Imagine parallel parking is available along the shoulder of a road, but no parking spaces are marked.
The first person to park picks a spot on the shoulder at random. Then another car also chooses a spot along the shoulder at random, with the constraint that the second car can’t overlap the first. This process continues until no more cars can park. How many people can park this way?
Assume all cars have the same length, and we choose our units so that cars have length 1. The expected number of cars that can park is a function M(x) of the length of the parking strip x. Clearly if x < 1 then M(x) = 0. Alfréd Rényi [1] found that for x ≥ 1, M(x) satisfies the equation
This is an unusual equation, difficult to work with because it defined M only implicitly. It’s not even clear that the equation has a solution. But it does, and the ratio of M(x) to x approaches a constant as x increases.
The number m is known as Rényi’s parking constant.
This say for a long parking strip, parking sequentially at random will allow about 3/4 as many cars to park as if you were to pack the cars end-to-end.
More posts on Rényi’s work
[1] Steven R. Finch. Mathematical Constants. Cambridge University Press, 2003.
The post Rényi’s parking constant first appeared on John D. Cook.