Mathematics of Parking: Varying Parking Rate

TL;DR

This paper generalizes the classical parking problem by analyzing the saturation behavior when intervals are placed with an exponential distribution, deriving asymptotic limits and variance behavior as the rate parameter varies.

Contribution

It introduces a mathematical framework for the exponential parking problem, deriving integral equations and asymptotic results for saturation limits and variance.

Findings

Saturation number satisfies a specific integral equation.

Asymptotic limits for large and small rate parameters are characterized.

Variance of intervals at saturation is analyzed asymptotically.

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.