Poisson Process

The Poisson Process

• The Poisson process, denoted as ${N(t), t \geq 0}$, counts events in the interval $[0, t]$.
• It has a rate $\lambda > 0$.

Definition of a Poisson Process

• $N(0) = 0$, meaning initially, there are no events.
• Independent increments: Events in non-overlapping time intervals are independent.
• For all $s, t \geq 0$, the number of events in an interval of length $t$ follows a Poisson distribution with mean $\lambda t$: $$P(N(t+s) - N(s) = n) = e^{-\lambda t} \frac{(\lambda t)^n}{n!}, \quad n = 0, 1, 2, \dots$$

Homogeneous Poisson Process

• Because $P(N(t+s) - N(s) = n)$ depends on $t$ only, the length of the interval, the Poisson process is homogeneous.

Memoryless Property

• Due to the independent increments assumption, the Poisson process is memoryless.

Inter-arrival Times

• Let $T_1$ be the time of the first event.
• Let $T_i$ be the time between event $i-1$ and event $i$, for $i \geq 2$.
• $T_i$ is called the $i$-th inter-arrival time.
• The inter-arrival times $T_1, T_2, \dots$ are independent and identically distributed exponential random variables with parameter $\lambda$.

Proof of Inter-arrival Times Proposition

• $P(T_1 > t) = P(N(t) = 0) = e^{-\lambda t}$.
• Therefore, $T_1 \sim Exp(\lambda)$.
• $P(T_2 > t \mid T_1 = s) = P(0 \text{ events in } (s, s+t] \mid T_1 = s) = P(0 \text{ events in } (s, s+t]) = P(N(s+t) - N(s) = 0) = e^{-\lambda t}$.
• Therefore, $T_2 \sim Exp(\lambda)$ and is independent of $T_1$, and a similar argument holds for $T_i$, $i > 2$.

Waiting Times

• $S_n$ is the time of the $n$-th event.
• $S_n$ is also called the waiting time until the $n$-th event.
• $S_n = \sum_{i=1}^n T_i \sim Gamma(n, \lambda)$.
• $S_n$ follows a Gamma distribution with parameters $n$ and $\lambda$ because it's the sum of $n$ independent and identically distributed exponential random variables with parameter $\lambda$.

Remarks on Distributions

• The exponential distribution is a Gamma distribution special case when $n = 1$.
• The Gamma distribution is a Chi-squared distribution special case when $\lambda = \frac{1}{2}$ and $n = \frac{r}{2}$, with $r \in \mathbb{Z}^+$.
• The Erlang distribution is another name for the Gamma distribution when $n \in \mathbb{Z}^+$.