Over the past decade, many works on the modeling of wireless networks using stochastic geometry have been proposed. Results about probability of coverage, throughput or mean interference, have been provided for a wide variety of networks (cellular, ad-hoc, cognitive, sensors, etc). These results notably allow to tune network protocol parameters. Nevertheless, in their vast majority, these works assume that the wireless network deployment is flat: nodes are placed on the Euclidean plane. However, this assumption is disproved in dense urban environments where many nodes are deployed in high buildings. In this letter, we derive the exact form of the probability of coverage for the cases where the interferers form a 3D Poisson Point Process (PPP) and a 3D Modified Matern Process (MMP), and compare the results with the 2D case. The main goal of this letter is to show that the 2D model, although being the most common, can lead either to an optimistic or a pessimistic evaluation of the probability of coverage depending on the parameters of the model.
From a more theoretical point of view, 3D networks have been investigated in terms of capacity @cite_0 @cite_14 and scaling laws have been provided. Nevertheless, in the present work we focus on the stochastic geometric approach to the study of wireless networks in the sense of @cite_16 @cite_13 . This approach has the advantage to provide tractable results for the probability of coverage in the case of the PPP model and good approximations for other models @cite_16 .
@cite_0 @cite_14 @cite_13 @cite_16
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Stochastic geometry models for wireless communication networks have recently attracted much attention. This is because the performance of such networks critically depends on the spatial configuration of wireless nodes and the irregularity of the node configuration in a real network can be captured by a spatial point process. However, most analysis of such stochastic geometry models for wireless networks assumes, owing to its tractability, that the wireless nodes are deployed according to homogeneous Poisson point processes. This means that the wireless nodes are located independently of each other and their spatial correlation is ignored. In this work we propose a stochastic geometry model of cellular networks such that the wireless base stations are deployed according to the Ginibre point process. The Ginibre point process is one of the determinantal point processes and accounts for the repulsion between the base stations. For the proposed model, we derive a computable representation for the coverage probability—the probability that the signal-to-interference-plus-noise ratio (SINR) for a mobile user achieves a target threshold. To capture its qualitative property, we further investigate the asymptotics of the coverage probability as the SINR threshold becomes large in a special case. We also present the results of some numerical experiments. Stochastic geometry proves to be a powerful tool for modeling dense wireless networks adopting random MAC protocols such as ALOHA and CSMA. The main strength of this methodology lies in its ability to account for the randomness in the nodes' location jointly with an accurate description at the physical layer, based on the SINR, that allows to consider also random fading on each link. Existing models of CSMA networks adopting the stochastic geometry approach suffer from two important weaknesses: 1) they permit to evaluate only spatial averages of the main performance measures, thus hiding possibly huge discrepancies in the performance achieved by individual nodes; 2) they are analytically tractable only when nodes are distributed over the area according to simple spatial processes (e.g., the Poisson point process). In this paper we show how the stochastic geometry approach can be extended to overcome the above limitations, allowing to obtain node throughput distributions as well as to analyze a significant class of topologies in which nodes are not independently placed. This volume bears on wireless network modeling and performance analysis. The aim is to show how stochastic geometry can be used in a more or less systematic way to analyze the phenomena that arise in this context. It first focuses on medium access control mechanisms used in ad hoc networks and in cellular networks. It then discusses the use of stochastic geometry for the quantitative analysis of routing algorithms in mobile ad hoc networks. The appendix also contains a concise summary of wireless communication principles and of the network architectures considered in the two volumes. It is shown that the capacity scaling of wireless networks is subject to a fundamental limitation which is independent of power attenuation and fading models. It is a degrees of freedom limitation which is due to the laws of physics. By distributing uniformly an order of n users wishing to establish pairwise independent communications at fixed wavelength inside a two-dimensional domain of size of the order of n, there are an order of n communication requests originating from the central half of the domain to its outer half. Physics dictates that the number of independent information channels across these two regions is only of the order of radicn, so the per-user information capacity must follow an inverse square-root of n law. This result shows that information-theoretic limits of wireless communication problems can be rigorously obtained without relying on stochastic fading channel models, but studying their physical geometric structure.