## 3 System Model
### Energy Model  
For the energy model of LoRa devices in the proposed scheme, we employed the same energy model as [38; 40]. Unlike class C LoRa devices, which are usually main-powered, the energy consumption of each LoRa device must be precisely stated. The energy consumption \(E_{tx,j}\) for transmitting a \(\beta\)-bit LoRa packet from a LoRa device LD(i) to the GW can be categorized as follows: the LoRa device waking up, low-power listening, radio preparation, signal transmission, radio off, and processing. Furthermore, the consumed energy for signal transmission is affected by the transmission power as well as the varying transmission time caused by the spreading factor. It is precisely defined as (1) [41],  
\[E_{tx,j}=P_{tx}\times T_{SF(i)} \tag{1}\]  
where \(P_{tx}\) denotes the transmit power with time on air for a LoRa device with \(SF(i)\), and \(T_{SF(i)}\) denotes the time on air for LD(i) expressed as (2) and (5) [38]. \[T_{SF(i)}=T_{Symbol}\times(N_{pr}+N_{pl}) \tag{2}\]  
\[N_{pr}=(n_{preamble+4.25})\times\frac{2^{SF_{i}}}{BW} \tag{3}\]  
\[N_{pl}=8+\max(\bigg{[}\frac{8L-4 SF_{i}+44}{4(SF_{i}-2 DE)}\bigg{]}CR,0))\times \frac{2^{SF_{i}}}{BW} \tag{4}\]  
\[T_{SF(i)}=(20.25+\max(\bigg{[}\frac{8L-4 SF_{i}+44}{4(SF_{i}-2 DE)}\bigg{]}CR,0)) \times\frac{2^{SF_{i}}}{BW} \tag{5}\]  
In Equation (2), \(N_{pr}\) and \(N_{pl}\) represent the number of symbols in the packet preamble and payload, respectively. Substituting for \(T_{symbol}\), \(N_{pr}\), and \(N_{pl}\) in (2), we get (5), if low data rate optimization is enabled, \(DE=1\), otherwise \(DE=0\), \(\text{CR}=4/5\) represents the code rate, and \(n_{preamble}=8\) is the preamble length. It is worth noting that the energy consumption of a LoRa device in the annulus (\(j\)) is proportional to airtime (\(\tau\) ), with large values for a high SF. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**SF** & **SNR** & **Receiver Sensitivity** & **Bit Rates** \\ (**i**) & **(dB)** & **qs (dBm)** & **(kbps)** \\ \hline
7 & \(-6\) & \(-123\) & \(5.47\) \\
8 & \(-9\) & \(-126\) & \(3.13\) \\
9 & \(-12\) & \(-129\) & \(1.76\) \\
10 & \(-15\) & \(-132\) & \(0.98\) \\
11 & \(-17.5\) & \(-135.5\) & \(0.54\) \\
12 & \(-20\) & \(-137\) & \(0.29\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: LoRa parameters based on SF. Subsequently, we evaluated the energy consumption \(E_{n-hops}(SF,P_{tx})\) using the multi-hop communication from a LoRa source device(i) to the GW through n hops. Then, the energy consumption for sending a packet was computed as follows:  
\[E_{n-hops}(SF,P_{tx})=\sum_{i=1}^{n}(E_{tx,i}+E_{rx,i}) \tag{6}\]  
where \(P_{tx}\) is the consumed power according to the supply current and supply voltage for the transmitter, \(E_{tx,i}\) and \(E_{rx,i}\) are the energy consumption of the transmitter and receiver in the \(i\)th hop according to its current configuration on the transmitted power, spreading factor, and bandwidth, respectively. Classs A and B LoRa devices on the LoRa network typically spend the bulk of their time in light-sleep mode due to their low energy consumption and battery power. Meanwhile, in order to calculate the total dissipated energy consumption \(E_{total\_N_{i}}(t)\) of a LoRa device \(N_{i}\) at a specific time (t), we considered two states, active and inactive modes, given as (7) and (8) and thus derived \(E_{total\_N_{i}}(t)\) as follows. \[E_{total\_N_{i}}(t)=E_{inactive\_N_{i}}(t)+E_{active\_N_{i}}(t) \tag{7}\]  
Here, \(E_{inactive\_N_{i}}(t)\) is the dissipated energies in the state where devices are in sleep mode (\(E_{sleep\_N_{i}}(t)\)) relative to 1.8 uA [42] and \(E_{active\_N_{i}}\) is the dissipated energies by the LoRa device during the transmission mode, reception mode, data measurement, processing, and wake up.