Knuth’s parking scheme is a model in computer science for hashing with linear probing. One may imagine a circular parking
lot 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 into this model by throwing Molotov cocktails on each
site at a smaller rate n
−α
, where 0 < α < 1 is a fixed parameter. When a car is hit by a Molotov cocktail, it burns and the fire propagates to the entire occupied
interval which turns vacant. We show that with high probability when n → ∞, the parking lot becomes saturated at a time close to 1 (i.e. as in the absence of fire) for α > 2/3, whereas for α < 2/3, the average occupation approaches 1 at time 1 but then quickly drops to 0 before the parking lot is ever saturated.
Our study relies on asymptotics for the occupation of the parking lot without fires in certain regimes which may be of independent
interest.
Communicated by F. Toninelli