Sentry Selection in Wireless Networks

Abstract

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