Working Vacation Policy for Load Sharing K-out-of-N: G System

  • Sudeep Kumar Department of Mathematics, AIAS, Amity University Uttar Pradesh, Noida, India
  • Ritu Gupta Department of Mathematics, AIAS, Amity University Uttar Pradesh, Noida, India
Keywords: Reliability, load sharing, working vacation, repairable and K-out-of-N: G system


In industrial systems, the K-out-of-N: G system is a prominent type of redundancy. The load sharing protects such system from malfunctioning/destroying and avoids overload problem that affects the system reliability in a significant manner. In this paper we develop a Markovian model of load-sharing K-out-of-N: G system having non-identical repairable components wherein the server may on working vacation. During his vacation period, the server repairs the failed components with different service rates rather than completely terminating service rate. The failed component gets immediately repaired by the server if not on vacation, and unequal load is distributed among remaining surviving components. The lifetime of each component is load dependent followed by non-identical exponential distribution with different failure rates. The system is failed down due to common cause with failure density which is also exponentially distributed. We suggest closed structure analytic expressions for reliability, cost estimation and other performance measures of the load-sharing K-out-of-N: G repairable system by incorporating the concept of working vacation. For the solution aspiration, Runge-Kutta method is utilized to solve the system of differential equations. Furthermore, we perform the numerical analysis for two illustrations 1-out-of-3: G system and 3-out-of-4: G system. The numerical simulation is carried out for the validation of analytical results which are exhibited and compared by giving numerical outcomes and neuro-fuzzy outcomes based on fuzzy interference system with the help of MATLAB.


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Author Biographies

Sudeep Kumar, Department of Mathematics, AIAS, Amity University Uttar Pradesh, Noida, India

Sudeep Kumar has received his MSc degree in 2017. He is presently conducting research leading to PhD Degree in the Department of Mathematics, Amity Institute of Applied Science, Amity University, Noida, India. He has attended and presented his work in International and National conferences/seminars. His areas of interest are software and hardware reliability and performance analysis of machine repair problems.

Ritu Gupta, Department of Mathematics, AIAS, Amity University Uttar Pradesh, Noida, India

Ritu Gupta is faculty in Department of Mathematics, Amity University, Noida, India. She received Ph.D. degree from Department of Mathematics, Institute of Basic Science, Dr. B. R. Ambedkar University, Agra, India. She has published several research papers in reputed journals/refereed proceedings. She has attended and presented her work in twenty-five International and National Conferences/Seminars. Her areas of interest are reliability modeling of software/hardware systems, performance modeling of fault tolerant systems, machine repairing systems and queueing problems.


S. Eryilmaz, ‘On reliability analysis of a k-out-of-n system with components having random weights’, Reliab. Eng. Syst. Saf., vol. 109, pp. 41–44, Jan. 2013, doi: 10.1016/j.ress.2012.07.010.

Y. Mo, L. Xing, S. V. Amari, and J. Bechta Dugan, ‘Efficient analysis of multi-state k-out-of-n systems’, Reliab. Eng. Syst. Saf., vol. 133, pp. 95–105, Jan. 2015, doi: 10.1016/j.ress.2014.09.006.

T. Yuge, M. Maruyama, and S. Yanagi, ‘Reliability of a k-out-of-n System with Common-cause Failures Using Multivariate Exponential Distribution’, Procedia Comput. Sci., vol. 96, pp. 968–976, Jan. 2016, doi: 10.1016/j.procs.2016.08.101.

R. A. Rahmani, M. Izadi, and B. E. Khaledi, ‘Importance of components in k-out-of-n system with components having random weights’, J. Comput. Appl. Math., vol. 296, pp. 1–9, Apr. 2016, doi: 10.1016/

J. E. Byun, H. M. Noh, and J. Song, ‘Reliability growth analysis of k-out-of-N systems using matrix-based system reliability method’, Reliab. Eng. Syst. Saf., vol. 165, pp. 410–421, Sep. 2017, doi: 10.1016/j.ress.2017.05.001.

K. O. Kim, ‘Optimal number of components in a load-sharing system for maximizing reliability’, J. Korean Stat. Soc., vol. 47, no. 1, pp. 32–40, Mar. 2018, doi: 10.1016/j.jkss.2017.08.003.

E. Nezakati and M. Razmkhah, ‘Reliability analysis of a load sharing k-out-of-n:F degradation system with dependent competing failures’, Reliab. Eng. Syst. Saf., vol. 203, p. 107076, Nov. 2020, doi: 10.1016/j.ress.2020.107076.

A. Dembinska, N. I. Nikolov, and E. Stoimenova, ‘Reliability properties of k-out-of-n systems with one cold standby unit’, J. Comput. Appl. Math., vol. 388, p. 113289, May 2021, doi: 10.1016/

K. Jenab and B. S. Dhillon, ‘Assessment of reversible multi-state k-out-of-n:G/F/Load-Sharing systems with flow-graph models’, Reliab. Eng. Syst. Saf., vol. 91, no. 7, pp. 765–771, Jul. 2006, doi: 10.1016/j.ress.2005.07.003.

T. Yinghui and Z. Jing, ‘New model for load-sharing k-out-of-n: G system with different components’, J. Syst. Eng. Electron., vol. 19, no. 4, pp. 748–842, Aug. 2008, doi: 10.1016/S1004-4132(08)60148-6.

W. Yamamoto, L. Jin, and K. Suzuki, ‘Optimal allocations for load-sharing k-out-of-n:F systems’, J. Stat. Plan. Inference, vol. 139, no. 5, pp. 1777–1781, May 2009, doi: 10.1016/j.jspi.2008.05.029.

J. V. Deshpande, I. Dewan, and U. V. Naik-Nimbalkar, ‘A family of distributions to model load sharing systems’, J. Stat. Plan. Inference, vol. 140, no. 6, pp. 1441–1451, Jun. 2010, doi: 10.1016/j.jspi.2009.12.005.

B. Singh and P. K. Gupta, ‘Load-sharing system model and its application to the real data set’, Math. Comput. Simul., vol. 82, no. 9, pp. 1615–1629, May 2012, doi: 10.1016/j.matcom.2012.02.010.

H. Z. Abou-Ziyan and A. F. Alajmi, ‘Effect of load-sharing operation strategy on the aggregate performance of existed multiple-chiller systems’, Appl. Energy, vol. 135, pp. 329–338, Dec. 2014, doi: 10.1016/j.apenergy.2014.06.065.

N. Zhang, M. Fouladirad, and A. Barros, ‘Maintenance analysis of a two-component load-sharing system’, Reliab. Eng. Syst. Saf., vol. 167, pp. 67–74, Nov. 2017, doi: 10.1016/j.ress.2017.05.027.

J. Zhang, Y. Zhao, and X. Ma, ‘Reliability modeling methods for load-sharing k-out-of-n system subject to discrete external load’, Reliab. Eng. Syst. Saf., vol. 193, p. 106603, Jan. 2020, doi: 10.1016/j.ress.2019.106603.

Y. Zhang, ‘Reliability Analysis of Randomly Weighted k-out-of-n Systems with Heterogeneous Components’, Reliab. Eng. Syst. Saf., vol. 205, p. 107184, Jan. 2021, doi: 10.1016/j.ress.2020.107184.

R. Gupta and Z. Tasneem, ‘Load sharing repairable system with imperfect coverage in fuzzy environment’, Int. J. Qual. Reliab. Manag., vol. 39, no. 7, pp. 1725–1744, Feb. 2022, doi: 10.1108/IJQRM-08-2021-0298.

C. H. Lin and J. C. Ke, ‘Multi-server system with single working vacation’, Appl. Math. Model., vol. 33, no. 7, pp. 2967–2977, Jul. 2009, doi: 10.1016/j.apm.2008.10.006.

L. Yuan and Z. D. Cui, ‘Reliability analysis for the consecutive-k-out-of-n: F system with repairmen taking multiple vacations’, Appl. Math. Model., vol. 37, no. 7, pp. 4685–4697, Apr. 2013, doi: 10.1016/j.apm.2012.09.008.

W. Wu, Y. Tang, M. Yu, and Y. Jiang, ‘Reliability analysis of a k-out-of-n:G repairable system with single vacation’, Appl. Math. Model., vol. 38, no. 24, pp. 6075–6097, Dec. 2014, doi: 10.1016/j.apm.2014.05.020.

M. Habib, F. Yalaoui, H. Chehade, N. Chebbo, and I. Jarkass, ‘Design optimization and redundant dependency study of series k-out-of-n: G repairable systems’, IFAC-Papers On Line, Jan. 2016, vol. 49, no. 28, pp. 126–131, doi: 10.1016/j.ifacol.2016.11.022.

H. Kim, ‘Optimal reliability design of a system with k-out-of-n subsystems considering redundancy strategies’, Reliab. Eng. Syst. Saf., vol. 167, pp. 572–582, Nov. 2017, doi: 10.1016/j.ress.2017.07.004.

Y. Zhang, W. Wu, and Y. Tang, ‘Analysis of an k-out-of-n:G system with repairman’s single vacation and shut off rule’, Oper. Res. Perspect., vol. 4, pp. 29–38, 2017, doi: 10.1016/j.orp.2017.02.002.

D. Y. Yang and C. L. Tsao, ‘Reliability and availability analysis of standby systems with working vacations and retrial of failed components’, Reliab. Eng. Syst. Saf., vol. 182, pp. 46–55, Feb. 2019, doi: 10.1016/j.ress.2018.09.020.

C. Shekhar, A. Kumar, and S. Varshney, ‘Load sharing redundant repairable systems with switching and reboot delay’, Reliab. Eng. Syst. Saf., vol. 193, p. 106656, Jan. 2020, doi: 10.1016/j.ress.2019.106656.

C. Wu, R. Pan, X. Zhao, and S. Cao, ‘Reliability evaluation of consecutive-k-out-of-n: F systems with two performance sharing groups’, Comput. Ind. Eng., vol. 153, p. 107092, Mar. 2021, doi: 10.1016/j.cie.2020.107092.

M. Jain and R. Gupta, ‘Load sharing M-out of-N: G system with non-identical components subject to common cause failure’, Int. J. Math. Oper. Res., vol. 4, no. 5, pp. 586–605, 2012, doi: 10.1504/IJMOR.2012.048932.