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We consider a single server Markovian queueing system with Bernoulli feedback, balking, server's states-dependent reneging, and retention of reneged customers under variant of multiple vacation policy. We obtain the steady-state solution of the considered queueing system based on the use of probability generating functions (PGFs). Then, the closed-form expressions of different system characteristics are derived. Finally, we present some numerical results in order to show the impact of the parameters of impatience timer rates on performance measures of the system.

This paper deals with a finite capacity multi-server Markovian queueing model with Bernoulli feedback, synchronous multiple vacation policy and customers’ impatience (balking and reneging). By empl...

The stationary distribution is the key of any queueing system; its determination is sufficient to infer the corresponding characteristics. This paper deals with the Er / M / 1 {\mathrm{Er}/M/1} queue. Bayesian inference is developed to estimate the system parameters, specially the root of the relative equation which allows the determination of the stationary distribution. A numerical study is performed with MCMC methods to support the results, and a comparison with another existing method in the literature (moments method) is provided.

In this investigation, a single server M/M/1/N feedback queueing system with vacation, balking, reneging and retention of reneged customers is analyzed. By considering the mathematical modeling, we derive the steady state probabilities of the number of customers in the system. We obtain important measures of effectiveness of the model by using the stationary distribution, and develop a cost model of the queueing system. Further, a numerical study and a cost profit analysis are carried out.

In this article we analyze the M=G=1 retrial queue with two-way communication and n types of outgoing calls from a stochastic comparison viewpoint. The main idea is that given a complex Markov chain that cannot be analyzed numerically, we propose to bound it by a new Markov chain, which is easier to solve by using a stochastic comparison approach. Particularly, we study the monotonicity of the transition operator of the embedded Markov chain relative to the stochastic and convex orderings. Bounds are also obtained for the stationary distribution of the embedded Markov chain at departure epochs. Additionally, the performance measures of the considered system can be estimated by those of an M=M=1 retrial queue with two-way communication and n types of outgoing calls when the service time distribution is NBUE (respectively, NWUE). Finally, we test numerically the accuracy of the proposed bounds.

This paper deals with a batch arrivals queue with general retrial time, breakdowns, repairs and reserved time. Here we assume that customers arrive according to compound Poisson processes. Any arri...

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.

In this paper, we aim at highlighting the influence of the density pole on the performances of its gamma-kernel estimator. To do this, we performed a comparative study for the performances of the gamma-kernel estimators with those provided by other bias effect correction techniques at the bounds, using the simulation technique. In conclusion, the results obtained confirm those provided in the literature and show that in some cases the normalization of the gamma estimators can considerably improve local and global performances of the gamma-kernel estimators.

Dans ce travail, nous nous sommes focalises sur les proprietes de monotonie qui permettent d’etablir quelques bornes stochastiques utiles dans la comprehension de modeles compliques et leur remplacement par des modeles plus simples pour lesquels, une evaluation peut etre faite. Nous avons considere la file d’attente M / G / 1 / 1 avec rappels. Nous avons derive differentes inegalites stochastiques par rapport aux ordres stochastique, convexe croissant et transformee de Laplace, qui assurent la monotonie de l’operateur de transition associe a la chaine de Markov induite. Les inegalites stochastiques obtenues fournissent des bornes simples pour la distribution stationnaire de la chaine de Markov incluse liee au modele d’attente etudie. De plus, des exemples numeriques sont donnes pour illustrer les resultats theoriques obtenus.

This paper considers a non-Markovian priority retrial queue which serves two types of customers. Customers in the regular queue have priority over the customers in the orbit. This means that the customer in orbit can only start retrying when the regular queue becomes empty. If another customer arrives during a retrial time, this customer is served and the retrial has to start over when the regular queue becomes empty again. In this study, a particular interest is devoted to the stochastic monotonicity approach based on the general theory of stochastic orders. Particularly, we derive insensitive bounds for the stationary joint distribution of the embedded Markov chain of the considered system.

The main goal in the present paper is to provide a technique that considers the stochastic comparison approach for investigating monotonicity and comparability of an $M_{1},M_{2}/G_{1},G_{2}/1$ retrial queues with two way communication. This approach is developed for comparing a non Markov process to Markov process with many possible stochastic orderings. Particularly, we show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and convex ordering, as well as the comparability of two transition operators. Bounds are also obtained for the stationary distribution of the number of customers at departure epochs. Additionally, the performance measures of the system considered can be estimated by those of an $M_1,M_2/M_1,M_2/1$ retrial queue with two way communication when the service time distribution is NBUE (respectively NWUE). Finally, we validate stochastic comparison results by presenting a numerical example illustrating the interest of the approach.

Resume Dans cette Note, nous etudions le probleme du choix du parametre de lissage pour un estimateur a noyau de l'operateur de transition d'une chaine de Markov. Pour ce faire, nous avons considere la file d'attente G I / M / 1 / N . Nous avons constate que l'estimateur du parametre de lissage choisi, par la minimisation d'une certaine norme matricielle, donne de meilleurs resultats, en termes de vitesse de convergence de l'erreur quadratique moyenne, que les alternatives classiques.

Dans ce papier, nous avons aborde le probleme de la non conservation de masse des estimateurs a noyaux gamma (gamma et gamma modie) pour des echantillons nis. Pour cela, deux approches (Macro-normalisation et Micro-normalisation) ont ete proposees pour remedier a ce probleme. An d'illustrer la necessite de ces approches, une etude de comparaison par simulation a ete realisee. Par consequent, les resultats obtenus montrent qu'il est preferable d'utiliser les estimateurs a noyaux gamma normalises. English Abstract In this paper, we study the nonunit mass of the gamma (gamma and the modied gamma) kernels estimator for the nite samples. For this, we propose to use the normalization approach (Macro-normalization and the Micro-Normalization) to correct the problem of the nonunit mass. In the aim to illustrate the necessity of the normalization approach a detailed simulation study investigates the local and global performances of the considered estimators. The obtained results show the good performances of the normalized gamma kernels estimators. Keywords: Propriete d'unite de masse; Approches de normalisation; Simulation

This paper focuses on stochastic comparison of the Markov chains to derive some qualitative approximations for anM/G/1retrial queue with a Bernoulli feedback. The main objective is to use stochastic ordering techniques to establish various monotonicity results with respect to arrival rates, service time distributions, and retrial parameters.

In this paper, we consider an M/G/1 retrial queue with vacations and we derive several stochastic comparison properties in the sense of strong stochastic ordering and convex ordering. The stochastic inequalities provide simple insensitive bounds for the stationary queue length distribution.

This paper deals with a flexible multi-station manufacturing system modelled by re-entrant queueing model. Our model incorporates classical queueing systems with exponential service times and controlled arrival process under a priority service discipline. The system is decomposed into N fundamental multi-productive stations and 2N − 1 classes, a part follows the route fixed by the system, where each one is processed by N stations requiring 2N − 1 services. We assume that there is an infinite supply of work available, so that there are always parts ready for processing step 1. Our purpose in this paper is to present a detailed theoretical and simulation analysis of this priority multi-station manufacturing system.

Dans ce travail, nous utilisons les outils mathématiques de l’analyse qualitative pour étudier les propriétés de monotonie et de comparabilité, du modèle d'attente avec rappels à un seul serveur avec Bernoulli feedback et clients négatifs, par rapport aux ordres stochastiques donnés. Les mesures de performance d'un tel système sont disponibles sous des formes explicites mais complexes. Elles ne sont donc pas faciles à interpréter pour que le praticien puisse en bénéficier. Pour pallier ces difficultés, les méthodes de comparaison stochastique ont été introduites pour qu'on puisse avoir des estimations qualitatives de ces mesures en les bornant par des mesures de performance d'autres modèles plus simples. Particulièrement, nous avons prouvé la monotonie de l'opérateur de transition de la chaîne de Markov incluse. Nous avons également établi des conditions pour lesquelles les opérateurs de transition ainsi que les probabilités stationnaires, associées à deux chaînes de Markov incluses, ayant la même structure mais avec des paramètres différents, sont comparables au sens des ordres stochastiques donnés. Des exemples numériques sont donnés pour illustrer les résultats théoriques obtenus.