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Paper details
Number 2 - June 2020
Volume 30 - 2020
On three methods for bounding the rate of convergence for some continuous-time Markov chains
Alexander Zeifman, Yacov Satin, Anastasia Kryukova, Rostislav Razumchik, Ksenia Kiseleva, Galina Shilova
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.
Keywords
inhomogeneous continuous-time Markov chains, weak ergodicity, Lyapunov functions, differential inequalities, forward Kolmogorov system