Approximation of an M/M/s queue by the M/M/∞ one using the operator mathod

In this paper, we provide an approximate analysis of an M/M/s queue using the operator method (strong stability method). Indeed, we use this approach to study the stability of the M/M/∞ system (ideal system), when it is subject to a small perturbation in its structure (M/M/s is the resulting perturbed system). In other words, we are interested in the approximation of the characteristics of an M/M/s system by those of an M/M/∞ one. For this purpose, we first determine the approximation conditions of the characteristics of the perturbed system, and under these conditions we obtain the stability inequalities for 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 obtained theoretical results and which is executed on the considered systems in order to compare its output results with those of simulation.