
RF #97RM102: RF1Aerospace Applications of Weibulland Monte Carlo Simulationwith Importance SamplingSalvatore J. Bavuso; NASA; HamptonKey Words: Weibull, Reliability, Monte Carlo, simulation, importance sampling, HARP (Hybrid Automated Reliability Prediction), Fault tree, HIRel (HARP integrated Reliability tool system)Abstract/Summary Recent developments in reliability modelingand computer technology have made it practicalto use the Weibull time to failure distribution tomodel the system reliability of complex faulttolerant computerbased systems. These systemmodels are becoming increasingly popular inspace systems applications as a result ofmounting data that support the decreasingWeibull failure distribution and the expectationof increased system reliability. This presentationintroduces the new reliability modelingdevelopments and demonstrates their applicationto a novel space system application. Theapplication is a proposed guidance, navigation,and control (GN&C) system for use in a longduration manned spacecraft for a possible Marsmission. Comparisons to the constant failure ratemodel are presented and the ramifications ofdoing so are discussed. The combination of modeling spacecraftsystems with the Weibull time to failuredistribution and the modeling of cold or warmspares to reflect reduced power usage presentsthe reliability modeler with a very difficultmathematical model to evaluate. Such reliabilitymodels are nonMarkovian because the modelrequires multiple clocks. One keeps track ofcomponent failures whose clock starts at theinitiation of the mission and the others start whencold or warm Weibull spares are fully powered,i.e., when switched on. The generalmathematical model which describes thesesystems is given by the ChapmanKolmogorovequations. Presently, the most general solutionmethodology for such models is the Monte Carlosimulationmethod. Although very powerful and general,the Monte Carlo simulation method has not beenused for highly reliable or long duration systemsbecause of the enormous computer resourcesrequired to evaluate these models. Two recentdevelopments have mitigated this shortcoming:The most obvious is the availability of fast andinexpensive microcomputer systems whosecomputational speed is ever increasing andwhose cost is increasing less proportionally. Thesecond development which is less known is theprobabilistic modeling technique calledimportance sampling. Although this techniquehas been available for at least two decades, itsuse has been limited. Importance sampling is avariance reductiontechnique. Thistechniqueallows the efficient sampling of failure eventsthat have very long mean times to failure andreduces the spread (variance) of predicted systemfailure events; thus giving greater confidence forthe predicted mean value (system reliability).The increased sampling efficiency reducescomputational time and cost and increases theprecision of the computedresults. Importance sampling requires the user tospecify certain biasing values. In the generalapplication of importance sampling, the bestassignment of these biasing values is difficult toderive, and the success of the technique hingescritically on the correct choice. This problemhas been an important factor in limiting the useof importance sampling. The Monte Carlointegrated HybridAutomated ReliabilityProgram (MCIHARP) has sidestepped thisdifficulty by requiring a specific system model.The system model is always a Markov chainwithout repair. TheMarkov chain can more