### Abstract

Language | English |
---|---|

Pages | 4883–4904 |

Number of pages | 22 |

Journal | Applied Mathematical Modelling |

Volume | 37 |

Issue number | 7 |

Early online date | 1 Nov 2012 |

DOIs | |

Publication status | Published - 1 Apr 2013 |

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### Keywords

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

### Cite this

*Applied Mathematical Modelling*,

*37*(7), 4883–4904. https://doi.org/10.1016/j.apm.2012.09.055

}

*Applied Mathematical Modelling*, vol. 37, no. 7, pp. 4883–4904. https://doi.org/10.1016/j.apm.2012.09.055

**Reliability of a multi-state system subject to shocks using phase-type distributions.** / Segovia Garcia, Maria Del Carmen; Labeau, Pierre-Etienne.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Reliability of a multi-state system subject to shocks using phase-type distributions

AU - Segovia Garcia, Maria Del Carmen

AU - Labeau, Pierre-Etienne

PY - 2013/4/1

Y1 - 2013/4/1

N2 - 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.

AB - 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.

KW - phase-type distributions

KW - multi-state system

KW - management science

UR - http://www.scopus.com/inward/record.url?scp=84872618382&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2012.09.055

DO - 10.1016/j.apm.2012.09.055

M3 - Article

VL - 37

SP - 4883

EP - 4904

JO - Applied Mathematical Modelling

T2 - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 7

ER -