  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. The steady-state behavior of the system is briefly
discussed using numerical simulations and a heuristic light-traffic
approximation.
