 Solution

The stationary balance equations on the closed class $\{1,\dots ,M\}$ 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 $1={\pi }_1(M+1)/2$, soThese 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).