I have a continuous birth-death chain $\{X(t)\}$ on state space $\{0, 1, \dots, n-1, n\}$, with birth rates $\lambda_i$ and death rates $\mu_i$, where state 0 is absorbing ($\mu_0 = \lambda_0 = 0$), and the right-most state is not absorbing (could be reflecting, doesn't matter).
What is the mean-time to absorption into state $0$ if I start the process at any state $n \geq j \geq 1$? All the resources I have found seem to apply to infinite processes, or to finite processes with both endpoints being absorbing.
EDIT: Looking for a result still.