Let's say I have $n$ machines in my system. The machines go down at rate $\lambda$ and once down, they come back up at rate $\mu$. We can ask, "how many machines are down at a given instant. And this can be modeled as a continuous time Markov chain. The states of this chain and rates at which transitions happen are shown below.
Now, I want to add another layer of complexity to this model. The number of machines in the system isn't fixed at $n$. That is just where the system starts. Machines come into the system at rate $\kappa$ and leave the system at rate $\nu$. I now want to model the number of machines that are down as a Markov chain. What would the transition rates look like?
Along the lines of the suggestion in your comment, I'd model this as a 2D state $S_t=(U_t, D_t)$ where $U_t$=#machines that are up and $D_t$=#machines that are down.
Then from a state $(u,d)$ there are 5 possible transitions:
(This diagram assumes that when a new machine enter the system it is up.) Here, one can go