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A unified model for Kakutani's interval splitting and Rényi's random packing

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Fumiyasu Komaki  and
Yoshiaki Itoh
Fumiyasu Komaki*
Affiliation:
Graduate University for Advanced Studies
Yoshiaki Itoh
Affiliation:
Graduate University for Advanced Studies
*
∗All correspondence should be addressed to this author.
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Abstract

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One-dimensional random packing, known as the car-parking problem, was first analyzed by Rényi (1958). A stochastic version of Kakutani's (1975) interval splitting is another typical model on a one-dimensional interval. We consider a generalized car-parking problem which contains the above two models as special cases. In the generalized model, one can park a car of length l, if there is a space not less than 1. We give the limiting packing density and the limiting distribution of the length of randomly selected gaps between cars. Our results bridge the two models of Rényi and Kakutani.

Keywords

GENERALIZED RANDOM PACKINGPACKING DENSITYDISTRIBUTION OF GAPSASYMPTOTIC BEHAVIOR

MSC classification

Secondary: 60K99: None of the above, but in this section
Type
Letters to the Editor
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 502 - 505
Copyright
Copyright © Applied Probability Trust 1992 

Footnotes

Postal address for both authors: Department of Statistical Science, The Graduate University for Advanced Studies, and the Institute of Statistical Mathematics, 4–6–7, Minami Azabu, Minato-ku, Tokyo, 106, Japan.

References

Bankövi, G. (1962) On gaps generated by a random space filling procedure. Publ. Math. Inst. Hungar. Acad. Sci. 7, 395–407.Google Scholar
Kakutani, S. (1975) A problem of equidistribution on the unit interval [0, 1]. Lecture Notes in Mathematics 541, 369–376.CrossRefGoogle Scholar
Rényi, A. (1958) On a one-dimensional problem concerning random space-filling problem. Publ. Math. Inst. Hungar. Acad. Sci. 3, 109–127.Google Scholar
Van Zwet, W. R. (1978) A proof of Kakutani's conjecture on random subdivision of longest intervals. Ann. Prob. 6, 133–137.Google Scholar