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Âûïóñê 7, 2006

ÓÄÊ 519.68

ÅÀËÈÇÀÖÈß È ÈÑÏÎËÜÇÎÂÀÍÈÅ ÇÀÄÀ× ÌÎÄÅËÈÎÂÀÍÈß ÂÇÀÈÌÎÄÅÉÑÒÂÈß ÂÎÄÎÎÄÀ Ñ ÒÂÅÄÛÌ ÒÅËÎÌ Â ÑÏÅÖÈÀËÈÇÈÎÂÀÍÍÎÌ ÏÎ ÀÌÌÍÎÌ ÊÎÌÏËÅÊÑÅ Â. Â. Ïîïîâ, Ï. À. Ñåìèí àññìàòðèâàþòñÿ ïðàêòè÷åñêèå âîïðîñû ïðîãðàììíîé ðåàëèçàöèè è èñïîëüçîâàíèÿ ðàçëè÷íûõ ìîäåëåé âçàèìîäåéñòâèÿ âîäîðîäà ñ òâåðäûì òåëîì â ñïåöèàëèçèðîâàííîì ïðîãðàììíîì êîìïëåêñå.

1.

Ââåäåíèå

 íàñòîÿùåå âðåìÿ â ìèðå ïðîâîäÿòñÿ øèðîêîìàñøòàáíûå èññëåäîâàíèÿ ïî ïîèñêó ýåêòèâíûõ ñïîñîáîâ õðàíåíèÿ è òðàíñïîðòèðîâêè âîäîðîäà, êîòîðûé ðàññìàòðèâàåòñÿ â êà÷åñòâå ïåðñïåêòèâíîãî ýíåðãîíîñèòåëÿ â ðàìêàõ êîíöåïöèè âîäîðîäíîé ýíåðãåòèêè [13℄. Ââèäó áîëüøîé ñòîèìîñòè è òðóäîåìêîñòè ïðîâåäåíèÿ ðåàëüíûõ ýêñïåðèìåíòîâ ïî èçó÷åíèþ âçàèìîäåéñòâèÿ âîäîðîäà ñ òâåðäûì òåëîì ÷àñòüþ èññëåäîâàòåëüñêèõ ðàáîò ÿâëÿåòñÿ ìàòåìàòè÷åñêîå ìîäåëèðîâàíèå.  ÷àñòíîñòè, ýòî ìîäåëèðîâàíèå ïðîöåññîâ ãèäðèðîâàíèÿ è äåãèäðèðîâàíèÿ ïîðîøêîâ ìåòàëëîâ â ñâÿçè ñ êîíñòðóèðîâàíèåì ãèäðèäíûõ íàêîïèòåëåé (àêêóìóëÿòîðîâ) âîäîðîäà, ìîäåëèðîâàíèå ïðîöåññîâ ïðîíèöàåìîñòè ÷åðåç êîíñòðóêöèîííûå ìàòåðèàëû è çàùèòíûå ïîêðûòèÿ  ïðîáëåìû âûáîðà ìàòåðèàëîâ è ïðîáëåìû âîäîðîäíîé êîððîçèè (õðóïêîñòè) ìåòàëëîâ, ðåøåíèå îáðàòíûõ çàäà÷ îïðåäåëåíèÿ ïàðàìåòðîâ ìîäåëåé ïî ýêñïåðèìåíòàëüíûì äàííûì. Òàêèå ðàáîòû âåäóòñÿ â ëàáîðàòîðèè ÌÏÒÑ ÈÏÌÈ ÊàðÍÖ ÀÍ íà ïðîòÿæåíèè ïîñëåäíèõ ëåò [48℄. Ââèäó ðàçíîîáðàçèÿ è ñïåöèè÷íîñòè çàäà÷, ìîäåëåé, ÷èñëåííûõ ìåòîäîâ è àëãîðèòìîâ çà ïîñëåäíèå ãîäû â ëàáîðàòîðèè ñîçäàíî ìíîæåñòâî ðàçëè÷íûõ ïðîãðàìì. Îñíîâíûì íåäîñòàòêîì ýòèõ



Â. Â. Ïîïîâ, Ï. À. Ñåìèí, 2006

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ïðîãðàìì, ïðåïÿòñòâóþùèì äàëüíåéøåìó èõ èñïîëüçîâàíèþ, ÿâëÿåòñÿ "íåäîîîðìëåííîñòü" äëÿ èñïîëüçîâàíèÿ êîíå÷íûì ïîòðåáèòåëåì  èçèêîì-èññëåäîâàòåëåì èëè èíæåíåðîì. Âîçíèêëà íåîáõîäèìîñòü óíèèöèðîâàòü è èíòåãðèðîâàòü èõ â åäèíûé ìàñøòàáèðóåìûé ïðîãðàììíûé êîìïëåêñ (ÏÊ) [9℄, êîòîðûé ïîçâîëèë áû ýåêòèâíî õðàíèòü, ðåäàêòèðîâàòü, îáðàáàòûâàòü ìîäåëüíûå è ýêñïåðèìåíòàëüíûå äàííûå.  ñòàòüå ðàññìàòðèâàþòñÿ âîïðîñû ðåàëèçàöèè â äàííîì ïðîãðàììíîì êîìïëåêñå çàäà÷ ìîäåëèðîâàíèÿ âçàèìîäåéñòâèÿ âîäîðîäà ñ òâåðäûì òåëîì è èõ èñïîëüçîâàíèÿ.

2.

Êîíöåïöèÿ ïðîãðàììíîãî êîìïëåêñà

Èññëåäîâàíèÿ âçàèìîäåéñòâèÿ âîäîðîäà ñ òâåðäûì òåëîì ïðåäïîëàãàþò ïðîâåäåíèå ýêñïåðèìåíòîâ. Ïðè ýòîì ïðîèñõîäÿùèå èçèêîõèìè÷åñêèå ïðîöåññû, âîçìîæíî îïèñàòü ñ íåêîòîðîé òî÷íîñòüþ â âèäå ìàòåìàòè÷åñêîé ìîäåëè.  çàâèñèìîñòè îò çàäà÷è, ýêñïåðèìåíòàëüíîãî ìåòîäà, ñòåïåíè äåòàëèçàöèè è ó÷åòà òåõ èëè èíûõ èçèêîõèìè÷åñêèõ ïðîöåññîâ, ìàòåìàòè÷åñêèå ìîäåëè ìîãóò áûòü ðàçëè÷íîé ñëîæíîñòè è ñîäåðæàòü îò íåñêîëüêèõ åäèíèö äî äåñÿòêîâ ìîäåëèðóåìûõ âåëè÷èí (óíêöèè êîíöåíòðàöèé, äåñîðáöèîííûå ïîòîêè è ïð.), êîòîðûå áóäóò îïðåäåëÿòüñÿ ìíîæåñòâîì ïàðàìåòðîâ ìîäåëè (ïàðàìåòðû, îïèñûâàþùèå ìàòåðèàëû è âåùåñòâà, èñïîëüçóåìûå â ýêñïåðèìåíòå) è ïðîâåäåíèÿ ýêñïåðèìåíòà. Äëÿ ìîäåëèðîâàíèÿ, èäåíòèèêàöèè ïàðàìåòðîâ ðàçëè÷íûõ ìîäåëåé è âèçóàëèçàöèè ðåçóëüòàòîâ ìîãóò èñïîëüçîâàòüñÿ ñîâåðøåííî ðàçëè÷íûå àëãîðèòìû. Òàêèì îáðàçîì, ìíîæåñòâî çàäàâàåìûõ ïàðàìåòðîâ, îñîáåííîñòè èñïîëüçîâàíèÿ è ðàáîòû òåõ èëè èíûõ àëãîðèòìîâ äåëàþò çàòðóäíèòåëüíûì ïðîâåäåíèå áîëüøîãî êîëè÷åñòâà ÷èñëåííûõ ýêñïåðèìåíòîâ è ïîâûøàþò âåðîÿòíîñòü îøèáîê ñîâåðøàåìûõ èññëåäîâàòåëåì.  êà÷åñòâå ïðèìåðà ðàññìîòðèì ìîäåëèðîâàíèå âîäîðîäîïðîíèöàåìîñòè çàùèòíîãî ïîêðûòèÿ, íàíåñåííîãî íà êîíñòðóêöèîííûé ìàòåðèàë (ìîäåëü [10℄, ÌÊÈ). Äëÿ ïðîâåäåíèÿ ðàñ÷åòà òðåáóåòñÿ çàäàòü:  12 ïàðàìåòðîâ, îïðåäåëÿþùèõ âîäîðîäîïðîíèöàåìîñòü äâóõñëîéíîé ñèñòåìû ìàòåðèàë-ïîêðûòèå (äëÿ êàæäîãî ñëîÿ çàäàþòñÿ êîýèöèåíòû àäñîðáöèè, äåñîðáöèè, äèóçèè, çàõâàòà è âûñâîáîæäåíèÿ âîäîðîäà èçèêî-õèìè÷åñêèìè äååêòàìè ñòðóêòóðû ìàòåðèàëà è êîýèöèåíòû íà ñòûêå ñëîåâ);

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 8 ïàðàìåòðîâ, çàäàþùèõ óñëîâèÿ ïðîâåäåíèÿ ýêñïåðèìåíòà (òîëùèíû ñëîåâ, äëèòåëüíîñòü ýêñïåðèìåíòà, äàâëåíèÿ â êàìåðàõ, ïåðèîä âêëþ÷åíèÿ äèññîöèàòîðà, óðîâåíü êîíöåíòðàöèè ïîä ïîâåðõíîñòüþ ïðè âêëþ÷åííîì äèññîöèàòîðå, íà÷àëüíîå ðàñïðåäåëåíèå âîäîðîäà â ìàòåðèàëå);  12 ïàðàìåòðîâ, îïðåäåëÿþùèõ ðàáîòó àëãîðèòìà è çàïèñü ðåçóëüòàòîâ (êîëè÷åñòâî øàãîâ ïî ïðîñòðàíñòâó è âðåìåíè, íàçâàíèÿ 2-õ àéëîâ ïîòîêîâ è 8-ìè àéëîâ êîíöåíòðàöèé). Òàêèì îáðàçîì, âñÿêèé ðàç ïåðåä ïðîâåäåíèåì ðàñ÷åòîâ òðåáóåòñÿ çàäàòü, êàê ìèíèìóì, 32 ïàðàìåòðà. Òàêîå êîëè÷åñòâî ïàðàìåòðîâ çàäàâàòü â êîìàíäíîé ñòðîêå òðóäîåìêî, ïîýòîìó ëó÷øå õðàíèòü èõ â àéëå íàñòðîåê, íî èññëåäîâàòåëü, ïðîâîäÿùèé ðàñ÷åòû, íà÷íåò ïóòàòüñÿ ïîñëå ïðîâåäåíèÿ äåñÿòêîâ è ñîòåí ðàñ÷åòîâ ñ èñïîëüçîâàíèåì ðàçëè÷íûõ àëãîðèòìîâ, òðåáóþùèõ ðàçëè÷íûõ àéëîâ íàñòðîåê. Îòìåòèì, ÷òî â ñëó÷àå åñëè òðåáóåòñÿ ïîñìîòðåòü èçìåíåíèå âûõîäíîãî äåñîðáöèîííîãî ïîòîêà ïðè èçìåíåíèè òîëüêî îäíîãî ïàðàìåòðà, çàäàíèå îñòàëüíûõ 31 ïàðàìåòðà áóäåò èçëèøíèì, ïîñêîëüêó 10 ïàðàìåòðîâ (èìåíà àéëîâ) ìîæíî ñãåíåðèðîâàòü àâòîìàòè÷åñêè, à çíà÷åíèÿ 20 ïàðàìåòðîâ âçÿòü èç ïðåäûäóùåãî ýêñïåðèìåíòà. Ïîÿâëÿåòñÿ íåîáõîäèìîñòü ñîçäàíèÿ áîëåå óäîáíîãî è áîëåå ñëîæíîãî ïîëüçîâàòåëüñêîãî èíòåðåéñà, ÷òî òðåáóåò çíà÷èòåëüíûõ çàòðàò ñèë è âðåìåíè. Çàêîíîìåðíûì ÿâëÿåòñÿ ñäåëàòü åãî óíèâåðñàëüíûì è íàñòðàèâàåìûì äëÿ ðàçëè÷íûõ çàäà÷, ò.å. ðàçäåëèòü ïðîãðàììíûé êîä êàæäîé çàäà÷è íà àëãîðèòìè÷åñêóþ è èíòåðåéñíóþ ÷àñòè. Òîãäà ðåàëèçàöèÿ íîâîé çàäà÷è ñâåäåòñÿ ê íàñòðîéêå èíòåðåéñíîé ÷àñòè è íàïèñàíèþ êîäà òîëüêî ïîäêëþ÷àåìîãî àëãîðèòìà.  ïðîöåññå ðåàëèçàöèè ðàçëè÷íûõ çàäà÷ âûðàáîòàëàñü ñëåäóþùàÿ êîíöåïöèÿ ïðîãðàììíîãî êîìïëåêñà. Ïðîãðàììíûé êîìïëåêñ ñîñòîèò èç äâóõ ÷àñòåé (ðèñ. 1):  âû÷èñëèòåëüíûå ïðîåêòû, êàæäûé èç êîòîðûõ âêëþ÷àåò â ñåáÿ ÁÄ (ñîäåðæèò ïàðàìåòðû ìîäåëåé, ýêñïåðèìåíòîâ, àëãîðèòìîâ è ïð.), àëãîðèòìû äëÿ ðàñ÷åòà è âèçóàëèçàöèè (íàáîð DLL è/èëè MATLAB ïðîãðàìì), àéëû íàñòðîåê ïîëüçîâàòåëüñêîãî èíòåðåéñà;  ÿäðî, âêëþ÷àþùåå â ñåáÿ ïîëüçîâàòåëüñêèé èíòåðåéñ (ðåäàêòîðû ïðîåêòîâ, ýêñïåðèìåíòîâ, ïîëåé, ÁÄ), ïðîãðàììíûé èíòåðåéñ (ñâÿçü ñ ÁÄ è àëãîðèòìàìè îîðìëåííûìè â âèäå DLL èëè MATLAB ïðîãðàìì), à òàêæå îáùóþ óíêöèîíàëüíîñòü äëÿ âñåõ âû÷èñëèòåëüíûõ çàäà÷.

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Ядро программного комплекса Программный интерфейс

Экспериментальный проект База данных Подключаемые MatLab-программы

Интерфейс с БД

Подключаемые DLL-библиотеки

Интерфейс с DLL

Параметры

Интерфейс с MatLab Конфигурация

Интерфейс пользователя Редактор проекта Редактор экспериментов

Вычислительная среда MatLab

Редактор полей

èñ. 1.

Ñòðóêòóðà ïðîãðàììíîãî êîìïëåêñà.

Ïîëüçîâàòåëüñêàÿ ðàáîòà ñ ïðîãðàììíûì êîìïëåêñîì âûãëÿäèò ñëåäóþùèì îáðàçîì. Ïîëüçîâàòåëü çàïóñêàåò ÏÊ, îòêðûâàåò â íåì íóæíûé âû÷èñëèòåëüíûé ïðîåêò, ïðè ýòîì ïîäêëþ÷àåòñÿ ÁÄ, íàñòðàèâàåòñÿ èíòåðåéñ. Òåïåðü â ïðîãðàììíîì êîìïëåêñå âîçìîæíî ïðîâîäèòü ðàñ÷åòû, ïðåäóñìîòðåííûå âûáðàííûì ïðîåêòîì (ðèñ. 5). Äàëåå ïîëüçîâàòåëü äîëæåí ñîçäàòü è çàïîëíèòü çàïèñè â ÁÄ. Êàæäàÿ çàïèñü ñîîòâåòñòâóåò âû÷èñëèòåëüíîìó ýêñïåðèìåíòó è õðàíèò âñå íåîáõîäèìûå ïàðàìåòðû. Ïðè ýòîì ÷àñòü ïîëåé ìîæåò çàïîëíÿòüñÿ àâòîìàòè÷åñêè ãåíåðèðóåìûìè çíà÷åíèÿìè èëè çíà÷åíèÿìè ïî óìîë÷àíèþ. Ïðè ðàáîòå ñ ÁÄ ïîëüçîâàòåëü ìîæåò íàñòðàèâàòü è èñïîëüçîâàòü ñåðâèñíûå óíêöèè (êîïèðîâàíèå è âñòàâêà çíà÷åíèé ãðóïï ïîëåé, çàïîëíåíèå ïî óìîë÷àíèþ, àâòîìàòè÷åñêàÿ ãåíåðàöèÿ çíà÷åíèé, ñîðòèðîâêà, ïîèñê, ãðóïïèðîâêè, îòáîðû). Ïîñëå çàäàíèÿ ïàðàìåòðîâ îñòàåòñÿ çàïóñòèòü àëãîðèòì äëÿ ðàñ÷åòà è ïðîñìîòðåòü ïîëó÷åííûå ðåçóëüòàòû íàæàòèåì ñîîòâåòñòâóþùèõ êíîïîê. Äëÿ ðàçðàáîò÷èêà ðåàëèçàöèÿ àëãîðèòìà ñâîäèòñÿ ê ñîçäàíèþ âû÷èñëèòåëüíîãî ïðîåêòà. Ñ ïîìîùüþ âñòðîåííûõ ñðåäñòâ ïðîãðàììíîãî êîìïëåêñà îí ñîçäàåò ÁÄ è íàñòðàèâàåò ïîëüçîâàòåëüñêèé èíòåðåéñ. Äàëåå ñëåäóåò ðåàëèçàöèÿ àëãîðèòìà â âèäå DLL èëè EXE àéëà íà ëþáîì ÿçûêå ïðîãðàììèðîâàíèÿ èëè íàïèñàíèå MATLAB ïðîãðàììû. Ïðè ýòîì ðàçðàáîò÷èê äîëæåí îðèåíòèðîâàòüñÿ íà ðàáîòó DLL ñ ïðîãðàììíûì êîìïëåêñîì ÷åðåç ïðîãðàììíûé èíòåðåéñ, êîòîðûé ïîçâîëÿåò óïðàâëÿòü ðàáîòîé àëãîðèòìà, ðåàëèçóåò ïåðåäà÷ó è âîçâðàò ïàðàìåòðîâ (ïåðåäàåòñÿ âñÿ âûáðàííàÿ çàïèñü, íàñòðîéêè ïðîåêòà è ÏÊ). Äëÿ èñïîëüçîâàíèÿ ñðåäñòâ âèçóàëèçàöèè ïðåäóñìîòðåííûõ â ÏÊ, ñëåäóåò îðìèðîâàòü àéëû äàííûõ íåêîòîðîãî ïðåäóñìîòðåííîãî îðìàòà.  ñëó÷àå íàïèñàíèÿ àëãîðèòìîâ â âèäå DLL

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(EXE) â ñðåäå Delphi èëè â âèäå MATLAB ïðîãðàìì ìîæíî èñïîëüçîâàòü ðàçðàáîòàííûå êëàññû è áèáëèîòåêè óíêöèé, ÷òî îáëåã÷àåò èñïîëüçîâàíèå ïðîãðàììíîãî èíòåðåéñà è ðàáîòó ñ àéëàìè. Òàêîé ïîäõîä ïîçâîëÿåò çíà÷èòåëüíî óìåíüøèòü êîëè÷åñòâî ðóòèííîé ðàáîòû ïðè ðåàëèçàöèè è èñïîëüçîâàíèè àëãîðèòìîâ. Ïîëüçîâàòåëü èìååò óäîáíûé åäèíîîáðàçíûé äëÿ âñåõ çàäà÷ ãðàè÷åñêèé èíòåðåéñ ñ áîãàòîé íàñòðàèâàåìîé óíêöèîíàëüíîñòüþ. àáîòà ïîëüçîâàòåëÿ ñîñòîèò èç ïîñëåäîâàòåëüíîñòè äåéñòâèé îäèíàêîâûõ äëÿ ëþáûõ ðàñ÷åòîâ. Îòïàäàåò íåîáõîäèìîñòü ïîìíèòü íàñòðîéêè îòäåëüíûõ àëãîðèòìîâ è ñïåöèèêó çàäà÷. Ïàðàìåòðû è ðåçóëüòàòû ðàñ÷åòîâ óäîáíî õðàíÿòñÿ, èìååòñÿ âîçìîæíîñòü îðèåíòàöèè â ïðîâåäåííûõ âû÷èñëèòåëüíûõ ýêñïåðèìåíòàõ (ïîèñê, ñîðòèðîâêà, îòáîðû, ãðóïïèðîâêè ïàðàìåòðîâ). àçðàáîò÷èê ñîñðåäîòà÷èâàåòñÿ òîëüêî íà ðàçðàáîòêå âû÷èñëèòåëüíîãî àëãîðèòìà è òàêèì îáðàçîì ñîêðàùàþòñÿ çàòðàòû âðåìåíè è ñèë. Îòïàäàåò íåîáõîäèìîñòü â ïåðåêîìïèëÿöèè âñåãî ïðîãðàììíîãî êîìïëåêñà è åñòü âîçìîæíîñòü ñêðûòü äåòàëè ðåàëèçàöèè àëãîðèòìîâ. Äëÿ ðåàëèçàöèè àëãîðèòìîâ âîçìîæíî èñïîëüçîâàíèå ëþáûõ ÿçûêîâ ïðîãðàììèðîâàíèÿ. 3.

Îñíîâíûå ýòàïû ðåàëèçàöèè àëãîðèòìà

Äëÿ ðåàëèçàöèè àëãîðèòìà â ïðîãðàììíîì êîìïëåêñå òðåáóåòñÿ ñîçäàòü âû÷èñëèòåëüíûé (ýêñïåðèìåíòàëüíûé) ïðîåêò, êîòîðûé áóäåò âêëþ÷àòü â ñåáÿ ÁÄ è àéëû íàñòðîåê, ïîñëå ÷åãî ñîçäàòü DLL (EXE) èëè MATLAB ïðîãðàììó. àññìîòðèì ïîñëåäîâàòåëüíî âñå ýòàïû.

Ñîçäàíèå ýêñïåðèìåíòàëüíîãî ïðîåêòà. Ñîçäàíèå ýêñïåðèìåíòàëüíîãî ïðîåêòà íà÷èíàåòñÿ ñ âûáîðà ïóíêòà ìåíþ File\New. Äàëåå â ïîÿâèâøåìñÿ äèàëîãîâîì îêíå âûáîðà àéëà ïðîåêòà (ðèñ. 2) òðåáóåòñÿ óêàçàòü êàòàëîã ðàçìåùåíèÿ ïðîåêòà è íàçâàíèÿ àéëà. Äàëåå îòêðîåòñÿ îêíî íàñòðîéêè ïðîåêòà (ðèñ. 2), â êîòîðîì ñëåäóåò çàïîëíèòü âñå ïîëÿ. Åñëè êàêèå-ëèáî óêàçàííûå àéëû íå ñóùåñòâóþò, òî îíè áóäóò àâòîìàòè÷åñêè ñîçäàíû. Ïðè ýòîì àâòîìàòè÷åñêè ñîçäàåòñÿ ïóñòàÿ áàçà äàííûõ, êîòîðóþ âîçìîæíî ðåäàêòèðîâàòü âî âñòðîåííîì ðåäàêòîðå (ðèñ. 3). Òðåáóåòñÿ çàäàòü è íàñòðîèòü âñå íåîáõîäèìûå ïîëÿ. Íàçâàíèÿ ïîëåé áóäóò ÿâëÿòüñÿ íàçâàíèÿìè ïàðàìåòðîâ â ñïèñêàõ ïàðàìåòðîâ ïåðåäàâàåìûõ â êîä àëãîðèòìà. Äàëåå íåîáõîäèìî óêàçàòü ïóòè ê àëãîðèòìàì è ê àéëàì äàííûõ íà çàêëàäêå Algorithms\Parameters (ðèñ. 3). Íàñòðîéêà íà

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èñ. 2.

Îêíà âûáîðà è íàñòðîéêè ïðîåêòà.

èñ. 3. Îêíà ðåäàêòîðà ÁÄ è íàñòðîéêè äîïîëíèòåëüíûõ ïàðàìåòðîâ ïðîåêòà.

äðóãèõ çàêëàäêàõ ãðóïï ïîëåé, ïåðåìåííûõ ïðîåêòà, ñâÿçåé è îãðàíè÷åíèé ïîëåé ÿâëÿåòñÿ íåîáÿçàòåëüíûì, íî ïîëåçíûì. Ïîñëå âûõîäà èç äèàëîãîâîãî îêíà íàñòðîéêè ïðîåêòà â óêàçàííîì ðàíåå êàòàëîãå ïðîåêòà áóäóò ñîðìèðîâàíû ñëåäóþùèå àéëû:  îñíîâíîé àéë ïðîåêòà (ðàñøèðåíèå .hip), â êîòîðîì õðàíÿòñÿ ññûëêè íà äðóãèå àéëû è íåêîòîðûå ïàðàìåòðû ïðîåêòà;  àéëû áàçû äàííûõ (.dbf,. dx,.fpt), â êîòîðûõ õðàíÿòñÿ äàííûå ÷èñëåííûõ ýêñïåðèìåíòîâ, èíäåêñû è êîììåíòàðèè;  àéë êîíèãóðàöèè ïîëåé ÁÄ (.hipdb), êîòîðûé ñîäåðæèò îïèñàíèå ïîëåé ÁÄ;  àéë ñâÿçåé è îãðàíè÷åíèé (.hiplinks), â êîòîðîì îïðåäåëÿþòñÿ ñâÿçè ìåæäó ïîëÿìè ÁÄ (è èõ çíà÷åíèÿìè), à òàêæå îãðàíè÷åíèÿ çíà÷åíèé äëÿ ïîëåé ÁÄ;  àéë îïèñàíèÿ àëãîðèòìîâ (.hipalgs), îïðåäåëÿþùèé íàñòðîéêè àëãîðèòìîâ è íåêîòîðûå äðóãèå ïàðàìåòðû;

åàëèçàöèÿ è èñïîëüçîâàíèå çàäà÷ ìîäåëèðîâàíèÿ ...

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 àéë âñïîìîãàòåëüíûõ íàñòðîåê (.hippref). Òåïåðü íàñòðîéêà ïðîåêòà çàêîí÷åíà, ïðè íåîáõîäèìîñòè ìîæíî îòêðûòü ïðîåêò (File\Open, âûáðàòü .hip àéë) è ïîìåíÿòü ëþáûå ïàðàìåòðû èëè èçìåíèòü ÁÄ ïðîåêòà (File\Proje t\Settings).

Âûçîâ àëãîðèòìîâ èç ÏÊ è ïåðåäà÷à ïàðàìåòðîâ. Ïðîãðàììíûé êîìïëåêñ ìîæåò èñïîëüçîâàòü àëãîðèòìû, îîðìëåííûå êàê èñïîëíÿåìûå àéëû, äèíàìè÷åñêè ïîäêëþ÷àåìûå áèáëèîòåêè (DLL) è MATLAB ïðîãðàììû. Êîìïëåêñ îïðåäåëÿåò ðåàëèçàöèþ àëãîðèòìà ïî ðàñøèðåíèþ àéëà (.exe,.dll,.m) è â çàâèñèìîñòè îò ýòîãî âàðüèðóåòñÿ ñïîñîá âûçîâà àëãîðèòìà è îáìåíà äàííûìè.  îáùåì ñëó÷àå âñåãäà ìîæåò èñïîëüçîâàòüñÿ îáìåí äàííûìè ÷åðåç àéëû îïðåäåëåííîãî îðìàòà. Ýòî ïîçâîëèëî óïðîñòèòü è ñäåëàòü ãèá÷å êîä, óñòðàíèòü ïîòåíöèàëüíóþ ïðîáëåìó íåñîâìåñòèìîñòè ñî ñðåäîé MATLAB äðóãèõ âåðñèé, îáëåã÷èòü îòëàäêó îòäåëüíûõ àëãîðèòìîâ. Ïðè âûçîâå àëãîðèòìà ïðîãðàììíûé êîìïëåêñ âûïîëíÿåò ñëåäóþùèå äåéñòâèÿ:  îçäàåò ñëåäóþùèå óíèêàëüíûå âðåìåííûå àéëû:  àéë ñ âõîäíûìè äàííûìè àëãîðèòìà;  àéë èíäèêàòîðà ðàáîòû àëãîðèòìà;  àéë äëÿ ñîõðàíåíèÿ ðåçóëüòàòîâ ðàáîòû àëãîðèòìà;  çàïóñêàåò âûïîëíåíèå àëãîðèòìà â îòäåëüíîì ïðîöåññå, ÷òî ïîçâîëÿåò âûïîëíÿòü ïàðàëëåëüíî íåñêîëüêî àëãîðèòìîâ;  ïåðåäàåò àëãîðèòìó èìåíà ñîçäàííûõ àéëîâ, à òàêæå íåêîòîðûå äðóãèå ïàðàìåòðû;  îæèäàåò çàâåðøåíèÿ ðàáîòû àëãîðèòìà. àáîòà àëãîðèòìà ìîæåò áûòü ïðåðâàíà ïî æåëàíèþ ïîëüçîâàòåëÿ;  â ñëó÷àå óñïåøíîãî çàâåðøåíèÿ, ñ÷èòûâàåò è ñîõðàíÿåò ðåçóëüòàòû ðàáîòû àëãîðèòìà, óäàëÿåò âðåìåííûå àéëû. Çàïóñê àëãîðèòìîâ, ðåàëèçîâàííûõ â âèäå èñïîëíÿåìûõ àéëîâ, ïðîèçâîäèòñÿ ñ ïîìîùüþ óíêöèè API CreatePro ess(). Íàçâàíèå àéëà ñ ïàðàìåòðàìè ïåðåäàåòñÿ â êîìàíäíîé ñòðîêå. Âûïîëíåíèå àëãîðèòìîâ, îîðìëåííûõ â âèäå DLL, èíèöèèðóåòñÿ âûçîâîì ýêñïîðòèðóåìîé óíêöèè TDLLMainPro ñòðîãî îïðåäåëåííîãî îðìàòà. Âñå íåîáõîäèìûå ïàðàìåòðû ïåðåäàþòñÿ ÷åðåç àðãóìåíòû ýòîé óíêöèè. Äëÿ âûïîëíåíèÿ MATLAB ïðîãðàìì êîìïëåêñ èñïîëüçóåò ìîäóëü MATLAB-laun her, îîðìëåííûé â âèäå DLL, êîòîðûé ÷åðåç MATLAB API çàïóñêàåò âû÷èñëèòåëüíóþ ñðåäó, âûçûâàåò óíêöèþ

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MatLabAlgorithmLaun her è ïåðåäàåò åé íàçâàíèÿ àéëà ïàðàìåòðîâ. Ôóíêöèÿ ñ÷èòûâàåò àéë ïàðàìåòðîâ, ïðîïèñûâàåò ïóòè â MATLAB è çàïóñêàåò óêàçàííóþ â àéëå óíêöèþ ñ àëãîðèòìîì (ðèñ. 4). Программный комплекс Параметры обработки данных

Экспериментальные данные

MatLab launcher (DLL)

Файл параметров рабочей функции

Файл с экспериментальными данными

Взаимодействие с вычислительной средой MatLab (MatLab API)

Параметры обработки данных

Управление MatLab

Среда MatLab èñ. 4.

Выполнение MatLab-программы

Экспериментальные данные

MatLab-программа (алгоритм)

Çàïóñê MATLAB ïðîãðàììû.

Ôàéë ïàðàìåòðîâ, êîòîðûé èñïîëüçóåòñÿ ñ MATLAB ïðîãðàììàìè è èñïîëíÿåìûìè àéëàìè, ñîñòîèò èç ñòðîê âèäà <èìÿ ïàðàìåòðà> <çíà÷åíèå ïàðàìåòðà>. Îáÿçàòåëüíûì ÿâëÿåòñÿ íàëè÷èå ïàðàìåòðà ñ èìåíåì Fun tion name, êîòîðûé çàäàåò MATLAB óíêöèþ ñ àëãîðèòìîì. Êðîìå ýòîãî â àéëå ìîãóò áûòü ïåðåäàíû ïàðàìåòðû, íåîáõîäèìûå äëÿ âûçûâàåìîé óíêöèè (ïðè çàïóñêå âû÷èñëèòåëüíîãî àëãîðèòìà ïåðåäàþòñÿ íàçâàíèÿ ïîëåé è èõ çíà÷åíèÿ äëÿ âûáðàííîé çàïèñè ÁÄ, à ïðè âèçóàëèçàöèè ðåçóëüòàòîâ  íàçâàíèå àéëà äàííûõ), è ïàðàìåòðû ñ èìåíàìè Path.<íîìåð> äëÿ óêàçàíèÿ ïóòåé, ïî êîòîðûì MATLAB áóäåò èñêàòü âñïîìîãàòåëüíûå óíêöèè, àéëû ïàðàìåòðîâ è äàííûõ. Ïðè çàâåðøåíèè ðàáîòû àëãîðèòì ìîæåò ïåðåïèñàòü ýòîò àéë ïàðàìåòðîâ, çàïèñàâ â íåãî â òîì æå îðìàòå íîâûå çíà÷åíèÿ ïàðàìåòðîâ. Ïðîãðàììíûé êîìïëåêñ ñ÷èòûâàåò ïàðàìåòðû è çàïèñûâàåò èõ â âûáðàííóþ çàïèñü ÁÄ. Åñëè àëãîðèòì â àéëå ïàðàìåòðîâ íå íàõîäèò íóæíûé ïàðàìåòð, òî ïðîèçîéäåò îøèáêà, êîòîðóþ îáðàáàòûâàåò àëãîðèòì. Ïðîãðàììíûé êîìïëåêñ èãíîðèðóåò îøèáêè, âîçíèêàþùèå ïðè ñ÷èòûâàíèè âîçâðàùàåìûõ ïàðàìåòðîâ. Ïîñêîëüêó ñèñòåìà MATLAB èìååò ìîùíûå ñðåäñòâà âèçóàëèçàöèè, òî ýòîò ìåõàíèçì âûçîâà MATLAB èñïîëüçóåòñÿ â ïðîãðàììíîì êîìïëåêñå äëÿ ïðîñìîòðà ðåçóëüòàòîâ ðàáîòû. Ïðè ýòîì â êà÷åñòâå íàçâàíèÿ óíêöèè ñ àëãîðèòìîì ïåðåäàåòñÿ íàçâàíèå óíêöèè äëÿ ïîñòðîåíèÿ ãðàèêîâ, à â êà÷åñòâå ïàðàìåòðîâ èìÿ àéëà äàííûõ.

Ñîçäàíèå àëãîðèòìà â âèäå DLL èëè EXE àéëà. Äëÿ îïðåäåëåííîñòè áóäåì ðàññìàòðèâàòü íàïèñàíèå àëãîðèòìà â ñðåäå Delphi 7. Ñîçäàíèå DLL â Delphi íà÷èíàåòñÿ ñ ïóíêòà ìåíþ (File\New\DLL), ïîñëå ÷åãî òðåáóåòñÿ íàïèñàòü îñíîâíîé ìîäóëü:

åàëèçàöèÿ è èñïîëüçîâàíèå çàäà÷ ìîäåëèðîâàíèÿ ...

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unit AlgDLL; interfa e uses MyDefsFile, Classes, SysUtils; //óêàçûâàåì ñâîè àéëû ñ êîäîì, à òàêæå // íåîáõîäèìûå ñòàíäàðòíûå áèáëèîòåêè pro edure TDLLMainPro (var Options: APOptions; Vars: TList; Params: TList; var flAbort: boolean; var Exe Status: integer; Exe Message: PChar); StdCall; exports TDLLMainPro ; implementation pro edure TDLLMainPro (var Options: APOptions; Vars: TList; Params: TList; var flAbort: boolean; var Exe Status: integer; Exe Message: PChar); StdCall; var Alg: TMyAlgorithm; Begin if (Exe Status <= 0)and(not flAbort) then Begin Try //Ñîçäàåì ýêçåìïëÿð êëàññà è èíèöèàëèçèðóåì ïåðåìåííûå Alg:=TMyAlgorithm.Create(Options, Vars, Params); Try Alg.Computation; //Âû÷èñëÿåì ex ept Alg.ErrorStatus := 1; Alg.ErrorMessage := 'Íåèçâåñòíîå èñêëþ÷åíèå'; end; Exe Status := Alg.ErrorStatus; StrCopy(Exe Message, PChar(Alg.ErrorMessage)); Params := Alg.GetParams(Params); //Äîïèñûâàåì â ñïèñîê Params, finally // âîçâðàùàåìûå çíà÷åíèÿ Alg.Free; //Óíè÷òîæàåì ýêçåìïëÿð êëàññà end; end; end; end.

Çäåñü îïèñûâàåòñÿ îñíîâíàÿ ýêñïîðòèðóåìàÿ èç DLL ïðîöåäóðà TDLLMainPro , êîòîðàÿ â êà÷åñòâå ïàðàìåòðîâ èìååò ñòðóêòóðó Options, ñïèñêè ïàðàìåòðîâ Vars è Params  ïåðåìåííûå ïðîåêòà è ïîëÿ âûáðàííîé çàïèñè â ÁÄ, ëàã ïðåðûâàíèÿ ðàáîòû flAbort (íå èñïîëüçóåòñÿ), êîä è ñîîáùåíèå îá îøèáêå: Exe Status, Exe Message. Îïèñàíèå êëàññà TMyAlgorithm, ðåàëèçóþùåãî àëãîðèòì, à òàêæå ñòðóêòóð APOptions è APVal, ÿâëÿþùèìèñÿ ïóíêòàìè ñïèñêîâ TList, íàõîäÿòñÿ â àéëå MyDefsFile. Íèæå ïðèâîäèì îïèñàíèå ñòðóêòóð, â êîòîðûõ ïåðåäàþòñÿ äàííûå ìåæäó ïðîãðàììíûì êîìïëåêñîì è DLL. APVal = re ord //(àíàëîã stru t â C++) Name: AnsiString; //íàçâàíèå ïàðàìåòðà èëè ïîëÿ ÁÄ Value: Variant; //çíà÷åíèå ïàðàìåòðà èëè ïîëÿ ÁÄ â âûáðàííîé çàïèñè

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ValueType: TFieldType; //òèï ïîëÿ ÁÄ end; APOptions = re ord SemaphoreName: AnsiString; ParamsFile: AnsiString; ProgressFile: AnsiString; LogFile: AnsiString; ResultFile: AnsiString; end;

//èìÿ ñåìàîðà äëÿ ïðåðûâàíèÿ àëãîðèòìà //àéë ïàðàìåòðîâ //àéë èíäèêàòîðà ðàáîòû //àéë ñîîáùåíèé, âîçíèêàþùèõ ïðè ðàáîòå //àéë ñ âîçâðàùàåìûìè ðåçóëüòàòàìè ðàáîòû

Êîä àëãîðèòìà íàõîäèòñÿ â êëàññå TMyAlgorithm, ðåàëèçàöèÿ êîòîðîãî îïðåäåëÿåòñÿ èñêëþ÷èòåëüíî ðàçðàáîò÷èêîì. Íåîáõîäèìûì ÿâëÿåòñÿ òîëüêî íàëè÷èå ìåòîäîâ: Create  äëÿ ñîçäàíèÿ ýêçåìïëÿðà êëàññà è èíèöèàëèçàöèè ïåðåìåííûõ, ÷àñòü çíà÷åíèé êîïèðóåòñÿ èç ïåðåäàííûõ ñòðóêòóð; Computation  äëÿ âûïîëíåíèÿ ðàñ÷åòîâ; GetParams  äëÿ âîçâðàùåíèÿ ïðåäóñìîòðåííûõ çíà÷åíèé â ïðîãðàììíûé êîìïëåêñ, ãäå îíè áóäóò çàïèñàíû â ñîîòâåòñòâóþùèå ïîëÿ âûáðàííîé çàïèñè ÁÄ; Free  äëÿ êîððåêòíîãî óíè÷òîæåíèÿ ýêçåìïëÿðà êëàññà. Òàêæå êëàññ äîëæåí èìåòü ïåðåìåííûå ErrorStatus è ErrorMessage äëÿ âîçâðàùåíèÿ êîäà è ñîîáùåíèÿ îá îøèáêå, âîçíèêøåé ïðè ðàáîòå. Äëÿ çàâåðøåíèÿ àëãîðèòìà ïî íàæàòèþ êíîïêè îñòàíîâêè èç ïðîãðàììíîãî êîìïëåêñà óíêöèÿ Computation äîëæíà ñîäåðæàòü ðàãìåíò êîäà, ïðåðûâàþùåãî àëãîðèòì (ïåðåìåííàÿ hndl îáúÿâëåíà êàê THandle): //èíèöèàëèçàöèÿ ñåìàîðà hndl:=OpenSemaphore(MUTANT_ALL_ACCESS,false,PChar(Options.SemaphoreName)); if hndl=0 then hndl:=CreateSemaphore(nil,1,1,PChar(Options.SemaphoreName)); //îñíîâíîé öèêë, â êîòîðîì ïðîèñõîäÿò âû÷èñëåíèÿ ïî ðàçíîñòíîé ñõåìå for j := 0 to StT do //êîä ðàçíîñòíîé ñõåìû ... //îïðîñ ñåìàîðà äëÿ îñòàíîâêè ðàñ÷åòîâ if j mod InquiryStep = 0 then //ïîçâîëÿåò ñ÷èòàòü áûñòðåå if WaitForSingleObje t(hndl,1) = WAIT_TIMEOUT then Begin ErrorStatus := 10; ErrorMessage := 'Âûïîëíåíèå ïðåðâàíî ïîëüçîâàòåëåì.'; end else ReleaseSemaphore(hndl,1,nil); //Âûéäåì, åñëè ïðîèçîøëà îøèáêà èëè ïðåðûâàíèå ïîëüçîâàòåëåì if ErrorStatus > 0 then break; end;

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Ñîçäàíèå èñïîëíÿåìîãî àéëà ñ àëãîðèòìîì ïðîèçâîäèòñÿ àíàëîãè÷íûì îáðàçîì. Ïðè ýòîì íå îáúÿâëÿåòñÿ óíêöèÿ TDLLMainPro . Äëÿ ïåðåäà÷è ïàðàìåòðîâ èç ïðîãðàììíîãî êîìïëåêñà ñ÷èòûâàåòñÿ è ðàçáèðàåòñÿ àéë ïàðàìåòðîâ, íàçâàíèå êîòîðîãî ïåðåäàåòñÿ â êîìàíäíîé ñòðîêå. Èíîðìèðîâàòü ïðîãðàììíûé êîìïëåêñ î òåêóùåì ñîñòîÿíèè âû÷èñëåíèé, î âîçíèêàþùèõ îøèáêàõ è ñîîáùåíèÿõ àëãîðèòìû ìîãóò ÷åðåç àéëû, èìåíà êîòîðûõ ñîäåðæàòñÿ â ïîëÿõ ProgressFile, LogFile ñòðóêòóðû APOptions.

Ñîçäàíèå àëãîðèòìà â âèäå MATLAB ïðîãðàììû. Ïðè çàïóñêå MATLAB ïðîãðàììû ñíà÷àëà áóäåò çàïóùåíà óíêöèÿ MatLabAlgorithmLaun her, êîòîðàÿ ïîñëå ñ÷èòûâàíèÿ àéëà ïàðàìåòðîâ è óñòàíîâêè íåîáõîäèìûõ ïóòåé çàïóñòèò óêàçàííóþ åé óíêöèþ ñ âû÷èñëèòåëüíûì àëãîðèòìîì. Êîä óíêöèè ñëåäóþùèé: fun tion Status = MatLabAlgorithmLaun her(varargin) strName = ''; if (nargin == 1) strName = varargin{1}; end; if strName == '' strName = 'Laun h.par'; end; [Status, Laun hPar℄ = ReadLaun hParams(strName); % ïðîïèñûâàåì ïóòè ê óíêöèÿì è àéëàì äàííûõ â MATLAB for i = 1:length(Laun hPar) if length(findstr(Laun hPar(i).Name,'Path.')) > 0 addpath(Laun hPar(i).Val); end end; if Status == 0 f = f n hk(GetParamValByName('Fun tion name', Laun hPar)); Status = feval(f, Laun hPar); end; fun tion [Status, Laun hPar℄ = ReadLaun hParams(FName) Status = 0; mes = ''; [F, mes℄ = fopen(FName, 'rt'); if F ~= -1 % Åñëè àéë îòêðûò, òî ïûòàåìñÿ ñ÷èòàòü çàãîëîâîê Str = ''; Str1 = ''; while (feof(F) == 0) lines = fgetl(F); line = strrep(lines,(','),('.')); if findstr('-', line) > 0 Str = har(Str, line); end end f lose(F); % Ïðåîáðàçóåì çàãîëîâî÷íóþ èíîðìàöèþ â äðóãîé âèä sz = size(Str); for i = 1:sz(1) p = findstr('-',Str(i,:)); s = length(Str(i,:)); Name = deblank(Str(i,1:p-1)); Vl = deblank(Str(i,p+2:s)); Laun hPar(i).Name = har(Name); Laun hPar(i).Val = har(Vl);

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else end;

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end % Ïîêàæåì ñîîáùåíèå îá îøèáêå disp(mes); Status = 1;

fun tion outParVal = GetParamValByName(Name, Par) outParVal = ''; fl = 0; Names = fieldnames(Par); for i=1:size(Par,2) if str mp(lower(deblank(getfield(Par(i), har(Names(1))))),lower(deblank(Name)))==1 outParVal = getfield(Par(i), har(Names(2))); fl =1; break; end end if fl == 0 disp(['ïàðàìåòð ' Name ' íå íàéäåí!'℄); end;

Ôóíêöèÿ ñ âû÷èñëèòåëüíûì àëãîðèòìîì äîëæíà â êà÷åñòâå àðãóìåíòà ïîëó÷àòü ñïèñîê ñ ïàðàìåòðàìè Laun hPar è âîçâðàùàòü êîä îøèáêè èëè 0 â ñëó÷àå íîðìàëüíîãî çàâåðøåíèÿ. Ïîëó÷åíèå çíà÷åíèé ïàðàìåòðîâ ïî èìåíè èç ñïèñêà Laun hPar ìîæåò îñóùåñòâëÿòüñÿ èñïîëüçîâàíèåì óíêöèè GetParamValByName. Èíîðìèðîâàòü ïðîãðàììíûé êîìïëåêñ î òåêóùåì ñîñòîÿíèè âû÷èñëåíèé, î âîçíèêàþùèõ îøèáêàõ è ñîîáùåíèÿõ àëãîðèòìû ìîãóò ÷åðåç àéëû, èìåíà êîòîðûõ ïåðåäàþòñÿ â ïàðàìåòðàõ ñ èìåíàìè Progress file, Log file.

Ôàéëû äàííûõ è èõ âèçóàëèçàöèÿ. àçðàáîò÷èê ïðîåêòà ìîæåò ïðåäóñìîòðåòü èñïîëüçîâàíèå àéëîâ äàííûõ ëþáîãî îðìàòà, â òîì ÷èñëå è ñâîåãî ñîáñòâåííîãî, â ýòîì ñëó÷àå îí äîëæåí òàêæå ðàçðàáîòàòü ïðîãðàììó äëÿ âèçóàëèçàöèè äàííûõ. Äëÿ èçáåæàíèÿ ýòèõ ðóòèííûõ äåéñòâèé âîçìîæíî èñïîëüçîâàòü àéëû äàííûõ ïðåäóñìîòðåííîãî ÏÊ îðìàòà. Ôàéë äàííûõ çàïèñûâàåòñÿ â îðìàòå ASCII è ñîñòîèò èç òåêñòîâîãî çàãîëîâêà, îïèñûâàþùåãî ïàðàìåòðû, ïðè êîòîðûõ àéë áûë ïîëó÷åí, çà êîòîðûì ñëåäóåò ÷àñòü ñ äàííûìè.  ïåðâîé ñòðîêå çàãîëîâêà óêàçûâàåòñÿ òèï äàííûõ (äåñîðáöèîííûé ïîòîê, êîíöåíòðàöèÿ â ëîâóøêàõ è ò.ä.), äàëåå ðàçäåëåííûå ïóñòûìè ñòðîêàìè ðàñïîëàãàþòñÿ ñåêöèè: Parameters, Variables, Additional parameters, Data part des ription. Êàæäàÿ èç ñåêöèé âêëþ÷àåò ïàðàìåòðû, çàïèñàííûå â âèäå <èìÿ ïàðàìåòðà> - <çíà÷åíèå ïàðàìåòðà>. Ñåêöèÿ Parameters ñîäåðæèò çíà÷åíèÿ ïîëåé âûáðàííîé çàïèñè ÁÄ.  ñåêöèþ Variables ïîìåùàþòñÿ ïåðåìåííûå ïðîåêòà. Ñåêöèÿ Additional parameters çàðåçåðâèðîâàíà. Ñåêöèÿ Data part des ription ñîäåðæèò ïàðàìåòðû, èñïîëüçóåìûå ïðè âèçóàëèçàöèè (çíà÷åíèÿ ïî îñÿì, íàçâàíèÿ îñåé è ïð.). Òåã îòäåëÿåò çàãîëîâîê îò ÷àñòè ñ äàííûìè. Äàííûå çàïèñûâàþòñÿ â âèäå îäíîìåðíîãî èëè äâóõìåðíîãî ìàññèâà.  ñëó÷àå

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îäíîìåðíîãî ìàññèâà êàæäîå ÷èñëî çàïèñûâàåòñÿ ñ íîâîé ñòðîêè, â äâóõìåðíîì ìàññèâå ÷èñëà â ñòðîêå ðàçäåëÿþòñÿ ";". Ïðè ðåàëèçàöèè àëãîðèòìîâ íà Delphi êîä äëÿ ñîçäàíèÿ è çàïèñè àéëîâ òàêîãî îðìàòà ìîæåò áûòü ñëåäóþùèì. unit MyDefsFile; ...

onst FT_ = 'Hydrogen on entration file'; Type TMyAlgorithm = lass AddParams: TList; //Ñïèñîê äîïîëíèòåëüíûõ ïàðàìåòðîâ Conz: array of double; //Ìàññèâ, õðàíÿùèé êîíöåíòðàöèþ ïðè ðàñ÷åòàõ ... publi f_ : TDataFile; //Îáúÿâëåíèå íàøåãî êëàññà, ðàáîòàþùåãî ñ àéëîì ...

onstru tor Create(var Options: APOptions; Vars: TList; Params: TList); destru tor Destroy(); end; ...

onstru tor TMyAlgorithm.Create(var Options:APOptions;Vars:TList;Params:TList); var ConzFN: string; begin ... ConzFN := GetValByName('CONZFILE', Params); //Ñ÷èòûâàíèå èìåíè àéëà äàííûõ //Ñîçäàíèå ýêçåìïëÿðà êëàññà f_ := TDataFile.Create(Params, Vars, AddParams, ConzFN, FT_ , 0); f_ .Clear; //Èíèöèàëèçàöèÿ, èñïîëüçóåòñÿ ïåðåä ëþáûì îòêðûòèåì àéëà f_ .OpenFileToWrite; //Îòêðûòèå àéëà, çäåñü ïèøåòñÿ çàãîëîâîê ... end; destru tor TMyAlgorithm.Destroy; begin f_ .CloseDataFile; //Çàêðûòèå àéëà f_ .Free; //Óíè÷òîæåíèå ýêçåìïëÿðà êëàññà ... end; pro edure TMyAlgorithm.Computation; begin //îñíîâíîé öèêë, â êîòîðîì ïðîèñõîäÿò âû÷èñëåíèÿ ïî ðàçíîñòíîé ñõåìå for j := 0 to StT do ... //çàïèñü äàííûõ â àéë

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end; end.

Â. Â. Ïîïîâ, Ï. À. Ñåìèí

end;

if j mod WriteStep = 0 then f_ .WriteNext(Conz);

Ïîñòðîåíèå äâóõ è òðåõìåðíûõ ãðàèêîâ äëÿ òàêèõ àéëîâ äàííûõ âîçìîæíî ñ ïîìîùüþ MATLAB óíêöèè ShowDatFile (ðèñ. 6). 4.

Èñïîëüçîâàíèå ïðîåêòà

Ïðè ðàáîòå ñ êîìïëåêñîì èññëåäîâàòåëü èñïîëüçóåò ïîëüçîâàòåëüñêèé èíòåðåéñ (ðèñ. 5), îñíîâíîé ÷àñòüþ êîòîðîãî ÿâëÿþòñÿ ðåäàêòîðû âû÷èñëèòåëüíûõ ýêñïåðèìåíòîâ è ïîëåé. Ïîñêîëüêó îïòèìàëüíûì ÿâèëîñü èñïîëüçîâàíèå ÁÄ â âèäå îäíîé áîëüøîé òàáëèöû, ãäå ñòðîêè ñîîòâåòñòâóþò âû÷èñëèòåëüíûì ýêñïåðèìåíòàì, à ñòîëáöû çàäàâàåìûì ïàðàìåòðàì, òî ðåäàêòîð ýêñïåðèìåíòîâ ïîçâîëÿåò ïðîñìàòðèâàòü ìíîæåñòâî çàïèñåé ÁÄ, íî ïðè ýòîì âèäíà òîëüêî ÷àñòü ïîëåé, à äëÿ ïðîñìîòðà âñåõ ïîëåé òåêóùåé (âûáðàííîé) çàïèñè èñïîëüçóåòñÿ ðåäàêòîð ïîëåé. Ñ ïîìîùüþ ïàíåëè óïðàâëåíèÿ ÁÄ è ëþáîãî èç ýòèõ ðåäàêòîðîâ âîçìîæíî ðåäàêòèðîâàòü äàííûå. Óïðàâëÿòü äàííûìè, âû÷èñëåíèÿìè è ïðîñìîòðîì ðåçóëüòàòîâ ïîëüçîâàòåëü ìîæåò ñ ïîìîùüþ ïóíêòîâ ìåíþ èëè ñ ïîìîùüþ êíîïîê íà ïàíåëÿõ èíñòðóìåíòîâ. Îñíîâíûå äåéñòâèÿ ïîëüçîâàòåëÿ ïðè èñïîëüçîâàíèè ïðîåêòà áûëè ïðèâåäåíû â ïóíêòå 2.  îêíå íàñòðîéêè ïðîåêòîâ ìîæíî äîïîëíèòåëüíî çàäàòü ãðóïïû êîïèðîâàíèÿ è âñòàâêè ïîëåé, à òàêæå ãðóïïû îòîáðàæåíèÿ ïîëåé â ðåäàêòîðå ïîëåé. 5.

åàëèçàöèÿ

Ïðîãðàììíûé êîìïëåêñ ðåàëèçîâàí â ñðåäå ïðîãðàììèðîâàíèÿ Borland C++Builder 6.0 ñ èñïîëüçîâàíèåì äîïîëíèòåëüíûõ êîìïîíåíòîâ: à) RXLib 2.75 äëÿ ðåàëèçàöèè ñïåöèàëèçèðîâàííûõ òèïîâ ïîëåé; á) TMS Component Pa k 3.7 äëÿ ðåàëèçàöèè èíòåðåéñà ïîëüçîâàòåëÿ; â) Advantage Database Server 8.0 äëÿ âçàèìîäåéñòâèÿ ñ òàáëèöàìè ÁÄ. Ïðîãðàììíàÿ ïîäñèñòåìà MATLAB-laun her îîðìëåíà â âèäå DLL, êîòîðàÿ íàïèñàíà íà ÿçûêå MS Visual Ñ++. Òàáëèöû ÁÄ ýêñïåðèìåíòàëüíûõ ïðîåêòîâ õðàíÿòñÿ â îðìàòå DBF/CDX.

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Îñíîâíîå îêíî ïðîãðàììíîãî êîìïëåêñà, ðåäàêòîðû ýêñïåðèìåíòîâ è ïîëåé.

èñ. 5.

13

6.4

x 10

−3

c(t,x), cm

6.3

19

x 10

6.2

6 6.1 6

4

5.9

l, cm

2

5.8

0 5.7

0 0

0.02 1000

2000

3000

t, s4000

0.04

5.6 5.5 0

100

200

300

400

500

600

700

800

Âèçóàëèçàöèÿ äàííûõ ñðåäñòâàìè MATLAB: îáúåìíàÿ êîíöåíòðàöèÿ è äåñîðáöèîííûé ïîòîê. èñ. 6.

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Â. Â. Ïîïîâ, Ï. À. Ñåìèí

Ëèòåðàòóðà 1. International Symposium on Metal Hydrogen Systems, Fundamentals and Appli ations. Âook of abstra ts // Krakow, Poland, 2004. 2. Hydrogen Materials S ien e and Chemistry of Carbon Nanomaterials // NATO S ien e Series II: Mathemati s, Physi s and Chemistry. Vol. 172. Kluwer A ademi Publishers, Netherlands. 2004. 3. Hydrogen Materials S ien e and Chemistry of Carbon Nanomaterials. Ñáîðíèê òåçèñîâ IX ìåæäóíàðîäíîé êîíåðåíöèè // åä. Ùóð Ä.Â., Çàãèíàé÷åíêî Ñ.Þ., Âåçèðîãëó Ò.Í. Kiev: AHEU, 2005. 4. Zaika Yury V. Identifi ation of a hydrogen transfer model with dynami al boundary

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Âûïóñê 7, 2006

ÓÄÊ 512.544.43

Î ÂÎÇÍÈÊÍÎÂÅÍÈÈ ÃÈÃÀÍÒÑÊÎÃÎ ϕÏÐÈÌÈÒÈÂÍÎÃÎ ÈÄÅÀËÀ ÑËÓ×ÀÉÍÎÃÎ ÀÂÒÎÌÎÐÔÈÇÌÀ ϕ ÊÎÍÅ×ÍÎÉ ÁÓËÅÂÎÉ ÐÅØÅÒÊÈ Å. Â. ÕÂÎÐÎÑÒßÍÑÊÀß Â ðàáîòå ðàññìàòðèâàåòñÿ ìíîæåñòâî An,N âñåõ àâòîìîðôèçìîâ ϕ áóëåâîé ðåøåòêè äëèíû n òàêèõ, ÷òî ÷èñëî âñåõ ϕïðèìèòèâíûõ èäåàëîâ ðàâíî N , íà êîòîðîì çàäàíî ðàâíîìåðíîå ðàñïðåäåëåíèå âåðîÿòíîñòåé. Äîêàçàíî, ÷òî ãèãàíòñêèé ϕïðèìèòèâíûé èäåàë ñëó÷àéíîãî àâòîìîðôèçìà ϕ ∈ An,N âîçíèêàåò òîëüêî â ñëó÷àå, êîãäà n, N → ∞ òàê, ÷òî N/ ln n → 0.

Ïóñòü Ln  áóëåâà ðåøåòêà äëèíû n [1]. Îáîçíà÷èì ÷åðåç Aut (Ln ) ìíîæåñòâî âñåõ àâòîìîðôèçìîâ áóëåâîé ðåøåòêè Ln . Èäåàë I ðåøåòêè Ln áóäåì íàçûâàòü ϕïðèìèòèâíûì, åñëè ϕ(I) = = I è èç ϕ(x) = x, x ∈ I , ñëåäóåò, ÷òî ëèáî x ðàâíî íàèìåíüøåìó ýëåìåíòó ðåøåòêè Ln , ëèáî x ðàâíî íàèáîëüøåìó ýëåìåíòó èäåàëà I . Îáîçíà÷èì ÷åðåç An,N ìíîæåñòâî âñåõ àâòîìîðôèçìîâ ϕ áóëåâîé ðåøåòêè Ln òàêèõ, ÷òî ÷èñëî âñåõ ϕïðèìèòèâíûõ èäåàëîâ ðàâíî N . Çàäàäèì íà Aut (Ln ) è An,N ðàâíîìåðíîå ðàñïðåäåëåíèå âåðîÿòíîñòåé.  ðàáîòå [1] ïîëó÷åíû ïðåäåëüíûå òåîðåìû äëÿ ÷èñëà ϕ ïðèìèòèâíûõ èäåàëîâ ñëó÷àéíîãî àâòîìîðôèçìà ϕ ∈ Aut (Ln ) ïðè n → ∞ è äëÿ ÷èñëà ϕïðèìèòèâíûõ èäåàëîâ çàäàííîé äëèíû ñëó÷àéíîãî àâòîìîðôèçìà ϕ ∈ An,N ïðè n, N → ∞. Ïðåäëîæåíèå 1 [1] ïîêàçûâàåò, ÷òî äàííûå ÷èñëîâûå õàðàêòåðèñòèêè àâòîìîðôèçìà c

Å. Â. Õâîðîñòÿíñêàÿ, 2006

96

Å. Â. Õâîðîñòÿíñêàÿ

îïðåäåëÿþò äëèíó ðåøåòêè Fix n (') íåïîäâèæíûõ ýëåìåíòîâ àâòîìîðèçìà ', ÷èñëî 'ïðèìèòèâíûõ êîíãðóýíöèé ðåøåòêè L , à òàêæå ÷èñëî 'ïðèìèòèâíûõ êîíãðóýíöèé  ðåøåòêè L ñ çàäàííîé äëèíîé àêòîð - ðåøåòêè L =. Áóäåì ãîâîðèòü, ÷òî ïðè äåéñòâèè ñëó÷àéíîãî àâòîìîðèçìà âîçíèêàåò ãèãàíòñêèé 'ïðèìèòèâíûé èäåàë, åñëè ïðè ' A äëèíà ìàêñèìàëüíîãî 'ïðèìèòèâíîãî èäåàëà èìååò ïîðÿn äîê n ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1, à ñëåäóþùèé ïî âåëè÷èíå 'ïðèìèòèâíûé èäåàë èìååò áåñêîíå÷íî ìàëóþ â ñðàâíåíèè ñ n äëèíó ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1. L

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ÄÎÊÀÇÀÒÅËÜÑÒÂÎ.  [1℄ äîêàçàíî, ÷òî ñóùåñòâóþò âçàèìíî îäíîçíà÷íûå ñîîòâåòñòâèÿ ìåæäó ìíîæåñòâîì Aut (L ) è ìíîæåñòâîì âñåõ ïîäñòàíîâîê nýëåìåíòíîãî ìíîæåñòâà A âñåõ àòîìîâ ðåøåòêè L , à òàêæå ìåæäó 'ïðèìèòèâíûìè èäåàëàìè ðåøåòêè L è öèêëàìè ïîäñòàíîâêè ' íà ìíîæåñòâå A .  [2℄ óñòàíîâëåíî, ÷òî â ñëó÷àéíîé ïîäñòàíîâêå ñòåïåíè n, èìåþùåé N öèêëîâ, âîçíèêàåò ãèãàíòñêèé öèêë (ò.å. äëèíà ìàêñèìàëüíîãî öèêëà èìååò ïîðÿäîê n ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1, à ñëåäóþùèé ïî âåëè÷èíå öèêë èìååò áåñêîíå÷íî ìàëóþ â ñðàâíåíèè ñ n äëèíó ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1 òîëüêî â ñëó÷àå, êîãäà n; N òàê, ÷òî N= ln n 0. Îòñþäà ñëåäóåò óòâåðæäåíèå òåîðåìû. n

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Ëèòåðàòóðà 1. Òàðàðèí Â. Ì., ×åðåïàíîâà Å. Â. Î äåéñòâèè ñëó÷àéíîãî àâòîìîðèçìà êîíå÷íîé áóëåâîé ðåøåòêè // Òðóäû ÈÏÌÈ ÊàðÍÖ ÀÍ, 2005, ñ.139147. 2. Êàçèìèðîâ Í. È. Âîçíèêíîâåíèå ãèãàíòñêîé êîìïîíåíòû â ñëó÷àéíîé ïîäñòàíîâêå ñ èçâåñòíûì ÷èñëîì öèêëîâ // Äèñêðåòíàÿ ìàòåìàòèêà, 2003, 15, 3, ñ.128144.

Òðóäû ÈÏÌÈ ÊàðÍÖ ÀÍ Âûïóñê 7, 2006

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Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ

Âûïóñê 7, 2006

ÓÄÊ 519.87.872 ÄÎÂÅÐÈÒÅËÜÍÎÅ ÎÖÅÍÈÂÀÍÈÅ ÂÅÐÎßÒÍÎÑÒÈ ÏÅÐÅÏÎËÍÅÍÈß ÁÓÔÅÐÀ ÍÀ ÎÑÍÎÂÅ ÓÑÊÎÐÅÍÍÎÃÎ ÐÅÃÅÍÅÐÀÒÈÂÍÎÃÎ ÌÎÄÅËÈÐÎÂÀÍÈß ÑÈÑÒÅÌÛ

À. Â. Á

, Å. Â. Ì

ÎÐÎÄÈÍÀ

M/M/1

ÎÐÎÇÎÂ

 ñòàòüå ïðåäëîæåíà ìîäèôèêàöèÿ óñêîðåííîãî ìåòîäà èìèòàöèîííîãî ìîäåëèðîâàíèÿ Splitting äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà (γ ). Ïîëó÷åíà ôîðìóëà âåðîÿòíîñòè ïåðåïîëíåíèÿ

γ

M/M/1 è ïðîâåäåíî ñðàâíåíèå γ , ïîëó÷åííîé ìåòîäîì Splitting.

äëÿ ñèñòåìû

ñ îöåíêîé âåðîÿòíîñòè

Ðàáîòà ïîääåðæàíà Ðîññèéñêèì ôîíäîì ôóíäàìåíòàëüíûõ èññëåäîâàíèé, ãðàíòû 04-01-00671, 04-07-90115.

1.  Øèðîêèé êëàññ ïðîöåññîâ, îïèñûâàþùèõ äèíàìèêó êîììóíèêàòèâíûõ ñåòåé, èìååò ðåãåíåðàòèâíóþ ñòðóêòóðó. Äëÿ ýôôåêòèâíîãî èíòåðâàëüíîãî îöåíèâàíèÿ ïàðàìåòðîâ òàêèõ ñèñòåì èñïîëüçóåòñÿ ðåãåíåðàòèâíûé ìåòîä èìèòàöèîííîãî ìîäåëèðîâàíèÿ [13]. Ïîñêîëüêó îöåíèâàåìûå âåðîÿòíîñòè, êàê ïðàâèëî, î÷åíü ìàëû, òî ñòàíäàðòíûå ìåòîäû èìèòàöèîííîãî ìîäåëèðîâàíèÿ (Ìîíòå-Êàðëî) íå ýôôåêòèâíû. Ñóùåñòâóåò ðÿä ìåòîäîâ óñêîðåííîãî ìîäåëèðîâàíèÿ [4, 5], ïîçâîëÿþùèõ ïîëó÷èòü îöåíêó ñ çàäàííîé òî÷íîñòüþ ñ ìåíüøèìè âû÷èñëèòåëüíûìè çàòðàòàìè (â ÷àñòíîñòè, ìåòîä Splitting è RESTART [6]).  äàííîé ñòàòüå ìû ïðåäëàãàåì íåêîòîðóþ ìîäèôèêàöèþ ìåòîäà Splitting [79] áåç îòñå÷åíèÿ òðàåêòîðèé (ò. å. ìîäåëèðîâàíèå òðàåêòîðèè îñòàíàâëèâàåòñÿ òîëüêî ïîñëå äîñòèæåíèÿ íóëåâîãî óðîâíÿ). Ñ ïîìîùüþ ìåòîäà Splitting ñòðîèòñÿ îöåíêà âåðîÿòíîñòè äîñòèæåíèÿ ñîáûòèÿ íà öèêëå ðåãåíåðàöèè (íàïðèìåð, ÂÅÄÅÍÈÅ

c

À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ, 2006

126

À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ

âåðîÿòíîñòü ïðåïîëíåíèÿ áóôåðà çàÿâîê äî ìîìåíòà îïóñòåíèÿ ñèñòåìû îáñëóæèâàíèÿ). Äîâåðèòåëüíûé èíòåðâàë îöåíèâàåìîé âåðîÿòíîñòè ñòðîèòñÿ íà îñíîâå ìåòîäà ðåãåíåðàòèâíîãî ìîäåëèðîâàíèÿ. 2. Ì S Ðàññìîòðèì ñëó÷àéíûé ïðîöåññ X = {X(t), t ≥ 0} ñ ïðîñòðàíñòâîì ñîñòîÿíèé E . Ïóñòü ìíîæåñòâî A ÿâëÿåòñÿ ïîäìíîæåñòâîì ìíîæåñòâà E . Äàëåå áóäåì íàçûâàòü ìíîæåñòâî A ìíîæåñòâîì ðåäêèõ ñîáûòèé. Òðåáóåòñÿ îöåíèòü âåðîÿòíîñòü γ = P (X ∈ A) ðåäêîãî ñîáûòèÿ èç ìíîæåñòâà A ⊂ E ÅÒÎÄ ÐÀÑÙÅÏËÅÍÈß

PLITTING

Z γ=

(1)

I(X∈A) dF (x) = EI(X∈A) ,

ãäå I(X∈A) ôóíêöèÿ-èíäèêàòîð ðåäêîãî ñîáûòèÿ èç ìíîæåñòâà A. 2.1. Îöåíèâàåìûå âåðîÿòíîñòè. Öåëüþ äèñêðåòíîãî ìîäåëèðîâàíèÿ ñèñòåì ìàññîâîãî îáñëóæèâàíèÿ îáû÷íî ÿâëÿåòñÿ ïîñòðîåíèå îöåíêè íåêîòîðûõ ñòàöèîíàðíûõ èëè ïåðåõîäíûõ õàðàêòåðèñòèê. Ïðèìåðîì òàêèõ õàðàêòåðèñòèê ÿâëÿþòñÿ ñòàöèîíàðíîå âðåìÿ îæèäàíèå çàÿâîê â î÷åðåäè, âåðîÿòíîñòü ïîòåðè çàÿâîê ïðè ïåðåïîëíåíèè, âåðîÿòíîñòü ïåðåïîëíåíèÿ áóôåðà äî òîãî, êàê ñèñòåìà ñòàíåò ïóñòîé è ò.ä. Îáîçíà÷èì TA ìîìåíò ïåðâîãî äîñòèæåíèÿ ïðîöåññîì ìíîæåñòâà A. Ïóñòü TB ìîìåíò äîñòèæåíèå ìíîæåñòâà B ⊂ E (îáû÷íî B = {0}). Äàëåå TB áóäåò èìåòü ñìûñë ìîìåíòà ñèëüíîé ðåãåíåðàöèè äëÿ ïðîöåññà X . (Íàïðèìåð, äëÿ ïðîöåññà èçìåíåíèÿ äëèíû î÷åðåäè â ñèñòåìå ìàññîâîãî îáñëóæèâàíèÿ TA - ìîìåíò ïåðåïîëíåíèÿ áóôåðà çàÿâîê, TB - ìîìåíò îïóñòåíèÿ î÷åðåäè.) Òîãäà ñîáûòèå {TA < TB } îòðàæàåò ïîïàäàíèå â ðåäêîå ìíîæåñòâî A íà öèêëå ðåãåíåðàöèè ïðîöåññà X . Äëÿ âû÷èñëåíèÿ îöåíêè âåðîÿòíîñòè γ = P (TA < TB ), ðàçäåëèì ïðîñòðàíñòâî ñîñòîÿíèé E ïðîöåññà X íà M + 1 âëîæåííûõ ïîäìíîæåñòâ Ci : (2)

A = CM +1 ⊂ CM ⊂ · · · ⊂ C1 .

Ïîäìíîæåñòâà Ci çàäàþòñÿ òàê íàçûâàåìîé f , çàäàííîé íà E

ôóíêöèåé çíà÷è-

ìîñòè

Ci = {x ∈ E | f (x) ≥ Ti }, i ∈ [1, M + 1],

(3)

Äîâåðèòåëüíîå îöåíèâàíèå âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà 127

ãäå {Ti } èçâåñòíûå çíà÷åíèÿ ôóíêöèè f , íàçûâàåìûå ïîðîãàìè èëè óðîâíÿìè. Âåðîÿòíîñòü γ ïðåäñòàâëÿåò ïðîèçâåäåíèå óñëîâíûõ âåðîÿòíîñòåé p1 = P (C1 ), . . . , pi+1 = P (Ci+1 |Ci ), i ∈ [1, M ], γ ≡ p1 p2 . . . pM +1 . Ïóñòü pˆ1 , . . . , pMˆ+1 îöåíêè äëÿ ñîîòâåòñòâóþùèõ óñëîâíûõ âåðîÿòíîñòåé p1 , . . . , pM +1 , ãäå êàæäàÿ îöåíêà pˆi âû÷èñëÿåòñÿ ñòàíäàðòíûì ìåòîäîì Ìîíòå-Êàðëî. Íèæå ïðèâåäåì îïèñàíèå àëãîðèòìà ìåòîäà Splitting. 2.2. Îïèñàíèå ìåòîäà. Îáîçíà÷èì êîëè÷åñòâî äîñòèæåíèé óðîâíÿ Ti âî âðåìÿ ìîäåëèðîâàíèÿ Ni (N0 = 1), i ∈ [1, M + 1] (NM +1 - êîëè÷åñòâî ðåäêèõ ñîáûòèé {TA < TB }). Ïóñòü Ri çàäàåò êîëè÷åñòâî ïîäòðàåêòîðèé, ãåíåðèðóåìûõ íà óðîâíå Ti , RM +1 = 1 (íà ïîñëåäíåì óðîâíå ïîäòðàåêòîðèè íå ãåíåðèðóþòñÿ). Ïîøàãîâàÿ ñõåìà àëãîðèòìà Splitting ñëåäóþùàÿ: (1) N0 = 1, Ni = 0 äëÿ âñåõ i ∈ [1, M + 1]. (2) Ìîäåëèðîâàòü òðàåêòîðèþ ïðîöåññà X ñòàíäàðòíûì ìåòîäîì Ìîíòå-Êàðëî. (3) Ìîäåëèðîâàòü òåêóùóþ òðàåêòîðèþ (ñãåíåðèðîâàííóþ íà óðîâíå i). Åñëè ïðîèçîøëî äîñòèæåíèå óðîâíÿ Ti+1 , òî ïåðåõîäèì ê øàãó 4. Åñëè äîñòèãëè óðîâíÿ T0 , òî ïðåõîäèì ê øàãó 2. Åñëè ïðîèçîøëî ñîáûòèå, çàâåðøàþùåå ìîäåëèðîâàíèå, òî ïåðåéòè íà øàã 5 àëãîðèòìà. (4) Åñëè äîñòèãëè óðîâíÿ Ti+1 , òî óâåëè÷èì ñ÷åò÷èê Ni íà 1, ðàñùåïèì òðàåêòîðèþ íà Ri+1 ïîäòðàåêòîðèé-ïîòîìêîâ. Êàæäàÿ òðàåêòîðèÿ ìîäåëèðóåòñÿ îòäåëüíî, ñîãëàñíî øàãó 3. (5) Âû÷èñëèì îöåíêè óñëîâíûõ âåðîÿòíîñòåé äîñòèæåíèÿ óðîâíÿ i ïðè óñëîâèè ñòàðòà ñ óðîâíÿ i − 1, pˆi =

Ni , Ri−1 Ni−1

(4)

ãäå Ri−1 Ni−1 ïðåäñòàâëÿåò îáùåå êîëè÷åñòâî òðàåêòîðèé, ñòàðòóþùèõ ñ óðîâíÿ i − 1. Ïåðåéòè íà øàã 6. (6) Ðåçóëüòèðóþùàÿ îöåíêà âåðîÿòíîñòè ðåäêîãî ñîáûòèÿ {TA < TB } ðàâíà γˆ =

M +1 Y i=1

pˆi .

(5)

128

À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ

Íåîáõîäèìî îòìåòèòü, ÷òî ïðåäëîæåííàÿ â äàííîé ðàáîòå ìîäèôèêàöèÿ ìåòîäà Splitting ðàáîòàåò ïî öèêëàì ðåãåíåðàöèè ìîäåëèðóåìîãî ïðîöåññà. Òàêàÿ ìîäèôèêàöèÿ ïåðâîíà÷àëüíîãî àëãîðèòìà (ñì. [79]) ïîçâîëÿåò ïðèìåíèòü àïïàðàò ðåãåíåðàòèâíîãî ìîäåëèðîâàíèÿ (êîòîðûé îïèðàåòñÿ íà ÖÏÒ) äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà îöåíèâàåìîé âåðîÿòíîñòè γ. Ïðèâåäåííûé âûøå àëãîðèòì ðåàëèçîâàí íà ÿçûêå C + + â âèäå ðåêóððåíòíîé ôóíêöèè, ïàðàìåòðàìè êîòîðîé ÿâëÿþòñÿ íîìåð óðîâíÿ íà êîòîðîì ïðîèñõîäèò ðàñùåïëåíèå è òåêóùåå çíà÷åíèå ôóíêöèè çíà÷èìîñòè f (x). Âîçâðàò èç ðåêóðñèè ïðîèñõîäèò ïðè äîñòèæåíèè ìíîæåñòâà B . Çàìå÷àíèå.

3. À P (TA < TB ) M/M/1 Ðàññìîòðèì îäíîñåðâåðíóþ ìîäåëü ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ M/M/1 ñ áåñêîíå÷íûì áóôåðîì è íåçàâèñèìî îäèíàêîâî ðàñïðåäåëåííûìè èíòåðâàëàìè ìåæäó ïðèõîäàìè çàÿâîê {τn }, èìåþùèìè ðàñïðåäåëåíèå ÍÀËÈÒÈ×ÅÑÊÀß ÎÖÅÍÊÀ

ÄËß ÑÈÑÒÅÌÛ

P (τn ≤ x) = 1 − e−λx ,

ãäå τn = tn+1 − tn è tn - ìîìåíò ïðèõîäà çàÿâêè ñ íîìåðîì n, n ≥ 1. Ïóñòü í.î.ð. âðåìåíà îáñëóæèâàíèÿ {Sn } èìåþò ðàñïðåäåëåíèå P (Sn ≤ x) = 1 − e−µx .

Ðàññìîòðèì ñëó÷àéíûé ïðîöåññ X = {ν(t), t ≥ 0} íà ïðîñòðàíñòâå ñîñòîÿíèé E = {0, 1, ...}, ãäå ν(t) - êîëè÷åñòâî çàÿâîê â ñèñòåìå â ìîìåíò âðåìåíè t. Ïîëó÷èì äëÿ ñèñòåìû M/M/1 àíàëèòè÷åñêóþ îöåíêó âåðîÿòíîñòè γ = P (TA < TB ). Äëÿ óäîáñòâà îïðåäåëèì äâà ñîáûòèÿ H+ = { ïðèõîä çàÿâêè ïðîèçîøåë ðàíüøå, ÷åì óõîä}, H− = { óõîä çàÿâêè ïðîèçîøåë ðàíüøå, ÷åì ïðèõîä}. Ïóñòü p = P (H+ ) - âåðîÿòíîñòü óâåëè÷åíèÿ äëèíû î÷åðåäè íà 1 (ñêà÷êà ââåðõ), q = P (H− ) - âåðîÿòíîñòü óìåíüøåíèÿ äëèíû î÷åðåäè íà 1 (ñêà÷êà âíèç), L - çàäàííîå çíà÷åíèå äëèíû î÷åðåäè (êàê ïðàâèëî L èìååò ñìûñë ïîðîãà äëèíû î÷åðåäè, ïîñëå êîòîðîãî ïðîèñõîäèò ïîòåðÿ çàÿâîê). Ïóñòü TA - ìîìåíò âðåìåíè, êîãäà äëèíà î÷åðåäè ñòàëà ðàâíà L, TB - ìîìåíò îïóñòåíèÿ î÷åðåäè.

Äîâåðèòåëüíîå îöåíèâàíèå âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà 129

Îáîçíà÷èì âåðîÿòíîñòü òîãî, ÷òî äëèíà î÷åðåäè äîñòèãíåò çíà÷åíèÿ L äî òîãî, êàê î÷åðåäü îïóñòååò, ïðè óñëîâèè, ÷òî â ìîìåíò âðåìåíè t äëèíà î÷åðåäè ðàâíà n Pn (t) = P (TA < TB | ν(t) = n), (6) ãäå 1 ≤ n < L. Ïî ñâîéñòâó ìàðêîâîñòè ðàññìàòðèâàåìîãî ïðîöåññà X ëåâàÿ ÷àñòü (6) íå çàâèñòè îò t, ïîýòîìó äàëåå âìåñòî Pn (t) áóäåì èñïîëüçîâàòü Pn . Îïðåäåëèì ãðàíè÷íûå óñëîâèÿ PL = P (TA < TB | ν(t) = L) = 1, P0 = P (TA < TB | ν(t) = 0) = 0. (7) Ïîëó÷èì ðåêóððåíòíîå ñîîòíîøåíèå äëÿ Pn P (TA < TB , ν(t) = n) = P (ν(t) = n) P (TA < TB , H+ , ν(t) = n) + P (TA < TB , H− , ν(t) = n) = P (ν(t) = n) P (TA < TB | H+ , ν(t) = n) · P (H+ , ν(t) = n) + P (ν(t) = n) P (TA < TB | H− , ν(t) = n) · P (H− , ν(t) = n) = P (ν(t) = n) P (TA < TB | ν(t + ∆t) = n + 1) · P (H+ )+ P (TA < TB | ν(t + ∆t) = n − 1) · P (H− ). Pn = P (TA < TB | ν(t) = n) =

(8)

Ñëåäîâàòåëüíî, Pn ñâÿçàíû ñëåäóþùèì óðàâíåíèåì (9) Îïðåäåëèì âåðîÿòíîñòè p è q äëÿ ñèñòåìû M/M/1, îïèñàííîé âûøå. Ðàññìîòðèì äâå íåçàâèñèìûå ïîñëåäîâàòåëüíîñòè í.î.â. {τn } è {Sn }. Çàÿâêà ñ íîìåðîì n óøëà èç ñèñòåìû (ïðîèçîøëî ñîáûòèå H− ), åñëè Sn < τn (ò. å. âðåìÿ îáñëóæèâàíèÿ çàÿâêè ñ íîìåðîì n ìåíüøå âðåìåíè äî ïðèõîäà ñëåäóþùåé çàÿâêè ñ íîìåðîì n + 1). Ñëåäîâàòåëüíî âåðîÿòíîñòü q óõîäà çàÿâêè èç ñèñòåìû M/M/1 ðàâíÿåòñÿ Pn = p · Pn+1 + q · Pn−1 , 1 ≤ n < L.

Z q = P (Sn < τn ) = 0

∞

λe−λt (1 − e−µt )dt = 1 −

λ µ = . λ+µ λ+µ

(10)

Ñîîòâåòñòâåííî ôîðìóëå (10) âåðîÿòíîñòü ïðèõîäà çàÿâêè â ñèλ ñòåìó (P (H+ )) ðàâíà p = 1 − q = . Ïóñòü ρ = λ/µ - èíòåíñèâλ+µ íîñòü òðàôôèêà, òîãäà èç (10) ñëåäóåò ρ = p/q .

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À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ

Óðàâíåíèå (9) òàêæå ìîæíî ïîëó÷èòü èñïîëüçóÿ çàäà÷ó î ðàçîðåíèèè èãðîêà (ñì. [10]). Ìîäèôèöèðóÿ ôîðìóëû, ïðèâåäåííûå â [10], ïîëó÷èì, ÷òî ðàçíîñòíîå óðàâíåíèå (9) ñ ãðàíè÷íûìè óñëîâèÿìè (7) èìååò ñëåäóþùåå ðåøåíèå. Ïðè ρ 6= 1 Pn =

ρL − ρL−n . ρL − 1

(11)

Ïîêàæåì, ÷òî (11) ÿâëÿåòñÿ ðåøåíèåì äëÿ (9). Ïîäñòàâèì ôîðìóëó (11) â (9) è ðàññìîòðèì ÷èñëèòåëü äðîáè â ïðàâîé ÷àñòè óðàâíåíèÿ p · ρL − p · ρL−n−1 + q · ρL − q · ρL−n+1 = ρL · (p + q) − ρL−n (p · ρ−1 + q · ρ) = ρL − ρL−n = Pn (ρL − 1).

(12)

 ñèëó òîãî, ÷òî ρ 6= 1 âûïîëíÿåòñÿ ρL 6= 1, ñëåäîâàòåëüíî, ôîðìóëà (11) åñòü ðåøåíèå óðàâíåíèÿ (9). Ðàññìîòðèì ρ = 1.  ýòîì ñëó÷àå ôîðìóëà (11) íå èìååò ñìûñëà. Ïîëó÷èì ðåøåíèå èç (11) ïåðåõîäÿ ê ïðåäåëó è âîñïîëüçîâàâøèñü ïðàâèëîì Ëîïèòàëÿ L · ρL−1 − (L − n) · ρL−n−1 n ρL − ρL−n = lim = . ρ→1 ρ→1 ρL − 1 L · ρL−1 L lim

(13)

Îöåíèâàåìàÿ âåðîÿòíîñòü γ = P (TA < TB ) ÿâëÿåòñÿ âåðîÿòíîñòüþ äîñòèæåíèÿ óðîâíÿ L ïðè åäèíè÷íîì íà÷àëüíîì çíà÷åíèè äëèíû î÷åðåäè, ò. å.  1   , ρ=1 LL γ = P (TA < TB | ν(t) = 1) = P1 = L−1 ρ −ρ   , ρ= 6 1. ρL − 1

(14)

ãäå ρ = λ/µ - èíòåíñèâíîñòü òðàôèêà. Íà ðèñóíêå ?? ïðèâåäåíû ðåçóëüòàòû ìîäåëèðîâàíèÿ âåðîÿòíîñòè γ = P (TA < TB ) äëÿ ñèñòåìû M/M/1 ìåòîäîì Splitting è ðåçóëüòàòû, ïîëó÷åííûå ïî ôîðìóëå (14). Ôóíêöèåé çíà÷èìîñòè äëÿ ìåòîäà Splitting (ñì. ÷àñòü 1) ÿâëÿåòñÿ f (t) = ν(t). Îöåíêà äëÿ γ ìåòîäîì Splitting ïðè ôèêñèðîâàííûõ çíà÷åíèÿõ ρ è L áûëà ïîëó÷åíà íà îñíîâå âûáîðêè îáúåìà 500 ñëó÷àéíûõ âåëè÷èí.

Äîâåðèòåëüíîå îöåíèâàíèå âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà 131

Ðèñ. 1.

Òî÷å÷íûå îöåíêè äëÿ âåðîÿòíîñòè γ â ñèñòåìå M/M/1 ïðè ρ = 0.6

4. Ð Êëàññè÷åñêèé ìåòîä ðåãåíåðàòèâíîãî ìîäåëèðîâàíèÿ îñíîâàí íà èñïîëüçîâàíèè öèêëîâ ñèëüíîé ðåãåíåðàöèè äëÿ ýôôåêòèâíîãî èíòåðâàëüíîãî îöåíèâàíèÿ. ÅÃÅÍÅÐÀÒÈÂÍÎÅ ÌÎÄÅËÈÐÎÂÀÍÈÅ

4.1. Ïîñòðîåíèå èíòåðâàëüíûõ îöåíîê. Ðàññìîòðèì X = {X(t), t ≥ 0} ðåãåíåðèðóþùèé ïðîöåññ (âîçìîæíî ñ çàäåðæêîé), çàäàííûé íà ïðîñòðàíñòâå ñîñòîÿíèé E ñ ìîìåíòàìè ðåãåíåðàöèè T (−1) = = 0 ≤ T (0) < T (1) < . . . . Ïóñòü αn = T (n) − T (n − 1), n ≥ 0 - äëèíà öèêëà ðåãåíåðàöèè. Ïóñòü Eτ1 < ∞ (ò. å. ïðîöåññ X ÿâëÿåòñÿ ïîëîæèòåëüíî âîçâðàòíûì). Äëÿ çàäàííîé èçìåðèìîé ôóíêöèè f : E → R îïðåäåëèì ïîñëåäîâàòåëüíîñòü ñëó÷àéíûõ âåëè÷èí Z

T (n)

f (X(s))ds, n ≥ 0.

Yn (f ) =

(15)

T (n−1) Óñëîâèå 1. Äëÿ ðåãåíåðàòèâíîãî ïðîöåññà X ñ Eα1 < ∞ ñóùåñòâóåò êîíñòàíòà γ òàêàÿ, ÷òî âûïîëíÿåòñÿ E[Y1 − γα1 ] = 0 è 0 < E(Y1 − γα1 )2 < ∞.

132

À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ

Åñëè ïðîöåññ X ÿâëÿåòñÿ ðåãåíåðèðóþùèì, òî {Zn , n ≥ 1}, ãäå Zn = Yn − γαn - ýòî ïîñëåäîâàòåëüíîñòü íåçàâèñèìûõ îäèíàêîâî ðàñïðåäåëåííûõ ñëó÷àéíûõ âåëè÷èí. Óñëîâèå 1 îçíà÷àåò, ÷òî E|Z1 | < ∞, E|Y1 | < ∞ è γ = EY1 /Eα1 . Îöåíêó äëÿ ïàðàìåòðà γ ìîæíî ïîëó÷èòü ìîäåëèðîâàíèåì íà âðåìåííîì èíòåðâàëå (0, t) ëèáî ìîäåëèðîâàíèåì ïî n öèêëàì ðåãåíåðàöèè. Áóäåì ðàññìàòðèâàòü ñëó÷àé ìîäåëèðîâàíèÿ ïî öèêëàì ðåãåíåðàöèè, òîãäà îöåíêà äëÿ γ èìååò ñëåäóþùèé âèä γn =

Pn n−1 k=1 Yk Yn P = . n αn n−1 k=1 αk

(16)

Èçâåñòíî, ÷òî îöåíêà γn äëÿ ïàðàìåòðà γ áóäåò ñòðîãî ñîñòîÿòåëüíîé, åñëè E|Y1 | < ∞ è Eα1 < ∞. Öåëü ðåãåíåðàòèâíîãî ìåòîäà ñîñòîèò â ïîñòðîåíèè àñèìïòîòè÷åñêè ïðàâèëüíîãî äîâåðèòåëüíîãî èíòåðâàëà äëÿ γ (äëÿ ïîñòðîåíèÿ èñïîëüçóåòñÿ ÖÏÒ). Òåîðåìà

1.

Ïóñòü

óñëîâèå

1

âûïîëíÿåòñÿ,

òîãäà

èìååò

ìåñòî

ñëàáàÿ ñõîäèìîñòü

n1/2 (γn − γ) ⇒ σN (0, 1), n → ∞, ãäå

σ 2 = EZ12 /(Eα1 )2 .

Ïðè ïîñòðîåíèè äîâåðèòåëüíîãî èíòåðâàëà äëÿ γ , íåîáõîäèìî èìåòü ñëàáî ñîñòîÿòåëüíóþ îöåíêó vn äèñïåðñèè σ2 , òîãäà 100(1 − − δ)-ïðîöåíòíûé äîâåðèòåëüíûé èíòåðâàë ñòðîèòñÿ ïî ôîðìóëàì  √  √ zδ vn zδ vn , γn − √ , γn + √ n n

(17)

ãäå zδ - êâàíòèëü, îïðåäåëÿåìàÿ âåðîÿòíîñòüþ P (N (0, 1) ≤ zδ ) = = 1 − δ/2. Îöåíêà vn ñòðîèòñÿ ñëåäóþùèì îáðàçîì Pn 1/n i=1 (Yi − γn αi )2 vn = , n ≥ 1. (18) α2n Òåîðåìà 2. Åñëè óñëîâèå

1 âûïîëíÿåòñÿ, òîãäà îöåíêà vn vn ⇒ σ22 , n → ∞.

ÿâëÿåòñÿ

ñëàáî ñîñòîÿòåëüíîé, ò. å.

Äëÿ ñòðîãîé ñîñòîÿòåëüíîñòè îöåíêè íåîáõîäèìû äîïîëíèòåëüíûå óñëîâèÿ EY12 < ∞ è Eα12 < ∞. Îäíàêî äëÿ ïðèìåíèìîñòè ðåãåíåðàòèâíîãî ìåòîäà äîñòàòî÷íî ñëàáîé ñõîäèìîñòè îöåíîê. Òàêèì îáðàçîì, óñëîâèå 1 ÿâëÿåòñÿ äîñòàòî÷íûì óñëîâèåì ïðèìåíèìîñòè ðåãåíåðàòèâíîãî ìåòîäà. Áîëåå ïîäðîáíî ñì. [1, 2, 11].

Äîâåðèòåëüíîå îöåíèâàíèå âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà 133

5. Ä P (TA < TB ) Ðàññìîòðèì ðåãåíåðèðóþùèé ïðîöåññ X = {ν(t), t ≥ 0} áåç çàäåðæêè, ãäå ν(t) - ýòî êîëè÷åñòâî çàÿâîê â ñèñòåìå M/M/1 â ìîìåíò âðåìåíè t. Äëÿ ìîäåëèðîâàíèÿ áóäåì èñïîëüçîâàòü n öèêëîâ ðåãåíåðàöèè ïðîöåññà X . Ïóñòü αi = T (i) − T (i − 1), i ∈ [1, n] äëèíû öèêëîâ ðåãåíåðàöèè ñ ñîîòâåòñòâóþùèìè ìîìåíòàìè ðåãåíåðàöèè T (i), i ∈ [0, n] . Äàëåå ïðèìåíèì ìåòîä èìèòàöèîííîãî ðåãåíåðàòèâíîãî ìîäåëèðîâàíèÿ (ñì. ÷àñòü 3) äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà äëÿ âåðîÿòíîñòè γ = P (TA < TB ). ÎÂÅÐÈÒÅËÜÍÎÅ ÎÖÅÍÈÂÀÍÈÅ ÂÅÐÎßÒÍÎÑÒÈ

Ïîñêîëüêó ïðîöåññ èçìåíåíèÿ äëèíû î÷åðåäè X äëÿ ìàðêîâñêèé, òî ëþáîå ïîïàäàíèå ïðîöåññà â ôèêñèðîâàííîå ñîñòîÿíèå ν(t) = k ÿâëÿåòñÿ ìîìåíòîì ðåãåíåðàöèè (k-ãî òèïà). Ìîäåëü M/M/1 õîðîøî èçó÷åíà, îäíàêî M/M/1 è ìîìåíòû ðåãåíåðàöèè 0-ãî òèïà (ïðè k = 0) èñïîëüçóþòñÿ â äàííîé ðàáîòå äëÿ èëëþñòðàöèè ìåòîäà è âîçìîæíîñòè ñðàâíåíèÿ ðåçóëüòàòîâ ìîäåëèðîâàíèÿ ìåòîäîì Splitting ñ àíàëèòè÷åñêîé ôîðìóëîé (14) (ñì. ÷àñòü 2). Ïóñòü â ôîðìóëå (15) çàäàííàÿ ôóíêöèÿ ñëó÷àéíîãî ïðîöåññà (i) (i) ðàâíà f = I{T , ãäå èíäèêàòîð I{T = 1, åñëè íà öèêëå A
Çàìå÷àíèå.

M/M/1

Z

Ti

(i)

Yn = T (i−1)

(i)

(i)

I{TA
Òîãäà ôîðìóëó (16) ìîæíî ïåðåïèñàòü â ñëåäóþùåì âèäå (i) i=1 I{TA
(1)

Pn γn =

=

(19)

(n)

I{TA
(20)

×èñëèòåëü ôîðìóëû (20) ÿâëÿåòñÿ ñóììîé äëèí öèêëîâ ðåãåíåðàöèè, íà êîòîðûõ ïðîèçîøëî ñîáûòèå {TA < TB }. Çíàìåíàòåëü ÿâëÿåòñÿ ñóììàðíîé äëèíîé âñåõ öèêëîâ ðåãåíåðàöèè. Ñ äðóãîé ñòîðîíû ðàññìîòðèì îöåíêó âåðîÿòíîñòè γ , ïîñòðîåííóþ â ìåòîäå Splitting. Ïåðåïèøåì ôîðìóëó (5) γˆ =

NM +1 . R0 R1 · · · R M

(21)

134

À. Â. Áîðîäèíà, Å. Â. Ìîðîçîâ

×èñëèòåëü ôîðìóëû (21) ïðåäñòàâëÿåò îáùåå êîëè÷åñòâî ñîáûòèé {TA < TB } (äîñòèæåíèé óðîâíÿ M + 1). Çíàìåíàòåëü ðàâåí îáùåìó ÷èñëó òðàåêòîðèé, ïîëó÷åííûõ ïðè ìîäåëèðîâàíèè ìåòîäîì Splitting. Êàæäàÿ òðàåêòîðèÿ ÿâëÿåòñÿ îòäåëüíûì öèêëîì ðåãåíåðàöèè ïðîöåññà X (è ìîãëà áûòü àíàëîãè÷íî ïîñòðîåíà ìåòîäîì êëàññè÷åñêîãî ìîäåëèðîâàíèÿ Ìîíòå-Êàðëî). Òàêèì îáðàçîì ìîäèôèêàöèÿ ìåòîäà Splitting (òðàåêòîðèÿ îáðûâàåòñÿ òîëüêî ïðè äîñòèæåíèè ìíîæåñòâà B ) ïîçâîëÿåò ñãåíåðèðîâàòü R0 · R1 · · · RM öèêëîâ ðåãåíåðàöèè. Ñëåäîâàòåëüíî â ôîðìóëå (20) ÷èñëî öèêëîâ ðåãåíåðàöèè n = R0 · R1 · · · RM . Ñîãëàñíî ñâîéñòâó PASTA (Poisson Arrival See Time Average) (ñì. [12]) äëÿ ñèñòåì, â êîòîðûõ ìîìåíòû ïðèõîäà çàÿâîê tn , n ≥ 1 ðàñïðåäåëåíû ïî çàêîíó Ïóàññîíà (ò. å. M/·/·), îöåíêà ïî ôîðìóëå (20) ýêâèâàëåíòíà îöåíêå, ïîëó÷åííîé ìåòîäîì Splitting (ôîðìóëà (21)) è âûðàæåííîé â òåðìèíàõ ñîáûòèé (ò. å. γˆ = γn ), òîãäà n X

n· αi =

i=1

(i) i=1 I{TA
Pn

NM +1

(22)

.

Ïîñòðîèì äîâåðèòåëüíûé èíòåðâàë äëÿ γ ïî ôîðìóëå (17). Äëÿ ýòîãî ïîëó÷èì îöåíêó vn äëÿ σ2 ïî ôîðìóëàì (18) è (22) (i) 2 i=1 (I{TA
n·

Pn

=

(23)

Ïîäñòàâèâ ôîðìóëó (23) â âûðàæåíèå (17), ïîëó÷èì äîâåðèòåëüíûé èíòåðâàë äëÿ âåðîÿòíîñòè γ , îöåíèâàåìîé ìåòîäîì Splitting. Äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà ñ ïîìîùüþ ðåãåíåðàòèâíîãî ìåòîäà òðåáóåòñÿ âíåñòè èçìåíåíèÿ â îïèñàííûé â ÷àñòè 1 àëãîðèòì.  ïðîöåññå ìîäåëèðîâàíèÿ íåîáõîäèìî âåñòè ïîäñ÷åò äëèí öèêëîâ αi , i ∈ [1, n], è õðàíèòü çíà÷åíèÿ (i) I{TA
Äîâåðèòåëüíîå îöåíèâàíèå âåðîÿòíîñòè ïåðåïîëíåíèÿ áóôåðà 135

6. Ç Â äàííîé ñòàòüå ïðåäëîæåíà ìîäèôèêàöèÿ ìåòîäà óñêîðåííîãî ìîäåëèðîâàíèÿ Splitting äëÿ ïîñòðîåíèÿ äîâåðèòåëüíîãî èíòåðâàëà âåðîÿòíîñòè ïåðåïîëíåíèÿ γ . Ìîäåëèðîâàíèå ïî öèêëàì ðåãåíåðàöèè ðàññìàòðèâàåìîãî ïðîöåññà äëèíû î÷åðåäè ïîçâîëÿåò ñòðîèòü èíòåðâàëüíûå îöåíêè, èñïîëüçóÿ ðåãåíåðàòèâíûé ïîäõîä. Ôîðìóëû, ïðèâåäåííûå â ÷àñòè 4, ïîçâîëÿþò ñòðîèòü äîâåðèòåëüíûé èíòåðâàë â ïðîöåññå óñêîðåííîãî ìîäåëèðîâàíèÿ ìåòîäîì Splitting. Êðîìå òîãî ïîëó÷åíû òî÷å÷íûå îöåíêè âåðîÿòíîñòè γ äëÿ ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ M/M/1 è ïðîâåäåíî ñðàâíåíèå ñ àíàëèòè÷åñêèì ðåøåíèåì (14). Ðèñóíîê ?? ïîêàçûâàåò, ÷òî îöåíêà, ïîëó÷åííàÿ ìîäåëèðîâàíèåì, ïðàêòè÷åñêè ñîâïàäàåò ñ àíàëèòè÷åñêîé îöåíêîé. ÀÊËÞ×ÅÍÈÅ

Ëèòåðàòóðà 1. P. of

W. the

Glynn,

D.

L.

regenerative

Iglehart. method

(1993). //

Conditions

Management

for

the

Science

applicability

39:

1108-1111.

http://citeseer.ist.psu.edu/36432.html 2. P. W. Glynn. Some topics in regenerative steady state simulation. 1993. 3. K. Sigman, R. Wolff. A review of regenerative processes. SIAM Review, Vol. 35, No. 2, pp.269-288. 1993. 4. P. E. Heegaard, A survey of Speedup simulation techniques. Workshop tutorial on Rare Event Simulation, Aachen, Germany. 1997. 5. P. Shahabuddin. Rare event simulation in stochastic models. Proceedings of the WSC 1995, IEEE Press., pp. 178-185. 6. A. Borodina, E. Morozov. Simulation of rare events with speed-up techniques: Splitting and RESTART. Proceedings of Finnish Data Processing Week at the Petrozavodsk State University (FDPW'2005), Vol. 7, 2006. 7. M. Garvels. The splitting method in rare event simulation. May 2000. 8. P. Glasserman, P. Heidelberger, P. Shahabuddin, and T. Zajic. A look at multilevel splitting. In H. Niederreiter, editor, Monte Carlo and Quasi Monte Carlo Methods 1996, Lecture Notes in Statistics, volume 127, pages 99-108. Springer Verlag, 1996. 9. P.

Glasserman,

P.

Heidelberger,

P.

Shahabuddin,

and

T.

Zajic.

Splitting

for

rare event simulation: analysis of simple cases. Proceedings of the 1996 Winter Simulation Conference. 10. Â. Ôåëëåð. Ââåäåíèå â òåîðèþ âåðîÿòíîñòèé è åå ïðèëîæåíèÿ. Òîì 1. Ì.:Ìèð, 1964. 11. P. W. Glynn, D. L. Iglehart. A joint central limit theorem for the sample mean and regenerative variance estimator. Annals of Operations Research 8, 1987, 41-55. 12. R. W. Wolff. Poisson Arrivals See Time Average. Opns. Res, 30, 223-231. 1982.

Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ

Âûïóñê 7, 2006

ÓÄÊ 512.56

ÎÁ ÎÄÍÎÉ ÌÎÄÅËÈ ÑËÓ×ÀÉÍÎÃÎ ÃÐÀÔÀ "ÈÍÒÅÐÍÅÒ" ÒÈÏÀ

Í.Ñ. ÊÈÑÅËÅÂÀ

Äàíî îïèñàíèå ñëó÷àéíûõ ãðàôîâ, ó êîòîðûõ ñòåïåíè âåðøèí  íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû, ðàñïðåäåëåííûå ïî çàêîíó P(D = d) = d−τ /ζ(τ ), ãäå D  ñëó÷àéíàÿ P âåëè÷èíà, ðàâíàÿ ñòåïåíè âåðøèíû ãðàôà, d = 1, 2, . . ., ζ(τ ) = ∞d=1 d−τ  äçåòà-ôóíêöèÿ Ðèìàíà, τ ∈ (2, 3) (êîíå÷íîå ìàòåìàòè÷åñêîå îæèäàíèå è áåñêîíå÷íàÿ äèñïåðñèÿ), à îáðàçîâàíèå ðåáåð ïðîèñõîäèò ðàâíîâåðîÿòíî. Ïðåäëîæåíî èññëåäîâàòü òàêèå ñëó÷àéíûå ãðàôû äëÿ îïèñàíèÿ òîïîëîãèè Èíòåðíåò. Ïîëó÷åíû ïðåäåëüíûå ðàñïðåäåëåíèÿ äëÿ êðàéíèõ ÷ëåíîâ âàðèàöèîííîãî ðÿäà, ñîñòàâëåííîãî èç íåóáûâàþùåé ïîñëåäîâàòåëüíîñòè ñòåïåíåé âåðøèí ãðàôà, òàêæå äîêàçàíà ëîêàëüíàÿ ïðåäåëüíàÿ òåîðåìà äëÿ ñóììû ñòåïåíåé âåðøèí ïðè N → ∞, ãäå N  ÷èñëî âåðøèí â ãðàôå. Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå ãðàíòà ÐÔÔÈ 05-0100007à. Cëó÷àéíûå ãðàôû â êà÷åñòâå ìîäåëåé äëÿ èçó÷åíèÿ îñíîâíûõ õàðàêòåðèñòèê òîïîëîãèè Èíòåðíåò èíòåíñèâíî èñïîëüçóþòñÿ óæå íåñêîëüêî ëåò. Îäíîé èç òàêèõ ìîäåëåé ïîñâÿùåíà ñòàòüÿ [1]. Ïóñòü ãðàô ñîäåðæèò N âåðøèí, èìåþùèõ íîìåðà 1, 2, . . . , N . Îáîçíà÷èì D1 , . . . , DN ïîñëåäîâàòåëüíîñòü íåçàâèñèìûõ îäèíàêîâî ðàñïðåäåëåííûõ ñëó÷àéíûõ âåëè÷èí, ïðèíèìàþùèõ çíà÷åíèÿ 1, 2, . . .. Áóäåì ñ÷èòàòü, ÷òî ñëó÷àéíàÿ âåëè÷èíà Di ðàâíà ÷èñëó ðåáåð, èíöèäåíòíûõ i-îé âåðøèíå.

c

Í.Ñ. Êèñåëåâà, 2006

Îá îäíîé ìîäåëè ñëó÷àéíîãî ãðàôà "Èíòåðíåò" òèïà

137

 ñòàòüå [1] èñïîëüçóåòñÿ ñëåäóþùèé çàêîí ðàñïðåäåëåíèÿ ñòåïåíåé âåðøèí:

P{Di ≥ d} = d−τ +1 ,

ãäå i = 1, . . . , N, d = 1, 2, . . ., τ ∈ (2, 3). Îòñþäà

P{Di = d} = d−τ +1 − (d + 1)−τ +1 . Íà ïåðâîì ýòàïå êîíñòðóèðîâàíèÿ ãðàôà ðàññìàòðèâàþòñÿ âåðøèíû ñ âûõîäÿùèìè èç íèõ ðåáðàìè è íå ó÷èòûâàþòñÿ âåðøèíû íà äðóãîì êîíöå êàæäîãî ðåáðà. Äëÿ òàêîé ìîäåëè ââîäèòñÿ ïîíÿòèå ïîëóðåáðà. Ïîëóðåáðî  ýòî ðåáðî, âûõîäÿùåå èç âåðøèíû ñ íåçàêðåïëåííîé âåðøèíîé íà äðóãîì êîíöå. Íà ñëåäóþùåì ýòàïå äàííûå ïîëóðåáðà ðàâíîâåðîÿòíî ñîåäèíÿþòñÿ ìåæäó ñîáîé, îáðàçóÿ ðåáðà ãðàôà. Åñëè ñóììà ñòåïåíåé âåðøèí íå÷åòíà, òî äîáàâëÿåòñÿ íîâàÿ âåðøèíà ñ íîìåðîì 0 è åäèíè÷íîé ñòåïåíüþ D0 = 1. Íîâàÿ ìîäåëü, ïðåäñòàâëåííàÿ â äàííîé ðàáîòå, ÿâëÿåòñÿ ñëó÷àéíûì ãðàôîì ñ àíàëîãè÷íûì ïîñòðîåíèåì, íî ñ èíûì çàêîíîì ðàñïðåäåëåíèÿ ñòåïåíåé:

P{Di = d} =

1 , ζ(τ )dτ

(1)

∞ P

ãäå d = 1, 2, . . ., τ ∈ (2, 3), ζ(τ ) = d−τ  äçåòà-ôóíêöèÿ Ðèìàíà. d=1 Â ñòàòüå ïîêàçàíî, ÷òî íîâàÿ ìîäåëü ñîõðàíÿåò îñíîâíûå ñâîéñòâà èçâåñòíîé ìîäåëè, óêàçàííûå â ñòàòüÿõ [1] è [3]. Ïðè ýòîì âûäâèãàåòñÿ ïðåäïîëîæåíèå, ÷òî ïðåäëîæåííàÿ ìîäåëü ìîæåò îêàçàòüñÿ áîëåå óäîáíîé äëÿ ðåøåíèÿ íåêîòîðûõ çàäà÷ â ñèëó áîëåå ïðîñòîãî âûðàæåíèÿ P{Di = d}. Èç (1) ñëåäóåò, ÷òî ïðè d → ∞

P{Di ≥ d} =

1 d−τ +1 (1 + o(1)). ζ(τ )(τ − 1)

(2)

Ðàñïðåäåëåíèå (1) íå èìååò êîíå÷íîé äèñïåðñèè è èìååò êîíå÷íîå ìàòåìàòè÷åñêîå îæèäàíèå. Îáîçíà÷èì D ñòåïåíü ïðîèçâîëüíîé âåðøèíû ãðàôà, òîãäà

ED =

∞ X d=1

d·

1 ζ(τ − 1) = . ζ(τ )dτ ζ(τ )

138

Í.Ñ. Êèñåëåâà

Îáîçíà÷èì . Òîãäà

F (x)

óíêöèþ ðàñïðåäåëåíèÿ ñëó÷àéíîé âåëè÷èíû

D

F (x)

=

Pf

g=

D < x

(

0 1  ( )(

1

1)

de x

ïðè  1, èíà÷å, x

 +1

(1 + o(1))

ãäå d e  íàèìåíüøåå öåëîå  . âåðøèíàìè, ó Ïóñòü ( N )  ìíîæåñòâî ãðàîâ ñ êîòîðûõ ïåðâàÿ âåðøèíà èìååò ñòåïåíü ðàâíóþ , âòîðàÿ è ò.ä. Îáîçíà÷èì îáùóþ ñóììó ñòåïåíåé âåðøèí N X n

x

x

G d1 ; : : : ; d

N

d1

l

=

d2

di :

i=1

Âåðîÿòíîñòü òîãî, ÷òî ñëó÷àéíûé ãðà ñ âåðøèíàìè ïðèíàäëåæèò ìíîæåñòâó ( N ) ðàâíà: g

N

G d1 ; : : : ; d

Pf 2

g= = Pf = g

Pf N = N g Ïîñêîëüêó âñå ñîåäèíåíèÿ ïîëóðåáåð ìåæäó ñîáîé â ïàðû ðàâíîâåðîÿòíû, åñòåñòâåííî ñ÷èòàòü, ÷òî âñå ãðàû â ìíîæåñòâå ( N ) ðàâíîâåðîÿòíû. Òàê êàê âñå ïîëóðåáðà çàíóìåðîâàíû, òî ñóùåñòâóåò ( 1)( 3)

1 ñïîñîáîâ ñîåäèíèòü èõ ìåæäó ñîáîé. Çíà÷èò, óñëîâíàÿ âåðîÿòíîñòü âûáîðà êîíêðåòíîãî ãðàà èç ýòîãî ìíîæåñòâà ðàâíà: g

g = Pf

G(d1 ; : : : ; dN )

D1

=

=

d1 ; : : : ; DN D1

d1

dN

:::

D

d

:

G d1 ; : : : ; d

l

:::

l

g

Pf j 2

g= (

1



1 Òîãäà âåðîÿòíîñòü ïîÿâëåíèÿ êîíêðåòíîãî ãðàà 2 ( N) ðàâíà: g

Pf N = N g Pf g = Pf (= 1)( 3)

1  ðàññìàòðèâàåìîé ìîäåëè íåò îãðàíè÷åíèé íà ñîåäèíåíèÿ ïîëóðåáåð â ïàðû, ïîýòîìó íåò îãðàíè÷åíèÿ íà ñóùåñòâîâàíèå êðàòíûõ ðåáåð è ïåòåëü.  ñòàòüå [1℄ äîêàçàíà ëåììà: g g

G(d1 ; : : : ; dN )

l

1)(l

3)

:::

g

g

D1

d1

l

:::

l

D

:::

d

:

G d1 ; : : : ; d

:

Îá îäíîé ìîäåëè ñëó÷àéíîãî ãðàà "Èíòåðíåò" òèïà

Ëåììà 1. Ïóñòü

U

è

F G=N

!1

X

V

äâà íåïåðåñåêàþùèõñÿ ìíîæåñòâà âåðøèí.

F

=

Îáîçíà÷èì

Åñëè

X

139

2

Di ;

G

i U

=

2

Dj :

j V

âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1, ïðè

N

!1

,

òî ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1, ñóùåñòâóåò âåðøèíà èç ìíîæåñòâà èç

U,

êîòîðàÿ ñâÿçàíà ðåáðîì ñ êàêîé-íèáóäü âåðøèíîé

V.

Èç ýòîé ëåììû ñëåäóåò, ÷òî âåðøèíû, ñòåïåíü êîòîðûõ íå ìåíüøå, ÷åì N 1=2+" , ãäå " > 0, ñîåäèíåíû ìåæäó ñîáîé õîòÿ áû îäíèì ñâîèì ðåáðîì ñ âåðîÿòíîñòüþ, ñòðåìÿùåéñÿ ê 1, ñëåäîâàòåëüíî, îíè îáðàçóþò ïîëíûé ãðà. Ïîêàæåì íèæå, ÷òî åñëè âûáðàòü " òàê, ÷òîáû âûïîëíÿëîñü íåðàâåíñòâî " < (3  )=2( 1), òî òàêèå âåðøèíû ñóùåñòâóþò. Îáîçíà÷èì  ìíîæåñòâî âåðøèí ãðàà, ó êîòîðûõ ñòåïåíè âåðøèí óäîâëåòâîðÿþò óñëîâèþ: Di  N 1=2+" , ãäå i = 1; : : : ; N . Òîãäà ìîùíîñòü ìíîæåñòâà îïðåäåëÿåòñÿ:

j j =

Xf N

I Di

i=1

N

1=2+"

g;

ãäå I(A) - èíäèêàòîð ñîáûòèÿ A. Ïðèìåíèâ òåîðåìó ÌóàâðàËàïëàñà ïðè ñëåäóþùèé ðåçóëüòàò:

Pfj j = kg =

p21 + (1(1) o

N p

)

p

N

! 1,

ïîëó÷èì

k Np)2 Np(1 p) ;


e

( 2

ãäå p = PfD  N 1=2+" g = ( ( )( 1)) 1
(N 1=2+" )  +1 (1 + o(1)). Íåòðóäíî âèäåòü, ÷òî N p ! 1, ñëåäîâàòåëüíî, ÷èñëî âåðøèí, óäîâëåòâîðÿþùèõ çàäàííîìó óñëîâèþ, ñòðåìèòñÿ ê áåñêîíå÷íîñòè. Äàëåå ïîñòðîèì âàðèàöèîííûé ðÿä D(1)

6 D 6 ::: 6 D N ; (2)

(

)

ðàñïîëîæèâ D1 ; D2 ; : : : ; DN â íåóáûâàþùåì ïîðÿäêå. Íàéäåì ïðåäåëüíûå ðàñïðåäåëåíèÿ äëÿ êðàéíèõ ÷ëåíîâ ýòîãî ðÿäà.

140

Í.Ñ. Êèñåëåâà

Òåîðåìà 1.

ëî, òîãäà Ä

Ïóñòü N ! 1 è k  èêñèðîâàííîå íàòóðàëüíîå ÷èñPfD(k) = 1g ! 1:

ÎÊÀÇÀÒÅËÜÑÒÂÎ.

Fk (x); k = 1; 2; : : : ; N , îáîçíà÷àåò óíêk -ãî ÷ëåíà âàðèàöèîííîãî ðÿäà (3), òîãäà

Ïóñòü

öèþ ðàñïðåäåëåíèÿ

= PfD(k) < xg = 1 PfD(k)  xg:

Fk (x) ßñíî, ÷òî

PfD(k)  xg = PfD1  x; : : :D; N  xg + CN1
PfD1 < x; D2  x; : : :D; N > xg+ + CN2
PfD1 < x; D2 < x; D3  x; : : :D; N > xg + : : : + + CNk 1
PfD1 < x; : : : ; Dk 1 < x; Dk  x; : : :D; N > xg = = PfD1 > x; : : :D; N > xg+ +

X

k 1 i=1

i CN

x>

1

1 +

X

k 1 i=1

(N

N!


i)!i!

1

< x; : : :D ; i < x; Di+1

PfD(k)

Fk (x), ñëåäóåò, ÷òî

Èç îïðåäåëåíèÿ Äàëåå ïðè


PfD

Fk (x)



=

de

1

x



de

x

 +1

 +1

i 

N

 xg:

g = 0 ïðè x  1.

< x

(1 + o(1))

(1 + o(1))

 x; : : :D;

N

de

x

+

 +1

(1 + o(1))

N

i

;

= ( ( )( 1)) 1 . Èñõîäÿ èç òîãî, ÷òî x > 1 è < 1, ïîëó÷èì, ÷òî 1 Fk (x) ! 0 ïðè N ! 1. Îòñþäà ñëåäóåò, ÷òî PfD(k)  xg ! 0 ïðè x > 1 è N ! 1. Óòâåðæäåíèå òåîðåìû äîêàçàíî. ãäå

Ïóñòü N ! 1 è k  èêñèðîâàííîå íåîòðèöàòåëüíîå öåëîå ÷èñëî, òîãäà

Òåîðåìà 2.

Pf( 1) ln D(N

ãäå = ( ( )( ãàììà-óíêöèÿ.

1))

k) 1

è

ln N

g=

<x

(k + 1; e x) (1 + o(1)); k!

R (m; x) = x1 tm

1

e t dt

(3)

 äîïîëíèòåëüíàÿ

Îá îäíîé ìîäåëè ñëó÷àéíîãî ãðàà "Èíòåðíåò" òèïà

Ä

141

ÎÊÀÇÀÒÅËÜÑÒÂÎ.

Pf( 1) ln D(N k) ln N < xg = PfD(N k) < (N ex )  1 g = 1 1 = PfD1 < (N ex )  1 ; : : : ; DN < (N ex )  1 g+ 1

k X 1 1 + CNi
PfD1 < (N ex )  1 ; : : : ; DN i < (N ex)  1 ; i=1

DN

i+1 > (N ex ) 

N > (N ex )  1 g = k  N i X

N! 1
(N ex )i = ( N i)!i! N ex i=0 1

1

1;:::;D

Ïðèìåíÿÿ îðìóëó Ñòèðëèíãà, ïîëó÷èì ðàâåíñòâî

k x X ( e x )i + o(1) = PfD(N k) < (N ex )  1 1 g = e e i=0 i! =

(k + 1; e x) (1 + o(1)); k!

÷òî è òðåáîâàëîñü äîêàçàòü. Ñëåäñòâèå 1.

Ïðè

k

= 0 ïðåäåëüíîå ðàñïðåäåëåíèå äëÿ ìàêñèìàëü-

íîé ñòåïåíè ãðàà èìååò âèä: x Pf( 1) ln D(N ) ln N < xg = e e (1 + o(1)):

(4)

Äëÿ íàõîæäåíèÿ ïðåäåëüíûõ ðàñïðåäåëåíèé ñóììû ñòåïåíåé âåðøèí ìîæíî âîñïîëüçîâàòüñÿ òåîðåìàìè 2.1.1, 2.2.2, 2.6.1 è 4.2.1 èç [2℄. Îäèíàêîâî ðàñïðåäåëåííûå íåçàâèñèìûå ñëó÷àéíûå âåëè÷èíû

D1 ; D2 ; : : : ; DN ÿâëÿþòñÿ ðåøåò÷àòûìè ñ ìàêñèìàëüíûì øà-

ãîì, ðàâíûì åäèíèöå, è óíêöèåé ðàñïðåäåëåíèÿ (1). Îáðàçóåì ñóììû:

SN

=

D1

+ D2 + : : : + DN BN

AN

;

AN è BN - íîðìèðóþùèå ïîñòîÿííûå. FN (x) ñóìì SN ñëàáî ñõîäèòñÿ ê íåêîòîðîé óíêöèè ðàñïðåäåëåíèÿ G(x), òî ãîâîðÿò, ÷òî ãäå

Åñëè óíêöèÿ ðàñïðåäåëåíèÿ

142

Í.Ñ. Êèñåëåâà

ïðèòÿãèâàåòñÿ ê G(x). Ìíîæåñòâî âñåõ ôóíêöèé, ïðèòÿãèâàþùèõñÿ ê G(x), íàçûâàåòñÿ îáëàñòüþ ïðèòÿæåíèÿ çàêîíà G(x). Èç òåîðåìû 2.1.1 êíèãè [2] ñëåäóåò, ÷òî ðàñïðåäåëåíèå ñóìì SN èìååò â êà÷åñòâå ïðåäåëüíîãî ðàñïðåäåëåíèÿ óñòîé÷èâûé çàêîí G(x), è âèä íîðìèðóþùèõ ïîñòîÿííûõ ïðè ìåäëåííî ìåíÿþùåéñÿ â ñìûñëå Êàðàìàòà ôóíêöèè h(x) = 1 áóäåò ñëåäóþùèì: F (x)

BN = N 1/(τ −1) ,

AN = N · E D,

ãäå 1 < τ − 1 < 2. Ôóíêöèÿ ðàñïðåäåëåíèÿ F (x) ïðèíàäëåæèò îáëàñòè ïðèòÿæåíèÿ óñòîé÷èâîãî çàêîíà ñ ïîêàçàòåëåì τ − 1 â ñèëó òåîðåìû 2.6.1 èç [2].Îòñþäà è èç òåîðåì 4.2.1 è 2.2.2 èç [2], ïîëó÷èì ñëåäóþùèé ðåçóëüòàò: Òåîðåìà 3.

Ïðè

ñïðàâåäëèâî:

N →∞   1/(τ −1) k · ζ(τ ) − N · ζ(τ − 1) sup N P{LN = k} − g → 0, N 1/(τ −1) ζ(τ ) k PN LN = i=1 Di g(x) G(x) τ −1 ϕ(x) G(x)

ãäå

(5)

,  ïëîòíîñòü óñòîé÷èâîãî ðàñïðåäåëåíèÿ ñ ïîêàçàòåëåì , ïðè÷åì õàðàêòåðèñòè÷åñêàÿ ôóíêöèÿ óñòîé÷èâîãî ðàñïðåäåëåíèÿ èìååò âèä: ϕ(t) =

 = exp iγt +

  Γ(3 − τ ) π(τ − 1) τ −1 t π(τ − 1) cos |t| 1 + i tg . ζ(τ )(τ − 1)(τ − 2) 2 |t| 2

Òàêèì îáðàçîì, ïîëó÷åííûå ðåçóëüòàòû ñâèäåòåëüñòâóþò î òîì, ÷òî ðàññìîòðåííûå ñâîéñòâà ïðåäëîæåííîé ìîäåëè àíàëîãè÷íû ñîîòâåòñòâóþùèì ñâîéñòâàì ìîäåëè, èçó÷àâøåéñÿ â ñòàòüÿõ [1] è [3].

Ëèòåðàòóðà

1. Reittu H., Norros I. On the power-low random graph model of massive data networks, Performance Evaluation, 55, 2004, 3-23. 2. Èáðàãèìîâ È.À., Ëèííèê Þ.Â. Íåçàâèñèìûå è ñòàöèîíàðíî ñâÿçàííûå âåëè÷èíû. Ì., 1965. 3. Ñòåïàíîâ Ì.Ì. Î ïðåäåëüíûõ ðàñïðåäåëåíèÿõ ñòåïåíåé óçëîâ â ñëó÷àéíûõ ãðàôàõ "Èíòåðíåò" òèïà // Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ, Ïåòðîçàâîäñê, 2005, âûï.6, ñ.235-242.

Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ

Âûïóñê 7, 2006

ÓÄÊ 519.144.1 + 519.16 ÊÎÌÏÎÇÈÒÍÛÉ ÝÂÎËÞÖÈÎÍÍÛÉ ÀËÃÎÐÈÒÌ ÄËß ÏÎÒÎÊÎÂÎÉ ÇÀÄÀ×È ØÒÅÉÍÅÐÀ.

Â.Ä. Ê

.

ÓÊÈÍ

Îáñóæäàåòñÿ êîìïîçèòíûé ýâîëþöèîííûé àëãîðèòì äëÿ ðåøåíèÿ çàäà÷è Øòåéíåðà ñ ïîòîêàìè è çàâèñÿùèìè îò íèõ âåñàìè. Ïðèâîäÿòñÿ îñíîâíûå ðåçóëüòàòû òåñòèðîâàíèÿ ýòîãî àëãîðèòìà.

Çàäà÷à Øòåéíåðà ñ ïîòîêàìè è çàâèñÿùèìè îò íèõ âåñàìè  îäíî èç îáîáùåíèé ïðîáëåìû Øòåéíåðà [1]. Äàëåå äëÿ êðàòêîñòè ýòà çàäà÷à íàçûâàåòñÿ ïîòîêîâîé. Îíà îòíîñèòñÿ ê êëàññó NPòðóäíûõ çàäà÷ [2], äëÿ ïîèñêà ðåøåíèé êîòîðûõ îáû÷íî ïðèìåíÿþò ïðèáëèæåííûå ìåòîäû, â ÷àñòíîñòè, ìåòîäû ýâîëþöèîííîãî ìîäåëèðîâàíèÿ.  ñòàòüå ðàññìàòðèâàåòñÿ îäèí èç êîìïîçèòíûõ ýâîëþöèîííûõ àëãîðèòìîâ, ðàçðàáîòàííûõ äëÿ ðåøåíèÿ çàäà÷è. 1. Ï Äàäèì êîìáèíàòîðíóþ ïîñòàíîâêó ïîòîêîâîé çàäà÷è Øòåéíåðà, èñïîëüçóÿ òåðìèíîëîãèþ òåîðèè ãðàôîâ. Ââåäåì ñëåäóþùèå îáîçíà÷åíèÿ: G = (V, D)  îðèåíòèðîâàííûé ê êîðíþ ïëîñêèé ãðàô, ãäå V  ìíîæåñòâî âåðøèí è D  ìíîæåñòâî äóã îðãðàôà. V = V0 ∪ Vs , |V | = |V0 | + |Vs | = n + s. V0  ìíîæåñòâî âåðøèí îðãðàôà, ðàñïîëîæåííûõ â ôèêñèðîâàííûõ òî÷êàõ ïëîñêîñòè, íàçûâàåìûõ òåðìèíàëüíûìè âåðøèíàìè. i0 ∈ V0  êîðåíü äåðåâà, èëè âûäåëåííàÿ òåðìèíàëüíàÿ âåðøèíà, â êîòîðîé íàõîäèòñÿ ñòîê. M0 = {mi ∈ R+ |i ∈ V0 \{i0 }}  ìíîæåñòâî ìîùíîñòåé èñòî÷íèêîâ â òåðìèíàëüíûõ âåðøèíàõ. ÎÑÒÀÍÎÂÊÀ ÇÀÄÀ×È

c

Â.Ä. Êóêèí., 2006

144

Â.Ä. Êóêèí.

Vs  ìíîæåñòâî âåðøèí îðãðàôà, ðàñïîëîæåííûõ â äîïîëíèòåëüíûõ ñâîáîäíî ðàçìåùàåìûõ òî÷êàõ ïëîñêîñòè, íàçûâàåìûõ òî÷êàìè Øòåéíåðà (ÒØ). X0 = {(xi , yi )|xi , yi ∈ R+ ; i ∈ V0 }  çàäàííîå ìíîæåñòâî êîîðäèíàò òåðìèíàëüíûõ âåðøèí. Xs = {(xi , yi )|xi , yi ∈ R+ ; i ∈ Vs }  ìíîæåñòâî êîîðäèíàò ÒØ. D = Dr  ìíîæåñòâî äóã îðãðàôà, ñîåäèíÿþùèõ åãî âåðøèíû â òîïîëîãèè τ ∈ T . T  ìíîæåñòâî òîïîëîãèé, îïðåäåëÿþùèõ ñïîñîáû ñîåäèíåíèÿ âåðøèí îðãðàôà äóãàìè. Èñòî÷íèêè ïîðîæäàþò ïîòîêè íà äóãàõ.  óçëàõ (âíóòðåííèõ âåðøèíàõ) âûïîëíÿþòñÿ óñëîâèÿ ñîõðàíåíèÿ ïîòîêîâ: ñóììèðóþòñÿ ïîòîêè èç çàõîäÿùèõ äóã è îáðàçóþò ïîòîêè íà èñõîäÿùèõ äóãàõ. Åñëè óçåë â òåðìèíàëüíîé âåðøèíå, òî ïîòîê èç íåå òîæå âõîäèò â ýòó ñóììó. f (q)  âåñîâàÿ ôóíêöèÿ; f (q(i,j) )  çíà÷åíèå âåñîâîé ôóíêöèè, çàâèñÿùåå îò âåëè÷èíû q(i,j) ïîòîêà íà äóãå (i, j), èëè âåñ ýòîé äóãè. l åâêëèäîâà ìåòðèêà; l(i,j) = ((xi − xj )2 − (yi − yj )2 )1/2  äëèíà äóãè (i, j) îðãðàôà. L(i,j) = l(i,j) · f (q(i,j) )  âçâåøåííàÿ äëèíà äóãè (i, j).  êîìáèíàòîðíîé ïîñòàíîâêå ïîòîêîâàÿ çàäà÷à Øòåéíåðà  ýòî ñëåäóþùèé îáúåêò: W ST P =< V0 , M0 , X0 , l, f (q), (τ, Xs ), SL >,

ãäå ïàðà (τ, Xs )  òàêîå ðåøåíèå çàäà÷è, äëÿ êîòîðîãî ïðè íà÷àëüíûõ óñëîâèÿõ (V0 , M0 , X0 , l, f (q)) è óñëîâèÿõ ñîõðàíåíèÿ ïîòîêîâ ñóììà âçâåøåííûõ äëèí äóã îðãðàôà ìèíèìàëüíà: SL =

X

L(i,j) → min,

τ ∈T :(i,j)∈Dτ

Ñîîòâåòñòâóþùèé ïàðå (τ, Xs ) îðãðàô G = (V, Dτ ) ÿâëÿåòñÿ äåðåâîì. Ïîñëå îïòèìèçàöèè êîîðäèíàò ÒØ îíî ñòàíîâèòñÿ äåðåâîì Øòåéíåðà (ÄØ), ò.å. îòíîñèòåëüíî ìèíèìàëüíûì äåðåâîì äëÿ äàííîé òîïîëîãèè τ ∈ T . Èùåòñÿ äåðåâî, ìèíèìàëüíîå ñðåäè ÄØ äëÿ âñåõ òîïîëîãèé (ÌÄØ). Ìû ðåøàåì ïîòîêîâóþ çàäà÷ó, ðàññìàòðèâàÿ òîëüêî äåðåâüÿ ñ ïîëíûìè òîïîëîãèÿìè [1], ò.å. ñ ìàêñèìàëüíûì ÷èñëîì ÒØ (s = n − 2). ×èñëî âåðøèí âñåãäà ðàâíî N = 2n − 2. Âûðîæäåííûå

Êîìïîçèòíûé ýâîëþöèîííûé àëãîðèòì

145

äóãè (íóëåâîé äëèíû), êîòîðûå ìîãóò ïîëó÷àòüñÿ ïðè îïòèìèçàöèè ïîëîæåíèÿ ÒØ, ìû ñ÷èòàåì ïîëíîïðàâíûìè ýëåìåíòàìè äåðåâüåâ. Ýòî íå íàðóøàåò îáùíîñòè ïîäõîäà, çàòî ïîçâîëèëî óíèôèöèðîâàòü ïðåäñòàâëåíèå òîïîëîãèè ñòðîêîé ôèêñèðîâàííîé äëèíû [3]. Ïîòîêîâàÿ çàäà÷à ðàññìàòðèâàåòñÿ â ïðèëîæåíèè ê ïðîåêòèðîâàíèþ òðàíñïîðòíûõ ñåòåé, äëÿ êîòîðûõ õàðàêòåðíà ïîëîæèòåëüíàÿ âåùåñòâåííàÿ âîãíóòàÿ âåñîâàÿ ôóíêöèÿ f (q). Îíà îáû÷íî èíòåðïîëèðóåòñÿ ñëåäóþùåé êóñî÷íî-ëèíåéíîé ôóíêöèåé: f (q) = K(q) + T r(q) · q,

ãäå: K(q), T r(q)  ñòóïåí÷àòûå (êóñî÷íî-ïîñòîÿííûå) ôóíêöèè îò ïîòîêà.  òðàíñïîðòíûõ ñåòÿõ K(q) è T r(q) ïðèíÿòî íàçûâàòü êàïèòàëüíîé è òðàíñïîðòíîé ñîñòàâëÿþùèìè ñîîòâåòñòâåííî.  ýòîì ñëó÷àå öåëåâàÿ ôóíêöèÿ çàäà÷è èìååò âèä: SL0 =

X

[K(q(i,j) ) + T r(q(i,j) ) · q(i,j) ] · l(i,j) → min .

τ ∈T :(i,j)∈Dτ

2. Ê Íà îñíîâå ìåòîäîëîãèè ýâîëþöèîííîãî ìîäåëèðîâàíèÿ è íàøåãî îïûòà ðåøåíèÿ ïîòîêîâîé çàäà÷è [3] áûëè ðàçðàáîòàíû îðèãèíàëüíûå ãåíåòè÷åñêèå îïåðàòîðû è äâóõóðîâíåâàÿ ýâîëþöèîííàÿ ìîäåëü. Îíè âìåñòå ñ áèîëîãè÷åñêèìè àíàëîãèÿìè ïðèâîäÿòñÿ â [4,5]. Íà âåðõíåì óðîâíå ìîäåëè èìèòèðóåòñÿ ïðîöåññ ñëó÷àéíîãî îáðàçîâàíèÿ íîâûõ âèäîâ îñîáåé áåç èõ äèâåðãåíöèè. Îñîáü ïðåäñòàâëÿåòñÿ ïàðîé (τ, Xs ), ãäå τ  òîïîëîãèÿ (õðîìîñîìà), Xs  êîîðäèíàòû ÒØ. Íà íèæíåì óðîâíå ìîäåëèðóåòñÿ ýâîëþöèÿ ïîïóëÿöèè îñîáåé ïîðîæäåííîãî âèäà êàê ñìåíà åå ïîêîëåíèé â äèñêðåòíûå ìîìåíòû âðåìåíè. Îñîáü îöåíèâàåòñÿ ôèòíåñôóíêöèåé, ñîîòâåòñòâóþùåé öåëåâîé ôóíêöèè ïîòîêîâîé çàäà÷è. Çà ñ÷åò ëó÷øèõ íàéäåííûõ â ïîïóëÿöèÿõ ðàçíûõ âèäîâ îñîáåé ôîðìèðóåòñÿ ýëèòíàÿ ãðóïïà ðåøåíèé. Íà ýòîé ìîäåëè ñòðîèëèñü êîìïîçèòíûå àëãîðèòìû, èñïîëüçóþùèå ðàçëè÷íûå ïðèåìû ýâîëþöèîííîãî ìîäåëèðîâàíèÿ. Âû÷èñÎÌÏÎÇÈÒÍÛÉ ÝÂÎËÞÖÈÎÍÍÛÉ ÀËÃÎÐÈÒÌ

146

Â.Ä. Êóêèí.

ëèòåëüíûé ýêñïåðèìåíò ïîêàçàë, ÷òî âñå îíè îáåñïå÷èâàþò ïîèñê õîðîøèõ ðåøåíèé äëÿ øèðîêîãî äèàïàçîíà óñëîâèé. Ýòî äîñòèãàåòñÿ çà ñ÷åò ïðèìåíåíèÿ îðèãèíàëüíûõ ãåíåòè÷åñêèõ îïåðàòîðîâ, ó÷èòûâàþùèõ ñïåöèôèêó çàäà÷è. Áûëî îòîáðàíî íåñêîëüêî àëãîðèòìîâ, èìåþùèõ ëó÷øèå âðåìåííûå õàðàêòåðèñòèêè, òîì ÷èñëå ðàññìàòðèâàåìûé íèæå àëãîðèòì. Ýòî àëãîðèòì ýâîëþöèîííîãî ïîèñêà, â êîòîðîì ïîïóëÿöèÿ ñîñòîèò èç îäíîé îñîáè, è îí ìîæåò èíòåðïðåòèðîâàòüñÿ êàê ñòðàòåãèÿ ýâîëþöèîííîãî ïîèñêà. Âíåøíèì öèêëîì ñëó÷àéíîãî âèäîîáðàçîâàíèÿ â àëãîðèòìå óïðàâëÿåò ïàðàìåòð Z , êîòîðûé çàäàåò ÷èñëî àêòîâ îáðàçîâàíèÿ íîâûõ âèäîâ. Ïåðâûé øàã ðàáîòû àëãîðèòìà  ãåíåðàöèÿ ðîäèòåëÿ ïîïóëÿöèè. Ïåðâûé ðîäèòåëü  ýòî ïðàðîäèòåëü: àâòîìàòè÷åñêè ôîðìèðóåìàÿ îñîáü ïðåäêîâîãî âèäà [4], à êàæäûé ñëåäóþùèé  ñëó÷àéíûé ðîäèòåëü, ïîðîæäàåìûé èç ïðåäûäóùåãî. Âî âíåøíèé öèêë âëîæåí öèêë ýâîëþöèè ïîïóëÿöèè ïîðîæäåííîãî âèäà, êîòîðàÿ ìîäåëèðóåòñÿ êàê ÷åðåäóþùèåñÿ ïàðû ñìåí ïîêîëåíèé  ìóòàíòîâ è ãèáðèäíûõ îñîáåé. Ýòîò öèêë óïðàâëÿåòñÿ êðèòåðèàëüíûì ïðèçíàêîì H . Ïåðåä êàæäîé íîâîé ïàðîé ñìåíû ïîêîëåíèé ïðèçíàê H = 0. Åñëè õîòÿ áû â îäíîì èç äâóõ ïîñëåäîâàòåëüíûõ ïîêîëåíèé íàéäåí ëó÷øèé ïîòîìîê, H = 1. Îñíîâíûå øàãè àëãîðèòìà âêëþ÷àþò äâå ïîñëåäîâàòåëüíûå îïåðàöèè: ïåðåñòðîéêó õðîìîñîìû (òîïîëîãèè τ ) è ëîêàëüíóþ îïòèìèçàöèþ ìíîæåñòâà Xs êîîðäèíàò ÒØ íà ïëîñêîñòè. Äëÿ ïåðåñòðîéêè èñïîëüçóþòñÿ: îïåðàòîð ñëó÷àéíûõ ìóòàöèé, îïåðàòîð ñåëåêòèâíûõ ìóòàöèé è îïåðàòîð êðîññèíãîâåðà [4]. Äåéñòâèÿìè ýòèõ îïåðàòîðîâ óïðàâëÿþò ñïåöèàëüíûå ïàðàìåòðû  íîðìû, óñòàíîâëåííûå â õîäå âû÷èñëèòåëüíîãî ýêñïåðèìåíòà. Êîìïîçèòíûé ýâîëþöèîííûé àëãîðèòì.

Øàã 1. Öèêë ñëó÷àéíîãî âèäîîáðàçîâàíèÿ: z = 1 ÷ Z Åñëè z = 1, òî ãåíåðàöèÿ ïðàðîäèòåëÿ. Èíà÷å ãåíåðàöèÿ ñëó÷àéíîãî ðîäèòåëÿ âèäà z . Ïåðåñòðîéêà õðîìîñîìû ïðåäûäóùåãî ðîäèòåëÿ îïåðàòîðîì ñëó÷àéíûõ ìóòàöèé. Îïòèìèçàöèÿ êîîðäèíàò ÒØ ðîäèòåëÿ. Çàïîìèíàíèå ðîäèòåëÿ. Çàäàíèå òåêóùåé îñîáè.

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Øàã 2. Ýâîëþöèÿ ïîïóëÿöèè îñîáåé âèäà z . Íîìåð ïîêîëåíèÿ ïîïóëÿöèè h = 1. 2.1. Êðèòåðèàëüíûé ïðèçíàê H = 0. Öèêë ãåíåðàöèè ìóòàíòîâ òåêóùåé îñîáè: v = 2÷(2n−2) Ïîðîæäåíèå ïîòîìêà v îò òåêóùåé îñîáè îïåðàòîðîì ñåëåêòèâíûõ ìóòàöèé. Îïòèìèçàöèÿ êîîðäèíàò ÒØ. Åñëè ïîòîìîê ëó÷øå, òåêóùàÿ îñîáü îáíîâëÿåòñÿ: H = 1.

Íîìåð ïîêîëåíèÿ ïîïóëÿöèè h = h + 1. 2.2. Öèêë ãåíåðàöèè ãèáðèäíûõ îñîáåé: v = (n + 1) ÷ (2n − 2) Ïîðîæäåíèå ïîòîìêà v îò òåêóùåé îñîáè îïåðàòîðîì êðîññèíãîâåðà. Îïòèìèçàöèÿ êîîðäèíàò ÒØ. Åñëè ïîòîìîê ëó÷øå, òåêóùàÿ îñîáü îáíîâëÿåòñÿ: H = 1.

Íîìåð ïîêîëåíèÿ ïîïóëÿöèè h = h + 1. Åñëè H = 1, ïåðåõîä ê øàãó 2.1. Øàã 3. Ýëèòíàÿ ãðóïïà G. Åñëè G íå ñîçäàíà, òåêóùàÿ îñîáü âêëþ÷àåòñÿ â G, èíà÷å, åñëè òåêóùàÿ îñîáü ëó÷øå õóäøåé îñîáè â G, çàìåíÿåò åå. Øàã 4. Åñëè z < Z , òî ïåðåõîä ê øàãó 1, èíà÷å  ê øàãó 5. Øàã 5. Âûáîð ðåøåíèÿ.  öèêëå v = 2÷(2n−2) èç òåêóùåé îñîáè ãåíåðèðóåòñÿ ïîêîëåíèå ìóòàíòîâ ñ ïîìîùüþ îïåðàòîðà ñåëåêòèâíûõ ìóòàöèé. Ãåîìåòðè÷åñêèé ñìûñë äåéñòâèÿ îïåðàòîðà  ïîïûòêè ïåðåáðîñèòü äóãó v íà äóãè äåðåâà èç åå áëèæàéøåé îêðåñòíîñòè äî ïåðâîé óäà÷íîé ïîïûòêè èëè äî èñ÷åðïàíèÿ ñôîðìèðîâàííîãî ñïèñêà ïåðåáðîñîê. Îïåðàòîð ïðèìåíÿåòñÿ êî âñåì äóãàì, êðîìå êîðíåâîé.  öèêëå v = (n + 1) ÷ (2n − 2) ïîðîæäàåòñÿ ïîêîëåíèå ãèáðèäíûõ îñîáåé ñ ïîìîùüþ îðèãèíàëüíîãî îïåðàòîðà êðîññèíãîâåðà. Îí ðåàëèçîâàí òàê, ÷òî â ðåêîìáèíàöèè ó÷àñòâóþò õðîìîñîìû

148

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òåêóùåé îñîáè è íåêîòîðîé âèðòóàëüíîé îñîáè. Äëÿ ýòîãî èñïîëüçóåòñÿ àëãîðèòì èñ÷åðïûâàþùåãî è íåèçáûòî÷íîãî ïåðåáîðà ïîëíûõ òîïîëîãèé [3]. Åñëè ïðè ïîèñêå íàéäåí ëó÷øèé ïîòîìîê (ëèáî ìóòàíò, ëèáî ãèáðèäíàÿ îñîáü), òåêóùàÿ îñîáü îáíîâëÿåòñÿ. Òîëüêî îíà "âûæèâàåò", è ñ íåå ïðîäîëæàåòñÿ ïîèñê. Ôàêòè÷åñêè, ýòî àëãîðèòì íàïðàâëåííîãî ýâîëþöèîííîãî ïîèñêà.  ñèëó ñëó÷àéíîñòè âèäîîáðàçîâàíèÿ â àëãîðèòìå, íå ñòîèò ïðîáëåìà ïðåæäåâðåìåííîé ñõîäèìîñòè. Îí òàêæå ïðîñìàòðèâàåò áîëüøîå ÷èñëî âàðèàíòîâ, ÷òî óâåëè÷èâàåò âåðîÿòíîñòü íàõîæäåíèÿ õîðîøèõ ðåøåíèé. Âìåñòå ñ îðèãèíàëüíûìè îïåðàòîðàìè, ó÷èòûâàþùèìè ñïåöèôèêó ïîòîêîâîé çàäà÷è, ýòî îáåñïå÷èâàåò ýôôåêòèâíîñòü ðàáîòû àëãîðèòìà, ÷òî ïîäòâåðæäàåòñÿ ðåçóëüòàòàìè åãî òåñòèðîâàíèÿ. 3.  Âû÷èñëèòåëüíûé ýêñïåðèìåíò ñîñòîÿë èç òåñòèðîâàíèÿ ðàçíûõ ðåàëèçîâàííûõ íà C++ àëãîðèòìîâ, êîòîðîå ïðîâîäèëîñü â îñíîâíîì íà ÏÊ ñ òàêòîâîé ÷àñòîòîé 800 ÌÃö. Èñêàëèñü ðåøåíèÿ êàê ïîòîêîâîé çàäà÷è Øòåéíåðà, òàê è çàäà÷è áåç ïîòîêîâ. Ðåøàëèñü ñåðèè ïî 15 òåñòîâûõ çàäà÷ äëÿ êàæäîé èç ðàçìåðíîñòåé: 50, 60, 70, 80, 90, 100, 250, 500 è 1000 òåðìèíàëüíûõ âåðøèí. Äëÿ çàäà÷è áåç ïîòîêîâ èñïîëüçîâàëèñü òåñòû èç Èíòåðíåò-áèáëèîòåêè [6], âõîäíûå äàííûå êîòîðûõ áûëè ïîëó÷åíû ñ ïîìîùüþ ãåíåðàòîðà ñëó÷àéíûõ ÷èñåë. Äëÿ ïîòîêîâîé çàäà÷è â íèõ äîïîëíèòåëüíî çàäàâàëèñü ìîùíîñòè èñòî÷íèêîâ â òåðìèíàëüíûõ âåðøèíàõ è âåñîâàÿ ôóíêöèÿ. Êðîìå òîãî, ðåøàëèñü ïîòîêîâûå çàäà÷è, äëÿ êîòîðûõ áûëè èçâåñòíû ðåøåíèÿ èç ïðàêòèêè ïðîåêòèðîâàíèÿ òðàíñïîðòíûõ ñåòåé. Äëÿ èõ âõîäíûõ äàííûõ õàðàêòåðíà íåêîòîðàÿ ðåãóëÿðíîñòü. Ïðè òåñòèðîâàíèè ðàññìîòðåííîãî àëãîðèòìà ïîëó÷åíû ñëåäóþùèå ðåçóëüòàòû. 1. Àëãîðèòì íàõîäèò ðåøåíèÿ îáåèõ çàäà÷ çà âðåìÿ, ïðèåìëåìîå äëÿ ïðàêòè÷åñêîãî ïðèëîæåíèÿ ê îïòèìèçàöèè ïàðàìåòðîâ òðàíñïîðòíûõ ñåòåé. Îí íàõîäèò ðåøåíèÿ õîðîøåãî êà÷åñòâà äëÿ øèðîêîãî äèàïàçîíà âõîäíûõ äàííûõ. Äëÿ çàäà÷è Øòåéíåðà áåç ïîòîêîâ íàéäåíû ðåêîðäíûå çíà÷åíèÿ öåëåâîé ôóíêöèè, óêàçàííûå â Èíòåðíåò-áèáëèîòåêå. Äëÿ ïîòîêîâîé çàäà÷è ïîâòîðåíû Û×ÈÑËÈÒÅËÜÍÛÉ ÝÊÑÏÅÐÈÌÅÍÒ

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èëè óëó÷øåíû ðåøåíèÿ, èçâåñòíûå èç ïðàêòèêè ïðîåêòèðîâàíèÿ. 2. Àëãîðèòì ñïîñîáåí íàõîäèòü ñóáîïòèìàëüíûå ðåøåíèÿ. Íà ðèñóíêàõ íèæå ïðèâåäåíû ñõåìû äëÿ çàäà÷è áåç ïîòîêîâ (ðèñ.1) è ñîîòâåòñòâóþùåé ïîòîêîâîé çàäà÷è (ðèñ.2) ðàçìåðíîñòüþ 250 òåðìèíàëüíûõ âåðøèí ñ êîðíåì â òî÷êå 1.

Ñõåìà ÄØ ñ 250 òåðìèíàëüíûìè âåðøèíàìè. Çàäà÷à áåç ïîòîêîâ. Äëÿ ðåøåíèé ïîòîêîâîé çàäà÷è õàðàêòåðíû ñëåäóþùèå îñîáåííîñòè.   òî÷êàõ Øòåéíåðà óãëû ìåæäó âõîäÿùèìè äóãàìè ñ áëèçêèìè âåñàìè âñåãäà ìåíüøå 120 ãðàäóñîâ. Óãëû ìåæäó âõîäÿùèìè äóãàìè ñ áîëüøîé ðàçíèöåé âåñîâ áëèçêè ê 90 ãðàäóñàì. ( çàäà÷å áåç ïîòîêîâ âñå óãëû ðàâíû 120 ãðàäóñàì.)  Âåðîÿòíîñòü ïîÿâëåíèÿ âûðîæäåííûõ äóã â ïîòîêîâîé çàäà÷å ìåíüøå, ÷åì â çàäà÷å áåç ïîòîêîâ. Ðèñ. 1.

150

Â.Ä. Êóêèí.

Ñõåìà ÄØ ñ 250 òåðìèíàëüíûìè âåðøèíàìè. Çàäà÷à ñ ïîòîêàìè.  ÄØ ñ ëó÷øèì çíà÷åíèåì öåëåâîé ôóíêöèè â îñíîâíîì èìåþò áîëüøóþ ñóììó äëèí äóã, ÷åì ÄØ ñ õîðîøåé öåëåâîé ôóíêöèåé  çà ñ÷åò óâåëè÷åíèÿ äëèí äóã ñ ìåíüøèìè âåñàìè.  ÄØ îáëàäàþò îòíîñèòåëüíîé óñòîé÷èâîñòüþ. Ñðåäè õîðîøèõ äåðåâüåâ, êàê ïðàâèëî, íàáëþäàåòñÿ ìàëîå ÷èñëî îñíîâíûõ ñõåì ðàñïîëîæåíèÿ äóã â ïðèêîðíåâîé îáëàñòè, ãäå äóãè èìåþò áîëüøèé âåñ. Ðàçëè÷èÿ íà÷èíàþòñÿ áëèæå ê ïåðèôåðèè äåðåâüåâ, ãäå äóãè èìåþò ìåíüøèé âåñ. (Òàêàÿ óñòîé÷èâîñòü ñîâñåì íå ñâîéñòâåííà çàäà÷å áåç ïîòîêîâ). 3. Àëãîðèòì ñîçäàâàëñÿ äëÿ ïîòîêîâîé çàäà÷è è ïðè ïîèñêå ðåøåíèé äëÿ íåå áîëåå ýôôåêòèâåí, ÷åì äëÿ çàäà÷è áåç ïîòîêîâ. Äëÿ íàõîæäåíèÿ ëó÷øèõ ðåøåíèé â ïîòîêîâîé çàäà÷å âñåãäà òðåáîâàëîñü ìåíüøåå ÷èñëî àêòîâ âèäîîáðàçîâàíèÿ, ÷åì â çàäà÷å áåç ïîòîêîâ. Íàëè÷èå âåñîâ ïîðîæäàåò óñòîé÷èâîñòü ðåøåíèé â ïðèêîðíåâîé îáëàñòè è â êàêîé-òî ñòåïåíè "ñóæàåò"ïðîñòðàíñòâî ðåøåíèé ïîòîêîâîé çàäà÷è. Ýòî äåìîíñòðèðóåò ãðàôèê íà ðèñ. 3. Îí ïîñòðîåí äëÿ êîíòðîëüíûõ òåñòîâûõ çàäà÷ ðàçíûõ ðàçìåðíîñòåé êàê ñ ïîòîêàìè, òàê è áåç íèõ. Ïîêàçàíà îòíîñèòåëüíàÿ Ðèñ. 2.

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ðàçíèöà ÷èñëà àêòîâ âèäîîáðàçîâàíèÿ â íèõ. Îòìåòèì, ÷òî â àáñîëþòíîì âûðàæåíèè íàáëþäàëñÿ îäíîâðåìåííûé ðåçêèé ñêà÷îê ÷èñëà âèäîîáðàçîâàíèé â çàäà÷àõ ðàçìåðíîñòè 80. Îáðàòíàÿ êàðòèíà  äëÿ çàäà÷ ðàçìåðíîñòè 100. Ýòî îáúÿñíÿåòñÿ îñîáåííîñòÿìè ðàñïîëîæåíèÿ òåðìèíàëüíûõ âåðøèí â ýòèõ ïðèìåðàõ.

Ðèñ. 3.

Ñðàâíåíèå ÷èñëà àêòîâ âèäîîáðàçîâàíèÿ.

X  ðàçìåðíîñòü çàäà÷è. Y  îòíîñèòåëüíîå ÷èñëî

âèäîîáðàçîâàíèé â çàäà÷å áåç ïîòîêîâ. Äëÿ ïîòîêîâîé çàäà÷è ÷èñëî âèäîîáðàçîâàíèé ïðèíÿòî çà 100% .  îáåèõ çàäà÷àõ íàáëþäàåòñÿ ïðèìåðíî îäèíàêîâûé äèàïàçîí ÷èñëà ïîêîëåíèé ïîïóëÿöèè  îò 4 äî 24. Ìèíèìàëüíîå è ìàêñèìàëüíîå ÷èñëî ïîêîëåíèé íàáëþäàåòñÿ êðàéíå ðåäêî  äî 1% ñëó÷àåâ. Íàèáîëåå âåðîÿòíîå çíà÷åíèå òàêæå ñîâïàäàåò.  çàäà÷àõ ðàçìåðíîñòüþ 50-100 òåðìèíàëüíûõ âåðøèí ÷àùå âñåãî âñòðå÷àþòñÿ 8; â çàäà÷àõ ðàçìåðíîñòüþ 100-250 âåðøèí  10; â çàäà÷àõ ðàçìåðíîñòüþ 250-500  12 ïîêîëåíèé. Îäíàêî ìàêñèìàëüíûé ïðîöåíò íàèáîëåå âåðîÿòíûõ çíà÷åíèé ÷èñëà ïîêîëåíèé â ïîòîêîâîé çàäà÷å ìåíüøå, ÷åì â çàäà÷å áåç ïîòîêîâ. 4. Äëÿ ñðàâíåíèÿ, íà ýâîëþöèîííîé ìîäåëè áûë òàêæå ïîñòðîåí àëãîðèòì, â êîòîðîì öèêë ýâîëþöèè ïîïóëÿöèè ðåàëèçîâàí ïî

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Â.Ä. Êóêèí.

ñõåìå ñòàíäàðòíîãî ãåíåòè÷åñêîãî àëãîðèòìà, íî ñ íàøèìè îðèãèíàëüíûìè îïåðàòîðàìè. Ðåøàëèñü òå æå ñàìûå òåñòîâûå çàäà÷è  ñ ïîòîêàìè è áåç íèõ. Ïðè ïîèñêå ðåøåíèé îáåèõ çàäà÷ ýòîò àëãîðèòì îêàçàëñÿ ìåíåå ýôôåêòèâåí, ÷åì àëãîðèòì íàïðàâëåííîãî ýâîëþöèîííîãî ïîèñêà. Ïðîöåññ çàòÿãèâàëñÿ è íå ãàðàíòèðîâàë óñòîé÷èâîãî íàõîæäåíèÿ ðåøåíèé. Ýòî ðåçóëüòàò ñëó÷àéíîãî äåéñòâèÿ ãåíåòè÷åñêèõ îïåðàòîðîâ, êîòîðîå ïðèâîäèò ê "õàîòè÷åñêîìó"áëóæäàíèþ ïî ïðîñòðàíñòâó ðåøåíèé. Òàê, â êîíòðîëüíîé ïîòîêîâîé çàäà÷å íà 80 òåðìèíàëüíûõ âåðøèí àëãîðèòì íàïðàâëåííîãî ýâîëþöèîííîãî ïîèñêà íàøåë õîðîøåå ðåøåíèå çà 9 àêòîâ âèäîîáðàçîâàíèÿ, à ýòîò àëãîðèòì  çà 365. Ïîèñê ëó÷øåãî ðåøåíèÿ ýòèì àëãîðèòìîì áûë ïðåêðàùåí, êîãäà âðåìÿ ïðåâûñèëî ïðàêòè÷åñêè ïðèåìëåìûå ïðåäåëû. Êðîìå òîãî, ïðèìåðíî â 50% ñëó÷àåâ ýâîëþöèÿ ïîïóëÿöèè ñâîäèòñÿ ê 1 ïîêîëåíèþ. Âñëåäñòâèå ýòîãî òåðÿåòñÿ êà÷åñòâåííîå ðàçíîîáðàçèå ïîïóëÿöèè. 5. Òåñòèðîâàëñÿ òàêæå àëãîðèòì, â êîòîðîì ÷èñëî ïîêîëåíèé ýâîëþöèîíèðóþùåé ïîïóëÿöèè ôèêñèðîâàííîå. Åãî ïðîèçâîäèòåëüíîñòü çàâèñèò îò âëîæåííûõ öèêëîâ. Ñëîæíîñòü ôîðìèðîâàíèÿ ïîêîëåíèÿ ìóòàíòîâ O(n2 ).  ýòîì öèêëå 2n − 4 ðàç âûïîëíÿåòñÿ êîíå÷íîå ÷èñëî îïåðàöèé ïî ïåðåñòðîéêå õðîìîñîìû è îïòèìèçèðóåòñÿ ïîëîæåíèå n − 2 ïàð êîîðäèíàò ÒØ. Àíàëîãè÷íî, ñëîæíîñòü ôîðìèðîâàíèÿ ïîêîëåíèÿ ãèáðèäíûõ îñîáåé O(n2 ). Âðåìåííàÿ ñëîæíîñòü öèêëà ýâîëþöèè ïîïóëÿöèè O(1). Ïðè íàõîæäåíèè ðåøåíèÿ îáåèõ çàäà÷ ìàêñèìàëüíîå ÷èñëî àêòîâ ñëó÷àéíîãî âèäîîáðàçîâàíèÿ íå ïðåâûøàëî kn, ãäå êîíñòàíòà k << n. Çíà÷èò, âðåìåííàÿ ñëîæíîñòü ýòîãî öèêëà O(n), à âðåìåííàÿ ñëîæíîñòü àëãîðèòìà ñ ôèêñèðîâàííûì ÷èñëîì ïîêîëåíèé ïîïóëÿöèè O(n3 ). Ïî-âèäèìîìó, ðàññìîòðåííûé âûøå àëãîðèòì èìååò òàêóþ æå âðåìåííóþ ñëîæíîñòü, èáî ýìïèðè÷åñêèå äàííûå ñâèäåòåëüñòâóþò, ÷òî îæèäàåìîå ÷èñëî ïîêîëåíèé ïîïóëÿöèè ñ óâåëè÷åíèåì ðàçìåðíîñòè çàäà÷è ðàñò¼ò î÷åíü ñëàáî. Òàêèì îáðàçîì, âû÷èñëèòåëüíûé ýêñïåðèìåíò ïîêàçàë, ÷òî êîìïîçèòíûé ýâîëþöèîííûé àëãîðèòì èìååò ïðàêòè÷åñêóþ öåííîñòü ïðè ðåàëèçàöèè íà ÏÊ.  ñâÿçè ñ ýòèì, îáùèé ñëó÷àé êîìïîçèòíîãî àëãîðèòìà, â êîòîðîì ðàçìåð ïîïóëÿöèè ðàâåí ÷èñëó ïðîöåññîðîâ ìíîãîïðîöåññîðíîé ñèñòåìû, ïåðñïåêòèâåí äëÿ ïîñòðîåíèÿ ïàðàëëåëüíîãî àëãîðèòìà.

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4. Ç Äëÿ ðåøåíèÿ ïîòîêîâîé çàäà÷è Øòåéíåðà èñïîëüçîâàëàñü ìåòîäîëîãèÿ ýâîëþöèîííîãî ìîäåëèðîâàíèÿ. Íà åå îñíîâå ðàçðàáîòàí ðÿä àëãîðèòìîâ, â òîì ÷èñëå àëãîðèòì íàïðàâëåííîãî ýâîëþöèîííîãî ïîèñêà, êîòîðûé è áûë ðàññìîòðåí âûøå. Òåñòèðîâàíèå ýòîãî àëãîðèòìà ïîçâîëèëî ñäåëàòü ñëåäóþùèå âûâîäû.  Àëãîðèòì îáåñïå÷èâàåò óñòîé÷èâûé ïîèñê õîðîøèõ è ñóáîïòèìàëüíûõ ðåøåíèé ïîòîêîâîé çàäà÷è è çàäà÷è Øòåéíåðà áåç ïîòîêîâ, íî äëÿ ïîñëåäíåé ðàáîòàåò ìåäëåííåå.  Àëãîðèòì áîëåå ýôôåêòèâåí äëÿ ïîèñêà ðåøåíèé çàäà÷ êàê ñ ïîòîêàìè, òàê è áåç ïîòîêîâ, ïî ñðàâíåíèþ ñ àëãîðèòìîì, â êîòîðîì öèêë ýâîëþöèè ïîïóëÿöèè áûë ðåàëèçîâàí ïî ñõåìå ñòàíäàðòíîãî ãåíåòè÷åñêîãî àëãîðèòìà, íî ñ èñïîëüçîâàíèåì îðèãèíàëüíûõ îïåðàòîðîâ.  Îáùèé ñëó÷àé àëãîðèòìà, êîãäà ïîïóëÿöèÿ ñîñòîèò èç êîíå÷íîãî ñðàâíèòåëüíî íåáîëüøîãî ÷èñëà îñîáåé, ïåðñïåêòèâåí äëÿ ïîñòðîåíèÿ ïàðàëëåëüíîãî àëãîðèòìà äëÿ ìíîãîïðîöåññîðíîé ñèñòåìû. Òàêèì îáðàçîì, âû÷èñëèòåëüíûé ýêñïåðèìåíò óáåäèòåëüíî ïîêàçàë, ÷òî ýâîëþöèîííûé ïîäõîä ê ðåøåíèþ ïîòîêîâîé çàäà÷è Øòåéíåðà ïîçâîëÿåò ñîçäàâàòü àëãîðèòìû, èìåþùèå ïðàêòè÷åñêóþ öåííîñòü. Õîòÿ îíè íå ãàðàíòèðóþò íàõîæäåíèÿ îïòèìàëüíîãî ðåøåíèÿ, îíè ñ áîëüøåé âåðîÿòíîñòüþ, ÷åì ìåòîäû ïîñëåäîâàòåëüíîãî ïîèñêà, ñïîñîáíû íàõîäèòü ñóáîòèìàëüíûå ðåøåíèÿ. ÀÊËÞ×ÅÍÈÅ

Ëèòåðàòóðà 1. Ãèëáåðò Ý.Í., Ïîëëàê Ã.Î. Ìèíèìàëüíûå äåðåâüÿ Øòåéíåðà // Êèáåðíåòè÷. ñáîðíèê, íîâàÿ ñåðèÿ, âûï. 8. Ì.: Ìèð, 1971, c. 19-50. 2. Ãýðè Ì., Äæîíñîí Ä. Âû÷èñëèòåëüíûå ìàøèíû è òðóäíîðåøàåìûå çàäà÷è. Ì.: Ìèð, 1982. 3. Êóêèí Â.Ä., Êóçèíà Â.È. Ìåòîäèêà ðåøåíèÿ çàäà÷è Øòåéíåðà ñ ïîòîêàìè è âåñàìè // Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ. Âûï. 2. Ïåòðîçàâîäñê, 2000, c. 143-150. 4. Êóêèí Â.Ä. Ãåíåòè÷åñêèå àëãîðèòìû è çàäà÷à Øòåéíåðà ñ ïîòîêàìè è âåñàìè: ïîäõîä ê ïðîáëåìå // Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ. Âûï. 3. Ïåòðîçàâîäñê, 2002, c. 170-177. 5. Êóêèí Â.Ä. Ýâîëþöèîííàÿ ìîäåëü äëÿ ðåøåíèÿ ïîòîêîâîé çàäà÷è Øòåéíåðà // Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ. Âûï. 5. Ïåòðîçàâîäñê, 2004, c. 200-211. 6. OR-LIBRARY: http://mscmga.ms.ic.ac.uk/info.html

Òðóäû ÈÏÌÈ ÊàðÍÖ ÐÀÍ

Âûïóñê 7, 2006

ÓÄÊ 681.142.2+519.2 ÎÏÒÈÌÀËÜÍÎÅ ÓÏÐÀÂËÅÍÈÅ ÒÐÅÌß ÑÒÅÊÀÌÈ Â ÑËÓ×ÀÅ ÏÀÐÀËËÅËÜÍÎÃÎ ÂÛÏÎËÍÅÍÈß ÎÏÅÐÀÖÈÉ

À. À. Ë

ÀÇÓÒÈÍÀ

 ðàáîòå ïðåäëîæåíû è èññëåäóþòñÿ ìàòåìàòè÷åñêèå ìîäåëè, îïèñûâàþùèå íåñêîëüêî ìåòîäîâ ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé.  êà÷åñòâå êðèòåðèÿ îïòèìàëüíîñòè ðàññìîòðåíî ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ ïàìÿòè. Ðàáîòà âûïîëíåíà ïðè ïîääåðæêå ãðàíòà ÐÔÔÈ (ïðîåêò  06-01-00303).

Ïîíÿòèå ñòåêà øèðîêî èñïîëüçóåòñÿ ïðè ðàçðàáîòêå ïðîãðàììíîãî è àïïàðàòíîãî îáåñïå÷åíèÿ [1]. Ïðè ýòîì ÷àñòî âîçíèêàåò ïðîáëåìà ðàçìåùåíèÿ íåñêîëüêèõ ñòåêîâ (ïðîöåññîâ) â ïàìÿòè îäíîãî óðîâíÿ  îïåðàòèâíîé èëè ðåãèñòðîâîé [2]. Ñóùåñòâóåò íåñêîëüêî ñïîñîáîâ îðãàíèçàöèè ðàáîòû ñî ñòåêàìè. Ðàññìîòðèì çàäà÷ó îá îïòèìàëüíîì ñïîñîáå ðàçìåùåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé. Ïóñòü â êàæäûé ìîìåíò âðåìåíè ìîãóò ïðîèñõîäèòü ïàðàëëåëüíûå âêëþ÷åíèÿ èëè èñêëþ÷åíèÿ â òðè ñòåêà â ïàìÿòè îäíîãî óðîâíÿ.  ïàìÿòè îáúåìà m > 0 óñëîâíûõ åäèíèö ðàñïîëîæåíû òðè ïàðàëëåëüíûõ ñòåêà, âåðîÿòíîñòíûå õàðàêòåðèñòèêè êîòîðûõ ñëåäóþùèå: pi  âåðîÿòíîñòü âêëþ÷åíèÿ â ñòåê ñ íîìåðîì i; qi  âåðîÿòíîñòü èñêëþ÷åíèÿ èç ñòåêà ñ íîìåðîì i, pij  âåðîÿòíîñòü îäíîâðåìåííîãî âêëþ÷åíèÿ â äâà ñòåêà ñ íîìåðàìè i è j ; qij  âåðîÿòíîñòü îäíîâðåìåííîãî èñêëþ÷åíèÿ èç äâóõ ñòåêîâ ñ íîìåðàìè i è j ; p123  âåðîÿòíîñòü âêëþ÷åíèÿ â òðè ñòåêà îäíîâðåìåííî; q123  âåðîÿòíîñòü èñêëþ÷åíèÿ èç òðåõ ñòåêîâ îäíîâðåìåííî, ãäå c

À. À. Ëàçóòèíà, 2006

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pi > 0, qi > 0, i = 1, 2, 3. Èñêëþ÷åíèå èç ïóñòîãî ñòåêà ñ âåðîÿòíîñòüþ qi , qij , q123 îçíà÷àåò îñòàíîâêó íà ìåñòå. Ïðåäïîëàãàåòñÿ, ÷òî p1 + p2 + p3 + q1 + q2 + q3 + p12 + p23 + p13 + q12 + q23 + q13 + p123 + q123 = 1. Òåêóùèå äëèíû ñòåêîâ îáîçíà÷èì ÷åðåç x1 , x2 , x3 .

1. Ï Ïóñòü â ïàìÿòè îáúåìà m åäèíèö ðàñïîëîæåíû òðè ñòåêà ïîñëåäîâàòåëüíî. Äâóì ñòåêàì, ðàñòóùèì íàâñòðå÷ó äðóã äðóãó, îòâåäåíî s åäèíèö ïàìÿòè, à òðåòüåìó ñòåêó  m − s åäèíèö (ñì. Ðèñ.1). ÎÑËÅÄÎÂÀÒÅËÜÍÎÅ ÏÐÅÄÑÒÀÂËÅÍÈÅ

Ðèñ. 1.

Ïîñëåäîâàòåëüíîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ.

1.1. Òðè ñòåêà, äîïóñêàþùèõ òîëüêî âêëþ÷åíèÿ. Äëÿ íà÷àëà ðàññìîòðèì áîëåå ïðîñòîé ñëó÷àé, êîãäà âîçìîæíû òîëüêî âêëþ÷åíèÿ â ñòåêè. Áóäåì ñ÷èòàòü, ÷òî â êàæäûé ìîìåíò ñ ðàçíûìè âåðîÿòíîñòÿìè ìîãóò ïðîèçîéòè ñëåäóþùèå ñîáûòèÿ: ñ âåðîÿòíîñòüþ p0 âñå ñòåêè îñòàþòñÿ íà ìåñòå, ñ âåðîÿòíîñòüþ pi ïðîèçîéäåò âêëþ÷åíèå òîëüêî â ñòåê ñ íîìåðîì i, ñ âåðîÿòíîñòüþ p12 ïðîèçîéäåò îïåðàöèÿ ïàðàëëåëüíîãî âêëþ÷åíèÿ â ñòåêè 1 è 2, ñ âåðîÿòíîñòüþ p13 ïðîèçîéäåò îïåðàöèÿ ïàðàëëåëüíîãî âêëþ÷åíèÿ â ñòåêè 1 è 3, ñ âåðîÿòíîñòüþ p23 ïðîèçîéäåò îïåðàöèÿ ïàðàëëåëüíîãî âêëþ÷åíèÿ â ñòåêè 2 è 3, ñ âåðîÿòíîñòüþ p123 ïðîèçîéäåò îïåðàöèÿ ïàðàëëåëüíîãî âêëþ÷åíèÿ âî âñå òðè ñòåêà. Òîãäà â çàâèñèìîñòè îò òîãî êàêîé ñòåê áóäåò ðàñïîëîæåí îòäåëüíî, ìû áóäåì èìåòü ðàçíûå âåðîÿòíîñòè ïåðåõîäîâ. Ïóñòü ñòåê îäèí è äâà îáúåäèíåíû â ïàðó, à òðåòèé ñòåê ðàñïîëîæåí îòäåëüíî. Ââåäåì îáîçíà÷åíèÿ: x = x1 + x2 , y = x3 , r = p0 , p1 = p1 + p2 , p2 = p3 , p3 = p13 + p23 , p4 = p12 , p5 = p123 . Òîãäà â êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè ìû áóäåì èìåòü äâóìåðíîå ñëó÷àéíîå áëóæäàíèå âíóòðè ïðÿìîóãîëüíèêà [3], x >= 0, y >= 0, x < s + 1, y < m − s + 1 ñ âåðîÿòíîñòÿìè ïåðåõîäîâ èç òî÷êè (x, y) â òî÷êó (x, y) ðàâíîé r, èç òî÷êè (x, y) â òî÷êó (x + 1, y) ðàâíîé p1 , â òî÷êó (x, y + 1) ðàâíîé p2 , â òî÷êó (x + 1, y + 1) ðàâíîé

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p3 ,

â òî÷êó (x + 2, y) ðàâíîé p4 , â òî÷êó (x + 2, y + 1) ðàâíîé p5 (ñì. Ðèñ.2).

Ðèñ. 2. Îáëàñòü áëóæäàíèÿ äëÿ ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå, êîãäà ïðîèñõîäÿò òîëüêî âêëþ÷åíèÿ. Ïðîöåññ âûõîäèò èç íà÷àëà êîîðäèíàò è ïîãëîùàåòñÿ íà ïðÿìûõ x = s+1, è y = m−s+1. Íàøåé çàäà÷åé ÿâëÿåòñÿ íàõîæäåíèå òàêîãî çíà÷åíèÿ s è îïðåäåëåíèå òîãî, êàêîé ñòåê ðàñïîëîæèòü îòäåëüíî, ÷òîáû ñðåäíåå âðåìÿ áëóæäàíèÿ äî ïîãëîùåíèÿ áûëî ìàêñèìàëüíûì. Òàêèì îáðàçîì çäåñü ìû ìîæåì óïðàâëÿòü òîëüêî ïàðàìåòðàìè îáëàñòè, à íå íà÷àëüíîé òî÷êîé áëóæäàíèÿ. Îáîçíà÷èì ÷åðåç T (x, y) ñðåäíåå âðåìÿ áëóæäàíèÿ äî ïîãëîùåíèÿ, åñëè íà÷àëüíîé òî÷êîé áëóæäàíèÿ áûëà òî÷êà (x, y). Òîãäà

T (x, y) = r ∗ T (x, y) + p1 ∗ T (x + 1, y) + p2 ∗ T (x, y + 1) + p3 ∗ T (x + 1, y + 1) + p4 ∗ T (x + 2, y) + p5 ∗ T (x + 2, y + 1) + 1.

Ïîïàâ íà ãðàíèöû îáëàñòè áëóæäàíèÿ, ìû óæå íå äâèíåìñÿ íè íà îäèí øàã è, ñëåäîâàòåëüíî, T (s + 1, y) = 0, T (x, m − s + 1) = 0, T (x, m − s + 2) = 0. Íàøåé çàäà÷åé ÿâëÿåòñÿ âûáîð ñòåêà, êîòîðûé áóäåò ðàñïîëîæåí îòäåëüíî, è íàõîæäåíèå òàêîãî çíà÷åíèÿ s, ÷òîáû T (0, 0) áûëî ìàêñèìàëüíûì. Áûë ðàçðàáîòàí àëãîðèòì è íàïèñàíà ïðîãðàììà íà C++. Çàäàâàÿ âåðîÿòíîñòè è m, íàõîäèòñÿ s è T (0, 0). Ðåçóëüòàòû äëÿ m = 100 ïðåäñòàâëåíû â Òàáëèöàõ 1-2. Òàáëèöà 1. p0 = p1 = p2 = p3 = p12 = p13 = p23 = p, p123 = 1 − 7 ∗ p. p 0.05 0.1

îòäåëüíî 1 îòäåëüíî 2 îòäåëüíî 3 T (0, 0) 41.542580 54.348526

S 66 66

T (0, 0) 41.542580 54.348526

S 66 66

T (0, 0) 41.542580 54.348526

S 66 66

Èç òàáëèöû 1 âèäíî, ÷òî åñëè âñå âåðîÿòíîñòè ðàâíû, òî s è íå çàâèñÿò îò òîãî, êàêîé ñòåê ðàñïîëàãàåòñÿ îòäåëüíî.

T (0, 0)

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

Òàáëèöà 2.

p123 = 0. p

0.05 0.1 0.15

157

p0 = p1 = p2 = p12 = p13 = p23 = p, p3 = 1 − 6 ∗ p,

îòäåëüíî 1 îòäåëüíî 2 îòäåëüíî 3 T (0, 0) 87.295486 79.837135 73.716087

S 84 73 64

T (0, 0) 87.295486 79.837135 73.716087

S 84 73 64

T (0, 0) 85.757118 79.023659 73.850960

S 29 50 68

Âåðîÿòíîñòü âêëþ÷åíèÿ âî âñå òðè ñòåêà îäíîâðåìåííî ðàâíà íóëþ, p3 âûðàæàåòñÿ ÷åðåç îñòàëüíûå âåðîÿòíîñòè. Èç Òàáëèöû 2 âèäíî, ÷òî ïðè ìàëåíüêîì p áîëüøàÿ âåðîÿòíîñòü âêëþ÷åíèÿ â òðåòèé ñòåê, ïîýòîìó ïîä íåãî îòâîäèòñÿ áîëüøàÿ ÷àñòü ïàìÿòè; ïðè áîëüøîì p ìàëåíüêàÿ âåðîÿòíîñòü âêëþ÷åíèÿ â òðåòèé ñòåê, ïîýòîìó ïîä íåãî îòâîäèòñÿ ìåíüøàÿ ÷àñòü ïàìÿòè. 1.2. Îáùèé ñëó÷àé. Ïóñòü â ïàìÿòè îáúåìà m > 0 óñëîâíûõ åäèíèö ðàñïîëîæåíû òðè ïàðàëëåëüíûõ ñòåêà ïîñëåäîâàòåëüíî (ñì. Ðèñ.1). Îáîçíà÷èì m1 = s, m2 = m − s. Çàäà÷à ýôôåêòèâíîãî ðàçìåùåíèÿ òðåõ ñòåêîâ â ïàìÿòè ñâîäèòñÿ ê îïðåäåëåíèþ òî÷êè ðàçáèåíèÿ ïàìÿòè íà ó÷àñòêè äëÿ ïàðû ñòåêîâ è äëÿ îòäåëüíî ðàñïîëîæåííîãî ñòåêà, ò. å. ê îïðåäåëåíèþ m1 è m2 òàê, ÷òîáû ñðåäíåå âðåìÿ ðàáîòû äî ïåðåïîëíåíèÿ áûëî áû ìàêñèìàëüíûì. Äîïîëíèòåëüíî âîçíèêàåò âîïðîñ, êàêîé èç ñòåêîâ ðàñïîëàãàòü îòäåëüíî.  êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè çàäà÷è ðàññìîòðèì ñëó÷àéíîå áëóæäàíèå òî÷êè, êîîðäèíàòû êîòîðîé îïðåäåëÿþòñÿ òåêóùèìè äëèíàìè ñòåêîâ (x1 , x2 , x3 ), ïî öåëî÷èñëåííîé ðåøåòêå â ïðîñòðàíñòâå âíóòðè òðåóãîëüíîé ïðèçìû ñ âåðøèíàìè ïåðâîãî îñíîâàíèÿ â òî÷êàõ (−1, −1, −1), (−1, m1 + 1, −1), (m1 + 1, −1, −1) è âåðøèíàìè âòîðîãî îñíîâàíèÿ â òî÷êàõ (−1, −1, m2 + 1), (−1, m1 + 1, m2 + 1), (m1 + 1, −1, m2 + 1) ñî ñëåäóþùèìè âåðîÿòíîñòÿìè ñäâèãîâ âî âíóòðåííèõ òî÷êàõ: p1 , p2 , p3  âåðîÿòíîñòè ñäâèãà â ïîëîæèòåëüíîì íàïðàâëåíèè âäîëü îñåé x1 , x2 , x3 ñîîòâåòñòâåííî, q1 , q2 , q3  âåðîÿòíîñòè ñäâèãà â îòðèöàòåëüíîì íàïðàâëåíèè âäîëü îñåé x1 , x2 , x3 ñîîòâåòñòâåííî, p12 , p13 , p23 , p123  âåðîÿòíîñòè ñäâèãà â ïîëîæèòåëüíîì íàïðàâëåíèè âäîëü îñåé x1 è x2 ,

158

À. À. Ëàçóòèíà

x1 è x3 , x2 è x3 , x1 , x2 è x3 , ñîîòâåòñòâåííî, q12 , q13 , q23 , q123  âåðîÿòíîñòè ñäâèãà â îòðèöàòåëüíîì íàïðàâëåíèè âäîëü îñåé x1 è x2 , x1 è x3 , x2 è x3 , x1 , x2 è x3 , ñîîòâåòñòâåííî (ñì. Ðèñ. 3).

Ðèñ. 3. Îáëàñòü áëóæäàíèÿ äëÿ ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â îáùåì ñëó÷àå.

Ïëîñêîñòè x1 + x2 = m1 + 1 è x3 = m2 + 1 áóäóò âûñòóïàòü â êà÷åñòâå ïîãëîùàþùèõ ýêðàíîâ, ÷òî ñîîòâåòñòâóåò ïåðåïîëíåíèþ îòðåçêîâ ïàìÿòè äëÿ äâóõ è îäíîãî ñòåêà ñîîòâåòñòâåííî. Ïëîñêîñòè x1 = −1, x2 = −1 è x3 = −1 áóäóò îòðàæàþùèìè ýêðàíàìè. Ïóñòü ðàññìàòðèâàåìîå â êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè ñëó÷àéíîå áëóæäàíèå èìååò n íåâîçâðàòíûõ ñîñòîÿíèé. Òîãäà ïîãëîùàþùèå ñîñòîÿíèÿ ýòîãî ñëó÷àéíîãî áëóæäàíèÿ íóìåðîâàòü íå áóäåì, à åãî íåâîçâðàòíûå ñîñòîÿíèÿ ïðîíóìåðóåì îò 1 äî n, ò. å. êàæäîìó íåâîçâðàòíîìó ñîñòîÿíèþ áóäåò âçàèìíî îäíîçíà÷íî ñîîòâåòñòâîâàòü íåêîòîðûé íîìåð i = 1, . . . , n. Êàæäîå íåâîçâðàòíîå ñîñòîÿíèå ýòîãî ñëó÷àéíîãî áëóæäàíèÿ ñ íîìåðîì i, ãäå i = 1, . . . , n, ñîîòâåòñòâóåò öåëî÷èñëåííîé òî÷êå â ïðîñòðàíñòâå âíóòðè òðåóãîëüíîé ïðèçìû ñ êîîðäèíàòàìè (i1 , i2 , i3 ), ãäå i1 , i2 , i3 > 0, i1 + i2 6 m1 è i3 6 m2 .  äàëüíåéøåì ýòî ñîîòâåòñòâèå áóäåì îáîçíà÷àòü i = (i1 , i2 , i3 ).

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

159

Âûáîð íóìåðàöèè ñîñòîÿíèé ñóùåñòâåíåí, òàê êàê îí âëèÿåò íà âèä ìàòðèöû ïåðåõîäíûõ âåðîÿòíîñòåé. Âûáåðåì íóìåðàöèþ íåâîçâðàòíûõ ñîñòîÿíèé ðàññìàòðèâàåìîãî ñëó÷àéíîãî áëóæäàíèÿ ñëåäóþùèì îáðàçîì. Çàôèêñèðóåì x3 = m2 , òîãäà îáëàñòüþ áëóæäàíèÿ ñòàíåò âíóòðåííîñòü òðåóãîëüíèêà ñ âåðøèíàìè â òî÷êàõ: (−1, −1, m2 ), (−1, m1 + 1, m2 ), (m1 + 1, −1, m2 ). Ïðîíóìåðóåì íåâîçâðàòíûå ñîñòîÿíèÿ, íà÷èíàÿ ñ òî÷êè (m1 , 0, 0) ñíèçó ââåðõ ñïðàâà íàëåâî. Ïðèìåð íóìåðàöèè ñîñòîÿíèé (ñì. Ðèñ. 4).

Ïðèìåð íóìåðàöèè ñîñòîÿíèé. Ïåðåéäåì ê x3 = m2 − 1, òîãäà îáëàñòüþ áëóæäàíèÿ ñòàíåò âíóòðåííÿÿ îáëàñòü òðåóãîëüíèêà ñ âåðøèíàìè â òî÷êàõ: (−1, −1, m2 − 1), (−1, m1 + 1, m2 − 1), (m1 + 1, −1, m2 − 1). Ïðîäîëæàåì íóìåðàöèþ àíàëîãè÷íûì îáðàçîì, íà÷èíàÿ ñ òî÷êè (m1 , 0, 1). È òàê äàëåå äî òðåóãîëüíèêà ñ âåðøèíàìè â òî÷êàõ (−1, −1, 0), (−1, m1 + 1, 0), (m1 + 1, −1, 0). Ðèñ. 4.

Òåîðåìà 1. Ïðè âûáðàííîé íóìåðàöèè ñîñòîÿíèé ïîäìàòðèöà

Q

ìàòðèöû ïåðåõîäíûõ âåðîÿòíîñòåé äàííîé öåïè, ñîîòâåòñòâóþùàÿ

ïåðåõîäàì

èç

íåâîçâðàòíûõ

ñîñòîÿíèé

â

íåâîçâðàòíûå,

èìååò âèä:



Qn×n

   =   

ãäå

W m2 +1 U m2 0 ... 0 0

V m2 W m2 U m2 −1 ... 0 0

n = n1 (m2 + 1),

0 V m2 −1 W m2 −1 ... 0 0

n1 =

... 0 ... 0 ... 0 ... ... ... W2 . . . U1

0 0 0 ... V1 W1

(m1 + 1)(m1 + 2) ; 2

    ,   

160

À. À. Ëàçóòèíà

 Unl21 ×n1

   =   

K m1 +1 0 ... 0 0 0

M m1 −1 Lm1 −1 ... 0 0 0

Lm1 K m1 ... 0 0 0

ãäå

   Kll11×l1 =   

Ll(l11 +1)×l1

q3 0 ... 0 0 

    =     

M(ll11 +2)×l1

     Vnl21 ×n1     

0 q3 ... 0 0 q13 q23 0 ... 0 0 0 0

 q123   0  =  ...  0   0 0 C m1 +1 Dm1 E m1 −1 ... 0 0 0

0 M m1 −2 ... 0 0 0

0 0 q123 ... 0 0 0 0 C m1 Dm1 −1 ... 0 0 0

0 0 ... L2 K2 0

0 0 ... M1 L1 K1

    ,   

l2 = 1, . . . , m2 ;

... 0 0 ... 0 0 ... ... ... . . . q3 0 . . . 0 q3 0 q13 q23 ... 0 0 0

... 0 ... 0 ... ... . . . K3 ... 0 ... 0

... 0 ... 0 ... 0 ... ... . . . q13 . . . q23 ... 0 ... 0 ... 0 ... 0 ... ... . . . q123 ... 0 ... 0 0 0 C m1 −1 ... 0 0 0

   ,   0 0 0 ... 0 q13 q23

ãäå

l1 = 2, . . . , m1 + 1;

     ,    

0 0 0 ... 0 q123 0

ãäå

l1 = 1, . . . , m1 ;

     ,    

ãäå

0 ... 0 0 ... 0 0 ... 0 ... ... ... 0 . . . C3 0 . . . D2 0 . . . E1

l1 = 1, . . . , m1 ;

0 0 0 ... 0 C2 D1

0 0 0 ... 0 0 C1

     ,    

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

ãäå

   Cll11×l1 =   

p3 0 ... 0 0



Dll11×(l1 +1)

p13  0  =  ...  0 0 

l1 E(l 1 +2)×l1



Wnl21 ×n1

     =     

l2 = 1, . . . , m2 ;

0 p3 ... 0 0

... 0 0 ... 0 0 ... ... ... . . . p3 0 . . . 0 p3

p23 p13 ... 0 0

0 p23 ... 0 0 0 0

0 0

H m1 +1 S m1 B m1 −1 0 ... 0 0 0

T m1 H m1 S m1 −1 B m1 −2 ... 0 0 0

Am1 −1 T m1 −1 H m1 −1 S m1 −2 ... 0 0 0

  Sll11×(l1 +1) =   

p2 p1 ... 0 0

0 p2 ... 0 0

ãäå

0 0 0 ... 0 p123 0

p123 ... 0 0 0

p1 0 ... 0 0

  ,  

0 0 ... p23 p13

... 0 ... 0 ... 0 ... ... . . . p123 ... 0 ... 0





... 0 ... 0 ... ... . . . p13 ... 0

0  p123   0  =  ...  0   0 0

ãäå

161

l1 = 2, . . . , m1 + 1;

0 0 ... 0 p23      ,    

m1 −2

T H m1 −2 ... 0 0 0

   ,  

ãäå

ãäå

l1 = 1, . . . , m1 ;

l1 = 1, . . . , m1 ;

... 0 ... 0 ... 0 ... 0 ... ... . . . H3 . . . S2 . . . B1

0 0 0 0 ... T2 H2 S1

0 0 0 0 ... A1 T1 H1

      ,     

l2 = 2, . . . , m2 + 1; ... 0 0 ... 0 0 ... ... ... . . . p1 p2 . . . 0 p1

0 0 ... 0 p2

   ,  

ãäå

l1 = 1, . . . , m1 ;

162

À. À. Ëàçóòèíà

 Bll11×(l1 +2)

0 0 ... 0 0 

  =  

    =    

T(ll11 +1)×l1



Al(l11 +2)×l1

 Hll11×l1

  =  

    =    

p12 0 ... 0 0

0 p12 ... 0 0

... 0 ... 0 ... ... . . . p12 ... 0

q1 q2 0 ... 0 0 0

0 q1 q2 ... 0 0 0

... 0 ... 0 ... 0 ... ... . . . q1 . . . q2 ... 0

0 q12 0 ... 0 0 0

0 0 q12 ... 0 0 0

... 0 ... 0 ... 0 ... ... . . . q12 ... 0 ... 0

q2 + q12 + q23 + q123 0 ... 0 0 ãäå

1 H1×1 =

 Wn11 ×n1

   =   

0 0 ... 0 0

 0 0 0 0   ... ...   , ãäå l1 = 1, . . . , m1 ; 0 0  p12 0  0 0   0   ...   , ãäå l1 = 1, . . . , m1 ; 0   q1  q2  0 0   0   ...   , ãäå l1 = 1, . . . , m1 ; 0   q12  0  ... 0 0  ... 0 0  , ... ... ...   ... 0 0 . . . 0 q1 + q12 + q13 + q123

l1 = 2, . . . , m1 + 1;

q1 + q2 + q12 + q13 + q23 + q123

F m1 +1 S m1 0 ... 0 0

T m1 F m1 S m1 −1 ... 0 0

0 T m1 −1 F m1 −1 ... 0 0

... 0 ... 0 ... 0 ... ... ... F2 . . . S1




; 0 0 0 ... T1 F1

    ;   

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

163

Fll11×l1 =      

q2 + q3 + q12 + q23 + q123 0 ... 0 0

0 q3 ... 0 0

ãäå

1 F1×1 =

... ... ... ... ...

0 0 ... q3 0

0 0 ... 0 q1 + q3 + q12 + q13 + q123

   ,  

l1 = 2, . . . , m1 + 1;

q1 + q2 + q3 + q12 + q13 + q23 + q123




.

Ìîæíî íàéòè ôóíäàìåíòàëüíóþ ìàòðèöó N = (I − Q)−1 è âû÷èñëèòü ñðåäíåå âðåìÿ áëóæäàíèÿ [4]. Áûëà íàïèñàíà ïðîãðàììà, êîòîðàÿ ðåàëèçóåò äàííûé àëãîðèòì. Ðåçóëüòàòû ÷èñëåííûõ ýêñïåðèìåíòîâ ïðèâîäÿòñÿ â òàáëèöàõ 3-4. Óñëîâíûå îáîçíà÷åíèÿ â òàáëèöàõ 3-4: W1  ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ, êîãäà ïåðâûé è âòîðîé ñòåêè îáúåäèíåíû â ïàðó, à òðåòèé ñòåê ðàñïîëîæåí îòäåëüíî; W2  ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ, êîãäà ïåðâûé è òðåòèé ñòåêè îáúåäèíåíû â ïàðó, à âòîðîé ñòåê ðàñïîëîæåí îòäåëüíî; W3  ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ, êîãäà âòîðîé è òðåòèé ñòåêè îáúåäèíåíû â ïàðó, à ïåðâûé ñòåê ðàñïîëîæåí îòäåëüíî W4  ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ. Òàáëèöà 3. Ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé äëÿ p1 = 0.35, p2 = p3 = p12 = p23 = p13 = p123 = 0.05,q1 = q2 = q3 = q12 = q13 = q23 = q123 = 0.05. m 10 20

W1 m1 7 15

T 11.15 20.68

W2 m1 7 15

T 11.15 20.68

W3 m1 4 8

T 10.78 19.88

W4 T 8.56 17.96

164

À. À. Ëàçóòèíà

Èç òàáëèöû 3 âèäíî, ÷òî åñëè âñå âåðîÿòíîñòè ðàâíû, êðîìå âåðîÿòíîñòè âêëþ÷åíèÿ â îäèí èç ñòåêîâ (íàïðèìåð, ïåðâûé), è ïðè ýòîì ýòà âåðîÿòíîñòü áîëüøå âñåõ îñòàëüíûõ âåðîÿòíîñòåé, òî ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ áóäåò áîëüøå â òîì ñëó÷àå, åñëè ýòîò ñòåê ïîìåñòèòü â ïàðó. Òàáëèöà 4. Ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé äëÿ p1 = p2 = p3 = 0.01, p13 = p12 = p23 = p123 = 0.22,q1 = q2 = q3 = q12 = q13 = q23 = q123 = 0.01. m 10 20

W1 m1 6 13

T 5.29 10.10

W2 m1 6 13

T 5.29 10.10

W3 m1 6 13

T 5.29 10.10

W4 T 3.3 5.89

Ðåçóëüòàòû òàáëèöû 4 ïîêàçûâàþò, ÷òî åñëè âåðîÿòíîñòè îäèíàêîâû, òî ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ äëÿ âñåõ òðåõ ñëó÷àåâ îäèíàêîâî. p13 , p12 , p23 , p123 ,

2. Ñ Ïóñòü ìû ðàáîòàåì â ïàìÿòè ðàçìåðà m åäèíèö ñ òðåìÿ ñòåêàìè, ðàñïîëîæåííûìè ñâÿçíî â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé (ñì. Ðèñ. 5). ÂßÇÀÍÍÎÅ ÏÐÅÄÑÒÀÂËÅÍÈÅ

Ñâÿçàííîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé.

Ðèñ. 5.

 êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè èìååì ñëó÷àéíîå áëóæäàíèå â ïèðàìèäå: x1 + x2 + x3 6 m2 (ñì. Ðèñ. 6), ò.ê. ïîëîâèíà ïàìÿòè òðàòèòñÿ íà ñâÿçè.

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

165

Îáëàñòü áëóæäàíèÿ äëÿ ñâÿçàííîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé.

Ðèñ. 6.

Ïðîíóìåðóåì ñîñòîÿíèÿ: íà÷èíàÿ ñ ïëîñêîñòè x3 = 0, x1 + x2 + m m 2 , x3 = 1, x1 + x2 + x3 6 2 è ò.ä. íàâåðõ, â êàæäîé èç ýòèõ ïëîñêîñòåé íóìåðàöèÿ ñîñòîÿíèé íà÷èíàåòñÿ âäîëü ïðÿìîé m x1 + x2 = m 2 − x3 îò îñè x1 ê îñè x2 , x1 + x2 = 2 − x3 − 1 è ò.ä. (ñì. Ðèñ. 7). x3 6

Ðèñ. 7.

Ïðèìåð íóìåðàöèè ñîñòîÿíèé äëÿ m = 4.

Òîãäà ìîæíî ïîñòðîèòü ìàòðèöó Q ïåðåõîäíûõ âåðîÿòíîñòåé îäíîðîäíîé ïîãëîùàþùåé öåïè Ìàðêîâà. Òåîðåìà 2. Ïðè âûáðàííîé íóìåðàöèè ñîñòîÿíèé ïîäìàòðèöà

Q

ìàòðèöû ïåðåõîäíûõ âåðîÿòíîñòåé äàííîé öåïè, ñîîòâåòñòâóþùàÿ

ïåðåõîäàì

èìååò âèä:

èç

íåâîçâðàòíûõ

ñîñòîÿíèé

â

íåâîçâðàòíûå,

166

À. À. Ëàçóòèíà

An+1 n∗(n+1)



2   C (n−1)∗n  2  0 Q=   ...   0 0

ãäå

n=

0

...

0

0



B (n−1)∗n

...

0

0

...

0

0

    ,    

B n∗(n+1) 2

An(n−1)∗n

2

2

An−1 (n−2)∗(n−1)

C (n−2)∗(n−1) 2

2

... 0 0

... 0 0

... ... . . . A23 . . . C1

... B3 A11

m 2,

ïîäìàòðèöà

Aln∗(n+1)

- êâàäðàòíàÿ ìàòðèöà ðàçìåðíîñòè

2

×

n∗(n+1) 2

n∗(n+1) × 2

èìååò âèä:

A n∗(n+1) l = 2



Dn×n

 Fn×(n−1)   Xn×(n−2)   ...   0   0 0

En×(n−1)

Gn×(n−2)

D(n−1)×(n−1) F(n−1)×(n−2) ... 0 0 0

E(n−1)×(n−2) D(n−2)×(n−2) ... 0 0 0 ãäå

... 0 ... 0 ... 0 ... ... . . . D3×3 . . . F3×2 . . . X3×1

0 0 0 ... E3×2 D2×2 F2×1

0 0 0 ... G3×1 E2×1 D1×1

     ,    

l = 1, ...., n, Dn×n =

     

q2 + q12 + q13 + q23 + q123 0 ... 0 0 ãäå

D1×1 =

0 ... 0 0 0 ... 0 0 ... ... ... ... 0 ... 0 0 0 . . . 0 q1 + q12 + q13 + q23 + q123
q1 + q2 + q 1 2 + q13 + q23 + q123 , A n∗(n+1) n+1 = 2

   ,  

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè



Hn×n

 Fn×(n−1)   Xn×(n−2)   ...   0   0 0

En×(n−1)

Gn×(n−2)

H(n−1)×(n−1) F(n−1)×(n−2) ... 0 0 0

E(n−1)×(n−2) H(n−2)×(n−2) ... 0 0 0

 Hn×n

  =  

q2 + q12 + t 0 0 0 t 0 ... ... ... 0 0 0 0 0 0 ãäå

ãäå

    =    

Fn×(n−1)

ïîäìàòðèöà èìååò âèä:

0 0 0 ... H3×3 F3×2 X3×1

0 0 0 ... E3×2 H2×2 F2×1

   =   

B n∗(n+1)

0 0 0 ... G3×1 E2×1 H1×1

     ,    



... 0 0 ... 0 0 ... ... ... ... t 0 . . . 0 q1 + q12 + t

  ,  

t = q3 + q13 + q23 + q123 ,

q1 + q2 + q3 + q12 + q13 + q23 + q123

H1×1 =

En×(n−1)

... ... ... ... ... ... ...

167

q1 0 0 q2 q1 0 0 q1 q2 ... ... ... 0 0 0 0 0 0

... 0 0 ... 0 0 ... 0 0 ... ... ... . . . q 2 q1 . . . 0 q2



p1 p 2 0 0 p 1 p2 0 0 p1 ... ... ... 0 0 0 0 0 0

... 0 0 ... 0 0 ... 0 0 ... ... ... . . . p2 0 . . . p 1 p2



- ìàòðèöà ðàçìåðíîñòè

2

B n∗(n+1) = 2




,

   ,   

   ,    n∗(n+1) 2

×

(n−1)∗n 2

168



À. À. Ëàçóòèíà

0 ... 0 Tn×n

0 ... 0 0

0

T(n−1)×(n−1) Z(n−2)×(n−1) ... 0 0 0

       Z(n−1)×(n)   V(n−2)×(n)   ...   0   0

0 ... 0 0 0 T(n−2)×(n−2) ... 0 0 0

   Tn×n =   

   Z(n−1)×(n) =   

   V(n−2)×(n) =   

ïîäìàòðèöà èìååò âèä:

p13 0 ... 0 0

0 ... 0 0 0 0 ... T3×3 Z2×3 V1×3

p3 0 . . . 0 0 0 p3 . . . 0 0 ... ... ... ... ... 0 0 . . . p3 0 0 0 . . . 0 p3 p23 p13 ... 0 0

0 p123 0 0 ... ... 0 0 0 0

C (n−1)∗n

... ... ... ... ... ... ... ... ... ...

0 p23 ... 0 0

... 0 ... 0 ... ... . . . p13 ... 0

0

... 0 ... 0 ... ... . . . p123 ... 0

p123 ... 0 0

C (n−1)∗n = 2

0 ... 0 0 0 0 ... 0 0 T1×1

        ,       

   ,  

0 0 ... p23 p13

0 0 ... 0 p23

0 0 ... 0 p123

- ìàòðèöà ðàçìåðíîñòè

2

0 ... 0 0 0 0 ... 0 T2×2 Z1×2

   ,  

0 0 ... 0 0

(n−1)∗n 2

   ,  

×

n∗(n+1) 2

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

0  0   ...   0  0 0 

. . . . . .

0 0 ... 0 0 0

Wn×n 0 ... 0 0 0

Yn×(n−1) W(n−1)×(n−1) ... 0 0 0

 Wn×n

  =  



Yn×(n−1)

Jn×(n−2)

Jn×(n−2) Y(n−1)×(n−2) ... 0 0 0

. . . . . .

0 0 ... W3×3 0 0

q3 0 ... 0 0 0 q3 . . . 0 0 ... ... ... ... ... 0 0 . . . q3 0 0 0 . . . 0 q3

    =    

q13 q23 0 ... 0 0 0



0

 p123   0  =  ...  0   0 0

0 q13 q23 ... 0 0 0

... 0 ... 0 ... 0 ... ... . . . q13 . . . q23 ... 0

... 0 p123 ... 0 0 0

0 0 ... 0 ... 0 ... ... . . . p123 ... 0 ... 0

0 0 ... Y3×2 W2×2 0

169

0 0 ... J3×1 Y2×1 W1×1

    ,  

   ,  

0 0 0 ... 0 q13 q23

     ,    

 0 0 ... 0 p123 0

    .    

Ïîñòðîèì ôóíäàìåíòàëüíóþ ìàòðèöó N = (I − Q)−1 . Ìîæíî íàéòè ñðåäíåå âðåìÿ áëóæäàíèÿ. Áûëà íàïèñàíà ïðîãðàììà, êîòîðàÿ ðåàëèçóåò äàííûé àëãîðèòì, ðåçóëüòàòû ïðåäñòàâëåíû â òàáëèöàõ 3-4. Èç òàáëèö 3-4 âèäíî, ÷òî â áîëüøèíñòâå ñëó÷àåâ ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ áîëüøå â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ ñòåêîâ, ÷åì â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ, íî íå

170

À. À. Ëàçóòèíà

âñåãäà. Çàìåòèì, ÷òî â ñëó÷àå, êîãäà âñå âåðîÿòíîñòè ðàâíû, ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé áîëüøå, ÷åì ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ. 3. Ñ

,

ËÓ×ÀÉ

ÊÎÃÄÀ ÄËÈÍÀ ÈÍÔÎÐÌÀÖÈÎÍÍÎÉ ×ÀÑÒÈ ÁÎËÜØÅ ÅÄÈÍÈÖÛ

Ïðåäïîëîæèì, ÷òî èíôîðìàöèîííàÿ ÷àñòü èìååò ïðîèçâîëüíûé ðàçìåð l, òîãäà ñðàâíèì äâà ñïîñîáà ñâÿçàííîå è ïîñëåäîâàòåëüíîå ïðåäñòàâëåíèÿ ñòåêîâ. Ñâÿçàííîå ïðåäñòàâëåíèå: ïîëå ñâÿçè òàêæå èìååò åäèíè÷íóþ äëèíó, ðàçìåð ïàìÿòè m êðàòåí l + 1. Òîãäà â êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè èìååì ñëó÷àéíîå áëóæäàíèå ïî öåëî÷èñëåííîé ðåøåòêå, ñ ðàññòîÿíèåì ìåæäó óçëàìè l + 1, â ïèðàìèäå: m x1 + x2 + x3 6 l+1 , ïðåäïîëàãàåòñÿ, ÷òî m êðàòíî l + 1. Ïðîíóìåðóåì ñîñòîÿíèÿ òàêæå, êàê è â çàäà÷å î ñâÿçàííîì ïðåäñòàâëåíèè òðåõ ñòåêîâ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé (ïî ïëîñêîñòÿì ïèðàìèäû). Ïîñòðîèì ìàòðèöó ïåðåõîäíûõ âåðîÿòíîñòåé äëÿ îäíîðîäíîé ïîãëîùàþùåé öåïè Ìàðêîâà. Ìàòðèöà Q áóäåò èìåòü òàêóþ æå ñòðóêòóðó, êàê ìàòðèöà Q äëÿ ñëó÷àÿ ñâÿçàííîé îðãàíèçàöèè ïàìÿòè. Íàéäåì ôóíäàìåíòàëüíóþ ìàòðèöó N = (I − Q)−1 . Âû÷èñëèì ñðåäíåå âðåìÿ áëóæäàíèÿ. Äàëåå ðàññìîòðèì ïîñëåäîâàòåëüíîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ñëó÷àå ðàçíîé äëèíû èíôîðìàöèîííîé ÷àñòè, ò.å. l > 1.  ýòîì ñëó÷àå ðàçìåð ïàìÿòè m äîëæåí áûòü êðàòåí l. Ïðîíóìåðóåì ñîñòîÿíèÿ òàêæå, êàê è â çàäà÷å î ïîñëåäîâàòåëüíîì ïðåäñòàâëåíèè òðåõ ñòåêîâ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé (ïî ïëîñêîñòÿì ïèðàìèäû). Ïîñòðîèì ìàòðèöó ïåðåõîäíûõ âåðîÿòíîñòåé äëÿ îäíîðîäíîé ïîãëîùàþùåé öåïè Ìàðêîâà. Ìàòðèöà Q áóäåò èìåòü òàêóþ æå ñòðóêòóðó, êàê ìàòðèöà Q äëÿ ñëó÷àÿ ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ. Âû÷èñëèì ñðåäíåå âðåìÿ. Ðåçóëüòàòû ñðàâíåíèÿ ýòèõ äâóõ ñïîñîáîâ ïðåäñòàâëåíû â òàáëèöàõ 5-6.

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

171

Òàáëèöà 5. Ñðàâíåíèå ñâÿçàííîãî è ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèè è ðàçìåðå èíôîðìàöèîííîé ÷àñòè áîëüøåì åäèíèöû, m = 30, p1 = 0.87, p2 = p3 = p12 = p13 = p23 =

p123 = 0.01, q1 = q2 = q3 = q12 = q13 = q23 = q123 = 0.01. l=2

Ñâÿçàííîå 11.54 Ïîñëåäîâàòåëüíîå 14.66

l=5 6.27 6.26

Òàáëèöà 6. Ñðàâíåíèå ñâÿçàííîãî è ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèè è ðàçìåðå èíôîðìàöèîííîé ÷àñòè áîëüøåì åäèíèöû, m = 30, p1 = p2 = p3 = p12 = p13 = p23 =

p123 = 0.01, q1 = 0.87, q2 = q3 = q12 = q13 = q23 = q123 = 0.01. l=2

l=5

Ñâÿçàííîå 212.97 93.98 Ïîñëåäîâàòåëüíîå 93.37 61.63 Èç òàáëèö 5-6 âèäíî, ÷òî ñ óâåëè÷åíèåì ðàçìåðà l äëÿ áîëüøèíñòâà âåðîÿòíîñòåé ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ òðåõ ïàðàëëåëüíûõ ñòåêîâ ëó÷øå, ÷åì â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ. 4. Ñ Ñâÿçàííîå ïðåäñòàâëåíèå ñòðóêòóð äàííûõ â ïàìÿòè ÿâëÿåòñÿ ÷àñòíûì ñëó÷àåì ñòðàíè÷íîãî ïðåäñòàâëåíèÿ ñòðóêòóð äàííûõ.  ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ ïàìÿòü ðàçäåëåíà íà ñòðàíèöû îäèíàêîâîãî ðàçìåðà. Íà ñâÿçü òðàòèòñÿ îäèí ýëåìåíò êàæäîé ñòðàíèöû, ïîýòîìó ïîòåðè íà ñâÿçè â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ ñòåêîâ áóäóò ìåíüøå, ÷åì â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ. Ìû ïðåäïîëàãàåì, ÷òî ñóùåñòâóåò ìåõàíèçì ðàáîòû ñî ñïèñêîì ñâîáîäíûõ ñòðàíèö, êîòîðûé ïîçâîëÿåò ïðè çàïîëíåíèè ñòðàíèöû ïðåäîñòàâèòü ñòåêó íîâóþ ñòðàíèöó. Åñëè æå âåðøèíû ñòåêà îñâîáîäèëè ñòðàíèöó, òî ýòà ñòðàíèöà âîçâðàùàåòñÿ â ñïèñîê ñâîáîäíûõ ñòðàíèö. ÒÐÀÍÈ×ÍÎÅ ÏÐÅÄÑÒÀÂËÅÍÈÅ

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À. À. Ëàçóòèíà

Ïóñòü â ñòðàíè÷íîé ïàìÿòè ðàçìåðà m åäèíèö (ñì. Ðèñ. 9) ìû ðàáîòàåì ñ òðåìÿ ñòåêàìè â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé.

Ñòðàíè÷íîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé.

Ðèñ. 9.

Ñíà÷àëà îïðåäåëèì îïòèìàëüíûé ðàçìåð ñòðàíèöû. Ïóñòü ðàçìåð ñòðàíèöû x åäèíèö ïàìÿòè. Òîãäà êîëè÷åñòâî ñòðàíèö mx . Îáîçíà÷èì ÷åðåç M (x) êîëè÷åñòâî åäèíèö ïàìÿòè, êîòîðîå òðàòèòñÿ íåïîñðåäñòâåííî ïîä äàííûå. Áóäåì ñ÷èòàòü, ÷òî M (x) = 2∗(x−1) m− m , ò.ê. mx åäèíèö ïàìÿòè òðàòÿòñÿ íà ñâÿçè; òàêæå x − 2 ïîñëå ïåðåïîëíåíèÿ îäíîãî èç ñòåêîâ, â òîì ñëó÷àå, êîãäà ñïèñîê ñâîáîäíîé ïàìÿòè óæå ïóñò, ðàáîòà çàêàí÷èâàåòñÿ, ìû ïðåäïîëàãàåì, ÷òî äâå äðóãèå ñòðàíèöû, ñ êîòîðûìè ðàáîòàëè äâà äðóãèõ ñòåêà, çàïîëíåíû íå ïîëíîñòüþ äàííûìè, ïðåäïîëîæèì, ÷òî íàïîëîâèíó, ò.å. 2∗(x−1) åäèíèö. 2 Òîãäà M 0 (x) = xm2 − 1, ò.å. îïòèìàëüíûé ðàçìåð ñòðàíèöû x = √ m. Åñëè ïðîâåðèòü çíàê âòîðîé ïðîèçâîäíîé, òî ìîæíî óáåäèòüñÿ, ÷òî ýòà òî÷êà ìàêñèìóìà. Ìû ðàáîòàåì ñ äèñêðåòíîé ìîäåëüþ, ïîýòîìó íåîáõîäèìî ïðîâåðèòü äâà çíà÷åíèÿ - îêðóãëèòü x äî öåëîãî ñíèçó è ñâåðõó.  êà÷åñòâå ìàòåìàòè÷åñêîé ìîäåëè èìååì ñëó÷àéíîå áëóæäàíèå â ïèðàìèäå: x1 + x2 + x3 6 m∗ , ãäå m∗ = m − mx äëÿ ñëó÷àÿ, êîãäà ïàìÿòü èñïîëüçóåòñÿ ïîëíîñòüþ (ìàêñèìàëüíî), ò.å. ïîñëå ïåðåïîëíåíèÿ èç îäíîãî èç ñòåêîâ, êîãäà â ñïèñêå ñâîáîäíîé ïàìÿòè íåò ñâîáîäíûõ ñòðàíèö, äâå äðóãèå ñòðàíèöû, ñ êîòîðûìè ðàáîòàëè äâà äðóãèõ ñòåêà, òîæå çàïîëíåíû ïîëíîñòüþ, è m∗ = m − m x − 2 ∗ (x − 2) äëÿ ñëó÷àÿ, êîãäà ïàìÿòü èñïîëüçóåòñÿ íå ïîëíîñòüþ (ìèíèìàëüíî), ò.å. ïîñëå ïåðåïîëíåíèÿ èç îäíîãî èç ñòåêîâ, êîãäà â ñïèñêå ñâîáîäíîé ïàìÿòè íåò ñâîáîäíûõ ñòðàíèö,

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

173

äâå äðóãèå ñòðàíèöû, ñ êîòîðûìè ðàáîòàëè äâà äðóãèõ ñòåêà, çàïîëíåíû ìèíèìàëüíî, â íèõ ñîäåðæèòñÿ òîëüêî ïî îäíîé åäèíèöå äàííûõ (ñì. Ðèñ.10).

Ðèñ. 10. Îáëàñòü áëóæäàíèÿ äëÿ ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé. Ïðîíóìåðóåì ñîñòîÿíèÿ òàêæå, êàê è â çàäà÷å î ñâÿçàííîì ïðåäñòàâëåíèè òðåõ ñòåêîâ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé (ïî ïëîñêîñòÿì ïèðàìèäû). Ïîñòðîèì ìàòðèöó ïåðåõîäíûõ âåðîÿòíîñòåé äëÿ îäíîðîäíîé ïîãëîùàþùåé öåïè Ìàðêîâà. Ìàòðèöà Q äëÿ ñëó÷àÿ ñòðàíè÷íîé îðãàíèçàöèè ïàìÿòè áóäåò èìåòü òàêóþ æå ñòðóêòóðó, êàê ìàòðèöà Q äëÿ ñëó÷àÿ ñâÿçàííîé îðãàíèçàöèè ïàìÿòè. Ïîñòðîèì ôóíäàìåíòàëüíóþ ìàòðèöó N = (I − Q)−1 . Âû÷èñëèì äâå îöåíêè äëÿ ñðåäíåãî âðåìåíè (ìàêñèìàëüíóþ è ìèíèìàëüíóþ), îáîçíà÷èì èõ Tmax è Tmin . Áûëà íàïèñàíà ïðîãðàììà, êîòîðàÿ ðåàëèçóåò äàííûé àëãîðèòì. Ðåçóëüòàòû ïðåäñòàâëåíû â òàáëèöàõ 7-8, òàì æå, äëÿ ñðàâíåíèÿ ïðèâîäèòñÿ ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ. Òàáëèöà 7. Ñòðàíè÷íîå è ñâÿçàííîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèè p1 = 0.87, p2 = p3 = p12 = p13 = p23 = p123 = 0.01,

q1 = q2 = q3 = q12 = q13 = q23 = q123 = 0.01. m 16 25

x 4 5

Tmin 9.42 15.79

Tmax 13.66 22.23

m 16 26

T 9.42 14.72

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À. À. Ëàçóòèíà

Òàáëèöà 8. Ñòðàíè÷íîå è ñâÿçàííîå ïðåäñòàâëåíèå òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèè p1 = p2 = p3 = p12 = p13 = p23 = p123 = 0.01,

q1 = 0.87, q2 = q3 = q12 = q13 = q23 = q123 = 0.01. m 16 25

x 4 5

Tmin 163.64 314.77

Tmax 263.51 470.45

m 16 26

T 163.64 289.08

Èç òàáëèö 7-8 âèäíî, ÷òî ñðåäíåå âðåìÿ äî ïåðåïîëíåíèÿ â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé áîëüøå, ÷åì ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ òàêèõ æå ñòåêîâ. Ñ óâåëè÷åíèåì ðàçìåðà ïàìÿòè m (m > 16) ñðåäíåå âðåìÿ äëÿ ñâÿçàííîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ áóäåò õóæå, ÷åì ìèíèìàëüíàÿ îöåíêà äëÿ ñðåäíåãî âðåìåíè â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ. Ïðîâåðèì, ñ êàêîãî m ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ áóäåò âñåãäà ìåíüøå, ÷åì ìèíèìàëüíàÿ îöåíêà äëÿ ñðåäíåãî âðåìåíè â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ. Äëÿ ýòîãî íåîáõîäèìî ñðàâíèòü îáëàñòè áëóæäàíèÿ. Êàê òîëüêî îáëàñòü áëóæäàíèÿ áóäåò áîëüøå, òàê ñðåäíåå âðåìÿ áóäåò áîëüøå: m−

√ m m − 2 ∗ (x − 2) > m − , x = m => x 2

m ⊂ (−∞, 4)

[

(16, +∞),

√

ò.å. íà÷èíàÿ ñ x = 17 ( x = 4 èëè x = 5 ) è áîëüøèõ, ìèíèìàëüíàÿ îöåíêà äëÿ ñðåäíåãî âðåìåíè â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ âñåãäà ëó÷øå, ÷åì ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ. √ Åñëè â êà÷åñòâå ðàçìåðà ñòðàíèöû ðàññìàòðèâàåòñÿ íå x = m, à ëþáîé äðóãîé ïðîèçâîëüíûé ðàçìåð, òî ýòî óòâåðæäåíèå òàêæå âåðíî: m−

m m − 2 ∗ x + 4 > , m ∗ (x − 2) > 4 ∗ x ∗ (x − 2), x 2

Îïòèìàëüíîå óïðàâëåíèå òðåìÿ ïàðàëëåëüíûìè ñòåêàìè

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äëÿ x > 2 ìèíèìàëüíàÿ îöåíêà äëÿ ñðåäíåãî âðåìåíè âñåãäà ëó÷øå, ÷åì ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçíîãî ïðåäñòàâëåíèÿ, åñëè x< m 4 , (x 6 2 - äëÿ íàøåé çàäà÷è íå èìååò ñìûñëà). 5. Ç Â äàííîé ðàáîòå ðàññìîòðåíî íåñêîëüêî ñïîñîáîâ ïðåäñòàâëåíèÿ òðåõ ñòåêîâ â ïàìÿòè îäíîãî óðîâíÿ â ñëó÷àå ïàðàëëåëüíîãî âûïîëíåíèÿ îïåðàöèé. Áûëè ïðåäëîæåíû ìàòåìàòè÷åñêèå ìîäåëè è ïîëó÷åíû ÷èñëåííûå ðåçóëüòàòû äëÿ îöåíêè ýòèõ ìåòîäîâ ñ òî÷êè çðåíèÿ ìàêñèìèçàöèè ñðåäíåãî âðåìåíè ðàáîòû äî ïåðåïîëíåíèÿ ïàìÿòè. Ñðåäíåå âðåìÿ â ñëó÷àå ïîñëåäîâàòåëüíîãî ïðåäñòàâëåíèÿ, äëÿ áîëüøèíñòâà íàáîðîâ âåðîÿòíîñòåé, ëó÷øå, ÷åì ñðåäíåå âðåìÿ âî âñåõ îñòàëüíûõ ñïîñîáàõ ïðåäñòàâëåíèÿ. Ñðåäíåå âðåìÿ â ñëó÷àå ñòðàíè÷íîãî ïðåäñòàâëåíèÿ òðåõ ñòåêîâ âñåãäà áîëüøå, ÷åì ñðåäíåå âðåìÿ â ñëó÷àå ñâÿçàííîãî ïðåäñòàâëåíèÿ. ÀÊËÞ×ÅÍÈÅ

Ëèòåðàòóðà 1. Êíóò Ä. 2. Koopman

Èñêóññòâî ïðîãðàììèðîâàíèÿ äëÿ ÝÂÌ. Ì.: Ìèð, 1976.T.1. P.

Stack

Computers.

Ellis

Horwood,

1989.

URL:

http://www.cs.cmu.edu/~koopman/stack_computers/ 3. Â. Ôåëëåð. Ââåäåíèå â òåîðèþ âåðîÿòíîñòåé è åå ïðèëîæåíèÿ. Ìèð, Ìîñêâà,1964. 4. Êåìåíè Äæ., Ñíåëë Äæ. Êîíå÷íûå öåïè Ìàðêîâà. Ì:Íàóêà, 1970.


   


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Òðóäû ÈÏÌÈ ÊàðÍÖ ÀÍ

Âûïóñê 7, 2006

ÓÄÊ 519.6, 539.2

ÀÇÍÎÑÒÍÀß ÑÕÅÌÀ ÄËß ÌÎÄÅËÈÎÂÀÍÈß ÒÄÑ-ÑÏÅÊÒÀ ÄÅ ÈÄÈÎÂÀÍÈß ÌÅÒÀËËÎÂ Ñ Ó×ÅÒÎÌ ÒÅÏËÎÏÎ ËÎÙÅÍÈß È ÑÆÀÒÈß

Í. È. ÎÄ×ÅÍÊÎÂÀ Ìîäåëèðóåòñÿ êèíåòèêà âûäåëåíèÿ âîäîðîäà èç ïîðîøêà ãèäðèäà ìåòàëëà ñ ó÷åòîì äèíàìèêè ìåæàçîâîãî ïåðåõîäà, ñæàòèÿ ÷àñòèö ïîðîøêà ãèäðèäà, äåñîðáöèîííûõ ïðîöåññîâ íà ïîâåðõíîñòè è òåïëîïîãëîùåíèÿ íà ãðàíèöå ðàçäåëà àç. Ïðåäñòàâëåíà ðàçíîñòíàÿ ñõåìà äëÿ êðàåâîé çàäà÷è ñ ïîäâèæíûìè ãðàíèöàìè ïðèìåíèòåëüíî ê ýêñïåðèìåíòàëüíîìó ìåòîäó òåðìîäåñîðáöèîííîé ñïåêòðîìåòðèè (ÒÄÑ). Ïðèâåäåíû ðåçóëüòàòû ÷èñëåííîãî ìîäåëèðîâàíèÿ.

1. ÂÂÅÄÅÍÈÅ Â [1℄ ïðåäñòàâëåíà äèóçèîííàÿ ìîäåëü ñ ó÷åòîì òåïëîïîãëîùåíèÿ íà ãðàíèöå ðàçäåëà àç è ýåêòà ñæàòèÿ. Äàííàÿ ïóáëèêàöèÿ ÿâëÿåòñÿ ïðîäîëæåíèåì ýòîé ðàáîòû.  ñòàòüå ðàññìàòðèâàåòñÿ ÷èñëåííûé ìåòîä (ïîñòðîåíèå ðàçíîñòíîé ñõåìû) ðåøåíèÿ ñîîòâåòñòâóþùåé êðàåâîé çàäà÷è ñ íåëèíåéíûìè äèíàìè÷åñêèìè ãðàíè÷íûìè óñëîâèÿìè è ïîäâèæíûìè ãðàíèöàìè. Ñîîòâåòñòâóþùàÿ áèáëèîãðàèÿ èìååòñÿ â [27℄. Ìîäåëü ïðåäñòàâëåíà ïðèìåíèòåëüíî ê ýêñïåðèìåíòàëüíîìó ìåòîäó òåðìîäåñîðáöèîííîé ñïåêòðîìåòðèè (ÒÄÑ). Îáðàçåö ïîðîøêîîáðàçíîãî ãèäðèäà ìåòàëëà ïîìåùàþò â êàìåðó ñ ïîñëåäóþùèì ïîñòîÿííûì âàêóóìèðîâàíèåì è îòíîñèòåëüíî ìåäëåííûì íàãðåâîì. Ïî ðåãèñòðèðóåìîìó ñ ïîìîùüþ ìàññ-ñïåêòðîìåòðà äåñîðáöèîííîìó ïîòîêó âîäîðîäà èç îáðàçöà ñóäÿò î òåõ èëè èíûõ õàðàêòåðèñòèêàõ



Í. È. îä÷åíêîâà, 2006

184

Í. È. îä÷åíêîâà

åãî âçàèìîäåéñòâèÿ ñ èññëåäóåìûì ìàòåðèàëîì. àññìîòðèì âûñîêîòåìïåðàòóðíûé ïèê ïîòîêà, êîãäà äèóçèÿ ÿâëÿåòñÿ îäíèì èç ëèìèòèðóþùèõ àêòîðîâ. 2. ÎÑÍÎÂÍÛÅ ÏÎÍßÒÈß È ÎÁÎÇÍÀ×ÅÍÈß Ïðè ðàñïàäå ðàññìàòðèâàåìîãî ìîäåëüíîãî ãèäðèäà ïðîèñõîäèò ñóùåñòâåííîå óâåëè÷åíèå ïëîòíîñòè (óìåíüøåíèå îáúåìà). Ýòî ñâÿçàíî ñ ïåðåñòðîéêîé ðåøåòêè. Äëÿ íåêîòîðûõ ãèäðèäîâ ýåêò äîñòèãàåò íåñêîëüêèõ äåñÿòêîâ ïðîöåíòîâ. Ïóñòü  êîýèöèåíò ñæàòèÿ: â ïðîöåññå ðàçëîæåíèÿ ãèäðèäà îáúåì ñ V óìåíüøàåòñÿ äî çíà÷åíèÿ V (0 < < 1, = onst, T 2 [T ; T + ℄). àçìåðû ÷àñòèö ñ÷èòàåì óæå äîñòàòî÷íî ìàëûìè, ÷òîáû âîçíèêàþùèå íàïðÿæåíèÿ íå ïðèâîäèëè ê äàëüíåéøåìó èçìåëü÷åíèþ. Áóäåì ñòðåìèòüñÿ ó÷åñòü ýåêò ìèíèìàëüíûìè ìàòåìàòè÷åñêèìè ñðåäñòâàìè. Îáîçíà÷èì L(t)  ðàäèóñ ñæèìàþùåéñÿ ñåðè÷åñêîé ÷àñòèöû ïîðîøêà ãèäðèäà, (t)  ðàäèóñ ñæèìàþùåãîñÿ ãèäðèäíîãî ÿäðà. Óìåíüøåíèå L(t) èíèöèèðóåòñÿ ¾ïðîïàâøèì¿ îáúåìîì â ðåçóëüòàòå ïåðåñòðîéêè ðåøåòêè íà ãðàíèöå ðàçäåëà àç. Ñîêðàùàÿ ìíîæèòåëü 4=3, èìååì ñâÿçü  è L â îðìå (1 )[03 3℄ = L30 L3. Çäåñü L0 = L(0), 0 = (0), íà÷àëüíûé ìîìåíò âðåìåíè t = 0 ñîîòâåòñòâóåò íà÷àëó âñïëåñêà äåñîðáöèè. ×òîáû ñóùåñòâåííî íå óñëîæíÿòü ìîäåëü, ïîëàãàåì, ÷òî âíåøíÿÿ êîðêà êàê-áû ¾îñåäàåò â öåëîì¿. Ïóñòü (t; r)  êîíöåíòðàöèÿ àòîìàðíîãî âîäîðîäà, ðàñòâîðåííîãî â êîðêå ìåòàëëà (â  àçå) òîëùèíû (L ) â ìîìåíò âðåìåíè t; Q  êîíöåíòðàöèÿ âîäîðîäà â ãèäðèäå ( àçå); b(T )  êîýèöèåíò äåñîðáöèè; D(T )  êîýèöèåíò äèóçèèè;   êèíåòè÷åñêàÿ êîíñòàíòà (  1:46
1021 1= m2
s
Torr); p(t)  äàâëåíèå ìîëåêóëÿðíîãî âîäîðîäà â êàìåðå; s  êîýèöèåíò ïðèëèïàíèÿ âîäîðîäà ê ïîâåðõíîñòè (äèññîöèàöèè ÷àñòè ìîëåêóë H2 íà àòîìû). Íàãðåâ íà ïîâåðõíîñòè îáû÷íî ëèíåéíûé: TL (t)  T (t; L) = vt + T0 . Ñ÷èòàåì, ÷òî ê íà÷àëüíîìó ìîìåíòó âðåìåíè (t = 0 ñîîòâåòñòâóåò íà÷àëó ïèêà ÒÄÑ-ñïåêòðà) ðàñòâîðåííûé â ãèäðèäå âîäîðîä ïîêèíóë îáðàçåö è Q =

rit = onst ÿâëÿåòñÿ êðèòè÷åñêîé êîíöåíòðàöèåé.  ïðîöåññå äàëüíåéøåãî íàãðåâà ïðîèñõîäèò ðàñïàä àçû è ñîîòâåòñòâóþùàÿ àêòèâèçàöèÿ äåñîðáöèè. Ïëîòíîñòü äåñîðáöèîííîãî ïîòîêà ìîäåëèðóåì êâàäðàòè÷íîé çàâèñèìîñòüþ îò êîíöåíòðàöèè àòîìîâ â ïðèïîâåðõíîñòíîì ñëîå (âàðèàíò îáúåìíîé

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

185

äåñîðáöèè): J (t) = b(T ) 2 (t; L). Çàâèñèìîñòü ïàðàìåòðîâ îò òåìïåðàòóðû ñ÷èòàåì àððåíèóñîâñêîé. Íàïðèìåð, äëÿ êîýèöèåíòà äåñîðáöèè b(T ) = b0 expf Eb =[RT ℄g; T 2 [T ; T + ℄; ãäå Eb  ýíåðãèÿ àêòèâàöèè, b0 = onst, R = onst. Äàâëåíèå ìîëåêóëÿðíîãî âîäîðîäà â êàìåðå èíèöèèðóåò ïëîòíîñòü ïîòîêà s(T )p(t) àòîìîâ H íà ïîâåðõíîñòü (â ïðèïîâåðõíîñòíûé îáúåì). Äåãèäðèðîâàíèå ïðåäïîëàãàåò sp < b 2L , L  (t; L). Ëèøü ÷àñòü äåñîðáèðîâàâøåãîñÿ âîäîðîäà âîçâðàùàåòñÿ îáðàòíî â ðåçóëüòàòå îòêà÷êè H2 . Ìîäåëü áóäåì èçëàãàòü â îäíî÷àñòè÷íîì ïðèáëèæåíèè. Äëÿ ìîäåëèðîâàíèÿ ïîðîøêà èñïîëüçîâàëîñü äèñêðåòíîå íîðìàëüíîå ðàñïðåäåëåíèå ÷àñòèö ïî ðàäèóñàì. Ýòî, â íåêîòîðîì ñìûñëå, ¾êîìïåíñèðóåò¿ ïîãðåøíîñòü ñåðè÷åñêîé àïïðîêñèìàöèè. Ïðè ÷èñëåííîì ìîäåëèðîâàíèè èñïîëüçîâàëîñü 10-15 çíà÷åíèé L0. 3. ÝÔÔÅÊÒ ÎÁÚÅÌÍÎ Î ÑÆÀÒÈß Îáå ãðàíèöû ïîäâèæíû:  = (t), L = L(t), (1 )[30 3 ℄ 3 L0 L3. Ïëîòíîñòü ïîòîêà àòîìîâ âîäîðîäà (ïî íàïðàâëåíèþ âäîëü ðàäèóñà, jr0j = 1) ìîäåëèðóåì âûðàæåíèåì

= r0

 + (t; r)v(t; r); r ãäå v (t; r) ¾âñòðå÷íàÿ ñêîðîñòü ñðåäû¿ (v < 0). Óðàâíåíèå äèj (t; r) = D(T )

óçèè ñ ó÷åòîì ñåðè÷åñêîé ñèììåòðèè ïðèìåò âèä:

   = r1 r r D(T ) r

(t; r)v(t; r) : Óòî÷íèì âûðàæåíèå äëÿ v (t; r).  ìîìåíò âðåìåíè t = t ìûñëåííî âûäåëèì øàð V (r), r > , è îïðåäåëèì ñêîðîñòü èçìåíåíèÿ åãî îáúåìà â ïðîöåññå ñæàòèÿ. Îáîçíà÷èâ ðàäèóñ ÷åðåç R(t) (R(t ) = r), ïîëó÷èì
d V_ (r) = V R(t) = 4r R_ (t ): dt t0

 t

2

2

0

0

2

0

 óêàçàííûõ ïðåäïîëîæåíèÿõ (íàïðÿæåíèÿ íåêðèòè÷íû, íåò ðàñòðåñêèâàíèÿ) äîëæíî áûòü V_ (r) = V_ (L), ò. å. r2 R_ = L2 L_ . Ïðîïàäàþùèé îáúåì êîìïåíñèðóåòñÿ âíåøíèìè ñëîÿìè. Ïîýòîìó ñêîðîñòü äâèæåíèÿ ñðåäû âäîëü ðàäèóñà ïðåäñòàâèì â îðìå

_ 2 = (1 )2=r v(t; r) = L2 L=r _ 2 ; r 2 [; L℄:

186

Í. È. îä÷åíêîâà

Îòäåëüíî îáîñíóåì v(t; ) = (1 )_ : ýòî èìåííî òà ÷àñòü ñêîðîñòè _ äâèæåíèÿ ãðàíèöû ðàçäåëà àç, êîòîðàÿ èíèöèèðóåòñÿ ñæàòèåì. Îáîçíà÷èì ÷åðåç S (r) ïëîùàäü ñåðû ðàäèóñà r, TL (t)  T (t; L(t)). Äèåðåíöèðóåì ïî âðåìåíè ñîîòíîøåíèå ìàòåðèàëüíîãî áàëàíñà:

QV () + 4 Q _ + 2



Z L(t)

Z t

(t)

0

r2 (t; r) dr +

S (L) b(TL) 2 (; L) d

= onst;

Z L 

r2 t (t; r) dr + (t; L)L2L_ (t; )2 _ + b(TL) 2 (t; L)L2 = 0; 

Q

(t; ) 2 _ 2 D(T [t; ℄)

  + b(TL ) 2L L2 + L2 D(TL ) = 0: r  r L

Ïåðâûå äâà ñëàãàåìûõ îòíîñÿòñÿ ê ïðîöåññàì íà ãðàíèöå ðàçäåëà àç r = , îñòàëüíûå  íà ïîâåðõíîñòè r = L. Âñëåäñòâèå áàëàíñà ïîòîêîâ íà âíåøíåé ãðàíèöå b(TL) 2 (t; L) = D(TL ) r (t; L) (íåò íàêîïëåíèÿ H íà ïîâåðõíîñòè) ïðèõîäèì ê óñëîâèþ Ñòåàíà 



Q

(t; ) _(t) = D(T [t; ℄)

 : r 

Óðàâíåíèå äâèæåíèÿ ãðàíèöû ðàçäåëà àç ìîæíî âûâåñòè è íåçàâèñèìî èç áàëàíñà

Q ÆV

(t; ) ÆV

=

S ()D(T [t; ℄) r (t; )
t + o(
t); ÆV

=
V:

Òîãäà óæå ãðàíè÷íîå óñëîâèå íåëèíåéíîå òðåòüåãî ðîäà îêàæåòñÿ ñëåäñòâèåì. 4. ÌÎÄÅËÜ ÒÅÏËÎÏÎ ËÎÙÅÍÈß Â êîíòåêñòå ìåòîäà ÒÄÑ ïðîãðåâ ÷àñòèö ãèäðèäà çíà÷èòåëüíî áûñòðåå äèóçèè. Ïîýòîìó ðàçóìíî èñêàòü ðàñïðåäåëåíèå òåìïåðàòóðû â ðàñòóùåé êîðêå ìåòàëëà ñ ðàñòâîðåííûì äèóíäèðóþùèì âîäîðîäîì â îðìå êâàçèñòàöèîíàðà T (t; r) = AÆ (t) + B Æ (t)=r, r 2 [; L℄. Ñêîðîñòü íàãðåâà ìàëà: ðàñïðåäåëåíèå òåìïåðàòóðû íå ñèëüíî îòëè÷àåòñÿ îò ðàâíîìåðíîãî, èìååòñÿ îòíîñèòåëüíî íåáîëüøîå ïîíèæåíèå T ïðè r =  ïî ñðàâíåíèþ ñ r = L. Ïîýòîìó B Æ (t) < 0 (óíêöèÿ T (t; r) âûïóêëà ââåðõ). àññìîòðèì íà÷àëüíûé âàðèàíò ó÷åòà ýíäîòåðìè÷íîñòè ðàñïàäà ãèäðèäà. Ïîëàãàåì, ÷òî â ÷àñòèöå ïîðîøêà çà âðåìÿ ðàñïàäà

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

187

ãèäðèä íå óñïåâàåò ñóùåñòâåííî èçìåíèòü ñâîþ òåìïåðàòóðó â îòëè÷èå îò îêðóæàþùåé êîðêè ìåòàëëà. Íà ìàëîì ïðîìåæóòêå âðåìåíè [t; t +
t℄ îáðàçóåòñÿ øàðîâîé ñëîé ðàñòâîðà îáúåìà  ÆV = 3 ( +
)3 4=3,
 < 0,
 = (t +
t) (t)  _(t)
t. Êîëè÷åñòâî ïîãëîùåííîãî òåïëà ÆV; ïðîïîðöèîíàëüíîå ýòîìó îáúåìó, ðàâíî âåëè÷èíå P 42
t ñ òî÷íî òüþ äî o(
t). Çäåñü îáîçíà÷èëè P = r T = B Æ =2  ïëîòíîñòü òåïëîâîãî ïîòîêà. Ïðèðàâíèâàÿ äâà âûðàæåíèÿ è äåëÿ íà
t ! 0, îïðåäåëÿåì êîýèöèåíò B Æ : B Æ (t) = { _ (t)2 (t), { = =.  ïðåäåëàõ ïèêà ÒÄÑ-ñïåêòðà âåëè÷èíó { , õàðàêòåðèçóþùóþ òåïëîïðîâîäÿùèå ñâîéñòâà ìåòàëëà è ýíåðãåòèêó ðàñïàäà, ñ÷èòàåì ïîëîæèòåëüíîé êîíñòàíòîé. Êîýèöèåíò AÆ (t) îïðåäåëÿåòñÿ êîíòðîëèðóåìîé òåìïåðàòóðîé TL (t) íà ïîâåðõíîñòè. Èòàê, ãèïåðáîëè÷åñêîå ïî r ðàñïðåäåëåíèå T (t; r) íåñêîëüêî óìåíüøàåòñÿ ïðè r !  + 0 ïî ñðàâíåíèþ ñ T (t; L) èç-çà ýíäîòåðìè÷íîñòè ðàñïàäà, à ÿäðî ïðîãðåòî ïðàêòè÷åñêè ðàâíîìåðíî. Èç-çà áîëüøîãî ðàçëè÷èÿ â òåïëîïðîâîäÿùåé ñïîñîáíîñòè (íàïðèìåð, ãèäðèä  ñîëåïîäîáíûé) èìååòñÿ ðàçðûâ òåìïåðàòóðû íà ãðàíèöå ðàçäåëà àç. Ïðè (t) ! 0 èìååì B Æ (t) ! 0 è ïðîöåññ îñòàòî÷íîé äåãàçàöèè íà÷èíàåòñÿ è ïðîòåêàåò ïðè ðàâíîìåðíîì ïðîãðåâå ÷àñòèöû.

5. ËÎÊÀËÜÍÎÅ ÀÂÍÎÂÅÑÈÅ È ÍÀ×ÀËÜÍÛÅ ÄÀÍÍÛÅ Â ðàâíîâåñíîé ñèòóàöèè êîíöåíòðàöèÿ àòîìîâ H â ìåòàëëå óñòàíàâëèâàåòñÿ íà óðîâíå  = Q < Q. Ïðè âûñîêîé òåìïåðàòóðå ãèäðèä ðàñïàäàåòñÿ äîñòàòî÷íî èíòåíñèâíî è â óñëîâèÿõ äèóçèîííîãî áàðüåðà ìîæíî ñ÷èòàòü, ÷òî íà ãðàíèöå ðàçäåëà àç èìååò ìåñòî ëîêàëüíîå ðàâíîâåñèå (t; ) = . Óðàâíåíèþ äèóçèè t = D(t)( rr +2 r =r) óäîâëåòâîðÿåò íåçàâèñÿùàÿ îò âðåìåíè óíêöèÿ '(r) = A + B=r. Ïðèìåì åå çà íà÷àëüíîå ðàñïðåäåëåíèå (0; r) â ñëîå r 2 [0 ; L0 ℄: äî íà÷àëà èíòåíñèâíîãî ðàñïàäà ãèäðèäà (t = 0) â ðåçóëüòàòå îáúåìíîé äåñîðáöèè ïðîèñõîäèò ìåäëåííûé ðîñò íà÷àëüíîé êîðêè ìåòàëëà ñ ðàâíîìåðíûì ïðîãðåâîì. Òàêèì îáðàçîì, êîãäà ãðàíèöà íà íà÷àëüíîì ýòàïå äâèæåòñÿ î÷åíü ìåäëåííî, âèä ðàñïðåäåëåíèÿ ðàñòâîðåííîãî âîäîðîäà (ãèïåðáîëè÷åñêèé êâàçèñòàöèîíàð) ïî÷òè íå áóäåò ìåíÿòüñÿ ñî âðåìåíåì. Îáîçíà÷èì (0; L0) = 0 , òîãäà: '(0 ) = ,

188

Í. È. îä÷åíêîâà

'(L0 ) = 0 <  ) '(r) = A +

B r

= LL

0 0 0

0 0

+ L 
L  0 0

0 1
: 0 r

0

Ïàðàìåòð 0 íàõîäèì èç ãðàíè÷íîãî óñëîâèÿ íà ïîâåðõíîñòè ïðè t = 0:

b(T0 ) 20 = D(T0 )

' r L0

= D(T )
L  0

0

0 0
: 0 L0

Ýòî êâàäðàòíîå óðàâíåíèå îòíîñèòåëüíî 0 , êîðíè âåùåñòâåííû è ðàçíûõ çíàêîâ, òàê ÷òî ïðèïîâåðõíîñòíàÿ êîíöåíòðàöèÿ îïðåäåëÿåòñÿ îäíîçíà÷íî. Îòìåòèì, ÷òî íà÷àëüíîå ðàñïðåäåëåíèå '(r) â òîíêîé êîðêå ìåòàëëà ïðàêòè÷åñêè íå âëèÿåò íà ìîäåëüíûé ÒÄÑ-ïèê, ïîñêîëüêó ïî÷òè âåñü âîäîðîä ñîñðåäîòî÷åí â ãèäðèäíîì ÿäðå.

 t

6. ÊÀÅÂÀß ÇÀÄÀ×À ÒÄÑÄÅ ÈÄÈÎÂÀÍÈß Êîìïàêòíî ïðåäñòàâèì ìîäåëü: 0  (0; L0 ), (0) = 0 ,  h  = r1 r r D T [t; r℄
r 2

(0; r) = A +

2

B r

v(t; r) = (1 ) D(TL)

0 0

= LL

0 0 0

2 _ r2

i

+ L 
L 

_ = Lr L ; Q 2

2




v ; t 2 (0; t ); r 2 (t); L(t) ; 0 0

0

0 1
; r 2 (0 ; L0); 0 r




(t; ) _(t) = D T [t; ℄

r

(t)

;


 = b(TL) 2 (t; L); t; (t) =  < Q; TL(t)  T (t; L); r L(t)

  T (t; r) = AÆ (t) + B Æ (t) r 1 ; r 2 (t); L(t) ;

(1



) 30 3



=L

3 0

L3 ;

AÆ (t) + B Æ (t)L 1 (t) = TL (t); B Æ (t) = {2 (t)_(t); TL (t) = T0 + vt: Âåëè÷èíû Q, , { , ñ÷èòàåì ïîñòîÿííûìè â ïðåäåëàõ ÒÄÑ-ïèêà T 2 [T ; T +℄, êîýèöèåíòû äèóçèè è äåñîðáöèè çàâèñÿò îò òåìïåðàòóðû ïî çàêîíó Àððåíèóñà: D(T ) = D0 expf ED =[RT ℄g, b(T ) = b0 expf Eb=[RT ℄g. Ïîñëå òîãî êàê ãèäðèäíîå ÿäðî èñ÷åçíåò ((t ) = 0), ïåðåõîäèì ê çàäà÷å äåãàçàöèè áåç ïîäâèæíûõ ãðàíèö (r 2 [0; L℄, L = L(t ))

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

189

ñ óñëîâèåì ñèììåòðèè r (t; 0) = 0 è ðàâíîìåðíûì ïðîãðåâîì ìåòàëëè÷åñêèõ ÷àñòèö T (t; r) = TL(t)  T (t):

 t

 2

= D (t)

r2

 + 2r
r



; t 2 (t ; t ); r 2

(0; L);

'(r) = (t ; r);

  2 = 0; D(t)  D(T (t)); b(t)  b(T (t)): = b(t) (t; L ); r L r r=0 Ìîìåíò t îïðåäåëÿåòñÿ óñëîâèåì J (t ) = b(t ) 2 (t ; L )  0. D(t)

7. ÏÅÎÁÀÇÎÂÀÍÈÅ ÊÀÅÂÎÉ ÇÀÄÀ×È Ââåäåì çàìåíó ïåðåìåííûõ: (t; r) $ (t; x), r = (t) + x[L(t) (t)℄, u(t; x) = (t; r(t; x)), w(t; x) = v(t; r(t; x)). Ñóòü çàìåíû ñîñòîèò â òîì, ÷òîáû ïåðåâåñòè ïîäâèæíóþ îáëàñòü (t) 6 r 6 L(t) â íåïîäâèæíóþ 0 6 x 6 1. Ýòà çàìåíà íåîáõîäèìà äëÿ àëãîðèòìà ÷èñëåííîãî ðåøåíèÿ, â ïðîòèâíîì ñëó÷àå âîçíèêàþò òðóäíîñòè ñ ïîñòðîåíèåì ñåòêè ïî ïðîñòðàíñòâåííîé ïåðåìåííîé. Êðàåâàÿ çàäà÷à ÒÄÑ-äåãèäðèðîâàíèÿ ïîñëå çàìåíû:

u t

= (L 1 )

  u  D ( T ) + f (t; x)
u ; t 2 (0; t ); x 2 (0; 1); (1) 2 x x x

h

f (t; x)  _ (1 x) + L_ x +

2  + x(L


D(T ) w(t; x)
L 1  ; ) i

2 _ L2L_ = ( + x(L ))2 ( + x(L ))2 ; u0 u u0 1 + L0 0

; 0 L0 0 0 + x(L0 0 )

w(t; x)  v(t; r(t; x)) = (1 )
u(0; x) =

L0u0 L0

(1 )  D(T ) u  

u(t; 0) _(t) =


u0  u(0; 1); u = ; 

Q

D

(T ) u 

L  x

x=1

3 0

L  x

= b(TL)u (t; 1); 2

3

x=0



=L

3 0

L3;

(2)

; (0) = 0 ; u(t; 0) = u < Q; (3)

TL (t)  T (t; L(t)) = T0 + vt;

T (t; x) = AÆ + B Æ ( + x(L )) 1 ; AÆ + B Æ L

1

= TL ;

(4)

B Æ = {2 :_

190

Í. È. îä÷åíêîâà

Ôîðìàëüíî ìîæíî çàáûòü î èçè÷åñêîì ñìûñëå (t) êàê ãðàíèöû ðàçäåëà àç, à L(t) êàê ðàäèóñà ÷àñòèöû, è ðàññìàòðèâàòü èõ êàê óíêöèîíàëüíûå ïàðàìåòðû, ïîä÷èíåííûå óñëîâèÿì (3) è (2). Ïîëó÷èëàñü êðàåâàÿ çàäà÷à äëÿ óðàâíåíèÿ äèóçèè (1) (ñ êîýèöèåíòîì, çàâèñÿùèì îò ïàðàìåòðîâ). Ïîëó÷åííàÿ êðàåâàÿ çàäà÷à äîñòàòî÷íî ãðîìîçäêà, íî îíà â íåïîäâèæíîé îáëàñòè è åå ìîæíî ðåøàòü, ïîëüçóÿñü îáû÷íûìè ìåòîäàìè. 8. ÏÎÑÒÎÅÍÈÅ

ÍÅßÂÍÎÉ ÀÇÍÎÑÒÍÎÉ ÑÕÅÌÛ

Ââåäåì ñåòêó xi = ih, i = 0; 1; : : : ; N , tn = n , ãäå h = 1=N  øàã ïî ðàäèóñó,   øàã ïî âðåìåíè. Ââåäåì ñåòî÷íûå àïïðîêñèìàöèè Uin  u(tn ; xi ), Ln  L(tn ), n  (tn ), _ n  _ (tn ), L_ n  L_ (tn ), ani  D(tn ; xi h=2). Èç íà÷àëüíûõ äàííûõ Ui0 = u(0; xi ). Äëÿ àïïðîêñèìàöèè óðàâíåíèÿ (1) ñ ïîðÿäêîì O(h2 +  ) â óçëå (xi ; tn ) èñïîëüçóåì ñòàíäàðòíûé ÷åòûðåõòî÷å÷íûé øàáëîí (xi1 ; tn+1 ), (xi ; tn ), (xi ; tn+1):

Uin+1 ) ani +1 (Uin+1 Uin+1 1 ) + 2 2 h (Ln+1 n+1 ) n+1 n+1 +Kin+1
2hU(iL+1 Ui 1 ) ; n+1 n+1 2ani +1 Kin+1  _n+1 (1 xi ) + L_ n+1 xi + win+1 : n+1 + xi (Ln+1 n+1 )

Uin+1 Uin 

n n = ai (Ui +1 +1

+1 +1

Óìíîæèì îáå ÷àñòè óðàâíåíèÿ íà (Ln+1 n+1 )2 h2 , îáîçíà÷èì sn+1 = (Ln+1 n+1 )2 h2 = è ïðèâåäåì ïîäîáíûå ñëàãàåìûå: +1 bi Uin+1

i Uin+1 + ai Uin+1 sn+1Uin ; 1 = +1 bi  ani+1 + h Kin+1(Ln+1 n+1); i  ani +1 + ani+1+1 + sn+1;

2

ai  ani +1

h

n 2 Ki (Ln +1

+1

n+1 ):

+1 n+1 +1 Ñèñòåìà ðåøàåòñÿ ìåòîäîì ïðîãîíêè: Uin+1 = ni+1 Ui+1 + in+1 , i = 1; : : : ; N 1, n > 1: Óñëîâèÿ óñòîé÷èâîñòè âûïîëíåíû (h << 1): +1 j i j > jai j + jbi j = ani +1 + ani+1 , sn+1 > 0. ðàíè÷íûå óñëîâèÿ èìåþò

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

âèä

3UNn 4UNn + UNn = b(T n )(U n ) : L N 2h(Ln n ) êîýèöèåíòû äëÿ i = 2; : : : ; N 1 îïðåäåëÿþòñÿ +1

U0n+1 = Q; D(TLn+1 )
Ïðîãîíî÷íûå îðìóëàìè +1 ni+1 =

+1 1

+1

+1 2

+1

sn+1 Uin + ani +1

h K n+1 i 2

+1 2

+1

+1 ani+1 + hh2 Kin+1(Ln+1 n+1) +1 sn+1 + ani +1 + ani+1 ani +1 h2 Kin+1 (Ln+1

h

+1 in+1 =

191

(Ln

+1

i

n+1 ) ni +1

;

i

n+1 ) in+1

i : ani +1 h2 Kin+1 (Ln+1 n+1 ) ni +1 àññìîòðèì ëåâóþ ãðàíèöó ïðè i = 1. Ïîäñòàâëÿÿ â óðàâíåíèå h i h sn+1 (U1n+1 U1n ) = an2 +1 + K1n+1 (Ln+1 n+1 ) U2n+1 +1 sn+1 + ani +1 + ani+1

h

2

h

i

(an + an )U n + an h2 K n (Ln n ) U n çíà÷åíèå U n = Q, ïîëó÷èì ïåðâûå ïðîãîíî÷íûå êîýèöèåí1

+1

0

òû

+1

2

1

+1

1

+1

1

+1

+1

+1

0

+1

+1

n2 +1 =

an2 +1 + h2 K1n+1 (Ln+1 n+1 ) ; an1 +1 + an2 +1 + sn+1 h

i

sn+1 U1n + an1 +1 h2 K1n+1(Ln+1 n+1 ) Q n +1 2 = : an1 +1 + an2 +1 + sn+1 Âû÷èñëÿåì çíà÷åíèÿ n2 +1 è 2n+1 . Äàëåå íàõîäèì îñòàâøèåñÿ ïðîãîíî÷íûå êîýèöèåíòû ni +1 è in+1 , i = 3; : : : ; N . Äëÿ íàõîæäåíèÿ UNn+1 èñïîëüçóåì ãðàíè÷íîå óñëîâèå 3U n+1 4UNn+11 + UNn+12 = b(T n+1)(U n+1 )2: D(TLn+1 ) N L N 2h(Ln+1 n+1) Ïîäñòàâèì ñâÿçè UNn+12 = nN+11 UNn+11 + Nn+11 , UNn+11 = nN+1 UNn+1 + Nn+1 . Ïîëîæèì A  3 4nN+1 + nN+11 nN+1 , B  nN+11 Nn+1 + Nn+11 4 Nn+1. ðàíè÷íîå óñëîâèå ñ ó÷åòîì îáîçíà÷åíèé ïåðåïèøåòñÿ â îðìå

2h b(TLn )(Ln +1

+1

n+1 )D

1

(TLn )(UNn ) + AUNn + B = 0: +1

+1 2

+1

192

Í. È. îä÷åíêîâà

Ýòî êâàäðàòíîå óðàâíåíèå îòíîñèòåëüíî UNn+1 . Âûáèðàåòñÿ ïîëîæèòåëüíûé êîðåíü. Çíà÷åíèå óíêöèîíàëüíîãî ïàðàìåòðà n+1  (tn+1 ) íàõîäèì èç ðàçëîæåíèÿ n+1 â îêðåñòíîñòè n ïî îðìóëå Òåéëîðà, ó÷èòûâàÿ òîëüêî ëèíåéíîå ñëàãàåìîå. Îñòàëüíûìè ñëàãàåìûìè ïðåíåáðåãàåì:

D(T (tn ; n ))f 3U0n + 4U1n U2n g : 2h(Q u)(Ln n)  Çíà÷åíèå ïàðàìåòðà Ln+1 íàõîäèì èç ñîîòíîøåíèÿ (1 ) 30  3 = L30 L3 ïðè óæå èçâåñòíîì n+1 . n+1  n +  _n  n +

8.1. Äèóçèîííàÿ äåãàçàöèÿ. èäðèäíîãî ÿäðà íåò (t ) = 0. Ýòî çàäà÷à äåãàçàöèè áåç ïîäâèæíûõ ãðàíèö (r 2 [0; L℄, L = L(t )) ñ óñëîâèåì ñèììåòðèè r (t; 0) = 0 è ðàâíîìåðíûì ïðîãðåâîì ìåòàëëè÷åñêèõ ÷àñòèö T (t; r) = TL (t)  T (t):

 t

 2

= D (t)

r2

 + 2r
r



; t 2 (t ; t ); r 2

(0; L);

'(r) = (t ; r);

  2 = 0; D(t)  D(T (t)); b(t)  b(T (t)): = b(t) (t; L ); r L r r=0 Êîýèöèåíòû äëÿ i = 1; : : : ; N 1 óïðîùàþòñÿ ñëåäóþùèì îáD(t)

ðàçîì

+1 ni+1 =

sn+1

1 + h=xi + 2 ni [1 +1

h=xi ℄

+1 ; in+1 =

[1

h=xi ℄ in+1 + sn+1 Uin : sn+1 + 2 ni +1 [1 h=xi ℄

Äëÿ òîãî, ÷òîáû íàéòè ïåðâûå ïðîãîíî÷íûå êîýèöèåíòû, ïîëüçóåìñÿ óñëîâèåì ñèììåòðèè â öåíòðå øàðà:

  (t; 0) = 0 ) 2r D(t) r r

   2 (t; 0) ( t; 0)  2D(t) : r r2 Òàêèì îáðàçîì óðàâíåíèå äèóçèè ïðè r ! +0 ïåðåïèøåòñÿ â

âèäå

= 2r D(t)  t

 

(t; r) r

 = 3D(t) r : 2

2

 ïåðåìåííûõ (t; x) ïîëó÷àåì óðàâíåíèå (r

(t; r(x))):

u t

= 3 DL(t)
xu : 2

 2

2

=

xL , u(t; x)

=

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

193

àçíîñòíàÿ àïïðîêñèìàöèÿ óðàâíåíèÿ äèóçèè

Uin+1 Uin 

n = 3 D(Ltn )
Ui

+1 +1

2Uin + Uin +1

+1 1

: h2  Óìíîæèì îáå ÷àñòè óðàâíåíèÿ íà L2 h2 , sn+1 = h2 L2 (3D(tn+1 )) 1 , è ïðèâåäåì ïîäîáíûå ñëàãàåìûå: +1 Uin+1 (2 + sn+1)Uin+1 + Uin+11 = sn+1Uin: ðàíè÷íûå óñëîâèÿ èìåþò âèä u 3U0n+1 + 4U1n+1 U2n+1 = 0; = 0 ) x x=0 2h n +1 n +1 3U 4UN 1 + UNn+12 = b(t )(U n+1 )2: D(tn+1 ) N n+1 N 2hL àññìîòðèì ëåâóþ ãðàíèöó ïðè i = 1. Ïîäñòàâëÿÿ â óðàâíåíèå U2n+1 (2 + sn+1 )U1n+1 + U0n+1 = sn+1U1n +1 2

ñîîòíîøåíèå ( 3U0n+1 +4U1n+1 U2n+1 )=2h = 0 èëè U2n+1 = 4U1n+1, ïîëó÷àåì ïåðâûå ïðîãîíî÷íûå êîýèöèåíòû

n1 +1 = 1

sn+1

2

; 1n+1 =

sn+1

2

3U n + 0

+1

U1n :

Âû÷èñëÿåì çíà÷åíèÿ n1 +1 è 1n+1 . Äàëåå íàõîäèì îñòàâøèåñÿ ïðîãîíî÷íûå êîýèöèåíòû ni +1 è in+1 , i = 2; : : : ; N . Äëÿ íàõîæäåíèÿ UNn+1 èñïîëüçóåì ãðàíè÷íîå óñëîâèå

D(tn+1 )

3UNn

+1

4UNn + UNn = b(t )(U n ) : n N 2hL +1 1

+1 2

+1

+1 2

Àëãîðèòì îïðåäåëåíèÿ UNn+1 ñîâïàäàåò ñ ïðèâåäåííûì âûøå. 9. ÀË ÎÈÒÌ ×ÈÑËÅÍÍÎ Î ÅØÅÍÈß àññìàòðèâàåì êðàåâóþ çàäà÷ó ÒÄÑ-äåãèäðèðîâàíèÿ èç ï. 7 â ïåðåìåííûõ (t; x). 1. Îïðåäåëÿåì íà÷àëüíûå äàííûå (x)  u(0; x). Äëÿ ýòîãî íàõîäèì çíà÷åíèå êîíöåíòðàöèè u0 = u(0; 1), èñïîëüçóÿ ïðè t = 0 ãðàíè÷íîå óñëîâèå

D(TL ) u = b(TL)u2(t; 1); TL(t) = T0 + vt; L  x x=1

194

Í. È. îä÷åíêîâà

è íà÷àëüíîå ãèïåðáîëè÷åñêîå ðàñïðåäåëåíèå ( (0; r) = A + B=r)

L0 0 (u u0 ) ; u =  < Q: 0 )[0 + x(L0 0 )℄ 0 0 Ïîëó÷àåòñÿ êâàäðàòíîå óðàâíåíèå ñ ïîëîæèòåëüíûì êîðíåì u0 : D(T0 )0 D(T0 )0 u u20 + u0 = 0: b(T0 )L0 (L0 0 ) b(T0)L0 (L0 0 ) 2. Âû÷èñëÿåì íà÷àëüíóþ ñêîðîñòü äâèæåíèÿ ãðàíèöû _ 0 = _ (0)

(x)  u(0; x) = L Lu

0 0

èç óñëîâèÿ Ñòåàíà

u0 0

+ (L

0 (0) = L0 (u u0 ) : 0 Çäåñü êîýèöèåò äèóçèè ÿâëÿåòñÿ íåëèíåéíîé óíêöèåé îò _0 . Íàïîìíèì, ÷òî òåìïåðàòóðà T (t; r) = AÆ (t)+ B Æ(t)=r â ïåðåìåííûõ (t; x) èìååò âûðàæåíèå T (t; x) = TL(t)+ {2 _(L )(1 x)[L( + x(L ))℄ 1; TL(t) = T0 + vt: Ñëåäîâàòåëüíî, T (0; 0) = T0 + { _ 0 0 (L0 0 )=L0  f0 (_ 0 ). Ñ ó÷åòîì àððåíèóñîâñêîé çàâèñèìîñòè D(T ) äëÿ _ 0 ïîëó÷àåì óðàâíåíèå  L0 (u u0 ) D0 exp ED =[Rf0(_0 )℄ : _ 0 = (Q u)(L0 0)0 Ýòî çàäà÷à î íåïîäâèæíîé òî÷êå (_ 0 = F0 (_ 0 )).  èçè÷åñêè ðåàëüíîì äèàïàçîíå ïàðàìåòðîâ ìîäåëè (ï. 10) îíà ýåêòèâíî ðåøàåòñÿ ìåòîäîì ïðîñòîé èòåðàöèè.  3 3. Âû÷èñëÿåì 

) 0 1 = 0 +  _ 0 è çíà÷åíèå L1  L( ) èç (1  3 3 3 1 = L0 L1 . Íà îòðåçêå r 2 [1 ; L1℄ îïðåäåëÿåì òåìïåðàòóðó T (; r), èñïîëüçóÿ â âûðàæåíèè B Æ ( ) = {_1 21 âìåñòî íåèçâåñòíîãî ïîêà _ 1 íà÷àëüíîå çíà÷åíèå _0 .  ïåðåìåííûõ (t; x) ïîëàãàåì {2 _ (L  )(1 x) T (; x) = TL( ) + 1 0 1 1 ; x 2 [0; 1℄: L1 [1 + x(L1 1 )℄ 4. Íåÿâíûì ìåòîäîì íà ñëîå t =  ðåøàåì äèóçèîííóþ çàäà÷ó. 5. Çíàÿ ïðèáëèæåííîå çíà÷åíèå ãðàäèåíòà êîíöåíòðàöèè ~r 

r (; 1 ), ïîäñòàâëÿåì â óðàâíåíèå äâèæåíèÿ ãðàíèöû (óñëîâèå Ñòåàíà) [Q ℄_1 = D(T (; 1 ))~ r âûðàæåíèå òåìïåðàòóðû T (; 1 ) ÷åðåç ïåðåìåííóþ _1 â ñèëó ñîîòíîøåíèé AÆ ( ) + B Æ ( )=L( ) = TL ( ); B Æ ( ) = {_1 21 : 



Q u _0 =

D(T (0; 0)) 0 (0); L0 0

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

195

 ïåðåìåííûõ (t; x) ïîëó÷àåì óðàâíåíèå äëÿ îïðåäåëåíèÿ _ 1 :

[Q u℄_ = DL(f (_ ))
u~x; u~x  u ; x x f (_ )  TL( ) + { _ (L  )L . Çàäà÷ó _ = F (_ ) ðåøàåì ïðîñòûìè èòåðàöèÿìè, _  íà÷àëüíîå çíà÷åíèå ( << 1). 6. Âîçâðàùàåìñÿ ê ïóíêòó 3, ãäå ìû âûíóæäåíû áûëè èñïîëüçîâàòü âìåñòî _ çíà÷åíèå _ . Òåïåðü óæå â B Æ ( ) = { _  ïîäñòàâëÿåì òåêóùåå ïðèáëèæåíèå _ è ïåðåõîäèì ê ïóíêòó 4. 1

1

1

1

1 1

1

1



1

=0

1

1

1

1

1

1

1

0

1

2 1 1

0

1

Òàêèõ èòåðàöèé öåëåñîîáðàçíî ñäåëàòü íåñêîëüêî. Äëÿ áîëåå ãðóáûõ ðàñ÷åòîâ ýòîò ïóíêò àëãîðèòìà ìîæíî îïóñòèòü. 7. Ïåðåõîäèì íà ñëåäóþùèé âðåìåííîé ñëîé t = 2 è äåéñòâóåì àíàëîãè÷íî 36: 2 = 1 +  _1 è ò. ä. Êðèòåðèåì îñòàíîâêè ñëóæèò (t )  0 ( < L0 =10). 8. Ïåðåõîäèì ê ýòàïó äåãàçàöèè ìåòàëëà ñ ðàñòâîðåííûì H . 10. ÂÛ×ÈÑËÈÒÅËÜÍÛÅ ÝÊÑÏÅÈÌÅÍÒÛ Áóäåì îðèåíòèðîâàòüñÿ íà ìîùíóþ âàêóóìíóþ ñèñòåìó, ïðåíåáðåãàÿ âîçâðàòîì íà ïîâåðõíîñòü äåñîðáèðîâàâøåãîñÿ âîäîðîäà ( sp  0). Íà ðèñóíêàõ, èëëþñòðèðóþùèõ âëèÿíèå ðàçëè÷íûõ êèíåòè÷åñêèõ ïàðàìåòðîâ, èçîáðàæåíû ãðàèêè ïëîòíîñòè äåñîðáöèè J (t) = b(T (t)) 2 (t; L) ïðè ëèíåéíîì íàãðåâå. Çíà÷åíèÿ ïàðàìåòðîâ ñîîòâåòñòâóþò óáûâàíèþ ìàêñèìóìà. Îñòàëüíûå ïàðàìåòðû, îáùèå äëÿ âñåõ ãðàèêîâ, áåðóòñÿ èç òàáëèöû:

b0 = 7
10 10 m4 =s Eb = 145 KJ Q = 4
1022 m 3 T_ = 0:3 K/s D0 = 8
10 3 m2 =s ED = 50 KJ L0 = 5
10 3 m  = 0:15 T (0) = 400 K

= 0:75

 = 
Q { = 105

Êðóæêîì îòìå÷àåòñÿ ìîìåíò îêîí÷àíèÿ ðàñïàäà ãèäðèäà. àñïðåäåëåíèå ÷àñòèö ïî ðàäèóñàì íîðìàëüíîå: ñðåäíèé ðàäèóñ ÷àñòèöû L = 5
10 3 m,  = 10 3 m. Ïðè ÷èñëåííîì ìîäåëèðîâàíèè èñïîëüçîâàëñÿ äèñêðåòíûé íàáîð èç 10 ðàäèóñîâ. Íà ðèñ. 1 ïðèâåäåí ãðàèê çàâèñèìîñòè J îò òåêóùåé òåìïåðàòóðû T = TL (t) (TL (t) $ t) è ìàñøòàáèðîâàííûå êðèâûå L
max J=[1:5 max L ℄, 
max J=[2 max  ℄. Íà ðèñ. 2 ïðèâåäåíû ãðàèêè J (T ) (Eb = 150). Èçìåíåíèÿ ïî óáûâàíèþ ìàêñèìóìà: L0 = Lmax = 8
10 3 ; ñ ðàñïðåäåëåíèåì ÷àñòèö ïîðîøêà ïî ðàäèóñàì; L0 = 5
10 3 ; L0 = Lmin = 2
10 3 . èñ. 3 ïîêàçûâàåò, ÷òî Eb

196

Í. È. îä÷åíêîâà

ñóùåñòâåííî âëèÿåò íà äèíàìèêó ïîòîêà, ìåíÿåòñÿ âèä êðèâîé. Âàðèàöèè: Eb = 170; 150; 140. èñ. 4 îòðàæàåò âëèÿíèå ýíåðãèè àêòèâàöèè êîýèöèåíòà ðàñïàäà ãèäðèäà Ek (k0 = 5
10 5 ). Âàðèàöèè: Ek = 5; 10; 15. ×åì ìåäëåííåå äèóçèÿ (ðèñ. 5, 6, { = 106), òåì íèæå è áîëåå ñìåùåí âïðàâî ãðàèê. èñ. 7 èëëþñòðèðóåò âëèÿíèå ñêîðîñòè íàãðåâà. èñ. 8 îòðàæàåò çàâèñèìîñòü ãðàèêà ïëîòíîñòè äåñîðáöèîííîãî ïîòîêà îò êîýèöèåíòà ñæàòèÿ. Òàêèì îáðàçîì, âàðèàöèè ïàðàìåòðîâ íàõîäÿòñÿ â ñîîòâåòñòâèè ñ èçè÷åñêèìè ïðåäñòàâëåíèÿìè è ìîäåëü ìîæåò áûòü èñïîëüçîâàíà äëÿ èäåíòèèêàöèè êèíåòè÷åñêèõ ïàðàìåòðîâ äåãèäðèðîâàíèÿ êîíêðåòíûõ ìàòåðèàëîâ. 17

3

17

2

x 10

x 10

−3

Lmax = 8⋅ 10

J

J

J for L

mean −3

L = 5⋅ 10

1.6

L

min

2 1.2

−3

= 2⋅ 10

L ρ

0.8

1

0.4 0 400

500

600

T, C

èñ. 1. ðàèêè J , L, .

0 400

500

600

T, C

èñ. 2. àñïðåä-å ïî ðàäèóñàì.

17

16

x 10

16

J

170 E = 150 b 140

3

x 10

J

5 Ek = 10 15

12

2 8

1

0 400

4

500

600 T, C

èñ. 3. Âëèÿíèå ïàðàìåòðà Eb .

0 400

600

800

T, C

èñ. 4. Âëèÿíèå ïàðàìåòðà Ek .

àçíîñòíàÿ ñõåìà äëÿ ìîäåëèðîâàíèÿ ÒÄÑ-ñïåêòðà

197

17

x 10

17

J

11⋅ 10−3 D0 = 8 ⋅ 10−3 −3 4 ⋅ 10

2

J

x 10

50 ED = 60 70

1.8

1.2

1 0.6

0 400

500

600 T, C

èñ. 5. Âëèÿíèå ïàðàìåòðà D0 .

0 400

17

J

T, C

800

17

x 10

J dTL

1.8

0.3 −− = 0.2 0.1 dt

0.6

0.6

T, C

600

èñ. 7. Âëèÿíèå ñêîðîñòè íàãðåâà.

0.7

γ = 0.8 0.95

1.2

500

x 10

1.8

1.2

0 400

600

èñ. 6. Âëèÿíèå ïàðàìåòðà ED .

0 400

500

T, C

600

èñ. 8. Âëèÿíèå êîýèöèåíòà

.

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 !   "$#%&(')&+*-,# ./10 '3254+* 6 # 7#98:;5&+<>= ?@;AB;?CAABD4+*E&+F = G4IH&+<J4KF L4(AB* :#M1?$&(254+<>= ?@N5*O4('3'P.Q 4PR SAB*ET .U"WV(X(Y# Y#Z&+= F &+0 &\[ # .K]1&+<>&+0 &+= ;4+;_^# N # .K^ `5`')`5;4+;a7#I/1;bF L4c&+;5&+0 :5')= 'dABef4(g5`5= 0 = h5*E= &i&+;2 h5&+*OH&+= ;5= ;H= ;&je = ')L4+* :H&+<J4jk(kl/1D5F = <>&+0-mnAB;5F *OAB0, L4(AB* :o&+;2p?CAB;AB<>= ?KM-;5&+0 :5')= 'P. ; # Y. "WV(X+q5.D #qBV(rBs5t+u(q5# r#vw&+xP&+0 ACGJ[ # [ # .(^U4+F F = 4PG&M# QJ#+Mye = ')L4+* :zH&+<J4<JA254+0R-= F LK&(H54J2= ')F *E= h5`5F{4(2zDABD5`50 &+F = AB; | *O4('34+* G4(2KF{4+*E*E= F{AB* :z&+D5D5*OAB&(?$LIk(kK6&+<J4 , L4(AB* :z&+;2KM-D5D50 = ?$&+F = AB;'P.BG9#V.Y+u(uBr.BD #qBtBs5}(Y# ~5#vw&+xP&+0 ACG [ # [ # .$^U4+F F = 4PG&1M# QJ#PMe = ')L4+* :H&+<J41<JA254+0R-= F LJ<>= H*E&+F = AB; |(*O4('34+* G4(2>F{4+*E*E= F{AB* : &+D5D5*OAB&(?$LIk(k@6&+<J4, L4(AB* :z&+;2KM-D5D50 = ?$&+F = AB;'P.BG9#U"3u5.Y+u(u(~5.BD #5V(}Bs€"3uBX# q5#w‚(ƒ$‚C„…(†K‡# ‡# .ˆ‰+Š+ŠC‹‰+†(‚JŒ# >#Ž z…(‘9…(’lƒ$‚C‚C“5‰>”•–5‚+†+„‰+‘9‹—K•…(•5”+„—˜‹‰+’lk(kKŽ …™ ƒ$–‰+‘9‹‰ •–5‹9š„5‚C‘9…(’j‹z•–…B›œn„‰+‘9‘9…(’j›‚+Š+‰+›‚+ŠC‹9š‹€.BžŸV.B†(œn•U# Y.Y+u(uBY. B#BY(V(rBs5r+uBt# t#w‚(ƒ$‚C„…(†K‡# ‡# .5ˆ‰+Š+ŠC‹‰+†(‚Œ# >#5ˆ5‚+†(‘9…(†B‰( $‹‰@•…I ¡C”l†zƒ$‚C‚C“‚C¢I…(¢–5‚+‘9œd…(š–”5£>‚+¤J™ ¥ ‰+’w $–‰Cœ¦k(k@w‚+Š+‰+›‚+ŠC‹“5‰( $š…‰›…(‰C„5‹9–…(†(‚+‘9‹‰.Š+…B›§"WX.žŸq5.Y+u(uBt. B#}(rBs5V+u5# }#¨ ‰+ŠC–… P—‘IŒ# Œ# .©‰+‘9š5‰+†(‹“z># Œ# .ªn‰+›‹9‘9‚>«€# Œ#B¬‰(…B–5‹—@‹9­3– #l… $š†(‚.€"WV(V(X# X#¨ …(‘9ŠC–—­3‹9‘l® # ª1# .5¯…C„5Š$—‘ $š‹9’°‡# ±9# .5±‚+›š–‰C„5‹5ƒC‰ˆ # ‡# .w‹ ¥ ‰+‘9š…z«€# ²z#w‚+Š+‰+›‚+ŠC‹ ™ “5‰( $š‚C—@Š+‰(…B–5‹—K…(•ŠC‹9›‚C„5³C‘9œ¢z•–…B˜9‰( C C…(†5#5# |1‚C”š‚.U"WV(}(t# V#ˆ…(‹…(‘9…(†´Œ# ‡#>¨ …(†(œnJ‰+‘9‹‰\¡+µ1µ1‰+šŠC‹9†(‘9… $ŠC‹¶„‰( C…(•…C„5³+ƒC…(†(‚+‘9‹—9#J¨ ‰+ŠC–…ƒ$‚+†B…(5 $š. Y+u(u(~5#

Òðóäû ÈÏÌÈ ÊàðÍÖ ÀÍ Âûïóñê 7, 2006

ÓÄÊ 512.544.43

ÏÈÌÅÍÅÍÈÅ ÏÀÀËËÅËÜÍÛÕ ÂÛ×ÈÑËÅÍÈÉ ÄËß ÅØÅÍÈß ÍÅÊÎÒÎÛÕ ÇÀÄÀ× ÌÅÕÀÍÈÊÈ ÒÂÅÄÎ Î ÄÅÔÎÌÈÓÅÌÎ Î ÒÅËÀ Ì.Ï. Ù ÅÁÀÊ

 ðàáîòå ïðåäñòàâëåí ïðèìåð ïðèìåíåíèÿ ïàðàëëåëüíûõ âû÷èñëåíèé äëÿ ðåøåíèÿ îäíîé èç çàäà÷ ìåõàíèêè òâåðäîãî äåîðìèðóåìîãî òåëà. å÷ü èäåò î çàäà÷å îïðåäåëåíèÿ óñèëèé è ïåðåìåùåíèé â ìåõàíè÷åñêîé ñèñòåìå, ñîñòîÿùåé èç íàáîðà àáñîëþòíî æåñòêèõ òåë, æåñòêèõ ñòåðæíåé è ïîäàòëèâûõ îäíîñòîðîííèõ ñâÿçåé. Îïðåäåëÿþùèå ñîîòíîøåíèÿ çàäà÷è ïðèâîäÿò ê çàäà÷å êâàäðàòè÷íîãî ïðîãðàììèðîâàíèÿ, ðåøàòü êîòîðóþ ïðåäëàãàåòñÿ ìåòîäîì Âóëà. Âñå âû÷èñëåíèÿ ïðîèçâîäÿòñÿ íà êëàñòåðå  íàèáîëåå äîñòóïíîì ñóïåðêîìïüþòåðå.  íàñòîÿùåé ðàáîòå ðàññìàòðèâàåòñÿ ðàñ÷åò ìåõàíè÷åñêîé ñèñòåìû àáñîëþòíî æåñòêèõ òåë, æåñòêèõ ñòåðæíåé è ïîäàòëèâûõ îäíîñòîðîííèõ ñâÿçåé íà ïàðàëëåëüíîì êîìïüþòåðå. àññìàòðèâàåìàÿ çàäà÷à íàõîäèòñÿ íà ñòûêå äâóõ àêòóàëüíûõ íàïðàâëåíèé ðàçíûõ îáëàñòåé çíàíèé: ìåõàíèêè òâåðäîãî äåîðìèðóåìîãî òåëà

è èíîðìàöèîííûõ

òåõíîëîãèé.

Äëÿ ïåðâîé

àêòóëüíûì

íà-

ïðàâëåíèåì ÿâëÿåòñÿ ðàñ÷åò ñèñòåì ñ îäíîñòîðîííèìè ñâÿçÿìè, êîòîðûé

â ïîñëåäíåå

âðåìÿ âûçûâàåò âñå áîëüøèé íàó÷íûé è

ïðàêòè÷åñêèé èíòåðåñ. Ïàðàëëåëüíûå âû÷èñëåíèÿ, ñ äðóãîé ñòîðîíû, ÿâëÿþòñÿ ñåãîäíÿ îäíèì èç ïðèîðèòåòíûõ íàïðàâëåíèé èññëåäîâàíèé îáëàñòè èíîðìàöèîííûõ òåõíîëîãèé â ñâÿçè ñ íà÷àëîì ìàññîâîãî ïðîèçâîäñòâà ìíîãîÿäåðíûõ ïðîöåññîðîâ. Îäíîñòîðîííåé ñâÿçüþ íàçûâàþò òàêîé îáúåêò, êîòîðûé ìîæåò âîñïðèíèìàòü óñèëèÿ òîëüêî îäíîãî çíàêà. Ïðèìåðîì òàêîé



Ì.Ï. Ùåðáàê, 2006

Ïàðàëëåëüíûå âû÷èñëåíèÿ

207

ñâÿçè ìîæåò ñëóæèòü îáû÷íàÿ íèòü: îíà ñîïðîòèâëÿåòñÿ ðàñòÿæåíèþ, íî íå ïðåïÿòñòâóåò ñáëèæåíèþ ñâîèõ êîíöîâ. Ïðî òàêóþ ñâÿçü ãîâîðÿò, ÷òî îíà ðàáîòàåò òîëüêî íà ðàñòÿæåíèå. Ó÷åò îäíîñòîðîííèõ ñâÿçåé ïðèâîäèò ê ïîÿâëåíèþ íåðàâåíñòâ â ìàòåìàòè÷åñêîé ìîäåëè ìåõàíè÷åñêîé ñèñòåìû. Åñëè ïðèíÿòü ïðàâèëî çíàêîâ äëÿ ïåðåìåùåíèÿ ïî íàïðàâëåíèþ îäíîñòîðîííåé ñâÿçè Z è óñèëèÿ â íåé X òàêèì, ÷òî X > 0 Z > 0 òî óñëîâèÿ ðàáîòû â ñâÿçè ìîæíî çàïèñàòü â âèäå: (1) X > 0 Z > 0 XZ = 0 Ïðèìåíåíèå îäíîãî èç èçâåñòíûõ ìåòîäîâ ñòðîèòåëüíîé ìåõàíèêè (ìåòîäà ñèë) ê ðàññìàòðèâàåìîé ìåõàíè÷åñêîé ñèñòåìå äàåò ñëåäóþùèå ñîîòíîøåíèÿ: ;

;

Z

:

;

N

=

=

sX

;

+

S;

> s
+ A;

aX

(2)

>0 >0 >0
>0
=0

=0 (3) Çäåñü âåêòîðû , èìåþò ñìûñë óñèëèé â ñâÿçÿõ ðàññìàòðèâàåìîé ìåõàíè÷åñêîé ñèñòåìû, à â âåêòîðàõ è
çàïèñàíû ïåðåìåùåíèÿ, ñîïðÿæåííûå ïî ðàáîòå ñ è ñîîòâåòñòâåííî. Óðàâíåíèÿ (2) è (3) ÿâëÿþòñÿ óñëîâèÿìè Êóíà-Òàêêåðà (íåîáõîäèìûìè è äîñòàòî÷íûìè óñëîâèÿìè ñóùåñòâîâàíèÿ ðåøåíèÿ) äëÿ ñëåäóþùåé êâàäðàòè÷íîé çàäà÷è: X

X

X

;

N

;

Z

Z

;

;

N

:

N

Z

X

f 12

N

+ j = + >0 >0g (4) Ìàòðèöà â ïðîãðàììå (4) ÿâëÿåòñÿ ñèììåòðè÷íîé ïîëîæèòåëüíî îïðåäåëåííîé, ÷òî ïîçâîëÿåò ïðèìåíÿòü äëÿ ðåøåíèÿ ýòîé çàäà÷è ëþáîé ìåòîä êâàäðàòè÷íîãî ïðîãðàììèðîâàíèÿ. Ïðåäïî÷òåíèå, îäíàêî, áûëî îòäàíî ìåòîäó Âóëà. Âî ïåðâûõ,  ýòî òî÷íûé ìåòîä, êîòîðûé ïîçâîëÿåò ïðèéòè ê öåëè çà êîíå÷íîå ÷èñëî øàãîâ. Âî-âòîðûõ, îí èñïîëüçóåò óäîáíóþ âû÷èñëèòåëüíóþ ñõåìó ñèìïëåêñíîãî ìåòîäà. min

a

X aX

AX

N

sX

S;

N

;

X

:

208

Ì.Ï. Ùåðáàê

Ìåòîä Âóëà ñîñòîèò èç òðåõ ýòàïîâ, íà êàæäîì èç êîòîðûõ âûïîëíÿåòñÿ ïðèìåðíî îäèíàêîâûé îáúåì âû÷èñëåíèé. Íà ïåðâûõ äâóõ ýòàïàõ îñóùåñòâëÿåòñÿ ïîèñê áàçèñíîãî ðåøåíèÿ, à íà ïîñëåäíåì  ñîáñòâåííî ðåøåíèÿ çàäà÷è. Ïðè ýòîì âåçäå ïðèìåíÿåòñÿ àëãîðèòì ñèìïëåêñ ìåòîäà ñ ó÷åòîì äîïîëíèòåëüíîãî ïðàâèëà âûáîðà âåäóùåãî ýëåìåíòà, êîòîðîå ãîâîðèò î òîì, ÷òî îáå âåëè÷èíû ïàðû X Z íå ìîãóò áûòü áàçèñíûìè îäíîâðåìåííî. Åñëè äëÿ ðåøàåìîé ñèñòåìû ìîæíî áûëî áû óêàçàòü áàçèñíîå ðåøåíèå çàðàíåå, òî ýòî ïîçâîëèëî áû ñîêðàòèòü òðåáóåìûé îáúåì âû÷èñëåíèé â íåñêîëüêî ðàç.  ðàáîòå [1℄ ïîêàçàíà òàêàÿ âîçìîæíîñòü äëÿ ðàññìàòðèâàåìîãî ñëó÷àÿ. Äëÿ ýòîãî îñóùåñòâëÿåòñÿ ïåðåõîä ê äâîéñòâåííîé çàäà÷å: â êà÷åñòâå óíêöèè öåëè âûáèðàåòñÿ ïàðàìåòð p âíåøíåé íàãðóçêè íà êîíñòðóêöèþ. Åñëè â èñõîäíîé çàäà÷å òðåáîâàëîñü îòûñêàòü ìèíèìóì ýíåðãåòè÷åñêîé óíöèè ïðè êîñâåííûõ îãðàíè÷åíèÿõ íà ïàðàìåòð p (p 6 1), òî òåïåðü ñòàâèòñÿ çàäà÷à íà ìàêñèìóì p ïðè êîñâåííûõ îãðàíè÷åíèÿõ íà ýíåðãåòè÷åñêèõ óíêöèÿõ: (5) maxfp j  = p p > 0; p > 0g:  ñòàòüå [2℄ ïîëó÷åíû óñëîâèÿ Êóíà-Òàêêåðà äëÿ ýòîé ïðîãðàììû: i

i

= sX + pS  + S 0; Z = aX s>
+ pA + A0 ; (6) X > 0; Z > 0; N > 0;
> 0; X
Z = 0 N

= 0: Âåêòîðû A0 è S 0, ñîäåðæàùèå âåëè÷èíû íà÷àëüíûõ óñèëèé è ïåðåìåùåíèé â îäíîñòîðîííèõ ñâÿçÿõ, áëàãîäàðÿ âûáðàííîìó ïðàâèëó çíàêîâ, âñåãäà íåîòðèöàòåëüíû. Òàêèì îáðàçîì, ñèìïëåêñ-òàáëèöà, çàïèñàííàÿ äëÿ ïðîãðàììû (5), âñåãäà ðåàëèçóåò áàçèñíîå ðåøåíèå: ïðè íóëåâûõ çíà÷åíèÿõ ïåðåìåííûõ p, X íè îäíî èç óñëîâèé Êóíà-Òàêêåðà (6) íå îêàçûâàåòñÿ íàðóøåííûì. àçìåð ñèìïëåêñíîé òàáëèöû, íàä êîòîðîé ïðîèçâîäÿòñÿ âû÷èñëåíèÿ  (2 + n + l)  (2 + n + l), ãäå n è l  äëèíû âåêòîðîâ Z N

Ïàðàëëåëüíûå âû÷èñëåíèÿ

209

è N ñîîòâåòñòâåííî. Ïðè ýòîì âåëè÷èíà m = n + l ðàâíà ÷èñëó îäíîñòîðîííèõ ñâÿçåé â ñèñòåìå. Àëãîðèòì íå ïîçâîëÿåò îïðåäåëèòü çàðàíåå òðåáóåìûé îáúåì âû÷èñëåíèé, ò. ê. ýòî çàâèñèò íå òîëüêî îò ðàçìåðà, íî åùå è îò ñîäåðæàíèÿ ñèìïëåêñíîé òàáëèöû. Îäíàêî ïðàêòèêà ðàñ÷åòîâ ìåõàíè÷åñêèõ çàäà÷ ïîêàçàëà, ÷òî îáû÷íî äëÿ íàõîæäåíèÿ ðåøåíèÿ òðåáóåòñÿ âûïîëíèòü îêîëî m øàãîâ (îïåðàöèé èñêëþ÷åíèÿ àóññà-Æîðäàíà). Èç ýòîãî ñëåäóåò îöåíêà ñëîæíîñòè âñåãî àëãîðèòìà O(m3 ). Ïðè òàêîé òðóäîåìêîñòè âðåìåííûå çàòðàòû íà âû÷èñëåíèÿ çàìåòíû óæå ïðè m = 1000 äàæå ïðè èñïîëüçîâàíèè ñîâðåìåííûõ ïåðñîíàëüíûõ êîìïüþòåðîâ. Òàê, íà êîìïüþòåðå ñ ïðîöåññîðîì Intel Celeron 2100 ïîèñê ðåøåíèÿ çàäà÷è ñ m = 1000 çàíÿë 8 ñåêóíä, à çàäà÷è ñ m = 2000  óæå 55. Ïðè ýòîì íà àëãîðèòì Âóëà ïðèõîäèëîñü áîëåå 95% îò îáùåãî îáúåìà âû÷èñëåíèé. Ïîäîáíûå ïîêàçàòåëè íåïðèåìëåìû â ñëó÷àÿõ, êîãäà òðåáóåòñÿ âûïîëíèòü ñåðèþ ðàñ÷åòîâ, íàïðèìåð, ïðè àíàëèçå äâèæóùåéñÿ ñèñòåìû. Îñîáåííî òðåáîâàòåëüíûìè ê ñêîðîñòè âû÷èñëåíèé ÿâëÿþòñÿ ïðèëîæåíèÿ, âûïîëíÿþùèå ñèìóëÿöèþ äâèæåíèÿ ìåõàíè÷åñêîé ñèñòåìû â ðåæèìå ðåàëüíîãî âðåìåíè. Äëÿ òàêèõ ïðîãðàìì íåîáõîäèìî ïðåäóñìîòðåòü ñïîñîáû óâåëè÷åíèÿ ïðîèçâîäèòåëüíîñòè, ïðè÷åì ñòåïåíü óâåëè÷åíèÿ äîëæíà áûòü, ïî âîçìîæíîñòè, êîíèãóðèðóåìîé. Ïîäîáíóþ çàäà÷ó ìîæíî ðåøèòü ïðèáåãíóâ ê ïàðàëëåëüíûì âû÷èñëåíèÿì, ðåàëèçóåìûõ íà ìàñøòàáèðóåìûõ ñóïåðêîìïüþòåðàõ. Áûëî ðåøåíî ðàçðàáàòûâàòü àëãîðèòì äëÿ êëàñòåðîâ  íàèáîëåå äîñòóïíîãî òèïà ñóïåðêîìïüþòåðîâ, êîòîðûé îòíîñèòñÿ ê êëàññó ïàðàëëåëüíûõ êîìïüþòåðîâ ñ ðàñïðåäåëåííîé ïàìÿòüþ. Ïðåäëàãàåìûé àëãîðèòì ó÷èòûâàåò ãëàâíûå îñîáåííîñòè ïðîãðàìì äëÿ ïîäîáíûõ êîìïüþòåðîâ, òàêèå êàê:  äàííûå çàäà÷è ðàñïðåäåëÿþòñÿ ïî âû÷èñëèòåëüíûì óçëàì;  êàæäûé óçåë ïðîèçâîäèò âû÷èñëåíèÿ òîëüêî ñ ëîêàëüíûìè äàííûìè;  îáìåí äàííûìè ìåæäó óçëàìè îñóùåñòâëÿåòñÿ ÷åðåç ëîêàëüíóþ âû÷èñëèòåëüíóþ ñåòü ïîñðåäñòâîì ïðîãðàììíîãî èíòåðåéñà MPI.

210

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Ïåðâîå îçíà÷àåò, ÷òî âñå äàííûå çàäà÷è ïî âîçìîæíîñòè ðàâíîìåðíî ðàñïðåäåëÿþòñÿ ìåæäó óçëàìè êëàñòåðà. Êàæäûé èç íèõ ïîëó÷àåò òîëüêî ÷àñòü çàäà÷è  òàêóþ, ÷òîáû ñ íåé ìîæíî áûëî âûïîëíèòü ìàêñèìàëüíî âîçìîæíûé îáúåì âû÷èñëåíèé áåç îáðàùåíèÿ ê äàííûì äðóãèõ óçëîâ. Ýòî òðåáîâàíèå ñâÿçàíî ñ òåì, ÷òî îáðàùåíèå ê óäàëåííûì äàííûì íàïðÿìóþ íåâîçìîæíî, òàê êàê âû÷èñëèòåëüíûé óçåë êëàñòåðà íå èìååò ïðÿìîãî äîñòóïà ê ëîêàëüíîé ïàìÿòè äðóãîãî óçëà. Îáìåí äàííûìè îñóùåñòâëÿåòñÿ êîñâåííî, ïîñðåäñòâîì ïîñûëêè è ïðèåìà ñîîáùåíèé ÷åðåç ëîêàëüíóþ ñåòü.
÷àñòü âû÷èñëèòåëüíîé  îáû÷íîì àëãîðèòìå Âóëà áîëüøàÿ ìîùíîñòè ïðèõîäèòñÿ íà ìíîãîêðàòíóþ îïåðàöèþ èñêëþ÷åíèÿ àóññà-Æîðäàíà íàä ñèìïëåêñíîé òàáëèöåé. Îñòàâøóþñÿ ÷àñòü çàíèìàåò ïðîöåäóðà âûáîðà âåäóùèõ ýëåìåíòîâ.  ñòàòüå [3℄ áûë ïîëó÷åí

ïàðàëëåëüíûé àëãîðèòì, êîòîðûé

âûïîëíÿåò äâå

ýòè

ïðîöåäóðû è îòâå÷àåò òðåáîâàíèÿì, ïðèâåäåííûì âûøå. Äëÿ âûÿâëåíèÿ ïàððàëåëëèçìà â ñóùåñòâóþùåì àëãîðèòìå Âóëà áûëà ïðèìåíåíà îäíà èç ìåòîäèê, ïðåäëîæåííûõ Â. Â. Âîåâîäèíûì [4℄. Èíîðìàöèîííûé ãðà (ãðà, ïîêàçûâàþùèé ïîñëåäîâàòåëüíîñòü èíîðìàöèîííûõ ñâÿçåé) îäíîãî èòåðàöèîííîãî øàãà àëãîðèòìà ñîâìåùàëñÿ ñ ðàçíûìè âàðèàíòàìè ñõåì ðàñïðåäåëåííîãî õðàíåíèÿ äàííûõ. Ñðåäè ìíîæåñòâà âàðèàíòîâ áûë âûáðàí ñïîñîá õðàíåíèÿ ïî áëîêàì-ñòîëáöàì. Íà ðèñ. 1 ïîêàçàíà âûáðàííàÿ ñõåìà.

èñ. 1

 èíäåêñíûõ îáîçíà÷åíèÿõ àëãîðèòì ìîæåò áûòü çàïèñàí ñëå-

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Èç ðèñ. 1 è èç ïðèâåäåííîãî àëãîðèòìà âèäíî, ÷òî íà êàæäîì øàãå ìåòîäà Âóëà íà ëþáîì âû÷èñëèòåëüíîì óçëå èñïîëüçóåòñÿ òîëüêî îäèí íåëîêàëüíûé ýëåìåíò  ñòîëáåö (ýòîò ýëåìåíò ëîêàëåí òîëüêî äëÿ îäíîãî óçëà , äëÿ êîòîðîãî 1 6 6 ). Òàê êàê äàííûå, òðåáóåìûå âñåìè óçëàìè, îäèíàêîâû, îíè ïåðåäàþòñÿ çà îäèí ðàç, øèðîêîâåùàòåëüíî ïåðåä íà÷àëîì âû÷èñëèòåëüíîãî øàãà. Ïðîöåäóðà ïàðàëëåëüíîãî ïîèñêà âåäóùåãî ýëåìåíòà ïîñòðîåíà íà òåõ æå ðàññóæäåíèÿõ, ÷òî è ïðåäûäóùàÿ îïåðàöèÿ. ëàâíîå îòëè÷èå çàêëþ÷àåòñÿ â òîì, ÷òî çäåñü ðîëü ñòîëáöà âûïîëíÿåò ñòîëáåö 1 , ò. å. ñòîëáåö ñ íåëîêàëüíûìè äàííûìè èçâåñòåí çàðàíåå. Ýòî ïîçâîëÿåò ðàññûëàòü ñòîëáåö îäíîêðàòíî äî íà÷àëà âû÷èñëåíèé.  ïðîöåññå âû÷èñëåíèé íà âñåõ óçëàõ íàä íèì âûïîëíÿþòñÿ îäèíàêîâûå îïåðàöèè, â ðåçóëüòàòå ÷åãî ëþáîé ïðîöåññ ïîñòîÿííî èìååò ëîêàëüíûé äîñòóï ê òåêóùèì çíà÷åíèÿì âåêòîðà ñâîáîäíûõ ÷ëåíîâ. Òàêèì îáðàçîì óäàåòñÿ èçáåæàòü ìåæóçëîâûõ êîììóíèêàöèé, ñâÿçàííûõ ñ ïåðåñûëêîé ìàññèâà . ail

p

mp

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212

Ì.Ï. Ùåðáàê

èñ. 2

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213

Íà ðèñ. 2 ïðåäñòàâëåíà áëîê-ñõåìà ïàðàëëåëüíîé ÷àñòè ïðîãðàììû. Îíà íà÷èíàåòñÿ ñ âûçîâà óíêöèè MPI_INIT, êîòîðàÿ èíèöèàëèçèðóåò êîììóíèêàöèîííóþ ñðåäó. Ïîñëå åå âûçîâà êàæäûé ïðîöåññ êëàñòåðà ïîëó÷àåò óíèêàëüíûé íîìåð-àäðåñ, êîòîðûé â òåðìèíîãèè MPI íàçûâàåòñÿ íîìåðîì ïðîöåññà. Ñðàçó ïîñëå èíèöèàëèçàöèè êîììóíèêàöèîííîé ñðåäû ïðîèñõîäèò âåòâëåíèå ïðîãðàììû: ïðîãðàììà ãëàâíîãî ïðîöåññà (óçëà) ñ íîìåðîì 0 äâèæåòñÿ ïî ëåâîé âåòâè àëãîðèòìà (ñì. ðèñ. 2), à ïðîãðàììû îñòàëüíûõ  ïî ïðàâîé. Ïåðåäà÷à ñîîáùåíèé ïðîèñõîäèò â îñíîâíîì ìåæäó ïðàâîé è ëåâîé âåòâÿìè àëãîðèòìà. Íà áëîêñõåìå ìåæóçëîâîé îáìåí äàííûìè ïîêàçàí ïóíêòèðíûìè ëèíèÿìè. Íàä êàæäîé èç íèõ ïðèâåäåíî èìÿ MPI-óíêöèè, ñ ïîìîùüþ êîòîðîé îñóùåñòâëÿåòñÿ ýòîò îáìåí. Îïèñàííûé âûøå àëãîðèòì áûë ðåàëèçîâàí â ïðîãðàììå ðàñ÷åòà ñèñòåì àáñîëþòíî æåñòêèõ òåë è ïîäàòëèâûõ îäíîñòîðîííèõ ñâÿçåé.  ñòàòüå [3℄ îïèñàí ÷èñëåííûé ýêñïåðèìåíò ñ ýòîé ðåàëèçàöèåé, ïî ðåçóëüòàòàì êîòîðîãî áûëè ïîñòðîåíû ãðàèêè ïðîèçâîäèòåëüíîñòè è ìàñøòàáèðóåìîñòè âû÷èñëèòåëüíîé ñèñòåìû. Âû÷èñëåíèÿ ïðîèçâîäèëèñü íà êëàñòåðå, ñîçäàííîì íà áàçå îäíîãî èç êîìïüþòåðíûõ êëàññîâ Ïåòðîçàâîäñêîãî óíèâåðñèòåòà. Êëàñòåð ñîñòîèò èç 6 êîìïüþòåðîâ, êîòîðûå èìåþò ñëåäóþùóþ êîíèãóðàöèþ: ïðîöåññîð  Intel Celeron 2100, îïåðàòèâíàÿ ïàìÿòü  256 Mb (DDR), ñåòåâîé àäàïòåð  Fast Ethernet 100 Mbit, ÎÑ  Linux Mandrake 10.0, áèáëèîòåêè MPI  MPICH 1.2.5, òîïîëîãèÿ ñåòè  çâåçäà.

èñ. 3

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Ì.Ï. Ùåðáàê

 êà÷åñòâå òåñòîâîé çàäà÷è áûëà âçÿòà çàäà÷à ñòðîèòåëüíîé ìåõàíèêè, â êîòîðîé òðåáóåòñÿ îïðåäåëèòü óñèëèÿ â îäíîñòîðîííèõ ñâÿçÿõ ìåõàíè÷åñêîé ñèñòåìû. Îíà ñîñòîèò èç äâóõ àáñîëþòíî æåñòêèõ òåë, êîòîðûå ñîåäèíåíû ìåæäó ñîáîé îäíîñòîðîííèìè ïîäàòëèâûìè ñâÿçÿìè, ðàáîòàþùèìè òîëüêî íà ðàñòÿæåíèå. Ñâÿçè óñòàíàâëèâàþòñÿ ñëó÷àéíûì îáðàçîì, à êîëè÷åñòâî èõ ïðîèçâîëüíî. Âñåãî áûëî ñãåíåðèðîâàíî 9 ðàñ÷åòíûõ ñõåì ñ ðàçíûì ÷èñëîì ñâÿçåé. Âñå òåñòîâûå ñèñòåìû áûëè ðàñ÷èòàíû â ÷åòûðåõ êîíèãóðàöèÿõ êëàñòåðà: 1 ðàáîòàþùèé óçåë, 2, 4 è 6 óçëîâ.  êàæäîì ýêñïåðèìåíòå èçìåðÿëîñü âðåìÿ, çàòðà÷åííîå íà âû÷èñëåíèÿ. Ïî ýòèì äàííûì áûëè ïîñòðîåíû ãðàèêè ïðîèçâîäèòåëüíîñòè è ìàñøòàáèðóåìîñòè. Ïðîèçâîäèòåëüíîñòü îäíîóçëîâîé óñòàíîâêè, äîñòèãíóâ íàñûùåíèÿ ïðè n > 1000, êîëåáàëàñü îêîëî îòìåòêè 430 MFLOPS. Ïðè ýòîì òåîðåòè÷åñêàÿ ïèêîâàÿ ïðîèçâîäèòåëüíîñòü ïðîöåññîðà, óñòàíîâëåííîãî âî âñåõ óçëàõ (Intel Celeron 2100), ðàâíà 2100 MFLOPS. åàëüíàÿ ñêîðîñòü âû÷èñëåíèé, òàêèì îáðàçîì, íà îäíîì ïðîöåññîðå ñîñòàâèëà âñåãî ëèøü 20,4% îò òåîðåòè÷åñêîé. Ïîäîáíûé ïîêàçàòåëü ìîæíî ñ÷èòàòü íîðìàëüíûì äëÿ íåîïòèìèçèðîâàííîãî êîäà, îïåðèðóþùåãî ñ áîëüøèìè ìàññèâàìè âåùåñòâåííûõ ÷èñåë. Äëÿ ñðàâíåíèÿ, â òåñòå LAPACK, ñîáðàííîì ñ íåîïòèìèçèðîâàííîé referen e-âåðñèåé áèáëèîòåêè íèçêîóðîâíåâûõ óíêöèé BLAS, ðåàëüíàÿ ïðîèçâîäèòåëüíîñòü îäíîãî ïðîöåññîðà â íåêîòîðûõ ðàñ÷åòàõ íèæå ïèêîâîé â 3 ðàçà. Ñ ïîìîùüþ îïòèìèçàöèè, ó÷èòûâàþùåé àðõèòåêòóðó âñåãî âû÷èñëèòåëüíîãî óçëà, ñòðîåíèå åãî ïðîöåññîðà, ðàçìåðû êýøåé è ò. ï., ðåàëüíóþ ñêîðîñòü óäàåòñÿ äîâåñòè äî 80  100% îò òåîðåòè÷åñêîé.

èñ. 4

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Âòîðûì ïîñëå ïðîèçâîäèòåëüíîñòè âàæíûì ïîêàçàòåëåì ýåêòèâíîñòè

êëàñòåðíûõ

ñèñòåì

ÿâëÿåòñÿ

ìàñøòàáèðóåìîñòü.

Ýòà õàðàêòåðèñòèêà ïîêàçûâàåò ñïîñîáíîñòü êëàñòåðà íàðàùèâàòü ñâîþ ïðîèçâîäèòåëüíîñòü â îäíîé è òîé æå çàäà÷å ñ ïîäêëþ÷åíèåì ê âû÷èñëåíèÿì íîâûõ óçëîâ. Îáû÷íî ìàñøòàáèðóåìîñòü ïðåäñòàâëÿþò â âèäå ãðàèêà çàâèñèìîñòè ïðîèçâîäèòåëüíîñòè îò ðàçìåðà êëàñòåðà (êîëè÷åñòâà óçëîâ). ÿäîì ñ ýòîé êðèâîé â òåõ æå îñÿõ ñòðîÿò ãðàèê ñóììû ïðîèçâîäèòåëüíîñòåé óçëîâ êëàñòåðà, êîòîðûé ñ÷èòàåòñÿ ëèíèåé èäåàëüíîé ìàñøòàáèðóåìîñòè. åçóëüòàòû ïðîâåäåííûõ òåñòîâ ïîêàçûâàþò ïðåèìóùåñòâà èñïîëüçîâàíèÿ ïàðàëëåëüíûõ âû÷èñëåíèé è ãîâîðÿò î òîì, ÷òî ïðåäëîæåííûé ïàðàëëåëüíûé àëãîðèòì ìîæåò áûòü ýåêòèâíî ïðèìåíåí äëÿ ðåøåíèÿ áîëüøèõ çàäà÷. Åãî ðåàëèçàöèÿ ïîêàçàëà õîðîøóþ ìàñøòàáèðóåìîñòü íà îáîðóäîâàíèè, íå ïðåäíàçíà÷åííîì äëÿ ïîñòðîåíèÿ êëàñòåðîâ, ÷òî óêàçûâàåò íà âûñîêèå ¾ïàðàëëåëüíûå¿ êà÷åñòâà àëãîðèòìà. Ïðèìåíåíèå ñîâðåìåííîé àïïàðàòíîé áàçû ïîçâîäèò èñïîëüçîâàòü ýòè êà÷åñòâà â ïîëíîé ìåðå è ñäåëàåò âîçìîæíûì ïðèìåíåíèå ïðåäëîæåííîãî àëãîðèòìà äëÿ ðåøåíèÿ çàäà÷ ñ î÷åíü áîëüøîé ðàçìåðíîñòüþ.

Ëèòåðàòóðà 1. Ïåðåëüìóòåð À. Â. Èñïîëüçîâàíèå ìåòîäîâ êâàäðàòè÷íîãî ïðîãðàììèðîâàíèÿ äëÿ ñèñòåì ñ îäíîñòîðîííèìè ñâÿçÿìè // Èññëåäîâàíèÿ ïî òåîðèè ñîîðóæåíèé. Âûï. XIX. Ì.: Ñòðîéèçäàò, 1972, ñ. 138147. 2. îëüäøòåéí Þ. Á., Ùåðáàê Ì. Ï. àçðåøàþùèå óðàâíåíèÿ äëÿ ñèñòåìû æåñòêèõ òåë è ïîäàòëèâûõ ñâÿçåé // óêîïèñü äåï. â ÂÈÍÈÒÈ 02.04.2003, 587-Â2003. 3. Ùåðáàê Ì. Ï. Ïàðàëëåëüíûé àëãîðèòì ìåòîäà Âóëà äëÿ ñóïåðêîìïüþòåðîâ êëàñòåðíîé àðõèòåêòóðû // óêîïèñü äåï. â ÂÈÍÈÒÈ 17.12.2005, 1609-Â2005. 4. Âîåâîäèí Â. Â., Âîåâîäèí Âë. Â. Ïàðàëëåëüíûå âû÷èñëåíèÿ. ÑÏá.: ÁÕÂÏåòåðáóðã, 2002.

ABSTRACTS

METHODS OF MATHEMATICAL MODELLING Abakumov Y.G., Banin M.V. On sets of convergences guaranteed by test type theorems. The sets in the space C[a, b] on which the sequence of some linear operator classes is convergent on the test set implies that the sequence is convergent on the whole set. It is stated that these sets do not increase with the preservation of convergences. The general scheme for the sequence of linear functionals in the Banach space is considered. Ivanov A.P., Ostov Yu.Ya. Duality and principle of expansion in the problem of dynamics of flight. The paper considers the problem of maximizing kinetic energy in the end point of the path of a flying object. The solution method is construction of sub-optimal control. Popov V.V. Modelling of heat-mass transfer processes in hydride accumulator of hydrogen. The one-dimensional model of heat-mass transfer processes in hydride accumulator of hydrogen is considered. The model coefficients depend on temperature and concentration. The chemical reactions of hydrogenation and dehydrogenation are taken into account. Popov V.V., Semin P.A. Implementation and use of modelling problems of hydrogen interaction with solids in special software. The practical aspects of implementation and use of various models of hydrogen interaction with solids in special software are considered. Section I.

Stafeev S.V. On the parameter identifiability of a factor analysis model with correlated residuals and correlated factors. A factor analysis model with correlated residuals and correlated factors is considered. Conditions of local parameter identifiability are obtained and the class of models satisfying these conditions is found.

Tararin V.M.,

Cheplyukova I.A.

mapping of a finite Boolean lattice.

On the action of a random s

217

smapping of a finite Boolean lattice L is the upper endomorphism of lattice Ln such that from h(x) 6 1 follows h(ϕ(x)) = h(x). In the paper we consider the action of a random smapping of a finite Boolean lattice with a length n as n → ∞. Falko A.A. A best-choice game with the possibility of an applicant refusing an offer and with redistribution of probabilities. A full-information best-choice game is considered. The optimal strategies in case of the possibility of an applicant refusing an offer and the redistribution of probabilities are obtained. Khvorostyanskaya E.V. On emergence of the giant ϕprimitive ideal of a random automorphism ϕ of a finite Boolen lattice. We consider the set An,N of all automorphisms ϕ of a Boolen lattice with a length n such that the number of all ϕ primitive ideals is equal to N . We specify the uniform distribution on An,N . It is proved that the giant ϕprimitive ideal of a random automorphism ϕ ∈ An,N can occur only if n, N → ∞ so that N/ ln n → 0. Chernov I.A. Convergence of the lattice approximations of the solution to the boundary-value problem with nonlinear boundary condition and moving bound. In the paper we consider the Stefan-type boundary-value problem with moving bound and nonlinear boundary condition. It appears as a model of hydriding under constant conditions. We√ present the numerical method; it provides lattice approximations of h precision, h is for the lattice step. We prove that these approximations converge to a weak continuous solution to the PDE; this solution also satisfies the boundary conditions. The moving bound belongs to C1 . So we can also claim the existence of a weak solution to the considered problem. ϕ

COMPUTER SCIENCE Borodina A.V., Morozov E.V. Overflow probability estimation with accelerated regeneration simulation in the queue M/M/1. This paper is devoted to a modification of speed-up method Splitting. This approach allows to use the regenerative simulation technics to construct the confidence interval for the overflow probability. Section II.

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On an "Internet" type, random graph model. Random graphs, where vertex degrees are independent random variables are considered. The distribution of the vertex degrees follows the P{D = d} = d−τ /ζ(τ ) law, wherePD is the random variable −τ denoting the vertex degree, ζ(τ ) = ∞ is Riemann zeta d=1 d function, d = 1, 2, . . ., τ ∈ (2, 3) (finite mean and infinite variance). Such graphs can be used as models for Internet topology. Limit theorems for the extreme terms of the variation series formed from the nondecreasing sequence of the vertex degrees and for the sum of the node degrees when N → ∞ are proved. Kukin V.D. A composite evolutionary algorithm for solving the weighted Steiner tree problem. A composite evolutionary algorithm for solving the weighted Steiner tree problem is suggested. Main results of testing the algorithm are reported. Lazutina A.A. On the problem of optimal control of three parallel stacks in bounded space. The paper considers issues related to building mathematical models and optimal algorithms of consecutive and connected three parallel stacks control in a single-level memory. These models were constructed as random walks. Pechnikov A.A. The problem of rational allocation of links in a controlled and localized Internet-resource system. The problem of rational allocation of cross-links to Internetresource within a some localized system is considered. Each participant is obliged to make a certain number of links to other participants to increase their importance so that the final values of the importance differed from the value averaged for all participants of the system as little as possible (a case of a "social" decision). Rodchenkova N.I. A differences scheme for modelling the TDSspectrum of dehydriding of metals with heat absorption and a size reduction effect. In the paper we consider the problem with moving bounds that models the kinetics of hydrogen desorption from hydrides of metals. Phase change, size reduction, desorption processes on the surface, and heat absorption on the phase bound are taken into consideration. The differences scheme for the corresponding boundary-value problem Kiseleva N.S.

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with moving bounds is presented for the experimental method of thermal desorption spectrometry (TDS). The results of numerical modeling are presented. Rettieva A.N.,Rodionov A.V. The model of dynamic game of forest resources management problem is constructed. The equilibrium controls of the participants are received. The numerical modeling for parameters is carried out according to statistics on Karelia. Shcherbak M.P. Parallel computing in application to analysis of some problems in mechanics. One problem of mechanics is solved in this paper by means of parallel computing. More precisely, in this paper one can find a description of the analysis of a mechanical system composed of rigid bodies and unilateral constrains. This analysis is performed with the help of parallel computing. Defining relations of this problem related to the quadratic programming problem, that is solved by Vulf method. Parallel calculations are executed on the cluster which is the most available type of scalable parallel computer.

Научное издание МЕТОДЫ МАТЕМАТИЧЕСКОГО МОДЕЛИРОВАНИЯ И ИНФОРМАЦИОННЫЕ ТЕХНОЛОГИИ

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Печатается по решению Ученого совета института прикладных математических исследований Карельского научного центра РАН

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