## 2 Parametric ALT for a System Operated by a Machinery
### Placing a Complete ALT Strategy  
Reliability can be expressed as the system's capability to run under specified situations for an expressed mean length of time [32]. It is typically illustrated with the \"bathtub curve\", which is described as the middle curve in Figure 3. The bathtub curve has three sections which can be stated according to the shape parameter on the Weibull plot. That is, in the first section, there is a decreasing rate of failure in the earlier portion of the mechanical product's lifetime (\(\beta<1\)). In the second section, there is the constant rate of failure (sometimes a slanted line, \(\beta=1\)) during the middle life of the product, which follows an exponential distribution. Eventually, there is an increasing rate of failure to the end of the merchandise's lifetime (\(\beta>1\)) that pursues a Weibull distribution function. There is no initial value of the shape parameter, \(\beta\), because there is no testing data or market statics. As an initial guess, \(\beta=2\) might be used in the wear-out section before it will be validated by parametric ALT. The reliability function, \(R(t)\), is defined as \(R(t)=1-F(t)(or\ X)\). The accumulative distribution function (CDF), \(F(t)(or\ X)=1-R(t)\), is defined by:  
\[F(t)=P(T<t) \tag{1}\]  
The failure rate, \(\lambda\), on the (slanted) bathtub curve in Figure 3 can be expressed as:  
\[\lambda(t)=f(t)/R(t)=\frac{dF(t)/dt}{R(t)}=\frac{\left(1-R(t)\right)^{\prime} }{R(t)}=\frac{-R^{\prime}(t)}{R(t)} \tag{2}\]  
where \(f\) is the failure density function. Figure 2: A multimodule system life is resolved by the newly conceived module. If Equation (2) is integrated with respect to time, the \(X\%\) cumulative failure, \(F(L_{B})\), at \(t=L_{B}\) can be evaluated. That is, \(F(L_{B})\) might be stated as follows:  
\[F(t)=\int\lambda(t)dt=-lnR(t) \tag{3}\]  
\[\text{Or}A(\textit{or}\,X)=\langle\lambda\rangle\cdot L_{B}=\int_{0}^{L_{B}} \lambda(t)\cdot dt=-lnR(L_{B})=-ln(1-F)\cong F(L_{B}) \tag{4}\]  
If \(T_{1}\) is assumed to be the time of the first failure in the second section of the bathtub, reliability, \(R(t)\), will be stated as follows:  
\[R(t)=P(T_{1}>t)=P(\textit{no}\,\textit{failure}\,\textit{in}\,(0,t])=\frac{(m )^{0}e^{-m}}{0!}=e^{-m}=e^{-\lambda t} \tag{5}\]  
where \(m\) is the Poisson parameter, which can be defined as \(\lambda t\) (mean). If the failure rate of a product follows the attributes of the bathtub curve, it will probably not be successful in the market. Because of design imperfections, large early failures could harm the brand name from the product launch. That is, high failure rates in the early life of a product incur the manufacturer to pay warranty expenses. The market share could be expected to eventually be negatively impacted by the high failure rates. The manufacturer would need to improve the product by (1) eliminating unanticipated early failures, (2) reducing random nonsuccess for its operation period, and (3) increasing its system life compared with that of other companies. As the design is improved, the product life from the market enhances, and its failure rate decreases. For such circumstances, the traditional bathtub curve should be transformed into an uncomplicated curve with a small failure rate and a lengthier lifetime. In the end, if the product is well designed, the accumulated failure rate of the product, \(F(t)\), is reduced until the desirable lifetime is achieved. The product's reliability curve would then resemble the line labeled the \"Hockey stick line\" in Figure 3.  
Figure 3: Reliability measure; BX life (\(L_{B}\)) on the bathtub. The reliability of a mechanical system in Equation (5) can be straightforwardly defined as the product of failure rate, \(\lambda\), and lifetime, \(L_{B}\). That is:  
\[R(L_{B})=1-F(L_{B})=e^{-\lambda L_{B}}\cong 1-\lambda L_{B} \tag{6}\]  
This relationship is satisfactory below approximately 20% of the accumulative failure rate [33]. As an example, consider a water dispenser in a domestic refrigerator. It involves a simple mechanical procedure.