Let $(X_n)_{n \geq 0}$ be an absorbing, discrete, time-homogeneous Markov chain defined on a finite state-space, and let $N$ be the associated fundamental matrix.
The fundamental matrix $N$ provides by itself the mean of the time spent by the process in a transient state. The row sum of $N$ gives us the mean time to absorption of the process.
Let $\delta_{i}t$ be an estimation of the mean (continuous) time to move into state $i$. This is a know parameter of the process
It is possible to combine the statistics provided by $N$, that is, the mean sojourn times, and the known parameters $\delta_{i}t$ to compute a continuous-time estimation of the time necessary to absorption?
Ex: Let $i$ be the initial state.
$$T_{\text{abs}} = \sum_{j} N_{ij} \delta_{i}t$$
