An Efficient Formula for Generalized Multi-State $k$ -Out-of- $n$ : G System Reliability

The generalized multi-state-k-out-of-n: G system [named GMS(k,n,G)] model was suggested by Huang et al. This system contains n components and is very useful for description of the practical systems so that the components, as well as the system, take the states: 0, 1, 2, ..., H. The GMS(k,n,G) is in state ≥ j, if k
_l
components or more are in state ≥ l, where l is an integer and j ≤ l ≤ H. This system has many practical applications; however, the existing methods of evaluation system-reliability with unequal components probabilities are suitable only for some special cases. In this paper, we will suggest efficient formulas to evaluate the exact reliability of GMS(k,n,G) with equal and unequal components probabilities. These formulas are based on the conditional probability and are suitable for all system types: increasing, constant, decreasing, and non-monotone k values system. Also, we will give the theoretical background, computer codes, and various numerical examples for the suggested formulas.