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November, 1992 A Continuous Polling System with General Service Times
Dirk P. Kroese, Volker Schmidt
Ann. Appl. Probab. 2(4): 906-927 (November, 1992). DOI: 10.1214/aoap/1177005580

Abstract

Consider a ring on which a server travels at constant speed. Customers arrive on the ring according to a Poisson process, at locations independently and uniformly distributed over the circle. Whenever the server encounters a customer, he stops and serves the client according to a general service time distribution. After the service is completed, the server removes the customer from the ring and resumes his round. The model is analyzed by means of point processes and regenerative processes in combination with some stochastic integration theory. This approach clarifies the analysis of the continuous polling model and provides the means for further generalizations. For every time $t$, the locations of customers that are waiting for service and the positions of clients that have been served during the last tour of the server are represented by random counting measures. These measures converge in distribution as $t \rightarrow \infty$. A recursive expression for the Laplace functionals of the limiting random measures is found, from which the corresponding $k$th moment measures can be derived.

Citation

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Dirk P. Kroese. Volker Schmidt. "A Continuous Polling System with General Service Times." Ann. Appl. Probab. 2 (4) 906 - 927, November, 1992. https://doi.org/10.1214/aoap/1177005580

Information

Published: November, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0772.60075
MathSciNet: MR1189422
Digital Object Identifier: 10.1214/aoap/1177005580

Subjects:
Primary: 60K25
Secondary: 60G55 , 60G57 , 60H05 , 90B22

Keywords: cluster representation , Cyclic server system , Laplace functionals , limit distributions , moment formulas , random counting measures , regnerative processes , stochastic decomposition , stochastic integration

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.2 • No. 4 • November, 1992
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