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:

  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 \(\lambda \times \delta t\).

Expressing this 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.

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:

  1. Two or more call-outs are possible in any hour, but multiple events are virtually impossible in a second (and impossible in a short enough time period).
  2. Knowing that there were call-outs in one hour does not affect the probability of call-outs in the next hour (or any other later time period).
  3. A rate of \(\lambda = 1.5\) per hour is equivalent to \(\lambda = \frac{1.5} {60}\) per minute or \(\lambda = \frac{1.5} {3,600}\) per second. Over very small intervals of time, \(\lambda\) can be interpreted as a probability; for example, the probability of a call-out in any second is approximately \(\frac{1.5} {3,600}\).

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.