Research On The Strategy Of Maintenance Replacement Of Cumulative Shock Model

Shock model, which is one of the important contents in the mathematical theory of reliability, are used for describing systems that are subject to shocks of random magnitudes in a stochastic environment. The study of the lifetime distribution of a shock model is the essential issues in the shock model theory. At the same time, the research of the optimal maintenance/replacement policy of shock model when the system is repairable is also an important problem to be discussed. Shock models mainly include extreme shock model, cumulative shock model and others. In this paper we investigate the preventive repair and replacement problem for a class of cumulative shock model.The existing studies of maintenance problem for the cumulative shock model are not sufficient. First, it is always assumed that the shocks to a system can be detected completely. However, in practice, some shocks may not be detected due to some uncontrollable factors or some random reasons. That is, the shocks to a system is incompletely observed. Secondly, many researches assume that the external shock is the only reason of the system failure. However, most systems deteriorate continuously with time, and may fail to work when the deterioration of the system is serious. It is reasonable to combine the information of system deterioration and the damage caused by shocks together to construct mathematical models for such systems. Finally, the shocks to system are assumed to yield homogenous Poisson process with intensity ? in shock models. Non-homogenous Poisson process shock stream is more general in applications.In this paper, we consider the following maintenance problems for shock models. First, we assume that the shocks come according to the homogeneous Poisson process, and each shock the system subjected is detected with probability p, and is undetected with probability 1-p. The preventive maintenance replacement problem of the system is discussed. Secondly, the linear deterioration mechanism is introduced to the cumulative shock model with homogeneous Poisson process shock resource. It is also supposed that the shock is incompletely observed. The optimal preventive maintenance and replacement policy of the system is discussed in this case. Finally, we extend the maintenance and replacement problem to the two-unit system with the shock comes according to a non-homogeneous Poisson process. In this model, we assume that preventively/failure maintenance is executed at some time to the main unit, while the other unit, which is easy to fail, is minimal repaired at failure.For the models suggested above, the average length of a renewal cycle of the systems are calculated, and the total costs of the failure maintenance and the preventive maintenance of the system are calculated, respectively. Then the long-run average cost rate of the systems are derived. The existence of the optimal preventive and replacement policies, which minimizes the total expected cost rate, are proved. The optimal policies are determined theoretically. Numerical example are provided to verify the effectiveness of the theoretical results obtained.