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Distribution of Cell Area in Bounded Poisson Voronoi Tessellations with Application to Secure Local Connectivity

Research output: Contribution to journalArticle

Original languageEnglish
Number of pages20
JournalJournal of Statistical Physics
Early online date21 Jun 2019
DOIs
DateAccepted/In press - 15 Jun 2019
DateE-pub ahead of print (current) - 21 Jun 2019

Abstract

Poisson Voronoi tessellations have been used in modeling many types of systems across different sciences, from geography and astronomy to telecommunications. The existing literature on the statistical properties of Poisson Voronoi cells is vast, however, little is known about the properties of Voronoi cells located close to the boundaries of a compact domain. In a domain with boundaries, some Voronoi cells would be naturally clipped by the boundary, and the cell area falling inside the deployment domain would have different statistical properties as compared to those of non-clipped Voronoi cells located in the bulk of the domain. In this paper, we consider the planar Voronoi tessellation induced by a homogeneous Poisson point process of intensity λ>0 in a quadrant, where the two half-axes represent boundaries. We show that the mean cell area is less than λ−1 when the seed is located exactly at the boundary, and it can be larger than λ−1 when the seed lies close to the boundary. In addition, we calculate the second moment of cell area at two locations for the seed: (i) at the corner of a quadrant, and (ii) at the boundary of the half-plane. We illustrate that the two-parameter Gamma distribution, with location-dependent parameters calculated using the method of moments, can be of use in approximating the distribution of cell area. As a potential application, we use the Gamma approximations to study the degree distribution for secure connectivity in wireless sensor networks deployed over a domain with boundaries.

    Research areas

  • Clipped Voronoi cells, physical layer security, Poisson Voronoi tessellation, Stochastic Geometry

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  • Full-text PDF (accepted author manuscript)

    Rights statement: This is the author accepted manuscript (AAM). The final published version (version of record) is available online via Springer at https://link.springer.com/article/10.1007%2Fs10955-019-02343-y . Please refer to any applicable terms of use of the publisher.

    Accepted author manuscript, 758 KB, PDF-document

    Embargo ends: 21/06/20

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