 Solution

All four permitted hops have rate one: $1\to 2$, $2\to 1$, $2\to 3$, and $3\to 1$. Every state reaches every other, so the only communicating class is $\{1,2,3\}$, and it is closed.

Solving $\pi Q=0$, ${\sum }_i{\pi }_i=1$, gives ${\pi }_1=2{\pi }_2$ and ${\pi }_3={\pi }_2$. ThusFor the transient probability, let $p_j(t)=P_{1j}(t)$. The forward equations and $p_1+p_2+p_3=1$ give $p_2^{\prime }=p_1-2p_2$, $p_1^{\prime }=1-2p_1$. From $p_1(0)=1$, $p_1(t)=1/2+e^{-2t}/2$. Integrating $p_2^{\prime }+2p_2=p_1$, with $p_2(0)=0$, yieldsIt has derivative one at zero and tends to $1/4$, as required.