Simple model for events in time

There are many situations in which 'events' happen at random times. For example,

The simplest kind of model for such events is a homogeneous Poisson process.

Definition

A homogeneous Poisson process is a series of events that occur over time in which:

  1. Multiple events cannot happen at exactly the same time.
  2. The occurrence of events during any interval of time is independent of whether events occurred in other non-overlapping intervals.
  3. The probability of an event happening in any infinitesimally small interval \((t, t+\delta t]\) is \(\lambda \times \delta t\).

Less formally,

  1. events happen singly,
  2. occurrence of events at different times are independent, and
  3. the chance of an event happening is the same at all times.

The parameter \(\lambda\) is the rate of events occurring and is expressed as "events per unit time". For example, a model for emergency requests for ambulances from a hospital might have \(\lambda = 1.5\) call-outs per hour.

This model is often an over-simplification of reality. It might be possible for two ambulances to be called out simultaneously after a bad road accident (not satisfying condition 1). Moreover, the rate of call-outs is likely to vary between different times of day (not satisfying condition 3). However a homogeneous Poisson is an approximation that might still be able to give reasonable insight.

The model 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 \(\lambda(t)\).

Single events

There are some scenarios in which only one such event may happen. For example,

The time until the event happens is also random, and can be modelled as the time to the first event in a homogeneous Poisson process.