Quasi-birth-and-death processes with an explicit rate matrix

We determine the equilibrium distribution for a class of quasi-birth-and-death (QBD) processes using the matrix-geometric method, which requires the determination of the rate matrix R. In contrast to most QBD processes, the class under consideration allows for an explicit description of R, by exploiting its probabilistic interpretation. We show that the problem of finding each element of R reduces to counting lattice paths in the transition diagram. For resolving this counting problem, we present an extended version of the classical Ballot theorem. We also present various examples of queueing models that fit into the class