A Steepest Descent Relocation Algorithm for Placement of Sinks in a Sensor Network

In this paper we use an information flow model for placement of traffic sinks in a wireless sensor network. Our mathematical model translates a communication network composed of countably many sensors into a continuum model described by a continuous vector field. This vector field models flow of information in a sensor network, and its magnitude is the intensity of the communication activity, and its orientation is the direction in which the traffic is forwarded. We use this formulation for design of a sensor network in which the traffic generated by the wireless sensors needs to be routed to one of the multiple traffic sinks in different locations of the network. We show that the optimal vector field satisfies a set of PDEs similar to Maxwell's equations in electrostatics, with Neumann boundary conditions. We prove that the orientation of the vector field at the location of each sink is the direction of the steepest descent of the total communication cost function of the network. This direction corresponds to the direction of force in an analogous electrostatics problem. We use this result in order to introduce iterations that relocate sinks and show that in the optimal placement of sinks, the value of the vector field is zero at the location of each sinks.