Phase-Type Approximations for Wear Processes in a Semi-Markov Environment

The reliability of a single-unit system experiencing degradation (wear) due to the influence of a general, observable environment process is considered. In particular, the failure time distribution is evaluated using only observations of the unit’s current operating environment which is characterized as a finite semi-Markov process (SMP). In order to impose the Markov property, generally distributed environment state sojourn times are approximated as phase-type (PH) random variables using observations of state holding times and transition rates. The use of PH distributions facilitates the use of existing analytical results for reliability evaluation of units subjectto an environment process that evolves as a continuous-time Markov chain. The procedure is illustrated through three numerical examples, and results are compared with those obtained via Monte Carlo simulation. The maximum absolute deviation in probability for failure time distributions was on the order of 0.004. The results of this thesis provide a novel approach to the reliability analysis of units operating in randomly evolving environments for which degradation or failure time observations are difficult or impossible to obtain.

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