Consider a set of n closed finite intervals with non-negative endpoints, symbolizing sensors on the real line. The sensors are placed randomly in such a way that the distances between the consecutive centres are independent random variables with expectation inversely proportional to n. Sensor's length is equal among all the sensors, and overlaps between sensors are possible. In this work we address two fundamental sensor allocation problems. The interference problem tries to reallocate the sensors from their initial positions so as to eliminate overlaps. The coverage problem, on the other hand, allows overlaps, but tries to eliminate gaps (uncovered spaces) between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. The paper investigates asymptotical behaviour of the optimal costs as the number of sensors increases to infinity. We propose a new unifying approach using tools from queueing theory, reflected random walks and weak convergence. We show that the results depend substantially on whether the sensor length is asymptotically greater, less or equal to the average distance between the original midpoints. The methodology is then used to address a more complicated, modified coverage problem, in which the overlaps between any two sensors can not exceed a certain parameter.
The authors of REF do not solve the coverage problem, but address the problem of reallocating n sensors on the halfinfinitive interval [0, ∞) to eliminate the overlaps and remove the gaps, securing uninterrupted coverage starting from the origin.
Sensor allocation problems on the real line
J. Applied Probability
J. Applied Probability
Mathematics Computer Science