International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Reliability Characterization of Binary-Imaged Multi-State 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.

Fares Ahmad Muhammad Ghaleb
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, 21589, Saudi Arabia.


Received on December 07, 2019
Accepted on March 26, 2020


A notable reliability model is the binary threshold system (also called the weighted-k-out-of-n system), which is a dichotomous system that is successful if and only if the weighted sum of its component successes exceeds or equals a particular threshold. The aim of this paper is to extend the utility of this model to the reliability analysis of a homogeneous binary-imaged multi-state coherent threshold system of (m+1) states, which is a non-repairable system with independent non-identical components. The paper characterizes such a system via switching-algebraic expressions of either system success or system failure at each non-zero level. These expressions are given either (a) as minimal sum-of-products formulas, or (b) as probability–ready expressions, which can be immediately converted, on a one-to-one basis, into probabilities or expected values. The various algebraic characterizations can be supplemented by a multitude of map representations, including a single multi-value Karnaugh map (MVKM) (giving a superfluous representation of the system structure function S), (m+1) maps of binary entries and multi-valued inputs representing the binary instances of S, or m maps, again of binary entries and multi-valued inputs, but now representing the success/failure at every non-zero level of the system. We demonstrate how to reduce these latter maps to conventional Karnaugh maps (CKMs) of much smaller sizes. Various characterizations are inter-related, and also related to pertinent concepts such as shellability of threshold systems, and also to characterizations via minimal upper vectors or via maximal lower vectors.

Keywords- System reliability, Probability-ready expression, Threshold system, Multi-state system, Multi-valued Karnaugh map, Minimal upper vector, Maximal lower vector.


Rushdi, A. M. A., & Ghaleb, F. A. M. (2021). Reliability Characterization of Binary-Imaged Multi-State Coherent Threshold Systems. International Journal of Mathematical, Engineering and Management Sciences, 6(1), 309-321.

Conflict of Interest

The authors assert that no conflict of interest exists.


The first-named author (AMR) benefited greatly from (and is sincerely grateful for) his earlier collaboration and enlightening discussions with Engineer Mahmoud Ali Rushdi, Munich, Germany.


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