Bounds on the rate of convergence for inhomogeneous $M/M/S$ systems with either state-dependent transitions, or batch arrivals and service, or both

TL;DR

This paper develops a unified method using logarithmic norms to compute convergence bounds for inhomogeneous multiserver queueing systems with complex features like state-dependent rates and batch processes, supported by extensive numerical analysis.

Contribution

It introduces a novel approach based on logarithmic norms to derive sharp convergence bounds for complex inhomogeneous queueing systems, including batch and state-dependent features.

Findings

The method provides sharp upper and lower bounds on convergence rates.

Numerical examples demonstrate the approach's effectiveness in approximating system characteristics.

The approach can be used for accurate estimation of limiting probabilities and mean queue length.

Abstract

In this paper one presents method for the computation of convergence bounds for four classes of multiserver queueing systems, described by inhomogeneous Markov chains. Specifically one considers inhomogeneous $M/M/S$ queueing system with possibly state-dependent arrival and service intensities and additionally possible batch arrivals and batch service. The unified approach based on 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 result, one shows by virtue of numerical examples that the approach based on logarithmic norm can also be used for approximation of limiting characteristics (idle probability and mean number of customers in the system) of the considered systems with given approximation error. Extensive numerical examples are provided.