# Reliability Evaluation of the Factors That Influenced COVID-19 Patients' Condition
## 4 Usage the Structure Function for System Evaluation  
The quantification of the system is an important step in risk analysis and system reliability. There are known indices and measures that allow a system evaluation based on its mathematical representation by the structure function. Let us consider the group of basic indices for the system reliability analysis based on this mathematical representation. For the definition of these indices, there is the following assumption: the system is considered in the stationary state and a time independent analysis can be implemented in this case. The two most important indices for the system evaluation are availability and unavailability. One of the important measures is availability \(A\), which is defined as system functioning probability [26]:  
\[A=Pr\{\phi(x)=1\}, \tag{2}\]  
and the system unavailability \(U\) is the probability that the system will not work [26]:  
\[U=Pr\{\phi(x)=0\}=1\text{-}\,A. \tag{3}\]  
The system availability and unavailability can be calculated if the probability of every system component failure (\(q_{i}\)) and functioning (\(q_{i}\)) is known:  
\[q_{i}=Pr\{x_{i}=0\}, \tag{4}\]  
\[q_{i}=Pr\{x_{i}=1\}. \tag{5}\]  
For example, the structure function of the laparoscopic surgery procedure presented in Table 1. Let the probabilities of these system components be defined as \(q_{1}=0.10\) (\(p_{1}=0.90\)), \(q_{2}=0.10\) (\(p_{2}=0.90\)), \(q_{3}=0.15\) (\(p_{3}=0.85\)), and \(q_{4}=0.05\) (\(p_{4}=0.95\)). We have to compute the availability and unavailability of this system. This availability and unavailability are computed according to (2) and (3). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**A\({}_{1}\)** & **A\({}_{2}\)** & **A\({}_{3}\)** & **A\({}_{4}\)** & **B** & **Confidence of Class Attribute B** \\ \hline
0 & 1 & 1 & 1 & 0 & 0.738 \\
1 & 0 & 0 & 1 & 0 & 0.707 \\
1 & 0 & 1 & 0 & 0 & 0.575 \\
1 & 0 & 1 & 1 & 1 & 0.528 \\
1 & 1 & 0 & 0 & 0 & 0.765 \\
1 & 1 & 0 & 1 & 1 & 0.605 \\
1 & 1 & 1 & 0 & 1 & 0.655 \\
1 & 1 & 1 & 1 & 1 & 0.655 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Structure function of the simplified laparoscopic surgery procedure. \[A=Pr\{\phi(\mathbf{x})=1\}=p_{1}\times q_{2}\times p_{3}\times p_{4}+p_{1}\times p _{2}\times q_{3}\times p_{4}+p_{1}\times p_{2}\times p_{3}\times q_{4}+p_{1}\times p _{2}\times p_{3}\times p_{4}=0.8766 \tag{6}\]  
\[U=1-A=0.1234 \tag{7}\]  
The system availability and system unavailability are very important reliability indices, which allow evaluating of the system as a whole. Beside these indices in reliability engineering, there are a group of indices that permit analysing the system components' influence on the system availability/reliability. The indices of this group are named \"importance measures\". These indices show the influence of every system component on system failure or functioning [40]. One of the simpler importance measures is Structure Importance (SI). SI of the \(i\)-th system component is defined as the probability of the system failure caused by the failure of this component. SI accumulates the system topological properties. This index is calculated based on the proportion of the system states that change their states depending on the i-th component's change in the space of all possible system states [40]. This index is computed based on the structure function as [40; 41]:  
\[SI_{i}=\left.N_{i}\middle/2^{n-1^{1}}\right. \tag{8}\]  
where \(N_{i}\) is number of the system state (state vectors) for which the change of the \(i\)-th system component state result in the change of the system state. According to study in [41], the number of the system states \(N_{i}\) in (8) can be defined based on Logical Differential Calculus (in particular, Direct Partial Logical Derivatives, DPLD). In Boolean algebra, the DPLD of the logical function allows indicating the function values for which the change of the \(i\)-th variable value from \(s\) to \(\overline{s}\) results in the change of the function value from \(j\) to \(\overline{j}\).