$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins again until an empty or partly loaded bin is met. The question is that what is the expectation of the toss times for the $n$ balls, and what is the distribution of the number of balls in the $m$ bins ($n \leq mk$). For the first question, a simplified version can also be: when there are already $n$ balls accumulated in the $m$ bins according to the above toss rule, what is the average toss times for another ball tossed into the bins (we could discuss the case in this stable state rather than the build process).
I know without the re-toss process, the number of balls in the bins must follow the binomial distribution; however, when the re-toss is added, it turns to be quite different.