On the Solution to QBD Processes with Finite State Space

In this article, we present a solution to a class of Quasi-Birth-and-Death processes with finite state space and show that the stationary probability vector has a matrix geometric representation. We show that such models have a level-dependent rate matrix. The corresponding rate matrix is given explicitly in terms of the model parameters. The resulting closed-form expression is proposed as a basis for efficient calculation of the stationary probabilities. The method proposed in this article can be applied to several queueing systems.