I know that for a CTMC, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent CTMCs evolving simultaneously. Each of the $N$ CTMCs can be in state 0 or 1 and the state of CTMC $k$ at time $t$ is given by $X_k(t)$. The rate matrix of CTMC $i$ is

$$Q_i=\left[ \begin{matrix} -\sum_{j=1}^N \lambda_{ji}I_{X_j(t)=1} & \sum_{j=1}^N \lambda_{ji}I_{X_j(t)=1}\\ \mu_i & -\mu_i \end{matrix}\right]$$

In such a coupled system of CTMCs, the forumula $P(X_i(t)=1\mid X_i(0)=0)$ is no longer the $(0,1)^{th}$ entry of $e^{Q_i t}$, since $Q_i$ in turn depends on the states in other CTMCs. How does one proceed to find $P(.)$ in such coupled CTMC systems?

I know that for a continuous-time Markov chain, the probability of transition from time $0$ to $t$ is given by $P(t)=e^{Q(t)t}$. I have a system of $N$ interdependent continuous-time Markov chains evolving simultaneously. Each of the $N$ Markov chains can be in state 0 or 1 and the state of Markov chain $k$ at time $t$ is given by $X_k(t)$. The rate matrix of Markov chain $i$ is

$$Q_i=\left[ \begin{matrix} -\sum_{j=1}^N \lambda_{ji}I_{X_j(t)=1} & \sum_{j=1}^N \lambda_{ji}I_{X_j(t)=1}\\ \mu_i & -\mu_i \end{matrix}\right]$$

In such a coupled system of Markov chains, the forumula $P(X_i(t)=1\mid X_i(0)=0)$ is no longer the $(0,1)^{th}$ entry of $e^{Q_i t}$, since $Q_i$ in turn depends on the states in other Markov chains. How does one proceed to find $P(.)$ in such coupled Markov chain systems?