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Mathematics of Parking: Varying Parking Rate

Abstract

In the classical parking problem, unit intervals (“car lengths”) are placed uniformly at random without overlapping. The process terminates at saturation, i.e. until no more unit intervals can be stowed. In this paper, we present a generalization of this problem in which the unit intervals are placed with an exponential distribution with rate parameter \(\lambda \). We show that the mathematical expectation of the number of intervals present at saturation satisfies a certain integral equation. Using Laplace transforms and Tauberian theorems, we investigate the asymptotic behavior of this function and describe a way to compute the corresponding limits for large \(\lambda \). Then, we derive another integral equation for the derivative of this function and use it to compute the above limits for small \(\lambda \) with the help of some asymptotic results for integral equations. We also show that the corresponding limits converge to the uniform case as \(\lambda \) vanishes, yielding the well-known Renyi constant. Finally, we reveal the asymptotic behavior of the variance of the intervals at saturation.

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Fig. 1

Notes

  1. 1.

    Let \(S:[0,\infty )\rightarrow {\mathbb {R}}\) be a nondecreasing function such that \(S(0)=0\) and let \(w(s)=\displaystyle \int _0^\infty e^{-xs}\,dS(x)\). Then, for \(\rho >0\),

    $$\begin{aligned} w(s)\sim \frac{C}{s^\rho }\quad \hbox { as}\ s\rightarrow 0^+ \end{aligned}$$

    if and only if

    $$\begin{aligned} S(x)\sim \frac{C}{\Gamma (\rho +1)}x^\rho \quad \hbox { as}\ x\rightarrow \infty . \end{aligned}$$
  2. 2.

    Let \(f:[0,\infty )\rightarrow {\mathbb {R}}\) and let \(F(s)=\int _0^\infty f(x)e^{-xs}\,dx\) be its Laplace transform. If F(s) has no poles, z, in the complex plane with Re\((z)\ge 0\), except at most one pole at \(z=0\), then

    $$\begin{aligned} \lim _{x\rightarrow \infty }f(x)=\lim _{s\rightarrow 0^+}sF(s). \end{aligned}$$

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Acknowledgements

The authors are thankful to Jake Hymowitz for the visualization of these results and to the Reviewer who suggested a possible improvement to the decay rate of the error terms in (11) and (14).

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Correspondence to Pavel B. Dubovski.

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Communicated by Eric A. Carlen.

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Dubovski, P.B., Tamarov, M. Mathematics of Parking: Varying Parking Rate. J Stat Phys 182, 22 (2021). https://doi.org/10.1007/s10955-020-02678-x

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Keywords

  • Asymptotic behavior
  • Integral equations
  • Parking problem
  • Percolation
  • Rényi constant
  • Saturation
  • Self-similar random variable
  • Tauberian theorems

Mathematics Subject Classification

  • 60K35
  • 82B43
  • 45D99