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A sample path analysis of the M/M/1 queue

Published online by Cambridge University Press:  14 July 2016

Francois Baccelli  and
William A. Massey
Francois Baccelli*
Affiliation:
INRIA
William A. Massey*
Affiliation:
AT&T Bell Laboratories
*
∗ Postal address: INRIA, Centre de Sophia Antipolis, 2004 route des Lucioles, 06565 Valbonne Cedex, France.
∗∗ Postal address: AT&T Bell Laboratories, Murray Hill, NJ 07974, USA.
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Abstract

The exact solution for the transient distribution of the queue length and busy period of the M/M/1 queue in terms of modified Bessel functions has been proved in a variety of ways. Methods of the past range from spectral analysis (Lederman and Reuter (1954)), combinatorial arguments (Champernowne (1956)), to generating functions coupled with Laplace transforms (Clarke (1956)). In this paper, we present a novel approach that ties the computation of these transient distributions directly to the random sample path behavior of the M/M/1 queue. The use of Laplace transforms is minimized, and the use of generating functions is eliminated completely. This is a method that could prove to be useful in developing a similar transient analysis for queueing networks.

Keywords

TRANSIENT ANALYSISBUSY PERIODBESSEL FUNCTIONSRANDOM WALKSREFLECTION PRINCIPLE
Type
Short Communications
Information
Journal of Applied Probability , Volume 26 , Issue 2 , June 1989 , pp. 418 - 422
Copyright
Copyright © Applied Probability Trust 1989 

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References

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