A Selection-Replacement Process on the Circle

Abstract

Given $N$ points on a circle, a selection-replacement operation removes one of the points and replaces it by another. To select the removed point, an extra point $P$, uniformly distributed, arrives at random 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 $d = d(N)$, the distance that the searching point $P$ must travel to select a point. After many repeated selection-replacements, the joint probability distribution of the $N$ points tends to a stationary limit. We examine the mean of $d$ in this limit. Exact means are found for $N \leq 3$. For large $N$, the mean grows like $(\log^{3/2} N)/N$. These means are larger than the means $1/(N + 1)$ that would be obtained with $N$ independent uniformly distributed points because the selection mechanism tends to cluster the $N$ points into clumps. In a computer application, the circle represents a track on a disk memory, $P$ is a read-write head, the $N$ points mark the beginnings of $N$ files and $d$ determines the time wasted as the head moves from the end of the last file processed to the beginning of the next. $N$ is a parameter of the service rule (the next service goes to one of the $N$ customers waiting the longest).