 Solution

Let $h_i={\mathbb{E}}_i{T}_0$ be the expected hitting time of state $0$. For $i\ge 1$, first-step analysis givesWith $d_i=h_i-h_{i-1}$ this becomesThe minimal nonnegative solution hasso

Equivalently, Kac's lemma gives ${\mathbb{E}}_0{T}_0^{+}=1/{\pi }_0$. A first step from $0$ either returns immediately with probability $1-p$ or moves to $1$ with probability $p$, soSubstituting ${\pi }_0=(q-1)/(q-1+q^{2}p)$ gives the same result.