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A state-dependent GI/G/1 queue

Published online by Cambridge University Press:  26 September 2008

Charles Knessl,
Charles Tier,
B. J. Matkowsky and
Z. Schuss
Charles Knessl
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, Chicago, IL 60607, USA
Charles Tier
Affiliation:
Department of Mathematics, Statistics and Computer Science, University of Illinoisat Chicago, Chicago, IL 60607, USA
B. J. Matkowsky
Affiliation:
Department of Engineering Sciences and Applied Mathematics, The Technological Institute, Northwestern University, Evanston, IL 60201, USA
Z. Schuss
Affiliation:
Department of Mathematics, Tel Aviv University, Tel Aviv, Israel
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Abstract

We consider a state-dependent GI/G/1 queueing system characterized by the unfinished work U(t) in the system at time t. We introduce state-dependence by allowing (i) the arrival process to depend on the instantaneous value of U(t), (ii) the service rate, that is, the rate at which U(t) decreases in the absence of arrivals, to depend on U(t), and (iii) the customer's service requirement to depend on U(t*) where t* denotes the instant in which that customer entered the system. We consider the limit of short inter-arrival times and small service requests and compute asymptotic approximations to the stationary density of the unfinished work, including the stationary probability of finding the system empty, using the WKB method and the method of matched asymptotic expansions.

Information

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 5 , Issue 2 , June 1994, pp. 217 - 241
Copyright
Copyright © Cambridge University Press 1994

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