Aicha Bareche

Kernel density in the study of the strong stability of the M/M/1 queueing system

We use kernel density with correction of boundary effects to study the strong stability of the M/M/1 system after perturbation of arrival flow (respectively service times), to evaluate the proximity of G/M/1 (respectively M/G/1) and M/M/1 systems when the distribution G is unknown. Simulation studies are performed to support the results.

Statistical techniques for a numerical evaluation of the proximity of G/G/1 and G/M/1 queueing systems

We study the strong stability of a G/M/1 queueing system after perturbation of the service times. We are interested in the determination of the proximity error between the corresponding service time distributions of G/G/1 and G/M/1 systems, the approximation error on the stationary distributions, and confidence intervals for the difference between the corresponding characteristics of the quoted systems in the stationary state, when the general distribution of service times G in the G/G/1 system is unknown and must be estimated by means of a nonparametric estimation method. We use the Student test to accept or reject the equality of the corresponding characteristics. The boundary effects are taken into account. Simulation studies are performed to support the results.

Interest of Boundary Kernel Density Techniques in Evaluating an Approximation Error of Queueing Systems Characteristics

We show the interest of nonparametric methods taking into account the boundary correction techniques for a numerical evaluation of an approximation error between the stationary distributions of and queueing systems, when the density function of the general arrivals law in the system is unknown and defined on a bounded support. To compute this error, we use two kinds of norms: the norm and the weight norm. Numerical examples based on simulation studies are presented for the two cases of considered norms. A comparative study of the results has been provided.

Quality of the Approximation of Ruin Probabilities Regarding to Large Claims

The aim of this work is to show, on the basis of numerical examples based on simulation results, how the strong stability bound on ruin probabilities established by Kalashnikov (2000) is affected regarding to different heavy-tailed distributions.

Statistical Methodology for Approximating G/G/1 Queues by the Strong Stability Technique

We consider a statistical methodology for the study of the strong stability of the M/G/1 queueing system after disrupting the arrival flow. More precisely, we use nonparametric density estimation with boundary correction techniques and the statistical Student test to approximate the G/G/1 system by the M/G/1 one, when the general arrivals law G in the G/G/1 system is unknown. By elaborating an appropriate algorithm, we effectuate simulation studies to provide the proximity error between the corresponding arrival distributions of the quoted systems, the approximation error on their stationary distributions and confidence intervals for the difference between their corresponding characteristics.

Approximate controllability of stochastic bounds of stationary distribution of an M/G/1 queue with repeated attempts and two-phase service

ABSTRACT This paper aims to study the monotonicity properties and the stochastic controllability of some performance measures of an M/G/1 queue with repeated attempts and two-phase service. First, we prove the monotonicity of the transition operator of the embedded Markov chain relative to convex ordering. Then, we obtain comparability conditions for the distribution of the number of customers in the system. Finally, we give insensitive bounds for the stationary distribution of the embedded Markov chain of the model under consideration. To do so, we use the partial information about the aging concepts of the first essential service time distribution and the second optional service time distribution. To highlight the different obtained theoretical results, numerical examples based on simulation are provided. More precisely, we discuss numerically the conditions under which the approximation of our considered model by an M/M/1 retrial queue with exponential two-phase service is valid.

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.

Impact of Nonparametric Density EstimationDensity estimation on the ApproximationApproximation of the G∕G∕1 Queue by the M∕G∕1 One

In this paper, we show the interest of nonparametric boundary density estimation to evaluate a numerical approximation of \(G/G/1\) and \(M/G/1\) queueing systems using the strong stability approach when the general arrivals law G in the \(G/G/1\) system is unknown. A numerical example is provided to support the results. We give a proximity error between the arrival distributions and an approximation error on the stationary distributions of the quoted systems.