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A Markov chain approach to periodic queues

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen  and
Hermann Thorisson
Søren Asmussen
Affiliation:
University of Copenhagen
Hermann Thorisson*
Affiliation:
University of Göteborg
*
∗∗Postal address: Department of Mathematics, Chalmers University of Technology and University of Göteborg, S-41296 Göteborg, Sweden.
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Abstract

We consider GI/G/1 queues in an environment which is periodic in the sense that the service time of the nth customer and the next interarrival time depend on the phase θ n at the arrival instant. Assuming Harris ergodicity of {θ n} and a suitable condition on the traffic intensity, various Markov chains related to the queue are then again Harris ergodic and provide limit results for the standard customer- and time-dependent processes such as waiting times and queue lengths. As part of the analysis, a result of Nummelin (1979) concerning Lindley processes on a Markov chain is reconsidered.

Keywords

PERIODICITYQUEUEHARRIS RECURRENCELINDLEY PROCESSWAITING TIMEQUEUE LENGTHCOUPLING
Type
Research Papers
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 215 - 225
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

∗

Present address: Institute of Electronical Systems, Aalborg University, Strandvejen, DK-9000 Aalborg, Denmark.

Supported by the Swedish Natural Science Research Council and by the Icelandic Science Foundation.

References

[1]Arndt, K. (1984) On the distribution of the supremum of a random walk on a Markov chain. Limit Theorems and Related Problems, ed. Borovkov, A. A., Optimizations Software, New York, 253–267.Google Scholar
[2]Asmussen, S. (1987) Applied Probability and Queues. Wiley, Chichester.Google Scholar
[3]Athreya, K. B. and Ney, P. (1978) A new approach to the limit theory of recurrent Markov chains. Trans. Amer. Math. Soc. 245, 493–501.10.1090/S0002-9947-1978-0511425-0Google Scholar
[4]Böhme, O. (1982) Periodic Markov transition functions, I. Math. Nachr. 108, 231–239.10.1002/mana.19821080118Google Scholar
[5]Griffeaths, D. (1979) Coupling methods for Markov processes. Studies in Probability and Ergodic Theory, Adv. Math. Supplementary Series 2, 1–43.Google Scholar
[6]Harrison, J. M. and Lemoine, A. J. (1977) Limit theorems for periodic queues. J. Appl. Prob. 14, 566–576.10.2307/3213459Google Scholar
[7]Jagers, P. and Nerman, O. (1985) Branching processes in a periodically varying environment. Ann. Prob. 13, 254–268.10.1214/aop/1176993079Google Scholar
[8]Lemoine, A. J. (1981) On queues with periodic Poisson input. J. Appl. Prob. 18, 889–900.Google Scholar
[9]Nummelin, E. (1978) A splitting technique for Harris recurrent Markov chains. Z. Wahrscheinlichkeitsth. 43, 309–318.10.1007/BF00534764Google Scholar
[10]Nummelin, E. (1979) A conservation property for general GI/G/1 queues with an application to tandem queues. Adv. Appl. Prob. 11, 660–672.Google Scholar
[11]Nummelin, E. (1984) General Irreducible Markov Chains and Non-negative Operators. Cambridge University Press, Cambridge.10.1017/CBO9780511526237Google Scholar
[12]Orey, S. (1971) Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities.Google Scholar
[13]Phatarfod, R. M. (1980) The bottomless dam with seasonal inputs. Austral. J. Statist. 22, 212–217.10.1111/j.1467-842X.1980.tb01169.xGoogle Scholar
[14]Revuz, D. (1975) Markov Chains. North-Holland, Amsterdam.Google Scholar
[15]Thorisson, H. (1983) The coupling of regenerative processes. Adv. Appl. Prob. 15, 531–561.10.2307/1426618Google Scholar
[16]Thorisson, H. (1985) Periodic regeneration. Stoch. Proc. Appl. 20, 85–104.10.1016/0304-4149(85)90018-3Google Scholar