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

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 ).