# Resource Allocation to Massive Internet of Things in LoRaWANs
## 3 Link-Level Model
### Link Performance Model
#### 3.3.2 Interference Model  
The LoRaWAN uses ALOHA as the channel access mechanism, where EDs do not listen to the channel before transmitting a packet. This mechanism causes a collision if multiple EDs transmit at the same time over the same channel. Therefore, the LoRaWAN network suffers from two types of collisions owing to multiple SFs, which are as follows: (1) Two or more packets with the same SF collide with each other over the same channel, and (2) packets with different SFs collide over the same channel [4]. These SF interferences cause a low SINR. By using the interference model described in [5], the SINR anticipated by the GW of \(\mathbb{C}\) (\(\gamma_{\mathbb{C}}\)), as given by:  
\[\gamma_{\mathbb{C}}=\frac{P_{\geq},r}{\sigma^{2}+\mathbb{P}}, \tag{8}\]  
where \(P_{\geq},r\) is the power of a packet under observation and \(\mathbb{P}\) is the cumulative power, as presented in [5].  
When two packets are received at a GW using \(SF_{(i,j)}\) (i.e., \(SF_{(i)}=9\)_and_\(SF_{(j)}=9\), as shown in Figure 1), it is considered successful if \(\gamma_{\mathbb{C}}\geq\beta_{(i,j)}\), where \(\beta_{(i,j)}\) (in dB unit) is a threshold corresponding to an acceptable SINR [47].  
The SINR margin elements contained in \(\beta_{(i,j)}\) (in dB) are calculated, assuming that the two packets are completely overlapped. However, in general, packets are not entirely overlapped. For this reason, the interfering power must be equalized in order to compute the \(\gamma_{\mathbb{C}}\) value [5; 48]. It is assumed that, in general, the interfering energy of the mutual signal and the interferer can \"spread out\" on the useful signal as computed using Equation (8). If the interference is concentrated on a few consecutive symbols, assuming that a good interleaver will spread it out and allow the error correction code (i.e., the coding rate employed by LoRa CSS modulation, 4/5 in our case) to eventually correct the error caused by the interferer [49].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**SF** & **BW [kHz]** & \(S_{g}\) & \(S_{e}\) \\ \hline
12 & 125 & \(-142.5\) & \(-137.0\) \\
11 & 125 & \(-140.0\) & \(-135.0\) \\
10 & 125 & \(-137.5\) & \(-133.0\) \\
9 & 125 & \(-135.0\) & \(-130.0\) \\
8 & 125 & \(-132.5\) & \(-127.0\) \\
7 & 125 & \(-130.0\) & \(-124.0\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Sensitivities (dBm) of end devices and gateway for 125 kHz bandwidth.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**SF** & **SNR (dB)** \\ \hline
12 & \(-20\) \\
11 & \(-17.5\) \\
10 & \(-15\) \\
9 & \(-12.5\) \\
8 & \(-10\) \\
7 & \(-7.5\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The LoRa demodulator signal to noise ratio for spreading factors.  
Based on the elements in \(\beta_{(i,j)}\), we observed that one of two signals employing the same SF with the corresponding received power could be correctly received if both signals overlap for a small amount of time and their respective \(\gamma_{\hat{C}}\) after equalization is greater than 6 dB. The same observation is realized in [50]. The term \"equalization\" refers to the process of multiplying SNR of an interferer signal with the overlapping time, divided by the duration of the desired signal [48].