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An invariant distribution for the G/G/1 queueing operator

Published online by Cambridge University Press:  01 July 2016

Nicholas Bambos  and
Jean Walrand
Nicholas Bambos*
Affiliation:
University of California, Berkeley
Jean Walrand*
Affiliation:
University of California, Berkeley
*
∗Postal address for both authors: Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA.
∗Postal address for both authors: Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley, CA 94720, USA.
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Abstract

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We consider the G/G/1 queue as an operator that maps inter-arrival times to inter-departure times of points, given the service times. For arbitrarily fixed statistics of service times, we are interested in the existence of distributions of inter-arrival times that induce identical distributions on the inter-departure times. In this note we prove, by construction, the existence of one of such distribution.

Keywords

QUEUEING THEORY
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 254 - 256
Copyright
Copyright © Applied Probability Trust 1990 

References

[1] Doob, J. L. (1953) Stochastic Processes. Wiley, New York.Google Scholar
[2] Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Comb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar