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A Poisson process, named after the French mathematician Siméon-Denis Poisson (1781 - 1840), is a stochastic process which is defined in terms of the occurrences of events. This counting process, given as a function of time N(t), represents the number of events since time t = 0 (see examples). Also, the number of events between time a and time b is given as N(b) − N(a) and has a Poisson distribution. Image File history File links Derived from public domain images featured at: http://commons. ...
Siméon Poisson. ...
In the mathematics of probability, a stochastic process is a random function. ...
It has been suggested that this article be split into multiple articles. ...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...
The Poisson process is a continuous-time process: its discrete-time counterpart is the Bernoulli process. The Poisson process is one of the most well-known Lévy processes. Poisson processes are also examples of continuous-time Markov processes. A Poisson process is a pure-birth process, the simplest example of a birth-death process. In probability and statistics, a Bernoulli process is a discrete-time stochastic process consisting of a sequence of independent random variables taking values over two letters. ...
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that has stationary independent increments -- this phrase will be explained below. ...
In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that satisfies the Markov property and takes values from a set called the state space. ...
This process is an example of a Markov process(a stochastic process) where the transitions are limited to the nearest neighbors only. ...
Types of Poisson processes
Homogeneous Poisson process A homogeneous Poisson process is characterized by a rate parameter λ, also known as intensity, such that the number of events in time interval (t,t + τ] follows a Poisson distribution with associated parameter λτ. This relation is given as Sample Homogeneous Poisson Process File links The following pages link to this file: Poisson process Categories: Images with unknown source ...
In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...
where N(t + τ) − N(t) describes the number of events in time interval (t, t + 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 number of "events" or "arrivals" that occur per unit time. In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...
N(t) is a sample homogeneous Poisson process, not to be confused with a density or distribution function.
Non-homogeneous Poisson process -
Main article: Non-homogeneous Poisson process In general, the rate parameter may change over time. 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 In probability theory, a non-homogeneous Poisson process is a Poisson process with rate parameter such that the rate parameter of the process is a function of time. ...
Thus, the number of arrivals in the time interval (a, b], given as N(b) − N(a), follows a Poisson distribution with associated parameter λa,b In probability theory and statistics, the Poisson distribution is a discrete probability distribution. ...
A homogeneous Poisson process may be viewed as a special case when λ(t) = λ, a constant rate.
Spatial Poisson process A further variation on the Poisson process, called the spatial Poisson process, introduces a spatial dependence on the rate function and is given as where for some vector space V (e.g. R2 or R3). For any set (e.g. a spatial region) with finite measure, the number of events occurring inside this region can be modelled as a Poisson process with associated rate function λS(t) such that In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ...
In mathematics, a measure is a function that assigns a number, e. ...
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 else we may scale and λ(t) appropriately. Now, represents the spatial probability density function 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. In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. ...
General characteristics of the Poisson process In its most general form, the only two conditions for a stochastic process to be a Poisson process are: In the mathematics of probability, a stochastic process is a random function. ...
- Orderliness: which roughly means
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- which implies that arrivals don't occur simultaneously (but this is actually a mathematically-stronger statement).
- Memorylessness (also called evolution without after-effects): the number of arrivals occurring in any bounded interval of time after time t is independent 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 random variables. For the homogeneous Poisson process, these inter-arrival times are exponentially-distributed with parameter λ. Also, the memorylessness property shows that the number of events in one time interval is independent from the number of events in an interval that is disjoint from the first interval. This latter property is known as the independent increments property of the Poisson process. In probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. ...
In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...
To illustrate the exponentially-distributed 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 ]. Consequently the probabilities of these events are the same:
- P(Tk > t) = P(N(t) < k).
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, and is thus memoryless. 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. In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ...
In probability theory, memorylessness is a property of certain probability distributions: the exponential distributions and the geometric distributions. ...
In probability theory, a sequence or other collection of random variables is independent and identically distributed (i. ...
Examples - The number of web page requests arriving at a server may be characterized by a Poisson process except for unusual circumstances such as coordinated denial of service attacks.
- The number of telephone calls arriving at a switchboard, or at an automatic phone-switching system, may be characterized by a Poisson process.
- The number of photons hitting a photodetector, when lit by a laser source, may be characterized by a homogeneous Poisson process. Other sources may show either a bunching or an antibunching of the photons.
- The number of particles emitted via radioactive decay by an unstable substance may be characterized by a non-homogeneous Poisson process, where the rate decays as the substance stabilizes.
- The number of raindrops falling over a wide spatial area may be characterized by a spatial Poisson process.
- The arrival of "customers" is commonly modelled as a Poisson process in the study of simple queueing systems.
A denial-of-service attack (also, DoS attack) is an attack on a computer system or network that causes a loss of service to users, typically the loss of network connectivity and services by consuming the bandwidth of the victim network or overloading the computational resources of the victim system. ...
Radioactive decay is the process in which an unstable atomic nucleus loses energy by emitting radiation in the form of particles or electromagnetic waves. ...
See also In probability theory, a compound Poisson distribution is the probability distribution of a Poisson-distibuted number of independent identically-distributed random variables. ...
This article or section should be merged with Poisson process A compound poisson process with rate and jump size distribution G is a parametrized family of random variables given by where, is a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which...
In probability theory, a continuous-time Markov process is a stochastic process { X(t) : t ≥ 0 } that satisfies the Markov property and takes values from a set called the state space. ...
There are very few or no other articles that link to this one. ...
In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions that represents the sum of exponentially distributed random variables, each of which has mean . ...
In queuing theory, Markovian arrival processes are used to model the arrival customers to queue. ...
Further reading - Snyder & Miller, Random Point Processes in Time and Space. Springer-Verlag. ISBN 0-387-97577-2.
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