Abstract
The paper investigates the asymptotic behavior of almost surely maximum length in queueing systems. For a system M / M / m, 1≤ m< ∞, a statement of the type of the law of iterated logarithm is established. We also consider the case m = ∞ for which the asymptotic behavior is much different.
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B. V. Gnedenko and I. N. Kovalenko, Introduction to Queuing Theory [in Russian], Nauka, Moscow (1966).
S. Karlin, A First Course in Stochastic Processes [Russian translation], Mir, Moscow (1971).
V. V. Anisimov, V. K. Zakusilo, and V. S. Donchenko, Elements of the Queuing Theory and Asymptotic Analysis of Systems [in Russian], Vysshaya Shkola, Kyiv (1987).
J. W. Cohen, “Extreme values distribution for the M / G /1 and G / I / M / 1 queueing systems,” Ann. Inst. H. Poincare., Sect. B, Vol. 4, 83–98 (1968).
C. W. Anderson, “Extreme value theory for a class of discrete distribution with application to some stochastic processes,” J. Appl. Prob., Vol. 7, 99–113 (1970).
D. L. Iglehart, “Extreme values in the GI / G /1 queue,” Ann. Math. Statist., Vol. 43, 627–635 (1972).
R. F. Serfozo, “Extreme values of birth and death processes and queues,” Stochastic Processes and Their Applications, Vol. 27, 291–306 (1988).
S. Asmussen, “Extreme values theory for queues via cycle maxima,” Extremes, Vol. 1, 137–168 (1998).
O. K. Zakusylo and I. K. Matsak, “On extreme values of some regenerative processes,” Teoriya Jmovirn. ta Mat. Statystyka, No. 97, 58–71 (2017).
I. K. Matsak, “Asymptotic behavior of extreme values of random variables. Discrete case,” Ukr. Math. J., Vol. 68, No. 7, 1077–1090 (2016).
J. Galambos, The Asymptotic Theory of Extreme Order Statistics, Wiley (1978).
W. Feller, An Introduction to Probability Theory and its Applications, Vol. 1, Wiley (1968).
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Translated from Kibernetika i Sistemnyi Analiz, No. 2, March–April, 2019, pp. 171–179.
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Dovhai, B.V., Matsak, I.K. Asymptotic Behavior of Extreme Values of Queue Length in M / M / m Systems. Cybern Syst Anal 55, 321–328 (2019). https://doi.org/10.1007/s10559-019-00137-4
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DOI: https://doi.org/10.1007/s10559-019-00137-4