Application of Splitting to Failure Estimation in Controllable Degradation System

We consider a regenerative degradation process composed by a sum of the successive phases, where preventive repair is used to prevent an instantaneous failure. For an optimal control of such a systems, calculation of the failure probability, the average length of the regeneration cycle with or without failure, etc., is critically important. If the degradation process is Markovian, then the required steady-state performance measures are analytically available, however in is not the case if the process is non-Markov, in which case simulation is used to estimate the unknown parameters of the system. In this work, the regenerative structure of the degradation process is used to calculate the mentioned above steady-state parameters. Moreover, provided the failure within a regeneration cycle is a rare event, we apply a regenerative variant of the splitting method to estimate the failure probability. It is shown that this approach is much less time-consuming in comparison with crude Monte Carlo simulation. The efficiency of the approach is demonstrated by a detailed analysis of the degradation process generated by the i.i.d. exponential phases. The explicit analytical results for this case are then compared with the corresponding simulation results obtained by crude Monte Carlo and splitting method.