Note sur la simulation d'une file M/G/1 selon la distribution du temps de service

International audienceDans cette note nous souhaitons mettre en lumière une difficulté particulière de la simulation liée aux types de distributions supposées dans le modèle. Nous appuyons notre étude sur un modèle simple de type file d'attente pour lequel une résolution analytique existe et peut donc servir de base de comparaison. Nos résultats suggèrent que l'usage de certaines distributions (dont celle de Pareto) dans les modèles tend à accentuer la complexité de leur résolution par simulation. Cet effet est d'autant plus prononcé que le coefficient de variation de la distribution est élevé. Ainsi, si dans un modèle une loi n'est connue que par ses deux (ou n- premiers) moments, il convient de choisir la distribution la plus simple à simuler

Breaking the dimensionality curse in multi-server queues

International audiencePh/Ph/c and and Ph/Ph/c/N queues can be viewed as a common model of multi-server facilities. We propose a simple approximate solution for the equilibrium probabilities in such queues based on a reduced state description in order to circumvent the well-known and dreaded combinatorial growth of the number of states inherent in the classical state description. The number of equations to solve in our approach increases linearly with the number of servers and phases in the service time distribution. A simple fixed-point iteration is used to solve these equations. Our approach applies both to open models with unrestricted buffer size and to queues with finite-size buffers. The results of a large number of empirical studies indicate that the overall accuracy of the proposed approximation appears very good. For instance, the median relative error for the mean number in the queue over thousands of examples is below 0.1% and the relative error exceeds 5% in less than 1.5% of cases explored. The accuracy of the proposed approximation becomes particularly good for systems with more than 8 servers, and tends to become excellent as the number of servers increases

A recurrent solution of Ph/M/c/N-like and Ph/M/c-like queues

International audienceWe propose an efficient semi-numerical approach to compute the steady-state probability distribution for the number of requests at arbitrary and at arrival time instants in Ph/M/c-like systems with homogenous servers in which the inter-arrival time distribution is represented by an acyclic set of memoryless phases. Our method is based on conditional probabilities and results in a simple computationally stable recurrence. It avoids the explicit manipulation of potentially large matrices and involves no iteration. Due to the use of conditional probabilities, it delays the onset of numerical issues related to floating- point underflow as the number of servers and/or phases increases. For generalized Coxian distributions, the computational complexity of the proposed approach grows linearly with the number of phases in the distribution

Une solution approchée pour les files Ph/Ph/1 et Ph/Ph/1/N

International audienceNous proposons une approximation simple pour évaluer les probabilités stationnaires du nombre de clients et les probabilités d'état trouvées à l'arrivée dans les files Ph/Ph/1 et Ph/Ph/1/N. Pour cette dernière, ceci inclut la probabilité du dépassement de capacité. Les distributions de type phase considérées ici sont acycliques. Notre méthode s'appuie sur une itération entre les solutions d'une file M/Ph/1 à taux d'arrivée dépendant de l'état et une file Ph/M/1 à taux de service dépendant de l'état. Nous résolvons ces deux files à l'aide d'une récurrence simple et efficace. En itérant entre ces deux modèles, notre approximation divise l'espace d'états, et peut ainsi traiter des distributions avec un grand nombre de phases (plus de 100) nécessaires à la représentation de distributions à queue lourde qui risquent de poser problème aux méthodes numériques classiques. La méthode proposée converge généralement en quelques dizaines d'itérations. Notre approximation est asymptotiquement exacte, et sa précision est bonne : généralement à quelques pourcents de la valeur exacte, sauf quand à la fois les distributions des inter-arrivées et du temps de service présentent une faible variabilité. Dans ce cas, en particulier pour des niveaux de charge modérés, nous ne recommandons pas l'usage de notre méthode. Keywords: File monoserveur, distribution de type phase, files Ph/Ph/1 et Ph/Ph/1/N, probabilités stationnaires, probabilités de dépassement de capacité, grand nombre de phases, solution approchée, stabilité numérique

Reducing the complexity of the performance analysis of a multi- server facilities

Systems with multiple servers are common in many areas and their correct dimensioning is in general a difficult problem under realistic assumptions on the pattern of user arrivals and service time distribution. We present an approximate solution for the underlying Ph/Ph/c/N queueing model. Our approximation decomposes the solution of the Ph/Ph/c/N queue into solutions of simpler M/Ph/c/N and Ph/M/c/N queues. It is conceptually simple, easy to implement and produces generally accurate results for the mean number in the system, as well as the loss probability. A significant speed advantage compared to the numerical solution of the full Ph/Ph/c/N queue can be gained as the number of phases representing the arrival process and/or the number of servers increases

Approximate Analysis of Production Systems

In this paper complex production systems are studied where a single product is manufactured and where each production unit stores its output in at most one buffer and receives its input from at most one buffer. The production units and the buffers may be connected nearly arbitrarily. The buffers are supposed to be of finite capacity and the goods flow is continuous. For such netwroks it is possible to estimate the throughput by applying repeated aggregation over the production units. The approximation appears to be best when the network shows some resemblance with a flow line

Manufacturing flow line systems: a review of models and analytical results

The most important models and results of the manufacturing flow line literature are described. These include the major classes of models (asynchronous, synchronous, and continuous); the major features (blocking, processing times, failures and repairs); the major properties (conservation of flow, flow rate-idle time, reversibility, and others); and the relationships among different models. Exact and approximate methods for obtaining quantitative measures of performance are also reviewed. The exact methods are appropriate for small systems. The approximate methods, which are the only means available for large systems, are generally based on decomposition, and make use of the exact methods for small systems. Extensions are briefly discussed. Directions for future research are suggested.National Science Foundation (U.S.) (Grant DDM-8914277