Let $n \in \mathbb{N}$ and consider a collection of $n$ continuous-time Markov processes such that $\{ X_k(t) \}_{t}$ is a continuous-time Markov process over the state space $\mathbb{B} = \{ 0, 1 \}$ with rate matrix $$Q_k = \begin{pmatrix} -\alpha_k & \alpha_k \\ \beta_k & -\beta_k \end{pmatrix}$$ and $\mathbb{P}(X_k(0) = 0) = 1$ for each $0 < k \le n$.
Now let $f : \mathbb{B}^n \to \mathbb{B}$ be arbitrary with $f(0, 0, \dots, 0) = 0$, what can we say about the process $\{ F(t) \}_t$ such that $F(t) = f(X_1(t), X_2(t), \dots, X_n(t))$? For each fixed $t \ge 0$ what is $\mathbb{P}(F(\tau) = 1 \text{ for some } \tau \le t)$?
Edit: Maybe it would help to illustrate with an example, this won't take away from the generality but might help make this question seem a little less contrived.
Say there are $n$ light switches that flicker on and off according to a Markov process. In other words if the $n$-th switch is "on" then the time until it turns "off" is distributed as $\operatorname{Exp}(\beta_n)$ and similarly when it is "off" the time until it turns "on" is distributed as $\operatorname{Exp}(\alpha_n)$.
Now say there are some particular and known switch combinations that cause something bad to happen. The two questions I'd like to answer are what is the probability that something bad is currently happening at time $t$ and what is the probability that something bad has already happened at time $t$. Lastly is there a nice way to express the expected time at which the first bad thing happens. This is clearly depend on what the bad states actually are but the hope is that there is some nice answer that can be given in terms of sum or product over the set of bad states.