Stationary Tessellations

 Let $Y=\{Y_1,Y_2,\ldots\}$ be an arbitrary, stationary (not necessarily Poisson) point process in $\RL^2$. Then with probability 1 the random Voronoi cells C_n are bounded and $C=\{C_1,C_2,\ldots\}$ is called a random Voronoi tessellation relative to Y; see Møller (1994), Stoyan et al. (1995). Note that C is stationary, i.e., the translated tessellations $C+x = \{C_1+x,C_2+x,\ldots\}$ have the same distribution for all $x \in \RL^2$.

Now, let $C=\{C_1,C_2,\ldots\}$ be an arbitrary stationary tessellation which is not necessarily a Voronoi tessellation related to a point process in $\RL^2$.Then, C induces further stationary random sets which can be used for analyzing communication networks: for example, the edge-set E which is a stationary segment process, and the stationary point processes X⁽⁰⁾, X⁽¹⁾, X⁽²⁾ of nodes, edge-midpoints, and cell-centroids respectively. By $\lambda_i$ we denote the intensity of X⁽ⁱ⁾, i.e., the expected number of points of X⁽ⁱ⁾ per unit area, i=1,2,3. Note that $\lambda_1 = \lambda_0 + \lambda_2$. The intensity of E is traditionally denoted by L_A which is the expected total length of segments per unit area.

Furthermore, let $\overline{n}_{02}$ denote the expected number of polygons touching the typical node of C, $\overline{l}_0$ the expected total length of the edges emanating from that node, $\overline{l}_1$the expected length of the edge through the typical edge-midpoint, $\overline{n}_{20}$ the expected number of nodes on the boundary of the cell containing the typical cell-centroid, $\overline{l}_2$and $\overline{a}_2$ the expected perimeter and area of that cell respectively.

All these expectations can be expressed by the three parameters $\lambda_0, \lambda_2$ and L_A:

$$\begin{array} {llllll} \overline{n}_{02} & = & 2 + \frac{\D ... ...ambda_0 + \lambda_2} = \frac{\D L_A}{\D \lambda_1};\end{array}$$

see Stoyan et al. (1995). Several of these formulae can be explained by the fact that each tessellation is associated with a dual tessellation which is obtained by connecting pairs of neighboring cell-centroids of the original tessellation. If C is generated by a homogeneous Poisson process, then it is possible to give formulae for higher-order moments which involve numerical integration; see Møller (1994), Stoyan et al. (1995) for further details and references.

Moreover, from the above formulae one obtains that

$$\frac{1}{\overline{n}_{02}} + \frac{1}{\overline{n}_{20}} = ... ...; , \; \; \overline{l}_{2} = \overline{n}_{20} \overline{l}_1$$

and

$$3 \leq \overline{n}_{02} \; , \; \; \overline{n}_{20} \leq 6... ...ambda_0 \leq 2 \lambda_2 \; , \; \; \lambda_2 \leq 2 \lambda_0.$$

An important special case is given when  

$$\overline{n}_{02} = 3$$ (2)

which is called the ordinary equilibrium state. For example, this equality holds for the Poisson-Voronoi tessellation - that is when C is induced by a homogeneous Poisson process; see also Section 5.3. However, simple examples where (5.2) is not true are tessellations formed by stationary line processes; see Figure 11 for a realization of such a process.

 

Figure 11:  Line process

In this case, we typically have $\overline{n}_{02} =4$.


$\mbox{ }$

In the ordinary equilibrium state we further have  

$$\overline{n}_{20} = 6 \; , \; \; 2 \lambda_2 = \lambda_0 \; , \; \; 2 \lambda_1 = 3 \lambda_0$$ (3)

which is an immediate consequence of (5.2). Thus, in this case the expected number of edges of the typical cell is 6 as in the traditional reference model with regular hexagons.

Relationships between characteristics of the typical cell of a stationary tessellation and its neighboring cells have been studied in Chiu (1994), Weiss (1995). Formulae for different types of contact distribution functions have been derived in Heinrich (1996), Last and Schassberger (1996, 1998) where also the chord length distribution of stationary (non-Poissonian) Voronoi tessellations is considered. In Heinrich and Muche (1997), representation formulae are given for the second factorial moment measure of the point process of nodes and the second moment of the number of edges of the typical cell associated with a stationary Voronoi tessellation in the ordinary equilibrium state.