Probabilistic (m, n)-Parking Functions

Pamela E. Harris

doi:10.37236/13864

Probabilistic (m, n)-Parking Functions

2025-11-03

0

PRE

AI

DOI：10.37236/13864

原文求助

摘要

En 中文

In this article, we establish new results on the probabilistic parking model (introduced by Durmic, Han, Harris, Ribeiro, and Yin) with m cars and n parking spots and probability parameter p is an element of [0,1]. For any m <= n and p is an element of [0,1], we study the parking preference of the last car, denoted am, and determine the conditional distribution of am 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<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 am converges to the uniform distribution on [n]. Even for small epsilon = 1-c, an epsilon-proportion of extra spots reduces the convergence rate from 1/root n to 1/n when p not equal 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 am and the uniform distribution on [n] decreases at least linearly in c.

Keyword:

PARKING FUNCTIONS

Probabilistic (m, n)-Parking Functions

摘要

期刊

机构

引用论文