On an estimate of the rate of convergence for regenerative processes in queuing theory and related problems

TL;DR

This paper discusses how to estimate the convergence rate of regenerative Markov processes in queuing and reliability systems using the adjunction method, providing strict upper bounds for their stationary distribution convergence.

Contribution

It demonstrates the application of the adjunction method to derive strict upper bounds for the convergence rate of ergodic Markov processes in queuing and reliability models.

Findings

The adjunction method effectively bounds the convergence rate.

Applicable to queuing systems, networks, and reliability models.

Provides a practical approach for analyzing system stability.

Abstract

In queuing theory and related problems, it is very important to know the numerical characteristics of an investigated system - both in stationary and non-stationary modes. In some cases, such characteristics can be calculated, but this is possible for a limited number of real models. However, in most cases, it is possible to calculate or estimate the stationary values of the characteristics of the studied models. We can use linear Markov processes to model how the overwhelming majority of queuing systems (QS), reliability systems, and queuing networks (QNS) operate; the processes are regenerative in many cases. In the case when the regeneration period of the Markov process has a finite average value, this process is ergodic. An adjunction method can be used to obtain strict upper bounds for the convergence rate of the Markov process (MP) distribution to a stationary distribution (naturally, in cases where the MP under consideration is ergodic). This method can be applied to analyze queuing systems, queuing networks, and reliability systems. The purpose of this paper is to show how to use the adjunction method to obtain strict upper bounds for the MP distribution convergence rate as the regenerating Markov process converges to a stationary distribution if the Markov process is ergodic. This method can be applied to QS, QNS, and reliability systems. The article is a summary of the plenary report of the authors at the VIII International Youth Scientific Conference "Mathematical and software support of information, technical, and economic systems"; (Tomsk, May 26-30, 2021). In Russian.