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q-SERIES IN MARKOV CHAINS WITH BINOMIAL TRANSITIONS

STUDYING A QUEUE WITH SYNCHRONIZATION

Published online by Cambridge University Press:  13 November 2008

Antonis Economou  and
Stella Kapodistria
Antonis Economou
Affiliation:
Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece E-mail: [email protected]; [email protected]
Stella Kapodistria
Affiliation:
Department of Mathematics, University of Athens, Panepistemiopolis, Athens 15784, Greece E-mail: [email protected]; [email protected]
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Abstract

We consider a single-server Markovian queue with synchronized services and setup times. The customers arrive according to a Poisson process and are served simultaneously. The service times are independent and exponentially distributed. At a service completion epoch, every customer remains satisfied with probability p (independently of the others) and departs from the system; otherwise, he stays for a new service. Moreover, the server takes multiple vacations whenever the system is empty.

Some of the transition rates of the underlying two-dimensional Markov chain involve binomial coefficients dependent on the number of customers. Indeed, at each service completion epoch, the number of customers n is reduced according to a binomial (n, p) distribution. We show that the model can be efficiently studied using the framework of q-hypergeometric series and we carry out an extensive analysis including the stationary, the busy period, and the sojourn time distributions. Exact formulas and numerical results show the effect of the level of synchronization to the performance of such systems.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 23 , Issue 1 , January 2009 , pp. 75 - 99
Copyright
Copyright © Cambridge University Press 2009

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