On Reliability Function of a Parallel System with Three Renewable Components

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10684)


Considered system consists of three renewable components that are connected in parallel. The components are described by continuous time independent alternating processes. The sojourn times in the operative state for all components have exponential distributions. The sojourn times in the failed state have arbitrary absolute continuous distributions. All sojourn times are independent. The system is working at time t if at least one component is working. We consider a problem of computation of system reliability on given time interval for the known initial states of the components. Non-stationary and stationary regimes are considered.


Alternating processes Recurrent event Renewal equation System reliability 



The publication was prepared with the support of the “RUDN University Program 5–100”, and was financially supported by the Russian Foundation for Basic Research according to the research projects Nos. 17-07-00142 and 17-01-00633.


  1. 1.
    Gnedenko, B.V., Belyaev, Y.K., Solovyev, A.D.: Mathematical Methods of Reliability. Academic Press, Cambridge (1969)zbMATHGoogle Scholar
  2. 2.
    Srinivasan, S.K., Gopalan, M.N.: Probabilistic analysis of a two-unit system with a warm standby and a single repair facility. Oper. Res. 21(3), 748–754 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Efrosinin, D., Rykov, V.: Sensitivity analysis of reliabilituy characneristic to the shape of the life and repare time distributions. Eur. J. Oper. Res. 176, 347–360 (2007)CrossRefGoogle Scholar
  4. 4.
    Rykov, V.: Multidimensional alternative processes reliability models. In: Dudin, A., Klimenok, V., Tsarenkov, G., Dudin, S. (eds.) BWWQT 2013. CCIS, vol. 356, pp. 147–156. Springer, Heidelberg (2013). CrossRefGoogle Scholar
  5. 5.
    Efrosinin, D., Rykov, V.: Sensitivity analysis of reliability characteristics to the shape of the life and repair time distributions. In: Dudin, A., Nazarov, A., Yakupov, R., Gortsev, A. (eds.) ITMM 2014. CCIS, vol. 487, pp. 101–112. Springer, Cham (2014). Google Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. 2. John Wiley and Sons Inc., Hoboken (1971)zbMATHGoogle Scholar
  7. 7.
    Andronov, A.M., Vishnevsky, V.M.: Algorithm of state stationary probability computing for continuous-time finite Markov chain modulated by semi-Markov process. In: Vishnevsky, V., Kozyrev, D. (eds.) DCCN 2015. CCIS, vol. 601, pp. 167–176. Springer, Cham (2016). CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematical Methods and ModelingTransport and Telecommunication InstituteRigaLatvia
  2. 2.Department of Applied Probability and InformaticsPeoples’ Friendship University of Russia (RUDN University)MoscowRussia
  3. 3.V.A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia

Personalised recommendations