Reliability and optimal replacement policy for a generalized mixed shock model

Abstract

A generalized mixed shock model, which mixes two run shock models, is developed and analyzed. According to the model, the system subject to both internal degradation and external shocks fails upon the occurrence of $k_1$ consecutive shocks whose magnitude is between predefined critical values of $d_1$ and $d_2$ such that $d_1<d_2$, or $k_2$ consecutive shocks whose magnitude is above $d_2$. The system’s reliability, mean time to failure, and mean residual lifetime are all calculated under the assumption that the lifetime of the system due to internal wear and external shock arrival times follows a phase-type distribution. The best policy for replacement is also discussed. There are also graphical representations and numerical examples for the proposed model, in which both lifetime distribution of internal degradation and the interarrival periods between external shocks follow the Erlang distribution.

Appendix

Proof of (13): Since continuous phase-type random variable R has representation $PH_c(\varvec{\alpha },\textbf{A})$ and from (1), we have

$$\begin{aligned} P(R>t)=\varvec{\alpha }\exp (\textbf{A}t)\textbf{e}'=\varvec{\alpha }\left( \sum _{n=0}^\infty \frac{t^n}{n!}\textbf{A}^n\right) \textbf{e}' \end{aligned}$$

and

$$\begin{aligned} E(min(R,t))&=\int _0^\infty P(min(R,t)>u)du \\&=\int _0^t P(R>u)du \\&=\int _0^t \varvec{\alpha }\left( \sum _{n=0}^\infty \frac{u^n}{n!}\textbf{A}^n\right) \textbf{e}' du \\&=\varvec{\alpha }\left( \textbf{A}^{-1}\sum _{n=0}^\infty \frac{t^{n+1}}{(n+1)!}\textbf{A}^{n+1}\right) \textbf{e}' \\&=\varvec{\alpha }\left( \textbf{A}^{-1}\sum _{n=-1}^\infty \frac{t^{n+1}}{(n+1)!}\textbf{A}^{n+1}-\textbf{I}\right) \textbf{e}' \\&=\varvec{\alpha }\left( \textbf{A}^{-1}(\exp (\textbf{A}t)-\textbf{I}) \right) \textbf{e}' \\&=\varvec{\alpha }\textbf{A}^{-1}\exp (\textbf{A}t)\textbf{e}' -\varvec{\alpha }\textbf{A}^{-1}\textbf{e}'. \end{aligned}$$