# Modeling the Energy Performance of LoRaWAN
## 4 Modeling LoRaWAN End-Device Current Consumption
### Unacknowledged Transmission
Therefore, the calculated end-device lifetime results provided in this paper provide an upper bound on the actual end-device lifetime that can be expected. Finally, another important performance parameter is the energy cost of data delivery, \(EC_{delivery\_unACK}\), which provides the energy consumed by the end-device per each delivered bit of data payload in unacknowledged mode, as shown below:  
\[T_{w2w}=RECEIVE\_DELAY\_2-RECEIVE\_DELAY\_1-T_{rx1w} \tag{4}\]0 \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(DR\)} & \multirow{2}{*}{\(SF\)} & \multirow{2}{*}{\(T_{sym}\)} & \multirow{2}{*}{\(T_{presemble}\)} & \multirow{2}{*}{\(T_{rx1w}\)} & \multirow{2}{*}{\(T_{rx2w}\)} & \multirow{2}{*}{\(DE\)} & \multicolumn{2}{c}{**FRM Payload**} & \multicolumn{2}{c}{\(T_{tx}\) Max} & \multicolumn{2}{c}{\(T_{tx}\) Min} \\ \cline{5-10}  & & & & & & & & **Max** & **Min** & **Uplink** & **Downlink** \\ \cline{3-10}  & & & **(ms)** & **(ms)** & **(ms)** & & & **(bytes)** & **(bytes)** & **(ms)** & **(ms)** \\ \hline
0 & 12 & 32.77 & 401.41 & 262.14 & 33.02 & 1 & 51 & 0 & 2793.5 & 991.8 \\
1 & 11 & 16.38 & 200.70 & 131.07 & 16.64 & 1 & 51 & 0 & 1560.6 & 577.5 \\
2 & 10 & 8.19 & 100.35 & 98.30 & 8.45 & 0 & 51 & 0 & 698.4 & 288.7 \\
3 & 9 & 4.10 & 50.18 & 49.15 & 4.35 & 0 & 115 & 0 & 676.9 & 144.4 \\
4 & 8 & 2.05 & 25.09 & 24.58 & 2.30 & 0 & 242 & 0 & 707.1 & 72.2 \\
5 & 7 & 1.02 & 12.54 & 12.29 & 1.28 & 0 & 242 & 0 & 399.6 & 41.2 \\
6 & 7 & 0.51 & 6.27 & 6.14 & 0.64 & 0 & 242 & 0 & 199.8 & 20.6 \\
7 & - & 0.02 & 0.48 & 1.28 & 1.28 & - & 242 & 0 & 42.4 & 3.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Summary of values for \(T_{tx}\), \(T_{rx1w}\) and \(T_{rx2w}\), along with relevant parameter settings. We have assumed an 8-symbol preamble length, a CR of 4/5 (except for the 20-bit physical header, for which a CR of 4/8 is used), and a bandwidth BW = 125 kHz (except for DR6, with a bandwidth BW = 250 kHz, and for DR7, which is based on FSK). \(T_{tx}\) max is obtained by considering the maximum frame payload (FRM Payload) size for each Data Rate (DR), while \(T_{tx}\) min corresponds to the time to transmit a data message that carries no data (e.g., an acknowledgment). In the latter case, the physical frame length is the contribution of the physical header (PHDR) and PHDR_CRC fields, the MAC Header (MHDR), the FHDR and the MIC fields (see Section 3.3.1), leading to a total length of 14.5 bytes plus 8 preamble symbols. Note that the CRC is only present in uplink transmissions (see Section 3.2.2). For DR7, an additional margin should be added to the \(T_{rx1w}\) and \(T_{rx2w}\) values to account for possible drifts of the oscillator used for the timer that controls a receive window start. where \(V\) denotes the voltage and \(E[l_{delivery\_unACK}]\) indicates the expected amount of data successfully delivered by the end-device per data frame transmitted. Note that in the previous equation, the numerator computes the energy consumed by the device during \(T_{notif}\). Let \(l_{phy}\) be the FRM Payload field size (i.e., the amount of data carried in the payload of the data message sent by the end-device), and let \(l_{Data}\) be the total size of the data message, including all headers. Let \(b\) denote the BER as introduced in the first paragraph of Section 4. Since in unacknowledged transmission there is a single transmission attempt, which may suffer bit errrors, \(E[l_{delivery\_unACK}]\) is determined as:  
\[E\Big{[}l_{delivery\_unACK}\Big{]}=\ l_{pay}\cdot(1-b)^{l_{Data}} \tag{14}\]  
Finally, let us assume that collisions may occur, i.e., an end-device message transmission may overlap with messages transmitted by other end-devices connected to the same gateway. Let \(p_{call}\) be the probability that a data message transmitted by an end-device collides with at least another message transmission.