精确计算有限长度道路停车的分布规律与聚集特性。
Exact Finite-Length Theory of Uniform Car Parking: Spatial Laws, Absorption, and Aggregation
- 将停车位置分解为间隔单元，用递归算法求解联合密度。
- 得到间隙统计和吸收数量的解析表达式，精度达有限长度。
- 适合研究随机吸附、物理建模与空间分布问题的研究者。
均匀汽车停车过程是一维随机序列吸附问题：单位长度车辆以均匀随机位置到达线段，仅在有足够空间时停放，直到无法再停。本文建立了有限长度 $s$ 下的精确理论。停车位置的联合密度被分解为若干“堵塞单元”，每个单元上为有理函数，并通过 $O(2^n n)$ 复杂度的子集递归计算；边际分布与间隙顺序统计量表示为超对数函数，其权重由积分坐标数决定；吸收计数及聚合量则通过来自 Rényi 的积分方程处理。
原文摘要 · Abstract (English)
The uniform car-parking process is the one-dimensional random sequential adsorption of unit cars on a segment of finite length $s$: cars arrive at uniformly random positions and park wherever they fit, until no gap admits another. This paper develops the exact finite-$s$ theory. The joint density of the parked positions is resolved into jamming cells, on each of which it is a rational function, and evaluated by a subset recursion in $O(2^n n)$ operations; the marginal and gap order statistics are obtained as hyperlogarithms whose weight is fixed by the number of coordinates integrated out; and the absorption count and the aggregate quantities are treated through the integral equation descending from Rényi.
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