International Journal of Mathematical, Engineering and Management Sciences

ISSN: 2455-7749

Signature of A-Within-B-From-D/G Sliding Window System

Akshay Kumar
Department of Applied Sciences, Tula's Institute, The Engineering and Management College, Uttarakhand, India.

S. B. Singh
Department of Mathematics, Statistics & Computer Science, G. B. Pant University of Agriculture & Technology, Pantnagar, Uttarakhand, India.


Received on June 14, 2018
Accepted on July 24, 2018


In this study, we have proposed a model of the sliding window coherent system in case of multiple failures. The considered model consists of G linearly required multi-state elements and G number of parallel elements in A-within-B-from-D/G for each multi-state. The system fails if at least A group elements out of B consecutive of D consecutive multi-state elements have performance lower than the weight w. We have evaluated the signature reliability, expected value and system sensitivity on the basis of the extended universal generating function of the considered system.

Keywords- Sliding window system, Universal generating function, Signature reliability, Sensitivity.


Kumar, A., & Singh, S. B. (2019). Signature of A-Within-B-From-D/G Sliding Window System. International Journal of Mathematical, Engineering and Management Sciences, 4(1), 95-107.

Conflict of Interest

The authors confirm that this article contents have no conflict of interest.


The authors would like to express her sincere thanks to the referees and for their valuable suggestions towards to the improvement of the paper.


Barlow, R. E., & Proschan, F. (1975). Importance of system components and fault tree events. Stochastic Processes and their Applications, 3(2), 153-173.

Bhattacharya, D., & Samaniego, F. J. (2008). On the optimal allocation of elements within coherent systems. Statistics & Probability Letters, 78(7), 938-943.

Boland, P. J. (2001). Signatures of indirect majority systems. Journal of Applied Probability, 38(2), 597-603.

Da Costa Bueno, V. (2013). A multistate monotone system signature. Statistics & Probability Letters, 83(11), 2583-2591.

Da, G., Zheng, B., & Hu, T. (2012). On computing signatures of coherent systems. Journal of Multivariate Analysis, 103(1), 142-150.

Eryılmaz, S. (2010). Mixture representations for the reliability of consecutive-k systems. Mathematical and Computer Modelling, 51(5), 405-412.

Eryilmaz, S. (2012). The number of failed elements in a coherent system with exchangeable elements. IEEE Transactions on Reliability, 61(1), 203-207.

Faghih-Roohi, S., Xie, M., Ng, K. M., & Yam, R. C. (2014). Dynamic availability assessment and optimal element design of multi-state weighted k-out-of-n systems. Reliability Engineering & System Safety, 123(2014), 57-62.

Franko, C., & Tütüncü, G. Y. (2016). Signature based reliability analysis of repairable weighted k-out-of-n: G systems. IEEE Transactions on Reliability, 65(2), 843-850.

Habib, A., Al-Seedy, R. O., & Radwan, T. (2007). Reliability evaluation of multi-state consecutive k-out-of-r-from-n: G system. Applied Mathematical Modelling, 31(11), 2412-2423.

Kumar, A. & Singh, S. B. (2017a). Computations of signature reliability of coherent system, International Journal of Quality & Reliability Management, 34(6), 785-797.

Kumar, A., & Singh, S. B. (2017b). Signature reliability of sliding window coherent system. In Mathematics Applied to Engineering, Elsevier, London, pp.83-95.

Levitin, G. (2002). Optimal allocation of elements in a linear multi-state sliding window system. Reliability Engineering & System Safety, 76(3), 245-254.

Levitin, G. (2003). Common supply failures in linear multi-state sliding window systems. Reliability Engineering & System Safety, 82(1), 55-62.

Levitin, G. (2005). The universal generating function in reliability analysis and optimization (p. 442). London: Springer. DOI 10.1007/1-84628-245-4.

Levitin, G., & Ben-Haim, H. (2011). Consecutive sliding window systems. Reliability Engineering & System Safety, 96(10), 1367-1374.

Levitin, G., & Dai, Y. (2012). k-out-of-n sliding window systems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 42(3), 707-714.

Li, Y. F., Ding, Y., & Zio, E. (2014). Random fuzzy extension of the universal generating function approach for the reliability assessment of multi-state systems under aleatory and epistemic uncertainties. IEEE Transactions on Reliability, 63(1), 13-25.

Mahmoudi, M., & Asadi, M. (2011). The dynamic signature of coherent systems. IEEE Transactions on Reliability, 60(4), 817-822.

Marichal, J. L., & Mathonet, P. (2013). Computing system signatures through reliability functions. Statistics & Probability Letters, 83(3), 710-717.

Navarro, J., & Hernandez, P. J. (2008). Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika, 67(3), 277-298.

Navarro, J., & Rubio, R. (2009). Computations of signatures of coherent systems with five elements. Communications in Statistics-Simulation and Computation, 39(1), 68-84.

Navarro, J., Ruiz, J. M., & Sandoval, C. J. (2007). Properties of coherent systems with dependent elements. Communications in Statistics-Theory and Methods, 36(1), 175-191.

Negi, S., & Singh, S. B. (2015). Reliability analysis of non-repairable complex system with weighted subsystems connected in series. Applied Mathematics and Computation, 262, 79-89.

Owen, G. (1975). Multilinear extensions and the Banzhaf value. Naval Research Logistics Quarterly, 22(4), 741-750.

Owen, G. (1988). Multilinear extensions of games. The Shapley Value. Essays in Honor of Lloyd S. Shapley, 139-151.

Peng, R., Xiao, H., & Liu, H. (2017). Reliability of multi-state systems with a performance sharing group of limited size.Reliability Engineering & System Safety, 166, 164-170.

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

Samaniego, F. J. (2007). System signatures and their applications in engineering reliability (Vol. 110). Springer Science & Business Media. Doi.10.1007/978-0-387-71797-5.

Shapley, L. S. (1953). A value for n-person games. In: Contributions to the Theory of Games, Vol. 2. In: Annals of Mathematics Studies, vol. 28. Princeton University Press, Princeton, NJ, 307–317.

Sun, Z. L., Xie, M., Ng, K. M., & Habibullah, M. S. (2012). A study of lifetime optimization of transportation system. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 42(4), 1013-1019.

Xiang, Y., & Levitin, G. (2012a). Combined m-consecutive and k-out-of-n sliding window systems. European Journal of Operational Research, 219(1), 105-113.

Xiang, Y., & Levitin, G. (2012b). Linear-gap sliding window systems. IEEE Transactions on Reliability, 61(2), 560-568.

Xiang, Y., Levitin, G., & Dai, Y. (2013). Optimal allocation of multistate elements in consecutive sliding window systems. IEEE Transactions on Reliability, 62(1), 267-275.

Xiao, H., Peng, R., & Levitin, G. (2015). Optimal replacement and allocation of multi‐state elements in k‐within‐m‐from‐r/n sliding window systems. Applied Stochastic Models in Business and Industry, 32(2), 184-198.

Xiao, H., Peng, R., Wang, W., & Zhao, F. (2014). Linear m-gap-consecutive k-out-of-r-from-n system with common supply failures. International Conference on Reliability, Maintainability and Safety (ICRMS), 2014 (pp. 348-353).

Yu, H., Yang, J., & Mo, H. (2014). Reliability analysis of repairable multi-state system with common bus performance sharing. Reliability Engineering & System Safety, 132, 90-96.

Yueqin, W., Yun, F., & Qihua, W. (2010). A universal generating function approach for reliability analysis of multi-state systems. 2010 Second WRI Global Congress on Intelligent Systems; Wuhan, China.los Alamitos; IEEE Computer Society; 2010.Vol. 3, pp. 207-210.

Navarro, J., & Rychlik, T. (2010). Comparisons and bounds for expected lifetimes of reliability systems. European Journal of Operational Research, 207(1), 309-317.

Privacy Policy| Terms & Conditions