What question did this study set out to answer?

The aim is to verify ergodicity and compute the stationary distribution for quasi-birth-and-death processes with varied transition rates.

April 1, 2026Open Access

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

Key Points

The aim is to verify ergodicity and compute the stationary distribution for quasi-birth-and-death processes with varied transition rates.
Review established results for level-independent and level-dependent QBD processes.
Propose an extension to LDQBD processes without specific asymptotic assumptions.
Analyze a multi-server retrial queueing system as a practical application.
The proposed method extends ergodicity verification to more general LDQBD processes.
Demonstrated application showed effective calculations for stationary distributions in the tested queueing system.

Abstract

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.

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

Key Points

Abstract

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