 Two-state excursion visit law

Let $a={\mathbb{P}}_i(T_j<T_i)$ and $b={\mathbb{P}}_j(T_i<T_j)$. Before returning to $i$, the chain visits $j$ zero times with probability $1-a$ and, for $k\ge 1$, exactly $k$ times with probabilityWhen $b>0$, the expected number of visits is $a/b$.