The field of wireless communications has been experiencing tremendous growth with the ever-increasing dependence on wireless services. In the operation of a communication network, the network coverage and node placement are of profound importance. The network performance metrics can be modeled as nonlinear functions of inter-node distances. Therefore, a geometric abstraction of the distance between wireless devices becomes a prerequisite for accurate system modeling and analysis. A geometrical probability approach is presented in this dissertation to characterize the probabilistic distance properties, for analyzing the location-critical performance metrics through various spatial distance distributions. Ideally, the research in geometrical probability shall give results for the distance distributions 1) over elementary geometries such as a straight line, squares and rectangles, and 2) over complex geometries such as rhombuses and hexagons. Both 1) and 2) are the representative topological shapes for communication networks. The current probability and statistics literature has explicit results for 1), whereas the results for 2) are not in existence. In particular, the absence of the distance distributions for rhombuses and hexagons has posed challenges towards the analytical modeling of location-critical performance metrics in complex geometries. This dissertation is dedicated to the application of existing results in 1) elementary geometries to the networking area, and the development of a new approach to deriving the distance distributions for complex geometries in 2), bridging the gap between the geometrical probability and networking research. The contribution of this dissertation is twofold. First, the one-dimensional Poisson point process in 1) is applied to the message dissemination in vehicular ad-hoc networks, where the network geometry is constrained by highways and city blocks. Second, a new approach is developed to derive the closed-form distributions of inter-node distances associated with rhombuses and hexagons in 2), which are obtained for the first time in the literature. Analytical models can be constructed for characterizing the location-critical network performance metrics, such as connectivity, nearest/farthest neighbor, transmission power, and path loss in wireless networks. Through both analytical and simulation results, this dissertation demonstrates that this geometrical probability approach provides accurate information essential to successful network protocol and system design, and goes beyond the approximations or Monte Carlo simulations by gracefully eliminating the empirical errors. / Graduate

Wireless devices and applications have been an unavoidable part of human lives in the past decade. In the past few years, the global mobile data traffic has grown considerably and is expected to grow even faster in future. Given the fact that the number of wireless nodes has significantly increased, the contention and interference on the license-free industrial, scientific, and medical band has become severer than ever. Cognitive radio nodes were introduced in the past decade to mitigate the issues related to spectrum scarcity. In this dissertation, we focus on the interference and performance analysis of networks coexisting with cognitive radio networks and address the design and analysis of spectrum allocation and routing for cognitive radio networks. Spectrum allocation enables nodes to construct a link on a common channel at the same time so they can start communicating with each other. We introduce a new approach for the modeling and analysis of interference and spectrum allocation schemes for cognitive radio networks with arbitrarily-shaped network regions. First, for the first time in the literature, we propose a simple and efficient approach that can derive the distribution of the distance between an arbitrary interior/exterior reference point and a random point within an arbitrary convex/concave irregular polygon. This tool is essential in analyzing important distance-related performance metrics in wireless communication networks. Second, considering the importance of interference analysis in cognitive radio networks and its important role in designing spectrum allocation schemes, we model and analyze a heterogeneous cellular network consisting of several cognitive femto cells and a coexisting multi-cell network. Besides the cumulative interference, important distance-related performance metrics have been investigated, such as the signal-to-interference ratio and outage probability. Finally, the spectrum allocation and routing problems in cognitive radio networks have been discussed. Considering a wireless cognitive radio network coexisting with a cellular network with irregular polygon-shaped cells, we have used the tools developed in this dissertation and proposed a joint spectrum allocation and routing scheme. / Graduate

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / O presente trabalho tem por finalidades: demonstrar o problema da agulha de Buffon, fazer uma pequena generalizaÃÃo do resultado obtido e apresentar aplicaÃÃes baseadas nos fundamentos do referido problema. O problema da agulha de Buffon està inserido no estudo da Teoria das Probabilidades, particularmente na subÃrea de probabilidade geomÃtrica. Para chegarmos à soluÃÃo desta questÃo, alÃm dos conceitos e propriedades atinentes à Teoria das probabilidades à necessÃrio o conhecimento de noÃÃes bÃsicas do cÃlculo integral. Nos capÃtulos 2, 3 e 4 à apresentado um estudo preliminar sobre probabilidade, com os conceitos bÃsicos, propriedades e a formulaÃÃo de alguns modelos probabilÃsticos. Durante o desenvolvimento do trabalho, sempre que possÃvel, os conceitos e definiÃÃes sÃo inseridos com o auxÃlio de um problema motivador e para fixaÃÃo dos mesmos sÃo mostrados exemplos resolvidos. O Ãltimo capÃtulo evidencia a importÃncia do problema de Buffon como mÃtodo para realizar estimativas e como fundamento para o processo de captaÃÃo de imagens pelos aparelhos de tomografia computadorizada, um grande avanÃo para a Medicina no que diz respeito ao diagnÃstico por imagens. / This paper has the objective of showing Buffon's needle problem, doing a minor generalization of the results obtained hereby, and also presenting some applications based upon the fundamentals of such problem. Buffon's needle problem has been inserted into the study of Theory of Probability, particularly in its sub-area of geometrical probability. In order to attain the solution to this question, in addition to the concepts and the properties concerning the theory of probabilities, it is necessary that one should have some basic knowledge about integral calculus. In chapters 2, 3, and 4 there is a preliminary study of probability, with the basic concepts, properties and formulation of some probabilistic models being presented. During the development of this paper, whenever it was possible, the concepts and definitions were inserted with the aid of a motivational problem and they were solved by means of fixing the same examples as shown. The final chapter presents the importance of Buffon's needle problem as a method of making estimates and as a foundation for the process of capturing images in CT (computerized tomography) scanning machines, such a great breakthrough in what concerns the diagnosis by means of imaging.