## 3 Transmission Scenarios
### UR-MCJST Scenario
Since \(Y_{C|I_{i4},\frac{\alpha}{2}}\) is typically assumed to be known at receiver \(D\), we can subtract the second and third terms from \(y_{d}\) in Equation (12) to yield:  
\[P_{\text{s}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right \rfloor}\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]+\frac{1}{K}\sum_{t_{2} =\left\lfloor\frac{\alpha}{2}K+1\right\rfloor}^{\left\lfloor\frac{\alpha}{2} K+1\right\rfloor}\mathbb{E}[X^{2}_{\{t_{2},\frac{\alpha}{2}\}}] \tag{2}\]6 \[P_{\text{s}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right \rfloor}\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]+\frac{1}{K}\sum_{t_{2} =\left\lfloor\frac{\alpha}{2}K+1\right\rfloor}^{\left\lfloor\frac{\alpha}{2} K+1\right\rfloor}\mathbb{E}[X^{2}_{\{t_{2},\frac{\alpha}{2}\}}] \tag{2}\]7  
We can calculate the \(SNR\) at \(D\) as follows:  
\[P_{\text{s}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right \rfloor}\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]+\frac{1}{K}\sum_{t_{2} =\left\lfloor\frac{\alpha}{2}K+1\right\rfloor}^{\left\lfloor\frac{\alpha}{2} K+1\right\rfloor}\mathbb{E}[X^{2}_{\{t_{2},\frac{\alpha}{2}\}}] \tag{2}\]8 \[P_{\text{s}} = \frac{1}{K}\sum_{t_{1}=1}^{\left\lfloor\frac{\alpha}{2}K\right \rfloor}\mathbb{E}[X^{2}_{\{t_{1},\frac{\alpha}{2}\}}]+\frac{1}{K}\sum_{t_{2} =\left\lfloor\frac{\alpha}{2}K+1\right\rfloor}^{\left\lfloor\frac{\alpha}{2} K+1\right\rfloor}\mathbb{E}[X^{2}_{\{t_{2},\frac{\alpha}{2}\}}] \tag{2}\]9  
where \(\gamma_{sr}=\frac{P_{s}|h_{sr}|^{2}}{N_{0}}\), \(\gamma_{sd}=\frac{P_{s}|h_{rd}|^{2}}{N_{0}}\), \(\gamma_{C|I,r}=\frac{P_{C|I,r}|h_{C|I,r}|^{2}}{N_{0}}\) and \(\gamma_{rd}=\frac{|h_{rd}|^{2}}{N_{0}}\). Similarly, from (7), we can derive the received \(SNR\) at the relay as:  
\[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]0 \[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]1  
The achievable rate at the destination \(R_{D}\) for the multiple jammers case can be computed as:  
\[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]2  
Similarly, the achievable rate at the untrusted relay is given by:  
\[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]3where \(\gamma_{R,CJ_{i}}\) is the \(SNR\) at the \(i\)th relay given by (15). The achievable S-D secrecy rate is the measure of how much information is transmitted from \(S\) to \(D\) without being revealed to the eavesdropper, which is given by:  
\[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]4  
Each CJ transmits an average power of \(\frac{\kappa_{1}}{\beta}\), as given by (6), during the third phase. Hence, the SNR at the CJ's destination is given by:  
\[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]5 \[+\frac{1}{K}\sum_{t_{3}=\left\lfloor\alpha K+1\right\rfloor}^{K }\mathbb{E}[X^{2}_{\{t_{3},1-\alpha\}}]\]6  
Here, \(h_{C_{i}D_{i}}\) denotes the fading channel coefficient between the \(i\)th jammer and the corresponding destination \(DJ_{i}\) of that jammer. Additionally, the power of the \(i\)th jammer, represented as \(P_{CJ_{i}}\) is subject to the power budget constraint \(\overline{P}CJ_{i}\), which ensures that \(PCJ_{i}\) does not exceed \(\overline{P}_{CJ_{i}}\). Notice that there is interference from other CJs, who transmit to their destinations at the same time. (The utility rate of a node functioning as a cooperative jammer is determined by its attainable reliable communication rate toward its destination \(DJ_{i}\), considering the time-bandwidth fraction \((1-\alpha)\). This utility rate is evaluated in relation to the cost associated with the total transmission power.)