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Well-posedness and Stability of the Repairable System with Three Units and Vacation

Author

Listed:
  • Han Xiaoshuang

    (College of Science & Technology, Yanbian University, Yanji133002, China)

  • Teng Mingyan

    (Department of Mathematics, Bohai University, Jinzhou121002, China)

  • Fang Ming

    (Department of Mathematics, Science College, Yanbian University, Yanji133002, China)

Abstract

The stability of the repairable system with three units and vacation was investigated by two different methods in this note. The repairable system is described by a set of ordinary differential equation coupled with partial differential equations with initial values and integral boundaries. To apply the theory of positive operator semigroups to discuss the repairable system, the system equations were transformed into an abstract Cauchy problem on some Banach lattice. The system equations have a unique non-negative dynamic solution and positive steady-state solution and dynamic solution strongly converges to steady-state solution were shown on the basis of the detailed spectral analysis of the system operator. Furthermore, the Cesáro mean ergodicity of the semigroup T(t) generated by the system operator was also shown through the irreducibility of the semigroup.

Suggested Citation

  • Han Xiaoshuang & Teng Mingyan & Fang Ming, 2014. "Well-posedness and Stability of the Repairable System with Three Units and Vacation," Journal of Systems Science and Information, De Gruyter, vol. 2(1), pages 54-76, February.
    • Handle: RePEc:bpj:jossai:v:2:y:2014:i:1:p:54-76:n:6
    DOI: 10.1515/JSSI-2014-0054
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