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On three methods for bounding the rate of convergence for some continuous-time Markov chains

Zeifman Alexander, Satin Yacov, Kryukova Anastasia, Razumchik Rostislav, Kiseleva Ksenia, Shilova Galina

International Journal of Applied Mathematics and Computer Science

2020

Vol. 30, no. 2

251--266

Consideration is given to 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 well 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 with 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.

Bounds on the rate of convergence for one class of inhomogeneous Markovian queueing models with possible batch arrivals and services

Zeifman A., Razumchik R., Satin Y., Kiseleva K., Korotysheva A., Korolev V.

International Journal of Applied Mathematics and Computer Science

2018

Vol. 28, no. 1

141--154

In this paper we present a method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically, we consider an inhomogeneous M/M/S queueing system with possible state-dependent arrival and service intensities, and additionally possible batch arrivals and batch service. A unified approach based on a logarithmic norm of linear operators for obtaining sharp upper and lower bounds on the rate of convergence and corresponding sharp perturbation bounds is described. As a side effect, we show, by virtue of numerical examples, that the approach based on a logarithmic norm can also be used to approximate limiting characteristics (the idle probability and the mean number of customers in the system) of the systems considered with a given approximation error.

On truncations for weakly ergodic inhomogeneous birth and death processes

Zeifman A., Satin Y., Korolev V., Shorgin S.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 3

503--518

We investigate a class of exponentially weakly ergodic inhomogeneous birth and death processes. We consider special transformations of the reduced intensity matrix of the process and obtain uniform (in time) error bounds of truncations. Our approach also guarantees that we can find limiting characteristics approximately with an arbitrarily fixed error. As an example, we obtain the respective bounds of the truncation error for an Mt/Mt/S queue for any number of servers S. Arbitrary intensity functions instead of periodic ones can be considered in the same manner.