For evanescent Markov processes with a single transient communicating class, it is often of interest to examine the limiting probabilities that the process resides in the various transient states, conditional on absorption not having taken place. Such distributions are known as quasi-stationary (or limiting-conditional) distributions. In this paper we consider the determination of the quasi-stationary distribution of a general level-independent quasi-birth-and-death process (QBD). This distribution is shown to have a form analogous to the matrix-geometric form possessed by the stationary distribution of a positive recurrent QBD. We provide an algorithm for the explicit computation of the quasi-stationary distribution.
"The quasi-stationary behavior of quasi-birth-and-death processes." Ann. Appl. Probab. 7 (1) 134 - 155, February 1997. https://doi.org/10.1214/aoap/1034625256