Reliability analysis of a system with gradually refilled time reserve (Q1407137)

Main reliability characteristics are derived for a system described by a semi-Markov model with discrete-continuous phase space of system states and gradually refilled time reserve. It means a system may spend some additional time (i.e. time reserve) during operation to restore its characteristics. The system includes the object presented by one structural element and the gradually replenished time reserve. The nonfailure operation time of the object is a random variable (RV) \(\alpha\) with given distribution function (DF), and the recovery time is a RV \(\beta\) with given DF. The time reserve equals \(\tau\) for \(t=0\). It has the rate \(1/\nu\) during the object recovery/operation. The growth of time reserve is limited by \(\tau\) (the volume of accumulator). The appereance of system failure starts when the time reserve is completely exhausted and lasts up to the object recovery. Suppose the RVs \(\alpha, \beta\) are independent with given densities and finite mathematical expectations. The system operation is described using the semi-Markov kernel of the process of Markov restoration in the differential form.