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Novel representations for a coherent threshold reliability system: a tale of eight signal flow graphs

A threshold system is a reliability system whose success/failure is a threshold switching function in the successes/failures of its components. A coherent system (CS) is one that is causal, monotone, and with relevant components. The coherent threshold system (CTS), typically called the weighted k-out-of-n system, is consequently described by strictly positive weights and threshold. This paper presents recursive relations as well as boundary conditions for eight entities pertaining to a CTS. These are (a) expressions of monoform literals as well as disjoint or probability- ready expressions for either system success or failure, and (b) all-additive formulas as well as inclusion-exclusion ones for either system reliability or unreliability. These entities are obtained according to the best policy of implementing the Boole{Shannon expansion with respect to a higher-weight component before it is made with respect to a lower-weight one. With this best policy, the success and failure expressions with monoform literals are both minimal and shellable. Each of the eight entities considered is represented by an acyclic (loopless) signal ow graph (SFG). The SFG for system success or failure is isomorphic to a reduced ordered binary decision diagram, which is the optimal data structure for a Boolean function. The interrelations between the SFGs demonstrate optimal procedures for implementing (a) the probability (real) transform of a Boolean function, (b) inversion or complementation of a Boolean function, and (c) disjointing or orthogonalization of a sum-of-products expression of a Boolean function. The SFGs discussed herein for a CT can be extended to a general coherent system. They reduce to elegant symmetric regular graphs for the special case of a partially redundant system (k-out-of-n system).