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On the random coverage of the circle: a discrete model

On the random coverage of the circle: a discrete model -- The problem on the coverage of the circle is considered when n groups, consisting of m particles each, are allocated into N cells on the circle in the following way: the particles of each group are placed into m adjacent cells and the initial particles of the groups occupy positions on the circle by simple random sampling without replacement. The number of empty segments, the number of such segments of a given length, the total number of empty cells, the lengths of the maximum and minimum empty segments and other characteristics are investigated. The emphasis is on the asymptotic analysis of distributions of these characteristics a&N,n, -- -+ and m = m(N). The basis of the investigations is the model of the conditional distribution of independent geometric random variables under the condition that their sum is fixed. The conditions under which the results known in the continuous model can be derived from their discrete analogues by means of the corresponding passage to the limit are determined. 1. INTRODUCTION In the well-known and popular problem on the random coverage of the circle (see [1, Vol.2], [3] and the bibliography in it) n arcs of length are placed at http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

On the random coverage of the circle: a discrete model

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1994.4.2.147
Publisher site
See Article on Publisher Site

Abstract

-- The problem on the coverage of the circle is considered when n groups, consisting of m particles each, are allocated into N cells on the circle in the following way: the particles of each group are placed into m adjacent cells and the initial particles of the groups occupy positions on the circle by simple random sampling without replacement. The number of empty segments, the number of such segments of a given length, the total number of empty cells, the lengths of the maximum and minimum empty segments and other characteristics are investigated. The emphasis is on the asymptotic analysis of distributions of these characteristics a&N,n, -- -+ and m = m(N). The basis of the investigations is the model of the conditional distribution of independent geometric random variables under the condition that their sum is fixed. The conditions under which the results known in the continuous model can be derived from their discrete analogues by means of the corresponding passage to the limit are determined. 1. INTRODUCTION In the well-known and popular problem on the random coverage of the circle (see [1, Vol.2], [3] and the bibliography in it) n arcs of length are placed at

Journal

Discrete Mathematics and Applications – de Gruyter

Published: Jan 1, 1994

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