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

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 x_k = x₀R^k, 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.