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This article considers an one-unit multi-state system with a general  α -series process and an order-replacement policy. It is supposed that the working times of the system after repair become shorter and shorter, while the repair times of the system become longer and longer. Further, an ordering policy  N  − 1 and a replacement policy  N  is adopted. After the  N th repair, the system is replaced by an identical new one, and the unit for replacement is immediately ordered at the end of theth repair. Some important properties of the system are derived by using the probability theory and the stochastic process theory. In addition, the explicit expression for the long-run expected cost function is developed to search the optimal order-replacement policy.

A repairable multi-state system with a general α -series process and an order-replacement policy