Scaling limits for shortest path lengths along the edges of stationary tessellations - Supplementary material

Details Scaling limits for shortest path lengths along the edges of stationary tessellations - Supplementary material Scaling limits for shortest path lengths along the edges of stationary tessellations - Supplementary material

2009-12-23

arxiv.org/abs/0912.4516v1

Mathematics

Probability

Scientific paper

We consider spatial stochastic models, which can be applied e.g. to telecommunication networks with two hierarchy levels. In particular, we consider two Cox processes concentrated on the edge set of a random tessellation, where the points can describe the locations of low-level and high-level network components, respectively, and the edge set the underlying infrastructure of the network, like road systems, railways, etc. Furthermore, each low-level component is marked with the shortest path along the edge set to the nearest high-level component. We investigate the typical shortest path length 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 its distribution converges to simple parametric limit distributions if a certain scaling factor converges to zero and infinity, respectively. This can be used to approximate the density of the typical shortest path length by analytical formulae.

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