We consider a huge hierarchical sensor network consisting of millions of sensors arranged in clusters for scalability and cost-performance. We address the problem of how a centralized gateway can estimate the number of sensors affected by a certain event. We propose a scheme for solving this problem in the most efficient way in terms of communication cost, and a complete mathematical analysis of the estimation error. We show that the error of the new scheme is very small even if the number of sensors experiencing an event is several million.

We present a generic scheme for estimating the size of a group of nodes affected by the same event in a large-scale network, such as a grid, a sensor network or a wireless broadband access network, while receiving only a small number of feedback messages from this group. Using the proposed scheme, a centralized gateway analyzes the transmission times of these feedback messages, defines a likelihood function for them, and then uses the Newton-Raphson method to find the number of affected nodes for which this function is maximized. We present complete mathematical analysis for the precision of the proposed algorithm and provide tight upper and lower bounds for the estimation error. These bounds allow us to improve the precision of our estimation, and to bring the error very close to 0.