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2 International Journal of Applied Mathematics and Computer Science

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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.