## 3 Analysis of Three-Stage Incremental Cooperative Communication
### EE Analysis  
We analyse the EE for three-stage incremental cooperation according to the energy model given in [14]. This model considers the energy required to run the circuitry of the transmitter and receiver for both data and ACK/NACK packets. The total energy consumed in the transmission of a data packet is computed below for three-stage relaying process. \[ \begin{split}& EE_{CC,DATA}^{(3)}=[(E_{TX_{dec}}+4E_{RX_{dec}}+ \frac{P_{T}}{R})(1-PER_{SD})\\ &+(2E_{TX_{dec}}+5E_{RX_{dec}}+2\frac{P_{T}}{R})PER_{SD}(1-PER_{ SR_{1}})\\ &+(2E_{TX_{dec}}+5E_{RX_{dec}}+2\frac{P_{T}}{R})PER_{SD}PER_{SR_{ 1}}(1-PER_{SR_{2}})\\ &+(2E_{TX_{dec}}+5E_{RX_{dec}}+2\frac{P_{T}}{R})PER_{SD}PER_{SR_{ 1}}PER_{SR_{2}}(1-PER_{SR_{3}})\\ &+(2E_{TX_{dec}}+5E_{RX_{dec}}+2\frac{P_{T}}{R})PER_{SD}(1-PER_{ SR_{1}})PER_{R_{1}D}\\ & PER_{SR_{2}}(1-PER_{SR_{3}})\\ &+(3E_{TX_{dec}}+6E_{RX_{dec}}+3\frac{P_{T}}{R})PER_{SD}(1-PER_{ SR_{1}})\\ & PER_{R_{1}D}(1-PER_{SR_{2}})PER_{R_{2}D}PER_{SR_{3}}\\ &+(4E_{TX_{dec}}+7E_{RX_{dec}}+4\frac{P_{T}}{R})PER_{SD}(1-PER_{ SR_{1}})\\ & PER_{R_{1}D}(1-PER_{SR_{2}})PER_{R_{2}D}(1-PER_{SR_{3}})\\ &+(E_{TX_{dec}}+4E_{RX_{dec}}+\frac{P_{T}}{R})PER_{SD}PER_{SR_{ 1}}PER_{SR_{2}}PER_{SR_{3}}](L+H),\end{split}  \tag{5}\]  
where, \(L\) is the packet size and \(H\) is the overhead size in bits. \(E_{TX_{dec}}\) and \(E_{RX_{dec}}\) are the energies required for the transmitter and receiver electronics in transmitting and receiving one bit, respectively. \(R\) is the data rate. We find the total energy consumption of all of the events in which packet transmission is successful:  
1. The probability of successful direct transmission is \((1-PER_{SD})\). Three relays overhear the packet, which consumes receiving energy, \((E_{TX_{dec}}+4E_{RX_{dec}}+\frac{P_{T}}{R})\). 2. The direct link \(S-D\) fails, and \(R_{1}\) correctly receives and decodes the message from the source. \(R_{1}\) forwards the packet to the destination with probability \(PER_{SD}(1-PER_{SR_{1}})\), which results in a total energy consumption per bit of \((2E_{TX_{dec}}+5E_{RX_{dec}}+2\frac{P_{T}}{R})\). 3. In case \(S-D\)and \(S-R_{1}\) links fail and the \(S-R_{2}\) link is error free. The energy consumption is the same as in (ii). * In case \(S-D\), \(S-R_{1}\) and \(S-R_{2}\) links fail and the \(S-R_{3}\) link is error free. The energy consumption is the same as in (ii). * The \(S-D\) link fails; the \(S-R_{1}\) link is error free; and \(R_{1}\) decodes and forwards the message to the destination. \(S-R_{2}\) and \(R_{1}-D\) links fail, and the \(S-R_{3}\) link is error free. The probability of this event is \(PER_{SD}(1-PER_{SR_{1}})PER_{R_{1}D}PER_{SR_{2}}(1-PER_{SR_{3}})\), and the energy consumption per bit is the same as in (ii). * The \(S-D\) link fails; \(S-R_{1}\) and \(S-R_{2}\) links are error free; and \(R_{1}-D\), \(R_{2}-D\) and \(S-R_{3}\) links are in error with a total probability of \(PER_{SD}(1-PER_{SR_{1}})PER_{R_{1}D}(1-PER_{SR_{2}})PER_{R_{2}D}PER_{SR_{3}}\). The energy consumption per bit is \((3E_{TX_{slice}}+6E_{RX_{slice}}+3\frac{P_{T}}{R})\). * The direct link fails; \(S-R_{1}\), \(S-R_{2}\) and \(S-R_{3}\) are error-free links; whereas, \(R_{1}-D\) and \(R_{2}-D\) links are in error with a total probability of \(PER_{SD}(1-PER_{SR_{1}})PER_{R_{1}D}(1-PER_{SR_{2}})PER_{R_{2}D}(1-PER_{SR_{3}})\). The energy consumption per bit is \((4E_{TX_{slice}}+7E_{RX_{slice}}+4\frac{P_{T}}{R})\). * All four links from the source to the destination and relays fail with probability \(PER_{SD}PER_{SR_{1}}PER_{SR_{2}}PER_{SR_{3}}\). The energy consumption per bit for this event is \((E_{TX_{slice}}+4E_{RX_{slice}}+\frac{P_{T}}{R})\).