Cumulative operating time distributions for a class of non-Markovian series systems

Consider a series system with separately and independently maintained components. The failure and repair process of each component is assumed to be an alternating renewal process. It is supposed that, for each component, the failure time distribution is exponential whereas the repair time distribution is general. We derive the Laplace–Stieltjes transform of the distribution of Uc(T), the cumulative time that the series system is operating in [c,c + T] for arbitrary c, T ≥ 0, with respect to T, and obtain the asymptotic distribution of Uc(T) as c, T → ∞.