Let $S = \{1, 2, 3, …, a, b\}$ be a finite set of states, where $a$ and $b$ are the absorption states. The transition probability from state $i$ to $j$ is $P_{ij}$, and the transition rate from state $i$ to state $j$ is $Q_{ij}$.
When we have only one absorption state, we can use the method explained in the answer of this question as the Markov Chain will only absorb in one state. Does anyone have any idea how we can calculate the time until absorption to a specific absorption state when we have more than one state to absorb into?
Best Answer
The solution I used, and it matched the simulations, is removing all the transitions that goes to the other absorbing states and keep the ones that goes to the desired state. For example, if we have two absorbing states $a$ and $b$, and we want to calculate the time until absorption at state $b$, we remove all the transitions that goes to $a$ and keep the ones that goes to $b$, then calculate the time until absorption using the method here.