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.
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Preprints, Working Papers, ...
2011
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https://hal.archives-ouvertes.fr/hal-00505206
Contributor : Jean Bertoin <>
Submitted on : Wednesday, January 26, 2011 - 6:12:27 AM
Last modification on : Wednesday, October 12, 2016 - 1:04:10 AM
Document(s) archivé(s) le : Friday, December 2, 2016 - 4:16:43 PM

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

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