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Parking Cars with Spin but no Length

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Abstract

The car parking problem is a one-dimensional model of random packing. Cars arrive to park on a block of length x, sequentially. Each car has, independently, spin up or spin down, w.p. 0 < p ≤ 1, for spin up and q = 1 − p for spin down, respectively. Each car tries to park at a uniformly distributed random point t ∈ [0, x]. If t is within distance 1 of the location of a previously parked car of the same spin, or within distance a of the location of a previously parked car of the opposite spin, then the new car leaves without parking and the next car arrives, until saturation. We study the problem analytically as well as numerically. The expected number of up spins c(p, a) per unit length for sufficiently large x is neither monotonic in p for fixed a, nor is it monotone in a for fixed p, in general. An intuitive explanation is given for this nonmonotonicity.

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Authors and Affiliations

  1. The Institute of Statistical Mathematics, 4-6-7 Minami-Azabu Minato-ku, Tokyo, 106-8569, Japan

    Yoshiaki Itoh

  2. Statistics Department, Rutgers University, Hill Center, Piscataway, New Jersey, 08903

    Larry Shepp

Authors
  1. Yoshiaki Itoh
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  2. Larry Shepp
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Itoh, Y., Shepp, L. Parking Cars with Spin but no Length. Journal of Statistical Physics 97, 209–231 (1999). https://doi.org/10.1023/A:1004619102562

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  • DOI: https://doi.org/10.1023/A:1004619102562

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