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 maintenance place is managed by a repairperson, who follows a multiple vacation policy dependent on the operational status of the units. The repairperson is responsible for two primary tasks: corrective repair and preventive maintenance. The time durations within the system follow phase-type distributions, and the model is constructed utilizing Markovian Arrival Processes with marked arrivals. A variety of performance measures, including transient and stationary distributions, are calculated using matrix-analytic methods. This methodology allows for the representation of significant outcomes and the general behavior of the system in a matrix-algorithmic structure. To enhance the model's efficiency, both costs and rewards are incorporated into the analysis. A numerical example is presented to showcase the model's flexibility and effectiveness in real-world applications.