## 3 Methods
### Fundamental Indicators of Reliability and Maintainability  
Fundamental indicators of reliability, as magnitudes that quantitatively express reliability, are expressed by:  
* Probability of good operation--reliability function \(R(t)\);
* Probability of failure--unreliability function \(F(t)\);
* Failure probability density \(f(t)\);
* Failure intensity or rate \(\lambda(t)\);
* Mean time between failures \(MTBF\);
* Mean time to repair \(MTR\);
* Repair rate \(\mu\). From quantitative point of view, reliability was defined as being the probability for a product (technical system) to fulfill its mission (fundamental function), for a pre-established period of time, in certain given conditions [36; 37; 38; 39]. According to this definition, the probability of good operation \(p(t)\), that is reliability \(R(t)\), is expressed by the equation:  
\[p(t)=R(t)=P(t>t_{i}), \tag{1}\]  
where \(t\)--random time variable (time of mission) and \(t_{i}\)--specified limit of the duration of good operation. In an experimental determination, in order to obtain an analytical form of function, it is supposed that a behavior mode in time of a statistic population made up of N\({}_{0}\) new identical products is followed, which perform in the same work conditions and which have been manufactured based on the same technology, with the same imposed conditions. Considering that, at the moment \(t=0\), all the other \(N_{0}\) products are in a state of operation, then at moment \(t_{i}\), found in the range of \([t,\,t+\Delta t]\), there are only \(N\) products in a state of operation. Thus, along \(\Delta t\) time, it is considered that \(\Delta N=N_{0}-N\) flawed products. The proportion of products in state of operation at moment \(t_{i}\), that is, reliability \(\hat{R}(t)\), is given by the ratio:  
\[\hat{R}(t)=\frac{N}{N_{0}}, \tag{2}\]The analytical form of the empirical function of reliability is expressed in this form:  
\[\hat{R}(t_{i})=\frac{N_{t_{i}}}{N_{0}}, \tag{3}\]  
where \(N_{t_{i}}\)--the number of products (elements) being in operation at moment \(t_{i}\). The non-reliability function expresses the failure probability of a product that should perform well for an established period of time \(t_{i}\), under certain given conditions. It is expressed by the equation:  
\[F(t)=P(t<t_{i}), \tag{4}\]  
Between the reliability and non-reliability functions, the following relationship exists:  
\[F(t)=1-R(t), \tag{5}\]  
The analytical form of the empirical function of non-reliability is expressed in the form:  
\[\hat{F}(t_{i})=\frac{N_{0}-N_{t_{i}}}{N_{0}}, \tag{6}\]  
By deriving the non-reliability function in time, a new function is obtained, called the density of probability of failures\(f(t)\), which is the function of frequency or density of distribution and which expresses the relative frequency of failures in a time interval \(dt\):  
\[f(t)=\frac{dF(t)}{dt}=-\frac{dR(t)}{dt}, \tag{7}\]  
The analytical form of the empirical function which expresses density of probability is:  
\[\hat{f}(t_{i})=\frac{\Delta N_  
The average time of good operation expresses the average operation time until failure occurs, in case of un-reparable components or between two consecutive failures, in the case of reparable failures.  
The value of the parameter is given by the average of operation times, which, for a continuous distribution, at limit, can be written as:  
\[m=MTBF=\int_{0}^{\infty}R(t)dt=\int_{0}^{\infty}[1-F(t)]dt=\int_{0}^{\infty}e^{ -\int_{0}^{t}\lambda(t)dt}, \tag{12}\]  
where \(MTBF\)--mean time between failures. In the case of a batch of \(N_{0}\) products, each of them has a certain duration of operation, \(t_{f_{i}}\). The average of a good operating time is determined, in this case, based on the discrete values in effect, as the arithmetic average of a good operating time \(t_{f_{1}},t_{f_{2}},\ldots,\ t_{N_{0}}\):  
\[\widehat{MTBF}=\frac{\sum_{i=1}^{N_{0}}t_{f_{i}}}{N_{0}}, \tag{13}\]  
The mean time of repair \(MTR\) gives information regarding the number of hours referring to a reparation.