= Entry count of a continuous-time Markov chain
{title2=$N_s(t)$}
= Entry count
{synonym}
= Markov-chain entry count
{synonym}
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>.
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