On the size-distribution of Poisson Voronoi cells

By Járai-Szabó Ferenc, Z. Néda. 2004

535 citations recorded by the source corpus.

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two- and three-dimensional (3D) spaces there is no exact result known for the size distribution of Voronoi cells. Motivated by the simple form of the distribution function in the 1D case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size-distribution function in the practically important 2D and 3D cases as well. Denoting the dimensionality of the space by d (d=1,2,3) the f(y)=Const*y(3d-1)/2exp(-(3d+1)y/2) compact form is suggested for the normalized cell-size distribution function. By using large-scale computer simulations the viability of the proposed distribution function is studied and critically discussed.

Subfields: Computer Graphics and Computer-Aided Design, Environmental Engineering, Signal Processing

View Paper: https://www.semanticscholar.org/paper/8f5b1e98b4fc8a79492080b9edf31163f70359ad

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Citations in the corpus

Tractable Resource Management With Uplink Decoupled Millimeter-Wave Overlay in Ultra-Dense Cellular Networks (2015, background): “Consider the Voronoi cell size distribution [45] where the cell size is normalized by BS density.”
Modeling and Analyzing Millimeter Wave Cellular Systems (2016, methodology): “1) Association area distribution: The distribution of association area of a typical BS in a mmWave cellular network is assumed to be same as the approximate distribution of a typical PV cell with the same mean area as proposed in [163].”
Direct numerical simulation of horizontal open channel flow with finite-size, heavy particles at low solid volume fraction (2013, background): “of the Voronoi areas (normalized by the mean value) of randomly positioned points is independent of the particle number density (Ferenc and Néda, 2007).”
Modeling and Analysis of Cellular Networks Using Stochastic Geometry: A Tutorial (2016, background): “57 is a constant defined for the Voronoi tessellation in the R2 [224].”
Tractable Model for Rate in Self-Backhauled Millimeter Wave Cellular Networks (2014, result): “tribution of association cells and the resulting load, the analytical approximation proposed in [26] is used, where the area of a typical association cell is ascribed a gamma distribution proposed in [40] for a Poisson Voronoi with the same mean area. Note that the user association area of the tagge”
The Palm measure and the Voronoi tessellation for the Ginibre process (2006, background): “inly due to its large range of applicability: crystallography [7], astrophysics [32] and telecommunications [1], to mention only a few. This is also due to the simplicity of the simulation procedures [11, 13, 31], and to the fact that several theoretical results related to its geometrical characteri”
Joint Resource Partitioning and Offloading in Heterogeneous Cellular Networks (2013, unknown): “Using the area distribution proposed in [31] for PV C(y), the following lemma characterizes the probability mass function (PMF) of the load seen by a typical user.”
Edwards Statistical Mechanics for Jammed Granular Matter (2016, background): “A Poisson point process, e.g., leads likewise to a distribution of Voronoi cell volumes that is well described by a Γ-distribution (Ferenc and Néda, 2007; Kumar et al., 1992; Lazar et al., 2013).”
Offloading in Heterogeneous Networks: Modeling, Analysis, and Design Insights (2012, methodology): “Much progress has been made in modeling the area of Poisson Voronoi, see [30]–[32] and references therein, however that of a general multiplicatively weighted PV is an open problem.”