Mean holding time from a transition intensity matrix ($E(T_r)=-1/q_{rr}$)

A state's holding time in a time-homogeneous continuous-time Markov chain is exponentially distributed with rate $-q_{rr}>0$. Its mean is $-1/q_{rr}$; the probability that its next jump is to state $s$ is $q_{rs}/(-q_{rr})$. An absorbing state has infinite holding time.

For an exponential distribution with positive rate $\lambda$, the mean is $1/\lambda$. A confidence interval for the rate transforms to one for the mean by reversing and inverting its endpoints. The monotonic transformation preserves the coverage event; it does not give a prediction interval for one realized holding time.