On the Path Coverage Properties of Random Sensor Networks

In a sensor network, the points in the operational area that are suitably sensed are a two-dimensional spatial coverage process. For randomly deployed sensor networks, typically, the network coverage of two-dimensional areas is analyzed. However, in many sensor network applications, e.g., tracking of moving objects, the sensing process on paths, rather than in areas, is of interest. With such an application in mind, we analyze the coverage process induced on a one-dimensional path by a sensor network that is modeled as a two-dimensional Boolean model. In the analysis, the sensor locations form a spatial Poisson process of density \lambda and the sensing regions are circles of i.i.d. random radii. We first obtain a strong law for the fraction of a path that is k{\hbox{-}}{\rm sensed}, i.e., sensed by (\geq k) sensors. Asymptotic path-sensing results are obtained under the same limiting regimes as those required for asymptotic coverage by a two-dimensional Boolean model. Interestingly, the asymptotic fraction of the area that is 1-sensed is the same as the fraction of a path that is 1-sensed. For k = 1, we also obtain a central limit theorem that shows that the asymptotics converge at the rate of \Theta(\lambda^{1/2}) for k = 1. For finite networks, the expectation and variance of the fraction of the path that is k{\hbox{-}}{\rm sensed} is obtained. The asymptotics and the finite network results are then used to obtain the critical sensor density to k{\hbox{-}}{\rm sense} a fraction \alpha_{k} of an arbitrary path with very high probability is also obtained. Through simulations, we then analyze the robustness of the model when the sensor deployment is nonhomogeneous and when the paths are not rectilinear. Other path coverage measures like breach, support, "length to first sense,? and sensing continuity measures like holes and clumps are also characterized. Finally, we discuss some generalizations of the results like characterization of the coverage process of m{\hbox{-}}{\rm dimensional} "straight line paths? by n{\hbox{-}}{\rm dimensional}, n > m, sensor networks.