# A System-Level Methodology for the Design of Reliable Low-Power Wireless Sensor Networks
## 5 Energy-Latency-Reliability Trade-Off
### Packet Success Probability
#### 5.2.3 ARQ-Based Retransmissions  
In this section, the notation \(\overline{z}\) is used to denote \(\overline{z}=1-z\), where the real valued variable \(z\in[0,1]\). In this analysis, the \(L\)-Rep-ACK [39] protocol, which achieves a better feedback channel reliability by retransmitting ACK/NACK packets, is considered. Case studies presented in Section 6 include examples and discussions about the matter. The transmission is considered successful only if the transmitter receives \(L\) ACK packets for one data packet where \(L>1\). It should be noted that the particular case where \(L=1\) is the regular SAW-ARQ protocol. The same bit error probability, \(P_{e}\), during transmission of ACK and non-acknowledgement (NACK) packets of \(N_{f}=40\) bits through the feedback channel is considered. The probability of successfully receiving a data packet at B after a maximum of \(R\)\(L\)-Rep-ACK transmissions is given by  
\[P_{R}^{L-Rep-ACK}=1-P_{out}^{L-Rep-ACK}, \tag{27}\]  
where \(L\) is the time diversity order (i.e., number of ACK/NACK transmissions) and \(P_{out}^{ARQ}\) is the outage probability [39] of the \(L\)-Rep-ACK protocol and given by  
\[P_{out}^{L-Rep-ACK}=\sum_{j=1}^{R-1}\overline{p}^{j}P_{pe}^{L}\big{(}1-P_{pe}^ {L}\big{)}^{j-1}+\overline{p}^{R}\big{(}1-P_{pe}^{L}\big{)}^{R-1}, \tag{28}\]  
where \(j\) is the index of the transmission attempt and \(P_{pe}=\overline{P}_{f}=1-(1-P_{e})^{N_{f}}\) is the failure probability of an ACK packet. Figure 19 shows the impact of the maximum allowed number of 1-Rep-ACK transmission attempts, \(R\), on the packet success probability.  
Figure 18: Impact of the number of blind transmission attempts on the packet success probability.  
Moreover, as illustrated in Figure 20, using a maximum of \(R\) ARQ transmissions does not mean that \(R\) is always reached and that is the reason why it is possible to achieve better energy and latency efficiencies when using acknowledgements. The probability that a packet will be received successfully at the \(j^{th}\) packet transmission attempt is given by  
\[P_{j}^{L-R\rho-ACK}=P_{R}^{L-R\rho-ACK}\times\left(1-P_{R}^{L-R\rho-ACK}\right)^ {j-1}. \tag{29}\]  
When using \(L\)-Rep-ACK retransmissions, the required time to send a packet is given by  
\[t_{TRX}^{ARQ}=\sum_{j=1}^{R}P_{j}^{ARQ}\bigg{[}j\bigg{[}\bigg{(}\frac{l_{data} }{D_{R}}+\frac{L\,l_{ack}}{D_{R}}\bigg{)}+L\,t_{sb}\bigg{]}+t_{id}(j-1)\bigg{]}, \tag{30}\]  
where \(t_{sb}\) is the time spent in standby mode after sending the packet and before receiving the acknowledgement and \(l_{ack}\) is the length of the acknowledgement packet. Therefore, the energy consumption of the transceiver can be expressed as follows  
\[E_{TRX}^{ARQ}=V\sum_{j=1}^{R}P_{j}^{ARQ}\bigg{[}j\bigg{[}\bigg{(}\frac{l_{data }}{D_{R}}+\frac{L\,l_{ack}}{D_{R}}\bigg{)}(I_{TX}+I_{RX})+2\,L\,I_{sb}t_{sb} \bigg{]}+2l_{id}t_{id}(j-1)\bigg{]}. \tag{31}\]  
Figure 19: Impact of the number of 1-Rep-ACK transmission attempts on the packet success probability.  
Figure 20: Packet success probability at the \(j^{th}\) packet.  
Using the previous analytical results, the three reliable data transfer schemes can be objectively compared while taking into account the achievable reliability, the energy, and latency costs.