Poisson process
Encyclopedia
A Poisson process, named after the French mathematician Siméon-Denis Poisson (1781–1840), is a stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 in which events occur continuously and independently of one another
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

 (the word event used here is not an instance of the concept of event
Event (probability theory)
In probability theory, an event is a set of outcomes to which a probability is assigned. Typically, when the sample space is finite, any subset of the sample space is an event...

 frequently used in probability theory). Examples that are well-modeled as Poisson processes include the radioactive decay
Radioactive decay
Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...

 of atoms, telephone calls arriving at a switchboard, page view requests to a website
Website
A website, also written as Web site, web site, or simply site, is a collection of related web pages containing images, videos or other digital assets. A website is hosted on at least one web server, accessible via a network such as the Internet or a private local area network through an Internet...

, and rainfall. A Poisson process is usually described as a function of time, although it need not be.

The Poisson process is a collection } of random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

s, where is the number of events that have occurred up to time (starting from time 0). The number of events between time and time is given as and 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...

. Each realization
Realization (probability)
In probability and statistics, a realization, or observed value, of a random variable is the value that is actually observed . The random variable itself should be thought of as the process how the observation comes about...

 of the process is a non-negative integer-valued step function that is non-decreasing, but for intuitive purposes it is usually easier to think of it as a point pattern on [0,∞) (the points in time where the step function jumps, i.e. the points in time where an event occurs).

The Poisson process is a continuous-time process; the Bernoulli process
Bernoulli process
In probability and statistics, a Bernoulli process is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables Xi are identical and independent...

 can be thought of as its discrete-time counterpart (although strictly, one would need to sum the events in a Bernoulli process to also have a counting process). A Poisson process is a pure-birth process, the simplest example of a birth-death process
Birth-death process
The birth–death process is a special case of continuous-time Markov process where the states represent the current size of a population and where the transitions are limited to births and deaths...

. By the aforementioned interpretation as a random point pattern on it is also a point process
Point process
In statistics and probability theory, a point process is a type of random process for which any one realisation consists of a set of isolated points either in time or geographical space, or in even more general spaces...

 on the real half-line.

Definition

The basic form of Poisson process, often referred to simply as "the Poisson process", is a continuous-time counting process  {N(t), t ≥ 0} that possesses the following properties:
  • N(0) = 0
  • Independent increments (the numbers of occurrences counted in disjoint intervals are independent from each other)
  • Stationary increments (the probability distribution of the number of occurrences counted in any time interval only depends on the length of the interval)
  • No counted occurrences are simultaneous.


Consequences of this definition include:
  • The probability distribution
    Probability distribution
    In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

     of N(t) is 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...

    .
  • The probability distribution of the waiting time until the next occurrence is an exponential distribution
    Exponential distribution
    In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

    .
  • The occurrences are distributed uniformly
    Uniform distribution (continuous)
    In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...

     on any interval of time. (Note that N(t), the total number of occurrences, has a Poisson distribution over (0, t], whereas the location of an individual occurrence on is uniform.)


Other types of Poisson process are described below.

Homogeneous

The homogeneous Poisson process is one of the most well-known Lévy process
Lévy process
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

es. This process is characterized by a rate parameter λ, also known as intensity, such that the number of events in time interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...

 (tt + τ] follows 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 associated parameter λτ. This relation is given as


where N(t + τ) − N(t) = k is the number of events in time interval (tt + τ].

Just as a Poisson random variable is characterized by its scalar parameter λ, a homogeneous Poisson process is characterized by its rate parameter λ, which is the expected
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

 number of "events" or "arrivals" that occur per unit time.

N(t) is a sample homogeneous Poisson process, not to be confused with a density or distribution function.

Non-homogeneous

In general, the rate parameter may change over time; such a process is called a non-homogeneous Poisson process or inhomogeneous Poisson process.
In this case, the generalized rate function is given as λ(t). Now the expected number of events between time a and time b is


Thus, the number of arrivals in the time interval (ab], given as N(b) − N(a), follows 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 associated parameter λa,b


A homogeneous Poisson process may be viewed as a special case when λ(t) = λ, a constant rate.

Spatial

An important variation on the (notionally time-based) Poisson process is the spatial Poisson process. In the case of a one-dimension space (a line) the theory differs from that of a time-based Poisson process only in the interpretation of the index variable. For higher dimension spaces, where the index variable (now x) is in some vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V (e.g. R2 or R3), a spatial Poisson process can be defined by the requirement that the random variables defined as the counts of the number of "events" inside each of a number of non-overlapping finite sub-regions of V should each have a Poisson distribution and should be independent of each other.

Space-time

A further variation on the Poisson process, the space-time Poisson process, allows for separately distinguished space and time variables. Even though this can theoretically be treated as a pure spatial process by treating "time" as just another component of a vector space, it is convenient in most applications to treat space and time separately, both for modeling purposes in practical applications and because of the types of properties of such processes that it is interesting to study.

In comparison to a time-based inhomogeneous Poisson process, the extension to a space-time Poisson process can introduce a spatial dependence into the rate function, such that it is defined as , where for some vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

 V (e.g. R2 or R3). However a space-time Poisson process may have a rate function that is constant with respect to either or both of x and t. For any set (e.g. a spatial region) with finite measure
Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...

 , the number of events occurring inside this region can be modeled as a Poisson process with associated rate function λS(t) such that

Separable space-time processes

In the special case that this generalized rate function is a separable function of time and space, we have:


for some function . Without loss of generality, let


(If this is not the case, λ(t) can be scaled appropriately.) Now, represents the spatial probability density function
Probability density function
In probability theory, a probability density function , or density of a continuous random variable is a function that describes the relative likelihood for this random variable to occur at a given point. The probability for the random variable to fall within a particular region is given by the...

 of these random events in the following sense. The act of sampling this spatial Poisson process is equivalent to sampling a Poisson process with rate function λ(t), and associating with each event a random vector sampled from the probability density function . A similar result can be shown for the general (non-separable) case.

Characterisation

In its most general form, the only two conditions for a counting process to be a Poisson process are:
  • Orderliness: which roughly means


which implies that arrivals don't occur simultaneously (but this is actually a mathematically stronger statement).

  • Memorylessness
    Memorylessness
    In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

    (also called evolution without after-effects): the number of arrivals occurring in any bounded interval of time after time t is independent
    Statistical independence
    In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

     of the number of arrivals occurring before time t.


These seemingly unrestrictive conditions actually impose a great deal of structure in the Poisson process. In particular, they imply that the time between consecutive events (called interarrival times) are independent
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...

 random variables. For the homogeneous Poisson process, these inter-arrival times are exponentially distributed
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 with parameter λ (mean 1/λ).

Proof :
Let be the first arrival time of the Poisson process. Its distribution satisfies

Also, the memorylessness property entails that the number of events in any time interval is independent of the number of events in any other interval that is disjoint from it. This latter property is known as the independent increments property of the Poisson process.

Properties

As defined above, the stochastic process {N(t)} is a Markov process
Markov process
In probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...

, or more specifically, a continuous-time Markov process.

To illustrate the exponentially-distributed
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

 inter-arrival times property, consider a homogeneous Poisson process N(t) with rate parameter λ, and let Tk be the time of the kth arrival, for k = 1, 2, 3, ... . Clearly the number of arrivals before some fixed time t is less than k if and only if the waiting time until the kth arrival is more than t. In symbols, the event [N(t) < k] occurs if and only if the event [Tk > t] occurs. Consequently the probabilities of these events are the same:


In particular, consider the waiting time until the first arrival. Clearly that time is more than t if and only if the number of arrivals before time t is 0. Combining this latter property with the above probability distribution for the number of homogeneous Poisson process events in a fixed interval gives


Consequently, the waiting time until the first arrival T1 has an exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...

, and is thus memoryless
Memorylessness
In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

. One can similarly show that the other interarrival times Tk − Tk−1 share the same distribution. Hence, they are independent, identically-distributed (i.i.d.) random variables with parameter λ > 0; and expected value 1/λ. For example, if the average rate of arrivals is 5 per minute, then the average waiting time between arrivals is 1/5 minute.

Examples

The following examples are well-modeled by the Poisson process:
  • The number of goals in a soccer match

  • The arrival of customers in a queue.

  • The number of raindrops falling within a specified area.

  • The number of telephone calls arriving at a switchboard.

  • The number of particles emitted via radioactive decay
    Radioactive decay
    Radioactive decay is the process by which an atomic nucleus of an unstable atom loses energy by emitting ionizing particles . The emission is spontaneous, in that the atom decays without any physical interaction with another particle from outside the atom...

     by an unstable substance. In this case the Poisson process is non-homogeneous in a predictable manner - the emission rate declines as particles are emitted.

  • The number of web page requests arriving at a server. Even unusual circumstances such as coordinated denial of service attacks or flash crowds
    Flash crowd
    Flash crowd describes a network phenomenon where a network or host suddenly receives a lot of traffic. This is sometimes due to the appearance of a web site on a blog or news column....

     can be incorporated into a non-homogeneous Poisson process as temporary large increases in the rate parameter.

Occurrence

The Palm–Khintchine theorem
Palm–Khintchine theorem
In probability theory, the Palm–Khintchine theorem, believed to be the work of Conny Palm and Aleksandr Khinchin, expresses that a large number of not necessarily Poissonian renewal processes combined will have Poissonian properties....

 provides a result that shows that the superimposition of many low intensity non-Poisson point processes will be close to a Poisson process.

See also

  • Compound Poisson distribution
    Compound Poisson distribution
    In probability theory, a compound Poisson distribution is the probability distribution of the sum of a "Poisson-distributed number" of independent identically-distributed random variables...

  • Compound Poisson process
    Compound Poisson process
    A compound Poisson process is a continuous-time stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution...

  • Renewal process
  • Cox process
    Cox process
    A Cox process , also known as a doubly stochastic Poisson process or mixed Poisson process, is a stochastic process which is a generalization of a Poisson process...

     (generalization)
  • Gamma distribution
  • Markovian arrival processes
    Markovian arrival processes
    In queueing theory, Markovian arrival processes are used to model the arrival of customers to a queue.Some of the most common include the Poisson process, Markov arrival process and the batch Markov arrival process.-Background:...

  • Poisson sampling
    Poisson sampling
    In the theory of finite population sampling, Poisson sampling is a sampling process where each element of the population that is sampled is subjected to an independent Bernoulli trial which determines whether the element becomes part of the sample during the drawing of a single sample.Each element...

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