= Holding rate of a continuous-time Markov chain
{title2=$q_i$}
The holding rate at state $i$ of a <continuous-time Markov chain> with <generator matrix> $Q=(q_{ij})$ is
$$
q_i=-q_{ii}=\sum_{j\ne i}q_{ij}.
$$
For a conservative chain with finite $q_i>0$, its <holding time> $T_i$ has <exponential distribution>: $\Pr(T_i>t)=e^{-q_it}$ and $\mathbb E T_i=1/q_i$. Conditional on a jump, the next state is $j$ with probability $q_{ij}/q_i$. Thus the rate determines when departure occurs, while normalized outgoing rates determine where it goes.
If the outgoing rates are $q_{i1}=2$ and $q_{i2}=3$ for two distinct other states, the total rate is five, the mean holding time is $1/5$, and departure chooses them with probabilities $2/5$ and $3/5$. A zero rate gives an <absorbing state> with infinite holding time. Statewise finite rates alone do not rule out <explosion of a continuous-time Markov chain>.
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