Advertisement

Vestnik St. Petersburg University, Mathematics

, Volume 51, Issue 4, pp 322–326 | Cite as

The Problem of Selfish Parking

  • S. M. AnanjevskiiEmail author
  • N. A. Kryukov
Mathematics
  • 1 Downloads

Abstract

One of the models of discrete analog of the Rényi problem known as the “parking problem” has been considered. Let n and i be integers, n ≥ 0, and 0 ≤ in–1. Open interval (i, i + 1), where i is a random variable taking values 0, 1, 2, …, and n–1 for all n ≥ 2 with equal probability, is placed on interval [0, n]. If n < 2, we say that the interval cannot be placed. After placing the first interval, two free intervals [0, i] and [i + 1, n] are formed, which are filled with intervals of unit length according to the same rule, independently of each other, etc. When the filling of [0, n] with unit intervals is completed, the distance between any two neighboring intervals does not exceed 1. Let Xn be the number of placed intervals. This paper analyzes the asymptotic behavior of moments of random variable Xn. Unlike the classical case, exact expressions for the first moments can be found.

Keywords

random fill parking problem asymptotic behavior of moments 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Rényi “On a one-dimensional problem concerning random space-filling,” Publ. Math. Inst. Hung. Acad. Sci. 3, 109–127 (1958).MathSciNetzbMATHGoogle Scholar
  2. 2.
    A. Dvoretzky and H. Robbins “On the “parking” problem,” Publ. Math. Inst. Hung. Acad. Sci. 9, 209–226 (1964).MathSciNetzbMATHGoogle Scholar
  3. 3.
    P. E. Ney “A random interval filling problem,” Ann. Math. Stat. 33, 702–718 (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Mannion “Random packing of an interval,” Adv. Appl. Probab. 8, 477–501 (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S. M. Anan’evskii “The “parking” problem for segments of different length,” J. Math. Sci. (N. Y.) 93, 259–264 (1999). https://doi.org/.doi 10.1007/BF02364808MathSciNetCrossRefGoogle Scholar
  6. 6.
    S. M. Ananjevskii and E. A. Shulgina “On the measure of the occupied part of a segment in the "parking” problem,” Vestn. S.-Peterb. Univ., Ser. 1: Mat. Mekh., Astron, No. 4, 3–12 (2013).Google Scholar
  7. 7.
    S. M. Ananjevskii “Generalizations of the parking problem,” Vestn. St. Petersburg Univ.: Math. 49, 299–304 (2016). https://doi.org/.doi 10.3103/S1063454116040026MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    M. P. Clay and N. J. Simanyi “Renyi’s parking problem revisited,” arxiv 1406. 1781v1Google Scholar
  9. 9.
    L. Gerin “The Page–Rényi parking process,” arxiv 1411.8002v1Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

Personalised recommendations