Simplification of transition state diagrams via Generalized Perturbation Theory in the Markovian reliability analysis of aging safety systems

This work presents a proposal to simplify state transition diagrams in Markovian reliability analyses for large systems and subsequent calculation of their mean unavailability, considering that at least one component is aging. Two proposals are presented to circumvent the loss of the Markovian property: the method of stages and the method of supplementary variables. The method chosen was the latter because it presents the advantage of a standard solution, requiring only the updating of system equations and initial condition for each new case. Failure rates will be modeled by the two parameter Weibull distribution. The criterion used for simplifying state transition diagrams is the derivative of the integral quantity in non-null elements of the state transition matrix. To do so, these derivatives are obtained by the Generalized Perturbation Theory (GPT), which makes extensive use of the principle of importance conservation. The simplification proposed using the GPT formalism can be considered a good approximation for the original diagram to calculate the mean system unavailability and its relative error is negligible.