## 4 Theoretical Analysis of the Proposed PCSDF Cooperation Scheme
### Analyses of Outage Probability in Proposed Scheme  
Outage probability (OP) is defined as the probability of transmission failure, which is one of the most crucial measures to evaluate the entire wireless communications setup. A transmission interruption occurs when the link capacity cannot attain the required user rate. Therefore, for the PCSDF scheme, an interruption occurs in the three situations described below. 1. The outage probability \(P_{1}\) that refers to that the source S re-transmits messages to the destination D when \(\Omega_{k}=\emptyset\). 2. The outage probability \(P_{2}\) relates to the failure of the optimal relay when \(\Omega_{k}\
eq\emptyset\) and \(\beta_{r_{opt}}/\beta_{sd}\leq\theta\). 3. The outage probability \(P_{3}\) of the optimal relay is used to forward messages when \(\Omega_{k}\
eq\emptyset\) and \(\beta_{r_{opt}}/\beta_{sd}>\theta\). Thus, the OP of the entire system is expressed as  
\[P=P_{1}+P_{2}+P_{3}. \tag{25}\]  
Subsequently, the above three situations can be analyzed as follows. Situation I. When only link S-D is used to transmit messages, an interruption occurs until the channel capacity \(C_{sd}\) is less than the message transmission rate \(V\); the probability of \(\Omega_{k}=\emptyset\) is calculated as  
\[P(\Omega_{k}=\emptyset)=\prod\limits_{i=1}^{N}\Pr(C_{sT_{i}}<V). \tag{26}\]  
Thus, the OP can be expressed as  
\[ \begin{array}{l}P_{1}=P(\Omega_{k}=\emptyset)P(C_{sd}<V)\\ =\prod\limits_{i=1}^{N}\Pr(C_{sT_{i}}<V)\Pr(C_{sd}<V)\\ =(1-\exp(-\frac{2^{V}-1}{\Gamma_{sr}}))^{N}(1-\exp(-\frac{2^{V}-1}{\Gamma_{sd} }))\end{array} , \tag{27}\]  
where \(\Gamma_{sr}\) and \(\Gamma_{sd}\) are the average SNR of links S-R and S-D, respectively. Situation II. When \(K\) relays are selected to be the candidate relays from all \(M\) relays, the probability of \(\Omega_{k}\
eq\emptyset\) is expressed as  
\[ \begin{array}{l}\Pr(\Omega_{k}\
eq\emptyset)=C_{M}^{K}\prod\limits_{i\in \Omega_{k}}\Pr(C_{sT_{i}}\geq V)\prod\limits_{i\
otin\Omega_{k}}\Pr(C_{sT_{i }}<V)\\ =C_{M}^{K}\exp(-\frac{2^{V}-1}{\Gamma_{sr}})^{K}(1-\exp(-\frac{2^{V}-1}{\Gamma_ {sr}}))^{M-K}\end{array} , \tag{28}\]  
where \(C_{M}^{K}=\frac{M!}{K!(M-K)!}\). Then, the probability of \(\beta_{r_{opt}}/\beta_{sd}\leq\theta\) can be computed as  
\[ \begin{array}{l}\Pr(\beta_{r_{opt}}/\beta_{sd}\leq\theta)=\Pr(P_{r}|h_{rd _{opt}}|^{2}/N_{0}\leq\theta P_{s}|h_{sd}|^{2}/N_{0})\\ =\Pr(\gamma_{rd_{opt}}\leq\theta\gamma_{sd})\\ =\prod\limits_{i=1}^{k}\Pr(\gamma_{rd}<\theta\gamma_{sd})\\ =(1-\exp(-\frac{\theta\gamma_{sd}}{\Gamma_{sd_{gt}}}))^{k}\end{array} . \tag{29}\]With an actual information transmission rate \(V\), \(C_{\text{PCSDF}}\) is less than \(V\) when the optimal relay is out of work. This can be written as \(\frac{1}{2}\times\log_{2}(1+\gamma_{sd(1)}+\gamma_{sd(2)})<V\). Thus, the OP \(P_{2}\) is expressed as  
\[ \begin{array}{l}P_{2}=\Pr(\Omega_{k}\
eq\varnothing\cap\beta_{r_{opt}}/\beta_ {sd}\leq\theta)\\ \qquad\times\Pr(\gamma_{sd(1)}+\gamma_{sd(2)}<2^{2V}-1)\\ \qquad=C_{M}^{K}\exp(-\frac{2^{V}-1}{\Gamma_{M}})^{K}(1-\exp(-\frac{2^{V}-1}{ \Gamma_{M}}))^{M-K}\\ \qquad\times(1-\exp(-\frac{\theta_{T_{sd}}}{\Gamma_{M_{opt}}}))^{k}(1-\exp(- \frac{2^{2V}-1}{\gamma_{sd(1)}+\gamma_{sd(2)}}))\end{array} . \tag{30}\]  
Situation III. When the optimal relay is selected to take part in cooperation, the channel capacity of the system can be rewritten as \(C_{\text{PCSDF}}=\frac{1}{2}\times\log_{2}(1+\gamma_{sd(1)}+\gamma_{rd_{opt}})\). Similarly, an interruption occurs when \(C_{\text{PCSDF}}\) is less than information transmission rate \(V\), i.e., \(C_{\text{PCSDF}}<V\). This can be rewritten as  
\[\gamma_{sd(1)}+\gamma_{rd_{opt}}<2^{2V}-1. \tag{31}\]  
Thus, the OP \(P_{3}\) can be calculated as  
\[