## 1 Introduction  
Machining centers are complex systems with several subsystems, so considering opportunistic maintenance when repairing subsystems may avoid frequent downtime during processing and save costs. Accurately evaluating subsystem reliability is essential for rational maintenance strategy development. Some scholars have studied the reliability modeling of machining center subsystems. [PERSON] et al. [1] proposed a probability prediction method for the key subsystems of machining centers. Their research focused on the conditional probability of failure for each key subsystem in the next failure event. However, their study did not consider the time correlation of subsystem failures. Based on the analysis of failure data and the assumption that the failure time follows the Weibull distribution, [PERSON] et al. [2] researched the distribution and reliability functions of each subsystem. The failure data used in their study were large samples. [PERSON] et al. [3] evaluated the reliability of the machining center subsystems and calculated the system MTBF based on the MTBF of the subsystems. In addition to the machining centers, some researchers are concerned about the reliability evaluation of the subsystems. [PERSON] et al. [4] applied the Pagerank algorithm to analyze the failure correlation of CNC lathe subsystems and optimize the impact of each subsystem, establishing a reliability model for CNC lathe subsystems. [PERSON] et al. [5] modeled the CNC grinding machine subsystem by different distributions, and the reliability of the CNC grinding machine was evaluated through Monte Carlo simulation. [PERSON] and [PERSON] [6] analyzed the performance of the subsystems of the SPVC line subsystem using the Markov method under the assumption of exponential distribution. [PERSON] et al. [7] modeled the reliability of repairable subsystems within the framework of competing and complementary risks under the assumption of exponential distribution. [PERSON] [8] evaluated the reliability of dragline subsystems. [PERSON] et al. [9] calculated the subsystem reliability of the Reactor Protection System using the Support Vector Regression (SVR) method. [PERSON] et al. [10] represented Man, Machines, Materials, Management, and Environment (4M1E) as subsystems in complex chemical production processes and similarly employed the SVR method to evaluate the reliability of these aspects. [PERSON] et al. [11] evaluated the performance of the NPK fertilizer production line subsystems through indicators such as reliability, availability, maintainability, and dependability. There is relatively little research on subsystem reliability modeling, especially in machining center subsystems, and there is no reasonable consideration of the time correlation between subsystems. This paper proposes an improved average rank method to model subsystem reliability, which considers the downtime of the studied subsystem caused by other subsystem failures as censored data. Compared to the average rank method in the previous study [12], it considers not only the order of censored data but also their positions [13]. Overall, the improved average rank method takes into account the time correlation between subsystems and is suitable for situations where the failure data of the machining center are a small sample. It can integrate the failure information of the entire machine and improve modeling accuracy. After rational modeling of the subsystems, this article emphasizes how to arrange reasonable maintenance plans to help save costs and improve equipment availability. Opportunistic maintenance serves as an effective strategy by utilizing the downtime of one subsystem to conduct maintenance on other subsystems that are nearly due for maintenance. [PERSON] et al. [14] proposed a risk-based opportunistic maintenance approach for multi-station manufacturing systems that considers the machine performance-workpiece quality co-effect. The feasibility and correctness are verified through a seal manufacturing system. [PERSON] et al.