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We consider spatial stochastic models, which can be applied to, e.g. telecommunication networks with two hierarchy levels. In particular, we consider Cox processes X L and X H concentrated on the edge set T (1) of a random tessellation T, where the points X L,n and X H,n of X L and X H can describe the locations of low-level and high-level network components, respectively, and T (1) the underlying infrastructure of the network, such as road systems, railways, etc. Furthermore, each point X L,n of X L is marked with the shortest path along the edges of T to the nearest (in the Euclidean sense) point of X H . We investigate the typical shortest path length C * of the resulting marked point process, which is an important characteristic in, e.g. 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 0 or ∞. This can be used to approximate the density of C * by analytical formulae for a wide range of κ.

We consider modulated Poisson--Voronoi tessellations, intended as models for telecommunication networks on a nationwide scale. By introducing an algorithm for the simulation of the typical cell of the latter tessellation, we lay the mathematical foundation for such a global analysis. A modulated Poisson--Voronoi tessellation has an intensity which is spatially variable and, hence, is able to provide a broad spectrum of model scenarios. Nevertheless, the considered tessellation model is stationary and we consider the case where the modulation is generated by a Boolean germ-grain model with circular grains. These circular grains may either have a deterministic or random but bounded radius. Furthermore, based on the introduced simulation algorithm for the typical cell and on Neveu's exchange formula for Palm probability measures, we show how to estimate the mean distance from a randomly chosen location to its nearest Voronoi cell nucleus. The latter distance is interpreted as an important basic cost characteristic in telecommunication networks, especially for the computation of more sophisticated functionals later on. Said location is chosen at random among the points of another modulated Poisson process where the modulation is generated by the same Boolean model as for the nuclei. The case of a completely random placement for the considered location is thereby included as a special case. The estimation of the cost functional is performed in a way such that a simulation of the location placement is not necessary. Test methods for the correctness of the algorithm based on tests for random software are briefly discussed. Numerical examples are provided for characteristics of the typical cell as well as for the cost functional. We conclude with some remarks about extensions and modifications of the model regarded in this paper, like modulated Poisson--Delaunay tessellations.

We provide a stochastic modelling approach for multi-hierarchical fixed-access telecommunication networks where cables are installed along the underlying road system. It constitutes an extension of network models consisting of only two hierarchy levels. We consider the effects of the introduction of an additional level of hierarchy on two functionals relevant in telecommunication networks, namely typical shortest-path lengths and total fibre lengths. Intuitively speaking, in the extended scenario, the typical shortest-path length gets longer whereas the total fibre length decreases. Both theoretical and numerical results are provided. The underlying infrastructure is assumed to be represented by a STIT tessellation which is particularly suitable for stochastic modelling of multi-hierarchical fixed-access telecommunication networks. In this context, we present a description of the Palm version of a planar STIT tessellation and give an appropriate simulation algorithm.

We explore real telecommunication data describing the spatial geometrical structure of an urban region and we propose a model fitting procedure, where a given choice of different non-iterated and iterated tessellation models is considered and fitted to real data. This model fitting procedure is based on a comparison of distances between characteristics of sample data sets and characteristics of different tessellation models by utilizing a chosen metric. Examples of such characteristics are the mean length of the edge-set or the mean number of vertices per unit area. In particular, after a short review of a stochastic-geometric telecommunication model and a detailed description of the model fitting algorithm, we verify the algorithm by using simulated test data and subsequently apply the procedure to infrastructure data of Paris.

We study the tessellation defined as the intersection of two independent planar Poisson-Voronoi tessellations and derive the means of its main geometrical characteristics. For this intersection tessellation, we distinguis- h between six types of cells depending on whether they contain the nuclei of the two initial tessellations or not. The intensity and the mean area of each type of cell are computed either in closed form or via asymptotic expansions. The model can be used to represent the local zones of two competing telecommunication operators. Then the interconnection of two subscribers induces a specific cost within each type of cell of the intersecti- on tessellation associated with the two systems of local zones.

We consider random tessellations T in \({\mathbb{R}}^{2}\) and Cox point processes whose driving measure is concentrated on the edges of T. In particular, we discuss several classes of Poisson-type tessellations which can describe for example the infrastructure of telecommunication networks, whereas the Cox processes on their edges can describe the locations of network components. An important quantity associated with stationary point processes is their typical Voronoi cell Z. Since the distribution of Z is usually unknown, we discuss algorithms for its Monte Carlo simulation. They are used to compute the distribution of the typical Euclidean (i.e. direct) connection length D o between pairs of network components. We show that D o converges in distribution to a Weibull distribution if the network is scaled and network components are simultaneously thinned in an appropriate way. We also consider the typical shortest path length C o to connect network components along the edges of the underlying tessellation. In particular, we explain how scaling limits and analytical approximation formulae can be derived for the distribution of C o .

We investigate Cox processes of random point patterns in the Euclidean plane, which are located on the edges of random geometric graphs. Such Cox processes have applications in the performance analysis and strategic planning of both wireless and wired telecommunication networks. They simultaneously allow to represent the underlying infrastructure of the network together with the locations of network components. In particular, we analyze the Palm version X* of stationary Cox processes X living on random graphs that are built by the edges of an iterated random tessellation T. We derive a representation formula for the Palm version T* of T which includes the initial tessellation T 0 and the component tessellation T 1 of T as well as their Palm versions T* 0 and T* 1 . Using this formula, we are able to construct a simulation algorithm for X* if both T 0 , T 1 and their Palm versions T* 0 , T* 1 can be simulated. This algorithm for X* extends earlier results for Cox processes on simpler (non-iterated) tessellations. It can be used, for example, in order to estimate the probability densities of various connection distances, which are important performance characteristics of telecommunication networks. In a numerical study we consider the particular case that T 0 is a Poisson-Voronoi tessellation and T 1 is a Poisson line tessellation.

We consider the stochastic subscriber line model as a spatial stochastic model for telecommunication networks and we are interested in the evaluation of the required capacities at different locations of the network in order to provide, in fine, an estimation of the cable system which has to be installed. In particular, we consider hierarchical telecommunication networks with higher–level components (HLC) and lower–level components (LLC) located on the road system underlying the network. The cable paths are modeled by shortest paths along the edge set of a stationary random tessellation, whereas both HLC and LLC are modeled by Cox processes concentrated on the edges of this tessellation. We then introduce the notion of capacity which depends on the length of some subtree on the edge set of the underlying tessellation. Moreover, we investigate estimators for the density and distribution function of the typical length of this subtree which can be computed based on Monte Carlo simulations of the typical serving zone. In a numerical study, the density of the typical subtree length is determined for different specific models.

Distributional properties and a simulation algorithm for the Palm version of stationary iterated tessellations are considered. In particular, we study the limit behaviour of functionals related to Cox–Voronoi cells (such as typical shortest-path lengths) if either the intensity γ0 of the initial tessellation or the intensity γ1 of the component tessellation converges to 0. We develop an explicit description of the Palm version of Poisson–Delaunay tessellations (PDT), which provides a new direct simulation algorithm for the typical Cox–Voronoi cell based on PDT. It allows us to simulate the Palm version of stationary iterated tessellations, where either the initial or component tessellation is a PDT and can furthermore be used in order to show numerically that the qualitative and quantitative behaviour of certain functionals related to Cox–Voronoi cells strongly depends on the type of the underlying iterated tessellation.