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On Queues with Interarrival Times Proportional to Service Times

Published online by Cambridge University Press:  27 July 2009

Israel Cidon ,
Roch Guréin ,
Asad Khamisy  and
Moshe Sidi
Israel Cidon
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
Roch Guréin
Affiliation:
IBM T. J. Watson Research Center, Yorktown Heights, New York 10598
Asad Khamisy
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
Moshe Sidi
Affiliation:
Department of Electrical Engineering Technion, Haifa 32000, Israel
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Abstract

We analyze a family of queueing systems where the interarrival time In+1 between customers n and n + 1 depends on the service time Bn of customer n. Specifically, we consider cases where the dependency between In+1 and Bn is a proportionality relation and Bn is an exponentially distributed random variable. Such dependencies arise in the context of packet-switched networks that use rate policing functions to regulate the amount of data that can arrive to a link within any given time interval. These controls result in significant dependencies between the amount of work brought in by customers/packets and the time between successive customers. The models developed in the paper and the associated solutions are, however, of independent interest and are potentially applicable to other environments.

Several scenarios that consist of adding an independent random variable to the interarrival time, allowing the proportionality to be random and the combination of the two are considered. In all cases, we provide expressions for the Laplace-Stieltjes Transform of the waiting time of a customer in the system. Numerical results are provided and compared to those of an equivalent system without dependencies.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 10 , Issue 1 , January 1996 , pp. 87 - 107
Copyright
Copyright © Cambridge University Press 1996

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