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Modified Lindley process with replacement: dynamic behavior, asymptotic decomposition and applications

Published online by Cambridge University Press:  14 July 2016

J. George Shanthikumar  and
Ushio Sumita
J. George Shanthikumar*
Affiliation:
University of California, Berkeley
Ushio Sumita*
Affiliation:
University of Rochester
*
∗Postal address: School of Business Administration, University of California, Berkeley, CA 94720, USA.
∗∗Postal address: William E. Simon Graduate School of Business Administration, University of Rochester, Rochester, NY 14627, USA.
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Abstract

We consider a discrete-time stochastic process {Wn, n≧0} governed by i.i.d random variables {ξ n} whose distribution has support on (–∞,∞) and replacement random variables {Rn} whose distributions have support on [0,∞). Given Wn, Wn+ 1 takes the value Wn + ζ n+ 1 if it is non-negative. Otherwise Wn+ 1 takes the value Rn +1 where the distribution of Rn+ 1 depends only on the value of Wn + ζn +1. This stochastic process is reduced to the ordinary Lindley process for GI/G/1 queues when Rn = 0 and is called a modified Lindley process with replacement (MLPR). It is shown that a variety of queueing systems with server vacations or priority can be formulated as MLPR. An ergodic decomposition theorem is given which contains recent results of Doshi (1985) and Keilson and Servi (1986) as special cases, thereby providing a unified view.

Keywords

QUEUES WITH SERVER VACATIONS OR PRIORITYERGODIC DECOMPOSITION THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 26 , Issue 3 , September 1989 , pp. 552 - 565
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

Partially supported by NSF Grant ECS-8601210.

Partially suported by NSF Grant ECS-8600992 and by the IBM Program of Support for Education in Management of Information Systems.

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