= Solution
The stationary balance equations on the closed class $\{1,\ldots,M\}$ are $\pi_j=\pi_{j+1}+\pi_1/M$, with $\pi_{M+1}=0$. Working backward gives $\pi_j=\pi_1(M-j+1)/M$.
Normalization yields $1=\pi_1(M+1)/2$, so
$$
\boxed{\pi_j=\frac{2(M-j+1)}{M(M+1)}\quad(1\leq j\leq M),\qquad\pi_j=0\quad(j>M).}
$$
These probabilities sum to one and satisfy each balance equation. In particular $\pi_1=2/(M+1)$ is the reciprocal of the mean return time $(M+1)/2$, consistent with part (d).
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