Simple model for events in time
There are many situations in which 'events' happen at random times. Examples are:
The simplest kind of model for events in time is called a homogeneous Poisson process.
Definition
A homogeneous Poisson process is a series of events that occur over time in which:
Expressing this less formally,
This can be generalised to a non-homogeneous Poisson process if we allow the chance of an event happening to vary over time, replacing the constant \(\lambda\) with a function of time \(\lambda(t)\).
Rate of occurrence of events
In a homogeneous Poisson process, the parameter \(\lambda\) is the rate of events occurring. For example, a model for emergency requests for ambulances from a hospital might have \(\lambda = 1.5\) call-outs per hour. The assumption that the call-outs occur as a homogeneous Poisson process means that:
This model would be an over-simplification of reality. It might be possible for two ambulances to be called out simultaneously after a bad road accident. Moreover, the rate of call-outs is likely to vary between different times of day. However a homogeneous Poisson is an approximation that might still be able to give reasonable insight.
Single events
We have introduced Poisson processes for situations where multiple events happen over time. Sometimes only one such event that may happen. Examples are:
The time until the event happens is random, and it can be modelled as the first event in a homogeneous Poisson process.