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On the interchangeability and stochastic ordering of ·/M/1 queues in tandem

Published online by Cambridge University Press:  01 July 2016

Pantelis Tsoucas  and
Jean Walrand
Pantelis Tsoucas*
Affiliation:
University of California, Berkeley
Jean Walrand*
Affiliation:
University of California, Berkeley
*
∗Postal address: Department of Electrical Engineering and Computer Sciences and Electronics Research Laboratory, University of California, Berkeley CA 94720, USA.
∗Postal address: 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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A probabilistic proof is given of the fact that the departure process from two initially empty. ·/M/1 queues in tandem is unaffected when the service rates are interchanged. As a consequence of this, we show that when the sum of the service rates at the two queues is held constant the departure process stochastically increases as the service rates become equal. The proofs are based on coupling of reflected random walks.

Keywords

TANDEM QUEUESRANDOM WALKSCOUPLINGNETWORKSSERVER ALLOCATION
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 19 , Issue 2 , June 1987 , pp. 515 - 520
Copyright
Copyright © Applied Probability Trust 1987 

Footnotes

This research was supported in part by N.S.F. under Grant No. ECS-8421128.

References

Anantharam, V. (1985) Probabilistic proof of the interchangeability of ·/M/1 queues in series. Submitted to Stoch. Proc. Appl. Google Scholar
Lehtonen, T. (1986) On the ordering of tandem queues with exponential servers. J. Appl. Prob. 23, 115–129.Google Scholar
Shiryayev, A. N. (1984) Probability. Graduate Texts in Mathematics, Springer-Verlag, New York.Google Scholar
Weber, R. R. (1979) the interchangeability of ·/M/1 queues in series. J. Appl. Prob. 16, 690–695.Google Scholar