Random Tessellations and Cox Processes

Abstract

We consider random tessellations T in ${\mathbb{R}}^{2}$ and Cox point processes whose driving measure is concentrated on the edges of T. In particular, we discuss several classes of Poisson-type tessellations which can describe for example the infrastructure of telecommunication networks, whereas the Cox processes on their edges can describe the locations of network components. An important quantity associated with stationary point processes is their typical Voronoi cell Z. Since the distribution of Z is usually unknown, we discuss algorithms for its Monte Carlo simulation. They are used to compute the distribution of the typical Euclidean (i.e. direct) connection length D ^o between pairs of network components. We show that D ^o converges in distribution to a Weibull distribution if the network is scaled and network components are simultaneously thinned in an appropriate way. We also consider the typical shortest path length C ^o to connect network components along the edges of the underlying tessellation. In particular, we explain how scaling limits and analytical approximation formulae can be derived for the distribution of C ^o.