Exact Analysis of the State-Dependent Polling Model

Authors:
Mahender P. Singh;Mandyam M. Srinivasan
Affiliations:
Chicago, IL, USA singhmp@yahoo.com;Management Science Program, The University of Tennessee, Knoxville, Tennessee, USA
Venue:
Queueing Systems: Theory and Applications
Year:
2002

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Abstract

We consider a polling model in which a number of queues are served, in cyclic order, by a single server. Each queue has its own distinct Poisson arrival stream, service time, and switchover time (the server's travel time from that queue to the next) distribution. A setup time is incurred if the polled queue has one or more customers present. This is the polling model with State-Dependent service (the SD model). The SD model is inherently complex; hence, it has often been approximated by the much simpler model with State-Independent service (the SI model) in which the server always sets up for a service at the polled queue, regardless of whether it has customers or not. We provide an exact analysis of the SD model and obtain the probability generating function of the joint queue length distribution at a polling epoch, from which the moments of the waiting times at the various queues are obtained. A number of numerical examples are presented, to reveal conditions under which the SD model could perform worse than the corresponding SI model or, alternately, conditions under which the SD model performs better than a corresponding model in which all setup times are zero. We also present expressions for a variant of the SD model, namely, the SD model with a patient server.