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Diffusion approximations for queues with server vacations

Published online by Cambridge University Press:  01 July 2016

Offer Kella  and
Ward Whitt
Offer Kella*
Affiliation:
Yale University
Ward Whitt*
Affiliation:
AT&T Bell Laboratories
*
∗Postal address: Department of Operations Research, Yale University, 84 Trumbull St, New Haven, CT 06520, USA.
∗∗Postal address: Room MH 2C-178, AT&T Bell Laboratories, 600 Mountain Avenue, Murray Hill, NJ 07974, USA.
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Abstract

This paper studies the standard single-server queue with unlimited waiting space and the first-in first-out service discipline, modified by having the server take random vacations. In the first model, there is a vacation each time the queue becomes empty, as occurs for high-priority customers with a non-preemptive priority service discipline. Approximations for both the transient and steady-state behavior are developed for the case of relatively long vacations by proving a heavy-traffic limit theorem. If the vacation times increase appropriately as the traffic intensity increases, the workload and queue-length processes converge in distribution to Brownian motion with a negative drift, modified to have a random jump up whenever it hits the origin. In the second model, vacations are generated exogenously. In this case, if both the vacation times and the times between vacations increase appropriately as the traffic intensity increases, then the limit process is reflecting Brownian motion, modified by the addition of an exogenous jump process. The steady-state distributions of these two limiting jump-diffusion processes have decomposition properties previously established for vacation queueing models, i.e., in each case the steady-state distribution is the convolution of two distributions, one of which is the exponential steady-state distribution of the reflecting Brownian motion obtained as the heavy-traffic limit without vacations.

Keywords

SERVICE INTERRUPTIONSLIMIT THEOREMSHEAVY TRAFFICSTOCHASTIC DECOMPOSITION
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 3 , September 1990 , pp. 706 - 729
Copyright
Copyright © Applied Probability Trust 1990 

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