Level‐Dependent Quasi‐Birth‐and‐Death Processes

Abstract

AbstractThis article defines and describes the level‐dependent quasi‐birth‐and‐death (LDQBD) process, a generalization of the homogeneous (or level‐independent) quasi‐birth‐and‐death (QBD) process. Like its level‐independent counterpart, the LDQBD process is a bivariate Markov process whose transition probability matrix (or infinitesimal generator matrix) exhibits a block tridiagonal structure. However, unlike the level‐independent QBD, its transitions are explicitly dependent on the level. This article defines discrete‐ and continuous‐time versions of the LDQBD process, discusses necessary and sufficient conditions for positive recurrence, and describes the limiting distribution for each case. Algorithmic approaches, extensions of the basic models, and suggestions for further reading are also provided.