My question is as follows:
I have a continuous Markov chain with stationary distribution $\pi_j$. I want to calculate the expected number of visits to state $j$ as $t\to\infty$. My intuition suggests that this number should be equal to $\pi_j\lambda_j$, where $\lambda_j$ is the intensity of (exponentially distributed) state $j$. However, I can't find any reference for this.
Could someone provide me with a proof or a reference?
Thank you