I have looked around the literature but I've not found a clean answer to the following. Imagine that you have a generalized Pólya urn (GPU) in the sense of Pemantle's survey (Section 2.1 in Pemantle07). What is the expected time until the ratio of white balls to black balls is equal to $p$ (assuming that the initial ratio is $p_0$)? An answer for the much simpler case of the GPU with a diagonal $A$ matrix would also be interesting. A non-trivial upper bound would also suffice.