We present hybrid, degradation-based reliability models for a single-unit system whose degradation is driven by a semi-Markov environment. The primary objective is to develop a mathematical framework and associated computational techniques that unite environmental data and stochastic failure models to assess the current or future health of the system. By employing phase-type distributions, we construct a surrogate environment process that is amenable to analysis by exact Markovian techniques to obtain reliability estimates. By way of two numerical experiments, we illustrate the viability of our approach and assess the quality of the approximations. The numerical results indicate that remarkably accurate lifetime distribution and moment approximations are attainable.

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