Polya urn waiting time distributions

From an urn containing 1 ball each of n different colours, a ball is drawn, its colour noted and then it is returned to the urn along with k balls of the same colour.

What is the distribution of the number of balls of each colour in a multicolour urn (after a certain number of draws)?

Additionally, what is the distribution of wait times before drawing the first ball of a given colour in the multicolour urn model?

EDIT

in simulations i've run, i've found that the number of balls of each colour is described by the negative binomial, and the time to the first ball for each colour given by the Lomax (pareto type II) distribution. Ideally I'd like to find a proof or a reference to confirm these results.

• Is this a homework problem? Could you show us some methods of solving the problem that you've attempted? – P Schnell Mar 6 '14 at 22:12
• it's not homework. I've run some simulations to empirically fit the distributions. I'd like a proof or a reference if possible. I'll update the question with the simulation results – dylan2106 Mar 6 '14 at 22:48