## 3 Methodology
### Battery Lifespan  
Choosing the optimum SF parameter, which enables the highest possible transmission speed, leads to a longer battery life [48]. Despite the better range, a sensor node will consume more energy when transmitting with a higher spreading factor [3]. Therefore, in addition to modifying the spreading factor value, the transmission power can be altered to increase the range further or decrease the energy consumption. Considering the limitation in the duty cycle, after transmitting a message, every sensor node needs to wait Toff seconds before transmitting data again in that band (periodicity) [54]:  
\[\mathrm{T_{off}}=\frac{\mathrm{T_{air}}}{\mathrm{T_{dc}}}-\mathrm{T_{air}}, \tag{15}\]  
In the case of sending a message with a payload size of 51 bytes and a spreading factor of 12 while respecting a duty cycle limit of 1% (which dictates that the time for data transmit/hour is 36 s), the time off is 4 min and 57 s.  
Since the LoRa gateway utilizes a downlink channel allowing for a higher duty cycle and higher transmit power operation, a loss due to lowering the spreading factor can be resolved by transmitting at a higher power [4]. However, an optimal combination of energy consumption regarding limited battery life should be achieved. To achieve a longer sensor node battery lifetime, e.g., in the order of years, the LoRaWAN nodes need batteries with greater capacity than typical button cell batteries, e.g., of AA type, but these are more expensive [47]. The results presented in [47] show that an appropriately configured LoRaWAN sensor node powered by a battery of 2400 mAh can \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Output** & **SF6** & **SF7** & **SF8** & **SF9** & **SF10** & **SF11** & **SF12** \\ \hline SF & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ DE & 0 & 0 & 0 & 0 & 0 & 1 & 1 \\ Tsym (ms) & 0.5 & 1.0 & 2.0 & 4.1 & 8.2 & 16.4 & 32.8 \\ Tpreamble (ms) & 6.3 & 12.5 & 25.1 & 50.2 & 100.4 & 200.7 & 401.4 \\ \hline Application payload size & \multicolumn{6}{c}{51 bytes} \\ \hline payloadSymNb (symbols) & 123 & 103 & 93 & 83 & 73 & 83 & 73 \\ Tpayload (ms) & 63.0 & 105.5 & 190.5 & 340.0 & 598.0 & 1359.9 & 2392.1 \\ Tpacket (ms) & 69.2 & 118.0 & 215.6 & 390.1 & 698.4 & 1560.6 & 2793.5 \\ TTN Fair Access Policy & \multicolumn{6}{c}{254} \\ (messages/day) & \multicolumn{6}{c}{139} \\ TNN Fair Access Policy & \multicolumn{6}{c}{5.8} \\ (messages/hour) & \multicolumn{6}{c}{3.2} \\ Duty cycle (s): 0.1\% & 69.2 & 118.0 & 215.6 & 390.1 & 698.4 & 1560.6 & 2793.5 \\ Duty cycle (s): 1\% & 6.9 & 11.8 & 21.6 & 39.0 & 69.8 & 156.1 & 279.3 \\ Duty cycle (s): 10\% & 0.7 & 1.2 & 2.2 & 3.9 & 7.0 & 15.6 & 27.9 \\ \hline Application payload size & \multicolumn{6}{c}{11 bytes} \\ \hline payloadSymNb (symbols) & 53 & 48 & 43 & 38 & 33 & 38 & 33 \\ Tpayload (ms) & 27.1 & 49.2 & 88.1 & 155.6 & 270.3 & 622.6 & 1081.3 \\ Tpacket (ms) & 33.4 & 61.7 & 113.2 & 205.8 & 370.7 & 823.3 & 1482.8 \\ TNN Fair Access Policy & \multicolumn{6}{c}{265} \\ (messages/day) & \multicolumn{6}{c}{20.3} \\ TNN Fair Access Policy & \multicolumn{6}{c}{20.3} \\ (messages/hour) & \multicolumn{6}{c}{11.0} \\ Duty cycle (s): 0.1\% & 33.4 & 61.7 & 113.2 & 205.8 & 370.7 & 823.3 & 1482.8 \\ Duty cycle (s): 1\% & 3.3 & 6.2 & 11.3 & 20.6 & 37.1 & 82.3 & 148.3 \\ Duty cycle (s): 10\% & 0.3 & 0.6 & 1.1 & 2.1 & 3.7 & 8.2 & 14.8 \\ \hline Application payload size & \multicolumn{6}{c}{6 bytes} \\ \hline payloadSymNb (symbols) & 48 & 38 & 38 & 33 & 28 & 33 & 28 \\ Tpayload (ms) & 24.6 & 38.9 & _77.8_ & 135.2 & 229.4 & 540.7 & 917.5 \\ Tpacket (ms) & 30.8 & 51.5 & 102.9 & 185.3 & 329.7 & 741.4 & 1318.9 \\ TNN Fair Access Policy & \multicolumn{6}{c}{583} \\ (messages/day) & \multicolumn{6}{c}{291} \\ TNN Fair Access Policy & \multicolumn{6}{c}{24.3} \\ (messages/hour) & \multicolumn{6}{c}{21.1} \\ Duty cycle (s): 0.1\% & 30.8 & 51.5 & 102.9 & 185.3 & 329.7 & 741.4 & 1318.9 \\ Duty cycle (s): 1\% & 3.1 & 5.1 & 10.3 & 18.5 & 33.0 & 74.1 & 131.9 \\ Duty cycle (s): 10\% & 0.3 & 0.5 & 1.0 & 1.9 & 3.3 & 7.4 & 13.2 \\ \hline \hline \end{tabular}
\end{table}