Calculation of reliability function and remaining useful life for a Markov failure time process

Reliability analysts are interested in calculating a reliability function (RF), e.g. in order to establish an optimal replacement policy. To implement this policy, it is often important to include measured condition information, such as those from oil or vibration analysis. Information from condition monitoring can be included in reliability analysis by considering the hazard rate function as a function of a stochastic covariate process. In this paper, the failure process along with the covariate process is represented by a discrete Markov process. Methods are designed for calculating the conditional and unconditional RFs and for computing the remaining useful life (RUL) as a function of the current conditions. It is shown that a function that appears in the computation can be obtained as a solution to a Kolmogorov-type system of differential equations. The product-integration method is suggested as the main general method for numerical calculation. The same method is also used to calculate the RUL. Illustration of the main concepts is given using field data from a transmission's oil analysis histories.