Poisson Process

a stochastic process describing the moments at which certain random events occur. In a Poisson process, the number of events occurring within any fixed interval of time has a Poisson distribution, and the numbers of the events occurring in nonoverlapping intervals of time are independent.

Suppose μ(s, t) is the number of events whose moments of occurrence τ_i satisfy the inequalities 0 ≤ s < τ_i ≤ t, and suppose λ (s, t) is the mathematical expectation of μ(s, t). In a Poisson process, for any 0 ≤ s₁ < t₁ ≤ s₂ < t₂ ≤ … ≤ s_r < t_r’ the random variables μ(s₁, t₁), μ(s₂, t₂), …, μ(s_r, t_r) are independent, and the equality μ(s, t) = η has probability

e^{-λ(s, t)}[λ(s, t)]ⁿ/n!

In a homogeneous Poisson process, λ (s, t) = a(t – s), where a is the mean number of events in a unit of time and the distances τ_n -τ_n-1 between neighboring moments τ_n are independent and have an exponential distribution with density ae^-at, t ≥ 0.

If there are many independent processes that describe the moments certain rare events occur, the total process yields in the limit a Poisson process under certain conditions.

The Poisson process is a convenient mathematical model that is often used in various applications of probability theory. In particular, it is used to describe a request flow in queuing theory—for example, calls arriving at a telephone exchange or ambulance trips in response to traffic accidents in a large city.

A generalization of the Poisson process is the Poisson random distribution of points in a plane or in space. The number of points here in any fixed region has a Poisson distribution (with mean proportional to the area or volume of the region), and the numbers of points in nonoverlapping regions are independent. This distribution is often used in calculations in such fields as astronomy, physics, ecology, and engineering.