Document Type

Article

Publication Date

2010

Keywords

Poisson process, coverage, partition

Abstract

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(ErkFrk) ≤ 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.

Publication Title

Advances in Applied Probability

Volume

42

Issue

1

First Page

1

Last Page

25

DOI

http://dx.doi.org/10.1239/aap/1269611141

Required Publisher's Statement

Published by Project Euclid

DOI: 10.1239/aap/1269611141

Comments

This is the authors' version of the paper. The publisher's version is here: http://projecteuclid.org/euclid.aap/1269611141

Subjects - Topical (LCSH)

Poisson processes; Partitions (Mathematics); Wireless sensor networks--Mathematical models

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

Included in

Mathematics Commons

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