteter Markov-Ketten bei den Wartesystemen M(Y)/G m /l und GIm/M(Y)/l. Unternehmensforschung 15,229-239. Sharda (1968). A queueing problem with intermittently available server and arrivals and departures in batches of variable size. Z. Angew. Math. Mech. 48,471-476. - - - - (1970). A discrete time limited space queueing problem with batch arrivals forming a Markovian chain and service in M parallel channels in batches of variable size. Cah. Cen. bUd. Rech. Oper. 12,178-194. - - - - (1979). A priority queueing problem with intermittently available phase type service. Cah. Cen. Etud. Rech. Oper. 21, 191-205. Suzuki, T., and Y. Yoshida (1971). Inequalities for many-server queue and other queues. l.

Oper. Res. Soc. lpn. 13,59-77. Syski, R. (1960). Introduction to congestion theory in telephone systems. Oliver and Boyd, London. Teghem, J., J. Loris-Teghem, and J. P. Lambotte (1969). Modeles d'attente M/G/1 et GI/M/l a arrivees et services en groupes. Lecture Notes in Operations Research and Mathematical Economics, 8, Springer-Verlag, New York. Watanabe, M., H. Miyahara, and T. Hasegawa (1977). An analysis of passenger queues at stations in series. Proceedings of the 7th International Symposium on Transportation and Traffic Theory, Japan, 717-742.

GLOSSARY OF SYMBOLS

AppendixC

bn = E(X) =

GLOSSARY OF SYMBOLS

n

b(a) b

C

c;

An(t) = interarrival time distribution with arrival rate depending on t,n Oft,n in An (t) are missing, it means arrival rate is constant) Xn arrival group size distribution with group size probability depending on n (if n in Xn is missing, it means group size probability is independent of n) Bn service time distribution with service rate depending on n (if n is missing in B n , it means service rate is constant) a quorum for service group b capacity for service group M storage capacity (if last descriptor is missing, it is assumed to be infinite) Most of the models are covered by the above notation. The only apparent exception to the above notation is the priority queuing system model discussed in Section 3.6. The notation for that model is confined to Section 3.6 and is not used elsewhere.

A(z) a (t)

Cumulative distribution function of interarrival (renewal) time U such that P(U';;;; t) = A(t) Probability generating function of {am} Probability density function of interarrival time, aU) = dA(t)/dt

ilea)

B(t)

356

f;

Laplace transform of aCt), ilea) = e-ata(t) dt P(arrival group size = m) Expected (mean or average) value of arrival group size, = L ma m Expected interarrival time, = f t dA(t) Cumulative distribution function of service time V such that P(V';;;; t) = B(t)

I

(k)Ph

h=n n

where Ph = P(X = k) Laplace transform of b (t), b(a) = fo= e -cxt b (t) dt Expected (mean or average) service time, = f t dB(t) Constant Square of the coefficient of variation of arrival

=O;!(a)2

= lip

grOUP

size

b

,

Square of the coefficient of variation of service"time, = (p Ob)2 1 Number of servers in multiserver queuing system 2 Constant 3

The above notation represents a e-server queue with:

A(t)

Probability density functi~n of service time, bet) = dB(t)ldt nth binomial moment of a random variable X,

bet)

bn

Following is a list of frequently occurring mathematical symbols together with their definitions. Symbols which occur infrequently, in isolated sections or problems, are not always included. It is possible that the symbols defined below may sometimes have other definitions in isolated portions of the text. English symbols precede Greek symbols. Symbols from each alphabet are given in their own alphabetical order. Nonliteral symbols are listed after both the English and the Greek symbols. A~n(t)!B:; b !e!M

357

e(z)

C c.dJ. D

d.L

EC·),E{o}, E[o] E 1 ,E

Eh

As superscript, denotes complementary function, FC(x) = 1 - F(x)

distribution

Probability that customer enters in the rth phase in a modified Erlangian distribution Probability generating fUnction of {c r } Expected value (mean or average) of cr = Lrer Same as d.L 1 Arbitrary constant 2 deterministic (constant) interarrival-time or service-time distribution Distribution function Expected value Exponential distribution

F'k(x)

Erlang-k, k fixed, distribution for interarrival or service time Erlang-J, J a random variable, distribution for interarrival or service time Exponential function, exp (x) = eX First-come, first-served queue discipline P(X';;;; x). The notation Fx(x) may be replaced by F(x) if it is clear what.!andom variable X is under consideration; similarly [x(x) and [x(a) may be replaced by [(x) and f(a), respectively Complementary distribution function, Fk(x) = 1 - F x(x)

[x (x)

dFx(x)ldx

lx(a)

E[e- aX ] =

EJ exp (0) FCFS Fx(x)

G

?eneral probability distribution of service times; independence usually assumed Greatest common divisor General independent probability distribution of inter arrival times

IS

GCD GI

f e- cxx dFx(x)

APPENDIXC

358

Actual idle-time random variable 2 Number of phases in system Virtual (including zero idle time) idle-time random variable

I

Iv iff Li.d.

If and only if LV.'S

L

L.T. limt-> = [(t) In (0)

L.-S.T. M N(t) -> N

O(x)

Independent identically distributed random variables Probability generating function of {k j } Probability of j arrivals (departures) during a service (interarrival) period E(N), steady-state expected (mean or average) number in system at random epoch E(Nq ), steady-state expected number in queue at random epoch E(W), steady-state expected number in system at (immediately after) departure epoch E(N-), steady-state expected number in system at (immediately before) arrival epoch Laplace transform Limit of a continuous function [(t) as t -> co Natural logarithmic function or logarithm to the base e Lap lace -S tieltjes transform 1 Exponential interarrival- or service-time distribution 2 Capacity of waiting room Number in system at time t -> steady-state number in system at a random epoch (This definition is not valid in Section 5.4.) Number in queue at time t -> steady-state number in queue at a random epoch Number in system at time t = an + 0 -+ steady-state number in system immediately after a departure epoch Number in system at time t = a~ - 0 -+ steady-state number in system immediately before an arrival epoch [(x) = 0 [g(x)] as x -> a if and only iflim x -+a [f(x)/g(x)J = K,

IKI
Pn(t) P~,P~,P~,Pn

[(x) =

0

GLOSSARY OF SYMBOLS

P(z, t) P(z; a) P(z, x) p(z,x;a)

Pn p.d.f. p.g.f. p.m.f. P(z), r(z), P+(z), pa(z)

Rea LV.

S

Wq(t),

[W;Ct)]

Wq , W;

Wq(t), [g(x)] asx -> a if and only if lim x -+a [f(x)/g(x)] = 0

Bold face denotes a matrix of one-step transition probabilities, = !Pij J or a vector One-step transition probability of going from state i to state j Steady-state joint probability for Erlang models of i customers (or phases) in system together with the customer being in the jth service (or arrival) phase 2 Steady-state joint probability of i in queue and j servers busy in multiserver queuing system Probability that n customers are in system at time t Steady-state probability of n in system immediately after an arrival epoch, after a departure epoch, before an arrival epoch,

[w;Ct)] w q , (t), [wq,(t)] Wq(a),

[w;Co:)]

w

q,

(0:),

[W;, (0:)]

z

359

at a random epoch, in that order Probability generating function of {Pn(t)} Laplace transform of P(z, t) Steady-state joint probability generating function Laplace transform of PCz, x; t) Steady-state number of phases in system at (immediately before) an arrival epoch Probability density function Probability generating function Probability mass function Probability generating function of {Pn }, {P,";"}, {P~}, {P,;'}, in that order Real part of a complex variable a Random variable Steady-state difference between service time and interarrival time,S= V-U Interarrival time between arrivals at a~ -1 and a~ Service time of kth customer Time spent in system in the steady-state case Time spent in queue by the first customer of a group in the steady-state case (subscript I is often omitted in case of single arrivals) Time spent in queue by a random customer of an arrival group in the steady-state case Cumulative distribution function of the virtual [actual] waiting time in queue of the random customer of an arrival group Expected virtual [actual] waiting time in queue of a random customer of an arrival group Expected virtual [actual] waiting time in queue of the first customer of an arrival group (subscript I is not used in case of single arrivals) Probability density function of virtual [actual] waiting time in queue of a random customer Probability density function of virtual [actual] waiting time of the first customer to be served in an arrival group (This definition is not valid in Section 3.6.) Laplace-Stieltjes transform of the distribution function of the virtual [actual] waiting time in queue of a random customer Laplace-Stieltjes transform of the distribution function of the virtual [actual] waiting time in queue of the first customer to be served in an arrival group (This definition is not valid in Section 3.6.) z -transform variable

360

APPENDIXC

GLOSSARY OF SYMBOLS

Greek Symbols Parameter of the Laplace or Laplace-Stieltjes transform Infinitesimal interval of time Dirac delta function 0 (x) = 0 for all x other than x = 0, is 0 (x) dx = I; as a consequence e -ax /) (x) dx such that =1 r = k I A Kronecker symbol, = { otherwise

Ct

/::"t /) (x)

f;;

A A(t)

Ae fJ. k

n

On

b1

bi

+ 0 (o~ -

0)

o~

0&

o.k

T

•

Traffic intensity, utilization factor, or congestion index Epochs (instants) of departures (arrivals) ordered by the temporal parameter n An epoch immediately after a departure (before an arrival) Variance of arrival group size Variance of service time Variance of a random variable X Time; sometimes T is used in dimensionless form as a ratio of mean service time to mean interarrival time, particularly in Chapter 6

Greatest integer contained in argument, [x J = n, n .;;; x n

+I

2 Simple brackets 3 Matrix notation min (0, 0), as in (X)- = min (0, X)

= max (0, 0), as in

(xt = =

*)

(

:3

Ix,

X.;;; 0

\ 0,

X>O

max (0, X)

f 0,

lX,

Usually denotes convolution Superscript parenthesis:

X.;;; 0

x>o

<

Order of differentiation Order of convolution

Bar over symbol indicates transform of a function "given," as in peA IB) = Prob(event A given event B) Division, as in alb = a .;. b Approximately equal to Set membership Such that Implies Implies and is implied by Denotes the number of combinations of n objects taken r at a time,

0,

= { n!/(r!(n -r)!),

n
Less than or equal to Greater than or equal to

b z ... b k (product notation)

Nonliteral Symbols

[oJ

E

Conditional service rate, 1)(x) = b (x)/(l - B(x» Expected (mean or average) arrival rate independent of t, = lla Expected (mean or average) arrival rate at time t Effective input rate Expected (mean or average) service rate per server, = lIb

1)(x)

i=l

1 2

f;;

0:

361

1 1

An interval (simply connected subset of the real line) open on the left but closed on the right, {x: c < x';;; ( [. , . [, [. , .J, and J. , .[ are interpreted similarly One of the definitions of the exponential function

AUTHOR INDEX

Authors referred to in the text by year generally are given here. Abol'nikov, L. M., 240, 249, 282, 292 Abramov, A. Kh., 203, 234 Abramowitz, M., 14, 17, 18,20,32,90, 104 Adiri, I., 175 Ahlfors, L. V., 344, 352 Alagaraja, K., 203, 234 Alfa, A. S., 353 Allan, S, E. A., 325, 340 Apostol, T. M., 347, 352 Arndt, U., 326, 341 Badii, L., 14, 33 Bagchi, T. P., 353 Bahadur, R. R., 125, 175 Bahary, E. S., 203, 234 Bailey, N. T. J., 20, 32,107,125,175,185, 186, 203, 209, 232 Baily, D. E., 275, 294 Barber, B., 121-122, 156, 175 Bartlett, M. S., 40, 104 Beightler, C. S., 3. 32 Bhalaik, H. S., 151, 177 Bharucha-Reid, A. T., 46, 104 Bhat, U. N., 84, 104, 150, 151, 177, 178, 187, 202, 226, 232, 234, 353, 354 Blake, F., 1,32 Bloemena, A. R., 185,232 Borovkov, A. A., 354 Borthakur, A., 203, 226, 228, 232, 233, 234, 275, 294, 354 Brockmeyer, E., 8, 32, 107, 175 Brockwell, P. J., 127, 175,295,340 Bunday, B. D., 32 Burington, R. S., 222, 232 Burke, P. J., 93, 104, 172, 175,249,284,292 Cadzow, J. A., 3, 32 Carignan, P. L. B., 325, 340 Chakravarthy, S., 151, 178 Champernowne, D. G., 125, 175 Chandra, M. J., 325, 341

Chang, W., 202, 234 Chaudhry, M. L., 16,33,151, 172, 176, 177, 185,202.226,230,233, ti4, 249, 267, 281,284,293,296,340,341,354 Chiamsiri, S., 354 Chung, K. L., 46, 104 Churchill, R. V., 344, 352 <;:inlar, E., 40, 84, 104 Clarke, A. B., 125, 176 Cobham, A., 141, 176 Cohen, J. W., 71, 84, 104, 114, 176,274, 292,354 Conolly, B. W., 71,84, 104, 121, 156, 158, 176, 304, 326, 340 Conte, S. D" 15, 32, 330, 340 Cooper, R. B., 71,105 Copson, E. T., 20, 33 Cosmetatos, G. P., 275, 294 Cox, D. R., 40, 57,73, 105,259,261,292, 295,340 Crabill, T. B., 129, 176 Crane, M. A., 202, 234 Craven, B. D., 203, 235 Cromie, M. V., 16,33.249,267,281,284, 293 Crommelin, C. D., 153, 176, 258, 293 Dafermos, S. C" 71, 105 Dagsvik, J., 354 Dave, H. B., 203, 235 Deb, R. K., 203, 235 de Boor, c., 15,32,330,340 Delbrouck, L. E. N .. 354 Dick, R. S., 190, 233 Doob, J. L., 41, 105 Douglas, J. B., 16,33 Downton, F., 186,205, 208, 233 Dynkin, E. B., 42, 105 Easton, G., 230, 233, 326, 340 Eilon, S., 172, 176

363

Erdelyi. A .. 14.33 Erlang, A. K., 153, 176, 258, 293 Ernst, M., 134

Holman. D. F., 223, 226, 233, 249, 293, 296, 341 Ivnitskiy, V. A., 133, 176,203,235

Fabens, A. J., 192,233,274,293,322,341 Fainberg, M. A., 138, 151, 176, 177 Fang Wu, see Wu Fang Feller, W., I, 16,23,33,46,54,73,78, 105, 342,352 Finch. P. D., 57, 105, 168, 176,202,209, 216,233,295,341 Forsyth, A. R .. 20, 33 Foster, F. G., 121, 151, 176, 177,216,233, 296, 326, 341 Frank, H .. 202, 203, 235 Frankel. E., 185, 234 Franken, P., 82, 105,326,341 Fujisawa, T., 232, 233 Fukuta, J., 202, 235 Gani, J., 40, 105 Garber, H. N., 134, 176 Gaur. R. S., 354 Gaver, D. P., 46,105,107, 1l3, 116, 176 Georganas. N. D .. 203, 235 Ghare, P. M .. 267, 293 Ghosal, A., 180.209,226,233.296,341 Gnedenko, B. V., 9, 33 Goldman, A. J., 354 Goyal, J. K., 151, 177,203,235 Grassmann, W. K., 16.33,249,281,284, 293 Green. S. L., 343, 352 Griffin, W. C., 222, 233 Gross, D .. 45, 54, 105, 164, 176,209,233, 295, 341 Gupta, S. K .. III, 151, 176, 177 Halstr0m, H. L.. 8, 32, 107, 175 Handa, J. M .. 203, 235 Hardy, G. H .. 350. 352 Harris, eM., 45. 54, 105, 132, 164, 176, 203, 209. 233, 235, 295, 341 Hasegawa, T., 355 Hawkes, A. G .. 150, 176,249,293 Henderson, J. C, 290, 293 Henderson, W" 71, 105 Henrici, p" 15,33 Hildebrand. F. B., 351, 352 Hille, E.. 85. 105 Hinkley, D. V., 259, 261, 292 Hirasawa, K., 354

Jackson, R. R. P., 134, 164, 165, 176,290, 293 Jain, J. L., 151, 177, 202, 235, 304, 341 Jaiswal, N. K., 46, 57, 63, 105, 150, 176, 185, 186,207,209,233 Jensen, A" 8, 32,107,175 Jewell, W. S., 172, 176 Jordan, c., 262, 293, 351, 352 Jury, E. I., 3, 33 Kabak, 1. W., 249, 283, 285, 293 Kambo, N. S., 151, 177 Karlin, S., 40, 54, 73, 79, 105 Kasbyap, B. R. K., 203, 235 Kaufman, H., 14, 33 Keilson, J., 151, 177,354 Kemeny, J. G., 46, 105 Kendall, M. G., 14,33,87, 105 Kerridge, D., 151, 177,295,341 Khintchine, A. Y., 62, 105,207,233,296, 341 Kingman, J. F. C, 2, 33, 84, 105 Kinney, J. R., 354 Kleinrock, L., 14, 33, 39, 40, 54, 55, 72, 84, 105,151. 177,343,352 Kolesar, P., 203, 234 Kolmogorov, A. N., 9, 33 Kosten, L., 57, 105 Kotiab, T. C. T., 203, 235 Krakowski, M., 72, 106, 172, 176. Kuczura, A., 258, 288, 293 Laha, R. G., 13,33 Lambotte, J. P., 354, 355 Ledermann, W., Ill, 125, 176 Lee, A. M., 202, 234 Le Gall, P., 354 Leonard, M. S., 354 Levy, P., 41,106 Lindley, D. V., 54, 57, 106, 168, 176 Lippman. J. A., 151, 177 Little, J. D. C, 176 Loeve, M., 2, 33 Loris-Tegbem, J., 354, 355 Love, R. F., 274, 293 Lukacs, E., 13, 33 Luke, Y. L., 20, 33 Lwin, T., 209, 233

McMahon, G. B., 203, 236 Madan, K. C, 354 Magnus, W., 14, 33 Markin, P. G., 203, 235 Marshall, K. T., 340, 341 Maxwell, W. L., 172, 176 May, D. C., 222, 232 Medhi, J., 226, 228, 233, 275, 294, 354 Meisling, T., 71, 106, 193, 197,233 Mercer, A., 203, 233 Miller, H. D., 40, 105, 150 Miller, R. G., 354 Milne-Thomson, L. M., 351, 352 Mitten, L. G., 3, 32 Miyahara, H., 355 Miyazawa, M., 326, 341 Moban, C., 203, 235 Mohanty, S. G., 151, 177,203,235 Moore, S. C, 151, 178 Morse, P. M., 114, 125, 134, 176, 216, 234 Muntz, R. R., 151, 177 Murao, H., 203, 235 Murao, Y., 202, 209, 234, 235 Murari, K., 151, 178,203,236,275,294,354 Nadarajan, R., 275, 294 Nair, S. S., 202, 228, 234, 235 Nakamura, G., 151, 178,202,203,235 Narasimham, G. V. L., 151, 178 Natarajan, R., 197,234 Nemhauser, G. L., 3, 32 Neuts, M. F., 13,33,71,73,78, 105. 106, 150, 151, 178,202,223,228,234,235, 275, 294 Neveu, J., 2, 33 Newell, G. F., 84, 106 Nickols, D. G., 134, 164, 176 Novaes, A., 185,234 Nyunt, K. M., 216, 233, 296, 341 Oberhettinger, F., 14, 33 Palm, C., 52, 106 Parzen, E., 13, 23, 33, 40, 45, 106, 285, 293, 297,341,342,352 Pearce, C, 275, 294 Pegden, CD., 125, 177 Perera, A. G. A. D., 192,233,274,293,296, 341 Piaggio, H. T. H., 20, 33 Pike, M. C, 122, 177 Pollaczek, F., 258, 293

Polya, G., 3, 33 Polyak, D. G., 275, 294 Powell, W. B., 355 Prabhu, N. U., 40, 54, 71, 73, 84, 106, 164, 177, 180, 2!1, 234, 304, 316, 317, 341, 342, 352, 355 Rainville, E. D., 90, 106 Ramaswami, V., 151, 178 Rana, R. K., 355 Rani, R. G., 304, 341 Rao, S. S., 355 Regis, R. C, 57, 106 Restrepo, R. A., 114, 177 Reuter, G. E. H., 111, 125, 176 Reynolds, J. F., 277, 293 Riordan, J., 283, 294 Roberts, G. E., 14, 33 Rodemich, E., 151, 177 Roes, P. B. M., 274, 294 Romanovsky, V. 1., 48, 106~ Rosenlund, S. I., 151, 178 Rosenshine, M., 125, 177 Ross, S. M., 116, 151, 177 Rudin, W., 347, 352 Runnenburg, J. Tb., 202, 235 Russell, A., 15, 33 Saaty, T. L., 164, 177,290,294 Sack, R. A., 111, 125, 177 Sahbazov, A. A., 172, 177,277,294 ScbelIhaas, V. H., 355 Schmidt, V., 325, 341 Seal, H. L., 40, 106 Seneta, E., 48, 106 Serfozo, R. F., 203, 235 Shanbhag, D. N., 122, 151, 177, 178,240, 294 Shanthikumar, J. G., 325, 341 Sharda, 151, 178,203,235,275,294,355 Sharma, S. D., 151, 178 Shyu Kwang-Huei, 274, 294. See also Xu Guang-Hui Sim, S. H., 275, 294 Singh, V. P., 190,234 Smith, W. L., 54, 78, 79, 106, 165, 177, 180, 234, 295, 340 Snell, J. L., 46, 105 Soriano, A., 151, 172, 177, 178 Stegun, l. A., 14, 17, 18,20,32,90, 104 Steinijans, V. W., 96, 106 Stidham, S., 172, 177, 180,234 Stuart, A., 14,33,87, 105

366 Stuart, L M., 203. 236 Suzuki, T., 122, 177, 355 Syski, R .. 355

Takacs, L., 40. 63, 64, 73, 78, 84, 106, Ill. 122,127,150,177,178,192,224,225, 234,275.292,294,296,304,311,317, 319,341 Taylor. H. M., 40, 54, 73, 79, 105 Taylor, R. G" 15,33 Taylor. S. ]" 2, 33 Teghem, J" 355 Templeton, J. G. C, 185,202,226,233, 234, 275, 294, 296, 340, 353, 354 Thompson, J. W., 203, 235 Titchmarsh, E, C, 345, 349,352 Tricomi. F. G .. 14, 33 Tsvirkyn, A, D., 203, 234

AUTHOR INDEX Van Dantzig, D., 185, 234 Van Hoorn, M, H., lSI, 178 Watanabe, M" 355 Watson. G. N" 344, 345, 350, 352 Waugh, W. A. O'N., 203, 235 Weiss, H. J., 203, 236 Whittaker, E, T., 344, 345, 350, 352 Widder, D. V., 349, 352 Wishart, D, M, G" 46, 71, 106, 196,234, 302, 304, 341 Wu Fang, 275, 294, 304, 341 Xu Guang-Hui, 275, 294. See also Shyu Kwang-Huei Yao, D. D, W., 151, 178 Yasnogoridskiy, R. M., 249, 292 Yoshida, Y., 355

SUBJECT INDEX

The presence of subscripts and superscripts makes it difficult to arrange the queuing models given below in an unambiguous alphabetical order. Recall that in queuing models the exponential distribution of interarrival time or service time may be denoted by M, E or E, interchangeably, Alternating renewal process, 82-84 Applications: biological science, 23, 32 bus, multichannel bulk-arrival queue, 282 bus, renewal process, 80 car ferry, 179,202-203 computers, 35, 216-219 computing systems, 202 dams, 180,211 equipment failure, 82 guided tours, 179 multichannel bulk queues, 237 processor-sharing, 151 taxis and customers, 203 telephone systems, 283 toll gate, 37 traffic, 23, 179 trains, 179 transit system, 218-221 transportation networks, 202 see also Examples Autocorrelation, 86 Bulk-arrival, bulk-service queues, 353-355 Bulk-arrival queues, 107-178 Bulk-service queues, 179-236 Busy and idle periods, expected, 322-325 Ek / G/I,325 GI/E,/I,323 GI'/M/I,322 M/E~,B11, 324-325 M/GB/I and M/Gk/l, 325 M/G k,BIl ,324 M X/G/I,324

Characteristic equation, 67 Characteristic function, 13 Collective marks, method of. 185 Coefficient of variation, of deterministic LV., 28 of Erlang r.v" 28 of exponential LV" 28 of hyperexponential LV., 28 Concentrated service, 202 Conservation principle, 71-72 Convolution, 4-8. 24 Cumulant generating function. 13

D'/M/I,156 D'/M/c,258-261 delay measure, 260 limiting behavior of No. 258-260 Differential equations: partial. 20, 31-32 solution by Lagrange's method, 20 Differentiation, definite integral, 342 Dirac delta function, 114 Dirichlet series. 13 Discrete time analysis, 71. See also M/GB/I, imbedded Markov chain Distribution: binomial, 22, 125 busy period, 39 chi-square, 8 deterministic, 30 Erlang: generalized, 24. 133 modified. 22, 30, 108

367

368

SUBJECT INDEX

ordinary, 7-8, 22, 25, 133, 139, 184, 228, 229, 324 exponential, 7, 9, 25 gamma, 8 geometric, 23, 113, 278 hyperexponential, 9, 27-28 idle period, 39 negative binomial, 22 Poisson, 11,23,35,75 regular, 30 renewal, 74-75 system length, 38 truncated, continuous, 9-10, 29 truncated, discrete, 10-11, 23, 29 waiting time, 38 see also Life Double-ended queue, 203 E/G k /l,309-316

queue size N at arbitrary instant, 310 queue size N- before arrival, 310 queue size N' after departure, 311 relations among p,g,L's of N-, N, N', 311-316 see also M/Gk/1 EklE,1 I, 308-309, 320-321, 332-334 EkG/I,316-317 expected busy and idle periods, 325 E,IG/I and E/Gk/l, 317-321 E.IGI/l, 321-322 EVE,/I, 155-156,301-302 Ef/E,/I: generalized Eriang input, 133-139, 161-166 limiting behavior of N(t), 133-137 L, Lq, 137-139 ordinary k-Erlang input, 139 special cases: E,/M/I,165 Ek/M/I, 163-165,331-332 E~/D/I, 139 Ef/D/I, 162 E§/E,/I,162 M X/D/I,168 MX/E,/I, 162, 166

E,/M k •B/l,230 Erlang: A, K, Erlang, 7 delay system, 285

distribution, 8, 23, 228, 229 generalized, 24, 133 modified, 22, 30, 108 ordinary, 7-8, 22, 25, 133, 139, 184, 228, 229, 324 loss system, 285 Examples: airport luggage handling, 282 computer core storage system, 216 equipment failure, 82 GI/G/I,53 GI/M/c, 50

imbedded Markov chains, 50 M/G/I, 50, 54-65 p.d.f, of largest r, v,, 88 p.d.L of smallest r. v., 89 Poisson process branching, 97 supplementary variable technique, 54-63 transit system, 219 see also Applications Function, 16-20 autocorrelation, 86 Bessel, 17-18 characteristic, 13 conditional rate of failure, 27 correlation coefficient, 86 Dirac delta, 114 gamma, 16-17 generating, see Generating functions hazard, 26-27 hypergeometric, generalized, 18-19 hypergeometric, ordinary, 19 intensity, 27 mean value, 75, 90 modified Bessel, 18 renewal, 75-78, 90, 91 Generalized argument principle, 344 Generating functions, 13-14 cumulant, 14 moment, 13 probability, see Probability generating function GEOM/GB/I, 193-198,211-212 constant service time, 194 geometric service time, 196 limiting behavior of N~, 193-198 waiting time (in queue), 198

SUBJECT INDEX GIID/c and GIID'/l, 202 GIIE,/l, 302-304

expected busy and idle periods, 323-324 waiting time (in queue), 327 GIIE,./I and GI'/M/I, 304-309, 327-329 GI/G/I, 53-54, 102-104 with delay for first customer in busy period, 56-57 Lindley's results for, 54 k

GIIG /l,339 GIIM/I, 65-71, 100-102,155,202

characteristic (operator) equation for, 67 number N- in system before arrival, 67-68 number N in system at random epoch, . 68-69 virtual and actual idle time for, 69-71, 102 waiting time (in queue), 68 GIIM B/l,292 B GIIM Ic, 267-275, 291-292 limiting behavior of N-, 267-274 waiting time (in queue), 274 GIIM'/l,202 GI'/E/I,296-302

expected busy and idle periods, 322-323 idle period, 337 queue size N at arbitrary instant, 298, 326, 336-337 queue size N- before arrival, 297 queue size N' after departure, 297 relations among p.g.f.'s of N-, N, N" 298-301 see also Gr/M/l GI'/M/l, 116-122, 156-158,334-338 limiting behavior of N~, 116-121, 297 limiting behavior of N~, 297 limiting behavior of N(t), 298, 336-337 limiting number in system after arrival, 334-336 waiting time (in queue), 121, 156, 329 see also GI'/E/l GIx/G/I, 168-169,338-339 Gl x/G k •B/I,334 Gl x/M/1, 72, 158-159

Glx/M/c, 275 Increments:

independent, 42 stationary, 42

369 Inversion:

of discrete transform, 351 of other transforms, 14 Kolmogorov equations, 109 Laplace-Stieltjes transform (L-S.T.), 23, 345 Leibnitz's rule, 124 Length-biased sampling, 92, 174 Life: current, 81 excess, 80 joint distribution of past and residual, 93-94 past, 81,91,171 remaining, 80 residual, 80, 91 total,91 Linear difference equations, 350-351 Little's formula, 165, 166, 172,208,247 MIElE, . . . Ek/l, 231 M/Ef!l, 184-185 M/E:/I,314-316 M/G/I, 54-65, 217

busy period, 63-65 expected busy and idle periods, 83-84 idle period, 63-65 relation between p.g.f.'s of N, N', 207 waiting time (in queue), 62 waiting time, Lindley's approach, 54-56 functional equation approach, 99 supplementary variable approach, 98 B M/G /l, 180-186,203-209,216-219 expected busy and idle periods, 325 imbedded Markov chain, 183, 203-206 limiting behavior of Nq(t), 180-185 special cases: M/D B /I,205

with variable capacity, 208 waiting time (in queue), 185-186 M/EP/I, 204, 207, 215 M/E~/l, 204. See also E,/E,/I M/ElVl,207

M/G/I, 206-207, 217, 219. See also M/G/I M/G'/1,217 M/M B/I,217 M/GB/I/M, 186-190,209

SUBJECT INDEX

370

limiting behavior of

N~,

186-190

special cases: MIDI 11M, 209

M/EkIl/M,21O MIMI 11M, 209-210, 216

M/G~/I, 202 M/G k /l, 191-193,223-225,340

expected busy and idle ;-eriods, 325 limiting behavior of N~ and V&nl, 191-193 waiting time (in queue), 192-193 see also E/G'/I M/G"B/I ,222-228

with balking, 190 expected busy and idle periods, 324 special cases: M/E:,BII, 324-325 M/M B /l,291-292 M/Mk/l, 225, 331 M/M k,B/ I ,225-231 M/Mk,oo 11, 226 M/G~,B/I, 202

Mk/Mic/c, 285

M/M B/1/MB, 198-202,212-216

limiting behavior of N~, 201-202 transient behavior of N~, 198-201 M/M"/c, 261-267, 289-290 limiting behavior of Nq(l), 264-265, 290 lransient behavior of Nq(t), 261-264 waiting time (in queue), 265-266, 290 waiting time (in system), 266-267 M/M"B/c , 275 M/M k ,oo/c,275 M'/M/I, 116, 120-127, 329 limiting behavior of N~, 125-126 moments, 160 number of batches served in a busy period, 126 transient behavior of N~, 122-125 waiting time (in queue), 126 MX/D/c, 252-258, 286-289 limiting behavior of N~, 252-254, 286-288 probability of no service delay, 255-257, 286 waiting time (in queue), 257-258, 288-289 MX/EI/I, Ill, 152, 154 MX/EJ/I, 107-116, 153, 155 limiting behavior of N(t), 112 L, Lq , W, 115-1I6

special cases: M/M/I, 100, 113, 115 M X/D/I,163 Mk/EJ/l, 113 MX/Ek/1, 112-113, 115-116, 151, 154,

stochastic matrices of, 48

transient (non-recurrent) state of, 48 Memoryless property of exponential, 37, 85, 93 Multichannel bulk queues, 237-294

162, 166 MX/EI/I, Ill, 113, 115, 152, 154

transient behavior of N(t), 107 waiting time (in queue), 114-115 M X/G/I,72, 130, 167-175 expected busy and idle periods, 324 by imbedded Markov chain, 174 by supplementary variable technique, 172 waiting time (in queue), 167-169,249 with warm-up time, 167-172 MX/Gn/l, 127-133, 161 limiting behavior of N~, 127-129 modifications, 130 M~/Gn/l, 133 MX/HE n /l,151 MI", M~2/GI, G,ll, 140-150, 166-167,249 limiting behavior of joint distribution, 145 transient behavior of joint distribution, 140 waiting time (in queue), 146 non-priority customer, 148 priority customer, 146 MX/M/c, 240-252, 277-285 busy period (when c 2), 250-252 limiting behavior of N(t), 243-245 special cases: MIMic, 277, 289-292 Mk/M/c, 278, 281, 283 X M /M/2(/,11' /',;), 285 transient behavior of N(t), 240-243 waiting time (in queue), 245-249

=

M X/M/c/c,283 MX/M/oo, 241, 277 M'(t)/M/oo, 237-240, 275-277

transient behavior of N(t), 237-240 MX/SM/I, 150

Markov chains, 41 aperiodic state of, 49 ergodic state of, 50 homogeneous, 48-52 irreducible, 48-49 periodic state of, 48-49 recurrent state of, 48

371

SUBJECT INDEX

Occupation time, 39,224,230-231 Order statistics, 86 Partial differential equations, 20, 31-32 Past life, 81, 91,171 Phases, 8, 22, 36, 45, 108, III, 133, 134 Pollaczek-Khintchine transform formula, 56 Priority queues, 140-150, 166-167 non-priority customer, 148 priority customer, 146 Probability generating function (p.g,f.), 2-3 multivariate, 5 Process, stochastic, 40-45 special cases: alternating renewal, 82 arrival, 34 bulk-arrival Poisson, 42-44, 89-90 completion time, 140 compound, 98 compound Poisson, 42, 98, 127 counting, 74 delayed renewal, 73 difference of two Poisson, 97 equilibrium renewal, 74, 96 general renewal, 73 Markov, 41 modified renewal, 73 non-Markovian, 45 ordinary renewal, 73, 75, 91, 96 Poisson, 35, 75, 84-88, 91 Poisson, by branching a given Poisson, 97 regenerative, 52, 91 stationary renewal, 74 sum of Poisson processes, 97 time-homogeneous Poisson, 42-45, 84, 107, 127 Queue discipline: FCFS, 36, 39, 53, 141 head-of-the-line priority, 141 LCFS,36

priority, 140 Quorum, 179 Random customer, 93 Random events, 88 Random variables (r,v.'s), addition of, 4, 27. 30 difference of, 24 independently and identically distributed (i.i,d.r,v.'s),4 largest, 88 lattice. 2-3 least, 89 mean of, 3, 12 moments of. 3, 12 non-lattice. 2-3 sum of continuous ij,d .. 6-8 sum of discrete i.i,d" 4-6 variance of, 3, 12 (X-Yf,25-26 Regeneration points, 52 Renewal theory, 72-84 density, 76 equation, 76, 90 function, 75, 78 process, 72 Residual life, 79-80 paradox of, 80-81 Residue calculus, 101,343 Riemann-Stieltjes integral. 345-348 Service mechanism, 36 Strong law of large numbers, 8 Techniques: imbedded Markov chain, 46-52 integral equation, 53-54 modified Erlangian, 46 phase, 45-46 supplementary variable, 57-62 Theorems: Abelian, 349 Abel's, 349 Blackwell's, 78-79, 94, 96 elementary renewal, 78 key renewal, 79, 80, 95, 96 Lagrange's, 345 Rouche's, 345

SUBJECT

372 Tauberian, 349 Tauber's, 349 Time marks, 193 Traffic intensity, 37 Transforms: Fourier, 13 Fourier-Stieltjes, 13 inversion of, 14 Laplace, J 1-13

Laplace-Stieltjes, 1 J residue calculus, 343 Waiting time, definition of: actual, 38-39 virtual, 38-39 Warm-up time, 167 Z-transform, 3

INDE~