Stochastic analysis of k-out-of-n: G type of repairable system in combination of subsystems with controllers and multi repair approach

Document Type : Review Paper


1 Department of Mathematics Yusuf Maitama Sule University, Kano, Nigeria

2 Department of General Requirement, Sur College of Applied Sciences, Sur, Oman



This paper describes the investigation of different reliability measures of a complex system consisting of two subsystems with controllers in a series configuration, which is a useful opportunity for specific design problems. Subsystem-1 consisting n units functioning under the policy k-out-of-n: G; policy, and subsystem-2 has m units and operating under r-out-of-m: G; policy. The system failure rates of both subsystems are constant and assumed to obey an exponential distribution; two types of distribution are allowed to repair: general distribution and Gumbel-Hougaard family copula distribution. The system's partially failed states/ completely failed states are repaired using General/ copula distribution. After repair, the units in both the subsystems are "as good as new." The controller control both subsystems and the failure of controllers brings the subsystem in the complete failed state. The operator may fail the system deliberately if not satisfied with the organization. The system is analyzed employing the supplementary variable technique, and Laplace transforms implications and traditional system reliability measures, such as the system's availability, system reliability, and profit analysis, have been computed for particular values of failure and repair parameters.

Graphical Abstract

Stochastic analysis of k-out-of-n: G type of repairable system in combination of subsystems with controllers and multi repair approach


  • Availability of the system has evaluated for two cases Copula repair and General Repair
  • Comparison of both performances for Copula repair and General repair has conclusively elaborated.
  • System reliability of different cases have evaluated, MTTF for system performance is computed.
  • Profit analysis have also computed for the both cases copula repair and general repair. Copula repair is predicted better performance has concluded.


Bai, D.S., Yun, W.Y., & Chung, S.W. (1991). Redundancy optimization of k-out-of-n systems with common-cause failures. IEEE Transactions on Reliability, 40(1), 56-59.
Cox, D.R. (1955). The analysis of non-Markov stochastic processes by the inclusion of supplementary variables. Mathematical Proceedings of the Cambridge Philosophical Society, 51(3), 433-441.
Davoudpour, H. (2019). A hierarchical Bayesian network to compare maintenance strategies based on cost and reliability: a case of onshore wind turbines. International Journal of Industrial Engineering: Theory, Applications, and Practice, 26(3), 1-11.
El-Damcese, M.A., & El-Sodany, N.H. (2014). Availability and Reliability Analysis for the k-out-of-n: G system with Three Failures using Markov Model. International Journal of Scientific & Engineering Research, 5(12), 383-389.
Fawzi, B.B., & Hawkes, A.G. (1991). Availability of an r-out-of-n system with spares and repairs. J. Appl. Probab., 28(2), 397-408.
Galikowsky, C., Sivazlian, B.D., & Chaovalitwongse, P. (1996). Optimal redundancies for reliability and availability of series systems. Microelectron Reliability, 36(10), 1537-1546.
Goel L.R., & Gupta, P. (1984). Stochastic analysis of a two-unit parallel system with partial and catastrophic failures and preventive maintenance. Microelectron Reliability, 24(3), 881-883.
Jieyu, S., & Pecht, M.G. (1992). Reliability of a k-out-of-n Warm-Stand by the system. IEEE Transactions on Reliability, 41(1), 72-75.
Kullstam, P.A. (1981). Availability, MTBF, MTTR for the repairable m-out-of-n system. IEEE Transactions on Reliability, R-30(4), 393-394
Kumar, A., Pant, S., & Singh, S.B. (2017). Availability and cost analysis of an engineering system involving subsystems in a series configuration. International Journal of Quality & Reliability Management, 34(6), 879-894.
Kumar, P., & Gupta, R. (2007). Reliability analysis of a single unit M|G|1 system model with helping unit. J. of Comb. Info. & System Sciences, 32(1-4), 209-219.
Lado, A.K., Singh, V.V., Kabiru, H.I., & Yusuf, I. (2018). Performance and cost assessment of the repairable complex system with two subsystems connected in a series configuration. International Journal of Reliability and Applications, 19(1), 27-42.
Levitin, G. (2001). Incorporating common-cause failures into non-repairable multi-state series-parallel system analysis. IEEE Transactions on Reliability, 50, 380–388.
Liang, X., Xiong, Y., & Li, Z. (2010). Exact reliability formula for consecutive k-out-of-n repairable systems. IEEE Transactions on Reliability, 59(2), 313-318.
Lin, Y.K., Huang, C.F., Lin, J.S., & Li, S.Y. (2019). Evaluation of system reliability for a freeway system with stochastic speed. International Journal of Industrial Engineering, 26(1), 12-33.
Moustafa, M.S. (1996). Availability of k-out-of-n: G systems with M failure modes. Microelectron Reliability, 36, 385-388.
Moustafa, M.S. (1996). Transient analysis of reliability with and without repair for the k-out-of-n system with two failure modes. Reliability Engineering and System Safety, 53, 31-35.
Ogata, K. (2009). Modern control engineering. 5th edition, Prentice Hall Publisher, ISBN-10: 0136156738, ISBN-13: 9780136156734.
Oliveira, E.A., Alvim, A.C.M., & Melo, P.F.F. (2005). Unavailability analysis of safety systems under aging by supplementary variables with the imperfect repair. Annals of Nuclear Energy, 32(2), 241-252.
Raghav, D., Poonia, P.K., Gahlot, M., Singh, V.V., Ayagi, H.I., & Abdullahi, A.H. (2020). Probabilistic analysis of a system consisting of two subsystems in the series configuration under copula repair approach. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math, 27(3), 137-155.
Ram, M., & Singh, S.B. (2008). Availability and Cost Analysis of a parallel redundant complex system with two types of failure under preemptive- resume repair discipline using Gumbel-Hougaard family copula in repair. Int. J. Reliability., Quality Safety Eng., 15(4), 341–365.
Singh, V.V., & Ram, M. (2014). Multi-state k-out-of-n; type system analysis. Mathematics in engineering, science, and aerospace, 5(3), 281-292.
Singh, V.V., Gahlot, M., Ayagi, H.I., & Goel, C.K. (2018). Performance assessment of repairable system in the series configuration under different types of failure and repair policies using copula linguistics. Int. J. Reliability and Safety, 12(4), 348–363.
Singh, V.V., Poonia, P.K., & Abdullahi, A.H. (2020). Performance analysis of a complex repairable system with two subsystems in a series configuration with an imperfect switch.  J. Math. Comput. Sci., 10(2), 359-383.
Singh, V.V., Ram, M., & Rawal, D.K. (2013a). Cost Analysis of an Engineering System involving subsystems in Series Configuration. IEEE Transactions on Automation Science and Engineering, 10, 1124-1130.
Singh, V.V., Singh, S.B., Ram, M., & Goel, C.K. (2013). Availability, MTTF, and cost analysis of a system having two units in a series configuration with the controller. International Journal of System Assurance and Management, 4(4), 341–352.
Yam, R.C.M., Zuo, M.J., & Zhang, Y. (2003). A method for evaluation of reliability indices for repairable circular consecutive-k-out-of-n: F-systems. Reliability Engineering and System Safety, 79(1), 1-9.