Reliability of a multi-state system subject to shocks using phase-type distributions

Maria Del Carmen Segovia Garcia, Pierre-Etienne Labeau

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40 Citations (Scopus)


The reliability of a multi-state system is considered. The system is subject to both internal wear-out and external shocks causing damage that cumulates as shocks follow one another. As a consequence of this cumulating damage, the system wear-out process can be affected. The study of the system is achieved by means of phase-type distributions, which are used to model: the inter-arrival times between shocks, the magnitude of the damage due to the shocks, and the lifetime distribution of the system between shocks. In the latter case, the phases of the phase-type distribution refer to the different degradation levels (‘states’) of the system. The lifetime distribution of the system is affected by the shocks in different ways: if the cumulated damage after a shock exceeds q predefined thresholds, the system can no longer evolve in the first q least degraded levels; the corresponding phase-type distribution is then defined on the remaining phases. But after each shock causing no threshold to be exceeded, only the initial probability vector of the phase-type distribution is modified in order to account for a decrease in the expected residual lifetime of the system. Two particular cases are studied. First, several thresholds on the cumulated damage can be exceeded following the occurrence of one shock; secondly, only one threshold at a time can be exceeded.
Original languageEnglish
Pages (from-to)4883–4904
Number of pages22
JournalApplied Mathematical Modelling
Issue number7
Early online date1 Nov 2012
Publication statusPublished - 1 Apr 2013


  • phase-type distributions
  • multi-state system
  • management science


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