 Solution

For $a>0$ the chain is irreducible, and the normalized stationary distribution in (i) makes it positive recurrent. The mean-return-time formula givesHere $T_0^{+}=\inf \{n\ge 1:X_n=0\}$ includes a return at time one when the chain stays at zero. This convention is essential. Equivalently the mean hitting time of zero from one is $1/(b-a)$, so a first-step decomposition gives $1+a/(b-a)$, the same result. For $a=0$ the population remains zero and the next zero is at time one, also agreeing with the formula.