Occupation time of a continuous-time Markov chain (${\int }_0^{t}1\{Z(u)=s\}du$)

The occupation time records the duration spent in a state of a continuous-time Markov chain. Conditional on $Z(0)=r$, its expected value is ${\int }_0^t{p}_{rs}(u)du$, where $p_{rs}$ is the transition probability.