This paper studies a spatial queueing system on a circle, polled at random locations by a myopic server that can only observe customers in a bounded neighborhood. The server operates according to a greedy policy, always serving the nearest customer in its neighborhood, and leaving the system unchanged at polling instants where the neighborhood is empty. This system is modeled as a measure-valued random process, which is shown to be positive recurrent under a natural 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 be geometrically ergodic. We also briefly discuss how the stationary mean number of customers behaves in light and heavy traffic.