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We provide a parametric modeling approach suitable for various kinds of hierarchical networks based on random geometric graphs. In these networks, we have two kinds of components, so-called high-level components (HLC) and low-level components (LLC). Each HLC is associated with a serving zone and all LLC within this area are connected to the corresponding HLC. So-called sparse LLC networks, where only a few LLC occur in the typical serving zone, are a non-negligible subdomain when investigating hierarchical networks. Therefore, we supply distributional results for structural characteristics where two LLC are independently and uniformly distributed along the segment system of the typical serving zone. In particular, we are interested in the joint distribution of three quantities, namely the length of the joint part of the shortest paths from the LLC to the HLC as well as the lengths of the corresponding disjoint remaining parts. In order to provide a parametric, three-dimensional distribution function for t...

This paper presents a methodology for fast and reliable cost estimation of fixed access network deployment on any urban area. It is based on theoretical results presented in former papers for distances and attenuation results, adding the most important part of cost estimation. In particular it is showed that good estimations can be obtained in a very short time. The territory to be deployed, the network architecture and the scenarios are defined by the user via two user-friendly prototypes. The first one helps defining the limit of the city and computes mathematical parameters representing its street system. The second one computes global information on the network deployment on this territory, namely the probability distributions of distances and attenuation from a node of the network to the final customer, and an evaluation of the deployment cost, once given an architecture and engineering rules. The results were compared on two real French urban territories (in Tours and Rouen) to those given by an optimization tool currently used by Orange.

We consider a spatial stochastic model for telecommunication networks, the stochastic subscriber line model, and we investigate the distribution of the typical shortest path length between network components. Therefore, we derive a representation formula for the probability density of this distribution which is based on functionals of the so-called typical serving zone. Using this formula, we construct an estimator for the density of the typical shortest path length and we analyze the statistical properties of this estimator. Moreover, we introduce new simulation algorithms for the typical serving zone which are used in a numerical study in order to estimate the density and moments of the typical shortest path length for different specific models.

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 stationary Poisson–Voronoi tessellations (PVT) in the Euclidean plane and study the properties of Voronoi tessellations induced by linear Poisson processes on the edges of the PVT. We are especially interested in simulation algorithms for the typical cell. Two different simulation algorithms are introduced. The first algorithm directly simulates the typical cell, whereas the second algorithm simulates cells from which distributional properties of the typical cell can be obtained. This second algorithm can also be used for simulating the typical cell of other Cox–Voronoi tessellations. The implementation of both algorithms is tested for their correctness using random software tests. Then different cell characteristics are studied by simulation and compared with the typical cell of PVT and Cox–Voronoi tessellations based on linear Poisson processes on the lines of Poisson line processes. Our results can be applied, for example, in the analysis of telecommunication networks and vesicle paths on c...

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 consider random geometric models for telecommunication access networks and analyse their serving zones which can be given, for example, by a class of so-called Cox–Voronoi tessellations (CVTs). Such CVTs are constructed with respect to locations of network components, the nucleii of their induced cells, which are scattered randomly along lines induced by a Poisson line process. In particular, we consider two levels of network components and investigate these hierarchical models with respect to mean shortest path length and mean subscriber line length, respectively. We explain point-process techniques which allow for these characteristics to be computed without simulating the locations of lower-level components. We sustain our results by numerical examples which were obtained through Monte Carlo simulations, where we used simulation algorithms for typical Cox–Voronoi cells derived in a previous paper. Also, briefly, we discuss tests of correctness of the implemented algorithms. Finally, we present a short outlook to possible extensions concerning multi-level models and iterated random tessellations.

The end points of a fixed segment in the Euclidian plane covered by a Poisson Voronoi tessellation belong to the same cell or to two distinct cells. This marks off one or two points of the underlying Poisson process that are the nucleus(i) of the cell(s). Our interest lies in the geometrical relationship between these nuclei and the segment end points as well as between the nuclei. We investigate their probability distribution functions conditioning on the number of nuclei, taking into account the length of the segment. The aim of the study is to establish some tools to be used for the analysis of a telecommunication problem related to the pricing of leased lines. We motivate and give accurate approximations of the probability of common coverage and of the length distributions that can be included in spreadsheet codes as an element of simple cost functions. Copyright © 2006 John Wiley & Sons, Ltd.

We consider asymptotic properties of two functionals on Euclidean shortest-path trees appearing in random geometric graphs in R 2 which can be used, for example, as models for fixed-access telecommunication networks. First, we determine the asymptotic bivariate distribution of the two backbone lengths inside a certain class of typical Cox–Voronoi cells as the size of this cell grows unboundedly. The corresponding Voronoi tessellation is generated by a stationary Cox process which is concentrated on the edges of the random geometric graph and whose intensity tends to 0 . The limiting random vector can be represented as a simple geometric functional of a decomposition of a typical Poisson–Voronoi cell induced by an independent random sector. Using similar methods, we consider the asymptotic bivariate distribution of the total lengths of the two subtrees inside the Cox–Voronoi cell.

Shortest-path trees play an important role in the field of optimising fixed-access telecommunication networks with respect to costs and capacities. Distributional properties of the corresponding two half-trees originating from the root of such a tree are of special interest for engineers. In the present paper, we derive parametric approximation formulas for the marginal density functions of the total lengths of both half-trees. Besides, a parametric copula is used in order to combine the marginal distributions of these functionals to a bivariate joint distribution as, naturally, the total lengths of the half-trees are not independent random variables. Asymptotic results for infinitely sparse and infinitely dense networks are discussed as well.

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.

This paper deals with the multiple scattering of electromagnetic waves by a homogeneous distribution of spherical Rayleigh scatterers. We use the Feynman diagrams symbolism and define the planar diagrams which describe the interactions with possible backscattering to a previously used scatterer (loops). Resummation over every level of overlapping loops is performed analytically. In terms of the absorption properties of the equivalent continuous medium, departure from Twersky's assumption is shown to occur only in dense media, and the first level of loops is a good approximation of the whole set. Results are compared with the scalar waves case.

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 .

Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of the traditional models, from random graphs, complex networks theory, or stochastic geometry, takes into account the geometrical, functional, and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating, and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division or extension of space. We propose a dynamical model that mimics this logic and that, from simple general rules and a few parameters, succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.

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.

5G millimeter wave (mmWave) is a promising solution to the spectrum scarcity problem since very large bandwidths are available. However, due to their propagation properties, namely their very short wavelength, this frequency band can severely impact transmission, leading to a new major challenge: indoor coverage. This requires new studies to answer three major questions: What is the impact of the various environment geometries (apartment, factory, etc.) on indoor mmWave propagation? What is the impact of the various environment materials (concrete, plasterboard, etc.) on indoor mmWave propagation? and what planning methods should we use for the various indoor 5G applications (fixed wireless access, industry 4.0, etc.).In this work, we address this challenge through a novel theoretical framework that combines stochastic indoor environment modeling with advanced physical propagation simulation. This approach is particularly adapted to investigate indoor-to-indoor 5G mmWave propagation. Its system implementation, so-called iGeoStat, generates parameterized typical environments that account for the indoor spatial variations, then simulates radio propagation based on the physical interaction between electromagnetic waves and material properties. This framework is not dedicated to a particular environment, material, frequency or use case and aims to statistically understand the influence of indoor environment parameters on mmWave propagation properties, especially coverage, SINR, and path loss.iGeoStat's system implementation raises numerous computational challenges that we solve by formulating an adapted link budget and designing new memory optimization algorithms. The first simulation results for two major 5G applications (fixed wireless access and industry 4.0) are validated with measurement data and show the efficiency of iGeoStat to simulate multiple diffusion in realistic environments, within a reasonable amount of time and memory resources. Generated output maps confirm that diffusion has a critical impact on indoor mmWave propagation and that proper physical modeling is of the utmost importance to generate relevant propagation models.Using iGeoStat, the main propagation parameters are investigated in various scenarios, showing that the complex refractive index of the indoor material has a moderate impact on the received power, while its surface roughness parameter has a major impact and may completely change the power profile in the environment. Various acceleration techniques are also presented in this work where we show through the simulation results that the performance of iGeoStat can be further enhanced and that we can achieve at least 50% reduction in time simulation and memory usage without impacting the physical aspect of propagation., La 5G introduit des nouvelles bandes de fréquences, les ondes millimétriques, qui est une solution prometteuse au problème de l'épuisement du spectre grâce à la disponibilité de larges bandes passantes. Cependant, en raison de leurs propriétés de propagation, notamment leur très courte longueur d'onde, cette bande de fréquences peut avoir un impact sévère sur la transmission, conduisant à un nouveau défi majeur : la couverture indoor. Cela nécessite de nouvelles études pour répondre à trois questions fondamentales : Quel est l'impact des diverses géométries d'environnement (appartement, usine, etc.) sur la propagation des ondes millimétriques en indoor ? Quel est l'impact des divers matériaux de l'environnement (béton, bois, etc.) sur la propagation des ondes millimétriques en indoor ? et quelles méthodes de planification doit-on utiliser pour les diverses applications 5G indoor (accès fixe, industrie 4.0, etc.). Dans ce manuscrit, nous abordons ce problème à travers un nouveau cadre théorique qui combine la modélisation stochastique de l'environnement indoor avec la simulation avancée de la propagation physique. Cette approche est particulièrement adaptée pour étudier la propagation des ondes millimétriques 5G dans le cas où l'émetteur et le récepteur sont tous les deux en indoor. L'implémentation informatique de cette approche, appelée iGeoStat, génère des environnements typiques paramétrés qui tiennent compte des variations spatiales en indoor, puis simule la propagation radio en fonction de l'interaction physique entre les ondes électromagnétiques et les propriétés des matériaux. Ce cadre n'est pas dédié à un environnement, un matériau, une fréquence ou un cas d'usage particulier et vise à comprendre statistiquement l'influence des paramètres de l'environnement indoor sur les propriétés de propagation des ondes millimétriques, en particulier la couverture, le SINR et le path loss.L'implémentation d'iGeoStat soulève de nombreux défis de calcul informatique que nous résolvons en formulant un bilan de liaison adapté et de nouveaux algorithmes d'optimisation de la mémoire. Les premiers résultats de simulation pour deux applications 5G majeures (accès fixe et industrie 4.0) sont comparés à des données de mesure et montrent l'efficacité d'iGeoStat pour simuler la diffusion multiple dans des environnements réalistes, dans un temps et des ressources mémoire raisonnables. iGeostat génère des cartes qui confirment que la diffusion a potentiellement un impact majeur sur la propagation des ondes millimétriques en indoor et qu'une modélisation physique appropriée est de la plus haute importance pour générer des modèles de propagation pertinents. En utilisant iGeoStat, les principaux paramètres de propagation sont étudiés dans divers scénarios, montrant que l'indice de réfraction complexe du matériau indoor a un impact modéré sur la puissance reçue, tandis que la rugosité de surface a un impact majeur, pouvant modifier profondément le profil de puissance mesurée dans l'environnement. Diverses techniques d'accélération sont également présentées dans ce travail où nous montrons à travers les résultats de simulations que les performances d'iGeoStat peuvent être améliorées davantage et qu'il est possible d'obtenir une réduction d'au moins 50% du temps de simulation et de l'utilisation de la mémoire sans impacter l'aspect physique de la propagation.