, Volume 56, Issue 2, pp 409–431 | Cite as

Performance analysis and control F-policy for fault-tolerant system with working vacation

  • Madhu Jain
  • Chandra Shekhar
  • Rakesh Kumar MeenaEmail author
Theoretical Article


This investigation presents a Markov model for the performance analysis of the fault tolerant machining system with failure-prone server and supported by warm standbys. To utilize the server’s idle time, provision of server’s working vacation has been done which make the system cost effective. The online and warm standby machines may fail and can be repaired by a single skilled repairman. Due to capacity constraint, when the system reaches its full capacity, no more jobs for repairing of failed machines are allowed until the workload of repair jobs reduces to a threshold level ‘F’. Before initiating the repair of the failed machines in case of coming back from the vacation state, the server requires the setup time. To make system fault tolerable, apart from standby provisioning and repairing of failed machines, the concepts of reboot and recovery are included for the formulation of Markov model. The various performance measures including the reliability indices are derived by using the transient probabilities which are computed using Runge–Kutta method. By taking a suitable numerical illustration, various system indices are examined with respect to different parameters. The computational tractability and sensitivity analysis carried out for the established metrics will provides valuable insights for the maintainability and up-gradation of the existing machining systems.


Fault tolerant system Machine repair Working vacation Unreliable server F-policy Imperfect coverage Reliability 



The authors would like to thank the editorial board and anonymous referees for the valuable constructive comments and suggestions on an earlier version of this paper. The author (CS) extends his sincere thanks to funding agency DST FIST for a financial grant to the department having number SR/FST/MSI-090/2013(C).


  1. 1.
    Sivazlian, B.D., Wang, K.H.: Economic analysis of the M/M/R machine repair problem with warm standby. Microelectron. Reliab. 29, 25–35 (1989)CrossRefGoogle Scholar
  2. 2.
    Wang, K.H., Sivazlian, B.D.: Cost analysis of the M/M/R machine repair problem with spares operating under variable service rates. Microelectron. Reliab. 32, 1171–1183 (1992)CrossRefGoogle Scholar
  3. 3.
    Jain, M.: Reliability prediction of repairable redundant system with imperfect switching and repair. Arab. J. Sci. Eng. 41, 3717–3725 (2016)CrossRefGoogle Scholar
  4. 4.
    Jain, M., Meena, R.K.: Fault tolerant system with imperfect coverage, reboot and server vacation. J. Ind. Eng. Int. 13, 171–180 (2016)CrossRefGoogle Scholar
  5. 5.
    Doshi, B.T.: Queueing systems with vacations—a survey. Queueing. Syst. 1, 29–66 (1986)CrossRefGoogle Scholar
  6. 6.
    Gupta, S.M.: Machine interference problem with warm spares, server vacations and exhaustive service. Perform. Eval. 29, 195–211 (1997)CrossRefGoogle Scholar
  7. 7.
    Ke, J.C., Wu, C.H.: Multi-server machine repair model with standbys and synchronous multiple vacation. Comput. Ind. Eng. 62, 296–305 (2012)CrossRefGoogle Scholar
  8. 8.
    Wang, K.H., Liou, C.D., Wang, Y.L.: Profit optimization of the multiple-vacation machine repair problem using particle swarm optimization. Int. J. Syst. Sci. 45, 1769–1780 (2014)CrossRefGoogle Scholar
  9. 9.
    Servi, L.D., Finn, S.G.: M/M/1 queues with working vacations (M/M/1/WV). Perform. Eval. 50, 41–52 (2002)CrossRefGoogle Scholar
  10. 10.
    Wang, K.H., Chen, W.L., Yang, D.Y.: Optimal management of the machine repair problem with working vacation: Newton’s method. J. Comput. Appl. Math. 233, 449–458 (2009)CrossRefGoogle Scholar
  11. 11.
    Jain, M., Upadhyaya, S.: Synchronous working vacation policy for finite-buffer multi server queueing system. Appl. Math. Comput. 217, 9916–9932 (2011)Google Scholar
  12. 12.
    Liu, B., Cui, L., Wen, Y., Shen, J.: A cold standby repairable system with working vacation and vacation interruption following Markovian arrival process. Reliab. Eng. Syst. Saf. 142, 1–8 (2015)CrossRefGoogle Scholar
  13. 13.
    Wartenhrost, P.: N parallel queueing system with server breakdown and repair. Eur. J. Oper. Res. 82, 302–322 (1995)CrossRefGoogle Scholar
  14. 14.
    Wang, K.H.: Optimal operation of a Markovian queueing system with a removable and non-reliable server. Microelectron. Reliab. 35, 1131–1136 (1995)CrossRefGoogle Scholar
  15. 15.
    Hassan, N.A., Ibrahim, H.: Analysis of multi-level queueing systems with server breakdown by using recursive solution technique. Appl. Math. Model. 37, 3714–3724 (2013)CrossRefGoogle Scholar
  16. 16.
    Kuo, C.C., Ke, J.C.: Comparative analysis of standby systems with unreliable server and switching failure. Reliab. Eng. Saf. 145, 74–82 (2016)CrossRefGoogle Scholar
  17. 17.
    Gupta, S.M.: Interrelationship between controlling arrival and service in queueing systems. Comput. Oper. Res. 22, 1005–1014 (1995)CrossRefGoogle Scholar
  18. 18.
    Wang, K.H., Yang, D.Y.: Controlling arrival for a queueing system with an unreliable server: Newton-Quasi method. Appl. Math. Comput. 213, 92–101 (2009)Google Scholar
  19. 19.
    Yang, D.Y., Wang, K.H., Wu, C.H.: Optimization and sensitivity analysis of controlling arrivals in the queueing system with single working vacation. J. Comput. Appl. Math. 234, 545–556 (2010)CrossRefGoogle Scholar
  20. 20.
    Huang, H.I., Hsu, P.C., Ke, J.C.: Controlling arrival and service of a two removable-server system using genetic algorithm. Expert Syst. Appl. 38, 10054–10059 (2011)CrossRefGoogle Scholar
  21. 21.
    Jain, M., Shekhar, C., Shukla, S.: Machine repair problem with an unreliable server and controlled arrival of failed machines. OPSEARCH 51, 416–433 (2015)CrossRefGoogle Scholar
  22. 22.
    Jain, M., Bhagat, A.: Transient analysis of finite F-policy retrial queues with delayed repair and threshold recovery. Natl. Acad. Sci. Lett. 38, 257–261 (2015)CrossRefGoogle Scholar
  23. 23.
    Trivedi, K.S.: Probability and Statistics with Reliability, Queueing and Computer Science Applications, 2nd edn. Wiley, New York (2002)Google Scholar
  24. 24.
    Wang, K.H., Chiu, L.W.: Cost benefits analysis of availability system with warm standby units and imperfect coverage. Appl. Math. Comput. 172, 1239–1256 (2006)Google Scholar
  25. 25.
    Wang, K.H., Yen, T.C., Jian, J.J.: Reliability and sensitivity analysis of a repairable system with imperfect coverage under service pressure condition. J. Manuf. Syst. 32, 357–363 (2013)CrossRefGoogle Scholar
  26. 26.
    Jain, M., Gupta, R.: Optimal replacement policy for a repairable system with multiple vacations and imperfect fault coverage. Comput. Ind. Eng. 66, 710–719 (2013)CrossRefGoogle Scholar
  27. 27.
    Jain, M., Shekhar, C., Rani, V.: N-policy for multi-component machining system with imperfect coverage, reboot and unreliable server. Prod. Manuf. Res. 2, 457–476 (2014)Google Scholar
  28. 28.
    Ke, J.C., Liu, T.H.: A repairable system with imperfect coverage and reboot. Appl. Math. Comput. 246, 148–158 (2014)Google Scholar
  29. 29.
    Jain, M., Shekhar, C., Meena, R.K.: Admission control policy of maintenance for unreliable server machining system with working vacation. Arab. J. Sci. Eng. 42, 2993–3005 (2017)CrossRefGoogle Scholar
  30. 30.
    Shekhar, C., Jain, M., Iqbal, J., Raina, A.A.: Threshold control policy for maintainability of manufacturing system with unreliable workstations. Arab. J. Sci. Eng. 42, 4833–4851 (2017)CrossRefGoogle Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  1. 1.Department of MathematicsIIT RoorkeeRoorkeeIndia
  2. 2.Department of MathematicsBirla Institute of Technology and Science, Pilani CampusPilaniIndia
  3. 3.Department of MathematicsHansraj College University of DelhiDelhiIndia

Personalised recommendations