Poisson Process¶

Definition¶

Let $\lambda>0$ $0$ . A process $\{N_t\colon t\ge 0\}$ with $N_0=0$ $0$ , in which $N_t$ counts the arrivals in $[0,t]$ , is a Poisson process with rate $\lambda$ if it has independent increments (arrival counts in disjoint time intervals are independent) and

N_{t+s}-N_s\sim\operatorname{Pois}(\lambda t)

for all $s,t\ge 0$ $0$ .

Interpretation¶

A Poisson process models random arrivals over time. It assumes arrivals occur independently and at a constant average rate.

A staircase counting arrivals over time, with arrival times T1 to T6 on the axis and the exponential gap between two arrivals marked.

The count $N_t$ records arrivals in $[0,t]$ . For a Poisson process of rate $\lambda$ , $N_{t+s}-N_s\sim\operatorname{Pois}(\lambda t)$ and disjoint increments are independent. The interarrival times, that is, the waiting times between consecutive arrivals, are i.i.d. exponential $\operatorname{Exp}(\lambda)$ variables, with $\lambda$ expected arrivals per unit time.

Each press of the button draws new arrivals at rate $\lambda$ . The number in the window $(s,s+t]$ varies from draw to draw like a $\operatorname{Pois}(\lambda t)$ variable.

A stochastic process consists of variables indexed by time. The Poisson distribution specifies each interval count, while independent random variables describe counts on disjoint intervals; both properties are needed to specify the process, not just its one-time law.

Key Results¶

A Poisson process has independent increments.
The number of arrivals in an interval depends only on the interval length.
Interarrival times in a Poisson process are independent exponential random variables.

Examples¶

Customer arrivals to a store can be approximated by a Poisson process during a stable period.
Particle emissions from a radioactive source are often modelled as a Poisson process.
Requests to a server may be modelled as a Poisson process over short intervals.

Prerequisites¶

Sources¶

Probability Theory for EOR · Week 6

Aliases: homogeneous Poisson process, Poisson arrival process Keywords: Poisson point process, Poisson arrivals, counting process Symbols: lambda

Last updated: 20 October 2026 03:02