Representations of a coherent reliability system via signal flow graphs

Abstract EN

-coherent reliability system (CRS) is one that is causal, monotone and with relevant components.

We restrict ourselves herein to the case of a two-state system with statistically independent two-state components.

One of the most prominent methods to study the reliability of such a system is to characterize it via recursive relations together with boundary conditions.

This paper presents recursive relations as well as boundary conditions for six entities pertaining to a CRS.

These are (a) expressions of monoform literals for either system success or failure (b) probability-ready expressions for either system success or failure, and (c) all-additive formulas for either system reliability or unreliability.

Each of the six entities considered is represented by an acyclic (loopless) Mason signal flow graph (SFG).

The SFG for system success or failure is isomorphic to a Reduced Ordered Binary Decision Diagram (ROBDD) 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 reduce to elegant symmetric graphs for the special cases of a partially-redundant system (k-out-of-n system) and a threshold system (weighted k-out-of-n system).

The results obtained suggest a renaissance of the use of signal flow graphs in the study of system reliability for both coherent and noncoherent systems and for particular classes thereof.