Imperfect Delayed Repair Model Under Geometric-Alpha Process

Abstract

Planning for the maintenance of deteriorating systems is more difficult when repair operations cannot always be started immediately after failure due to operational and resource limitations. In this study, a stochastic maintenance model is presented for a deteriorating system whose successive operating times are modeled by a geometric–alpha process, where the deterioration of the system is modeled as a structured change in the failure behaviour in successive cycles. System failure is followed by imperfect delayed repair, meaning that repair is either immediately started or delayed, and repair action brings the system back to a state that is not as-good-as-new. It is assumed that an N-failure replacement policy is used, and the system is substituted with a new one when the N-th failure occurs. With the help of renewal reward arguments, an explicit expression of the long-term expected cost per unit time is obtained and the optimum replacement number $N^*$ is obtained analytically. To demonstrate the use of the proposed model and to explore the impact of the model parameters on the optimal maintenance decision, a numerical study is presented. The proposed Geometric–Alpha Process is further illustrated using the Aircondit7 dataset, a benchmark reliability dataset containing operating times between failures of an aircraft air-conditioning system. For this dataset, the Geometric–Alpha Process yields a higher maximized log-likelihood and a lower Akaike Information Criterion (AIC) than the classical Geometric Process, indicating a better fit to the data. The analytical results, numerical study, and data illustration demonstrate the applicability of the proposed maintenance framework for deteriorating systems with imperfect delayed repair.