Analysis of the Threshold for the Displacement to the Power of Random Sensors
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We address the fundamental problem of energy efficient displacement of random sensors to provide good communication within the network, i.e., to ensure complete coverage without interference. Consider n mobile sensors placed independently at random with the uniform distribution in the unit interval [0,1]. The sensors have identical sensing range, say r. We are interested in moving the sensors from their initial random positions to new locations so that every point in the [0,1] is within the range of at least one sensor, while at the same time no two sensors are placed at distance less than s apart. Further, for some fixed constant a> 0 if a sensor is displaced a distance equal to d it consumes energy proportional to d^a. Suppose the displacement of the i-th sensor is a distance d_i. As a cost measure for the displacement of a set of n sensors we consider the a-total displacement defined as the sum ∑_i=1^n d_i^a, for some constant a> 0. In this paper we discover a threshold around the sensing radius equal to 1/2n and the interference distance s=1/n for the expected minimum a-total displacement.
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