Choosing optimal service frequency for a system with outage delay (Q1842455)

We consider a system with time to failure distribution function \(F(t)\). The form of the distribution function (d.f.) \(F(t)\) is unknown, and we only have its mean \(s_ 1\) and the second moment \(s_ 2\), i.e., \(F\in K\), where \(K\) is the class of d.f.s of nonnegative random variables with given moments \(s_ i= \int^ \infty_ 0 x^ i dF(x)\), \(i= 1,2\). We assume that failure is detected instantaneously. After failure, the system is repaired in a time with d.f. \(G_ 1(t)\) with mean \(m_ 1\). Preventive maintenance is scheduled a certain time \(T\) after the system begins operating. The preventive maintenance is carried out if the system did not fail before the time \(T\). The preventive maintenance time is distributed according to the d.f. \(G_ 2(t)\) with mean \(m_ 2\). The system is characterized by the following specific feature. When failure occurs or when preventive maintenance starts, the system continues performing its normal functions for a certain short time \(x\) (the so-called outage delay). The mean idle time (after the expiry of the outage delay, assuming that the system could not be repaired during this short time) in systems with preventive maintenance is less than in systems where repair is undertaken only after failure. For such a system we determine the optimal service frequency.