Abstract
The multi-state consecutive linear (circular) [Formula: see text]-out-of-[Formula: see text]-from-[Formula: see text] system consists of [Formula: see text] linear (circular) ordered multi-state components. Both the system and its components can have [Formula: see text] different states: from complete failure (zero state) up to perfect functioning (([Formula: see text]1) state). In this paper we suggest, for the first time, exact reliability for these models. The system is at state below [Formula: see text] if and only if at least [Formula: see text] components out of any [Formula: see text] consecutive are in state below [Formula: see text] [Formula: see text]. Recent efforts in these branches have focused on simple situations or approximation bands for their reliability in two-state or multi-state models but closed form and exact amount not gained. In the continuation, there are the matlab programs of linear (circular) reliability system and [Formula: see text] state probability for [Formula: see text] in system. In the following, we applied comparative and numerical results and calculated the exact reliability of this strategic systems. Finally, we calculated the exact reliability for two real-world practical examples.
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