Bounds on the Rate of Convergence for $M^X_t/ M^X_t/1$ Queueing Models

TL;DR

This paper develops a method using differential inequalities to estimate how quickly certain Markov chain queueing models converge to their steady state, focusing on bounds for the convergence rate.

Contribution

It introduces a novel approach applying differential inequalities to derive upper bounds on convergence rates for a specific class of Markov chain queueing models.

Findings

Derived explicit upper bounds for convergence rates

Applied the method to homogeneous and inhomogeneous Markov chains

Provided insights into the speed of convergence for queueing systems

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. To obtain these estimates, we investigate the corresponding forward system of Kolmogorov differential equations.