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

• Main Authors: Zeifman Alexander, Razumchik Rostislav, Satin Yacov, Kiseleva Ksenia, Korotysheva Anna, Korolev Victor
• Format: Article
• Language: English
• Published: Sciendo 2018-03-01
• Series: International Journal of Applied Mathematics and Computer Science
• Subjects: inhomogeneous birth and death processes; weak ergodicity; rate of convergence; sharp bounds; logarithmic norm; forward kolmogorov system
• Online Access: https://doi.org/10.2478/amcs-2018-0011

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.