Karim Abbas

A functional approximation for the M/G/1/N queue

This paper presents a new approach to the functional approximation of the M/G/1/N built on a Taylor series approach. Specifically, we establish an approximative expression for the remainder term of the Taylor series that can be computed in an efficient manner. As we will illustrate with numerical examples, the resulting Taylor series approximation turns out to be of practical value.

A functional approximation for retrial queues with two way communication

In this paper we consider retrial queues with two way communication and finite orbit. We will develop a functional approximation of the stationary performances of this queue where the parameter of interest is the outgoing rate. Specifically, we will apply the Taylor series expansion method to propagate parametric uncertainty to performance measures. We provide an analysis of the

A parametric uncertainty analysis method for queues with vacations

\hbox {M}_{1},\, \hbox {M}_{2}/\hbox {G}_{1},\, \hbox {G}_{2}/1

A Taylor series approach to the numerical analysis of the M/D/1/N queue

M1,M2/G1,G2/1 retrial queue with two way communication and finite orbit. The sensitivity analysis is carried out by considering several numerical examples. More specifically, this paper proposes a numerical approach based on Taylor series expansion with a statistical aspect for analyzing the stationary performances of the considered queueing model, where we assume that the outgoing rate is not assessed in a perfect manner. Additionally, approximate expressions of the probability density functions, the expectation and the variance of the performance measures are obtained and compared to the corresponding Monte Carlo simulations results.

A weak perturbation theory for approximations of invariant measures in M/G/1 model

This paper studies perturbations of the single server queue with breakdowns, where the perturbations consist of introducing dependent breakdowns. Problems that are modeled by queueing models with breakdowns and repairs are often very complicated, and they are resolved only through approximations. Therefore, it is very important to justify these approximations and to estimate the resultant error. For this purpose, we use the strong stability method to approximate the characteristics of the M/G/1 queue with dependent breakdowns by those of the M/G/1 queue with classical (constant) breakdowns. This latter queue is simpler and more exploitable in practice. Thus, we prove the stability conditions and next obtain stability inequalities with exactly computing of the constants. These results give with precision the error, on the queue size stationary distribution, due to the approximation. For this, we elaborate from the obtained theoretical results an algorithm allowing one to verify the approximation conditions and to provide the made numerical error. The accuracy of the approach is evaluated by comparison with true distance values.

A Numerical Analysis of the M/G/1/N Queue with Vacations

In this paper, we propose a numerical method, based on Taylor series expansion, for computing the uncertainty in performance measures of vacation queueing model due to epistemic uncertainties in its parameters. Specifically, we consider the M/G/1/N queue with vacations, where we assume that vacation parameter is derived from finite number of observations, they themselves will have epistemic uncertainty associated with them. Using this method, we compute the expected value and the variance of the stationary distribution associated with the M/G/1/N queue with vacations. In order to illustrate the applicability of the proposed method, a series of numerical examples are discussed and the obtained results are compared to the corresponding Monte Carlo simulations ones.

Perturbation Analysis of M/M/1 Queue

We discuss the application of an efficient numerical algorithm to sensitivity analysis of the GI/M/1 queue. Specifically, we use a numerical approach based on the Taylor series expansion to examine the robustness of the GI/M/1 queue to some specific perturbations in the arrival process: linear and non-linear perturbations. For each kind of perturbation we approximately compute the sensitivity of the main characteristics of the GI/M/1 queue corresponding to the case where the arrival processes are lightly different from that of the nominal queue. Numerical examples are presented to illustrate the accuracy of the proposed approach.

Approximation of Performance Measures in an M/G/1 Queue with Breakdowns

Abstract This paper presents a functional approximation of the M/D/1/N built on a Taylor series approximation. Numerical examples are carried out to illustrate the performance of our approach.