Some properties of the rate operators in level dependent quasi-birth-and-death processes with a countable number of phases

Abstract

In this paper we consider systems which are generalizations of the standard quasi-birth-and-death processes (QBDs) in two directions. We allow transition probabilities to be level-dependent and there to be possibly infinitely many phases at each level. Our first results are explicit expressions for the matrices Gk, containing the probabilities that a process first reaches level k - 1 in phase j, given that it starts in phase i of level k, and Rk, containing the expected number of visits to phase j of level k +1 before first return to level k given that the process starts in phase i of level k. From these expressions, a generalization of the well-known matrix-geometric stationary distribution..