Appendix A: Generating Random Numbers from a Distribution

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

The Weibull reliability function is given by:

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

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


The Weibull conditional reliability function is given by:

[math]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]t\,\![/math] for [math]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.


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  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.