# Solutions of the basic matrix equation for m/g/1 and g/m/1 type markov chains

## Abstract

Let [formula ommitted] be a sequence of nonnegative matrices such that [formula ommitted] is a substochastic matrix. The unique minimal nonnegative solution of the matrix equation [formula ommitted] has been shown by M. F. Neuts to play a key role in the analysis of M/G/1 type Markov chains. In this paper, all of the power-bounded, matrix solutions of this equation are classified. Among these solutions, the subsets of nonnegative, substochastic and stochastic solutions are identified. In particular, the exact conditions under which the equation has infinitely many power-bounded solutions (infinitely many stochastic solutions) are given. Similar results are obtained for the solutions of the matrix equation [formula ommitted], which appears in the analysis of G/M/1 type Markov chains. © 1994, Taylor & Francis Group, LLC. All rights reserved.