# Modeling the Energy Performance of LoRaWAN
## 5 Evaluation
### Energy Cost of Data Delivery
In acknowledged transmission, this effect is emphasized, since retransmissions are needed. For example, for DR5, the energy cost of data delivery in acknowledged transmission for \(T_{notif}\) = 1 min increases by 60% when \(p_{call}\) increases from 0.1 to 0.3, while in unacknowledged mode, the energy cost per delivered bit increases by 12.8%. Figure 19: Energy cost of data delivery as a function of \(T_{Notif}\), for different DR settings, BER = 0, \(p_{call}\) = 0, and for 1-byte payload and maximum-sized payload that corresponds to each DR.  
Figure 18: Energy cost of data delivery as a function of \(T_{Notif}\), for different DR settings, and for BER = 0 and \(p_{call}\) = 0. Results for DR4, DR5 and DR6 overlap in the figure. Finally, we analyze and discuss the impact of payload (and thus, message) size on the energy cost per delivered bit in a dense LoRaWAN network, where collisions may occur. Assuming that end-devices perform acknowledged transmissions, network behavior can be modeled by an Aloha access protocol, as an approximation [7]. Under these conditions, the energy cost per delivered bit can be computed as the result of dividing the energy cost of a message transmission (including its retransmissions) by the number of payload bits carried. The energy consumed in a message transmission (including retransmissions) is roughly proportional to the number of transmission attempts per message. In fact, the amount of energy consumed in each retransmission (which includes the energy consumed over _ACK_TIMEOUT_) is much larger than the energy consumed during the actual transmission state, regardless of the message size, with an accuracy that increases with the DR since transmission time decreases, and with the load offered to the network. On the other hand, in Aloha, the expected number of transmission attempts per message is \(e^{2G}\), where \(G\) is the total load offered to the network, and thus the expected energy consumed to deliver a message is \(a\cdot e^{2G}\), where \(a\) is the total amount of energy consumed in each transmission attempt. Note that \(G\) is proportional to the message size, i.e., \(l_{pay}+l_{hand}\), where \(l_{hand}\) denotes the total size of the message headers. Therefore, the energy cost of data delivery under the described conditions can be approximated by (32):  
\[EC_{delivery}\approx\frac{a\cdot e^{2G}}{l_{pay}}=\frac{a\cdot e^{k\cdot(l_{pay }+l_{head})}}{l_{pay}}=\,\alpha\cdot\frac{e^{k\cdot l_{pay}}}{l_{pay}} \tag{32}\]  
Figure 21: Impact of \(p_{call}\) on the energy cost of data delivery, as a function of \(T_{Neif}\), for both unacknowledged and acknowledged transmission, for BER = 0, and for DR0 and DR5. Figure 20: Impact of BER on the energy cost of data delivery, as a function of \(T_{Neif}\), for both unacknowledged and acknowledged transmission, for \(p_{call}\) = 0, and for DR0 and DR5. where \(\alpha\) is a constant expressed in Joules, that can be computed by using (33), and \(k\) is a constant expressed in bit\({}^{-1}\), which depends on the transmission bit rate and on the total message rate offered to the network. \[\alpha=a\cdot e^{k\cdot l_{band}} \tag{33}\]  
Note that the model presented cannot accurately capture the DR decrease mechanism for retransmissions (see Section 3.3.2), when used, since Aloha assumes all packet transmissions have the same duration. Nevertheless, it allows to qualitatively capture behavior of the energy cost of data delivery as a function of packet size. As shown in Figure 22, impact of payload size on the energy cost per delivered bit depends on the value of \(k\). All curves follow a \"U\" shape with an optimal payload size that minimizes the energy cost per delivered bit. The \"U\" shape of the curves can be explained, on the one hand, by the fact that very large messages will lead to high energy cost per delivered bit due to a high number of collisions.