430 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 58, NO. 1, JANUARY 2012 Coverage Properties of the Target Area in Wireless Sensor Networks Xiaoyun Li, David K. Hunter, Senior Member, IEEE, and Sergei Zuyev Abstract—An analytical approximation is developed for the probability of sensing coverage in a wireless sensor network with randomly deployed sensor nodes each having an isotropic sensing area. This approximate probability is obtained by considering the properties of the geometric graph, in which an edge exists between any two vertices representing sensor nodes with over- lapping sensing areas. The principal result is an approximation to the proportion of the sensing area that is covered by at least one sensing node, given the expected number of nodes per unit area in a two-dimensional Poisson process. The probability of a specified region being completely covered is also approximated. Simulation results corroborate the probabilistic analysis with low error, for any node density. The relationship between this approximation and noncoverage by the sensors is also examined. These results will have applications in planning and design tools for wireless sensor networks, and studies of coverage employing computational geometry. Index Terms—Coverage, dimensioning, geometric graph, net- works, Poisson process, sensor. I. INTRODUCTION A WIRELESS sensor network (WSN) monitors some specific physical quantity, such as temperature, humidity, pressure or vibration. It collates and delivers the sensed data to at least one sink node, usually via multiple wireless hops. To en- sure sensing coverage, the subject of this paper, the WSN must sense the required physical quantity over the entire area being monitored—while doing this, both power consumption and the efficiency of data aggregation are crucial considerations. We assume ideal conditions where each sensor node has an isotropic sensing area defined by a circle of radius , although in practice it may be directional to some extent because of phys- ical obstacles. Although the analysis could be extended to cope with scenarios where a node’s sensing range depends on the en- vironment, the results in this paper nevertheless have practical significance for many deployments. They will be useful when Manuscript received September 25, 2008; revised August 25, 2011; accepted September 01, 2011. Date of current version January 06, 2012. X. Li is with the Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, University Town of Shenzhen, 518055, P. R. China (e-mail: xy.li@siat.ac.cn). D. K. Hunter is with the School of Computer Science and Electronic Engineering, University of Essex, Colchester CO4 3SQ, U.K. (e-mail: dkhunter@essex.ac.uk). S. Zuyev is with the Department of Mathematical Sciences, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden (e-mail: sergei.zuyev@chalmers.se). Communicated by U. Mitra, Associate Editor At Large. Color versions of one or more of the figures in this paper are available online at http://ieee.ieeexplore.org. Digital Object Identifier 10.1109/TIT.2011.2169300 estimating the sensor density required, or when determining the likelihood of holes in the sensing coverage. It is also assumed that the distribution of sensor nodes over the target sensing area is described by a homogeneous Poisson process, suggesting that the results are most relevant to applications with randomly scat- tered nodes. A point in the plane is said to be tricovered if it lies inside some triangle formed by three edges in the geometric graph. In this graph, each active sensor node is represented by a vertex, and an edge exists between any two vertices representing nodes with overlapping sensing areas; with the isotropic coverage as- sumed here, this happens when the corresponding nodes are less than units apart (Fig. 1). The clustering and graph parti- tioning properties of geometric graphs have already been in- vestigated [1]–[3], with applications for example in the design of frequency partitioning algorithms for wireless broadcast net- works. Furthermore, an area is said to be tricovered if every point within it is tricovered. A bound is determined for the prob- ability that all points in the target area which are further than units from its boundary are tricovered. Tricoverage is closely related to sensing coverage. If an area is not tricovered, there must be points inside it which are not covered by the sensing area of any node (white space in Fig. 1); see the proof in Section IV. The connected components of areas which are not tricovered are called large holes. However, a point may still be tricovered, but nevertheless not be covered by any node’s sensing area, as in Fig. 2. Such points are said to lie inside a trivial hole. An estimate obtained below shows that the propor- tion of space in homogeneous systems occupied by trivial holes is less than 0.03% regardless of the sensor node density, so they can in practice be ignored when calculating coverage. Hence the analytical calculations for the probability of full tricoverage pro- posed here provide a good approximation to the real probability of sensing coverage, albeit in an idealised scenario, although no assumptions are made about the shape of the overall area to be covered. This will be a useful guide for making network plan- ning and design decisions, especially as our analytical method generates results much more quickly than can be achieved by simulation. The coverage problem for sensor networks has been investi- gated in previous studies [4]–[6], with mathematical methods having been developed for the calculation or estimation of sensing coverage [7]–[10]. Although tricoverage provides a useful way of approximating overall coverage, the analysis presented here using the geometric graph is also directly rele- vant to a general class of distributed algorithms which use only local connectivity information in order to determine the extent of coverage; for examples see [11] and [12]. 0018-9448/$31.00 © 2012 IEEE