Poisson Processes

A Poisson process is a probabilistic model that counts events occurring over time (or space) at random. It is a counting process with a rate or intensity λ, and it is defined by precise assumptions: events occur independently of one another, the number of events in non-overlapping intervals is independent, and the probability of an event in a very small interval is proportional to its length.

The formal definition

Under those assumptions, the number of events in an interval of length t follows a Poisson distribution with mean λt, and the times between consecutive events follow an exponential distribution with mean 1/λ. No two simultaneous events are allowed. The homogeneous case, with a constant rate λ, is distinguished from the non-homogeneous case, in which the rate λ(t) varies over time—useful, for instance, when arrivals have rush hours.

A model with assumptions, not a description of reality

Applications should be treated with care, because its assumptions often fail. In neuroscience, the “Poisson spike train” is an idealized model of neuronal firing, not a literal description: real neurons have a refractory period that prevents immediately consecutive spikes, something the Poisson process does not capture. In financial markets, orders tend to arrive in clusters, which violates independence; that is why self-exciting extensions such as Hawkes processes are used, in which one event raises the probability of the next. The rule is to always present these cases as models with explicit assumptions.

What they are used for

Poisson processes are the basis for modeling event arrivals and queues—customers, network packets, requests to a server—for simulating neuronal activity in the study of spiking networks, and as an analytical starting point on which more realistic models that relax its assumptions are built.