Bounds on the Rate of Convergence for MtX/MtX/1 Queueing Models

Alexander Zeifman,Yacov Satin,Alexander Sipin

doi:10.3390/math9151752

Abstract

We apply the method of differential inequalities for the computation of upper bounds for the rate of convergence to the limiting regime for one specific class of (in)homogeneous continuous-time Markov chains. Such an approach seems very general; the corresponding description and bounds were considered earlier for finite Markov chains with analytical in time intensity functions. Now we generalize this method to locally integrable intensity functions. Special attention is paid to the situation of a countable Markov chain. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1.

Highlights

In this paper we consider the problem of finding the upper bounds for the rate of convergence for somehomogeneous continuous-time Markov chains.To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations.Consideration is given to classic inhomogeneous birth–death processes and to special inhomogeneous chains with transitions intensities, which do not depend on the current state
We investigate the corresponding forward system of Kolmogorov differential equations as a differential equation in the space of sequences l1
Studying models with continuous time from the theory of queues, biology, physics and other sciences, and obtaining guaranteed estimates of the rate of convergence, we can both make sure that the influence of the initial conditions of the system disappears with increasing time, and build the main characteristics of the system to control them

Summary

Introduction

In this paper we consider the problem of finding the upper bounds for the rate of convergence for some (in)homogeneous continuous-time Markov chains. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations. Let { X (t), t ≥ 0} be an inhomogeneous continuous-time Markov chain with the state space X = {0, 1, 2, . Throughout the paper it is assumed that in a small time interval h the possible transitions and their associated probabilities are qij (t)h + αij (t, h), if j 6= i, pij (t, t + h) =. As in [3], the four classes of of Markov chains X (t) with the following transition intensities:.

Preliminaries

Explicit Forms of the Reduced Intensity Matrices

Upper Bounds on the Rate of Convergence

Conclusions