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.

DOI https://doi.org/10.33889/IJMEMS.2021.6.1.020

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

Abstract

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.

Citation

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. https://doi.org/10.33889/IJMEMS.2021.6.1.020.

Conflict of Interest

The authors assert that no conflict of interest exists.

Acknowledgements

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.

References

Ansell, J.I., & Bendell, A. (1987). On alternative definitions of multistate coherent systems. Optimization, 18(1), 119-136.

Barlow, R.E., & Wu, A.S. (1978). Coherent systems with multi-state components. Mathematics of Operations Research, 3(4), 275-281.

Boedigheimer, R.A., & Kapur, K.C. (1994). Customer-driven reliability models for multistate coherent systems. IEEE Transactions on Reliability, 43(1), 46-50.

Ding, Y., Zio, E., Yanfu, L., 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.

El-Neweihi, E., Proschan, F., & Sethuraman, J. (1978). Multistate coherent systems. Journal of Applied Probability, 15(4), 675-688.

Eryilmaz, S. (2015). Capacity loss and residual capacity in weighted k-out-of-n: G systems. Reliability Engineering & System Safety, 136, 140-144.

Eryilmaz, S., & Bozbulut, A.R. (2019). Reliability analysis of weighted-k-out-of-n system consisting of three-state components. Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability, 233(6), 972-977.

Griffith, W.S. (1980). Multistate reliability models. Journal of Applied Probability, 17(3), 735-744.

Huang, J., & Zuo, M.J. (2004). Dominant multi-state systems. IEEE transactions on Reliability, 53(3), 362-368.

Hudson, J.C., & Kapur, K.C. (1983). Reliability analysis for multistate systems with multistate components. AIIE Transactions, 15(2), 127-135.

Janan, X. (1985). On multistate system analysis. IEEE Transactions on Reliability, 34(4), 329-337.

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.

Kumar, A., & Singh, S.B. (2018). Signature reliability of linear multi-state sliding window system. International Journal of Quality & Reliability Management, 35(10), 2403-2413.

Li, X., You, Y., & Fang, R. (2016). On weighted k-out-of-n systems with statistically dependent component lifetimes. Probability in the Engineering and Informational Sciences, 30(4), 533-546.

Lisnianski, A., & Levitin, G. (2003). Multi-state system reliability: assessment, optimization and applications (Vol. 6). World Scientific Publishing Company, Singapore.

Meenkashi, K., Singh, S.B., & Kumar, A. (2019). Reliability analysis of multi-state complex system with multi-state weighted subsystems. International Journal of Quality & Reliability Management, 36(4), 552-568.

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.

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

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

Rushdi, A.M.A. & Al-Amoudi, M.A. (2019), Reliability analysis of a multi-state system using multi-valued logic. IOSR Journal of Electronics and Communication Engineering, 14(1), 1-10.

Rushdi, A.M. (1987). A switching-algebraic analysis of consecutive-k-out-of-n: F systems. Microelectronics and Reliability, 27(1), 171-174.

Rushdi, A.M. (1988). A switching-algebraic analysis of circular consecutive-k-out-of-n: F systems. Reliability Engineering & System Safety, 21(2), 119-127.

Rushdi, A.M. (1990). Threshold systems and their reliability. Microelectronics and Reliability, 30(2), 299-312.

Rushdi, A.M. (1997). Karnaugh map, Encyclopedia of Mathematics, Supplement Volume I: pp. 327-328, In: Hazewinkel, M. (ed) Kluwer Academic publishers, Boston, USA.

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

Rushdi, A.M., & Abdulghani, A.A. (1993). A comparison between reliability analyses based primarily on disjointness or statistical independence: The case of the generalized INDRA network. Microelectronics and Reliability, 33(7), 965-978.

Rushdi, A.M., & Alturki, A.M. (2018). 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., & Goda, A.S. (1985). Symbolic reliability analysis via Shannon's expansion and statistical independence. Microelectronics and Reliability, 25(6), 1041-1053.

Rushdi, A.M., & Rushdi, M.A. (2017). Switching-algebraic analysis of system reliability. In: Ram, M. & Davim, J.P. (eds) Advances in Reliability and System Engineering. Springer, Cham, pp. 139-161.

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. (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 Science, 4(2), 306-326.

Rushdi, A.M.A., & Al-Amoudi, M.A. (2018). Switching-algebraic analysis of multi-state system reliability. Journal of Engineering Research and Reports, 3(3), 1-22.

Rushdi, A.M.A., & Alsayegh, A.B. (2019). Reliability analysis of a commodity-supply multi-state system using the map method. Journal of Advances in Mathematics and Computer Science, 31(2), 1-17.

Rushdi, A.M.A., & Alturki, A.M. (2015). Reliability of coherent threshold systems. Journal of Applied Sciences, 15(3), 431-443.

Rushdi, A.M.A., & Bjaili, H.A. (2016). An ROBDD algorithm for the reliability of double-threshold systems. British Journal of Mathematics and Computer Science, 19(6), 1-17.

Rushdi, A.M.A., & Hassan, A.K. (2015). Reliability of migration between habitat patches with heterogeneous ecological corridors. Ecological Modelling, 304, 1-10.

Rushdi, A.M.A., & Hassan, A.K. (2016). An exposition of system reliability analysis with an ecological perspective. Ecological Indicators, 63, 282-295.

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.

Salehi, M., Shishebor, Z., & Asadi, M. (2019). On the reliability modeling of weighted k-out-of-n systems with randomly chosen components. Metrika, 82(5), 589-605.

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.

Wood, A.P. (1985). Multistate block diagrams and fault trees. IEEE Transactions on Reliability, 34(3), 236-240.

Zuo, M.J., Huang, J., & Kuo, W. (2003). Multi-state k-out-of-n systems. In Handbook of Reliability Engineering (pp. 3-17). Springer, London.

Privacy Policy| Terms & Conditions