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Journal of Applied Sciences

Year: 2015 | Volume: 15 | Issue: 3 | Page No.: 431-443
Research Article

Reliability of Coherent Threshold Systems

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia
Alaa Mohammad Alturki
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia
A Threshold System (TS) 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 both monotone and with relevant components and hence its success function is expressible without any complemented literals. The Coherent Threshold System (CTS) is consequently described by strictly positive weights and threshold. It is a useful model for many decision or supply systems and being a natural generalization of the k-out-of-n system, it is typically called the weighted k-out-of-n system. This study lists fundamental properties of the CTS and presents two novel methods of deriving its weights and threshold. The first method is called the unit-gap method and proceeds by writing a set of 2n linear inequalities and then reducing this set utilizing symmetry and the elimination of dominated inequalities. The reduced set is then solved subject to the unit-gap restriction. The second method is called the fair-power method since it insists that the system weights be representative of component importance or voting power. This is achieved by making the weight of each component proportional to its Banzhaf index which is the weight of the Boolean derivative or difference of the system success with respect to the component success. The study further presents the recursive relations governing the success of the CTS, transforms these relations to the probability domain and then utilizes them together with appropriate boundary conditions to derive a recursive algorithm for computing the reliability of the CTS. The algorithm is given two pictorial interpretations in term of signal flow graphs and probability maps. An illustrative example demonstrates the implementation of the algorithm and the optimal order of the components to be followed during the algorithm implementation. The study is concluded with a general discussion of its findings compared to those of previously-published studies and an overview of potential future work.
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