I am reading the queueing theory volume 1 by Kleinrock. In the chapter on Continuous Time Markov Chain(CTMC), the author defines the infinitesimal generator, $Q(t)$ as having the following elements:

$ q_{ii}(t) = lim_{\Delta t \to 0 } \frac{p_{ii}(t, t+ \Delta t) - 1}{\Delta t}$

$ q_{ij}(t) = lim_{\Delta t \to 0 } \frac{p_{ij}(t, t+ \Delta t)}{\Delta t}$ , $i \neq j$

where $p_{ij}(t, t+ \Delta t)$ is the probability of moving to state j from i in a small interval $\Delta t$.

The author says that - "if the system at time $t$ is in the state $E_i$ then the probability that a transition occurs ( to any state other than $E_i$) during the interval $(t + \Delta t)$ is given by $- q_{ii} \Delta t + o(\Delta t) $. Thus we may say that $-q_{ii}$ is the rate at which the process departs from the state $E_i$ when it in the state."

I am unable to understand how the author arrives at the above interpretation from the limit definition. Can someone please provide clarification or point me to resources where there is an elaborate explanation?


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