A second row Parking Paradox

TL;DR

This paper analyzes two variations of a discrete car parking problem with two parking lines, demonstrating their solvability via finite-dimensional ODEs and comparing long-term density behaviors.

Contribution

It introduces and solves two new parking models with different obstruction rules, providing a mathematical framework and numerical comparison of their long-term densities.

Findings

Both models are solvable using finite-dimensional ODEs.

Model a) shows increased second line density over time.

Model b) also shows increased second line density, but less pronounced.

Abstract

We consider two variations of the discrete car parking problem where at every vertex of the integers a car arrives with rate one, now allowing for parking in two lines. a) The car parks in the first line whenever the vertex and all of its nearest neighbors are not occupied yet. It can reach the first line if it is not obstructed by cars already parked in the second line (screening). b) The car parks according to the same rules, but parking in the first line can not be obstructed by parked cars in the second line (no screening). In both models, a car that can not park in the first line will attempt to park in the second line. If it is obstructed in the second line as well, the attempt is discarded. We show that both models are solvable in terms of finite-dimensional ODEs. We compare numerically the limits of first and second line densities, with time going to infinity. While it is not surprising that model a) exhibits an increase of the density in the second line from the first line, more remarkably this is also true for model b), albeit in a less pronounced way.