Given $n$ balls and $m$ bins, let us consider an infinite process, where in each time slot we throw a ball at a random bin. When all $n$ balls are thrown, we take the balls from the bin with the maximum load, and throw them to random bins until all $n$ balls are thrown and we repeat the process.
What the expected number of balls in the bin with the maximum load is converges to?
A variation of the problem is described in https://www.cs.tau.ac.il/~azar/box.pdf, where instead of taken the bin with the maximum load, a random ball is selected at random.
In addition, by my experiments, it seems that for $n=\Theta(m)$ the bin with the maximum load converges $2n/m+$ balls. I am trying to prove it.
