Looking for some sort of formula for this, not really sure where to begin. Essentially I want to find some sort of condition where this number is equal to 1.
2026-04-04 06:21:33.1775283693
Given a DTMC, how do you find the expected number of visits to state $j$ before the first revisit to state $i$ (starting from state $i$)?
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If state $j$ is not reachable from state $i$, the expected number of visits is $0$.
If state $j$ is reachable from state $i$ and state $i$ is not positive recurrent, things might be complicated.
If state $j$ is reachable from state $i$ and state $i$ is positive recurrent, then it follows that state $j$ is also positive recurrent. In this case, the expected number of visits to state $j$ between visits to state $i$ is just the ratio of the expected return times to the two states, which is the inverse ratio of their stationary occupation probabilities. So this is $1$ exactly if the two states have the same stationary occupation probabilities.