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

Abstract

Abstract: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 probabilityready 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 flow 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).

Reliability Evaluation of Rooftop Solar Photovoltaics Using Coherent Threshold Systems

“…The system success of the solar PV system under consideration is investigated via two techniques to verify that it is indeed a threshold switching function, and consequently to determine a specific (non-unique) set of values for the weights and threshold of this function. The techniques used are the unit-gap method and the fair-power method, adapted from Rushdi and Alturki [7]. These two techniques seem quite suitable among the plethora of existing methods for the synthesis of threshold Boolean functions.…”

“…Once the threshold nature of the proposed system success is established and characterized, recursive relations and a recursion-based algorithm are forwarded for the reliability analysis of the corresponding CTS model [6][7][8].…”

“…2is analyzed by the K-map to get its prime-implicants and its prime implicates (complements of the prime implicants of the function's complement). We can find the prime implicants and prime implicates using properties of coherent functions, where a primeimplicant loop extends to the farthest cell from the all-true (all-1) cell and a prime-implicate loop extends to the farthest cell from the all-false (all-0) cells [7,11]. The prime implicants and the prime implicates of the system success are shown in Fig.…”

“…This paper investigates the reliability of a rooftop solar PV system using the coherent threshold system (CTS) model. This model has been widely discussed and applied in several scientific fields [6][7][8][9]. The CTS model is also widely known in the open literature by the controversial name of a weighted k-out-of-n:G system [7], since it might be viewed as a non-symmetric extension of the partially-redundant system.…”

“…This model has been widely discussed and applied in several scientific fields [6][7][8][9]. The CTS model is also widely known in the open literature by the controversial name of a weighted k-out-of-n:G system [7], since it might be viewed as a non-symmetric extension of the partially-redundant system. The coherent threshold system is a coherent reliability system, whose success is a causal monotonicallyincreasing threshold switching function, for which no component is dummy or irrelevant [6][7][8][9][10].A threshold switching function of n variables is characterized by (n + 1) real numbers: n weightsW and a threshold T. The success of the system is satisfied if the total sum of the products W X of a component successX by its weight W is equal to or larger than the threshold, and otherwise the system fails.…”