A continuously monitored system is considered, that gradually and stochastically deteriorates according to a bivariate non decreasing Lévy process. The system is considered as failed as soon as its bivariate deterioration level enters a failure zone, assumed to be an upper set. A preventive maintenance policy is proposed, which involves a delayed replacement, triggered by the reaching of some preventive zone for the system deterioration level. The preventive maintenance policy is assessed through a cost function on an in…nite horizon time.

A two-component system is considered, which is subject to accumulative deterioration. Due to common stress, the components are dependent. Their joint deterioration is modelled with a bivariate non decreasing Lévy process. The deterioration level of both components is known only through perfect and periodic inspections. By an inspection, components with deterioration level beyond a speci…c threshold are instantaneously replaced by new ones (corrective or preventive replacements). Otherwise, they are left as they are. Between inspections, failures remain unrevealed. This replacement policy is classical in a univariate setting, with deterioration modelled by a Gamma process. In the bivariate case, it leads to imperfect repairs at the system level, which highly complicates the study. The replacement policy is assessed through cost functions on both …nite and in…nite horizons, which take into account some economical dependence between components. Markov renewal theory is used to study the behaviour of the system, in a continuous and bivariate setting. Numerical experiments illustrate the study, considering a speci…c Lévy process with univariate Gamma processes as margins. Though technical details are not provided here for the numerical computations, the paper shows that there is a technical gap between the traditional one-dimensional studies and the present two-dimensional one, especially for the computation of the asymptotic distribution of the underlying Markov chain. Hence there is a need for further development in the bivariate (or multivariate) setting.

A system is considered, which is subject to external and possibly fatal shocks, with dependence between the fatality of a shock and the system age. Apart from these shocks, the system suffers from competing soft and sudden failures, where soft failures refer to the reaching of a given threshold for the degradation level, and sudden failures to accidental failures, characterized by a failure rate. A non-fatal shock increases both degradation level and failure rate of a random amount, with possible dependence between the two increments. The system reliability is calculated by four different methods. Conditions under which the system lifetime is New Better than Used are proposed. The influence of various parameters of the shocks environment on the system lifetime is studied.