A Second-row Parking Paradox

We consider two variations of the discrete car parking problem where at every vertex of a"currency sign cars (particles) independently arrive with rate one. The cars can park in two lines according to the following parking (adsorption) rules. In both models a car which arrives at a given vertex tries to park in the first line first. It parks (sticks) whenever the vertex and all of its nearest neighbors are not occupied yet. A car that cannot 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. In the screening model a) a car cannot pass through parked cars in the second line with midpoints adjacent to its vertex of arrival. In the model without screening b) cars park according to the same rules, but parking in the first line cannot be obstructed by parked cars in the second line. 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.