Counting intervals in the packing process

We consider a sequential interval packing process similar to Rényi's "car parking problem" but with a generator of random intervals which allows for arbitrarily small lengths. Embedding the process in continuous time, we view it as a self-similar interval splitting process. We determine the asymptotical behavior of the quantities such as the number of intervals packed to some instant and obtain convergence results in the context of the more general splitting model.