Poisson-Voronoi Tessellation

 In this section, we assume that $C=\{C_1,C_2,\ldots\}$ is a random Voronoi tessellation which is induced by a homogeneous Poisson process $Y=\{Y_1,Y_2,\ldots\}$ with intensity $\lambda_Y$ (see Figure 12). Then, the parameters $\lambda_0, \lambda_2$ and L_A are given by  

$$\lambda_0 = 2 \lambda_Y\; , \; \; \lambda_2 = \lambda_Y \; , \; \; L_A = 2 \sqrt{\lambda_Y};$$ (4)

see e.g. Miles (1970). In particular, $\lambda_1 = 3 \lambda_Y$, $\overline{n}_{02} = 3$ and $\overline{n}_{20}=6$.Further characteristics of the typical cell of a Poisson-Voronoi tessellation can be obtained by using numerical integration and by Monte Carlo simulation. For example, in Møller (1994) and Stoyan et al. (1995), see also Brakke (1985), Gilbert (1962), Muche and Stoyan (1992), the following results are given. Let $\Var(Z) = \Exp (Z- \Exp Z )^2$ and $\CV(Z) = \sqrt{Var(Z)}/\Exp Z$denote the variance and the coefficient of variation respectively of the positive random variable Z. Then

 

Figure 12:  Tessellation

$$\begin{array} {lll} \Var(N_{02}) = 1.7808, & \Var(L_2) = 0.9... ...) = 0.222, & \CV(L_2) = 0.243, & \CV(A_2) = 0.529,\end{array}$$

where N₀₂, L₂, A₂ denote the number of edges, the perimeter and the area respectively of the typical Poisson-Voronoi cell. Furthermore  

$$\overline{l^1} = \frac{\pi}{4} \lambda^{-\frac{1}{2}}_Y \,, ... ... \, , \quad \overline{l^3} = 0.960 \, \lambda^{-\frac{3}{2}}_Y$$ (5)

where $\overline{l^{i}}$ denotes the i-th moment of the length of the typical chord generated by an intersection of the Poisson-Voronoi tessellation with an arbitrary but fixed line. Note that alternatively one can consider the moments of the length of the chord generated by a `randomly chosen' test line and the typical Poisson-Voronoi cell. Miles and Maillardet (1982) determined the distribution of the number of edges of the typical cell which can be useful in connection with the problem of cochannel interference between cells (and optimal channel assignment to the cells) of a cellular wireless communication system. Note that this distribution does not depend on the intensity $\lambda_Y$ of the underlying Poisson process Y. In Table 1, the probability p_n that the typical cell has n edges is given for some values of n.

 

n 3 4 5 6 7 8 9 10

p_n 0.011 0.107 0.259 0.295 0.199 0.090 0.030 0.007

In Hinde and Miles (1980) Monte Carlo simulation has been used to obtain estimates for various further characteristics of the typical cell of a Poisson-Voronoi tessellation. For recent results on distributional properties related to the typical Poisson-Voronoi cell, we refer to Mecke and Muche (1995), Muche (1993, 1996, 1997), and Muche and Stoyan (1992). The density function of the half length $\frac{1}{2} \tilde{L}$ of the typical Delaunay edge is given in Møller (1994):

$$f(r) = \frac{4}{3} cr \left(1-\Phi(cr) + c^2 r \phi(cr)\right)$$

for $r \geq 0$, where $c= \sqrt{2 \pi \lambda_Y}$ and $\phi(t)$,$\Phi(t)$ denote the density and distribution functions respectively of the standard normal distribution. In particular

These results on characteristics of $\tilde{L}$ can be used, for instance to study the cable length connecting two typical neighboring stations.