A second row Parking Paradox

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.

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyis parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.