Example 1.1.1 (Deterministic models). In a deterministic model the output is fixed completely by the inputs, with no element of chance.

(i)

Mechanics. An object with initial speed \(u\) and constant acceleration \(a\) has travelled \(s = ut + \frac 12at^2\) after time \(t\).

(ii)

Geometry. A sphere of radius \(r\) has volume \(V = \frac 43\pi r^3\).

(iii)

Ohm’s law. The voltage across a conductor is \(V = IR\), the product of the current and the resistance.

Example 1.1.2 (Non-deterministic (stochastic) models). In a stochastic model the same conditions can lead to different outcomes.

(i)

Telecommunications. The number of data packets arriving at a server during a peak hour.

(ii)

Reliability. The lifetime of a semiconductor before it fails.

(iii)

Finance and insurance. The daily change in a stock index, or the number of claims made to an insurer in a quarter.

(iv)

Public health. The number of new infections in a population over six months.