The Annals of Applied Probability

A Continuous Polling System with General Service Times

Dirk P. Kroese and Volker Schmidt

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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.

Article information

Ann. Appl. Probab., Volume 2, Number 4 (1992), 906-927.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 90B22: Queues and service [See also 60K25, 68M20] 60G55: Point processes 60G57: Random measures 60H05: Stochastic integrals

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


Kroese, Dirk P.; Schmidt, Volker. A Continuous Polling System with General Service Times. Ann. Appl. Probab. 2 (1992), no. 4, 906--927. doi:10.1214/aoap/1177005580.

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