given a continuous time Markov chain $(X_t)_{t \geq 0}$ with Q-Matrix $Q$ and countable state space $S$, is there a formula for calculating the expected first return time of a state i.e a formula for $E_z[R_z]$ where
$$ R_z := \inf(t> \tau_z : X_t = z) \quad\text{and}\quad \tau_z := \inf(s\geq 0 : X_s \neq z). $$
If there is non such formula, is there a proof of concept how to show if $E_z[R_z]$ diverges or converges?
Thanks in advance for the help!