# Burning cars in a parking

Abstract : 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.
Keywords :
Type de document :
Pré-publication, Document de travail
2011
Domaine :

Littérature citée [17 références]

https://hal.archives-ouvertes.fr/hal-00505206
Contributeur : Jean Bertoin <>
Soumis le : mercredi 26 janvier 2011 - 06:12:27
Dernière modification le : jeudi 11 janvier 2018 - 06:12:29
Document(s) archivé(s) le : vendredi 2 décembre 2016 - 16:16:43

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Burning.pdf
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• HAL Id : hal-00505206, version 2

### Citation

Jean Bertoin. Burning cars in a parking. 2011. 〈hal-00505206v2〉

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