Suppose a box contains $m$ balls, numbered from 1 to $m$. A random number of balls are drawn from the box, their numbers are noted and the balls are then returned to the box. This is done repeatedly, with the sample sizes being iid. Let $X$ be the number of samples needed to see all the balls. This paper uses Markov-chain coupling to derive a simple but typically very accurate approximation for $EX$ in terms of the sample size distribution. The approximation formula generalizes the formula found by Polya for the special case of fixed sample sizes.
"How Many IID Samples Does it Take to See all the Balls in a Box?." Ann. Appl. Probab. 5 (1) 294 - 309, February, 1995. https://doi.org/10.1214/aoap/1177004841