# A fairly straightforward balls-and-urns problem

I am interested in the following question, which is stated in the abstract of [1]:

An urn contains $$r$$ different balls. Balls are drawn with replacement until any $$k$$ balls have been obtained at least $$m$$ times each. How many draws are necessary?

The cited paper solves this problem using Poissonization for various special cases (e.g. $$k=1$$ or $$k=r$$ and so on) but I cannot find a reference for the most general case, asymptotic or otherwise. Is there a name for this general problem, and are there any known solutions (or upper bounds) for it, assuming that $$r\to\infty$$?

[1] Holst, Lars. "On birthday, collectors', occupancy and other classical urn problems." International Statistical Review/Revue Internationale de Statistique (1986): 15-27.