Performance Evaluation of Aggregated Markov repairable Systems with Multi-Operating Levels

Many multi-state systems settle on more than one operating levels and the elementary states having a common operating level can be treated as one. In this paper, one kind of such Markov repairable systems is introduced, which is named aggregated Markov repairable systems with multi-operating levels. In the system, the functioning states are lumped together according to their membership as a common operating level and each operating level is characterized by a performance rate. The systems degrade from a higher operating level to a lower one when no preventive maintenances (PMs) are performed. While the system is in some deteriorated operating levels that are easy to be recognized PMs are carried out. The PMs may be perfect or imperfect. If the system fails, the repairs may restore it to any of its operating levels. A multivariate semi-Markov process is build to describe performance properties of the system. Several reliability indices such as the availability, frequencies of repairs and failures are presented. Furthermore, the up time, the length of a cycle, the sojourn times in various operating levels and visiting numbers to them, the times that the system satisfies demands of customers and the output during a cycle are studied. Semi-Markov process theory, Laplace transform and matrix methods are employed in the study. A numerical example is given to illustrate the results in the paper. The impact of PM on the system is considered through the numerical illustration.