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Losses per cycle in a single-server queue

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Ronald W. Wolff
Ronald W. Wolff*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA 94720, USA. Email address: wolff@ieor.berkeley.edu
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Abstract

Several recent papers have shown that for the M/G/1/n queue with equal arrival and service rates, the expected number of lost customers per busy cycle is equal to 1 for every n ≥ 0. We present an elementary proof based on Wald's equation and, for GI/G/1/n, obtain conditions for this quantity to be either less than or greater than 1 for every n ≥ 0. In addition, we extend this result to batch arrivals, where, for average batch size β, the same quantity is either less than or greater than β. We then extend these results to general ways that customers may be lost, to an arbitrary order of service that allows service interruption, and finally to reneging.

Keywords

Loss systemsrenegingPoisson arrivalsbusy cycleWald's equation

MSC classification

Primary: 60K25: Queueing theory
Type
Short Communications
Information
Journal of Applied Probability , Volume 39 , Issue 4 , December 2002 , pp. 905 - 909
Copyright
Copyright © Applied Probability Trust 2002 

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References

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