The curious shape of Poisson-Voronoi cells

In this blog we are exploring the shape of two kinds of cells in the Poisson-Voronoi tessellation on the plane, namely the 0-cell and the typical cell. The 0-cell is the cell containing the origin, while the typical cell is the cell obtained by conditioning on a Poisson point to be at the origin (which is the same as adding the origin to the PPP).

The cell shape has an important effect on the signal and interference powers at the typical user (in the 0-cell) and at the user in the typical cell. For instance, in the 0-cell, which contains the typical user at a uniformly random location, about 1/4 of the cell edge is at essentially the same distance to the base station as the typical user on average). Hence it is not the case that edge users necessarily suffer from larger signal attenuation than the typical user (who resides inside the cell).

The cell shape is determined by the directional radii of the cells when their nucleus is at the origin. To have a well-defined orientation, we select a location uniformly in the cell and rotate the cell so that this location falls on the positive x-axis. In the 0-cell, this involves first a translation of the cell’s nucleus to the origin, followed by a rotation until the original origin (which is uniformly distributed in the cell) lies on the positive x-axis. This is illustrated in Movie 1 below. In the typical cell, it involves adding a Poisson point, selecting a uniform location, and a rotation so that this uniform location lies on the positive x-axis. This is illustrated in Movie 2.

Movie 1. Rotated and translated 0-cell.

Movie 2. Rotated typical cell.

As indicated in the movies, the distances from the nucleus to the uniformly random location are denoted by D0 and D, respectively, and the directional radii by R0(ϕ) and R(ϕ), respectively. This way, the boundary of the cells is described in polar coordinates as (R0(ϕ),ϕ) and (R(ϕ),ϕ), ϕ ∈ [0,2π). In a cellular network model, the uniform random location could be that of a user, while the PPP models the base stations. In this case D0 is the link distance from the typical user to its serving base station, while D is the link distance from the typical base station to a randomly located user it serves. The distinction between the typical user’s and the typical base station’s point of view is explained in this blog.

Let λ denote the density of the PPP. Three results are well known:

  • The distribution of D0 follows from the void probability of the PPP. It is Rayleigh with mean 1/(2√λ).
  • Since the mean area of the typical cell is 1/λ, we have ∫ 0π 𝔼(R(ϕ)2) dϕ = 1/λ.
  • The minimum of R(ϕ) is distributed with pdf f(r)=8λπr exp(-4λπr2). This is half the distance to the nearest neighboring Poisson point (base station).

In contrast, there is no closed-form expression for the distribution of D. Due to size-biased sampling, the area of the 0-cell stochastically dominates that of the typical cell and, in turn, D0 dominates D.

Analyzing the directional radii, we obtain these new insights on the cell shapes:

  • If Ψ is uniform in [0,π], R(Ψ) is again Rayleigh with mean 1/(2√λ).
  • R0(π) is also Rayleigh with the same mean. In fact, R0(π) and D0 are iid.
  • R0(0) has mean 3/(4√λ) and is distributed as

\displaystyle f_{R_0(0)}(y)=2(\lambda\pi)^2 y^3 \exp(-\lambda\pi y^2).

  • Hence R0(0) is on average exactly 50% larger than R0(π). For the typical cell, simulation results indicate that R(0) is about 55% larger on average than R(π).
  • The difference R0(0)-D0 is distributed as f(r)=π√λ erfc(r √(πλ)). Its mean is 1/(4√λ). Hence the typical user is no further from the cell edge than the base station on average.
  • The joint distribution of D0 and R0(ϕ) can be given in exact analytical form.
  • 3/4 of the typical cell is further away from the nucleus than the nearest point on the cell edge (i.e., the minimum directional radius). Expressed differently, a uniformly random user in the typical cell has a 75% chance of being further away from the base station than the nearest edge user. By simulation, D on average is 2.7 times larger than the minimum of the directional radii.

In conclusion, the 0-cell and the typical cell are quite asymmetric around the nucleus (base station) and the uniformly random point (user). In the direction away from the base station, the user is about 4 times closer to the cell edge than in the direction towards the base station, and many locations on the cell edge are closer to the base station than the user inside the cell. These results have implications on the design of efficient cellular network transmission schemes, such as beamforming, NOMA, and base station cooperation, in both down- and uplink.

More details are available in Section II of this paper.

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