<abstract xmlns="http://www.w3.org/1999/xhtml"><p>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 <italic>M/M/S</italic> 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.</p></abstract>

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.

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

International Journal of Applied Mathematics and Computer Science

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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...

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

Publié en ligne: 04 juil. 2020

Pages: 251 - 266

Reçu: 08 oct. 2019

Accepté: 20 janv. 2020

DOI: https://doi.org/10.34768/amcs-2020-0020

Mots clés
inhomogeneous continuous-time Markov chains, weak ergodicity, Lyapunov functions, differential inequalities, forward Kolmogorov system

© 2020 Alexander Zeifman et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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

Publié en ligne: 04 juil. 2020

Pages: 251 - 266

Reçu: 08 oct. 2019

Accepté: 20 janv. 2020

DOI: https://doi.org/10.34768/amcs-2020-0020

Mots clésinhomogeneous continuous-time Markov chains, weak ergodicity, Lyapunov functions, differential inequalities, forward Kolmogorov system

© 2020 Alexander Zeifman et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Mots clés
inhomogeneous continuous-time Markov chains, weak ergodicity, Lyapunov functions, differential inequalities, forward Kolmogorov system