## 3 One-Dimensional Degradation Models Based on Stochastic Processes
### Degradation Models Based on Univariate Wiener Process with Random Effects  
The Wiener process [20] is generally suitable for modeling continuous and fluctuating degradation products. For a linear Wiener process \(X(t)\), \(X(t)\) has independent increments and is defined as the performance value of the product at a time \(t\), which satisfies:  
\[X(t)=X(0)+\lambda t+\sigma B(t) \tag{1}\]  
Among them, \(X(0)\) represents the initial value of the product, generally \(X(0)=0\), and \(B(t)\) is the standard Brownian motion. \(\lambda\) and \(\sigma\) are unknown parameters, where \(\lambda\) is the drift parameter while \(\sigma\) is the diffusion parameter. Furthermore, the increments \(\Delta X(t)\) follow a standard normal distribution, that is, \(\Delta X(t)\sim N\big{(}\lambda\Lambda t,\sigma^{2}\Delta t\big{)}\).  
Suppose the failure threshold of the product's parameter is \(l\), and the life \(T=\inf\{t|X(t)\geq l\}\) is defined as the first arrival time with \(X(t)\) degradation to the failure threshold \(l\). According to the statistical characteristics of the Wiener process, the life \(T\) obeys the inverse Gaussian distribution, that is:  
\[T\sim IG\bigg{(}\frac{l}{\lambda},\frac{l^{2}}{\sigma^{2}}\bigg{)} \tag{2}\]  
Therefore, the PDF and CDF of the life \(T\) are:  
\[f(t)=\frac{l}{\sqrt{2\pi\sigma^{2}t^{3}}}exp\left(-\frac{\left(l-\lambda t \right)^{2}}{2\sigma^{2}t}\right) \tag{3}\]  
\[F(t)=1-\Phi\bigg{(}\frac{l-\lambda t}{\sigma\sqrt{t}}\bigg{)}+exp\bigg{(} \frac{2\lambda l}{\sigma^{2}}\bigg{)}\bigg{[}1-\Phi\bigg{(}\frac{l+\lambda t }{\sigma\sqrt{t}}\bigg{)}\bigg{]} \tag{4}\]  
The reliability function corresponding to a one-dimensional degradation parameter which is described by the Wiener process can be expressed as:  
\[\begin{array}{l}R(t)=1-F(t)\\ =\Phi\bigg{(}\frac{l-\lambda t}{\sigma\sqrt{t}}\bigg{)}-exp\bigg{(}\frac{2 \lambda l}{\sigma^{2}}\bigg{)}\bigg{[}1-\Phi\bigg{(}\frac{l+\lambda t}{\sigma \sqrt{t}}\bigg{)}\bigg{]}\end{array} \tag{5}\]  
Assume that there are individual differences between samples in the process of manufacturing and assembling, let the drift parameter, \(\lambda\), be a random variable that describes random effects following a normal distribution [21], that is, \(\lambda\sim N\big{(}\mu,\sigma_{\lambda}^{2}\big{)}\). Then, the distribution function of the Wiener process with random effects follows the normal distribution with the mean, \(\mu t\), and variance, \(\big{(}\sigma^{2}t+\sigma_{\lambda}^{2}\big{)}\).  
The CDF of the product that takes into consideration random effects can be calculated by the following formula:  
\[F_{\lambda}(t)=\Phi\left(-\frac{l-\mu t}{\sqrt{\sigma^{2}t+\sigma_{\lambda}^{2}t^ {2}}}\right)+exp\left(\frac{2\mu l}{\sigma^{2}}+\frac{2\sigma_{\lambda}^{2}l^{2} }{\sigma^{4}}\right)\Phi\left(-\frac{2\sigma_{\lambda}^{2}lt+\sigma^{2}(l+\mu t )}{\sigma^{2}\sqrt{\sigma^{2}t+\sigma_{\lambda}^{2}t^{2}}}\right) \tag{6}\]  
Thus, the reliability function considering random effects can be deduced as follows:  
\[\begin{array}{l}R_{\lambda}(t)=1-F_{\lambda}(t)\\ =\Phi\left(\frac{l-\mu t}{\sqrt{\sigma^{2}t+\sigma_{\lambda}^{2}t^{2}}}\right) -exp\left(\frac{2\mu l}{\sigma^{2}}+\frac{2\sigma_{\lambda}^{2}l^{2}}{\sigma^ {4}}\right)\Phi\left(-\frac{2\sigma_{\lambda}^{2}lt+\sigma^{2}(l+\mu t)}{ \sigma^{2}\sqrt{\sigma^{2}t+\sigma_{\lambda}^{2}t^{2}}}\right)\end{array} \tag{7}\]