Self-stabilizing repeated balls-into-bins

We study the following synchronous process that we call repeated balls-into-bins. The process is started by assigning n balls to n bins in an arbitrary fashion. In every subsequent round, one ball is extracted from each non-empty bin according to some fixed strategy (random, FIFO, etc), and re-assigned to one of the n bins uniformly at random. We define a configuration legitimate if its maximum load is $$\mathcal {O}(\log n)$$O(logn). We prove that, starting from any configuration, the process converges to a legitimate configuration in linear time and then only takes on legitimate configurations over a period of length bounded by any polynomial in n, with high probability (w.h.p.). This implies that the process is self-stabilizing and that every ball traverses all bins within $$\mathcal {O}(n\log ^2 n)$$O(nlog2n) rounds, w.h.p.