Balanced reliability systems under Markov processes

Research on reliability for balanced systems has become an important issue in the reliability field; it is being intensively studied both in theory and applications. The concept of balance has many implications, which results in various balanced systems needing to be studied in reliability modelling, analysis and optimization and so on. In the present article, two balanced reliability systems are introduced and their reliability models are developed, then formulas for system reliability, probability density functions, and moments of lifetimes under two models are obtained using the results in aggregated stochastic processes or phase-type distributions for two-dimensional and high-dimensional situations. In addition, a matrix operator, the Kronecker product operator, is used to create concise forms for the related formulas. An algorithm for computing these reliability indexes is given to facilitate use. Some examples including symbolic and numerical ones for the simplest case, the two-dimensional situation, are shown to illustrate the presented results. The presented work may shed light on future research on reliability and safety for balanced systems under different concepts of balance.