Unique coverage in Boolean models

Consider a wireless cellular network consisting of small, densely scattered base stations. A user $u$ is {\em uniquely covered} by a base station $b$ if $u$ is the only user within distance $r$ of $b$. This makes it possible to assign the user $u$ to the base station $b$ without interference from any other user $u^{\mathrm{\prime }}$. 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.