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This paper studies a spatial queueing system on a circle, polled at randomlocations by a myopic server that can only observe customers in a boundedneighborhood. The server operates according to a greedy policy, always servingthe nearest customer in its neighborhood, and leaving the system unchanged atpolling instants where the neighborhood is empty. This system is modeled as ameasure-valued random process, which is shown to be positive recurrent under anatural stability condition that does not depend on the server's scan range.When the interpolling times are light-tailed, the stable system is shown to begeometrically ergodic. The steady-state behavior of the system is brieflydiscussed using numerical simulations and a heuristic light-trafficapproximation.

Stability of a spatial polling system with greedy myopic service