Open Access
August, 1993 A Selection-Replacement Process on the Circle
E. G. Coffman Jr., E. N. Gilbert, P. W. Shor
Ann. Appl. Probab. 3(3): 802-818 (August, 1993). DOI: 10.1214/aoap/1177005365

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 N3. For large N, the mean grows like (log3/2N)/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).

Citation

Download Citation

E. G. Coffman Jr.. E. N. Gilbert. P. W. Shor. "A Selection-Replacement Process on the Circle." Ann. Appl. Probab. 3 (3) 802 - 818, August, 1993. https://doi.org/10.1214/aoap/1177005365

Information

Published: August, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0780.60037
MathSciNet: MR1233627
Digital Object Identifier: 10.1214/aoap/1177005365

Subjects:
Primary: 60F99
Secondary: 05C70 , 60J10

Keywords: Matching , Probabilistic analysis of algorithms , Selection-replacement process

Rights: Copyright © 1993 Institute of Mathematical Statistics

Vol.3 • No. 3 • August, 1993
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