## 2 Imperfect Maintenance Model
\begin{cases}w_{m-1}^{+}=(1-\varepsilon)t_{m-1},\\ w_{m}^{-}=w_{m-1}^{+}+(t_{m}-t_{m-1}),\\ M_{m}=(t_{m}-t_{m-1})=w_{c}\cdot\varepsilon^{m-1}.\end{cases}  \tag{7}\]  
An overall expression of \(w_{m}^{+}\) can be obtained by analyzing the evolution of \(w_{3}^{+}\):  
\[\begin{array}{rcl}w_{3}^{+}&=&w_{2}^{+}+\left(w_{cAs it has been stated before, a component will be substituted or upgraded into a GAN situation when \(M_{m+1}<M_{min}\) is satisfied. Given \(w_{c}\) and \(M_{min}\), the amount of PM activities \(e\) executed before a component is replaced is calculated as follows:  
\[ \begin{split} w_{c}\varepsilon^{e-1}&\leq M_{min} \Rightarrow(e-1)\,ln\,\varepsilon\leq ln\left(\frac{M_{min}}{w_{c}}\right) \\ \Rightarrow e-1&\leq\frac{ln\left(\frac{M_{min}}{w_{ c}}\right)}{ln\,\varepsilon}\Rightarrow e\leq 1+ln\bigg{(}\frac{M_{min}}{w_{c}} \bigg{)}^{1/\varepsilon}.\end{split}  \tag{10}\]  
Under these conditions, it is possible to obtain an age-dependent reliability model in which the induced or conditional failure rate in the period \(m\) after the maintenance number \(m\) is given by:  
\[h_{m}(w)=h(w(t,\varepsilon))+h_{0},\quad w\geq w_{m}^{+}, \tag{11}\]  
where \(h_{0}\) represents the initial failure rate of the component, that is to say, when it was first installed. The age of the component after the maintenance number \(m\) is given by Equation (9). If we adopt a Weibull model for the failure rate, the expression of the induced failure rate after the maintenance number \(m\) can be written as:  
\[h_{m}(w)=\lambda^{\gamma}\cdot\gamma\cdot(w_{m}(t,\varepsilon))^{\gamma-1}+h_ {0},\quad w\geq w_{m-1}^{+}, \tag{12}\]  
where \(\lambda\) is the scale parameter, and \(\gamma\) is the shape parameter. For this Weibull distribution, the accumulated failure rate is defined by:  
\[H_{m}(w)=H(w_{m})=(\lambda w_{m}(t,\varepsilon))^{\gamma}+h_{0}w_{m}. \tag{13}\]  
For the specific cases of \(w_{m}^{-}\) and \(w_{m-1}^{+}\), the accumulated failure rate is given by:  
\[H_{m}^{-}=H_{m}(w_{m}^{-})=\big{(}\lambda w_{m}^{-}\big{)}^{\gamma}+h_{0}w_{m }^{-}=(\lambda w_{c})^{\gamma}+h_{0}w_{c} \tag{14}\]  
and:  
\[ \begin{array}{rcl}H_{m-1}^{+}&=&H_{m-1}(w_{m-1}^{+})=\big{(}\lambda w_{m-1}^ {+}\big{)}^{\gamma}+h_{0}w_{m-1}^{+}-=\\ &=&\Big{(}\lambda w_{c}(1-\varepsilon)\sum_{k=0}^{m-2}\varepsilon^{k}\Big{)}^ {\gamma}+h_{0}w_{c}(1-\varepsilon)\sum_{k=0}^{m-2}\varepsilon^{k}=\\ &=&\Big{(}\lambda w_{c}(1-\varepsilon)\sum_{k=1}^{m-1}\varepsilon^{k-1}\Big{)} ^{\gamma}+h_{0}w_{c}(1-\varepsilon)\sum_{k=1}^{m-1}\varepsilon^{k-1}.\end{array}  \tag{15}\]  
Next, we will define the averaged failure rate between \(t_{m}-t_{m-1}\) (\(h_{m}^{*}\)), which is necessary to calculate unavailability rates:  
\[ \begin{array}{rcl}h_{m}^{*}&=&\frac{1}{w_{m}^{-}-w_{m-1}^{+}} \left(H_{M}^{-}-H_{m-1}^{+}\right)=\frac{1}{w_{c}-\left(w_{c}(1-\varepsilon) \sum_{k=1}^{m-1}\varepsilon^{k-1}\right)}=\\ &=&\frac{\lambda^{\gamma}w_{c}^{\gamma-1}\left[1-\left((1-\varepsilon)\sum_{k =1}^{m-1}\varepsilon^{k-1}\right)^{\gamma}\right]}{1-(1-\varepsilon)\sum_{k=1 }^{m-1}\varepsilon^{k-1}}+h_{0}.\end{array}  \tag{16}\]  
Finally, using the averaging process given by Equation (16), the averaged failure rate over the lifetime of a component subject to \(e\) PM activities can be obtained:  
\[h^{*}=\frac{\sum_{1}^{t}h_{e}^{*}M_{e}}{M_{e}}. \tag{17}\]