Markov chains with applications in queueing theory, which have a matrix-geometric invariant probability vector
- 1 March 1978
- journal article
- research article
- Published by Cambridge University Press (CUP) in Advances in Applied Probability
- Vol. 10 (01) , 185-212
- https://doi.org/10.1017/s0001867800029542
Abstract
It is shown that a class of infinite, block-partitioned, stochastic matrices has a matrix-geometric invariant probability vector of the form (x 0, x 1,…), where xk = x 0 Rk , 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
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