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This paper presents a tractable analytical framework for the exact calculation of probability of node isolation when N sensor nodes are independently and uniformly distributed inside a finite square region. The proposed framework can accurately account for the boundary effects by partitioning the square into sub regions, based on the transmission range and the node location. The authors show that for each sub region, the probability that a random node falls inside a disk centered at an arbitrary node located in that sub region can be expressed analytically in closed-form. Using the results for the different sub regions, they obtain the exact probability of node isolation and an upper bound for the probability of connectivity.
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