 Solution

The stationary equation $\pi Q=0$ reads $r^T-R\pi =0$, because $\pi 1=1$. ThereforeIt is the unique stationary law and vanishes on the open classes. For distinct $i,j$, ${\pi }_i{q}_{ij}=r_i{r}_j/R={\pi }_j{q}_{ji}$. Thus the chain is reversible in its equilibrium law, including the reducible cases; on the closed class this is the usual reversible irreducible chain. This structure is the destination-dependent reset chain.