Consider a one-dimensional random walk with
$p_{i,i+1}=p$,
$p_{i+1,i}=q$,
$p_{i,i}=r$.
The expected time $k_i$ to hit the boundary $\{0,n\}$ starting at $i$ satisfies the recurrence equation
$k_i= 1 + p\, k_{i+1} + r\, k_i+q\, k_{i-1}$
and the boundary conditions $k_0=k_n=0$. If $p\neq q$ the solutions have the form
$\displaystyle k_i\ =\ \frac{i}{q-p}+A +B\bigg(\frac{q}{p}\bigg)^{\!i}$
where $A$ and $B$ are determined by the boundary conditions.
Question Is this correct? If correct, why does it not depend on $r$?