1. **Problem statement:** We want to approximate the expected time spent in states of a continuous-time Markov chain that has two phases: it uses transition rate matrix $A_1$ until time $T$, and then switches to matrix $A_2$ after time $T$. 2. **Key concepts:** The expected time in states for a continuous-time Markov chain can be found using the matrix exponential of the rate matrix. For a single phase with rate matrix $A$, the state probability vector at time $t$ is $\pi(t) = \pi(0) e^{At}$, where $\pi(0)$ is the initial distribution. 3. **Two-phase approach:** Since the chain changes at time $T$, we consider two intervals: - From $0$ to $T$, the chain evolves with $A_1$. - From $T$ onward, the chain evolves with $A_2$. 4. **Expected time calculation:** The expected time spent in each state up to some time $t > T$ is the sum of expected times in each phase: $$\text{Expected time} = \int_0^T \pi(0) e^{A_1 s} ds + \int_T^t \pi(0) e^{A_1 T} e^{A_2 (s-T)} ds$$ 5. **Evaluating integrals:** Using the formula for matrix exponentials integrals, $$\int_0^T e^{A_1 s} ds = A_1^{-1} (e^{A_1 T} - I)$$ and similarly for the second integral, $$\int_T^t e^{A_2 (s-T)} ds = A_2^{-1} (e^{A_2 (t-T)} - I)$$ 6. **Putting it all together:** $$\text{Expected time} = \pi(0) A_1^{-1} (e^{A_1 T} - I) + \pi(0) e^{A_1 T} A_2^{-1} (e^{A_2 (t-T)} - I)$$ 7. **Interpretation:** This formula gives the expected time spent in each state up to time $t$ considering the two-phase transition matrices. 8. **Summary:** To approximate expected times in a two-phase continuous-time Markov chain, compute the matrix exponentials and their integrals for each phase separately, then combine them as shown.