Scaling Limits for Shortest Path Lengths along the Edges of Stationary Tessellations

We consider spatial stochastic models, which can be applied e.g. to telecommunication networks with two hierarchy levels. In particular, we consider Cox processes XL and XH concentrated on the edge set T (1) of a random tessellation T , where the points XL,n and XH,n of XL and XH can describe the locations of low–level and high–level network components, respectively, and T (1) the underlying infrastructure of the network, like road systems, railways, etc. Furthermore, each point XL,n of XL is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of XH . We investigate the typical shortest path length C∗ of the resulting marked point process, which is an important characteristic e.g. in performance analysis and planning of telecommunication networks. In particular, we show that the distribution of C∗ converges to simple parametric limit distributions if a scaling factor κ converges to zero and infinity, respectively. This can be used to approximate the density of C∗ by analytical formulae for a wide range of κ.