International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Utilization of Symmetric Switching Functions in the Symbolic Reliability Analysis of Multi-State k-out-of-n 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.


Received on January 01, 2019
Accepted on January 24, 2019


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.


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.

Conflict of Interest

The author declares that no competing interests exist.


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.


Amari, S. V., Xing, L., Shrestha, A., Akers, J., & Trivedi, K. S. (2010). Performability analysis of multistate computing systems using multivalued decision diagrams. IEEE Transactions on Computers, 59(10), 1419-1433.

Arnold, R. F., & Harrison, M. A. (1963). Algebraic properties of symmetric and partially symmetric Boolean functions. IEEE Transactions on Electronic Computers, EC-12(3), 244-251.

Born, R. C., & Scidmore, A. K. (1968). Transformation of switching functions to completely symmetric switching functions. IEEE Transactions on Computers, C-17(6), 596-599.

Brown, F. M. (1990). Boolean Reasoning: The Logic of Boolean Equations, Kluwer Academic Publishers, Boston, MA, USA.

Caldwell, S. H. (1954). The recognition and identification of symmetric switching functions. Transactions of the American Institute of Electrical Engineers, Part I: Communication and Electronics, 73(2), 142-147.

Canteaut, A., & Videau, M. (2005). Symmetric Boolean functions. IEEE Transactions on Information Theory, 51(8), 2791-2811.

Cunkle, C. H. (1963). Symmetric Boolean functions. The American Mathematical Monthly, 70(8), 833-836.

Ding, Y., Zio, E., Li, Y., Cheng, L., & Wu, Q. (2012). Definition of multi-state weighted k-out-of-n: F systems. International Journal of Performability Engineering, 8(2), 217-219.

Fadhel S. F., Alauldin N. A., & Ahmed Y. Y. (2014). Reliability of dynamic multi-state oil supply system by structure function. International Journal of Innovative Research in Science, Engineering and Technology, 3(6), 13548-13555.

Hill, F. J., & Peterson, G. R. (1993). Computer aided logical design with emphasis on VLSI. 4th Edition, Wiley, New York, NY, USA.

Khorshidi, H. A., Gunawan, I., & Ibrahim, M. Y. (2015). On reliability evaluation of multistate weighted k-out-of-n system using present value. The Engineering Economist, 60(1), 22-39.

Kim, B. G., & Dietmeyer, D. L. (1991). Multilevel logic synthesis of symmetric switching functions. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 10(4), 436-446.

Kravets, V. N., & Sakallah, K. A. (2000). Generalized symmetries in Boolean functions. In Proceedings of the 2000 IEEE/ACM International Conference on Computer-Aided Design (pp. 526-532). IEEE Press.

Lee, S. C. (1978). Modern switching theory and digital design, Prentice-Hall, Englewood Cliffs, New Jersey, NJ, USA.

Levitin, G. (2013). Multi-state vector-k-out-of-n systems. IEEE Transactions on Reliability, 62(3), 648-657.

Levitin, G., Lisnianski, A., & Ushakov, I. (2003). Reliability of multi-state systems: a historical overview. In Lindqvist, B. H. & Doksum, K. A. (Editors), World Scientific, Mathematical and Statistical Methods in Reliability (pp. 123-137).

Li, C. Y., Chen, X., Yi, X. S., & Tao, J. Y. (2011). Interval-valued reliability analysis of multi-state systems. IEEE Transactions on Reliability, 60(1), 323-330.

Li, S., Sun, S., Si, S., Zhang, S., & Dui, H. (2014). Decision diagram based methods and reliability analysis for k-out-of-n: G systems. Journal of Mechanical Science and Technology, 28(10), 3917-3923.

Li, Z., & Kapur, K. C. (2011). Models and measures for fuzzy reliability and relationship to multi-state reliability. International Journal of Performability Engineering, 7(3), 241-250.

Marcus, M. P. (1956). The detection and identification of symmetric switching functions with the use of tables of combinations. IRE Transactions on Electronic Computers, EC-5(4), 237-239.

Maurer, P. M. (2015). Symmetric Boolean functions. International Journal of Mathematics, Game Theory, and Algebra, 24(2/3), 159-202.

Mishchenko, A. (2003). Fast computation of symmetries in Boolean functions. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 22(11), 1588-1593.

Mo, Y. (2014). A multiple-valued decision-diagram-based approach to solve dynamic fault trees. IEEE Transactions on Reliability, 63(1), 81–93.

Mo, Y., Xing L., & Amari, S. V. (2014). A multiple-valued decision diagram based method for efficient reliability analysis of non-repairable phased-mission systems. IEEE Transactions on Reliability, 63(1), 320–330.

Mo, Y., Xing, L., Amari, S. V., & Dugan, J. B. (2015). Efficient analysis of multi-state k-out-of-n systems. Reliability Engineering & System Safety, 133, 95-105.

Muroga, S. (1979). Logic design and switching theory. John Wiley & Sons, New York, NY, USA.

Muselli, M. (2005). Approximate multi-state reliability expressions using a new machine learning technique. Reliability Engineering & System Safety, 89(3), 261-270.

Ram, M. (2013). On system reliability approaches: a brief survey. International Journal of System Assurance, Engineering, and Management, 4(2), 101-117.

Ren, Y., Zeng, C., Fan, D., Liu, L., & Feng, Q. (2018). Multi-state reliability assessment method based on the MDD-GO model. IEEE Access, 6, 5151-5161.

Rushdi A. M. A. (2010). Partially-redundant systems: examples, reliability, and life expectancy. International Magazine on Advances in Computer Science and Telecommunications. 1(1), 1-13.

Rushdi M. A. M., Ba-Rukab O. M. & Rushdi A. M. (2016). Multidimensional recursive relation and mathematical induction techniques: The case of failure frequency of k-out-of-n systems. Journal of King Abdulaziz University: Engineering Science, 27(2), 15-31.

Rushdi, A. M. & Rushdi, M. A. (2017). Switching-algebraic analysis of system reliability, Chapter 6 in M. Ram and P. Davim (Editors), Advances in Reliability and System Engineering. Cham, Switzerland: Springer International Publishing, pp. 139-161.

Rushdi, A. M. (1983). How to hand-check a symbolic reliability expression. IEEE Transactions on Reliability, R-32(5), 402-408.

Rushdi, A. M. (1986). Utilization of symmetric switching functions in the computation of k-out-of-n system reliability. Microelectronics and Reliability, 26(5), 973-987.

Rushdi, A. M. (1993). Reliability of k-out-of-n systems, Chapter 5 in Misra, K. B. (Editor), New Trends in System Reliability Evaluation, Vol. 16, Fundamental Studies in Engineering, Elsevier Science Publishers, Amsterdam, The Netherlands, pp. 185-227.

Rushdi, A. M. A. (2018). Utilization of Karnaugh maps in multi-value qualitative comparative analysis. International Journal of Mathematical, Engineering and Management Sciences, 3(1), 28-46.

Rushdi, A. M. A., & Al-Qwasmi, M. A. (2016). Exposition and comparison of two kinds of a posteriori analysis of fault trees. Journal of King Abdulaziz University: Computing and Information Technology, 5(1), 55-74.

Rushdi, A. M. A., & Alturki, A. M. (2017). Computation of k-out-of-n system reliability via reduced ordered binary decision diagrams. British Journal of Mathematics & Computer Science, 22(3), 1-9.

Rushdi, A. M. A., & Alturki, A. M. (2018b). Unification of mathematical concepts and algorithms of k-out-of-n system reliability: A perspective of improved disjoint products. Journal of Engineering Research. 6(4), 1-31.

Rushdi, A. M. A., & Ghaleb, F. A. M. (2014). The Walsh spectrum and the real transform of a switching function: A review with a Karnaugh-map perspective. Journal of Engineering and Computer Sciences, Qassim University, 7, 73-112.

Rushdi, A. M., & Alturki, A. M. (2018a). Novel representations for a coherent threshold reliability system: a tale of eight signal flow graphs. Turkish Journal of Electrical Engineering & Computer Sciences, 26(1), 257-269.

Rushdi, A. M., Zagzoog, S., & Balamesh, A. S. (2019). Derivation of a scalable solution for the problem of factoring an n-bit integer. Journal of Advances in Mathematics and Computer Science, 30(1), 1-22.

Rushdi, R. A., & Rushdi, A. M. (2018). Karnaugh-map utility in medical studies: The case of Fetal Malnutrition. International Journal of Mathematical, Engineering and Management Sciences, 3(3), 220-244.

Shrestha, A., Xing, L., & Dai, Y. (2007). MBDD versus MMDD for multistate systems analysis. In Dependable, Autonomic and Secure Computing, 2007. DASC 2007. Third IEEE International Symposium on (pp. 172-180). IEEE.

Shrestha, A., Xing, L., & Dai, Y. (2010). Decision diagram-based methods and complexity analysis for multi-state systems. IEEE Transactions on Reliability, 59(1), 145-161.

Singh, V. V., & Ram, M. (2014). Multi-state k-out-of-n type system analysis. Mathematics in Engineering Science & Aerospace, 5(3), 281-292.

Song, X., Zhai, Z., Guo, Y., Zhu, P., & Han, J. (2017). Approximate analysis of multi-state weighted k-out-of-n systems applied to transmission lines. Energies, 10(11), 1740, doi:10.3390/en10111740.

Tian, Z., Zuo, M. J., & Yam, R. C. (2008). Multi-state k-out-of-n systems and their performance evaluation. IIE Transactions, 41(1), 32-44.

Unger, S. H. (1989). The essence of logic circuits. Englewood Cliffs, NJ, Prentice Hall, 1989, pp. 317.

Veeraraghavan, M., & Trivedi, K. S. (1994). A combinatorial algorithm for performance and reliability analysis using multistate models. IEEE Transactions on Computers, 43(2), 229-234.

Zang, X., Wang, D., Sun, H., & Trivedi, K. S. (2003). A BDD-based algorithm for analysis of multistate systems with multistate components. IEEE Transactions on Computers, 52(12), 1608-1618.

Zhao, X., & Cui, L. (2010). Reliability evaluation of generalised multi-state k-out-of-n systems based on FMCI approach. International Journal of Systems Science, 41(12), 1437-1443.

Zuo, M. J., Tian, Z., & Huang, H. Z. (2007). An efficient method for reliability evaluation of multistate networks given all minimal path vectors. IIE Transactions, 39(8), 811-817.

Privacy Policy| Terms & Conditions