Balancing reliability and maintenance cost rate of multi-state components with fault interval omission

In practice many components and systems exhibit more than two output performances, these systems are called multi-state systems (MSSs) [6, 9, 10, 22]. Since the mid-1970s, numerous researches have been conducted which focus on MSS reliability [2]. Four commonly used approaches about MSS reliability have been formed gradually: the extension of Boolean model [26], stochastic process theory [18], universal generating function (UGF) technology [13,14] and MonteCarlo simulation [23]. As to the stochastic process theory used in reliability analysis of MSSs, when the numbers of failures between arbitrary time intervals can be described as a Poisson process, Markov processes are often introduced to solve these questions [16, 20, 21]. When the operating time and repair time are non-exponentially distributed, a SemiMarkov process is often considered [7]. Besides Markov processes and Semi-Markov processes, the Wiener process [15, 19], the Gamma process [27] and the cumulative exposure process [10] are also considered in MSS reliability modelling. Research about MSS reliability has been a highlight topic in recent years and many new achievements are constantly emerging [17]. Studying on Markov repairable systems has always been an active branch in reliability theory. Jinhua Cao [11] studied the general model of Markov repairable systems, concluded the reliability analysis steps and deduced reliability indexes of voting systems, cold standby systems and warm standard systems. Cui et al. [5] proposed the definition of aggregated stochastic processes and applied into reliability analysis of repairable systems. Lisnianski [17] constructed a Markov reward model for reliability assessment of a multi-state system with variable demand. In his study, the process was assumed to be a homogenous Continuous Time Markov Chain (CTMC) with different possible states and corresponding transition possibility intensities. Other studies of reliability of multi-state systems using stochastic processes can be found in [25] and [29]. For a Markov repairable component containing N different output states, i.e., whose output performance is