Random covering of the circle: the size of the conneced components

Consider a circle of circumference 1. Throw n points at random onto this circle and append to each of these points a clockwise arc of lengths. The resulting random set is a union of a random number of connected components, each with specific size. Using tools designed by Steutel, we compute the joint distribution of the lengths of the connected components. Asymptotic results are presented when n goes to infinity and s to 0 jointly according to different regimes.