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M/G/∞ POLLING SYSTEMS WITH RANDOM VISIT TIMES

Published online by Cambridge University Press:  18 December 2007

M. Vlasiou  and
U. Yechiali
M. Vlasiou
Affiliation:
Georgia Institute of TechnologyH. Milton Stewart School of Industrial & Systems Engineering Atlanta, GA 30332-0205USA E-mail: vlasiou@gatech.edu
U. Yechiali
Affiliation:
Department of Statistics and Operations Research School of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel Aviv 69978, Israel E-mail: uriy@post.tau.ac.il
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Abstract

We consider a polling system where a group of an infinite number of servers visits sequentially a set of queues. When visited, each queue is attended for a random time. Arrivals at each queue follow a Poisson process, and the service time of each individual customer is drawn from a general probability distribution function. Thus, each of the queues comprising the system is, in isolation, an M/G/∞-type queue. A job that is not completed during a visit will have a new service-time requirement sampled from the service-time distribution of the corresponding queue. To the best of our knowledge, this article is the first in which an M/G/∞-type polling system is analyzed. For this polling model, we derive the probability generating function and expected value of the queue lengths and the Laplace–Stieltjes transform and expected value of the sojourn time of a customer. Moreover, we identify the policy that maximizes the throughput of the system per cycle and conclude that under the Hamiltonian-tour approach, the optimal visiting order is independent of the number of customers present at the various queues at the start of the cycle.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 22 , Issue 1 , January 2008 , pp. 81 - 106
Copyright
Copyright © Cambridge University Press 2008

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