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.
Type de document :
Pré-publication, Document de travail
2011
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https://hal.archives-ouvertes.fr/hal-00505206
Contributeur : Jean Bertoin <>
Soumis le : mercredi 26 janvier 2011 - 06:12:27
Dernière modification le : jeudi 27 avril 2017 - 09:46:21
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

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UPMC | INSMI | USPC | PMA

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Jean Bertoin. Burning cars in a parking. 2011. <hal-00505206v2>

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