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A server moves at a constant rate around a closed tour, stopping to perform services wherever requests are encountered. Requests appear according to a Poisson process at locations independently and uniformly distributed over the tour, and the service times are all taken to be one unit. Discrete versions of this model have applications ranging from machine repair to computer/communication polling systems. The distributions of the number served in a cycle and the number of waiting requests and their waiting times are calculated, and defections are studied. Because our continuous model is simpler than the discrete models, more easily interpreted formulas and some results that have yet to be obtained for discrete models are acquired.