Disclaimer: I am not a mathematician, although I have some appreciation for its intrinsic beauty, and admiration for those who are. This is a practical question, not an academic one.

Also, even if a solution isn't apparent, help in reformulating or the problem in language that will be more broadly familiar to others would be helpful. I have already done so as far as I am able - what appears here is a portion of a larger problem.

We have a large number NU + 1 of urns. (Large means that the relative difference between NU and NU+1 is well within the error bounds that I care about. The reason for the +1 will be apparent momentarily. Designate them Usub1...UsubNU, plus UsubE for the extra one.

Each urn has a capacity of CU balls.

There are a fixed number NB balls in the system, NB less than or equal to NU x NC.

Assume to begin that the "extra" urn UsubE, is empty. At a fixed rate, a ball is chosen at random and moved to UsubE. This continues until UsubE is full, at which point the urn with the fewest balls is chosen as the new UsubE and the process continues.

I want to find an expression or estimate for the expectation of the number of balls in the new UsubE at steady state.

Any solution or thoughts on a better formulation and/or tagging of the problem would be greatly appreciated.

Best, J