Example 1.1.1. Deterministic Models A deterministic model is one in which the output is uniquely determined by the initial conditions and parameter values, without any element of randomness.

(i)

Classical Mechanics: The displacement \(s\) of an object under constant acceleration \(a\) after time \(t\) is given by \(s = ut + \frac {1}{2}at^2\).

(ii)

Geometry: The volume \(V\) of a sphere is a direct function of its radius \(r\), defined by \(V = \frac {4}{3}\pi r^3\).

(iii)

Ohm’s Law: The voltage \(V\) across a conductor is the product of the current \(I\) and the resistance \(R\), such that \(V = IR\).

Example 1.1.2 (Non-deterministic (Stochastic) Models). These models account for inherent randomness or uncertainty; even with identical initial conditions, the observed outcomes may vary.

(i)

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

(ii)

Reliability Engineering: The operating lifespan of a semiconductor before failure under standard thermal conditions.

(iii)

Finance and Insurance: The daily fluctuations of a stock market index or the number of claims filed with an insurer in a fiscal quarter.

(iv)

Public Health: The transmission rate and number of new infections of a pathogen within a specific population over a six-month period.