The process itself was discovered independently and repeatedly in several settings, including experiments on radioactive decay, telephone call arrivals and actuarial science.

It is used, for example, in queueing theory[15] to model random events distributed in time, such as the arrival of customers at a store, phone calls at an exchange or occurrence of earthquakes.

[24] In the first case, the constant, known as the rate or intensity, is the average density of the points in the Poisson process located in some region of space.

[30] The Poisson point process can be defined, studied and used in one dimension, for example, on the real line, where it can be interpreted as a counting process or part of a queueing model;[31][32] in higher dimensions such as the plane where it plays a role in stochastic geometry[1] and spatial statistics;[33] or on more general mathematical spaces.

[25] There have been many applications of the homogeneous Poisson process on the real line in an attempt to model seemingly random and independent events occurring.

It has a fundamental role in queueing theory, which is the probability field of developing suitable stochastic models to represent the random arrival and departure of certain phenomena.

, we can give the finite-dimensional distribution of the homogeneous Poisson point process by first considering a collection of disjoint, bounded Borel (measurable) sets

In recent years, it has been frequently used to model seemingly disordered spatial configurations of certain wireless communication networks.

The previous homogeneous Poisson point process immediately extends to higher dimensions by replacing the notion of area with (high dimensional) volume.

has the finite-dimensional distribution:[67] Homogeneous Poisson point processes do not depend on the position of the underlying space through its parameter

, namely On the real line, the inhomogeneous or non-homogeneous Poisson point process has mean measure given by a one-dimensional integral.

In the case of point processes with refractoriness (e.g., neural spike trains) a stronger version of property 4 applies:[73]

[21][22] The intensity measure of this point process is dependent on the location of underlying space, which means it can be used to model phenomena with a density that varies over some region.

[84] If the intensity function is sufficiently simple, then independent and random non-uniform (Cartesian or other) coordinates of the points can be generated.

For example, simulating a Poisson point process on a circular window can be done for an isotropic intensity function (in polar coordinates

, accepting if it is smaller than the probability density function, and repeating until the previously chosen number of samples have been drawn.

Over the following years others used the distribution without citing Poisson, including Philipp Ludwig von Seidel and Ernst Abbe.

[91] [2] At the end of the 19th century, Ladislaus Bortkiewicz studied the distribution, citing Poisson, using real data on the number of deaths from horse kicks in the Prussian army.

[2][3] In Sweden 1903, Filip Lundberg published a thesis containing work, now considered fundamental and pioneering, where he proposed to model insurance claims with a homogeneous Poisson process.

Erlang derived the Poisson distribution in 1909 when developing a mathematical model for the number of incoming phone calls in a finite time interval.

Erlang unaware of Poisson's earlier work and assumed that the number phone calls arriving in each interval of time were independent of each other.

[95] A number of mathematicians started studying the process in the early 1930s, and important contributions were made by Andrey Kolmogorov, William Feller and Aleksandr Khinchin,[2] among others.

[3] Feller worked from 1936 to 1939 alongside Harald Cramér at Stockholm University, where Lundberg was a PhD student under Cramér who did not use the term Poisson process in a book by him, finished in 1936, but did in subsequent editions, which his has led to the speculation that the term Poisson process was coined sometime between 1936 and 1939 at the Stockholm University.

the generating functional is given by: which in the homogeneous case reduces to For a general Poisson point process with intensity measure

[139] There a number of methods that can be used to justify, informally or rigorously, approximating the occurrence of random events or phenomena with suitable Poisson point processes.

Stein's method can be used to derive upper bounds on probability metrics, which give way to quantify how different two random mathematical objects vary stochastically.

[108] Techniques based on Stein's method have been developed to factor into the upper bounds the effects of certain point process operations such as thinning and superposition.

For mathematical models the Poisson point process is often defined in Euclidean space,[1][36] but has been generalized to more abstract spaces and plays a fundamental role in the study of random measures,[148][149] which requires an understanding of mathematical fields such as probability theory, measure theory and topology.

[150] In general, the concept of distance is of practical interest for applications, while topological structure is needed for Palm distributions, meaning that point processes are usually defined on mathematical spaces with metrics.

The generality and tractability of Cox processes has resulted in them being used as models in fields such as spatial statistics[154] and wireless networks.