Sentry selection in wireless networks
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. © 2010 Applied Probability Trust.
Advances in Applied Probability
Balister, P., Bollobás, B., Sarkar, A., & Walters, M. (2010). Sentry selection in wireless networks. Advances in Applied Probability, 42 (1), 1-25. https://doi.org/10.1239/aap/1269611141