Consider a student who is working on a coding project in which we observe how many times the student updates a given file in a given week. A student is either "busy" or "not busy", and when a student is busy, they update the file with a mean rate of $2$ times a week, and when a student is not busy, they update the file with a mean rate of $6$ times a week. Given this structure, I would like to find the limiting observation distribution of the number of times the student will update the file in the week.
I can see that we have two hidden states, "busy" and "not busy" (hidden states $1$ and $2$ respectively), and I see that I have two discrete emission distributions based on these two hidden states (which I'm willing to assume are Poisson, but you can correct me if I am wrong). The question I am running into is how to calculate the observation distribution at time $t$, given that I have what seems to be two emisison distributions with non-finite support (a student can update the file an arbitrary amount of times). How might this affect the way we find the limiting observation distribution? Note that for a standard first-order Markov Chain with initial state distribution $\lambda$ and transition matrix $P$, The limiting distribution was the $\lambda_t$ such that
$$\lambda_t = \lambda_{t-1} = \lambda P^{t-2}.$$
However, we are working with a hidden markov model, and we are looking for the limiting distribution over a given week's observed number of updates.