Limit Theorems for Degree of Coverage and Lifetime in Large Sensor Networks

In this paper, we investigate the fundamental limits of sensor network lifetime that any algorithm can achieve. In our settings, n nodes are deployed as a Poisson point process with density lambda in a region of size S and each sensor node can cover a unit-area disk. For any k and lambda, let V(k, lambda) be the random variable (r.v.) of the size of the region that is covered by at most k - 1 nodes. Under these assumptions, we first show that for any function omega satisfying 1 Gtomega (k) Lt k^1/2 the r.v. V(k, k- omega(k)k^1/2) converges almost surely to S. In contrast, if the intensity is set to lambda = k + omega(k)k^1/2 we obtain that V(k, k +omega(k)k ^1/2) converges almost surely to 0. These limit theorems extend the results of Zhang and Hou in [21], [22] where the authors worked with fixed degree of coverage (k = O(1)) and lambda = log S+O(k) log log S. Assume that each sensor has the same lifetime T. As consequences of our analytical results, we derive randomized algorithms (working with high probability) that can maintain constantly high degrees of coverage while prolonging the lifetime of the network.