Burning cars in parkings

Jean Bertoin (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

Knuth's parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking with $n$ sites; cars arrive at each site with unit rate. When a car arrives at a vacant site, it parks there; otherwise it turns clockwise and parks at the first vacant site which is found. We incorporate fires to this model by throwing Molotov cocktails on each site at a smaller rate $n^{-\alpha}$ where $0<\alpha<1$ is a fixed parameter. When a car is hit by a Molotov cocktails, it burns and the fire propagates to the entire occupied interval which turns vacant. We show that with high probability when $n\to \infty$, the parking becomes saturated at a time close to $1$ (i.e. as in the absence of fire) for $\alpha>2/3$, whereas for $\alpha<2/3$, the mean occupation approaches $1$ at time $1$ but then quickly drops to $0$ before the parking is ever saturated. Our study relies on asymptotics for the occupation of the parking without fires in certain regimes which may be of independent interest.

Mots clés: Parking scheme; Forest fire; Phase transition

Domaines: Probabilités [math.PR]

Jean Bertoin. Burning cars in parkings. 2010. ⟨hal-00505206v1⟩