Exponential waiting times
If events occur at constant rate \(a>0\), the waiting time \(T\) until the next event has an exponential distribution.
\[f_T(t)=ae^{-at},\qquad t\ge0,\]\[P(T>t)=e^{-at},\qquad E[T]=\frac1a.\]Memoryless property
\[P(T>s+t\mid T>s)=P(T>t).\]Thus, after waiting for time \(s\) without an event, the remaining waiting-time distribution is unchanged.
Simulation
If \(U\sim\operatorname{Uniform}(0,1)\), an exponential waiting time can be generated by
\[T=-\frac{\ln U}{a}.\]Key idea. In a CTMC, exponential waiting times arise naturally when the event rate remains constant while the process stays in its current state.