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On queueing systems with renewal departure processes

Published online by Cambridge University Press:  01 July 2016

Mark Berman  and
Mark Westcott
Mark Berman*
Affiliation:
CSIRO Division of Mathematics and Statistics, Sydney
Mark Westcott*
Affiliation:
CSIRO Division of Mathematics and Statistics, Canberra
*
∗Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 218, Lindfield, NSW 2070, Australia.
∗∗Postal address: CSIRO Division of Mathematics and Statistics, P.O. Box 1965, Canberra City, ACT 2601, Australia.
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Abstract

It is proved that, for a large class of stable stationary queueing systems with renewal arrival processes and without losses, a necessary condition for the departure process also to be a renewal process is that its interval distribution be the same as that of the arrival process. This result is then applied to the classical GI/G/s queueing systems. In particular, alternative proofs of known characterizations of the M/G/1 and GI/M/1 systems are given, as well as a characterization of the GI/G/∞ system. In the course of the proofs, sufficient conditions for the existence of all the moments of the stationary queue-size distributions of both the GI/G/1 and GI/G/∞ systems are derived.

Keywords

RENEWAL PROCESSPOISSON PROCESSREGENERATIVE PROCESSSTATIONARITYSTABILITYGI/G/s QUEUEING SYSTEMINTERVAL DISTRIBUTIONPROBABILITY GENERATING FUNCTION
Type
Research Article
Information
Advances in Applied Probability , Volume 15 , Issue 3 , September 1983 , pp. 657 - 673
Copyright
Copyright © Applied Probability Trust 1983 

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