Coupon Collector's Problem

Let $n$ objects be picked repeatedly with probability $p_i$ that object $i$ is picked on a given try, with

$$\sum_i p_i=1.$$

Find the earliest time at which all $n$ objects have been picked at least once.

References

Hildebrand, M. V. ``The Birthday Problem.'' Amer. Math. Monthly 100, 643, 1993.