Distributional properties of euclidean distances in wireless networks involving road systems

Stochastic models for hierarchical telecommunication networks are considered, which can be applied to the analysis and planning of large wireless networks. The network geometry is modelled by random geometric graphs, and the locations of network nodes by point processes on the edges of these graphs. In particular, the locations of high-level components (HLC) are modeled by Cox processes concentrated on the edge sets of random graphs, where their serving zones are the cells of Voronoi tessellations induced by these Cox processes. The locations of low-level components (LLC) are either modeled by planar Poisson processes or by Cox processes concentrated on the same edge sets as the HLC. Distributional properties of distances between the locations of network nodes are closely related with the interference geometry and, consequently, the performance of wireless networks. Representation formulas are derived for the distribution function and density of the typical Euclidean connection distance between LLC and HLC. They lead to suitable estimators of these characteristics, which can be computed by Monte Carlo simulation of the typical serving zone and the typical segment system in it, respectively.