Threshold phenomena for interference with randomly placed sensors

This paper investigates the problem of efficient displacement of random sensors where good communication within the network is provided, there is no interference between sensors and the movement cost is minimized in expectation. Fix d ∈ N \ {0}. Assume n sensors are initially placed in the hyperoctant [0,∞)d according to d identical and independent Poisson processes each with arrival rate n1/d. Let 0 < s ≤ v be given real numbers. We are allowed to move the sensors, so that every two consecutive sensors are placed at distance greater than or equal to s and less than or equal to v. When a sensor is displaced a distance equal to m, the cost of movement is proportional to some (fixed) power a > 0 of the m distance travelled. We consider a−total movement (see Definition 2) for one dimension and d−dimensional a−total movement (see Definition 3) for the higher dimension. Motivation for using this extended cost metric arises from the fact that the parameter a in the exponents represents various conditions on the barrier: lubrication, obstacles, etc. The main contribution of this paper can be summarized as follows. 1) If the sensors are displaced in the half-infinite interval [0,∞), the tradeoffs between the interference distances s, v and the expected minimum a−total movement are derived. We discover a sharp decline and a sharp increase in the expected minimum a−total movement around the interference distance s, v equal to 1 n . 2) For the sensors in the hyperoctant [0,∞)d (d > 1), the tradeoffs between the interference distances s, v and the expected minimum d−dimensional a−total movement are obtained. We also discover a sharp decline and a sharp increase in the expected minimum d−dimensional a−total movement around the interference distance s, v equal to 1 n1/d .