 Solution

The total eventual mass of the closed class $\{4,5\}$ is ${\lambda }_4+{\lambda }_5+(1-p){\lambda }_1$. The other closed class cannot contribute to state five. Since $|1-a-b|<1$, the two-state chain within the last class converges to its stationary distribution, with state-five mass $a/(a+b)$. Therefore