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The Waiting-Time Distribution for the GI/G/1 Queue under the D-Policy

Published online by Cambridge University Press:  27 July 2009

Jingwen Li  and
Shun-Chen Niu
Jingwen Li
Affiliation:
Department of Decision SciencesNational University of Singapore, 10 Kent Ridge Crescent Singapore 0511
Shun-Chen Niu
Affiliation:
School of ManagementUniversity of Texas at Dallaas P. O. Box 830688 Richardson, Texas, 75083-0688
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Abstract

We study a generalization of the GI/G/l queue in which the server is turned off at the end of each busy period and is reactivated only when the sum of the service times of all waiting customers exceeds a given threshold of size D. Using the concept of a “randomly selected” arriving customer, we obtain as our main result a relation that expresses the waiting-time distribution of customers in this model in terms of characteristics associated with a corresponding standard GI/G/1 queue, obtained by setting D = 0. If either the arrival process is Poisson or the service times are exponentially distributed, then this representation of the waiting-time distribution can be specialized to yield explicit, transform-free formulas; we also derive, in both of these cases, the expected customer waiting times. Our results are potentially useful, for example, for studying optimization models in which the threshold D can be controlled.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 6 , Issue 3 , July 1992 , pp. 287 - 308
Copyright
Copyright © Cambridge University Press 1992

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