Sentry Selection in Wireless Networks

Let P be a Poisson process of intensity one in the infinite plane R2. We surround each point x ofP by the open disc of radius r centred at x. Now let Sn be a fixed disc of area n, and let Cr(Sn) be the set of discs which intersect Sn. Write E r for the event that Cr(Sn) is a k-cover of Sn, and F r for the event that Cr(Sn) may be partitioned into k disjoint single covers of Sn. We prove that P(E r \ F r ) ≤ ck/log n, and that this result is best possible. We also give improved estimates for P(E r ). Finally, we study the obstructions to k-partitionability inmore detail. As part of this study, we prove a classification theorem for (deterministic) covers ofR2 with half-planes that cannot be partitioned into two single covers.