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Abstract
In an M/M/s queueing system a server that completes service and finds no waiting units in line leaves for a vacation of an exponentially distributed duration. At the end of the vacation the server returns to the main system. Two models are analysed. In the first, a server returning to an empty queue takes immediately another vacation. In the second, only a single vacation is taken each time. For model 1, formulas for the distribution of the number of busy servers and the mean number of units in system, L, are derived. Numerical calculations indicate that L is very closely a linear function of the mean vacation time. Finally it is shown that model 2 may be analysed similarly to model 1.
Résumé
Dans une systéme de file d’attente M/M/s si, aprés avoir complété le service d’une unité, un serveur se retrouve devant une file d’attente vide, alors il se retire pour un congé d’une durée ayant une distribution exponentielle. Une fois son congé termine, ce dernier revient au systéme. Deux modéles sont étudiés. Dans le premier modéle, si au retour du serveur la file d’attente est vide, ce dernier se retire pour un nouveau congé. Dans le second modéle, un serveor ne peut se permettre deux périodes de congé successives. Pour le premier modéle, on dérive des formules pour la distribution du nombre de serveurs au travail et pour le nombre moyen d’unités dans le systéme, L. Des expériences numériques indiquent que L varie presque linéairement avec la durée moyenne d’un congé. Finalement on démontre qu’on peut réaliser une analyse similaire du second modéle.
Additional information
Notes on contributors
Yonatan Levy
YoNATAN LEVY is a PH D candidate at Johns Hopkins University. He received a B SC in Mathematics and Statistics and an MA in Operations Research, both from Tel-Aviv University. Mr Levy is interested in optimal control of queues and in stochastic decision processes.
Uri Yechiali
URI YECHIALI is a senior lecturer of Operations Research in the Department of Statistics at Tel-Aviv University. He earned B sc and M sc from the Technion, Israel, and Doctor of Eng. Science from Columbia University. His articles have appeared in various professional journals of Operations Research and Statistics, and he served as the Program Chairman of TIMS xx International Meeting. Dr Yechiali’s current research interests are in optimal control of queueing systems, and in optimal design of teletraffic networks.