Markov chains with transition delta-matrix: ergodicity conditions, invariant probability measures and applications

Abstract

A large class of Markov chains with so-called Δm,n-and Δ′m,n-transition matrices (“delta-matrices”) which frequently occur in applications (queues, inventories, dams) is analyzed.The authors find some structural properties of both types of Markov chains and develop a simple test for their irreducibility and aperiodicity. Necessary and sufficient conditions for the ergodicity of both chains are found in the article in two equivalent versions. According to one of them, these conditions are expressed in terms of certain restrictions imposed on the generating functions Ai(z) of the elements of the ith row of the transition matrix, i=0,1,2,…; in the other version they are connected with the characterization of the roots of a certain associated function in the unit disc of the complex plane. The invariant probability measures of Markov chains of both kinds are found in terms of generating functions. It is shown that the general method in some important special cases can be simplified and yields convenient and, sometimes, explicit results.As examples, several queueing and inventory (dam) models, each of independent interest, are analyzed with the help of the general methods developed in the article.