Appendix A: Generating Random Numbers from a Distribution

Simulation involves generating random numbers that belong to a specific distribution. We will illustrate this methodology using the Weibull distribution.

Generating Random Times from a Weibull Distribution

The cdf of the 2-parameter Weibull distribution is given by:

[math]\displaystyle{ F(T)=1-{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}\,\! }[/math]

The Weibull reliability function is given by:

[math]\displaystyle{ \begin{align} R(T)= & 1-F(t) \\ = & {{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}} \end{align}\,\! }[/math]

To generate a random time from a Weibull distribution, with a given [math]\displaystyle{ \eta \,\! }[/math] and [math]\displaystyle{ \beta \,\! }[/math], a uniform random number from 0 to 1, [math]\displaystyle{ {{U}_{R}}[0,1]\,\! }[/math] , is first obtained. The random time from a weibull distribution is then obtained from:

[math]\displaystyle{ {{T}_{R}}=\eta \cdot {{\left\{ -\ln \left[ {{U}_{R}}[0,1] \right] \right\}}^{\tfrac{1}{\beta }}}\,\! }[/math]

Conditional

The Weibull conditional reliability function is given by:

[math]\displaystyle{ R(t|T)=\frac{R(T+t)}{R(T)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\! }[/math]

The random time would be the solution for [math]\displaystyle{ t\,\! }[/math] for [math]\displaystyle{ R(t|T)={{U}_{R}}[0,1]\,\! }[/math].

BlockSim's Random Number Generator (RNG)

Internally, ReliaSoft's BlockSim uses an algorithm based on L'Ecuyer's [RefX] random number generator with a post Bays-Durham shuffle. The RNG's period is approximately 10^18. The RNG passes all currently known statistical tests, within the limits the machine's precision, and for a number of calls (simulation runs) less than the period. If no seed is provided the algorithm uses the machines clock to initialize the RNG.

References

1. L'Ecuyer, P., 1988, Communications of the ACM, vol. 31, pp.724-774
2. L'Ecuyer, P., 2001, Proceedings of the 2001 Winter Simulation Conference, pp.95-105
3. William H., Teukolsky, Saul A., Vetterling, William T., Flannery, Brian R., Numerical Recipes in C: The Art of Scientific Computing, Second Edition, Cambridge University Press, 1988.
4. Peters, Edgar E., Fractal Market Analysis: Applying Chaos Theory to Investment & Economics, John Wiley & Sons, 1994.
5. Knuth, Donald E., The Art of Computer Programming: Volume 2 - Seminumerical Algorithms, Third Edition, Addison-Wesley, 1998.