A Note on an Efficient Approach for Solving Quasi-Birth-and-Death Processes Exhibiting Special Structures

We consider a class of Markov chains, called quasi-birth-and-death (QBD) processes, for which the stationary probability distribution, when it exists, is of the matrix-geometric form. An essential step in most algorithms for computing such a distribution is the evaluation of a rate matrix R which is a solution of a matrix quadratic equation. In this paper, we show how the eigenvalues of R can be determined explicitly when the infinitesimal generator of the QBD process has a special structure. Under certain conditions, the stationary probability vector is obtained in terms of the unique eigenvalue without computing R.