A continuous polling system with general service times

Dirk Kroese, Volker Schmidt

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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.
Original languageEnglish
Pages (from-to)906-927
Number of pages22
JournalAnnals of applied probability
Volume2
Issue number4
DOIs
Publication statusPublished - 1992

Keywords

  • Laplace functionals
  • cluster representation
  • limit distributions
  • regnerative processes
  • random counting measures
  • IR-70358
  • moment formulas
  • stochastic integration
  • Cyclic server system
  • METIS-140669
  • stochastic decomposition

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