Suppose an infinite number of balls are thrown into $N$ bins (uniformly distributed).
What is the expectancy of the number of balls needed in order to fill all bins with at least $K$ balls in each bin.
I found answers to this problem with $K=1$ and even a partial answer to $K=2$ but nothing for the generalized form.
I think I found the answer in the following paper:
https://faculty.wharton.upenn.edu/wp-content/uploads/2012/04/Double-dixie-cup-problem.pdf
the expectancy of collecting $m$ balls each bins (having $n$ bins) is:
$$n(\log(n) + (m-1)\log\log n + o(1) )$$
*** this analysis is done for $m$-fixed and $n$-large