Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания
Розглянуто систему обслуговування M/M/1/0 з повторенням та комбінованою дисципліною обслуговування, а саме: заявки з орбіти обслуговуються у порядку черги, але при наявності вільного каналу заявка, що прийшла з первинного потоку, негайно відправляється на обслуговування.. Виведено формули для ймовір...
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Інститут кібернетики ім. В.М. Глушкова НАН України
2017
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| Schriftenreihe: | Кибернетика и системный анализ |
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| Zitieren: | Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания / Е.В. Коба // Кибернетика и системный анализ. — 2017. — Т. 53, № 3. — С. 67–72. — Бібліогр.: 18 назв. — рос. |
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irk-123456789-1447302019-01-03T01:23:12Z Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания Коба, Е.В. Системний аналіз Розглянуто систему обслуговування M/M/1/0 з повторенням та комбінованою дисципліною обслуговування, а саме: заявки з орбіти обслуговуються у порядку черги, але при наявності вільного каналу заявка, що прийшла з первинного потоку, негайно відправляється на обслуговування.. Виведено формули для ймовірностей станів і умову ергодичності. Проведено порівняння системи, що розглядається, з чистою системою Лакатоша. Рассмотрена система обслуживания M/M/1/0 с повторением и комбинированной дисциплиной обслуживания, а именно: заявки с орбиты обслуживаются в порядке очереди, но при наличии свободного канала заявка, пришедшая с первичного потока, сразу отправляется на обслуживание. Выведены формулы для вероятностей состояний и условие эргодичности. Проведено сравнение рассматриваемой системы с чистой системой Лакатоша. The paper considers the retrial queueing system M/M/1/0 with combined discipline of service, namely, a customer from the orbit is served in its turn, but in case of a free channel an arrival from the original flow is serviced immediately. The author obtained the expressions for state probabilities as well as the ergodicity conditions. The system is compared with the Lakatos type system. 2017 Article Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания / Е.В. Коба // Кибернетика и системный анализ. — 2017. — Т. 53, № 3. — С. 67–72. — Бібліогр.: 18 назв. — рос. 0023-1274 http://dspace.nbuv.gov.ua/handle/123456789/144730 519.872 ru Кибернетика и системный анализ Інститут кібернетики ім. В.М. Глушкова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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Russian |
| topic |
Системний аналіз Системний аналіз |
| spellingShingle |
Системний аналіз Системний аналіз Коба, Е.В. Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания Кибернетика и системный анализ |
| description |
Розглянуто систему обслуговування M/M/1/0 з повторенням та комбінованою дисципліною обслуговування, а саме: заявки з орбіти обслуговуються у порядку черги, але при наявності вільного каналу заявка, що прийшла з первинного потоку, негайно відправляється на обслуговування.. Виведено формули для ймовірностей станів і умову ергодичності. Проведено порівняння системи, що розглядається, з чистою системою Лакатоша. |
| format |
Article |
| author |
Коба, Е.В. |
| author_facet |
Коба, Е.В. |
| author_sort |
Коба, Е.В. |
| title |
Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания |
| title_short |
Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания |
| title_full |
Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания |
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Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания |
| title_full_unstemmed |
Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания |
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система обслуживания м /м /1/0 с повторением и комбинированной дисциплиной обслуживания |
| publisher |
Інститут кібернетики ім. В.М. Глушкова НАН України |
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2017 |
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Системний аналіз |
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http://dspace.nbuv.gov.ua/handle/123456789/144730 |
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Система обслуживания М /М /1/0 с повторением и комбинированной дисциплиной обслуживания / Е.В. Коба // Кибернетика и системный анализ. — 2017. — Т. 53, № 3. — С. 67–72. — Бібліогр.: 18 назв. — рос. |
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Кибернетика и системный анализ |
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| fulltext |
ÓÄÊ 519.872
Å.Â. ÊÎÁÀ
ÑÈÑÒÅÌÀ ÎÁÑËÓÆÈÂÀÍÈß M /M /1/0 Ñ ÏÎÂÒÎÐÅÍÈÅÌ
È ÊÎÌÁÈÍÈÐÎÂÀÍÍÎÉ ÄÈÑÖÈÏËÈÍÎÉ ÎÁÑËÓÆÈÂÀÍÈß
Àííîòàöèÿ. Ðàññìîòðåíà ñèñòåìà îáñëóæèâàíèÿ M /M /1/0 ñ ïîâòîðåíèåì è
êîìáèíèðîâàííîé äèñöèïëèíîé îáñëóæèâàíèÿ, à èìåííî: çàÿâêè ñ îðáèòû îá-
ñëóæèâàþòñÿ â ïîðÿäêå î÷åðåäè, íî ïðè íàëè÷èè ñâîáîäíîãî êàíàëà çàÿâêà,
ïðèøåäøàÿ ñ ïåðâè÷íîãî ïîòîêà, ñðàçó îòïðàâëÿåòñÿ íà îáñëóæèâàíèå. Âûâå-
äåíû ôîðìóëû äëÿ âåðîÿòíîñòåé ñîñòîÿíèé è óñëîâèå ýðãîäè÷íîñòè. Ïðîâå-
äåíî ñðàâíåíèå ðàññìàòðèâàåìîé ñèñòåìû ñ ÷èñòîé ñèñòåìîé Ëàêàòîøà.
Êëþ÷åâûå ñëîâà: ñèñòåìà îáñëóæèâàíèÿ, ñèñòåìà îáñëóæèâàíèÿ ñ ïîâòîðå-
íèåì, îðáèòà, ñèñòåìà îáñëóæèâàíèÿ ñ öèêëè÷åñêèì âðåìåíåì îæèäàíèÿ,
êîìáèíèðîâàííàÿ äèñöèïëèíà îáñëóæèâàíèÿ, óñëîâèå ýðãîäè÷íîñòè ñèñòåìû.
ÂÂÅÄÅÍÈÅ
Ñèñòåìû ìàññîâîãî îáñëóæèâàíèÿ (ÑÌÎ) ñ ïîâòîðåíèåì çàÿâîê ñòàëè àêòèâíî
èçó÷àòüñÿ, íà÷èíàÿ ñ 80-õ ãîäîâ ïðîøëîãî ñòîëåòèÿ. Äëÿ íèõ õàðàêòåðíî ñëåäó-
þùåå: åñëè ïðèøåäøàÿ çàÿâêà ïîñòóïàåò â ïåðåïîëíåííóþ ñèñòåìó, ò.å. îáñëó-
æèâàþùèé êàíàë (èëè îáñëóæèâàþùèå êàíàëû) è âñå ìåñòà â î÷åðåäè (ïðè åå
íàëè÷èè â ñèñòåìå) çàíÿòû, îíà ïîêèäàåò ñèñòåìó, íî íå òåðÿåòñÿ, à ÷åðåç íåêî-
òîðîå âðåìÿ ïîâòîðÿåò ñâîþ ïîïûòêó ïîïàñòü íà îáñëóæèâàíèå (â êàíàë èëè
â î÷åðåäü).  ýòîì ñëó÷àå âîçíèêàåò íåóïîðÿäî÷åííàÿ î÷åðåäü ïîâòîðíûõ çàÿ-
âîê, êîòîðóþ ïðèíÿòî íàçûâàòü îðáèòîé. Â íàñòîÿùåå âðåìÿ â ñâÿçè ñ ðàçâèòè-
åì èíôîðìàöèîííûõ òåõíîëîãèé ñèñòåìû îáñëóæèâàíèÿ ñ ïîâòîðåíèåì çàÿâîê
êàê ìîäåëè ôóíêöèîíèðîâàíèÿ èíôîðìàöèîííûõ ñèñòåì ïîëó÷èëè øèðîêîå ðàñ-
ïðîñòðàíåíèå.
Íàèáîëåå îáùàÿ ìîäåëü ÑÌÎ ñ ïîâòîðåíèåì (âîçâðàùåíèåì) çàÿâîê ïðåä-
ñòàâëåíà â îáçîðíîé ñòàòüå T. ßíãà è Äæ. Teìïëòîíà [1]. Êëàññèôèêàöèÿ ÑÌÎ
ñ âîçâðàùåíèåì çàÿâîê ââåäåíà â [2]. Ñðàâíèòåëüíûé àíàëèç êëàññè÷åñêèõ ñèñ-
òåì îáñëóæèâàíèÿ è ñèñòåì îáñëóæèâàíèÿ ñ âîçâðàùåíèåì çàÿâîê ïðîâåäåí
â [3], à ñòàòüè [4, 5] — ýòî èñ÷åðïûâàþùèå áèáëèîãðàôèè ïî èññëåäîâàíèþ
òàêèõ ñèñòåì.
 Óêðàèíå âàæíûå ðåçóëüòàòû îòíîñèòåëüíî ñèñòåì ñ ïîâòîðíûìè çàÿâêàìè
áûëè ïîëó÷åíû Â.Â. Àíèñèìîâûì [6, 7], È.Í. Êîâàëåíêî [8, 9], Å.Â. Êîáîé [10],
Å.À. Ëåáåäåâûì [11, 12] è èõ ó÷åíèêàìè. Íàïðàâëåíèå èõ èññëåäîâàíèÿ — ýðãî-
äè÷íîñòü íåìàðêîâñêèõ ÑÌÎ ñ âîçâðàùåíèåì, ñèñòåìû ñ öèêëè÷åñêèì âðåìåíåì
îæèäàíèÿ (ñèñòåìû òèïà Ëàêàòîøà), ìíîãîöèêëè÷åñêèå ñèñòåìû, ñèñòåìû ñ ïå-
ðåìåííûìè ïàðàìåòðàìè, òåëåôîííûå ñèñòåìû (â ÷àñòíîñòè, Call-öåíòðû),
ñèñòåìû ñ ðàçíîòèïíûìè çàÿâêàìè, òðàíñïîðòíûå ñèñòåìû è ò.ä.
ÎÁÙÅÅ ÎÏÈÑÀÍÈÅ ÑÈÑÒÅÌÛ
 1994 ã. â æóðíàëå íàó÷íûõ ðàáîò óíèâåðñèòåòà èì. Ëîðàíäà Ýòâåøà (ã. Áó-
äàïåøò) áûëà îïóáëèêîâàíà ñòàòüÿ âåíãåðñêîãî ìàòåìàòèêà Ëàñëî Ëàêàòîøà,
â êîòîðîé èçó÷àëèñü ñèñòåìû îáñëóæèâàíèÿ ñ ïîâòîðåíèåì è äèñöèïëèíîé îá-
ñëóæèâàíèÿ FIFO (ïåðâûé ïðèøåë — ïåðâûì îáñëóæèëñÿ) [13,14]. Ýòè ñèñòå-
ìû ïîëó÷èëè íàçâàíèå ñèñòåìû ñ öèêëè÷åñêèì âðåìåíåì îæèäàíèÿ èëè ñèñòå-
ìû òèïà Ëàêàòîøà.
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2017, òîì 53, ¹ 3 67
© Å.Â. Êîáà, 2017
 ýòèõ ñèñòåìàõ â îòëè÷èå îò êëàññè÷åñêèõ ÑÌÎ ñ ïîâòîðåíèåì âíîâü ïðèøåä-
øàÿ çàÿâêà ïðè íàëè÷èè çàÿâîê íà îðáèòå íå ìîæåò ïîïàñòü íà îáñëóæèâàíèå â ñâî-
áîäíûé êàíàë; îíà çàíèìàåò î÷åðåäü íà îðáèòå è îáñëóæèâàåòñÿ â ïîðÿäêå î÷åðåäè.
Ïóñòü, íàïðèìåð, tn — n-é ìîìåíò ïîÿâëåíèÿ çàÿâêè â ñèñòåìå. Åñëè n-ÿ çà-
ÿâêà ïîñòóïàåò â ñâîáîäíóþ ñèñòåìó (ò.å. çàÿâêè îòñóòñòâóþò êàê â êàíàëå îáñëó-
æèâàíèÿ, òàê è íà îðáèòå), îíà ñðàçó íàïðàâëÿåòñÿ â êàíàë îáñëóæèâàíèÿ. Åñëè
æå êàíàë çàíÿò, òî n-ÿ çàÿâêà îòïðàâëÿåòñÿ íà îðáèòó è âîçâðàùàåòñÿ îòòóäà ÷åðåç
ñëó÷àéíîå (â îáùåì ñëó÷àå) âðåìÿ, ïðåäïðèíèìàÿ ïîïûòêó îáñëóæèòüñÿ. Ïðè ýòî
åñëè êàíàë ñâîáîäåí, òî îíà îáñëóæèâàåòñÿ.  ïðîòèâíîì ñëó÷àå çàÿâêà âîçâðà-
ùàåòñÿ íà îðáèòó è ïðîöåññ ïîâòîðÿåòñÿ. Òàêèì îáðàçîì, çàÿâêè ìîãóò ïîñòóïàòü
íà îáñëóæèâàíèå ñ îðáèòû â ìîìåíòû t tn n n n n� � �� � �1 1 2, , ... , ãäå
�nk k, , ,�1 2 3, ... , — íåçàâèñèìûå îäèíàêîâî ðàñïðåäåëåííûå ñëó÷àéíûå âåëè÷è-
íû. Â êàíàë îáñëóæèâàíèÿ n-ÿ çàÿâêà ïîñòóïàåò ñðàçó æå ïîñëå âîçâðàùåíèÿ
ñ îðáèòû, íî íå ðàíüøå, ÷åì îáñëóæèòñÿ ( )n � �1 ÿ çàÿâêà.
Ë. Ëàêàòîø ðàññìàòðèâàë ýòè ñèñòåìû â ñâÿçè ñ ìîäåëèðîâàíèåì ïðîöåññà
ïîñàäêè âîçäóøíîãî ñóäíà, ãäå êàíàë îáñëóæèâàíèÿ — âçëåòíî-ïîñàäî÷íàÿ ïî-
ëîñà, à îðáèòà — çîíà îæèäàíèÿ.
Ñèñòåìû îáñëóæèâàíèÿ òèïà Ëàêàòîøà ïîëó÷èëè ðàçâèòèå â íàó÷íûõ
òðóäàõ È.Í. Êîâàëåíêî [15] è Å.Â. Êîáû [16]. Ñëåäóåò îòìåòèòü, ÷òî â íàñòî-
ÿùåå âðåìÿ ñèñòåìû òèïà Ëàêàòîøà èçó÷àþòñÿ òàêæå êàê ìîäåëè îïòè÷åñêèõ
áóôåðîâ [ñì., íàïðèìåð 17].
Ñëåäóåò îòìåòèòü, îäíàêî, ÷òî ÷èñòàÿ ìîäåëü Ëàêàòîøà íå âñåãäà ðàöèî-
íàëüíà. Âîçìîæíà ñèòóàöèÿ, êîãäà âðåìÿ ïðåáûâàíèÿ íà öèêëå îðáèòû çíà÷è-
òåëüíî áîëüøå, ÷åì âðåìÿ îáñëóæèâàíèÿ â êàíàëå, íàïðèìåð â àâèàöèè, êîãäà
âðåìÿ ïðåáûâàíèÿ âîçäóøíîãî ñóäíà íà êðóãå çîíû îæèäàíèÿ (öèêë îðáèòû)
çíà÷èòåëüíî ïðåâûøàåò âðåìÿ ïðåáûâàíèÿ âîçäóøíîãî ñóäíà ïðè ïîñàäêå íà
âçëåòíî-ïîñàäî÷íîé ïîëîñå (êàíàë îáñëóæèâàíèÿ). Âûäåëèì èç ìîäåëè Ëàêàòî-
øà ðàöèîíàëüíîå, à èìåííî: îáñëóæèâàíèå ñ îðáèòû áóäåò ïðîèñõîäèòü â ïîðÿä-
êå î÷åðåäè, íî ïðè ýòîì äîïóñêàåòñÿ âîçìîæíîñòü îáñëóæèâàíèÿ âíå î÷åðåäè
çàÿâîê èç ïåðâè÷íîãî ïîòîêà ïðè íàëè÷èè ñâîáîäíîãî êàíàëà.
Ðàññìîòðèì ñèñòåìó M M m/ / / 0 ñ âîçâðàùåíèåì è áåñêîíå÷íîé îðáèòîé, ò.å. âõî-
äÿùèé ïîòîê â ÑÌÎ — ïóàññîíîâñêèé (ïàðàìåòð �), âðåìÿ îáñëóæèâàíèÿ ðàñïðåäåëåíî
ïî ýêñïîíåíöèàëüíîìó çàêîíó (ïàðàìåòð �). Äàííàÿ ñèñòåìà, â îòëè÷èå îò ñèñòåìû Ëàêà-
òîøà, ôóíêöèîíèðóåò ñ âîçìîæíûì íàðóøåíèåì ïîðÿäêà îáñëóæèâàíèÿ FIFO â ñëó÷àå,
êîãäà â ìîìåíò ïîñòóïëåíèÿ çàÿâêè èç ïåðâè÷íîãî ïîòîêà õîòÿ áû îäèí èç m êàíàëîâ îá-
ñëóæèâàíèÿ ñâîáîäåí.  ýòîé ñèòóàöèè çàÿâêà íà÷èíàåò îáñëóæèâàòüñÿ íåìåäëåííî.
Îïèøåì äåòàëè ôóíêöèîíèðîâàíèÿ ñèñòåìû. Ïóñòü M — íàòóðàëüíîå ÷èñ-
ëî. Çàÿâêè, êîòîðûå íàõîäÿòñÿ íà îðáèòå, ìîæíî ðàçäåëèòü íà äâå êàòåãîðèè. Çà-
ÿâêè ïåðâîé êàòåãîðèè îáñëóæèâàþòñÿ â ïåðâóþ î÷åðåäü. Åñëè æå íà îðáèòå
ìåíüøå, ÷åì M çàÿâîê, òî âíîâü ïîñòóïèâøàÿ íà îðáèòó çàÿâêà ñ÷èòàåòñÿ ïåðâî-
î÷åðåäíîé è îòíîñèòñÿ ê ïåðâîé êàòåãîðèè. Åñëè íà îðáèòå M èëè áîëüøå çàÿ-
âîê, òî âíîâü ïîñòóïèâøàÿ çàÿâêà ïðèñîåäèíèòñÿ ê î÷åðåäè è îòíîñèòñÿ êî âòî-
ðîé êàòåãîðèè. Êàê òîëüêî êàêàÿ-íèáóäü çàÿâêà èç ïåðâîé êàòåãîðèè ïîïàäàåò
â êàíàë îáñëóæèâàíèÿ, òî íåêàÿ çàÿâêà èç âòîðîé êàòåãîðèè ïåðåõîäèò â ïåðâóþ
êàòåãîðèþ. Îðáèòà ïîäàåò íà îáñëóæèâàíèå çàÿâêó ïî ïðèíöèïó M -êàíàëüíîé
ñèñòåìû. Êàæäàÿ çàÿâêà ïåðâîé êàòåãîðèè ïîñûëàåò ïóàññîíîâñêèé ïîòîê ïîïû-
òîê ñ ïàðàìåòðîì �, ÷òîáû âîéòè â ÑÌÎ. Åñëè â ÑÌÎ ìåíåå m çàíÿòûõ êàíàëîâ,
òî çàÿâêà ïåðåõîäèò â ÑÌÎ, â ïðîòèâíîì ñëó÷àå âîçâðàùàåòñÿ íà îðáèòó.
Êàê áûëî îòìå÷åíî, îðáèòà ñîäåðæèò çàÿâêè ïåðâîé è âòîðîé êàòåãîðèé. Äî-
ïóñòèì, ÷òî âñåãî íà íåé k çàÿâîê, òîãäà ÷èñëî çàÿâîê ïåðâîé êàòåãîðèè ñîñòàâëÿ-
åò min( , )k M , îñòàëüíûå çàÿâêè îòíîñÿòñÿ êî âòîðîé êàòåãîðèè.
68 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2017, òîì 53, ¹ 3
Ñîñòîÿíèå ÑÌÎ ñ îðáèòîé îïèñûâàåòñÿ îäíîðîäíûì ìàðêîâñêèì ïðîöåññîì
( ( ), ( ))X t Y t ñ öåëî÷èñëåííûìè êîìïîíåíòàìè, ãäå 0 � �X t m( ) , 0 � � �Y t( ) .
Âåëè÷èíà X t( ) îáîçíà÷àåò ÷èñëî çàÿâîê, îáñëóæèâàåìûõ â ìîìåíò t, à âåëè÷èíû
Y t( ) — ÷èñëî çàÿâîê, íàõîäÿùèõñÿ íà îðáèòå â ìîìåíò t.
 ñèñòåìå âîçìîæíû ñëåäóþùèå ïåðåõîäû èç îäíîãî ñîñòîÿíèÿ â äðóãîå:
à) ïåðåõîäû, ñâÿçàííûå ñ ïîñòóïëåíèåì çàÿâîê:
� �( , ; , )i j i j� �1 ïðè 0 � �i m,
� �( , ; , )i j i j� �1 ïðè i m� ;
á) ïåðåõîäû ñ îðáèòû â ñèñòåìó:
� �( , ; , ) min( , )i j i j j M� � �1 1 ïðè i m� ;
â) ïåðåõîäû, ñâÿçàííûå ñ îáñëóæèâàíèåì çàÿâîê:
� �( , ; , )i j i j i� �1 ïðè i � 0.
Òàêèì îáðàçîì, êîãäà X t i m( ) � � è Y t j( ) � , òî çà âðåìÿ dt íà ñâîáîäíûé êà-
íàë ìîæåò ïîñòóïèòü îäíî òðåáîâàíèå ñ îðáèòû ñ âåðîÿòíîñòüþ M dt� ïðè j M
èëè ñ âåðîÿòíîñòüþ j dt� ïðè 0 � �j M . Êîãäà X t m( ) � , òî ñ èíòåíñèâíîñòüþ �
âåëè÷èíà Y t( ) óâåëè÷èâàåòñÿ íà åäèíèöó. Îñòàëüíûå èíòåíñèâíîñòè ïåðåõîäîâ
èç îäíîãî ñîñîòîÿíèÿ â äðóãîå î÷åâèäíû.
ÑÈÑÒÅÌÀ M M/ / /1 0 C ÂÎÇÂÐÀÙÅÍÈÅÌ
Îãðàíè÷èìñÿ ñëó÷àåì, êîãäà m M� �1. Íàéäåì óñëîâèå ýðãîäè÷íîñòè ÑÌÎ ñ
îðáèòîé è áóäåì âûâîäèòü ôîðìóëû äëÿ ñòàöèîíàðíûõ âåðîÿòíîñòåé ñîñòîÿ-
íèé. Äëÿ óïðîùåíèÿ îáîçíà÷èì p P X Y jj � � �( , )0 , q P X Y jj � � �( , )1 , ãäå
X Y, — ñòàöèîíàðíûå âåðñèè X t Y t( ), ( ) â ïðåäïîëîæåíèè, ÷òî ñòàöèîíàðíûé
ðåæèì ñóùåñòâóåò.
Íà ðèñ. 1 ïîêàçàíà íà÷àëüíàÿ ÷àñòü ãðàôà ïåðåõîäîâ.
Èñõîäÿ èç äàííîãî ãðàôà ïåðåõîäîâ èìååì
� �p q0 0� ,
( )� � � �� � �q p p0 0 1,
( ) ,� � �� � �p q nn n 0,
( ) ,� � � � �� � � � �
� �
q p q p nn n n n1 1 0.
Ââåäåì ïðîèçâîäÿùèå ôóíêöèè
P z p z Q z q zn
n
n
n
n
n
( ) , ( )� �
�
�
�
�
0 0
â ïðåäïîëîæåíèè, ÷òî ýòè ðÿäû ñõîäÿòñÿ
ïðè äàííûõ z. Èìååì
( ) ( ) ( )� � � �� � �P z Q z p0 ,
( ) ( ) ( ) ( )� � � � � �z P z z z Q z p� � � � � 0 .
Âìåñòî òðåõ ïàðàìåòðîâ � � �, , ââåäåì
äâà ïàðàìåòðà:
�
�
�
�
�
�
� �, .
ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2017, òîì 53, ¹ 3 69
0
p
0
q
1
q
1
p
2
p
2
q
3
p
3
q
Ðèñ.1. Ãðàô ïåðåõîäîâ ñèñòåìû
�
�
�
�
�
�
�
�
�
�
�
�
�
�
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ïîëó÷èì ñèñòåìó óðàâíåíèé
( ) ( ) ( ) ,
( ) ( ) ( ) ( ) .
� � �
� � � � �
� � �
� � � � �
�
�
P z Q z p
z P z z z Q z p
0
01
Îïðåäåëèòåëü ñèñòåìû èìååò âèä
� ( )
( )
( )( ( ) )z
z z z
z z�
� �
� � �
�
�
� �
�
�
� � � �
� �
� � � �
� � � �
1
1
1 .
Ïîëó÷èì îïðåäåëèòåëè ïåðåìåííûõ:
� P z p
z z
p z z( )
( )
( )( )�
�
� �
�
�
� �
�
�
� � ��
� �
� �0 0
1 1
1 1
1 1 ,
� Q z p
z
p z( ) ( )�
�
�
�
�
� �
�
�
� ��
� �
� �
� �0 0
1
1
1 .
Îòñþäà
P z
z
z
p z
z
P
( )
( )
( )
( )
( )
� �
�
� �
�
�
� �
� � � �
0 1
,
Q z
z
z
p
z
Q
( )
( )
( ) ( )
� �
� �
�
�
� �
� � � �
0
.
Èç óñëîâèÿ íîðìèðîâêè P Q( ) ( )1 1 1� � íàõîäèì � � � �p0
2
1� � �( ) . Îêîí÷à-
òåëüíûå ôîðìóëû äëÿ ïðîèçâîäÿùèõ ôóíêöèé èìåþò âèä
P z
z
z
( )
( ( ) )( )
( )
�
� � �
� �
� � � �
� � � �
1 1
2
,
Q z
z
( )
( ( ) )
( )
�
� �
� �
� � � �
� � � �
1
2
.
Èç ýòèõ ôîðìóë âèäíî, ÷òî íåîáõîäèìîå è äîñòàòî÷íîå óñëîâèå ýðãîäè÷íîñ-
òè ñèñòåìû èìååò âèä � � � �� �( ) , ò.å. � � �� �
2
1/ ( ) . Îáðàùàÿ ïðîèçâîäÿùèå
ôóíêöèè P z Q z( ), ( ) , ïîëó÷àåì ÿâíûå ôîðìóëû äëÿ p qn n, :
p0
2
1� � ��
�
�
,
pn
n
� � �
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�1
2
�
�
�
� � �
�
�
� � �
�
( ) ( )
�
�
�
�
�
�
�
�
�
�n 1
, n � 0 ,
qn
n
� � �
�
�
�
�
�
�
�
�
��
�
�
�
�
�
�
�
� �
�
�
� � �
�
1
2
( )
, n 0 .
Çàìåòèì, ÷òî Q P X( ) ( )1 1� � � �. Òàêèì îáðàçîì, ïðè óñëîâèè ýðãîäè÷íîñòè
âñå ïîñòóïàþùèå çàÿâêè îáñëóæèâàþòñÿ ñ âåðîÿòíîñòüþ åäèíèöà. Áîëåå òîãî,
âñå ìîìåíòû ðàñïðåäåëåíèÿ âåëè÷èíû Y êîíå÷íû, òàê êàê ðàäèóñ ñõîäèìîñòè
ôóíêöèé P z Q z( ), ( ) ðàâåí � � � �/ ( )� �1.
70 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2017, òîì 53, ¹ 3
ÑÐÀÂÍÅÍÈÅ ÐÀÑÑÌÎÒÐÅÍÍÎÉ ÑÈÑÒÅÌÛ Ñ ÑÈÑÒÅÌÎÉ ÒÈÏÀ ËÀÊÀÒÎØÀ
Ðàññìîòðèì ñèñòåìó M M/ /1 òèïà Ëàêàòîøà [13, 14] ñ ýêñïîíåíöèàëüíûì
ðàñïðåäåëåíèåì ñ ïàðàìåòðîì � âðåìåíè öèêëà íà îðáèòå, ïóàññîíîâñêèì âõî-
äÿùèì ïîòîêîì ñ ïàðàìåòðîì �, ýêñïîíåíöèàëüíûì âðåìåíåì îáñëóæèâàíèÿ ñ
ïàðàìåòðîì �. Ñðàâíèì óñëîâèÿ ýðãîäè÷íîñòè òàêîé ñèñòåìû ñ óñëîâèÿìè
ïðåäëîæåííîé íàìè ñèñòåìû.
Ïóñòü Wn — âðåìÿ îæèäàíèÿ n-é çàÿâêè â ñèñòåìå òèïà Ëàêàòîøà. Ïðè äîñ-
òàòî÷íî áîëüøîì çíà÷åíèè Wn âûïîëíÿåòñÿ ñîîòíîøåíèå
E W W EY EZ EUn n n n n( ) ~
� � �
� � �1 1 1,
ãäå Yn — âðåìÿ îáñëóæèâàíèÿ n-é çàÿâêè, Zn�1— âðåìÿ ìåæäó ïîñòóïëåíèåì
n-é è n �1-é çàÿâîê, U n�1 — öèêë n �1-é çàÿâêè íà îðáèòå.
Òàêèì îáðàçîì,
E W Wn n( ) ~
�
� � �1
1 1 1
� � �
. (1)
Ñîãëàíî êðèòåðèþ Ìóñòàôû [18] óñëîâèå ýðãîäè÷íîñòè ñèñòåìû ñîñòîèò
â òîì, ÷òîáû âûðàæåíèå (1) áûëî îòðèöàòåëüíûì, ò.å.
�� �� ��� � � 0, �� � �� � � 0, �
�
�
�
�1
.
Ïîëó÷åííîå óñëîâèå áîëåå ñèëüíîå, ÷åì óñëîâèå � � �� �
2
1/ ( ) äëÿ íàøåé ñèñ-
òåìû.
ÇÀÊËÞ×ÅÍÈÅ
Ðàññìîòðåíà ñèñòåìà îáñëóæèâàíèÿ M M/ / /1 0 ñ âîçâðàùåíèåì è êîìáèíèðî-
âàííîé äèñöèïëèíîé îáñëóæèâàíèÿ, ò.å. çàÿâêè ñ îðáèòû îáñëóæèâàþòñÿ â ïî-
ðÿäêå î÷åðåäè, íî ïðè íàëè÷èè ñâîáîäíîãî êàíàëà çàÿâêà, ïðèøåäøàÿ ñ ïåð-
âè÷íîãî ïîòîêà, íåìåäëåííî îòïðàâëÿåòñÿ íà îáñëóæèâàíèå. Äëÿ òàêîé ñèñòå-
ìû âûâåäåíû ôîðìóëû äëÿ âåðîÿòíîñòåé ñîñòîÿíèé, à òàêæå óñëîâèå
ýðãîäè÷íîñòè. Ïðîâåäåíî ñðàâíåíèå ðàññìàòðèâàåìîé ñèñòåìû ñ ÷èñòîé ñèñòå-
ìîé Ëàêàòîøà [13, 14].
ÑÏÈÑÎÊ ËÈÒÅÐÀÒÓÐÛ
1. Yang T., Templeton J.G.C. A survey on retrial queues. Queueing Systems. 1987. N 3. P. 201–233.
2. Êîáà Å.Â., Êîâàëåíêî È.Í. Ê êëàññèôèêàöèè ñèñòåì ìàññîâîãî îáñëóæèâàíèÿ ñ ïîâòîðåíèåì
âûçîâîâ. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç. 2010. ¹ 3. C. 84–91.
3. Artalejo J., Falin G. Standart and retrial queueing systems: a comparative analysis. Revista
matema' tica complutense. 2002. Vol. XV, N 1. Ð. 101–129.
4. Artalejo J. A classified bibliography of research in retrial queueing. Progress in 1990-1999. Top.
1999. N 7. P. 187–211.
5. Artalejo J. A classified bibliography of research in retrial queueing. Progress in 2000-2009.
Mathematical and Computer Modeling. 2010, Vol 51. P. 1071–1081.
6. Anisimov V.V. Analysis of retrial queueing systems in asymptotically aggregated environment.
Third International Workshop on Retrial Queues. Amsterdam, 2000. P. 43–46.
7. Àíèñèìîâ Â.Â., Êóðòóëóø Ì. Íåêîòîðûå ìàðêîâñêèå ìîäåëè ñèñòåìû îáñëóæèâàíèÿ ñ ïî-
âòîðíûìè âûçîâàìè â óñëîâèÿõ ìàëîé çàãðóçêè. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç. 2010. ¹ 6.
C. 110–126.
8. Êîâàëåíêî È.Í. Âåðîÿòíîñòü ïîòåðè â ñèñòåìå îáñëóæèâàíèÿ M G m/ / ñ T-ïîâòîðåíèåì âûçî-
âîâ â ðåæèìå ìàëîé íàãðóçêè. Äîïîâ³ä³ ÍÀÍ Óêðà¿íè. 2002. ¹ 5. C. 77–80.
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9. Êîâàëåíêî È.Í. Î äâóõöèêëè÷åñêîé ñèñòåìå îáñëóæèâàíèÿ. Êèáåðíåòèêà è ñèñòåìíûé àíà-
ëèç. 2015. Ò. 51, ¹ 1. C. 59–65.
10. Êîáà Å.Â. Óñëîâèå óñòîé÷èâîñòè íåêîòîðûõ òèïîâûõ ñèñòåì îáñëóæèâàíèÿ ñ âîçâðàùåíèåì
çàÿâîê. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç. 2005. ¹ 1. C. 124–127.
11. Lebedev E.A. On the first passage time of removing level for retrial queues. Äîïîâ³ä³ ÍÀÍ Óêðà¿íè.
2002. ¹ 3. C. 47–50.
12. Óñàð È.ß., Ëåáåäåâ Å.À. Ñèñòåìû ñ ïîâòîðíûìè âûçîâàìè è ïåðåìåííîé èíòåíñèâíîñòüþ âõî-
äÿùåãî ïîòîêà. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç. 2013. ¹ 3. C. 151–159.
13. Lakatos L. On a simple continuous cyclic-waiting problem. Annales Univ. Sci. Budapest, Sect.
Comp. 1994. N 14. P. 105–113.
14. Lakatos L. A probability model connected with landing of airplanes. Safety and reliability.
(Ed. A.A. Balkoma). Rotherdam: Brookfieled, 1999. P. 151–154.
15. Êîâàëåíêî È.Í., Êîáà Å.Â. Òðè ñèñòåìû îáñëóæèâàíèÿ ñ ïîâòîðíûìè âûçîâàìè, îòðàæàþùèå
íåêîòîðûå îñîáåííîñòè ïðîöåññà ïîñàäêè âîçäóøíûõ ñóäîâ. Ïðîáëåìû óïðàâëåíèÿ è èíôîð-
ìàòèêè. 2002. ¹ 2. C. 78–82.
16. Koba E.V. On a GI G/ / 1 retrial queueing system with a FIFO queueing discipline. Theory of
Stochastic Processes. 2002. N 8. P. 201–207.
17. Ñåðåáðÿêîâà Ñ.Â. Çàñòîñóâàííÿ öèêë³÷íèõ ñèñòåì îáñëóãîâóâàííÿ. Äîïîâ³ä³ ÍÀÍ Óêðà¿íè.
2016. ¹ 3. C. 32–37.
18. Áî÷àðîâ Ï.Ï., Ïå÷èíêèí À.Â. Òåîðèÿ ìàññîâîãî îáñëóæèâàíèÿ. Ìîñêâà: Èçä. ÐÓÄÍ, 1995. 529 ñ.
Íàä³éøëà äî ðåäàêö³¿ 01.11.2016
Î.Â. Êîáà
ÑÈÑÒÅÌÀ ÎÁÑËÓÃÎÂÓÂÀÍÍß M M/ / /1 0 Ç ÏÎÂÒÎÐÅÍÍßÌ
ÒÀ ÊÎÌÁ²ÍÎÂÀÍÎÞ ÄÈÑÖÈÏ˲ÍÎÞ ÎÁÑËÓÃÎÂÓÂÀÍÍß
Àíîòàö³ÿ. Ðîçãëÿíóòî ñèñòåìó îáñëóãîâóâàííÿ M M/ / /1 0 ç ïîâòîðåííÿì òà
êîìá³íîâàíîþ äèñöèïë³íîþ îáñëóãîâóâàííÿ, à ñàìå: çàÿâêè ç îðá³òè îáñëó-
ãîâóþòüñÿ ó ïîðÿäêó ÷åðãè, àëå ïðè íàÿâíîñò³ â³ëüíîãî êàíàëó çàÿâêà, ùî
ïðèéøëà ç ïåðâèííîãî ïîòîêó, íåãàéíî â³äïðàâëÿºòüñÿ íà îáñëóãîâóâàííÿ..
Âèâåäåíî ôîðìóëè äëÿ éìîâ³ðíîñòåé ñòàí³â ³ óìîâó åðãîäè÷íîñò³. Ïðîâåäå-
íî ïîð³âíÿííÿ ñèñòåìè, ùî ðîçãëÿäàºòüñÿ, ç ÷èñòîþ ñèñòåìîþ Ëàêàòîøà.
Êëþ÷îâ³ ñëîâà: ñèñòåìà îáñëóãîâóâàííÿ, ñèñòåìà îáñëóãîâóâàííÿ ç ïîâòî-
ðåííÿì, îðá³òà, ñèñòåìà îáñëóãîâóâàííÿ ç öèêë³÷íèì ÷àñîì î÷³êóâàííÿ,
êîìá³íîâàíà äèñöèïë³íà îáñëóãîâóâàííÿ, óìîâà åðãîäè÷íîñò³ ñèñòåìè.
E.V. Koba
RETRIAL QUEUEING SYSTEM M M/ / /1 0 WITH COMBINED DISCIPLINE OF SERVICE
Abstract. The paper considers the retrial queueing system M M/ / /1 0 with
combined discipline of service, namely, a customer from the orbit is served in
its turn, but in case of a free channel an arrival from the original flow is
serviced immediately. The author obtained the expressions for state probabilities
as well as the ergodicity conditions. The system is compared with the Lakatos
type system.
Keywords: queueing system, retrial queueing system, orbit, cycling-waiting
queueing system, combined discipline of service, stability condition of system.
Êîáà Åëåíà Âèêòîðîâíà,
äîêòîð ôèç.-ìàò. íàóê, äîöåíò, âåäóùèé íàó÷íûé ñîòðóäíèê Èíñòèòóòà êèáåðíåòèêè ÍÀÍ Óêðàèíû
èì. Â.Ì. Ãëóøêîâà, Êèåâ, e-mail: e-koba@yandex.ru.
72 ISSN 0023-1274. Êèáåðíåòèêà è ñèñòåìíûé àíàëèç, 2017, òîì 53, ¹ 3
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken voor kwaliteitsafdrukken op desktopprinters en proofers. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /NoConversion
/DestinationProfileName ()
/DestinationProfileSelector /NA
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure true
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles true
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /NA
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /LeaveUntagged
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [2834.646 2834.646]
>> setpagedevice
|