# Improving Energy Efficiency in LoRaWAN Networks with Multiple Gateways
## 4 System Model  
### Network Model  
Let us consider a LoRaWAN network with \(r\) gateways regularly distributed in an urban square area and an average of \(N_{t}\) nodes distributed following a Poisson point process. Let \(f_{1}\) be the fraction of nodes located indoors and \(f_{2}=1-f_{1}\) be the fraction of nodes located outdoors. Table 4 lists the required SNR to correctly decode messages at different SFs [23]. Figure 4 illustrates the distribution of \(N_{t}=500\) nodes among different SFs in a 4-gateway LoRaWAN deployment in a square area with a side length of 7 km.  
Table 5 lists numerical results related to Figure 4. Each node in this example uses the maximum allowed TP (14 dBm) and selects the lowest SF, according to Table 4, given its own radio conditions. Note that if a node is covered by multiple gateways, it will be assigned the lowest SF based on the highest SNR received by one of the gateways. We denote by \(D(s)\) the number of nodes selecting the SF \(s\) and by \(N(s)\) the number of nodes that use SF \(\geq s\).  
Figure 3: Time diagram of different LoRaWAN classes.  
We assume that all nodes are configured with the same modem settings (bandwidth, coding rate, preamble length, etc.) and transmit packets on a single channel with the same payload size \(k\). Therefore, the time on air to transmit a packet depends only on the used SF and is denoted by \(\mathit{ToA}(s)\). A detailed calculation of the time on air is given in [2]. Transmit attempts for all nodes are performed according to a Poisson distribution with a mean arrival rate of \(\lambda\) packets per second. Then, given a duty cycle limitation of \(d=1\%\), the packet generation rate for each node using SF \(s\) must verify \(\lambda.\mathit{ToA}(s)\leq d\). We suppose that all nodes in the network are of class A with unconfirmed transmission mode [5]; then, each node opens two receive windows \(RX1\) and \(RX2\) after RECEIVE_DELAY1 and RECEIVE_DELAY2 from the transmitting uplink message even if no acknowledgment is expected, as shown in Figure 5. These two receive windows are the only opportunity for the network server to send notifications and new setup to nodes. Note that \(RX2\) is only opened if neither data nor acknowledgment is received during \(RX1\). \(R\_D1\) and \(R\_D2\) refer to RECEIVE_DELAY1 and RECEIVE_DELAY2, respectively. Let \(T=1/\lambda\) be the time period between two successive uplink transmissions, and let \(T_{rx1}(s)\) and \(T_{rx2}(s)\) be the first and second receive window durations, respectively. Table 6 lists various variables and parameters used in our model.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**SF** & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline Required SNR [dB] & \(-7.5\) & \(-10\) & \(-12.5\) & \(-15\) & \(-17.5\) & \(-20\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: SNR thresholds of different SFs for 868 MHz band, bandwidth 125 kHz [21].  
Figure 4: Example of distribution of 500 nodes in a 4-gateway LoRaWAN network.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\mathbf{s}\) & **7** & **8** & **9** & **10** & **11** & **12** \\ \hline \(D(s)\) & 474 & 15 & 4 & 4 & 2 & 1 \\ \hline \(N(s)\) & 500 & 26 & 11 & 7 & 3 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Example of node distribution at different SF when \(N_{t}=500\) nodes.