A Geometric Theorem for Wireless Network Design Optimization

Source: OAI

ABSTRACT Consider an infinite square grid G. How many discs of given radius r, centered at the vertices of G, are required, in the worst case, to completely cover an arbitrary disc of radius r placed on the plane? We show that this number is an integer in the set (3.4; 5.6) whose value depends on the ratio of r to the grid spacing. This result can be applied at the very early design stage of a wireless cellular network to determine, under the recent International Telecommunication Union (ITU) proposal for a traffic load model, and under the assumption that each client is able to communicate if it is within a certain range from a base station, conditions for which a grid network design is cost effective, for any expected traffic demand.

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    ABSTRACT: Although most of the studies on coverage and connectivity in wireless sensor networks (WSNs) considered two-dimensional (2D) settings, such networks can in reality be accurately modeled in a three-dimensional (3D) space. The concepts of continuum percolation theory best fit the problem of connectivity in WSNs to find out whether the network provides long-distance multihop communication. In this paper, we focus on percolation in coverage and connectivity in 3D WSNs. We say that the network exhibits a coverage percolation (respectively, connectivity percolation) when a giant covered region (respectively, giant connected component) almost surely spans the entire network for the first time. Because of the dependency between coverage and connectivity, the problem is not only a continuum percolation problem but also an integrated continuum percolation problem. Thus, we propose an integrated-concentric-sphere model to address coverage and connectivity in 3D WSNs in an integrated way. First, we compute the critical density lambdaC con above which coverage percolation in 3D WSNs will almost surely occur. Second, we compute the critical density lambdac con above which connectivity percolation in 3D WSNs will almost surely occur. Third, we compute the critical density lambdac cov-con above which both coverage and connectivity percolation in 3D WSNs will almost surely occur. For each of these three problems, we also compute their corresponding critical network degree. Our results can be helpful in the design of energy-efficient topology control protocols for 3D WSNs in terms of coverage and connectivity.
    IEEE Trans. Parallel Distrib. Syst. 01/2009; 20:872-885.
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    ABSTRACT: Connectivity, primarily a graph-theoretic concept, helps define the fault tolerance of wireless sensor networks (WSNs) in the sense that it enables the sensors to communicate with each other so their sensed data can reach the sink. On the other hand, sensing coverage, an intrinsic architectural feature of WSNs plays an important role in meeting application-specific requirements, for example, to reliably extract relevant data about a sensed field. Sensing coverage and network connectivity are not quite orthogonal concepts. In fact, it has been proven that connectivity strongly depends on coverage and hence considerable attention has been paid to establish tighter connection between them although only loose lower bound on network connectivity of WSNs is known. In this article, we investigate connectivity based on the degree of sensing coverage by studying k-covered WSNs, where every location in the field is simultaneously covered (or sensed) by at least k sensors (property known as k-coverage, where k is the degree of coverage). We observe that to derive network connectivity of k-covered WSNs, it is necessary to compute the sensor spatial density required to guarantee k-coverage. More precisely, we propose to use a model, called the Reuleaux Triangle, to characterize k-coverage with the help of Helly's Theorem and the analysis of the intersection of sensing disks of k sensors. Using a deterministic approach, we show that the sensor spatial density to guarantee k-coverage of a convex field is proportional to k and inversely proportional to the sensing range of the sensors. We also prove that network connectivity of k-covered WSNs is higher than their sensing coverage k. Furthermore, we propose a new measure of fault tolerance for k-covered WSNs, called conditional fault tolerance, based on the concepts of conditional connectivity and forbidden faulty sensor set that includes all the neighbors of a given sensor. We prove that k-covered WSNs can sustain a large number of sensor failures provided that the faulty sensor set does not include a forbidden faulty sensor set.
    TAAS. 01/2009; 4.
  • 06/2011: pages 249 - 262; , ISBN: 9781119970422

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