Mixing time for the repeated balls into bins dynamics

We consider a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in L bins rL balls, where r is a fixed positive constant. At each time step a ball is removed from each non-empty bin. Then all these removed balls are uniformly reassigned into bins. We prove that the mixing time of the RBB process is of order L. Furthermore we show that if the initial configuration has o(L) balls per site the equilibrium is attained in o(L) steps.