Inter-arrival Times

Definition

Let $\{N(t),t\ge 0\}$ be a Poisson process of rate $\lambda$.

The Inter-arrival Times $\{T_n,n=1,2,\dots \}$ are defined as:

• $T_1$ = time of the first event
• $T_n$ = elapsed time between the $(n-1)$-th and $n$-th event, for $n>1$

Interpretation

Each event “resets the clock.” The time until the next event is always exponential with the same rate, regardless of how long you’ve already waited.

This follows from the memoryless property: the process has no memory of the past.

Distribution

Let $T_n$ be Inter-arrival Times

Then $T_n\stackrel{\text{i.i.d.}}{\sim }\text{Exp}(\lambda )$.

Proof Sketch

$P(T_1>t)=P(N(t)=0)=e^{-\lambda t}$

So $T_1\sim \text{Exp}(\lambda )$. For $T_2$, conditioning on $T_1=s$:

$P(T_2>t\mid T_1=s)=P(\text{0 events in }(s,s+t]\mid T_1=s)=e^{-\lambda t}$

By independent and stationary increments, this does not depend on $s$, so $T_2$ is independent of $T_1$ and also $\text{Exp}(\lambda )$. By induction, all $T_n$ are i.i.d. $\text{Exp}(\lambda )$.

Related

• Poisson Process
• Waiting Times
• Memoryless Property