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Transient behavior of the M/M/1 queue via Laplace transforms

Published online by Cambridge University Press:  01 July 2016

Joseph Abate  and
Ward Whitt
Joseph Abate*
Affiliation:
At & T Bell Laboratories
Ward Whitt*
Affiliation:
At & T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories, LC 2W-E06, 184 Liberty Corner Road, Warren, NJ 07060, USA.
∗∗Postal address: AT&T Bell Laboratories, MH 2C-178, Murray Hill, NJ 07974, USA.
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Abstract

This paper shows how the Laplace transform analysis of Bailey (1954), (1957) can be continued to yield additional insights about the time-dependent behavior of the queue-length process in the M/M/1 model. A transform factorization is established that leads to a decomposition of the first moment as a function of time into two monotone components. This factorization facilitates developing approximations for the moments and determining their asymptotic behavior as . All descriptions of the transient behavior are expressed in terms of basic building blocks such as the first-passage-time distributions. The analysis is facilitated by appropriate scaling of space and time so that regulated or reflected Brownian motion (RBM) appears as the special case in which the traffic intensity ρ equals the critical value 1. An operational calculus is developed for obtaining M/M/1 results directly from corresponding RBM results as well as vice versa. The analysis thus provides useful insight about RBM approximations for queues.

Keywords

APPROACH TO STEADY STATERELAXATION TIMESBIRTH-AND-DEATH PROCESSESQUEUESBROWNIAN MOTIONFIRST-PASSAGE TIMESBUSY PERIODMOMENTS
Type
Research Article
Information
Advances in Applied Probability , Volume 20 , Issue 1 , March 1988 , pp. 145 - 178
Copyright
Copyright © Applied Probability Trust 1988 

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