Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-13T00:47:21.089Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abate, J. and Whitt, W. (1987) Transient behavior of the M/M/1 queue: starting at the origin. Queueing Systems: Theory and Applications 2, 41–65.CrossRefGoogle Scholar
Baccelli, F. and Massey, W. A. (1988) A transient analysis of the two-node series Jackson network. Math. Operat. Res. To appear.Google Scholar
Champernowne, D. G. (1956) An elementary method of solution of the queueing problem with a single server and constant parameters. J. R. Statist. Soc. B18, 125–128.Google Scholar
Clarke, A. B. (1956) A waiting line process of Markov type. Ann. Math. Statist. 27, 452–459.CrossRefGoogle Scholar
Lederman, W. and Reuter, G. E. H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. R. Soc. London A246, 321–369.Google Scholar
Parthasarathy, P. R. (1987) A transient solution to an M/M/1 queue: A simple approach. Adv. Appl. Prob. 19, 997–998.CrossRefGoogle Scholar
Takács, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar