Suppose that there is a time-continuous markov process with 2 states, $A$ and $B$. The transition rate from $A$ to $B$ is $\beta$ and from $B$ to $A$ is $\alpha$. I'm interested in finding the distribution of the fraction of time that "particles" spend in state $B$ as the system evolves over some time $t$, assuming that the system is always at steady state.
Because the system is in steady state, its obvious that the expected fraction of time spent in $B$ is always going to be $\frac{\beta}{\alpha + \beta}$.
At very small times $t$ there have been no transitions, and so $\frac{\beta}{\alpha + \beta}$ particles will have been in state $B$ for the entire time, while $\frac{\alpha}{\alpha + \beta}$ particles will have been in state $A$ for the entire time, averaging out to our expectation value. On the other hand, at very long times $t$ every single particle will have been in state $B$ for $\frac{\beta}{\alpha + \beta}$ of the time, so we again get our expectation value but with a very different distribution for the fraction of time spent in state $B$.
I'm trying to figure out how to describe the distribution for intermediate times. I've been reading a fair amount about Markov processes, but I haven't found anything that discusses this sort of idea. I would really appreciate if anyone could give any pointers on how to proceed with this or where to look/what to read.