Abstract
A stress-strength model usually has more than one failure mode since the component suffers at least two types of stresses, complicating the expression of the likelihood function and increasing the computational complexity of the parameter estimation for general distributions (non-exponential distributions). A phase-type distribution (also known as a PH distribution) is dense and has closure properties, which makes it suitable to reduce the computational complexity of the stress-strength model. The traditional expectation-maximization (EM) method for estimating the parameters of the PH distribution cannot be used directly when the strength changes over time since the PH distribution is a continuous-time Markov process that must satisfy the relevant properties of the infinitesimal generator in the Markov state-space. Therefore, a parameter estimation method based on extending the Markov state-space with variable transition rates for the stress-strength model is proposed. Both failure and censored samples are considered. First, the stress-strength model based on the PH distribution is briefly introduced, and the likelihood functions for different failure modes are derived. Subsequently, the principle of the method is described in detail, the derivation process of the relevant equations is provided, and the limitations of the method are discussed. The performance of the method is evaluated using two simulation cases.
Highlights
In the reliability theory, the stress-strength model is typically used to investigate a system’s reliability and describe the degradation of the strength and the accumulated damage simultaneously
In Ref. [6], the mean remaining strength of a parallel system consisting of strength and stress components in the stress-strength model was estimated; the components followed an exponential distribution
Since the PH distribution is obtained based on the continuous-time Markov process, the transition rate of the phases in the Markov state-space must satisfy the relevant properties of the infinitesimal generator
Summary
The stress-strength model is typically used to investigate a system’s reliability and describe the degradation of the strength and the accumulated damage simultaneously. Since the PH distribution is obtained based on the continuous-time Markov process, the transition rate of the phases in the Markov state-space must satisfy the relevant properties of the infinitesimal generator. To avoid changing the form of the model and reflect the continuous change of the strength, we propose a method that extends the Markov state-space of the original PH distribution in which the transition rate of the phases changes over time. Both censored and failure samples are considered.
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