Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T18:32:23.708Z Has data issue: false hasContentIssue false
  • English
  • Français

On point processes defined by angular conditions on Delaunay neighbors in the Poisson–Voronoi Tessellation

Part of: Global differential geometry Graph theory General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 November 2021

François Baccelli Open the ORCID record for François Baccelli [Opens in a new window]  and
Sanket S. Kalamkar Open the ORCID record for Sanket S. Kalamkar [Opens in a new window]
François Baccelli*
Affiliation:
INRIA
Sanket S. Kalamkar*
Affiliation:
INRIA
*
*Postal address: INRIA Paris, France, and University of Texas at Austin, USA. Email address: [email protected]
**Postal address: INRIA Paris, France.
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a homogeneous Poisson point process of the Euclidean plane and its Voronoi tessellation. The present note discusses the properties of two stationary point processes associated with the latter and depending on a parameter $\theta$ . The first is the set of points that belong to some one-dimensional facet of the Voronoi tessellation and such that the angle with which they see the two nuclei defining the facet is $\theta$ . The main question of interest on this first point process is its intensity. The second point process is that of the intersections of the said tessellation with a straight line having a random orientation. Its intensity is well known. The intersection points almost surely belong to one-dimensional facets. The main question here concerns the Palm distribution of the angle with which the points of this second point process see the two nuclei associated with the facet. We will give answers to these two questions and briefly discuss their practical motivations. We also discuss natural extensions to three dimensions.

Keywords

Stochastic geometrypoint processrandom polytopeDelaunay graphPalm calculusmass transport principleintegral geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry 05C80: Random graphs
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 4 , December 2021 , pp. 952 - 965
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aldous, D. and Lyons, R. 2007. Processes on unimodular random networks. Electron. J. Prob. 12, 1454–1508.10.1214/EJP.v12-463CrossRefGoogle Scholar
Andrews, J. G., Baccelli, F. and Ganti, R. K. 2011. A tractable approach to coverage and rate in cellular networks. IEEE Trans. Commun. 59, 3122–3134.10.1109/TCOMM.2011.100411.100541CrossRefGoogle Scholar
Baccelli, F. and Błaszczyszyn, B. 2009. Stochastic Geometry and Wireless Networks, vol. I, Theory (Foundations and Trends in Networking). Now Publishers.10.1561/1300000006CrossRefGoogle Scholar
Baccelli, F. and Brémaud, P. 2003. Elements of Queueing Theory: Palm Martingale Calculus and Stochastic Recurrences (Applications of Mathematics: Stochastic Modelling and Applied Probability 26). Springer.Google Scholar
Baccelli, F. and Zuyev, S. 1996. Stochastic geometry models of mobile communication networks. In Frontiers in Queueing: Models and Applications in Science and Engineering, ed. J. Dshalalow, Chapter 8, pp. 227–243. CRC Press.Google Scholar
Kalamkar, S. S., Baccelli, F., Abinader, F. M. Jr, Marcano Fani, A. S. and Uzeda Garcia, L. G. 2020. Beam management in 5G: a stochastic geometry analysis. To appear in IEEE Trans. Wireless Commun. Available at https://arxiv.org/abs/2012.03181.10.1109/TWC.2021.3110785CrossRefGoogle Scholar
Møller, J. 1989. Random tessellations in $\mathbb{R}^{d}$ . Adv. Appl. Prob. 21, 37–73.10.2307/1427197CrossRefGoogle Scholar
Muche, L and Stoyan, D. 1992. Contact and chord length distributions of the Poisson Voronoi tessellation. J. Appl. Prob. 29, 467–471.10.2307/3214584CrossRefGoogle Scholar