In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.
Consider a CTMC with an absorbing state; let $Q$ be its infinitesimal generator and $Q_a$ be the reduced generator after the removal of the absorbing state. As long as $Q_a$ is irreducible, it is well-known in literature that a probability vector $q$ satisfying $q^\top Q_a=\theta q$ gives the limiting conditional distribution and QSD.
van Doorn and Pollett consider the case when $Q_a$ is reducible. They state and prove the conditions for the existence of a QSD measure in such a case. My question is on how to interpret one of their results for a simple death process. Let the death process $X$ have states $\{2,1,0\}$; it starts from 2 and gets absorbed at 0. The generator is $\begin{bmatrix}-\mu_2 & \mu_2 & 0\\0 & -\mu_1 & \mu_1 \\ 0 & 0 & 0\end{bmatrix}$. If $P(X(0)=2)=1$ and $\mu_2<\mu_1$, the QSD is $q_1=\frac{\mu_2}{\mu_1}$ and $q_2=1-q_1$, which is obtained by solving an equation of the form $q^\top Q_a=\theta q$.
How does one make sense of these probabilities? I understand the QSD probability has a (Yaglom) limit interpretation: $q_i=\lim_{t\to \infty} P(X(t)=i\mid T>t)$, where $T$ is the time of absorption. In the case of a positive recurrent CTMC, the stationary probability $\pi_i$ is the limit of the fractional time one spends in state $i$. However in this case, the expected fractional time one spends in state 2, conditioned on non-absorption is $\frac{1/\mu_2}{1/\mu_1+1/\mu_2}$, which is different from $q_2$. What is a way to understand this metric? Were I to run a simple simulation, which quantity will equal $q_1=\frac{\mu_2}{\mu_1}$?
