Renewals in Systems with Two Serially Connected Components

In this paper, we study renewals in a system with two components connected in series. Both components can undergo corrective maintenance (i.e., either full replacement/perfect repair or minimal repair) when a failure occurs. When one of the units fails and is correctively maintained, the other one is either preventively replaced (if cost-feasible) or simply left working as is. When a component is minimally repaired or left working as is, its remaining lifetime is reevaluated (“memory effect”) and is taken into consideration in calculating the next renewal. A “coupled” lifetime that combines the lifetimes of the two serially connected components is proposed to represent the joint lifetime of the system. We develop renewal functions based on the coupled lifetimes and show that they follow the classical or generalized renewal theory depending on whether the components work without memory or not. Approximation formulas for the new renewal functions are also obtained and validated by Monte Carlo simulations for various combinations of distributions, and a comparative cost analysis is conducted.