STOCHASTIC PROCESSES 12
Waiting Times
In a continuous-time Markov chain, the waiting time until the next jump is exponentially distributed.
Exponential waiting time
If the total transition rate from the current state is \(a>0\), then
with density
\[f_\tau(t)=ae^{-at},\qquad t\ge0.\]1. Survival probability
The probability that no event has occurred by time \(t\) is
2. Mean waiting time
\[E[\tau]=\frac1a.\]A larger event rate therefore produces a shorter average waiting time.
3. Memorylessness
The exponential distribution satisfies
\[P(\tau>s+t\mid\tau>s)=P(\tau>t).\]After waiting \(s\) units without an event, the remaining waiting-time distribution is unchanged.
4. Why exponential waiting times arise
A constant instantaneous event rate together with the Markov property leads to exponential waiting times.
5. Simulating a waiting time
If \(U\sim U(0,1)\), then
has an exponential distribution with rate \(a\).
6. Several event types
If rates are \(a_1,\ldots,a_m\), the total rate is
\[a=\sum_k a_k.\]The next waiting time uses this total rate, and event type \(k\) is chosen with probability \(a_k/a\).