Optimal placement of nodes in large sensor networks under a general physical layer model

We study a setting in which 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, transporting them over a wireless channel to the sink locations, and delivering them to the sinks. The problem is to find the optimal placement of these nodes, so that the minimum number of nodes is needed. Our approach is to assume a massively dense network, in which there are so many sources, sinks, and wireless nodes, that it does not make sense to model the network in terms of microscopic parameters, such as their individual placements, but rather in terms of macroscopic parameters, such as their spatial densities. We use calculus of variations to describe the optimal spatial density of nodes, that minimizes their total number but can still support the whole traffic, in terms of a scalar, nonlinear, partial differential equation. The derivation is under a general model for the physical layer, that encompasses many cases of interest. Our work can help in the design of networks that are deployed in the most efficient manner, not only avoiding the formation of bottlenecks, but also striking the optimal balance between avoiding congestion and having the data packets follow short routes.

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