Introducing a g-renewal component in system reliability analysis

Several models have been developed for imperfect repairs that assume that component is "better than old but worse than new" after repair. One of the most popular is the g-renewal process introduced by Kijima & Sumita [1]. They established an effectiveness parameter q, defining a virtual age of the system component at a given time after several repairs. Unfortunately, there is no closed form solution of corresponding g-renewal equation except special cases. Different approximate methods have been developed for other cases, however most of them are devoted to calculating the Expected Number of Failures (Renewal Function) and are not efficient enough to apply them in system reliability analysis. The main input parameters for system components (for example represented as basic events in a Fault Tree) are Failure Frequency (Failure Intensity) and Unavailability. Unavailability and Frequency of Failures calculation is only a preprocessing part of system reliability analysis. It should be done very fast, because the system (Fault Tree) can contain hundreds of components (basic events) [2, 3]. We suggest an efficient algorithm based on representation of g-renewal process as a continuous time semi-Markov Chain, whose state space is defined as a set of states of a component between i-th and (i+1)-th failures (i=0, 1, 2, ...). We applied the Monte Carlo method, which is a generalization of the approach [4], to the Markov Chain for calculation of Failure Frequency and Unavailability of a system component. The accuracy of this approach increased dramatically compared to the raw Monte Carlo simulation. This allowed implementing the algorithm in Fault Tree analysis software.