On the rate of convergence for some birth and death processes (Q1781662)

Evaluation of the rate of convergence for birth and death processes has been studied by a number of authors. There is also some interest in time-nonhomogeneous Markov chains; such chains model a variety of queueing systems. One of the interesting and important problems in this area is the problem on estimating the rate of convergence to the limit regime for ``weakly'' nonhomogeneous queues. The first such results were obtained by \textit{B. V. Gnedenko} and \textit{I. P. Makarov} [Differ. Uravn. 7, 1696--1698 (1971; Zbl 0269.60071)] and \textit{B. V. Gnedenko} and \textit{A. D. Solov'ev} [Math. Operationsforsch. Stat. 4, 379--390 (1973; Zbl 0292.60112)]. A new method of such problem solution was developed by the authors in a series of articles. It is based on two main ingredients: the logarithmic norm of semigroup generated by a linear operator and a similarity transformation of the reduced matrix of intensities of the considered Markov chain. In this article the authors deal with the classes of ``weakly'' nonhomogeneous birth and death processes and there are obtained more accurate bounds of the rate of convergence to a stationary distribution.