The probabilistic significance of the rate matrix in matrix-geometric invariant vectors

Abstract

Many queueing models have embedded Markov chains, whose invariant probability vector is of a (modified) matrix-geometric form. The rate matrix R is shown to have the following probabilistic interpretation:The element R_vj is the expected number of visits to the state (i + 1, j), before the first return to the set i = {(i, 1), …,(i, m)}, in a chain starting in the state (i, v).