Exponential waiting times

In a continuous-time stochastic model, an event rate determines a probability distribution for the waiting time until the next event.

Core idea. If the total event rate remains constant at \(a>0\) while the process stays in its current state, then the waiting time \(T\) satisfies \(T\sim\operatorname{Exp}(a)\).

Mean waiting time

\[\boxed{E[T]=\frac1a}.\]

A larger event rate gives a shorter mean wait, but \(1/a\) is not a fixed event time.

Survival function

\[\boxed{P(T>t)=e^{-at}}.\]

This is the probability that no event has yet occurred by time \(t\).

Cumulative probability

\[\boxed{P(T\le t)=1-e^{-at}}.\]

Thus the probability that the event occurs within the first \(t\) units of time is \(1-e^{-at}\).

Probability density

\[\boxed{f_T(t)=ae^{-at}},\qquad t\ge0.\]

Because \(T\) is continuous, \(P(T=t)=0\) for any exact value. Probabilities are obtained over intervals.

Probability between two times

For \(0\le s\[\boxed{P(sWhy the exponential form appears

Over a very short interval \(\Delta t\),

\[P(\text{event})\approx a\Delta t,\qquad P(\text{no event})\approx1-a\Delta t.\]

If \(t=n\Delta t\), then approximately

\[P(T>t)\approx(1-a\Delta t)^n.\]

Writing \(\Delta t=t/n\) and taking \(n\to\infty\),

\[\left(1-\frac{at}{n}\right)^n\to e^{-at}.\]

Memoryless property

\[\boxed{P(T>s+t\mid T>s)=P(T>t)}.\]

If no event has occurred during the first \(s\) units of time, the remaining waiting-time distribution is the same as when waiting began.

Connection to CTMCs. This memoryless property is compatible with the Markov property: while the state is unchanged, no extra record of how long the process has already waited is required.

Several competing events

If event types have current rates \(a_1,\ldots,a_m\), define

\[a_0=\sum_{j=1}^m a_j.\]

Then

\[\boxed{T\sim\operatorname{Exp}(a_0)},\]

and, conditional on an event occurring next,

\[\boxed{P(\text{event }j\text{ next})=\frac{a_j}{a_0}}.\]

SIS example

For

\[b(i)=\beta\frac{(N-i)i}{N},\qquad d(i)=\gamma i,\]

the total rate is

\[a(i)=b(i)+d(i),\]

\[T\sim\operatorname{Exp}(a(i)).\]

When the waiting time ends, infection is selected with probability \(b(i)/a(i)\) and recovery with probability \(d(i)/a(i)\).

Generating a waiting time

If \(U\sim\operatorname{Uniform}(0,1)\), then

\[\boxed{T=-\frac{\ln U}{a}}.\]

This is inverse-transform sampling.

Do not replace the random waiting time by \(1/a\) in a stochastic simulation. That would replace a random event time by its mean and remove the timing randomness.

Connection to the Poisson process

For a homogeneous Poisson process with rate \(\lambda\), consecutive inter-event times are independent exponential random variables with rate \(\lambda\). A state-dependent CTMC differs because its total rate may change after each jump.

Key idea. Exponential waiting times translate constant current event rates into random next-event times. Their mean is \(1/a\), their survival function is \(e^{-at}\), and their memoryless property is fundamental to continuous-time Markov chains.