# Survival Signature Approach for Reliability Evaluation of Linear Consecutive \(k\)-out-of-\(n\):\(G\) System

## DOI:

https://doi.org/10.26713/cma.v15i1.2439## Keywords:

Reliability, System signature, Survival signature, Linear consecutive k-out-of-n : G system, Hazard rate function## Abstract

The present study concentrates on computing the reliability function for linear consecutive \(k\)-out-of-\(n:G\) system with independent and identically distributed components. For systems where \(2k \ge n\), we establish a formulation for the system survival signature. This is subsequently utilized to find a non-recursive representation of system reliability. The attained closed-form representation of system reliability empowers us to easily evaluate the performance of higher-order consecutive systems. The system signature is also evaluated with the assistance of the survival signature. Additionally, a method for calculating the system hazard rate function in light of its components’ hazard rate functions is suggested. Both exponential and Pareto distributions are considered in assessing the reliability function for such systems. A numerical example related to a quality control system provides a concrete illustration of the results achieved through the proposed method.

### Downloads

## References

R. Bollinger and A. Salvia, Consecutive-k-out-of-n:F networks, IEEE Transactions on Reliability R-31(1) (1982), 53 – 56, DOI: 10.1109/TR.1982.5221227.

J. C. Chang and F. K. Hwang, Reliabilities of consecutive-k systems, in: Handbook of Reliability Engineering, H. Pham (editor), Springer, London (2003), DOI: 10.1007/1-85233-841-5_3.

D. T. Chiang and S.-C. Niu, Reliability of consecutive-k-out-of-n:F system, IEEE Transactions on Reliability R-30(1) (1981), 87 – 89, DOI: 10.1109/TR.1981.5220981.

G. Chopra and D. Kumar, Comparison of bridge systems with multiple types of components, Reliability: Theory and Applications 17 (2022), 282 – 296.

T. Cluzeau, J. Keller and W. Schneeweiss, An efficient algorithm for computing the reliability of consecutive-k-out-of-n:F systems, IEEE Transactions of Reliability 57(1) (2008), 84 – 87, DOI: 10.1109/TR.2008.916879.

F. P. A. Coolen and T. Coolen-Maturi, Generalizing the signature to systems with multiple types of components, in: Complex Systems and Dependability. Advances in Intelligent and Soft Computing, W. Zamojski, J. Mazurkiewicz, J. Sugier, T. Walkowiak and J. Kacprzyk (editors), Vol. 170, Springer, Berlin — Heidelberg, DOI: 10.1007/978-3-642-30662-4_8.

G. Da, M. Xu and P. Chan, An efficient algorithm for computing the signatures of systems with exchangeable components and applications, IISE Transactions 50(7) (2018), 584 – 595, DOI: 10.1080/24725854.2018.1429694.

W. Ding, R. Fang and P. Zhao, An approach to comparing coherent systems with ordered components by using survival signatures, IEEE Transactions on Reliability 70(2) (2020), 495 – 506, DOI: 10.1109/TR.2020.3023827.

S. Eryilmaz, Review of recent advances in reliability of consecutive k-out-of-n and related systems, Proceedings of the Institution of Mechanical Engineers, Part O: Journal of Risk and Reliability 224(3) (2010), 225 – 237, DOI: 10.1243/1748006XJRR332.

S. Eryilmaz, Mixture representations for the reliability of consecutive-k systems, Mathematical and Computer Modelling 51(5-6) (2010), 405 – 412, DOI: 10.1016/j.mcm.2009.12.007.

S. Eryilmaz, F. P. A. Coolen and T. Coolen-Maturi, Marginal and joint reliability importance based on survival signature, Reliability Engineering & System Safety 172 (2018) 118 – 128, DOI: 10.1016/j.ress.2017.12.002.

S. Eryilmaz and A. Tuncel, Generalizing the survival signature to unrepairable homogeneous multi-state systems, Naval Research Logistics 63 (2016), 593 – 599, DOI: 10.1002/nav.21722.

G. Feng, E. Patelli and M. Beer, Reliability analysis of systems based on survival signature, in: Proceedings of the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12), T. Haukaas (editor), Vancouver, Canada, July 12-15 (2015), http://hdl.handle.net/2429/53289.

J. C. Fu and B. Hu, On reliability of a large consecutive-k-out-of-n:F system with (k − 1)-step Markov dependence, IEEE Transactions on Reliability R-36(1) (1987), 75 – 77, DOI: 10.1109/TR.1987.5222299.

J. C. Fu, L. Wang and W. Y. W. Lou, On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials, Journal of Applied Probability 40(2) (2003), 346 – 360, DOI: 10.1239/jap/1053003548.

A. E. Gera, A consecutive k-out-of-n:G system with dependent elements – a matrix formulation and solution, Reliability Engineering & System Safety 68(1) (2000), 61 – 67, DOI: 10.1016/S0951-8320(00)00005-3.

X. Jia, J. Shen, F. Xu, R. Ma and X. Song, Modular decomposition signature for systems with sequential failure effect, Reliability Engineering & System Safety 189 (2019) 435 – 444, DOI: 10.1016/j.ress.2019.05.003.

J. Kontoleon, Reliability determination of a r-successive-out-of-n:F system, IEEE Transactions on Reliability R-29(5) (1980), 437, DOI: 10.1109/TR.1980.5220921.

W. Kuo, W. Zhang and M. Zuo, A consecutive-k-out-of-n:G system: The mirror image of a consecutive-k-out-of-n:F system, IEEE Transactions on Reliability 39(2) (1990), 244 – 253, DOI: 10.1109/24.55888.

W. Kuo and M. J. Zuo, Optimal Reliability Modeling: Principles and Applications, John Wiley & Sons, Inc., New Jersey, xvi + 543 pages (2003).

M. Lambiris and S. Papastavridis, Exact reliability formulas for linear & circular consecutive-k-out-of-n:F systems, IEEE Transactions on Reliability R-34(2) (1985), 124 – 126, DOI: 10.1109/TR.1985.5221969.

Y. Li, F. P. A. Coolen, C. Zhu and J. Tan, Reliability assessment of the hydraulic system of wind turbines based on load-sharing using survival signature, Renewable Energy 153 (2020), 766 – 776, DOI: 10.1016/j.renene.2020.02.017.

M.-S. Lin, An O(k/sup 2//spl middot/log(n)) algorithm for computing the reliability of consecutive-k-out-of-n:F systems, IEEE Transactions on Reliability 53(1) (2004), 3 – 6, DOI: 10.1109/TR.2004.823845.

J. Navarro, Likelihood ratio ordering of order statistics, mixtures and systems, Journal of Statistical Planning and Inference 138(5) (2008), 1242 – 1257, DOI: 10.1016/j.jspi.2007.04.022.

J. Navarro, J. M. Ruiz and C. J. Sandoval, Properties of coherent systems with dependent components, Communications in Statistics – Theory and Methods 36(1) (2007), 175 – 191, DOI: 10.1080/03610920600966316.

J. Navarro and S. Eryilmaz, Mean residual lifetimes of consecutive-k-out-of-n systems, Journal of Applied Probability 44(1) (2007), 82 – 98, DOI: 10.1239/jap/1175267165.

J. Navarro and T. Rychlik, Reliability and expectation bounds for coherent systems with exchangeable components, Journal of Multivariate Analysis 98(1) (2007), 102 – 113, DOI: 10.1016/j.jmva.2005.09.003.

J. Navarro and T. Rychlik, Comparisons and bounds for expected lifetimes of reliability systems, European Journal of Operational Research 207(1) (2010), 309 – 317, DOI: 10.1016/j.ejor.2010.05.001.

J. Navarro, F. J. Samaniego and N. Balakrishnan, The joint signature of coherent systems with shared components, Journal of Applied Probability 47(1) (2010), 235 – 253, DOI: 10.1239/jap/1269610828.

J. Qin and F. P. A. Coolen, Survival signature for reliability evaluation of a multi-state system with multi-state components, Reliability Engineering & System Safety 218(A) (2022), 108129, DOI: 10.1016/j.ress.2021.108129.

F. J. Samaniego, On closure of the IFR class under formation of coherent systems, IEEE Transactions on Reliability R-34(1) (1985), 69 – 72, DOI: 10.1109/TR.1985.5221935.

F. J. Samaniego and J. Navarro, On comparing coherent systems with heterogeneous components, Advances in Applied Probability 48(1) (2016), 88 – 111, DOI: 10.1017/apr.2015.8.

Y. L. Tong, A rearrangement inequality for the longest run, with an application to network reliability, Journal of Applied Probability 22(2) (1985), 386 – 393, DOI: 10.2307/3213781.

I. Triantafyllou, Consecutive-type reliability systems: An overview and some applications, Journal of Quality and Reliability Engineering 2015 (2015), 212303, 20 pages, DOI: 10.1155/2015/212303.

H. Yi, L. Cui and N. Balakrishnan, Computation of survival signatures for multistate consecutive-k systems, Reliability Engineering & System Safety 208 (2021), 107429, DOI: 10.1016/j.ress.2021.107429.

W. Zhang, Theory and Analysis of Consecutive-k-out-of-n:G Systems Reliability, PhD Thesis, Department of Industrial and Manufacturing Systems Engineering, Iowa State University, USA, (1988), DOI: 10.31274/RTD-180813-11863.

M. Zuo, Reliability and component importance of a consecutive-k-out-of-n system, Microelectronics Reliability 33(2) (1993), 243 – 258, DOI: 10.1016/0026-2714(93)90485-H.

## Downloads

## Published

## How to Cite

*Communications in Mathematics and Applications*,

*15*(1), 329–344. https://doi.org/10.26713/cma.v15i1.2439

## Issue

## Section

## License

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.