Analysis of the Threshold for Energy Consumption in Displacement of Random Sensors

Consider $n$ mobile sensors placed randomly in $m-$dimensional unit cube for fixed $m\in\{1,2\}.$ 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 unit cube is within the range of at least one sensor, while at the same time each pair of sensors is placed at interference distance greater or equal to $s.$ Suppose the displacement of the $i-$th sensor is a distance $d_i$. As a \textit{energy consumption} for the displacement of a set of $n$ sensors we consider the $a-$total displacement defined as the sum $\sum_{i=1}^n d_i^a,$ for some constant $a> 0.$ The main contribution of this paper can be summarized as follows. For the case of unit interval we \textit{explain a threshold} around the sensing radius equal to $\frac{1}{2n}$ and the interference distance equal to $\frac{1}{n}$ for the expected minimum $a-$total displacement. For the sensors placed in the unit square we \textit{explain a threshold} around the square sensing radius equal to $\frac{1}{2 \sqrt{n}}$ and the interference distance equal to $\frac{1}{\sqrt{n}}$ for the expected minimum $a-$total displacement