# Reliability Analysis of C\({}^{4}\)ISR Systems Based on Goal-Oriented Methodology
## 2 Methodology
 \tag{1}\]  
The reliability analysis is to find the minimum cut set of the GO model. For a system with \(M\) operators, find out the smallest cut set of the system until the states of other operators are 0 and only one operator whose state value is 1, calculate the probability that the system can run successfully according to the probability analysis method. After the first-order minimum failure sets of the system had been ascertained, repeat the above step until the second-order minimum failure set of the system is obtained. The qualitative analysis flowchart of the GO model is shown in Figure 1.  
Figure 1: The qualitative analysis flowchart of the GO model. For a complicated system such as a C\({}^{4}\)ISR system, common cause failures and common signals introduce uncertainty to the results concluded by the GO model. Accordingly, data pretreatments, also known as data corrections, are required. Generally, data correction of GO models, for both common signals and common cause failures, obeys the following rules:  
Regarding the correction of reliability data with common signals. The common signal is a signal that the input signal of multiple operators simultaneously. Hence, the state probability of which can be included in all subsequent signal flows related. For a system with \(M\) common signals, \(S_{j}\), \(j=1,2,\cdots,M\) and only one output signal R. The state probability of the output signal is:  
\[P_{R}=f(P_{S1},P_{S2},\cdots,P_{SM}) \tag{2}\]  
where, \(P_{R}\) is the success probability of the output signal, \(P_{S1},P_{S2},\cdots,P_{SM}\) denote \(M\) success probabilities of common signals. For one common signal, denote \(P_{S1}=0\) (The common signal S1 fails) and the success probability of the system output signal \(P_{R0}\), \(P_{S1}=1\) (The common signal S1 succeeds) and the success probability of the system output signal \(P_{R1}\). Hence, the success probability of the output signal is:  
\[P_{R}=(1-P_{S1})P_{R0}+P_{S1}P_{R1} \tag{3}\]  
For two common signals S1 and S2, whose success probabilities of the common signals are \(P_{S1}\) and \(P_{S2}\), respectively. The cumulative probability of the state of the output signal R is \(P_{R}\). According to the GO operation rule, \(P_{R}\) can be calculated by:  
\[P_{R}=c_{0}+c_{1}P_{S1}+c_{2}P_{S2}+c_{3}P_{S1}P_{S2} \tag{4}\]  
where, \(c_{0},c_{1},c_{2},c_{3}\) are constant parameters. Furtherly, Equation (4) can be reformed to to:  
\[P_{R}=(1-P_{S1})(1-P_{S2})P_{R00}+(1-P_{S1})P_{S2}P_{R01}+P_{S1}(1-P_{S2})P_{R 10}+P_{S1}P_{S2}P_{R11} \tag{5}\]  
In which, \(P_{R00}=c_{0}\), \(P_{R01}=c_{0}+c_{1}\), \(P_{R10}=c_{0}+c_{2}\), and \(P_{R11}=c_{0}+c_{1}+c_{2}+c_{3}\), and \(P_{R00},P_{R01},P_{R10},P_{R11}\) are probabilities of the system output signal in the success state under the condition that the common signals S1 and S2 are in failure-fault (\(P_{S1}=0,P_{S2}=0\)), failure-success (\(P_{S1}=0,P_{S2}=1\)), success-failure (\(P_{S1}=1,P_{S2}=0\)) and success-success (\(P_{S1}=1,P_{S2}=1\)) states, respectively. For \(M\) common signals \(S_{m}\), \(m=1,2,\cdots,M\), the success probability of each signal is \(P_{Sm}\), the success probability of the output signal is \(P_{R}\), The cumulative probability of the state of the output signal can be obtained by:  
\[P_{R}=\sum\
olimits_{K_{1}=0}^{1}\sum\
olimits_{K_{2}=0}^{1}\cdots\sum \
olimits_{K_{M}=0}^{1}P_{RK_{1}K_{2}\cdots K_{M}}\prod\limits_{m=1}^{M}[(1-P_ {Sm})(1-K_{m})+P_{Sm}K_{m}] \tag{6}\]  
where, \(P_{RK_{1}K_{2}\cdots K_{M}}\) represents the probability of the success state of the output signal under \(M\) common signals; \(K_{m}\) represents the correction coefficient of the \(m\)-th common signal, \(K_{m}\)= 0, denotes fault state, while, \(K_{m}\)= 1, reflects success state. Moreover, the probability of a single signal can be calculated as:  
\[\prod\limits_{m=1}^{M}[(1-P_{Sm})(1-K_{m})+P_{Sm}K_{m}] \tag{7}\]  
As for the reliability data correction with common cause failures. Common cause failures denote that several failures share the same common cause in a system.