Stochastic block–monotonicity in the approximation of the stationary distribution of infinite markov chains

Abstract

Markov chains with block-structured transition matrices find many applications in various areas. Such Markov chains are characterized by having a state space which is partitioned into levels, each level consisting of a number of stages. Examples include Markov chains of GI/M/1 type and M/G/l type, and, more generally, Markov chains of Toeplitz type. The level-dependent quasi-birth-and-death (LDQBD) process provides an additional example; the transition matrix does not have repeating blocks in this case. In the analysis of such Markov chains, a number of properties and/or measures which relate to transitions among levels play a dominant role, while transitions between stages within the same level are less important In this paper, we introduce the concept of block-monotonicity and apply this notion to the analysis of Markov chains possessing block structure. In particular, the problem of approximating the stationary probability vectors is successfully treated