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

Abstract

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.

Keywords

• inhomogeneous birth and death processes
• weak ergodicity
• rate of convergence
• sharp bounds
• logarithmic norm
• forward kolmogorov system