Operator-geometric stationary distributions for markov chains, with application to queueing models

This paper considers a class of Markov chains on a bivariate state space , whose transition probabilities have a particular ‘block-partitioned' structure. Examples of such chains include those studied by Neuts [8] who took E to be finite; they also include chains studied in queueing theory, such as ( N n , S n ) where N n is the number of customers in a GI / G /1 queue immediately before, and S n the remaining service time immediately after, the n th arrival. We show that the stationary distribution Πfor these chains has an ‘operator-geometric' nature, with , where the operator S is the minimal solution of a non-linear operator equation. Necessary and sufficient conditions for Πto exist are also found. In the case of the GI / G /1 queueing chain above these are exactly the usual stability conditions. G 1/ G /1 QUEUE; PHASE-TYPE; INVARIANT MEASURE; FOSTER'S CONDITIONS