## 4 Mission Reliability Modeling, Allocation, and Analysis Method Design
### Reliability Analysis Methodology Design
#### 4.3.2 Quantitative Calculation of Reliability Based on GO Method  
The unit mission reliability is calculated based on the running time and failure rate of each unit mission. The system mission reliability can be obtained by combining the reliability of the unit's completed mission with the transfer relationship. The unit reliability is calculated using Equation (17) as follows. Since the software itself does not fail, the software reliability is calculated as 1:  
\[R(t)=e^{-\lambda t} \tag{17}\]  
where \(\lambda\) is the unit failure rate; \(R(t)\) is the reliability of the unit at the \(t\) moment. The state value of the signal flow in the GO method is defined as \(0,\cdots,N\), with 0 representing advance, \(1,\cdots,N-1\) representing multiple states, and \(N\) representing fault. The state probability of the input signal is defined as \(P_{\mathrm{S}}(i)\) when the state value is \(i\), the \begin{table}
\begin{tabular}{c c c} \hline \hline
**AADL Unit** & **GO Operator Types** & **Operator** \\ \hline RF integrated equipment & 5 & Single input signal \\ Optical frequency radio frequency & 5 & Single input signal \\ Sonar & 5 & Single input signal \\ Display device & 1 & Two state unit \\ Computing service equipment & 1 & Two state unit \\ Remote air defense equipment & 1 & Two state unit \\ Short range air defense equipment & 1 & Two state unit \\ Electronic interference & 1 & Two state unit \\ Torpedo & 1 & Two state unit \\ Against underwater defense weapons & 1 & Two state unit \\ Short range weapons to land & 1 & Two state unit \\ Remote weapons to land & 1 & Two state unit \\ Dot generation software & 6 & Signal conducting unit \\ Target recognition software & 6 & Signal conducting unit \\ Situation display software & 6 & Signal conducting unit \\ Track generation software & 6 & Signal conducting unit \\ Situation recognition software & 6 & Signal conducting unit \\ Threat assessment software & 6 & Signal conducting unit \\ Decision allocation software & 6 & Signal conducting unit \\ Target acacusion software & 6 & Signal conducting unit \\ Network Device 1 & 1 & Two state unit \\ Network Device 2 & 1 & Two state unit \\ \hline \hline \end{tabular}
\end{table} 
Table 6: GO method operators corresponding to AADL units. Figure 8: Mapping relationship between the AADL and GO graph models. state probability of the output signal as \(P_{R}(i)\), and the sum of the state probability of signal flow as 1. The state cumulative probability of the input signal is defined as \(A_{S}(i)\), and the state cumulative probability of the output signal as \(A_{R}(i)\). The state cumulative probability calculation formulas of the input and output signals are shown in Equation (18). \[ \begin{array}{l}A_{S}(i)=\sum\limits_{j=0}^{i}P_{S}(j)\;\mathrm{i}=0,\cdots, \mathrm{N}-1,\,A_{S}(N)=1\\ A_{R}(i)=\sum\limits_{j=0}^{i}P_{R}(j)\;\mathrm{i}=0,\cdots,\mathrm{N}-1,\,A_{ R}(N)=1\end{array}  \tag{18}\]  
The state cumulative probability \(A(i)\) represents the sum of all state probabilities of the signal flow. Through the definition and operation rules of the standard operator types of the GO method, the state accumulation probability formulas for the operators of type I (two-state unit), type II (or gate), type V (signal generator), and type VI (unit that conducts with an action signal) required for modeling the GO diagram model of the ship electronic system are derived by using Equation (18). The derived state cumulative probability formulas are presented below. \[A_{R}(i)=A_{S}(i)\cdot P_{C}(1),\;\mathrm{i}=0,1,\cdots,\mathrm{N}-1 \tag{19}\]  
\[ \begin{array}{l}P_{R}(0)=A_{R}(0)\\ P_{R}(i)=A_{R}(i)-A_{R}(i-1),\;\mathrm{i}=1,\cdots,\mathrm{N}\end{array}  \tag{20}\]  
\[ \begin{array}{l}A_{R}(i)=\sum\limits_{j=0}^{i}P_{R}(j),\;\mathrm{i}=0,\cdots, \mathrm{N}-1\end{array}  \tag{21}\]  
\[