Journal of Mathematical Sciences, Vol.218, No. 2, October, 2016 ON THE CONVERGENCE RATE FOR QUEUEING AND RELIABILITY MODELS DESCRIBED BY REGENERATIVE PROCESSES* L. G. Afanasyeva 1 and A. V. Tkachenko 2 Convergence rates in total variation are established for some models of queueing theory and reliability theory. The analysis is based on renewal technique and asymptotic results for the renewal function. It is shown that convergence rate has an exponential asymptotics when the distribution function of the regeneration period satisfies Cram´ er’s condition. Results concerning polynomial convergence are also obtained. 1. Introduction We study the rate of convergence of some models of queueing theory and reliability theory to a limit regime. This work focuses on two results concerning the estimates of the convergence rate for regenerative stochastic processes. It is shown that the convergence rate has an exponential asymptotics when the distribution function F (t) of the regeneration period satisfies Cram´ er’s condition. Another result deals with polynomial convergence. In particular, if 1 − F (t) ∼ ct −α as t →∞ for some α> 1 and c< ∞, then under a weak additional assumption the total variation distance ||P t − P|| TV between the distribution P t of the regenerative process at time t and the limit distribution P as t →∞ satisfies the inequality c 1 t −(α−1)  ||P t − P|| TV  c 2 t −(α−1) , where 0 <c 1  c 2 < ∞ are constants and t is large enough (see Corollary 1 below). Establishing bounds for the rate of convergence is a very important problem from a practical point of view. Any valuable characteristic of the quality of a queueing system is computed for the stationary regime in practice. Even if there is no exact formula or a suitable algorithm for its computation, one may construct a simpler majorizing model and use limit characteristics of this model as upper bounds for the characteristics of the system under consideration. However, if the rate of convergence is unknown, then the error is also unknown. Bounds for the rate of convergence to limit regimes were subjects of study in many publications. For a long time, the estimates of the convergence rate (mostly for exponential decay) were known only for the simplest cases where the service times had exponential distributions [2]. The exponential convergence rate for an infinite server system with a Poisson input and with non-exponential distribution of the service times may be found in [11]. The total variation asymptotics for a wide class of regenerative processes was studied in the fundamental monograph [15]. The applied approach is based on the key result for the so-called distributional coupling time. For certain particular models see also [7, 8] and [16]. It is a hard task to mention all important publications where the convergence rates for the processes emerging in the queueing models were considered. Some of the works can be found among the references in [18]. The list of major contributors includes Kalashnikov [10], Borovkov [6], Thorisson [15], and Tweedie [17]. The goal of this paper is to demonstrate the wide potential of the renewal technique in the asymptotic analysis of the convergence rate. It is well known that stochastic processes arising in the queueing theory turn out to be regenerative in many cases. The regeneration points are the moments of time when the 1 Lomonosov Moscow State University, Moscow, Russia, e-mail: afanas@mech.math.msu.su 2 National Research University Higher School of Economics, Moscow, Russia, e-mail: tkachenko_av@hse.ru * This paper is partially supported by grant of the Russian Foundation For Basic Research 13–01–00653. Proceedings of the XXXII International Seminar on Stability Problems for Stochastic Models, Trondheim, Norway, June 16–21, 2014. 1072-3374/16/2182-0119 2016 Springer Science+Business Media New York 119 DOI 10.1007/s10958-016-3015-7