Conditional hitting time and state-dependent measures in semi-Markov models

The first hitting time to a set of certain events has been considered by the assumption of having some information about the status of the semi-Markov processes (SMPs) at given time t. The distribution function and expected value for the purposed hitting time have been separately formulated based on the conditional distribution of the excess life. In similar circumstances, the conditional probability of standing in a specific set, along with the corresponding mean sojourn time has also been formulated for additional leading time s, separately. The technique of prediction for SMPs and Laplace inverse techniques are the main tools. The steady-state behavior of the results as s→∞ has also been examined. By using theoretical results, the state-dependent version of several widely used reliability metrics has been introduced. It is pointed out that the results of Limnios can be derived from these measurements when t equals to zero, however, the techniques of derivations are completely different. Finally, as a typical example, a degraded system with preventive maintenance has been considered as a semi-Markov (SM) model. To clarify and investigate the validity of the results, a graphical analysis of the state-dependent reliability (SDR) measures of the system has been performed.