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For a continuous-time Markov chain with absorbing endpoints $0,N$ and rates $q_{j,j\pm 1}={\lambda }_j/2$, the jump chain is a simple symmetric random walk. On leaving $i$, absorption before returning has probability $p_i=\frac{1}{2}(1/i+1/(N-i))$, by gambler's ruin on each side. The geometric return count before absorption and independent rate-${\lambda }_i$ holding times give ${\mathbb{E}}_{i}I=1/(p_i{\lambda }_i)$. The rates at other states affect elapsed excursion times but not their return probabilities.

Use the positive-integer convention for the geometric distribution:Conditional on $K=k$, independence gives $\mathbb{E}[S\mid K=k]=k/\lambda$. Taking expectations givesThis is the convention required below because a chain starting at $i$ makes at least one visit to $i$. If instead $K$ counts failures on $\{0,1,\dots \}$, its mean is $(1-p)/p$, and the corresponding answer for that different random sum is $(1-p)/(p\lambda )$.

For the chain of part (b), let $K_i$ count visits to $i$, including the initial visit, before absorption. After a departure to $i-1$, gambler's ruin on $\{0,\dots ,i\}$ gives probability $1/i$ of reaching zero before returning to $i$. After a departure to $i+1$, the analogous probability of reaching $N$ before returning is $1/(N-i)$. The departure is equally likely in either direction, so the probability of absorption before a return isAt every return to $i$, the Strong Markov property resets these alternatives. Hence the geometric return count before absorption gives $K_i\sim \mathrm{Geom}(p_i)$ on positive integers. One may construct the chain from its embedded jump chain and a separate sequence of fresh independent exponential clocks. Then its successive holding times at $i$ are independent and identically distributed rate-${\lambda }_i$ exponential random variables, independently of $K_i$, andUsing the first calculation gives the occupation time at an interior state of a variable-rate symmetric walk:For $N=2,i=1$, $p_i=1$ and the only visit is the initial one; the degenerate positive-integer geometric distribution at one handles this case. Other states' rates affect the time of each excursion, but not its probability of return, which explains why only ${\lambda }_i$ appears in this expectation.