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Operator-geometric stationary distributions for markov chains, with application to queueing models

Published online by Cambridge University Press:  01 July 2016

R. L. Tweedie
R. L. Tweedie*
Affiliation:
CSIRO Division of Mathematics and Statistics
*
∗Present address: SIROMATH Pty Ltd, 1 York St., Sydney, NSW 2000, Australia.
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Abstract

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] who took E to be finite; they also include chains studied in queueing theory, such as (Nn, Sn) where Nn is the number of customers in a GI/G/1 queue immediately before, and Sn the remaining service time immediately after, the nth arrival.

We show that the stationary distribution Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI/G/1 queueing chain above these are exactly the usual stability conditions.

G1/G/1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS

Type
Research Article
Information
Advances in Applied Probability , Volume 14 , Issue 2 , June 1982 , pp. 368 - 391
Copyright
Copyright © Applied Probability Trust 1982 

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