SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS, VOL.24, NO.1, PP.281-291, 
2002.

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

AUTHORS: Tugrul Dayar and Franck Quessette

ABSTRACT: 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
on three examples.

KEY WORDS: Markov chains, quasi-birth-and-death processes, geometric
distributions