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Some explicit formulas for the steady-state behavior of the queue with semi-Markovian service times

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts
Affiliation:
Purdue University

Abstract

This paper discusses a number of explicit formulas for the steady-state features of the queue with Poisson arrivals in groups of random sizes and semi-Markovian service times. Computationally useful formulas for the expected duration of the various busy periods, for the mean numbers of customers served during them, as well as for the lower order moments of the queue lengths, both in discrete and in continuous time, and of the virtual waiting time are obtained. The formulas are recursive matrix expressions, which generalize the analogous but much simpler results for the classical M/G/1 model.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 1977 

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Footnotes

This paper was prepared for presentation as an invited address at the Statistics Days at Ball State University, Muncie, Indiana, 9-10 April 1976. This research was sponsored by the Air Force Office of Scientific Research Air Force Sytems Command USAF, under Grant No. AFOSR-72-2350 B.

References

[1] Çinlar, E. (1967) Time dependence of queues with semi-Markovian service times. J. Appl. Prob. 4, 356364.CrossRefGoogle Scholar
[2] Çinlar, E. (1969) Markov renewal theory. Adv. Appl. Prob. 1, 123187.CrossRefGoogle Scholar
[3] Gaver, D. (1963) A comparison of queue disciplines when service orientation times occur. Naval Res. Logist. Quart. 10, 219235.CrossRefGoogle Scholar
[4] Hunter, J. J. (1969) On the moments of Markov renewal processes. Adv. Appl. Prob. 1, 188210.CrossRefGoogle Scholar
[5] Loynes, R. M. (1962) A continuous-time treatment of certain queues and infinite dams. J. Austral. Math. Soc. 2, 484498.CrossRefGoogle Scholar
[6] Neuts, M. F. (1966) The single server queue with Poisson input and semi-Markov service times. J. Appl. Prob. 3, 202230.CrossRefGoogle Scholar
[7] Neuts, M. F. (1974) The Markov renewal branching process. In Proc. Conf. Math. Meth. in Theory of Queues 1973, Kalamazoo MI, Springer-Verlag, New York, 121.Google Scholar
[8] Neuts, M. F. (1976) Moment formulas for the Markov renewal branching process. Adv. Appl. Prob. 8, 690711.CrossRefGoogle Scholar
[9] Neuts, M. F. (1977) The M/G/1 queue with several types of customers and changeover times. Adv. Appl. Prob. To appear.Google Scholar
[10] Purdue, P. (1975) A queue with Poisson input and semi-Markov service times: busy period analysis. J. Appl. Prob. 12, 353357.CrossRefGoogle Scholar
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