Markov chains with transition delta-matrix: Ergodicity conditions, invariant probability measures and applications (Q1180693)

A general analytical method is proposed for the analysis of discrete Markov processes with socalled \(\Delta_{m,n}\) or \(\Delta'_{m,n}\) transition matrices (``delta-matrices''); then applications occur in: queues, inventories, dams [see the authors, J. Appl. Math. Simulation 1, No. 1, 13-24 (1987; Zbl 0622.60076)]. The authors study some properties of stochastic \(\Delta_{m,n}\) and \(\Delta'_{m,n}\) transition matrices and their applications to the analysis of the corresponding discrete Markov processes. For processes with an infinite number of states necessary and sufficient conditions are obtained in two equivalent versions (Section 3, 4). In Section 5 the authors consider that the invariant probability measures of Markov chains of both kinds are found in terms of generating functions with transition delta-matrices; it is shown that the general method can be simplified. In Section 6 the authors analyze the problem of finding the invariant probability measure of a Markov chain with transition \(\Delta'_{m,n}\) matrix.