# Classification of Systems and Maintenance Models
## 6 Example of Deterioration Process Modeling  
The PDF \(\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)\) of the stochastic deterioration process of the system state must be determined to calculate the probabilities of correct and incorrect decisions included in the maintenance effectiveness indicators. Equations (40)-(45) and (53)-(58) are valid for any deterioration process; however, the PDF \(\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)\) must be determined for a specific deterioration process.  
We assume that the following monotone stochastic function characterizes the degradation of the system:  
\[X(t)=A_{0}+A_{1}t^{\mu} \tag{125}\]  
where \(A_{0}\) is the initial random value of the system state parameter \(X(t)\) with values ranging from \(0\) to \(FT\), \(A_{1}\) is the random deterioration rate of the system state parameter specified in the range from \(0\) to \(\infty\), and \(\mu\) is the time exponent.  
We derive the following equation for the conditional PDF of random variables \(\Delta_{1},\ldots,\Delta_{n}\) under the condition that H = \(\eta\), assuming an additive relationship between random variables \(X(t_{n})\) and \(Y(t_{n})\) in (18) and using the approach described in [4]:  
\[\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)=\mu\int\limits_{0}^{FT}f(a_{0}) \omega(\eta|a_{0})\left(\frac{FT-a_{0}}{\eta}\right)^{n}\prod\limits_{i=1}^{n }\left|\left(\frac{\delta_{i}+\eta}{\eta}\right)^{\mu-1}\right|\Omega\bigg{[} RT_{i}-a_{0}-\left(FT-a_{0}\right)\left(\frac{\delta_{i}+\eta}{\eta}\right)^{\mu} \bigg{]}\frac{da_{0}}{\omega(\eta)} \tag{126}\]  
where \(f(a_{0})\) is the PDF of random variable \(A_{0},\ \omega(\eta|a_{0})\) is the conditional PDF of random variable H provided that \(A_{0}=a_{0}\), and \(\Omega(y_{i})\) is the PDF of the random measurement error \(Y(t_{i})\), \(i=1,\ldots,n\).  
If \(A_{0}=a_{0}=constant\), then:  
\[\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)=\mu\left(\frac{FT-a_{0}}{\eta} \right)^{n}\prod\limits_{i=1}^{n}\left|\left(\frac{\delta_{i}+\eta}{\eta} \right)^{\mu-1}\right|\Omega\bigg{[}RT_{i}-a_{0}-\left(FT-a_{0}\right)\left( \frac{\delta_{i}+\eta}{\eta}\right)^{\mu}\bigg{]} \tag{127}\]Using the Gaussian PDF of the measurement error \(Y(t_{i})\), we obtain - from (126):  
\[\begin{array}{l}\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)=\mu\bigg{(}\frac{1} {\sigma_{y}\sqrt{2\pi}}\bigg{)}^{n}\int\limits_{0}^{FT}f(a_{0})\omega(\eta|a_{0 })\left(\frac{FT-a_{0}}{\eta}\right)^{n}\prod\limits_{i=1}^{n}\left|\left(\frac {\delta_{i}+\eta}{\eta}\right)^{n-1}\right|\exp\biggl{\{}-\frac{1}{2\sigma_{y}^ {2}}[RT_{i}-a_{0}-\\ (FT-a_{0})\left(\frac{\delta_{i}+\eta}{\eta}\right)^{n}]^{2}\biggr{\}}^{2} \frac{da_{0}}{\omega(\eta)}\] (128) - from (127):  
\[\psi_{0}(\delta_{1},\ldots,\delta_{n}|\eta)=\mu\left(\frac{1}{\sigma_{y}\sqrt{ 2\pi}}\right)^{n}\left(\frac{FT-a_{0}}{\eta}\right)^{n}\prod\limits_{i=1}^{n} \left|\left(\frac{\delta_{i}+\eta}{\eta}\right)^{n-1}\right|\exp\left\{-\frac {1}{2\sigma_{y}^{2}}[RT_{i}-a_{0}-\ \left(FT-a_{0}\right)\left(\frac{\delta_{i}+\eta}{ \eta}\right)^{n}]^{2}\right\} \tag{129}\]  
where \(\sigma_{y}\) is the standard deviation of the random measurement error \(Y(t_{i}),\ i=1,\ldots,n\). When \(A_{0}=a_{0}=\textit{constant}\) and \(A_{1}\) is a normal random variable with PDF \(f(a_{1})\), the PDF of time-to-failure is given by [96]:  
\[\omega(\eta)=\frac{\mu\sigma_{A1}^{2}\eta^{2\mu-1}[m_{A1}\eta^{\mu}+(FT-a_{0}- m_{A1}\eta^{\mu})]}{\sqrt{2\pi}\big{(}\sigma_{A1}^{2}\eta^{2\mu}\big{)}^{1} 5\int\limits_{0}^{\infty}f(a_{1})da_{1}}\exp\biggl{[}-\frac{(FT-a_{0}-m_{A1} \eta^{\mu})^{2}}{2\sigma_{A1}^{2}\eta^{2\mu}}\biggr{]} \tag{130}\]