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

TL;DR

This paper analyzes the asymptotic behavior of shortest path lengths in spatial stochastic models of telecommunication networks, showing convergence to simple distributions under certain scaling limits, aiding network performance analysis.

Contribution

It introduces a framework for understanding the limiting distribution of shortest path lengths in Cox process models on tessellations, providing analytical approximations.

Findings

Distribution converges to simple parametric forms as scaling factors tend to zero or infinity.

Provides analytical formulas for approximating shortest path length densities.

Applicable to performance analysis of hierarchical telecommunication networks.

Abstract

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.