Reliability Evaluation for Multi-State Markov Repairable Systems with Redundant Dependencies

In multi-state systems with load-sharing, the load will be redistributed among working units while some units go to failure. Hence dependencies exit among units. To model the evolution of these systems, a multi-state Markov repairable system with redundant dependencies is introduced in this paper. More than two states are allowed for each unit of the system, including perfect working, deterioration and complete failure. The state transition rate of each working unit is a function of the number of functioning units and is quantified by a redundant dependence function. A two-dimensional vector, whose elements denote respectively the number of units that are in perfect and degraded operating states, is presented to describe accurately the performance of the system. The state space of the system is divided into distinct state sets according to the value of the vector. The visit probability to a specified state set (the acceptable, excellent, operating and warning state sets), the time to the first system failure and the steady-state distribution of sojourn time in the acceptable state set are discussed. Markov theory and aggregated stochastic process theory are used to get these reliability indices. A numerical example is presented to illustrate the results obtained in the paper. The impact of redundant dependencies in the system is also considered.