Light-Traffic Analysis for Queues with Spatially Distributed Arrivals

We consider the following continuous polling system: Customers arrive according to a homogeneous Poisson process or a more general stationary point process and wait on a circle in order to be served by a single server. The server is “greedy,” in the sense that he always moves with constant speed towards the nearest customer. The customers are served according to an arbitrary service time distribution, in the order in which they are encountered by the server. First-order and second-order Taylor-expansions are found for the expected configuration of customers, for the mean queue length, and for expectation and distribution function of the workload. It is shown that under light traffic conditions the greedy server works more efficiently than the cyclically polling server.

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