1Optimal Deployment of Large Wireless Sensor Networks

Abstract — A spatially distributed set of sources is creating data that must be delivered to a spatially distributed set of sinks. A network of wireless nodes is responsible for sensing the data at the sources, transporting them over a wireless channel, and delivering them to the sinks. The problem is to find the optimal placement of nodes, so that a minimum number of them is needed. The critical assumption is made that the network is massively dense, i.e., there are so many sources, sinks, and wireless nodes, that it does not make sense to discuss in terms of microscopic parameters, such as their individual placements, but rather in terms of macroscopic parameters, such as their spatial densities. Assuming a particular interference-limited, capacity-achieving physical layer, and specifying that nodes only need to transport the data (and not to sense them at the sources, or deliver them at the sinks once their location is reached), the optimal node placement induces a traffic flow that is identical to the electrostatic field created if the sources and sinks are replaced by a corresponding distribution of positive and negative charges. Assuming a general model for the physical layer, and specifying that nodes must not only transport the data, but also sense them at the sources and deliver them at the sinks, the optimal placement of nodes is given by a scalar nonlinear partial differential equation found by calculus of variations techniques. The proposed formulation and derived equations can help in the design of large wireless sensor networks that are deployed in the most efficient manner, not only avoiding the formation of bottlenecks, but also striking the optimal balance between reducing congestion and having the data packets follow short routes