# 4 Monte Carlo simulation

As seen in Chapter 3, modeling the propagation of variability in a system can be mathematically complex. Monte Carlo simulation is a relatively easy method for predicting the variation and bias in a system. The procedure for performing a Monte Carlo simulation is:

- define the system to be simulated,
- determine the variation behavior of the inputs to the system,
- for each input to the system, select a random number representative of the input's variation,
- compute the system output(s), and
- repeat the steps 3 and 4 until steady state is achieved.

## 4.1 Determining variation of the inputs

The variation of the inputs is determined by selecting a probability density function that adequately represents the behavior of the input and then determining the parameters of the distribution.

There are several options available for determining the variation of the inputs.

- Use the tolerance window and assume a distribution and a process capability index (
*C*_{p}) of 1.33.^{1}Consider a 100-ohm 5% resistor. The tolerance window is 100 ohms ± 5 ohms. A*C*_{pk}of 1.33 is equivalent to ± 4 standard deviations resulting in a standard deviation of 1.25 (5/4) ohms. Since resistance varies in a symmetrical pattern, the normal distribution can be assumed. The result is a normal distribution with a mean of 100 and a standard deviation of 1.25. For characteristics that are not symmetrical, a lognormal or Weibull distribution may be used. For example, roundness is bounded at zero and is ...

Get *Probabilistic Design for Optimization and Robustness for Engineers* now with O’Reilly online learning.

O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers.