Unique coverage in Boolean models

Consider a wireless cellular network consisting of small, densely scattered base stations. A user uu is uniquely covered   by a base station bb if uu is the only user within distance rr of bb. This makes it possible to assign the user uu to the base station bb without interference from any other user u′u′. We investigate the maximum possible proportion of users who are uniquely covered. We solve this problem completely in one dimension and provide bounds, approximations and simulation results for the two-dimensional case.