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

Abstract

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

Dynamic assessment of multi-state systems using phase-type modeling

“…Then, the phase type representation for the survival function of the series system is obtained from (16). If for example j = 2, then …”

“…2 = M (2) in (16). After simple matrix manipulations, the mean time spent by the system in state "2" or above is found to be E(T ≥2 S−P ) = 10.8343.…”

“…Phase-type distributions have been effectively used in the field of reliability. They have been utilizied for the study of reliability shock models in [14]- [16]. Neuts and Meier [17] used phase-type distributions in reliability modeling of systems with two components.…”

Reliability assessment of system under a generalized cumulative shock model

“…A host of researches has demonstrated the efficiency of phase-type distribution in constructing reliability models in recent literatures. Segovia and Labeau 16 developed a reliability model for a system subject to both internal wear-out and external shocks causing cumulating damage. Eryilmaz 17 discussed an extreme shock model with two thresholds, which generate a multi-state system.…”

Stochastic modeling of quality of systems operating in a heterogeneous environment

“…Shocks, affecting systems during operation, are common in practice (eg, electrical, thermal, and seismic shocks). On the other hand, from the modeling point of view, they constitute a useful tool for describing an effect of environment on reliability characteristics of engineering systems (see, eg, the works of Castro, Castro et al (2011), Montoro‐Cazorla et al, Frostig and Kenzin, Montoro‐Cazorla and Pérez‐Ocón, Huynh et al, van der Weide and Pandey, Nakagawa, Finkelstein and Cha, Mercier and Pham, Segovia and Labeau, and Sheu et al to name a few).…”