On the Maximum Movement of Random Sensors for Coverage and Interference on a Line

Abstract

This paper addresses the fundamental problem of time-efficient reallocation of mobile random sensors to provide good communication within the network without interference. Consider n mobile sensors placed on the half-infinite interval [0, ∞) according to the arrival times of the Poisson process with arrival rate λ > 0. Each sensor has identical sensing range r thus covering an interval of length 2r. Assume that there is interference between the signals of two sensors if their distance is less than s. If a sensor is displaced a distance equal to d, it takes the time that is proportional to the distance d traveled. Thus the time required for displacement of a system of n sensors is defined as the maximum of the displacement of individual sensors. We focus on the problem of time-efficient reallocation of the sensors so that in their final placement the sensor system: • There are no uncovered points from the origin to the last rightmost sensor, • Every two sensors are at distance greater or equal to s, and • The time required for the movement of the sensors to these final positions is minimized in expectations. In this paper the tradeoffs between the sensing radius r, the interference distance s, the number of sensors n, the arrival rate λ and the minimal maximum of the expected sensor's displacements are all derived. We discover a threshold around the interference distance s = 1/λ and the sensing radius r = 1/2λ for the minimal maximum of the expected sensor's displacements to avoid interference, and to ensure that there are no uncovered points between the origin and the location of the last rightmost sensor.