A Selection-Replacement Process on the Circle

Give N points on a circle, a selection-replacement operation removes one if the points and replaces it by another. To select the removed point, an extra point P arrives at random, uniformly distributed, and starts moving counterclockwise around the circle; P removes the first point it encounters. A new random point, uniformly distributed, then replaces the removed point. The quantity of interest is d approximately d (N), the distance that the searching point P must travel to select a point. After many repeated selection-replacements, the N points have a stationary limiting joint probability distribution and we examine the mean of d. Exact means are found for N