Development of a Constructive Method for Verification of Ergodicity and Calculation of the Stationary Distribution of Quasi-Birth-And-Death Processes with Spatially Inhomogeneous Transitions

The problem of verifying ergodicity and calculating the stationary distribution for continuous-time multidimensional Markov chains with a block tridiagonal generator (Quasi-Birth-and-Death (QBD) processes) is investigated. This work reviews established results for QBD processes with spatially inhomogeneous transitions (level-independent QBD processes—LIDQBD processes) and level-dependent QBD processes (LDQBD processes) that require specific asymptotic assumptions. We propose an extension of these results to LDQBD processes that do not rely on such assumptions. The practical application of the proposed methodology is demonstrated through the analysis of a multi-server retrial queueing system characterized by dependent arrival and retrial processes.