Estimation of the Rate of Convergence in the Limit Theorem for Extreme Values of Regenerative Processes

Zakusylo, O. K.; Matsak, I. K.

doi:10.1007/s11253-020-01855-1

Cited by 3 publications

(8 citation statements)

References 6 publications

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“…Iíøèé ïiäõiä äî ïîäiáíèõ çàäà÷, ÿêèé ãðóíòóâàâñÿ íà äåÿêîìó íåâèïàäêîâîìó ïåðåòâîðåííi ÷àñó, áóâ çàïðîïîíîâàíèé ó ñòàòòÿõ [11], [12]. Ïðè öüîìó ãðàíè÷íèì âèÿâèâñÿ åêñïîíåíöiéíèé ðîçïîäië.…”

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“…Ïðè öüîìó ãðàíè÷íèì âèÿâèâñÿ åêñïîíåíöiéíèé ðîçïîäië. Áiëüøå òîãî, äëÿ ÑÌÎ M/M/1 ó ðîáîòi [12] áóëà çíàéäåíà íàñòóïíà îöiíêà øâèäêîñòi çáiaeíîñòi.…”

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“…äå C -äåÿêà êîíñòàíòà, çàëåaeíà âiä ïàðàìåòðiâ λ, µ, i íå çàëåaeíà âiä u. Òóò ìè ñïðîáó¹ìî îòðèìàòè îöiíêè, àíàëîãi÷íi (3) äëÿ áiëüø çàãàëüíîãî âèïàäêó ïðîöåñiâ íàðîäaeåííÿ òà çàãèáåëi. Òàêi îöiíêè ðîçãëÿäàëèñü i ó ïðàöi [12]. Òîìó äàíó ðîáîòó ñëiä ðîçãëÿäàòè, ÿê ïðîäîâaeåííÿ ðîáîòè [12] òà äåÿêå óòî÷íåííÿ¨¨ðåçóëüòàòiâ.…”

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“…Òàêi îöiíêè ðîçãëÿäàëèñü i ó ïðàöi [12]. Òîìó äàíó ðîáîòó ñëiä ðîçãëÿäàòè, ÿê ïðîäîâaeåííÿ ðîáîòè [12] òà äåÿêå óòî÷íåííÿ¨¨ðåçóëüòàòiâ. Äàëi áóäóòü íàâåäåíi òàêîae çàñòîñóâàííÿ îòðèìàíèõ ðåçóëüòàòiâ äî ïðîöåñó Q(t), ÿêèé îïèñó¹ äîâaeèíó ÷åðãè â ÑÌÎ.…”

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“…Âåëè÷èíà q(u) çàäà¹òüñÿ ôîðìóëîþ (5), a 2 = ET 2 -öå äðóãèé ìîìåíò òðèâàëîñòi öèêëó ðåãåíåðàöi¨äëÿ ïðîöåñó íàðîäaeåííÿ òà çàãèáåëi âiäîìà (äèâ. [12]). Íàñëiäîê 1. ßêùî X(t) ïðîöåñ íàðîäaeåííÿ òà çàãèáåëi, ÿêèé çàäîâîëüíÿ¹ óìîâè…”

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See 4 more Smart Citations

On Extreme Values of Birth and Death Processes

Matsak

2021

BMJ

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We establish the convergence rate to exponential distribution in a limit theorem for extreme values of birth and death processes. Some applications of this result are given to processes specifying queue length.). We establish uniform estimates for the convergence rate in the exponential distribution in a limit theorem for extreme values of birth and death processes. This topic is closely related to the problem on the time of ﬁrst intersection of some level u by a regenerating process. Of course, we assume that both time t and level u grow inﬁnitely. The proof of our main result is based on an important estimate for general regenerating processes. Investigations of the kind are needed in diﬀerent ﬁelds: mathematical theory of reliability, queueing theory, some statistical problems in physics. We also provide with examples of applications of our results to extremal queueing problems M/M/s. In particular case of queueing M/M/1, we show that the obtained estimates have the right order with respect to the probability q(u) of the exceeding of a level u at one regeneration cycle, that is, only improvement of the corresponding constants is possible.

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On Extreme Values of Birth and Death Processes

Matsak

2021

BMJ

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Estimates of the convergence rate in a limit theorem for geometric sums and some of their applications

2023

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On extreme values of the queue length in some queuing systems

Akbash,

Doronina,

Matsak

2023

Georgian Mathematical Journal

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In this paper, the rate of convergence to the exponential distribution in the limit theorem for extreme values of the queue length in queuing systems M / G / 1 {M/G/1} and G / M / 1 {G/M/1} are studied. In proving these results, we first prove one general boundary theorem for extremal values of regenerative processes, which is of independent interest. This topic is closely related to the task of the first passage times to rare sets by the regenerative process. The need for such research arises in many areas: the mathematical theory of reliability, queuing theory, some statistical problems of physics.

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Estimation of the Rate of Convergence in the Limit Theorem for Extreme Values of Regenerative Processes

Cited by 3 publications

References 6 publications

On Extreme Values of Birth and Death Processes

On Extreme Values of Birth and Death Processes

Estimates of the convergence rate in a limit theorem for geometric sums and some of their applications

On extreme values of the queue length in some queuing systems

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