Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-22T13:18:32.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Random packing of an interval

Published online by Cambridge University Press:  01 July 2016

David Mannion
David Mannion*
Affiliation:
Royal Holloway College, London
Get access Rights & Permissions [Opens in a new window]

Abstract

At each stage of the packing of a closed interval K, a random number of random open intervals (the packing objects) are placed in that part of K which is as yet unoccupied. No overlapping between the packing objects is allowed. The packing prescription is such that the packing process terminates after at most a finite number of stages. Attention is focused on the final configuration, K = K– + G, where G is a random open subset of K, and is that part of K which is eventually occupied by packing objects, while K–, a random closed subset of K, is that part of K which remains unoccupied.

Keywords

RANDOM SPACE-FILLINGRANDOM PACKINGCAR-PARKINGCASCADE PROCESSES
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 3 , September 1976 , pp. 477 - 501
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Akeda, Y. and Hori, M. (1975) Numerical test of Palásti's conjecture on two-dimensional random packing density. Nature 254, 318–319.Google Scholar
[2] Akeda, Y. and Hori, M. (1976) Three-dimensional random packing. Biometrika. To appear.Google Scholar
[3] Ambartsumian, R. (1963) A problem concerning the filling of a straight line by random sequential addition of intervals. Proceedings of the VIIth All-Union Conference on Mathematics, Statistics and Probability, Tbilisi. Google Scholar
[4] Bánkövi, G. (1962) On gaps generated by a random space-filling procedure. Publ. Math. Inst. Hung. Acad. Sci. 7, 395–407.Google Scholar
[5] Blaisdell, B. E. and Solomon, H. (1970) On random sequential packing and a conjecture of Palasti. J. Appl. Prob. 7, 667–698.Google Scholar
[6] Downton, F. (1961) A note on vacancies on a line. J. R. Statist. Soc. B 23, 364–374.Google Scholar
[7] Dvoretzky, A. and Robbins, H. (1964) On the ‘parking’ problem. Publ. Math. Inst. Hung. Acad. Sci. 9, 209–225.Google Scholar
[8] Halmos, P. R. (1944) Random alms. Ann. Math. Statist. 15, 182–189.Google Scholar
[9] Kolmogorov, A. (1941) Uber das logarithmisch normale Verteilungsgesetz der Dimension der Teilchen bei Zerstuckelung. Dokl. Acad. Nauk. SSSR. 31, 99–101.Google Scholar
[10] Mackenzie, J. K. (1962) Sequential filling of a line by intervals placed at random and its application to linear absorption. J. Chem. Phys. 37, 723–728.Google Scholar
[11] Mannion, D. (1964) Random space-filling in one dimension. Publ. Math. Inst. Hung. Acad. Sci. 9, 143–154.Google Scholar
[12] Mannion, D. (1968) Unpublished Ph.D. Thesis, University of Cambridge.Google Scholar
[13] Mullooly, J. P. (1968) A one-dimensional random space-filling problem. J. Appl. Prob. 5, 427–435.Google Scholar
[14] Ney, P. E. (1962) A random interval filling problem. Ann. Math. Statist. 33, 702–718.Google Scholar
[15] Page, E. S. (1959) The distribution of vacancies on a line. J. R. Statist. Soc. B 21, 364–374.Google Scholar
[16] Palásti, I. (1960) On some random space-filling problems. Publ. Math. Inst. Hung. Acad. Sci. 5, 353–360.Google Scholar
[17] Rényi, A. (1958) On a one-dimensional problem concerning random space-filling. Publ. Math. Inst. Hung. Acad. Sci. 3, 109–127.Google Scholar
[18] Ulam, S. and Everett, C. J. (1948) Multiplicative systems. Proc. Natn. Acad. Sci. 34, 403–405.Google Scholar
[19] Widder, D. V. (1941) The Laplace Transform. Princeton University Press.Google Scholar
[20] Widom, B. (1966) Random sequential addition of hard spheres to a volume. J. Chem. Phys. 44, 3888–3894.CrossRefGoogle Scholar