This paper studies the fundamental problem of energy consumption in the movement of mobile random sensors ensuring k-weak coverage on the domain. In particular, we analyze two notions of k-weak coverage on the unit square, namely (1) $(k, x)$-weak coverage in which every straight-line path across the width of the unit square passes through the sensing range of at least k sensors; and (2) $(k, x, y)$-weak coverage in which every straight-line path across the width and the length of the unit square passes through the sensing range of at least k sensors. The number of reliable and p-reliable sensors (i.e., each sensor is reliable with probability p independently from other sensors) in energy-efficient reallocation is optimized to provide the desired k-weak coverage. For reliable sensors, we discover sharp decrease in the expected minimal energy consumption in the movement to provide the desired k-weak coverage as a function of the number of sensors. For p-reliable sensors, we design and analyze two energy-efficient algorithms to ensure the k-weak coverage. The novelty of the paper is in the theoretical understanding and analysis.

This article deals with reliable and unreliable mobile sensors having identical sensing radius r , communication radius R , provided that r ≤ R and initially randomly deployed on the plane by dropping them from an aircraft according to general random process. The sensors have to move from their initial random positions to the final destinations to provide greedy path k 1 -coverage simultaneously with k 2 -connectivity. In particular, we are interested in assigning the sensing radius r and communication radius R to minimize the time required and the energy consumption of transportation cost for sensors to provide the desired k 1 -coverage with k 2 -connectivity. We prove that for both of these optimization problems, the optimal solution is to assign the sensing radius equal to r = k 1 || E [S]||/2 and the communication radius R = k 2 || E [S]||/2, where || E [S]|| is the characteristic of general random process according to which the sensors are deployed. When r < k 1 || E [S]||/2 or R < k 2 || E [S]||/ 2, and sensors are reliable, we discover and explain the sharp increase in the time required and the energy consumption in transportation cost to ensure the desired k 1 -coverage with k 2 -connectivity.