Probabilistic $(m,n)$-Parking Functions

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 $c$.