Let P be a Poisson process of intensity one in the infinite plane R2. We surround each point x of P 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 Erk for the event that Cr(Sn) is a k-cover of Sn, and Frk for the event that Cr(Sn) may be partitioned into k disjoint single covers of Sn. We prove that P(Erk ∖ Frk) ≤ ck / logn, and that this result is best possible. We also give improved estimates for P(Erk). 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.
Advances in Applied Probability
Required Publisher's Statement
Published by Project Euclid
Balister, Paul; Bollobás, Béla; Sarkar, Amites; Walters, Mark. Sentry selection in wireless networks. Adv. in Appl. Probab. 42 (2010), no. 1, 1--25. doi:10.1239/aap/1269611141. http://projecteuclid.org/euclid.aap/1269611141.