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Stability condition for a single-server retrial queue

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Huei-Mei Liang  and
V. G. Kulkarni
Huei-Mei Liang*
Affiliation:
National Central University, Taiwan
V. G. Kulkarni*
Affiliation:
University of North Carolina, Chapel Hill
*
∗ Postal address: Department of Mathematics, National Central University, Taiwan, ROC.
∗∗ Postal address: Department of Operations Research, University of North Carolina, Chapel Hill, NC 27599–3180, USA.
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Abstract

A single-server retrial queue consists of a primary queue, an orbit and a single server. Assume the primary queue capacity is 1 and the orbit capacity is infinite. Customers can arrive at the primary queue either from outside the system or from the orbit. If the server is busy, the arriving customer joins the orbit and conducts a retrial later. Otherwise, he receives service and leaves the system.

We investigate the stability condition for a single-server retrial queue. Let λ be the arrival rate and 1/μ be the mean service time. It has been proved that λ/μ < 1 is a sufficient stability condition for the M/G/1/1 retrial queue with exponential retrial times. We give a counterexample to show that this stability condition is not valid for general single-server retrial queues. Next we show that λ /μ < 1 is a sufficient stability condition for the stability of a single-server retrial queue when the interarrival times and retrial times are finite mixtures of Erlangs.

Keywords

ORBITING QUEUESM/G/1/1 RETRIAL QUEUEPOISSON ARRIVALS

MSC classification

Primary: 60K25: Queueing theory
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 3 , September 1993 , pp. 690 - 701
Copyright
Copyright © Applied Probability Trust 1993 

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References

ÇInlar, E. (1975) Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Falin, G. I. (1986) Single-line repeated orders queueing systems. Math. Operat. forsch. Statist. Ser. Optim. 5, 649–667.Google Scholar
Falin, G. I. (1990) A survey of retrial queues. QUESTA 5, 127–167.Google Scholar
Foster, F. G. (1953) On the stochastic matrices associated with certain queueing processes. Ann. Math. Statist. 24, 355–360.CrossRefGoogle Scholar
Kamae, T., Krengel, U. and O'Brien, G. (1977) Stochastic inequalities on partially ordered spaces. Ann. Prob. 5, 899–912.CrossRefGoogle Scholar
Keilson, J., Cozzolino, J. and Young, H. (1968) A service system with unfilled requests repeated. Operat. Res. 16, 1126–1137.CrossRefGoogle Scholar
Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
Tijms, H. J. (1985) Stochastic Modeling and Analysis: A Computational Approach. Wiley, New York.Google Scholar
Yang, T. and Templeton, J. G. C. (1987) A survey on retrial queues. QUESTA 2, 201–233.Google Scholar