Sensor allocation problems on the real line

A large number n of sensors (finite connected intervals) are placed randomly on the real line so that the distances between the consecutive midpoints are independent random variables with expectation inversely proportional to n. In this work we address two fundamental sensor allocation problems. Interference problem tries to reallocate the sensors from their initial positions so as to eliminate overlaps. Coverage problem, on the other hand, allows overlaps, but tries to eliminate uncovered spaces between the originally placed sensors. Both problems seek to minimize the total sensor movement while reaching their respective goals. Using tools from queueing theory, Skorokhod reflections and weak convergence, we investigate the asymptotic behaviour of the optimal costs as n increases to infinity. The introduced 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.