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

Abstract

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

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

“…Remark 1 One can note that the assumption of regularity of infinitesimal matrix is close to the necessary for essential non-negativity of B * (t). However, there are examples for which infinitesimal matrix is not regular and B * (t) is essential non-negative, see (Ammar and Alharbi, 2018;Dharmaraja, 2000;Zeifman et al 2019).…”

On the Rate of Convergence and Limiting Characteristics for a Nonstationary Queueing Model

“…Such transformation has been applied in a series of papers for general Markovian queueing models (see, for example, [7]). As it was mentioned above, the analysis of the rate of convergence to the limiting regime (a detailed description of this approach and its generalization can be found, for example, in [8,17,18]) was based on the logarithmic norm of an operator function from l 1 to itself, which can be computed by the simple formula:…”

“…One can find an overview of the approaches for the performance evaluation of time-dependent queueing systems up to 2016 in [1]. The papers [2][3][4] are devoted to the construction of the main performance characteristics and papers [5][6][7][8] deal with the estimation of the convergence rate and approximations. The general framework for the study of time-dependent queueing system systems is described in detail in the recent paper [8].…”

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

“…The paper [51] contained a detailed study of the stability of essentially non-stationary birth-death processes with respect to conditionally small perturbations. Convergence rate estimates in terms of weight norms and hence, the corresponding bounds for new classes of Markov chains were considered in [61,66,41,68].…”