Spatial stochastic network models - scaling limits and Monte-Carlo methods

Abstract

The present thesis is motivated by a joint research project of the Institute of Stochastics at Ulm University and Orange Labs in Issy les Moulineaux, Paris, which deals with the analysis of telecommunication networks. During the last years the Stochastic Subscriber Line Model (SSLM) has been developed in this cooperation and it is still extended. The SSLM is a flexible model for telecommunication networks, especially designed for access networks in urban areas. It utilizes tools from Stochastic Geometry in order to represent the different parts of the network by spatial stochastic models which only depend on a small number of parameters. The aim of this thesis is to analyze so-called typical connection lengths in the SSLM in various ways. One part of the thesis focuses on the estimation of their densities and distribution functions via Monte-Carlo simulation. The developed estimators are all based on samples of the so-called typical serving zone. Therefore, simulation algorithms for the typical serving zone are derived for various network models. Another part of the thesis deals with limit theorems for the typical connection length. In particular, it is shown that the distribution of the typical connection length converges to well-known distributions if the parameters of the underlying network model tend to extremal cases. Both results, the estimated distributions and the asymptotic distributions, are used in order to obtain approximative parametric densities for the typical connection length. These parametric densities are finally compared to empirical distributions computed from huge databases without using any spatial information. It is shown that the obtained parametric densities fit quite well to real network data. In the final part of the thesis, required capacities at different locations of the network are analyzed.