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Journal of Systems Science and Complexity

, Volume 28, Issue 5, pp 1015–1032 | Cite as

Reliability analysis of a cold standby repairable system with repairman extra work

  • Xin ZhangEmail author
Article

Abstract

This paper deals with a cold standby repairable system with two identical units and one repairman who can do extra work in idle time. The authors are devoted to studying the unique existence and exponential stability of the system solution. C 0-semigroup theory is used to prove the existence of a unique nonnegative time-dependent solution of the system. Then by using the theory of resolvent positive operator, the authors derive that dynamic solution of the system exponentially converges to its steady-state one which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Some reliability indices of the system are discussed with a different method from traditional one. The authors also make a profit analysis to determine the optimal service time outside the system to maximize the system profit.

Keywords

C0-semigroup exponential stability repairable system resolvent positive operator 

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References

  1. [1]
    Papageorgiou E and Kokolakis G, Reliability analysis of a two-unit general parallel system with (n - 2) warm standbys, European Journal of Operational Research, 2002, 138: 127–141.MathSciNetCrossRefGoogle Scholar
  2. [2]
    Zhang Y L and Wang G J, A geometric process repair model for a repairable cold standby system with priority in use and repair, Reliability Engineering and System Safety, 2009, 94: 1782–1787.CrossRefGoogle Scholar
  3. [3]
    Bieth B, Hong L, and Sarkar J, A standby system with two repair persons under arbitrary life-and repair times, Mathematical and Computer Modelling, 2010, 51: 756–767.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Gurov S V and Utkin L V, The time-dependent availability of repairable m-out-of-n and cold standby systems by arbitrary distributions and repair, Microelectronics and Reliability, 1995, 35(11): 1377–1393.CrossRefGoogle Scholar
  5. [5]
    Gurov S V and Utkin L V, Cold standby systems with inperfect and noninstantaneous switch-over mechanism, Microelectronics and Reliability, 1996, 36(10): 1425–1438.CrossRefGoogle Scholar
  6. [6]
    Gopalan M N and Kumar U D, On the transient behaviour of a repairable system with a warm standby, Microelectronics and Reliability, 1996, 36(4): 525–532.CrossRefGoogle Scholar
  7. [7]
    Kea J B, Lee W C, and Wang K H Reliability and sensitivity analysis of a system with multiple unreliable service stations and standby switching failures, Physica A, 2007, 380(4): 455–469.CrossRefGoogle Scholar
  8. [8]
    Zhang Y L, Yamb R C M, and Zuo M J, A bivariate optimal replacement policy for a multistate repairable system, Reliability Engineering and System Safety, 2007, 92: 535–542.CrossRefGoogle Scholar
  9. [9]
    Zhang Y L and Wang G J, A deteriorating cold standby repairable system with priority in use, European Journal of Operational Research, 2007, 183: 278–295.CrossRefzbMATHGoogle Scholar
  10. [10]
    Jia J S and Wu S M, Optimizing replacement policy for a cold-standby system with waiting repair times, Applied Mathematics and Computation, 2009, 214: 133–141.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Wang W L and Xu G Q, The well-posedness and stability of a repairable standby human-machine system, Mathematical and Computer Modelling, 2006, 44: 1044–1052.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Wang W L and Xu G Q, Stability analysis of a complex standby system with constant waiting and different repairman criteria incorporating environmental failure, Applied Mathematical Modelling, 2009, 33: 724–743.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Shen Z F, Hu X X, and Fan W F, Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure, J. Math. Anal. Appl., 2008, 341: 457–466.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    Tang Y H, On the transient departure process of M*/G/1 queueing system with single server vacation, Journal of Systems Science and Complexity, 2007, 20: 562–571.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Tang Y H, Yu M M, Yun X, and Huang S J, Reliability indices of discrete-time GeoX/G/1 queueing system with unreliable service station and multiple adaptive delayed vacations, Journal of Systems Science and Complexity, 2012, 25: 1122–1135.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Yue D Q and Zhang F, A discrete-time Geo/G/1 ketrial queue with J-vacation policy and general retrial times, Journal of Systems Science and Complexity, 2013, 26: 556–571.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Liu R B and Tang Y H, One-unit repairable system with multiple delay vacations, Chinese Journal of Engineering Mathematics, 2006, 23(4): 721–724.MathSciNetzbMATHGoogle Scholar
  18. [18]
    Rakhee J M and Singh M, Bilevel control of degraded machining system with warm standbys, setup and vacation, Applied Mathematical Modelling, 2004, 28: 1015–1026.CrossRefzbMATHGoogle Scholar
  19. [19]
    Ke J C and Wang K H, Vacation policies for machine repair problem with two type spares, Applied Mathematical Modelling, 2007, 31: 880–894.CrossRefzbMATHGoogle Scholar
  20. [20]
    Guo L L, Xu H B, Gao C, and Zhu G T, Stability analysis of a new kind n-unit series repairable system, Applied Mathematical Modelling, 2011, 35: 202–217.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    Zhang X, Reliability analysis of two supplemental equipments cold standby system of power station, Systems Engineering — Theory & Practice, 2013, 33(10): 2615–2622 (in Chinese).Google Scholar
  22. [22]
    Adams R, Sololev Spaces, Academic Press, New York, 1975.Google Scholar
  23. [23]
    Wolfgang A, Resolvent positive operators, Proc. Landon Math. Soc., 1987, 54: 321–349.zbMATHGoogle Scholar
  24. [24]
    Cao J H and Cheng K, Introduction to Reliability Mathematics, Higher Education Press, Beijing, 1986.Google Scholar
  25. [25]
    Dunford N and Schwartz J T, Linear Operators, Part I, Wiley, New York, 1958.Google Scholar
  26. [26]
    Nagel R, One-parameter semigroup of positive operator, Lecture Notes in Mathematics, Springer-Verlag, New York, 1986.CrossRefGoogle Scholar
  27. [27]
    Taylor A E and Lay D C, Introduction to Functional Analysis, John Wiley & Sons, New York Chichester Bresbane Toronto, 1980.zbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Basic CoursesBeijing Union UniversityBeijingChina
  2. 2.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingChina

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