Probabilistic $(m,n)$-Parking Functions

TL;DR

This paper analyzes the probabilistic $(m,n)$-parking model, deriving explicit formulas for the last car's parking preference distribution, studying asymptotic behaviors, and showing how extra parking spots influence convergence rates.

Contribution

It provides new explicit formulas and asymptotic analysis for the distribution of the last car's parking preference in the probabilistic parking model, highlighting the impact of extra spots.

Findings

Explicit formulas for the distribution of the last car's parking preference.

Asymptotic behavior of the distribution as $n$ grows large.

Extra parking spots accelerate convergence to uniform distribution.

Abstract

In this article, we establish new results on the probabilistic parking model (introduced by Durm\'ic, Han, Harris, Ribeiro, and Yin) with $m$ cars and $n$ parking spots and probability parameter $p\in[0,1]$. For any $ m \leq n$ and $p \in [0,1]$, we study the parking preference of the last car, denoted $a_m$, and determine the conditional distribution of $a_m$ and compute its expected value. We show that both formulas depict explicit dependence on the probability parameter $p$. We study the case where $m = cn $ for some $ 0 < c < 1 $ and investigate the asymptotic behavior and show that the presence of ``extra spots'' on the street significantly affects the rate at which the conditional distribution of $ a_m $ converges to the uniform distribution on $[n]$. Even for small $ \varepsilon = 1 - c $, an $ \varepsilon $-proportion of extra spots reduces the convergence rate from $ 1/\sqrt{n} $ to $ 1/n $ when $ p \neq 1/2 $. Additionally, we examine how the convergence rate depends on $c$, while keeping $n$ and $p$ fixed. We establish that as $c$ approaches zero, the total variation distance between the conditional distribution of $a_m$ and the uniform distribution on $[n]$ decreases at least linearly in $c$.