On the method of finding of non-stationary states probabilities of Markov chains with continuous time and finite number of state and their application
The introduction indicates the object of research - a Markov chain (MC) with continuous time and finite number of states and the relevance of this topic. The intensities of transitions between the states of the MC and the intensity of the exit of the MC from the states depend on these states and do independ on time. In the first part of the article, a system of differential equations (DE) for the probabilities of states of the above MC is derived. The method of successive approximations, combined with the method of series, is proposed for solving this system of equations and its advantages are indicated in comparison with the previously known methods for solving systems of differential equations, as well as in comparison with the previously known generalization for open Markov QN. The properties of successive approximations are proved: each successive approximation over time converges to a stationary solution of the DE system, and the sequence of approximations itself converges to a single solution of the DE system; any successive approximation can be represented as a power series with an infinite radius of convergence. The last property is convenient for calculating the probabilities of states on a computer. In the second part of the article, applications of this method are presented for the analysis of closed Markov queueing networks (QN) with bounded waiting times for customers in queues of service systems and unreliable queueing systems (QS). However, the method proposed by the authors can be used to find the probabilities of states of closed QS with other features, for example, with bypasses of customers of service systems, with different types of customers, etc. Model examples are calculated on a computer.