Saturation in a Makovian Parking Process

Abstract

We consider $\mathbb{Z}$ as an infinite lattice street where cars of integer length $m \geq 1$ can park. The parking process is described by a 0–1 interacting particle system such that a site $z \in \mathbb{Z}$ is in state 1 whenever a car has its rear end at z and 0 otherwise. Cars attempt to park after exponential times with parameter $\lambda$, leave after exponential times with parameter 1 and are not allowed to touch nor overlap. We define and study a jamming occupation density for this parking process, using the quasi-stationary distribution of a Markov chain related to the reversible measure of the particle system. An extension to a strip in $\mathbb{Z}^2$ is also investigated.