Poisson process

A Poisson process, named after the French mathematician Siméon-Denis Poisson (1781–1840), is the stochastic process in which events occur continuously and independently of one another (the word event used here is not an instance of the concept of event frequently used in probability theory). Examples that are well-modeled as Poisson processes include the radioactive decay of atoms, telephone calls arriving at a switchboard, page view requests to a website, and rainfall.

The Poisson process is a collection {N(t) : t ≥ 0} of random variables, where N(t) is the number of events that have occurred up to time t (starting from time 0). The number of events between time a and time b is given as N(b) − N(a) and has a Poisson distribution. Each realization of the process {N(t)} 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: its discrete-time counterpart is the Bernoulli process. 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. By the aforementioned interpretation as a random point pattern on [0, ∞) it is also a point process on the real half-line.

Definition

A 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 the definition include:

The probability distribution of N(t) is a Poisson distribution.
The probability distribution of the waiting time until the next occurrence is an exponential distribution.

Types

Homogeneous

The homogeneous Poisson process is one of the most well-known Lévy processes. This 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

P[(N(t+\tau )-N(t))=k]={\frac {e^{-\lambda \tau }(\lambda \tau )^{k}}{k!}}\qquad k=0,1,\ldots ,

where N(t + τ) − N(t) is 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.

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

\lambda _{a,b}=\int _{a}^{b}\lambda (t)\,dt.

Thus, the number of arrivals in the time interval (a, b], given as N(b) − N(a), follows a Poisson distribution with associated parameter λ_a,b

P[(N(b)-N(a))=k]={\frac {e^{-\lambda _{a,b}}(\lambda _{a,b})^{k}}{k!}}\qquad k=0,1,\ldots .

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

Spatial

A further variation on the Poisson process, called the spatial Poisson process, introduces a spatial dependence on the rate function and is given as $\lambda ({\vec {x}},t)$ where ${\vec {x}}\in V$ for some vector space V (e.g. R² or R³). For any set $S\subset V$ (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 λ_S(t) such that

\lambda _{S}(t)=\int _{S}\lambda ({\vec {x}},t)\,d{\vec {x}}.

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

\lambda ({\vec {x}},t)=f({\vec {x}})\lambda (t)

for some function $f({\vec {x}})$ . Without loss of generality, let

\int _{V}f({\vec {x}})\,d{\vec {x}}=1.

(If this is not the case, λ(t) can be scaled appropriately.) Now, $f({\vec {x}})$ 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 ${\vec {X}}$ sampled from the probability density function $f({\vec {x}})$ . A similar result can be shown for the general (non-separable) case.

Properties

In its most general form, the only two conditions for a counting process to be a Poisson process are:

Orderliness: which roughly means

\lim _{\Delta t\to 0}P(N(t+\Delta t)-N(t)>1\mid N(t+\Delta t)-N(t)\geq 1)=0

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 (mean 1/λ).

Proof : Let $\tau _{1}$ be the first arrival time of the Poisson process. Its distribution satisfies

{\begin{aligned}Pr[\tau _{1}=x]=&\lim _{dt\to 0}{\frac {Pr[N_{x+dt}>0,N_{x}=0]}{dt}}\\=&\lim _{dt\to 0}{\frac {1-Pr[N_{dt}=0]}{dt}}Pr[N_{x}=0]\\=&\lim _{dt\to 0}{\frac {1-(1-\lambda dt+O(dt^{2}))}{dt}}\exp(-\lambda x)\\=&\lambda \exp(-\lambda x)\end{aligned}}

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.

To illustrate the exponentially-distributed inter-arrival times property, consider a homogeneous Poisson process N(t) with rate parameter λ, and let T_k 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 [T_k > t] occurs. Consequently the probabilities of these events are the same:

P(T_{k}>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

P(T_{1}>t)=P(N(t)=0)=P[(N(t)-N(0))=0]={\frac {e^{-\lambda t}(\lambda t)^{0}}{0!}}=e^{-\lambda t}.

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 T_k − T_k−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 arrival of "customers" in a queue.

The number of raindrops falling over an area.

The number of photons hitting a photodetector.

The number of telephone calls arriving at a switchboard, or at an automatic phone-switching system.

The number of particles emitted via radioactive decay by an unstable substance, where the rate decays as the substance stabilizes.

The long-term behavior of the number of web page requests arriving at a server, except for unusual circumstances such as coordinated denial of service attacks or flash crowds. Such a model assumes homogeneity as well as weak stationarity.