Optimization and Engineering

, Volume 13, Issue 4, pp 545–562 | Cite as

Performance analysis and optimization of a machine repair problem with warm spares and two heterogeneous repairmen

  • Dequan Yue
  • Wuyi Yue
  • Hongjuan Qi


In this paper, we study a machine repair problem with warm spares and two heterogeneous repairmen from the view points of both queueing and reliability. In this system, the first repairman is always available for serving the failed units, while the second repairman leaves for a vacation of random length when the number of failed units is less than N. We obtain expressions for the steady-state probabilities of the system by solving the steady-state probability equations iteratively. Then, we obtain some performance measures for the system. We also obtain some performance measures of reliability for the system such as the steady-state availability, the steady-state failure frequency and the mean time between the system failures. Moreover, we present derivations of the mean time taken until the first failure of the system. A cost model is developed to determine the optimum value N while the system availability is maintained at a certain level. A sensitivity analysis is also performed.


Repairable system Warm spare Vacation Availability Failure frequency Performance optimization 



The authors wish to thank the anonymous referee’s comments leading to a much improved presentation. This work was supported in part by the National Natural Science Foundation of China (No. 70671088), and was supported in part by GRANT-IN-AID FOR SCIENTIFIC RESEARCH (No. 21500086) and MEXT, Japan.


  1. Cao J, Cheng K (2006) Introduction to reliability mathematics. Advanced Education Press, Beijing (in Chinese) Google Scholar
  2. Cheng K (1999) Classes of life distributions and mathematical theory of reliability. Science Press, Beijing (in Chinese) Google Scholar
  3. Doshi B (1986) Single server queues with vacation: a survey. Queueing Syst 1:29–66 MathSciNetMATHCrossRefGoogle Scholar
  4. Doshi B (1990) Single server queues with vacations. In: Takagi H (ed) Stochastic analysis of computer and communication systems. North-Holland, Amsterdam, pp 217–265 Google Scholar
  5. Gupta SM (1997) Machine interference problem with warm spares, server vacations and exhaustive service. Perform Eval 29:195–211 CrossRefGoogle Scholar
  6. Haque L, Armstrong MJ (2007) A survey of the machine interference problem. Eur J Oper Res 179:469–482 MATHCrossRefGoogle Scholar
  7. Hsieh YC, Wang KH (1995) Reliability of a repairable system with spares and a removable repairman. Microelectron Reliab 35:197–208 CrossRefGoogle Scholar
  8. Jain M, Jain A (2010) Working vacations queueing model with multiple types of server breakdowns. Appl Math Model 34:1–13 MathSciNetMATHCrossRefGoogle Scholar
  9. Jain M, Rakhee, Maheshwari S (2004a) N-policy for a machine repair system with spares and reneging. Appl Math Model 28:513–531 CrossRefGoogle Scholar
  10. Jain M, Rakhee, Singh M (2004b) Bilevel control of degraded machining system with warm standbys, setup and vacation. Appl Math Model 28:1015–1026 MATHCrossRefGoogle Scholar
  11. Ke JC (2007) Operating characteristic analysis on M[X]/G/1 system with a variant vacation policy and balking. Appl Math Model 31:1321–1337 MATHCrossRefGoogle Scholar
  12. Ke JC, Wang KH (2007) Vacation policies for machine repair problem with two type spares. Appl Math Model 31:880–894 MATHCrossRefGoogle Scholar
  13. Ke JC, Lin CH, Yang JY, Zhang ZG (2009a) Optimal (d, c) vacation policy for a finite buffer M/M/c queue with unreliable servers and repairs. Appl Math Model 33:3949–3962 MathSciNetMATHCrossRefGoogle Scholar
  14. Ke JC, Liou CH, Lee SL (2009b) Machine repair problem in production systems with spares and server vacations. PAIRO Oper. Res. 43:35–54 MathSciNetMATHCrossRefGoogle Scholar
  15. Wang KH, Ke JC (2003) Probabilistic analysis of a repairable system with warm standbys plus balking and reneging. Appl Math Model 27:327–336 MATHCrossRefGoogle Scholar
  16. Wang KH, Sivazlian BD (1989) Reliability of a system with warm standbys and repairmen. Microelectron Reliab 29:849–860 CrossRefGoogle Scholar
  17. Yue D, Yue W (2009) Analysis of an M/M/c/N queueing system with balking, reneging, and synchronous vacations. In: Yue W, Takahashi Y, Takagi H (eds) Advances in queueing theory and network applications. Springer, New York, pp 165–180 CrossRefGoogle Scholar
  18. Zhang ZG, Tian N (2003) Analysis on queueing systems with synchronous vacations of partial servers. Perform Eval 52:269–282 CrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Statistics, College of ScienceYanshan UniversityQinhuangdaoP.R. China
  2. 2.Department of Intelligence and InformaticsKonan UniversityKobeJapan

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