Reliability of a k-out-of-n System with a Repair Facility – Essential and Inessential Services

  • M. K. Sathian
  • Viswanath C. NarayananEmail author
  • Vladimir Vishnevskiy
  • Achyutha Krishnamoorthy
Conference paper
Part of the Communications in Computer and Information Science book series (CCIS, volume 678)


In this paper we study reliability analysis of a k-out-of-n system with a repair facility which provides an essential and several inessential service with given probabilities. At the epoch the system starts, all components are in operational state. Service to failed components are in the order of their arrival. When a component is selected for repair, we assume that the server may select it either for a service that turns out to be different from what is exactly needed for it, which we call the inessential service with the probability p or for desired service, called essential service with probability (1-p). Once the inessential service process starts, a random clock is assumed to start ticking which decides the event to follow: if the clock realises first (still inessential service going on) the components ongoing service is stopped and it is replaced with a new component. On the other hand if the inessential service gets completed before the realisation of the random clock, then the component moves for the essential service immediately. The life-time of a component, the essential service time and the random clock time have independent exponential distributions and the inessential service time is assumed to follow a phase type distribution. The steady state distribution of the system has been obtained explicitly and several important performance measures derived and verified numerically. The extension of the results reported to the case of more than one essential service is worth examining. This has applications in medicine, biology and several other fields of activity.


k-out-of-n system Essential service Inessential service 



Second and fourth authors acknowledge Department of Science & Technology, Government of India under grant INT/RUS/RMES/P-3/2014 dated 15-4-2015; This research has been financially supported by the Russian Science Foundation (research project No.16-49-02021)


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Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • M. K. Sathian
    • 1
  • Viswanath C. Narayanan
    • 2
    Email author
  • Vladimir Vishnevskiy
    • 3
  • Achyutha Krishnamoorthy
    • 4
  1. 1.Department of MathematicsPanampilly Memorial Government CollegeThrissurIndia
  2. 2.Department of MathematicsGovernment Engineering CollegeThrissurIndia
  3. 3.V. A. Trapeznikov Institute of Control Sciences of Russian Academy of SciencesMoscowRussia
  4. 4.Department of MathematicsCochin University of Science and TechnologyKochiIndia

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