I am modeling a biological process using the Moran model which effectively describes a 1D random walk.
Starting at $i = 1$, the transition probabilities are:
$p_{i, i+1} = \frac{ri}{ri+N} \frac{N-i}{N}$,
$p_{i, i-1} = \frac{N-i}{ri+N-i} \frac{i}{N}$,
$p_{i, i} = 1 - p_{i, i+1} - p_{i, i-1}$,
where if $r > 1$ the random walk is biased towards larger values and the boundaries are $i = 0$ and $i = N$.
Now, given that the random walk doesn't get absorbed at $i = 0$, I'm interested in the expected number of steps until it is absorbed at $i=N$.
Unfortunately, I can't seem to find a derivation of this in the literature and can't derive it myself. Thanks for your help.