## 2 Theoretical Study
### Listening Nodes Competing for Signal Retransmission  
In the first stage of cooperative communication, the source node broadcasts a signal to the surrounding nodes, and other nodes, in addition to the best relay node, receive a copy of the signal as well. We define other nodes as listening nodes. During the second stage of forwarding the signal, the listening node also has the ability to retransmit the signal. Based on this, the listening node cooperative communication automatic retransmission request (LCARQ) policy is proposed. Figure 2 shows a schematic diagram of the process of retransmission involving the listening node. In the second stage of cooperative communication, after the relay node fails to forward the signal, the listener node sends the retransmission signal instead of the relay node. When the destination node sends the NACK signal, the relay node and the listener node determine the power needed for the signal to occur by competing. The process of retransmission competition is an innovative design and the competition process for retransmission power is a complex discussion process. The competitive power is determined by the channel quality of the node.  
The automatic retransmission request policy based on listening nodes was determined based on different assumptions.  
During the second stage of communication, if the RD channel quality of the listening node is less than the RD channel quality of the relay node, the retransmission signal is sent by the best relay node. It works in the same way as the signal-forwarding process of the cooperative communication. At this time, the node calculation process of signal retransmission can be obtained by the formula.  
\[\mathrm{R_{rd}}=\mathrm{max}_{i}\Big{(}|h_{i,j}|^{2}\Big{|}\mathrm{min}\Big{(} |h_{S,i}|^{2},|h_{i,D}|^{2}\Big{)}\Big{)},\,i\,=\,(1,2,3,\cdots,n_{1}) \tag{6}\]  
In the second stage of the retransmission process. A listening node is defined as a signaling retransmission node if the RD channel quality of the listening node is higher than the channel quality of the relay node. First, the RD communication link of the listening node is calculated and the listening node, whose value is greater than that of the relay node, is selected using the channel quality as an indicator. At least one listener node replaces the relay node. When all nodes with higher channel quality are selected, the forwarding nodes are identified in descending order. The defined forwarding node replaces the relay node to send a retransmission signal and the second phase of the cooperative communication ends.  
\[\mathrm{R_{rd}}=\mathrm{max}_{j}\Big{(}|h_{dj,D}|^{2}\Big{|}\mathrm{min}\Big{(} |h_{S,i}|^{2},|h_{j,D}|^{2}\Big{)}\Big{)},\,j\,=\,(1,2,\cdots,n_{2}). \tag{7}\]  
In summary, the automatic request retransmission policy for cooperative communication based on listening nodes is defined as follows. The listening node sends a retransmission signal when the listening node in the communication has competing power. When the listening node in the communication does not have competing power, the best relay node forwards the signal. The conditions for the end of the retransmission process contain  
Figure 2: Diagram of the retransmission request control policy based on listening nodes.  
retransmission four times or the signal is successfully decoded. The competition theoretic payoff of LCARQ can be summarized as follows.  
\[\text{R}_{\text{rd}}=\left\{\begin{array}{ll}\max_{i}\left(\left|\text{h}_{r,i,D}\right|^{2}\right|\text{min}\left(\left|\text{h}_{S,i}\right|^{2},\left| \text{h}_{i,D}\right|^{2}\right)\right),i\in\{1,2,\cdots,N_{1}\}&D=0\\ \max_{j}\left(\left|\text{h}_{dj,D}\right|^{2}\right|\text{min}\left(\left| \text{h}_{S,j}\right|^{2},\left|\text{h}_{j,D}\right|^{2}\right)\right),j\in\{ 1,2,\cdots,N_{2}\}&D\
eq 0\end{array}\right. \tag{8}\]