Algorithmic modelling of a complex redundant multi-state system subject to multiple events, preventive maintenance, loss of units and a multiple vacation policy through a MMAP

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.