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On the quasi-stationary distributions of the GI/M/1 queue

Published online by Cambridge University Press:  14 July 2016

E. K. Kyprianou
E. K. Kyprianou*
Affiliation:
University of Manchester
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Abstract

This paper studies the existence, in a stable GI/M/1 queue, of the limit as t → ∞ of the distribution of the virtual waiting time process at time t conditioned on the event that at no time in the interval [0, t] the queue has become empty. The conditional limit distribution obtained when the traffic intensity is strictly less than one is the weighted sum of an exponential and a gamma distribution. Similar conditional limit distributions are obtained for the queue size process and the waiting time process as defined by Prabhu (1964).

Keywords

GI/M/1 QUEUEDUAL M/G/1 QUEUEPRABHU'S WAITING TIME PROCESSVIRTUAL WAITING TIME PROCESSQUEUE SIZEBUSY PERIODLAPLACE-STIELTJES TRANSFORMS OF DISTRIBUTIONSASYMPTOTIC EXPANSIONS OF DISTRIBUTION FUNCTIONSQUASI-STATIONARY DISTRIBUTIONS
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 1 , March 1972 , pp. 117 - 128
Copyright
Copyright © Applied Probability Trust 1972 

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References

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[6] Kyprianou, E. (1971) The quasi-stationary distribution of the virtual waiting time in queues with Poisson arrivals. J. Appl. Prob. 8, 494–507.Google Scholar
[7] Prabhu, N. U. (1964) A waiting time process in the queue GI/M/1. Acta Math. Acad. Sci. Hung. 15, 363–371.CrossRefGoogle Scholar