Exact Finite-Length Theory of Uniform Car Parking: Spatial Laws, Absorption, and Aggregation
2026-08-24 • Robotics
RoboticsData Structures and AlgorithmsSymbolic Computation
AI summaryⓘ
The authors study a scenario where cars park randomly along a line until no more can fit, called the uniform car-parking process. They develop exact mathematical formulas for any finite road length that describe the positions of the parked cars and the sizes of gaps between them. Their approach breaks the problem into parts called jamming cells and uses recursion methods to calculate probabilities efficiently. They also connect their results to known integral equations introduced by Rényi.
random sequential adsorptionuniform car-parking processjammingjoint densitymarginal distributiongap statisticshyperlogarithmsintegral equationsRényi's parking problemrecursive algorithms
Authors
Ganesh P Kumar
Abstract
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.