mathematics//probability//stochastic process//Poisson process
A Poisson process is a stochastic process that counts events arriving one at a time, independently of each other and at a constant average rate \(\lambda\), and it is the default model of random arrivals in engineering: requests reaching a server, robots queueing at a charger, random shocks breaking a component, alarms from a large fleet. It is the simplest point process, and every richer one is described by how it departs from it.
A Poisson process is a stochastic process that counts events arriving one at a time, independently of each other and at a constant average rate λ\lambdaλ, and it is the default model of random arrivals in engineering: requests reaching a server, robots queueing at a charger, random shocks breaking a component, alarms from a large fleet. It is the simplest point process, and every richer one is described by how it departs from it.
Two facts carry the whole model. The number of events in a window of length ttt follows the Poisson distribution, and the gap between consecutive events follows the exponential distribution with the same rate:
P(N(t)=k)=(λt)kk!e−λt,P(gap>s)=e−λs.P(N(t)=k)=\frac{(\lambda t)^k}{k!}e^{-\lambda t},\qquad P(\text{gap}>s)=e^{-\lambda s}.P(N(t)=k)=k!(λt)ke−λt,P(gap>s)=e−λs.
The mean count in the window is λt\lambda tλt, and so is its variance. A depot whose 40 robots each come to charge about once every two hours sees λ=20\lambda=20λ=20 arrivals per hour; in a 6-minute slot it expects 2, and the chance of 5 or more in that slot is about 5%, which is what sizes the number of chargers.
Independent arrivals look clumped.
With exponential gaps, short gaps are the most likely ones, so bursts of three or four events in quick succession are what a Poisson process normally produces. A burst alone proves nothing about a common cause; a variance clearly above the mean, measured over many windows, does.
It emerges by itself from many small independent sources. Summing the arrivals of many units, each of which rarely produces an event, gives a process that is very nearly Poisson whatever each unit does, which is why the failures of a large fleet of electronics or the calls of a big population fit it well. The sum of independent Poisson processes is Poisson with the rates added.
Its constant rate is an assumption to check. Traffic has daily cycles (a rate that changes with time, the non-homogeneous version), and failures that trigger other failures, or trades that trigger trades, cluster in a way the model cannot produce; the Hawkes process adds that self-excitation. A count whose variance is much larger than its mean is the usual sign.
It is the arrival half of queueing theory: the M of the M/M/1 queue means Poisson arrivals, and with them waiting times grow sharply as the load approaches capacity. In reliability, a constant hazard function makes the failures of a repaired unit a Poisson process, the flat bottom of the bathtub.