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A generalization of little's law to moments of queue lengths and waiting times in closed, product-form queueing networks

Published online by Cambridge University Press:  14 July 2016

James McKenna
James McKenna*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: AT&T Bell Laboratories Murray Hill, NJ 07974, USA.
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Abstract

Little's theorem states that under very general conditions L = λW, where L is the time average number in the system, W is the expected sojourn time in the system, and λ is the mean arrival rate to the system. For certain systems it is known that relations of the form E((L)l) = λ lE((W)l) are also true, where (L)l = L(L – 1)· ·· (L – l + 1). It is shown in this paper that closely analogous relations hold in closed, product-form queueing networks. Similar expressions relate Nji and Sji, where Nji is the total number of class j jobs at center i and Sji is the total sojourn time of a class j job at center i, when center i is a single-server, FCFS center. When center i is a c-server, FCFS center, Qji and Wji are related this way, where Qji is the number of class j jobs queued, but not in service at center i and Wji is the waiting time in queue of a class j job at center i. More remarkably, generalizations of these results to joint moments of queue lengths and sojourn times along overtake-free paths are shown to hold.

Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 1 , March 1989 , pp. 121 - 133
Copyright
Copyright © Applied Probability Trust 1989 

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