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SERVER WAITING TIMES IN INFINITE SUPPLY POLLING SYSTEMS WITH PREPARATION TIMES

Published online by Cambridge University Press:  09 December 2015

Jan-Pieter L. Dorsman ,
Nir Perel  and
Maria Vlasiou
Jan-Pieter L. Dorsman
Affiliation:
Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands E-mail: j.l.dorsman@math.leidenuniv.nl
Nir Perel
Affiliation:
Department of Industrial Engineering and Management, Shenkar - Engineering, Design and Art, Ramat Gan, Israel E-mail: perelnir@shenkar.ac.il
Maria Vlasiou
Affiliation:
EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands Stochastics, Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands E-mail: m.vlasiou@tue.nl
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Abstract

We consider a system consisting of a single server serving a fixed number of stations. At each station, there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queueing networks, to an extension of polling systems and surprisingly to random graphs. We are interested in the waiting time of the server. For the case where the server polls the stations cyclically, we give a sufficient condition for the existence of a limiting waiting-time distribution and we study the tail behavior of the stationary waiting time. Furthermore, assuming that preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We also investigate a model variation where the server does not necessarily poll the stations in a cyclic order, but always serves the customer with the earliest completed preparation phase. We show that the mean waiting time under this dynamic allocation never exceeds that of the cyclic case, but that the waiting-time distributions corresponding to both cases are not necessarily stochastically ordered. Finally, we provide extensive numerical results investigating and comparing the effect of the system's parameters to the performance of the server for both models.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 30 , Issue 2 , April 2016 , pp. 153 - 184
Copyright
Copyright © Cambridge University Press 2015 

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