Your problem falls under the general category of "coupon collector's problems". There is a large literature on such problems, mainly concerned with formulas for the mean and variance of the random variable $T$.
The exact distribution of the time $T$ to have at least one bean ($N=1$) in each box
is known in terms of Stirling numbers of the second kind: see Henry's answer
here, or below
$$P(\mbox{ every box has at least one bean after }T \mbox{ seconds}) = P^{-T}\ P\ !\ \left\lbrace {T\atop P} \right\rbrace.$$
In general, I think you should take Alekk's advice and use the Poisson approximation.
