In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, a non-homogeneous Poisson process is a Poisson process

Poisson process

A Poisson process, named after the French mathematician Siméon-Denis Poisson , is a stochastic process in which events occur continuously and independently of one another...

 with rate parameter such that the rate parameter of the process is a function of time. Non-homogeneous Poisson process have been shown to describe numerous random phenomena including cyclone

Cyclone

In meteorology, a cyclone is an area of closed, circular fluid motion rotating in the same direction as the Earth. This is usually characterized by inward spiraling winds that rotate anticlockwise in the Northern Hemisphere and clockwise in the Southern Hemisphere of the Earth. Most large-scale...

 prediction, arrival times of calls to a call centre in a hospital laboratory, arrival times of aircraft to airspace around an airport and database transaction time

Transaction time

Transaction time , a concept originated by Richard T. Snodgrass and his doctoral student, is used in temporal databases. It denotes the time period during which a database fact is/was stored in the database....

s.

Definition

Write N(t) for the number of events by time t. A stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 is a non-homogeneous Poisson process for some small value h if:

1. N(0)=0
2. Non-overlapping increments are independent
3. P(N(t+h)-N(t)=1) = λ(t)h + o(h)
4. P(N(t+h)-N(t)>1) = o(h)

for all t and where, in big O notation

Big O notation

In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann-Landau notation, or...

, .

Properties

Write N(t) for the number of events by time t and for the mean. Then N(t) has a Poisson distribution

Poisson distribution

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...

with mean m(t), that is for k = 0, 1, 2, 3….

Simulation

To simulate a non-homogeneous Poisson process with intensity function λ(t), choose a sufficiently large λ so that λ(t) = λ p(t) and simulate a Poisson process with rate parameter λ. Accept an event from the Poisson simulation at time t with probability p(t).