A Semi-Geometric Process Model for a Deteriorating System and its Optimal Order-Replacement Policy

Abstract

This paper proposes a semi-geometric process (SGP) model to generalize the geometric process (GP) model. The SGP differs from the GP in that the failed system after repair may not degenerate. By assuming that the system is not degenerative at the $n$ th repair with probability $p_{n}$ and is geometrically degenerative with probability $1 - p_{n}$ , we give explicit expressions of the distribution function and the mathematical expectation of the system working times after repair. In addition, an order-replacement model is developed for a deteriorating system based on SGP. The policy $N$ is adopted, that is the system will be replaced after the $N$ th failure and the spare system is ordered at the end of the $(N - 1)$ th repair. The long-run average availability $A(N)$ and cost rate $C(N)$ are derived, and the optimal policy $N^{\ast}$ such that $C(N)$ is minimized is obtained under the premise of meeting the availability requirement. Finally, we give a numerical example to illustrate the model, and carry through some discussions and sensitivity analysis of the model.