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.

In this paper we study a non-stationary Markovian queueing model of a twoprocessor heterogeneous system with time-varying arrival and service rates which was firstly investigated in [21], see also time-dependent analysis of this model in the recent paper [20]. In general, non-stationary queueing models have been actively studied during some decades, see, for instance [3,5,6,11,16,18] and the references therein.In the paper [20] the authors deal with the so-called "time-dependent analysis", in other words, they try to find the state probabilities on a finite interval under some initial conditions (as a rule, initially, the number of customers in the queue is zero), see for instance [2]. 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]. In [23] we proved that there are four classes of Markovian queueing models for which the reduced forward Kolmogorov system can be transformed to

In this paper, we display a method for the computation of convergence bounds for a non-stationary two-processor heterogeneous system with catastrophes, server failures and repairs when all parameters varying with time. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence and the main limiting characteristics of the queue-length process are obtained. Finally a numerical example is presented to show the effect of parameters.

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