IFAC-PapersOnLine 49-12 (2016) 863–868 ScienceDirect ScienceDirect Available online at www.sciencedirect.com 2405-8963 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Control. 10.1016/j.ifacol.2016.07.883 © 2016, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Infinite-servers queue, Embedded Markov chain, Perturbation, Strong stability. 1. INTRODUCTION Infinite servers queues (multi-servers queues) are common in real life. Indeed, in practice, many situations are mod- eled by an infinite-servers queue, for instance: the pro- duction systems, computer systems, computer communi- cations and telecommunications, aerodrome management of an airport, ... However, the analytical results of such systems are generally expressed in terms of Laplace trans- forms or/and generating functions, which are not available in a closed form. For this reason there exists, when a practical study is performed in queueing theory, a common technique for substituting the real but complicated elements governing a queueing system by simpler ones in some sense close to the real elements. The queueing model so constructed represents an idealization of the real queueing one, and hence the stability problem arises. The stability problem in queueing theory is concerned with the domain within which the ideal queueing model may be taken as a good approximation of the real queueing system under consid- eration. In other words, we clarify the conditions for which the proximity in one way or another of the parameters of the system involves the proximity of the studied charac- teristics. In recent years, the practical needs have driven the re- search towards the determination of estimates and quan- titative performance measurement methods of stability. It is why we will place more emphasis, in this work, on the strong stability method (see A¨ ıssani and Kar- tashov (1983); Kartashov (1996)) which allows us to make both qualitative and quantitative analysis helpful in understanding complicated models by more simpler ones for which an evaluation can be made. This method, also called ”method of operators” can be used to investigate the ergodicity and stability of the stationary and non- stationary characteristics of the imbedded Markov chains (see Kartashov (1996)). In contrast to other methods, it supposes that the perturbations of the transition kernel are small with respect to some norms in the operators space. This stringent condition gives better stability estimates and enables us to find precise asymptotic expansions of the characteristics of the perturbed system. Besides, note that, in practice, all model parameters are imprecisely known because they are obtained by means of statistical methods. That is why the strong stability inequalities will allow us to numerically estimate the uncertainty shown during this analysis. The applicability of the strong stability method is well proved and documented in various fields and for different purposes. In particular, it has been applied to several queueing models (see for example Bareche and A¨ ıssani (2008); Benaouicha and A¨ ıssani (2005); Berdjoudj et al. (2012); Bouallouche-Medjkoune and A¨ ıssani (2006a,b)). On the other hand, note that there are some classical works, in the literature, about the approximation of the infinite servers queues. A traditional trend is the use of the heavy traffic approximation based on a diffusion process (see Gross et al. (2013); Glynn and Whitt (1991) for Aicha Bareche * Mouloud Cherfaoui ** Djamil A¨ ıssani *** * Research Unit LaMOS (Modeling and Optimization of Systems), University of Bejaia, 06000 Bejaia, Algeria, (e-mail: aicha bareche@yahoo.fr). ** Department of Mathematics, University of Biskra, 07000 Biskra, Algeria (e-mail: mouloudcherfaoui2013@gmail.com) *** Research Unit LaMOS (Modeling and Optimization of Systems), University of Bejaia, 06000 Bejaia, Algeria, (e-mail: lamos bejaia@hotmail.com) Abstract: In this work, we are interested in the approximation of the stationary characteristics of the GI/M/∞ system by those of an M/M/∞ system. In other words, we propose to study the strong stability of the M/M/∞ system (ideal system) when the arrivals flow is subject to a small perturbation (the GI/M/∞ is the resulting perturbed system). For this purpose, we first determine the approximation conditions of the characteristics of the perturbed queuing system, and under these conditions we obtain the stability inequalities of the stationary distribution of the queue size. To evaluate the performance of the proposed method, we develop an algorithm which allows us to compute the various theoretical results and which is executed on some systems (Coxian 2 /M/∞ and E 2 /M/∞) in order to compare its output results with those of simulation. Approximate analysis of an GI/M/∞ queue using the strong stability method