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Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector

Published online by Cambridge University Press:  01 July 2016

Marcel F. Neuts
Marcel F. Neuts*
Affiliation:
University of Delaware
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Abstract

It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x0, x1,…), where xk = x0Rk, for k ≧ 0. The rate matrix R is an irreducible, non-negative matrix of spectral radius less than one. The matrix R is the minimal solution, in the set of non-negative matrices of spectral radius at most one, of a non-linear matrix equation.

Applications to queueing theory are discussed. Detailed explicit and computationally tractable solutions for the GI/PH/1 and the SM/M/1 queue are obtained.

Keywords

STOCHASTIC MATRICESMARKOV CHAINSMARKOV RENEWAL PROCESSESCOMPUTATIONAL PROBABILITYPROBABILITY DISTRIBUTIONS OF PHASE TYPEGI/PH/1SM/M/1NON-LINEAR MATRIX EQUATIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 1 , March 1978 , pp. 185 - 212
Copyright
Copyright © Applied Probability Trust 1978 

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