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Saturation in a Markovian parking process. (English) Zbl 1012.60087

Summary: 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J27 Continuous-time Markov processes on discrete state spaces
60K30 Applications of queueing theory (congestion, allocation, storage, traffic, etc.)
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