Application of the coupling method for the asymptotic analysis of regenerative processes in queuing theory and related problems

In queuing theory (QT) and related problems, it is very important to know the numerical characteristics of an investigated system – both in stationary and non-stationary modes. Sometimes they can be calculating, but this is not possible for all models. However, it is often possible to calculate or estimate the stationary values of the characteristics of the models under study. If for a certain characteristic the rate of convergence (or an upper estimate of the rate of convergence) to a stationary value is known, then its value can be estimated at any time. At the same time, the behavior of many processes in QT and in related fields can be describing by linear Markov processes, which are often regenerative Markov processes (RMPs). If the regeneration period of RMP has a finite average value, then RMP is ergodic. To obtain upper bounds for the rate of convergence of RMP distribution to a stationary distribution, the coupling method may be used. The purpose of this paper is to show the application of the coupling method. This article is a small review of currently actively developing methods for obtaining upper bounds of the rate of convergence of the distribution of regenerative processes and fills the existing gap in the domestic literature.