ISSN: 2455-7749

**
Ali Muhammad Ali Rushdi **
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P. O. Box 80204, Jeddah, 21589, Saudi Arabia.

DOI https://dx.doi.org/10.33889/IJMEMS.2019.4.2-025

Received on January 01, 2019

;
Accepted on January 24, 2019

**Abstract**

Symmetric switching functions (SSFs) play a prominent role in the reliability analysis of a binary k-out-of-n: G system, which is a dichotomous system that is successful if and only if at least k out of its n components are successful. The aim of this paper is to extend the utility of SSFs to the reliability analysis of a multi-state k-out-of-n: G system, which is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of non-repairable multi-state k-out-of-n: G systems with independent non-identical components. The paper utilizes algebraic techniques of multiple-valued logic (together with known properties of SSFs) to evaluate each of the multiple levels of the system output as an individual binary or propositional function of the system multi-valued inputs. The formula of each of these levels is then written as a probability–ready expression, thereby allowing its immediate conversion, on a one-to-one basis, into a probability or expected value. The symbolic reliability analysis of a commodity-supply system (which serves as a standard gold example of a multi-state k-out-of-n: G system) is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.

**Keywords-** System reliability, Probability-ready expression, k-out-of-n system, Multi-state system, Multiple-valued logic, Boolean quotient, Checking symbolic reliability, Variable-entered Karnaugh map.

**Citation**

Rushdi, A. M. A. (2019). Utilization of Symmetric Switching Functions in the Symbolic Reliability Analysis of Multi-State k-out-of-n Systems. *International Journal of Mathematical, Engineering and Management Sciences*, *4*(2), 306-326. https://dx.doi.org/10.33889/IJMEMS.2019.4.2-025.

**Conflict of Interest**

The author declares that no competing interests exist.

**Acknowledgements**

The author benefited from (and is grateful for) his earlier collaboration and enlightening discussions with Engineer Mahmoud Ali Rushdi, Research Scientist at fortiss (Forschungsinstitut des Freistaats Bayern für softwareintensive Systeme und Services (“Research Institute of the Free State of Bavaria for software-intensive Systems and Services”)), Munich, Germany.

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