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Markov processes whose steady state distribution is matrix-exponential with an application to the GI/PH/1 queue

Published online by Cambridge University Press:  01 July 2016

Bhaskar Sengupta
Bhaskar Sengupta*
Affiliation:
AT & T Bell Laboratories, Holmdel
*
∗Postal address: HO 3L-309, AT & T Bell Laboratories, Holmdel, NJ 07733, USA.
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Abstract

This paper is concerned with a bivariate Markov process {Xt, Nt; t ≧ 0} with a special structure. The process Xt may either increase linearly or have jump (downward) discontinuities. The process Xt takes values in [0,∞) and Nt takes a finite number of values. With these and additional assumptions, we show that the steady state joint probability distribution of {Xt, Nt; t ≧ 0} has a matrix-exponential form. A rate matrix T (which is crucial in determining the joint distribution) is the solution of a non-linear matrix integral equation. The work in this paper is a continuous analog of matrix-geometric methods, which have gained widespread use of late. Using this theory, we present a new and considerably simplified characterization of the waiting time and queue length distributions in a GI/PH/1 queue. Finally, we show that the Markov process can be used to study an inventory system subject to seasonal fluctuations in supply and demand.

Keywords

BIVARIATE MARKOV PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 1 , March 1989 , pp. 159 - 180
Copyright
Copyright © Applied Probability Trust 1989 

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