Poisson Processes

The Poisson process is the model of points scattered completely at random in time or space, in which counts over disjoint regions are independent and Poisson distributed, making it the canonical description of random arrivals.

Definition

A Poisson process is a counting process whose numbers of events in disjoint regions are independent and Poisson distributed with mean proportional to the size of the region, equivalently a process of points with independent and stationary increments.

Scope

The topic covers the homogeneous Poisson process on the line defined by independent exponential interarrival times, its equivalent characterization through independent Poisson-distributed increments, the inhomogeneous and spatial Poisson point processes, the superposition and thinning operations, the order-statistics property of conditioned arrival times, and the Poisson process as the simplest continuous-time Markov counting process.

Core questions

• What independence and distributional properties characterize completely random points?
• Why are the waiting times between Poisson events exponentially distributed and memoryless?
• How do superposition and thinning combine and split Poisson processes?
• How are the arrival times distributed once the number of arrivals is known?

Key concepts

• independent increments
• exponential interarrival times
• superposition and thinning
• inhomogeneous intensity
• spatial point process

Key theories

Defining properties of the Poisson process

Independent Poisson-distributed counts over disjoint sets, exponential memoryless interarrival times, and the limit of many rare independent events all describe the same process, three equivalent characterizations that explain its universality.

Superposition, thinning, and the order-statistics property

Merging independent Poisson processes adds their rates, independently keeping each point with a fixed probability gives a thinned Poisson process, and conditioned on the count the arrival times are distributed as ordered uniform samples, a toolkit for manipulating Poisson points.

Clinical relevance

The Poisson process is the standard model for arrival streams in queueing and telecommunications, for the timing of radioactive decays and photon detections, for insurance claim arrivals, and as the spatial point-process model for the locations of stars, trees, or cellular events, where its thinning and superposition rules make analysis tractable.

History

Poisson derived the limiting law of rare events in 1837. Erlang applied Poisson arrivals to telephone traffic in the early twentieth century, founding queueing theory, and Kingman gave the modern measure-theoretic treatment of Poisson point processes on general spaces.

Key figures

• Simeon Denis Poisson
• Agner Krarup Erlang
• John Kingman

Seminal works

• kingman1993

Frequently asked questions

Why are the times between Poisson events exponential?

Because the process has no memory: the chance of an event in the next instant does not depend on how long one has already waited, and the exponential distribution is the unique continuous distribution with this memoryless property.

What does thinning a Poisson process do?

If each point of a Poisson process is independently kept with some fixed probability, the retained points again form a Poisson process with the rate scaled by that probability, and the kept and discarded points are independent.