Asymptotics in Knuth's parking problem for caravans

We consider a generalized version of Knuth’s parking problem, in which caravans consisting of a random number of cars arrive at random on the unit circle. Then each car turns clockwise until it finds a free lot to park. Extending a recent work by Chassaing and Louchard [9], we relate the asymptotics for the sizes of blocks formed by occupied lots with the dynamics of the additive coalescent. According to the behavior of the caravan’s size tail distribution, several qualitatively different versions of eternal additive coalescent are involved.