Reliabilities For -Out-Of- Systems

Abstract The ) , ( g f -out-of) ), , (( n j i :F ) ( g f > system consists of n components ordered in a line or a circle, while the system fails if, and only if, there are at least f failed components in the system or at least g failed components among components j j i i , 1 , , 1 , − + L (Here the component index has module n property in the circular case, i.e., components i and n+i indicate the same one, g i j ≥ − , if i j > ; g i j n ≥ + − + 1 , if i j < ). In this paper, we present the system reliability formulae with the forms of recursive equations and product of matrices by means of a probability argument and a two-stage finite Markov chain imbedding approach, respectively, for the ) , ( g f -out-ofF n j i : ) ), , (( systems.