Journal of Applied Mathematics and Stochastic Analysis
Volume 4 (1991), Issue 4, Pages 333-355
doi:10.1155/S1048953391000254
Markov chains with transition delta-matrix: ergodicity
conditions, invariant probability measures and applications
1Loyola Marymount University, Department of Mathematics, Los Angeles 90045, CA, USA
2San Francisco State University, Department of Mathematics, San Francisco 94132, CA, USA
Received 1 March 1991; Revised 1 August 1991
Copyright © 1991 Lev Abolnikov and Alexander Dukhovny. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.