Unreliable sensor grids: coverage, connectivity and diameter

We consider an unreliable wireless sensor grid-network with n nodes placed in a square of unit area. We are interested in the coverage of the region and the connectivity of the network. We first show that the necessary and sufficient conditions for the random grid network to cover the unit square region as well as ensure that the active nodes are connected are of the form p(n)r²(n) ~ log(n)/n, where r(n) is the transmission radius of each node and p(n) is the probability that a node is "active" (not failed). This result indicates that, when n is large, even if each node is highly unreliable and the transmission power is small, we can still maintain connectivity with coverage. We also show that the diameter of the random grid (i.e., the maximum number of hops required to travel from any active node to another) is of the order √{n/log(n)}. Finally, we derive a sufficient condition for connectivity of the active nodes (without necessarily having coverage). If the node success probability p(n) is small enough, we show that connectivity does not imply coverage.