A complex multi-state redundant system undergoing preventive maintenance and experiencing multiple events is being considered in a continuous time frame. The online unit is susceptible to various types of failures, both internal and external in nature, with multiple degradation levels present, both internally and externally. Random inspections are continuously monitoring these degradation levels, and if they reach a critical state, the unit is directed to a repair facility for preventive maintenance. The repair facility is managed by a single repairperson, who follows a multiple vacation policy dependent on the operational status of the units. The repairperson is responsible for two primary tasks: corrective repairs and preventive maintenance. The time durations within the system follow phase-type distributions, and the model is constructed using Markovian Arrival Processes with marked arrivals. A variety of performance measures, including transient and stationary distributions, are calculated using matrix-analytic methods. This approach enables the expression of key results and overall system behaviour in a matrix-algorithmic format. In order to optimize the model, costs and rewards are integrated into the analysis. A numerical example is presented to showcase the model's flexibility and effectiveness in real-world applications.

翻译：本文在连续时间框架下研究一个经历预防性维护并受多重事件影响的复杂多状态冗余系统。在线单元易受多种类型故障影响（包括内部与外部性质），且存在多级退化现象（涵盖内部与外部退化）。随机检测持续监控这些退化水平，当达到临界状态时，单元将被引导至维修设施进行预防性维护。维修设施由单一维修人员管理，其遵循基于单元运行状态的多重休假策略。维修人员主要负责两项核心任务：纠正性维修和预防性维护。系统内各时段服从相位型分布，模型通过带标记到达的马尔可夫到达过程构建。利用矩阵解析方法计算了包括瞬态分布与稳态分布在内的多种性能指标。该方法使得关键结果与整体系统行为能以矩阵算法形式表达。为优化模型，分析中引入了成本与奖励机制。通过数值算例展示了该模型在实际应用中的灵活性与有效性。