Quasi-Birth-and-Death Processes with Level-Geometric Distribution

A special class of homogeneous continuous-time quasi-birth-and-death (QBD) Markov chains (MCs) which possess level-geometric (LG) stationary distribution is considered. Assuming that the stationary vector is partitioned by levels into subvectors, in an LG distribution all stationary subvectors beyond a finite level number are multiples of each other. Specifically, each pair of stationary subvectors that belong to consecutive levels is related by the same scalar, hence the term level-geometric. Necessary and sufficient conditions are specified for the existence of such a distribution, and the results are elaborated in three examples.

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