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.

Two Approaches to the Construction of Perturbation Bounds for Continuous-Time Markov Chains

“…In the proofs of the following theorems we use the notion of the logarithmic norm of a linear operator function and related estimates of the norm of the Cauchy operator of a linear differential equation. The corresponding results were described in detail in our preceding works, see [17,52,66]. Here we restrict ourselves only to the necessary minimum.…”

“…This chain describes, for instance, the number of customers in a queueing system in which the customers arrive in groups, but are served one by one (in this case a k (t) is the arrival intensity of a group of k customers, and µ i (t) is the service intensity of the ith customer). The simplest models of this type were considered in [37], also see [65,66].…”

“…Note that it is convenient to use the results formulated above for the construction of perturbation bounds for Markov chains of the first four classes, see, e. g., [51,60,61,66].…”

On obtaining sharp bounds of the rate of convergence for a class of continuous-time Markov chains

“…Consider the following four classes of finite inhomogeneous continuoustime Markov chains, see (Zeifman et al 2018a), the detailed consideration of the corresponding countable chains is given in (Zeifman et al 2018c).…”

On limiting characteristics for a non-stationary two-processor heterogeneous system

“…Another approach is connected with the determination of the limiting mode, see [1].Essentially more information about queue-length process can be obtained using ergodicity and the corresponding estimates of the rate of convergence. A general approach to obtaining sharp bounds on the rate of convergence via the notion of the logarithmic norm of an operator function wsa recently discussed in details in our papers [23,24,25]. The first studies in this direction were published since 1980-s for birth-death models, see [14,15].…”