How we can calculate mean time to absorption if we have mean times the process spends in each state before making a transition into a different state and transition probabilities?
PS: I am aware of the regular way of using W = (I-Q)^(-1) to calculate mean time to absorption, however I do not want to use this way. I need an approach that takes into account the mean times the process spends in each state.
Intuitively, the idea is that you first identify the different possible state paths that the process can take from initial state $A$ to the absorbing state $B$, calculate the probabilities for each of these paths to occur using the state transition probabilities, and then determine the expected time until absorption of each of the individual paths using the mean time spent in each of the states in each path.
The average time until absorption is then just the average of each of those expected path times, weighted by the probability they occur.