# Reliability Analysis of Technical Means of Transport
## 2 Methodology
### Theoretical Grounds for Developing a Mathematical Model  
Theoretically, each object can be considered as a system in the course of tests. There is no objective division of objects into systems and elements. The developed formal model for this object constitutes grounds for determining whether a given object, the reliability of which is to be tested, will be treated as a system. In the case of reliability, the objective is to study the relationships between the system element reliability and the entire system reliability [10]. A mathematical model of a non-renewable technical object that describes its reliability, understood as the ability to execute tasks under specific conditions and within a known time interval, is a non-negative and constant random variable \(T\)[33; 34; 35; 36; 37; 38; 39]. The basic measure of the reliability of an object \(R(t)\) within a time interval \([0,t]\) is the object probability described by the following formula in Equation (1):  
\[R(t)=P(T\geq t)\text{ for }t\geq 0 \tag{1}\]  
The reliability function of an object \(R(t)\) for each \(t\geq 0\) has a value equal to the probability of an event involving object failure-free operation at least until \(t\), which is the probability of an object being in a state of fitness until \(t\). The function, which for each established \(t\geq 0\) adopts the value of the probability of an event that the object at moment \(t\) is damaged, is referred to as an unreliability function, described by the following formula in Equation (2):  
\[F(t)=P(T<t)=1-R(t) \tag{2}\]  
If the reliability function is absolutely continuous, then:  
\[R(t)=\int\limits_{t}^{\infty}f(u)du\text{ for }t\geq 0 \tag{3}\]  
Function \(f\) satisfying the condition in Equation (3) is called a probability density. At all continuity points, the probability density can be expressed as a derivative, using the following relationship in Equation (4):  
\[f(t)=\frac{d}{dt}[F(t)]=-\frac{d}{dt}[R(t)] \tag{4}\]  
If the probability density and reliability are known, then it is also possible to determine the failure rate with Equation (5):  
\[\lambda(t)=-\frac{d}{dt}[\ln R(t)]=\frac{f(t)}{R(t)} \tag{5}\]  
Using the [PERSON]'s formula, the relationships in Equations (6) and (7) can be noted as:  
\[f(t)\approx\frac{R(t)-R(t+\Delta t)}{\Delta t} \tag{6}\]  
\[\lambda(t)\approx\frac{R(t)-R(t+\Delta t)}{R(t)\Delta t} \tag{7}\]The formula in Equation (6) indicates an interpretation of the probability density, which is understood as a decline in reliability over a short time interval of \(\Delta t\), whereas the relationship in Equation (7) means that the failure rate is a relative decline in the reliability of an object per unit of time. A measure of the depletion of an object task execution reserve is a leading function or cumulative failure rate described by the formula:  
\[\Lambda(t)=-\ln[R(t)]=\int\limits_{0}^{t}\lambda(u)du\text{ for }t\geq 0 \tag{8}\]  
Each of the presented functions characterizing the object fitness time can be expressed by any different one of them [40; 41; 42; 43]. If the reliability function \(R(t)\) is known, the expected operating time before failure is calculated from the relationship in Equation (9):  
\[ET=\int\limits_{0}^{\infty}R(t)dt \tag{9}\]  
In relation to non-renewable objects, the expected value of the random variable \(T\) is also often called an expected fitness time [43], and its measure can be calculated using numerical integration methods. If an integrand is not an elementary function, the analytical determination of the integral can be very difficult or even impossible. Numerical formulas for the integration of functions with single independent variables are called quadratures [44]. The following designations were introduced to precisely explain the applied numerical formula:  
a--lower integration limit;  
\(b\)--upper integration limit;  
\(t_{i}\)--equidistant quadrature nodes, such as \(t_{i}=a+\frac{(b-a)}{n^{*}}i\), \(i=0,\ldots,n^{*}\);  
\(h\)--length of the smallest subdivision (applies to compound quadratures).