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

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

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Article ID: iaor1990728
Country: Israel
Volume: 26
Issue: 2
Start Page Number: 1
End Page Number: 7
Publication Date: Jun 1989
Journal: Journal of Applied Probability
Authors: ,
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), combinatorial arguments (Champernovne), to generating functions coupled with Laplace transforms (Clarke). In this paper, the authors 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.

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