, Volume 23, Issue 4, pp 875–889 | Cite as

Asymptotic behavior of repairable device systems under warranty and beyond warranty periods

  • B. El HabilEmail author


In this paper, we present a new mathematical model generalizing the model of a single unit repairable device system, with preventive maintenance under warranty period, and the model of a repairable multi-state device system. To study the time dependance solution of the resulting system, we present in this paper a new analytic approach. So, by using the \(C_0\)-semigroup theory of linear operators and some spectral properties, we proof that the dynamic solution of the system converges to steady state solution when the time go to infinity.


Repairable system \(C_0\)-semigroup Well-posedness Positivity Irreducibility Spectral theory Stability 

Mathematics Subject Classification

47D06 34G10 49K40 



  1. 1.
    Ramniwas, R., Kaydan, M.S., Kumar, J.: Stochastic modeling of single-unit repairable system with preventive maintenace under warranty. Int. J. Comput. Appl. 75(13), 36–41 (2013)Google Scholar
  2. 2.
    Dhillon, B.S., Singh, C.: Engineering Reliability-New Techniques and Applications. Wiley, Hoboken (1981)zbMATHGoogle Scholar
  3. 3.
    Weiwei, H., Houbao, X., Yu, J., Zhu, G.: Exponential stability of reparable multi-state device. J. Syst. Sci. Complex. 20, 437–443 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Xu, H., Hu, W.: Availability optimisation of repairable system with prentive maintenance policy. Int. J. Syst. Sci. 39(6), 655–664 (2008)CrossRefzbMATHGoogle Scholar
  5. 5.
    Engel, K.J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations, Graduate Text in Mathematics, vol. 194. Springer, New York (2000)zbMATHGoogle Scholar
  6. 6.
    Schaefer, H.H.: Banach Lattices and Positive Operators. Springer, New York (1974)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Laboratory of Engineering Sciences and Energies (Lab.SIE)Ibn Zohr UniversityAgadirMorocco

Personalised recommendations