Static mathematical model with example

A static mathematical model is a set of algebraic relationships between variables that describes the system at a given instant or at equilibrium. Time does not appear in the equations, so the output depends only on the present value of the inputs. It has no memory and needs no initial conditions.

General form

y = f(x1, x2, ..., xn), where y is the output and the x are inputs and parameters.

Characteristics

• Algebraic equations (linear or nonlinear).
• Gives steady-state or one-time behaviour.
• Easy to solve and can be evaluated directly or by solving simultaneous equations.
• Cannot predict transient behaviour or how the system evolves.

Examples

1. Ohm's law: V = I R. If R = 20 ohm and I = 0.5 A, V = 10 V. No time term appears.
2. Break-even model: profit P = (s - v) x - F. With selling price s = Rs 50, variable cost v = Rs 30, fixed cost F = Rs 10,000, break-even quantity x = F/(s - v) = 10,000/20 = 500 units.
3. Economic order quantity: Q = sqrt(2 D S / H). For D = 1,000 units/year, S = Rs 100 per order, H = Rs 5 per unit per year, Q = sqrt(40,000) = 200 units.
4. Static queue result (steady state): M/M/1 utilization rho = lambda/mu; with lambda = 4 and mu = 6 per hour, rho = 0.667.

Uses and limits

Static models are used for design, sizing and what-if analysis when the time pattern is unimportant. When the time behaviour matters (response after a disturbance, growth, oscillation) a dynamic model is needed. A static model can be seen as a dynamic model in its steady state, where all derivatives are zero.