Entry count of a continuous-time Markov chain ($N_s(t)$)

The entry count $N_s(t)$ counts jumps of a continuous-time Markov chain into state $s$ by time $t$, excluding initial occupation. It is a counting process. For homogeneous transition intensities, its mean has derivative ${\sum }_{i\ne s}{p}_{ri}(t)q_{is}$ conditional on initial state $r$, so its expected value is the sum of incoming rates times the expected occupation times of a continuous-time Markov chain. This density is an expected count rate, not necessarily a normalized probability density function.