On the Three Methods for Bounding the Rate of Convergence for some Continuous-time Markov Chains

TL;DR

This paper compares three analytical methods for bounding the convergence rate of certain continuous-time Markov chains, especially in queueing systems, and demonstrates their effectiveness through numerical examples and theoretical analysis.

Contribution

It introduces less restrictive conditions for applying three methods to bound convergence rates of specific Markov chains, including queueing models, and compares their results.

Findings

All methods produce the same sharp bounds for finite homogeneous birth-death processes.

Two numerical examples illustrate the application of the methods.

Less restrictive conditions expand the applicability of the methods.

Abstract

Consideration is given to the three different analytical methods for the computation of upper bounds for the rate of convergence to the limiting regime of one specific class of (in)homogeneous continuous-time Markov chains. This class is particularly suited to describe evolutions of the total number of customers in (in)homogeneous $M/M/S$ queueing systems with possibly state-dependent arrival and service intensities, batch arrivals and services. One of the methods is based on the logarithmic norm of a linear operator function; the other two rely on Lyapunov functions and differential inequalities, respectively. Less restrictive conditions (compared to those known from the literature) under which the methods are applicable, are being formulated. Two numerical examples are given. It is also shown that for homogeneous birth-death Markov processes defined on a finite state space with all transition rates being positive, all methods yield the same sharp upper bound.