# CONTINUOUS-IN-TIME VS. DISCRETE-IN-TIME MODELS - Quantitative Techniques for management

Mathematical models of time dependent processes can be split into two categories depending on how the time variable is to be treated. A continuous-in-time mathematical model is based on a set of equations that are valid for any value of the time variable. A discrete-in-time mathematical model is designed to provide information about the state of the physical system only at a selected set of distinct times.

The solution of a continuous-in-time mathematical model provides information about the physical phenomenon at every time value. The solution of a discrete-in-time mathematical model provides information about the physical system at only a finite number of time values. Continuous-in-time models have two advantages over discrete-in-time models:

1. they provide information at all times and
2. they more clearly show the qualitative effects that can be expected when a parameter or an input variable is changed.

On the other hand, discrete in time models have two advantages over continuous in time models:

1. they are less demanding with respect to skill level in algebra, trigonometry, calculus,
differential equations, etc. and
2. they are better suited for implementation on a computer.

Some Examples of Mathematical Models

Problem

Rotating all or part of a space station can create artificial gravity in the station. The resulting centrifugal force will be indistinguishable from gravitational force. Develop a mathematical model that will determine the rotational rate of the station as a function of the radius of the station (distance from the center of rotation) and the desired artificial gravitational force. Use this model to answer the question: What rotational rate is needed if the radius of the station is 150 m and Earth surface gravity is desired.

Problem

A stretch of Interstate 25 is being widened to accommodate increasing traffic going north and south. Unfortunately, the Department of Transportation is going to have to bring out the orange barrels and close all but one lane at the “big I” intersection. The department would like to have traffic move along as quickly as possible without additional accidents. What speed limit would provide for maximum, but safe, traffic flow?