Distance from the Nucleus to a Uniformly Random Point in the 0-cell and the Typical Cell of the Poisson-Voronoi Tessellation

Abstract

Consider the distances R o and Ro from the nucleus to a uniformly random point in the typical and Crofton cells, respectively, of the d-dimensional Poisson-Voronoi (PV) tessellation. The main objective of this paper is to characterize the exact distributions of R o and Ro . First, using the well-known relationship between the Crofton cell and the typical cell, we show that the random variable Ro is equivalent in distribution to the contact distance of the Poisson point process. Next, we derive a multi-integral expression for the exact distribution of R o . Further, we derive a closed-form approximate expression for the distribution of R o , which is the contact distribution with a mean corrected by a factor equal to the ratio of the mean volumes of the Crofton and typical cells. An additional outcome of our analysis is a direct proof of the well-known spherical property of the PV cells having a large inball.

Throughput and Age of Information in a Cellular-Based IoT Network

“…where c 2 = 9 7 [25]. Finally, by substituting A( Φd ) and B(Φ d ) in ( 6) and then averaging over the above distribution of R b , we obtain the b-th moment of p b (y, Φ) as presented in the following lemma.…”