page_title: Stochastic geometry models of wireless networks
section_title: Geometrical objects of interest in wireless networks
hierarchical_section_title: Stochastic geometry models of wireless networks / Overview / Geometrical objects of interest in wireless networks
caption_attribution_description: English: This shows the simulation of 4 Poisson Boolean models as the density of points increases (top left has the smallest density, bottom right the largest). Radii are constant (also known as the Gilbert disk model). The red clusters are the largest components in the sample window.
context_page_description: In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.
  In the early 1960s a stochastic geometry model was developed to study wireless networks. This model is considered to be pioneering and the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks.
context_section_description: There are number of examples of geometric objects that can be of interest in wireless networks. For example,  consider a collection of points in the Euclidean plane. For each point, place in the plane a disk with its center located at the point.  The disks are allowed to overlap with each other and the radius of each disk is random and (stochastically) independent of all the other radii. The mathematical object consisting of the union of all these disks is known as a Boolean (random disk) model and may represent, for example, the sensing region of a sensor network.  If all the radii are not random, but common positive constant, then the resulting model is known as the Gilbert disk (Boolean) model.
  Instead of placing disks on the plane, one may assign a disjoint (or non-overlapping) subregion to each node. Then the plane is partitioned into a collection of disjoint subregions. For example, each subregion may consist of the collection of all the locations of this plane that are closer to some point of the underlying point pattern  than any other point of the point pattern. This mathematical structure is known as a Voronoi tessellation and may represent, for example, the association cells in a cellular network where users associate with the closest base station.
  Instead of placing a disk or a Voronoi cell on a point, one could place a cell defined from the information theoretic channels described above. For instance, the point-to-point channel cell of a point was defined  as the collection of all the locations of the plane where a receiver could sustain a point-to-point channel with a certain quality from a transmitter located at this point. This, given that the other point is also an active transmitter, is a point-to-point channel in its own right .
  In each case,  the fact that the underlying point pattern is random (for example, a point process) or deterministic (for example, a lattice of points) or some combination of both, will influence the nature of the Boolean model, the Voronoi tessellation, and other geometrical structures such as the point-to-point channel cells constructed from it.
caption: Simulation of four Poisson–Boolean (constant-radius or Gilbert disk) models as the density increases with largest clusters in red.
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