The sojourn time is the time from when a Markov Chain reaches a certain state to when in leaves that state and it is usually denoted as $\tau_i$ for state $i$.
$$
\tau_i \sim \text{Geom}(1 - q_{ii})
$$
and therefore
$$
E(\tau_i) = \frac{1}{1 - q_{ii}}
$$
Counting Process