Probabilistic $(m,n)$-Parking Functions

TL;DR

The paper analyzes probabilistic $(m,n)$-parking functions with $m\le n$ cars and a bias parameter $p$, focusing on the last-car parking preference $a_m$. It develops explicit $p$-dependent formulas for the distribution and mean of $a_m$ using combinatorial constructs such as parking-function multi-shuffles and Abel's multinomial theorem, augmented by a Poisson large-deviation lemma. It extends previous results from the $m=n$ setting to general $m\le n$, derives asymptotics in the $m=cn$ regime, and reveals a sharp effect of extra spots on convergence to the uniform distribution, including a linear-in-$c$ decay of the total variation distance as $c\to0$ and a slower $1/n$ vs $1/\sqrt{n}$ rate when $p\neq 1/2$. The convergence-rate results are complemented by clear symmetry properties relating $p$ and $1-p$, and by upper/lower bounds for the distance to uniform that quantify how the ratio of cars to spots drives statistical behavior. Together, these findings illuminate how probabilistic parking dynamics depend on both the supply of spots and the forward-move bias, with potential implications for generalized parking models and related combinatorial probability problems.

Abstract

In this article, we establish new results on the probabilistic parking model (introduced by Durmíc, 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 $c$.

Figures (2)

• Figure 1: The conditional distribution $Q_{m,n,p}$ of $a_m$ (parking preference of the last car) in $100,000$ samples of preference lists of size $n=100$ chosen uniformly at random. Here the forward-moving probability $p=0.7$. The left plot is for $m=20$ cars and the right plot is for $m=100$ cars.
• Figure 2: The total variation distance $\|Q_{cn,n,p} - {\rm Uni}_n\|_{\mathrm{TV}}$ as a function of $c$. Here $n=100$, $m=cn$, $p=1$, and $c$ ranges from $0.1$ to $0.99$.

Theorems & Definitions (29)

• Theorem 2.1: Theorem 2 in DHHRY
• Definition 2.2
• Example 2.3
• Theorem 2.4: Abel's multinomial theorem, derived from Pitman Pitman and Riordan Riordan
• Lemma 2.5
• proof
• Remark 2.6
• Theorem 2.7
• Remark 2.8
• Remark 2.9