On extreme values of some regenerative processes
Authors:
O. K. Zakusylo and I. K. Matsak
Translated by:
N. N. Semenov
Journal:
Theor. Probability and Math. Statist. 97 (2018), 57-71
MSC (2010):
Primary 60K25, 60F05
DOI:
https://doi.org/10.1090/tpms/1048
Published electronically:
February 21, 2019
MathSciNet review:
3745999
Full-text PDF
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Additional Information
Abstract: A general limit theorem is proved for extreme values of regenerative processes. Some applications of this result are given for birth and death processes that determine the length of the queue in a queueing system.
References
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References
- B. V. Gnedenko and I. N. Kovalenko, Introduction to Queueing Theory, 2nd ed., Mathematical Modeling, vol. 5, Birkhäuser, Boston, 1989. MR 1035707
- S. Karlin, A First Course in Stochastic Processes, Academic Press, New York–London, 1966. MR 0208657
- V. V. Anisimov, O. K. Zakusylo, and V. S. Donchenko, Elements of the Queueing Theory and Asymptotic Analysis of Systems, “Vyshcha shkola”, Kiev, 1987. (Russian)
- J. W. Cohen, Extreme values distribution for the M/G/1 and G/M/1 queueing systems, Ann. Inst. H. Poincare., Sect.B 4 (1968), 83–98. MR 0232466
- C. W. Anderson, Extreme value theory for a class of discrete distribution with application to some stochastic processes, J. Appl. Prob. 7 (1970), 99–113. MR 0256441
- D. L. Iglehart, Extreme values in the GI/G/1 gueue, Ann. Math. Statist. 43 (1972), 627–635. MR 0305498
- R. F. Serfozo, Extreme values of birth and death processes and queues, Stoch. Process. Appl. 27 (1988), 291–306. MR 931033
- S. Asmussen, Extreme values theory for queues via cycle maxima, Extremes 1 (1998), 137–168. MR 1814621
- J. Galambos, The Asymptotic Theory of Extreme Order Statistics, John Wiley & Sons, New York–Chichester–Brisbane, 1978. MR 489334
- M. R. Leadbetter, G. Lindgren, and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Series in Statistics, Springer-Verlag, New York–Berlin, 1983. MR 691492
- D. Cox and W. Smith, Renewal Theory, “Sovetskoye Radio”, Moscow, 1967. (Russian) MR 0224173
- I. I. Gikhman, A. V. Skorokhod, and M. I. Yadrenko, Theory of Probability and Mathematical Statistics, “Vyshcha shkola”, Kiev, 1988. (Russian)
- I. K. Matsak, A lemma for random sums and its applications, Ukrain. Mat. Zh. (2017). (Ukrainian) (to appear)
- E. Yu. Barzilovich, Yu. K. Belyaev, V. A. Kashtanov, I. N. Kovalenko, A. D. Solovyev, and I. A. Ushakov, Problems of the Mathematical Theory of Reliability (B. V. Gnedenko, ed.), “Radio i Svyaz”, Moscow, 1983. (Russian) MR 758792
- B. V. Gnedenko, Yu. K. Belyayev, and A. D. Solovyev, Mathematical Methods of Reliability Theory, Academic Press, New York–London, 1969. MR 0345234
- W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, 2nd ed., John Wiley & Sons, New York–London–Sydney, 1971.
- B. V. Dovgaĭ and I. K. Matsak, On a redundant system with renewals, Theory Probab. Math. Statist. 94 (2017), 63–76. MR 3553454
- S. Karlin and J. McGregor, The classification of birth and death processes, Trans. Amer. Math. Soc. 86 (1957), 366–400. MR 0094854
- L. Takács, Some probability questions in the theory of telephone traffic, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl. 8 (1958), 151–210. (Hungarian) MR 0096319
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Additional Information
O. K. Zakusylo
Affiliation:
Department of Operation Research, Faculty of Computer Science and Cybernetics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
zakusylo@univ.net.ua
I. K. Matsak
Affiliation:
Department of Operation Research, Faculty of Computer Science and Cybernetics, Kyiv Taras Shevchenko National University, Volodymyrs’ka Street, 64/13, Kyiv 01601, Ukraine
Email:
ivanmatsak@univ.kiev.ua
Keywords:
Extreme values,
regenerative processes,
birth and death processes,
queueing systems
Received by editor(s):
August 4, 2017
Published electronically:
February 21, 2019
Article copyright:
© Copyright 2019
American Mathematical Society
