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][1].  

In general, I think you should take Alekk's advice and use the Poisson approximation.


  [1]: http://math.stackexchange.com/questions/26528/m-balls-n-boxes-probability-problem