Palm calculus for stationary Cox processes on iterated random tessellations

Abstract

We investigate Cox processes of random point patterns in the Euclidean plane, which are located on the edges of random geometric graphs. Such Cox processes have applications in the performance analysis and strategic planning of both wireless and wired telecommunication networks. They simultaneously allow to represent the underlying infrastructure of the network together with the locations of network components. In particular, we analyze the Palm version X* of stationary Cox processes X living on random graphs that are built by the edges of an iterated random tessellation T. We derive a representation formula for the Palm version T* of T which includes the initial tessellation T0 and the component tessellation T1 of T as well as their Palm versions T*0 and T*1. Using this formula, we are able to construct a simulation algorithm for X* if both T0, T1 and their Palm versions T*0, T*1 can be simulated. This algorithm for X* extends earlier results for Cox processes on simpler (non-iterated) tessellations. It can be used, for example, in order to estimate the probability densities of various connection distances, which are important performance characteristics of telecommunication networks. In a numerical study we consider the particular case that T0 is a Poisson-Voronoi tessellation and T1 is a Poisson line tessellation.