# Reliability Analysis of Technical Means of Transport
## 3 Mathematical Model--Implementation  
The characteristics resulting from a group randomly selected from a given object population are called empirical characteristics. The determination of empirical characteristics, both functional and numerical, can be presented in the case of observing the fitness time to failure of \(n\) studied objects. The following designations were adopted for the development of the vehicle reliability model:  
(a) The number of studied vehicles:  
\[n=n(t)+m(t), \tag{15}\]  
where:  
\(n(t)\)--the number of fit for the use of vehicles until moment \(t\);  
\(m(t)\)--the number of unserviceable vehicles until moment \(t\);  
(b) The reliability empirical function:  
\[\overline{R}(t)=\frac{n(t)}{n}=\frac{n-m(t)}{n}=1-\frac{m(t)}{n}, \tag{16}\]  
\begin{table}
\begin{tabular}{c c c} \hline
**Quadrature (Rule)** & **Disadvantages** & **Advantages** \\ \hline Rectangular & \begin{tabular}{c} Very low accuracy resulting from \\ node selection; high-error method \\ \end{tabular} & Calculation simplicity \\ \hline Trapezoidal & Accuracy depends on node selection & \begin{tabular}{c} Calculation simplicity; higher \\ accuracy than rectangular rule; need \\ for fewer calculations in order to \\ obtain better result accuracy \\ \end{tabular} \\ \hline Simpson’s & \begin{tabular}{c} Complicated calculation formula \\ based on three nodes \\ \end{tabular} & \begin{tabular}{c} Higher accuracy than rectangular and \\ trapezoidal rules \\ \end{tabular} \\ \hline Compound trapezoidal & \begin{tabular}{c} Sensitive to a number of nodes; \\ complex calculation formula \\ \end{tabular} & \begin{tabular}{c} Rapid convergence rate; possibility to \\ complete the calculation process after \\ reaching a set accuracy \\ \end{tabular} \\ \hline Compound Simpson’s & \
\begin{tabular}{c} Sensitive to a number of nodes; \\ extended numerical formula \\ \end{tabular} &
\begin{tabular}{c} Very rapid convergence rate; \\ possibility to complete calculations \\ after obtaining a set accuracy; method \\ accuracy increases with an increase in \\ the number of nodes \\ \end{tabular} \\ \hline \end{tabular}
\end{table}
Table 1: Advantages and disadvantages of the applied interpolation quadratures.  
3. The unreliability empirical function: \[\overline{Q}(t)=1-\overline{R}(t)=\frac{n-n(t)}{n}=1-\frac{n(t)}{n},\] (17)
4. The failure probability density empirical function: \[\overline{f}(t)=\frac{n(t)-n(t+\Delta t)}{n\Delta t}=\frac{\overline{R}(t)- \overline{R}(t+\Delta t)}{\Delta t},\] (18)
5. The failure rate empirical function: \[\overline{\lambda}(t)=\frac{n(t)-n(t+\Delta t)}{n(t)\Delta t}=\frac{ \overline{R}(t)-\overline{R}(t+\Delta t)}{\overline{R}(t)\Delta t},\] (19)
6. The empirical leading function: \[\overline{\Lambda}(t)=\sum_{i}\overline{\lambda}(t_{i})\Delta t.\] (20)  
[PERSON]'s theorem indicates that at each continuity point of a distribution function of a random variable describing the fitness time within a population, in which the sample was randomly selected, for each \(\varepsilon>0\), the following boundary property occurs:  
\[\lim_{n\rightarrow\infty}P\biggl{\{}\sup_{t}\bigl{|}\overline{Q}(t)-Q(t) \bigr{|}<\varepsilon\biggr{\}}=1. \tag{21}\]  
It justifies applying the empirical distribution function to evaluate the reliability of the objects selected from a given population [43].  
The knowledge of the failure rate function's waveform enables the obtaining of important information in the fields of:  
- Properly equipping the objects with tools and spare parts;  
- Determining the spare part production output;  
- Determining the usefulness of applying the preventive part or assembly replacements;  
- Determining the object operating periods;  
- Determining the optimum object durability;  
- Planning the recovery of scarce elements or assemblies from repaired and scrapped objects [42].