Renyi's Parking Problem Revisited

TL;DR

This paper revisits Rényi's 1958 parking problem, analyzing a symmetric discretized version with recursion formulas, series, and simulations, confirming the limiting filling density as Rényi's known parking constant.

Contribution

It introduces a symmetric discretized model of Rényi's parking problem, derives recursion formulas, and provides series and simulations to confirm the parking constant.

Findings

Expected filling density converges to Rényi's parking constant.

Derived recursion formulas for expected gaps.

Provided fast converging series and extensive simulations.

Abstract

R\'enyi's parking problem (or $1D$ sequential interval packing problem) dates back to 1958, when R\'enyi studied the following random process: Consider an interval $I$ of length $x$, and sequentially and randomly pack disjoint unit intervals in $I$ until the remaining space prevents placing any new segment. The expected value of the measure of the covered part of $I$ is $M(x)$, so that the ratio $M(x)/x$ is the expected filling density of the random process. Following recent work by Gargano {\it et al.} \cite{GWML(2005)}, we studied the discretized version of the above process by considering the packing of the $1D$ discrete lattice interval $\{1,2,...,n+2k-1\}$ with disjoint blocks of $(k+1)$ integers but, as opposed to the mentioned \cite{GWML(2005)} result, our exclusion process is symmetric, hence more natural. Furthermore, we were able to obtain useful recursion formulas for the expected number of $r$-gaps ($0\le r\le k$) between neighboring blocks. We also provided very fast converging series and extensive computer simulations for these expected numbers, so that the limiting filling density of the long line segment (as $n\to \infty$) is R\'enyi's famous parking constant, $0.7475979203...$.