# Reliability Analysis of C\({}^{4}\)ISR Systems Based on Goal-Oriented Methodology
## 2 Methodology  
The GO method is the combination of graph theory and probability methodology [32]. The operator of the GO method connects the signal flow to simulate units of a system. Overall, in this method, 17 operators have been defined, as shown in Table 1. The operator type determines characteristics of the operator including operation rules, function, etc. [33]. Moreover, signal flow in the GO model represents the logical relationship between input and output operators. State value and state probability are basic attributes of operators and signal flow [34]. The steps of the establishment of reliability GO graph model for a system based on the GO method, generally, are as follows [14; 16; 17; 18]: (i) System identification and analysis. Identify the system that to be analyzed including its scope, failure mode, function, and relationship among each unit; (ii) Input and output determination. Ascertain the inputs and outputs of the system that already identified in Step (i); (iii) the successful operation criteria determination. A successful state includes the degraded operation of the system and it can give the minimum output signal set; (iv) GO model construction. To create a GO diagram based on the structural diagram of the system by connecting the operators according to the signal flow direction, which includes several steps:  
* Select the corresponding operator according to the function of the system unit. * Connect signal flow to operators selected. Specifically, the essence of the signal flow is the direction of the signal in the system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Operators** & **Function** & **Category** & **Operators** & **Function** & **Category** \\ \hline \hline \(\epsilon\) & 1 & Simulate two-state units & Function & 10 & AND & Logical \\ \hline \(\epsilon\) & 2 & OR & Logical & 11 & K-out-of-M & Logical \\ \hline \(\epsilon\) & 3 & Simulate three state units & Function & 12 & Choose different output path by input signal & Function \\ \hline \(\epsilon\) & & Multiple input Unit & Function & 13 & Multiple inputs and output signals & Function \\ \hline \(\epsilon\) & & Single input Unit & Function & 14 & Linear relation of multiple (input) to one (output) & Logical \\ \hline \(\epsilon\) & & Open after receiving signals & Function & 15 & Output signal depends on the probability of input signal & Logical \\ \hline \(\epsilon\) & & Close after receiving signals & Function & 16 & The unit state resumed close by receiving the control signal & Function \\ \hline \(\epsilon\) & & Delay state & Function & 17 & The unit state resumed open by receiving the control signal & Function \\ \hline \(\epsilon\) & & Output signal depends on two input signals’ state & Logical & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Standard operators in the GO method. * Number the operators and signal flow in the GO diagram. * Checking the GO model until it complies with the drawing rules of the GO method, otherwise, modify the model (repeat the above steps). The qualitative analysis of the GO method is to seek the minimum failure unit within the system, while, the quantitative analysis calculates the overall reliability of the system. The state accumulation analysis method is used to quantitatively analyze the system. Accordingly, the state value of the signal flow in the GO method is defined as \(0,\cdots,N\), where 0 represents the advanced state and \(N\) represents the failure state. Other values correspond to multiple states between 0 and \(N\). Specifically, \(P_{S}(i)\) represents the input signal with the state \(i\), and \(P_{R}(i)\) is the state probability of \(P_{S}(i)\); \(A_{S}(i)\) denotes the cumulative probability of \(P_{S}(i)\), and \(A_{R}(i)\) reflects the cumulative probability of the state of the output signal flow. The calculation of \(A_{S}(i)\) and \(A_{R}(i)\) are listed in Equation (1):  
\[ \begin{split} A_{S}(i)=\sum_{j=0}^{i}P_{S}(j),i=0,\cdots,N-1,A_{S} (N)=1\\ A_{R}(i)=\sum_{j=0}^{i}P_{R}(j),i=0,\cdots,N-1,A_{R}(N)=1\end{split}