# Joint Spreading Factor and Channel Assignment in Multi-Operator LoRaWAN Deployments
## 3 System Model and Specifications  
We consider multiple LoRaWAN deployments belonging to different operators. Each network \(i\in\mathcal{N}\) has \(N^{i}\) nodes and \(r^{i}\) gateways. The gateways for the different operators are coincident, i.e., they are present at the same location sites. The devices are scattered across a square shaped area, as shown in Figure 1.  
All of the nodes generate packets with a rate \(\lambda^{i}\). Transmission attempts are done according to a Poisson process with packet size \(l\) being constant. Let \(T_{s}\) be the time that is needed to transmit a packet of size \(l\) while using spreading factor \(s\). LoRa supports SFs that range from 7 to 12. The latter represent a trade-off between coverage and data rate. For instance, SF7 yields the highest data rate, but it covers the shortest distance. The SFs are orthogonal meaning that packets transmitted during the same time frame with different SFs will be successfully received. LoRa utilizes forward error correction to detect and correct transmission errors with the coding rate set to 4/(C+4), where \(C\in\{\)1,2,3,4\(\}\). Table 1 shows the variation of data rate, sensitivity, and SNR thresholds as a function of the SFs that are utilized in the 868 MHz band for \(C=4\).  
The upper layer protocol LoRaWAN uses a star topology. Each network connects to its own network server with one hop communications, where all of the gateways forward the packets to the server. Figure 2 presents a typical architecture of such a network.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**SF** & **Data Rate [kbps]** & **Sensitivity [dBm]** & **SNR [dB]** \\ \hline
7 & 5.458 & \(-123\) & \([-7.5,\infty]\) \\
8 & 3.125 & \(-126\) & \([-10,-7.5]\) \\
9 & 1.757 & \(-129\) & \([-12.5,-10]\) \\
10 & 0.976 & \(-132\) & \([-15,-12.5]\) \\
11 & 0.537 & \(-134.5\) & \([-17.5,-15]\) \\
12 & 0.293 & \(-137\) & \([-20,-17.5]\) \\ \(\phi\) & 0 & Not covered & \(<\!-20\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: LoRa in the 868 MHz band.  
Figure 1: A scenario with two LoRaWAN gateways and four operators.  
Because LoRaWAN uses pure Aloha as a channel access scheme, the duty cycle in the LoRa band is limited to \(d\)=1%. As such, the packet generation rate must verify that \(\lambda T_{s}\)\(\leq\)\(d\). LoRaWAN supports multiple frequency bands within the unlicensed bands around 433, 868, and 915 MHz. In Europe, the 868 MHz band is used with 125, 250, and 500 kHz bandwidth channels.  
Figure 3 shows a typical network stack of a LoRa node. LoRa devices can be divided into three classes based on the application:  
* Class A: the device wakes up to send a packet. After sending the packet, the device listens for a short period of time before going back to sleep. This class of LoRa devices consumes the least amount of energy.
* Class B: the device has regularly scheduled listening periods.
* Class C: under this configuration, a device is always listening. This leads to maximum energy consumption.  
An additional advantage for our proposals in this paper is that, contrary to many others in the state-of-the-art, they do not require additional synchronization to be added to current LoRaWAN specifications for class A. The network server can notify the nodes of their SF and channel assignments within their receiving windows. Finally, Table 2 has a summary of the notations concerning our system model.  
Figure 3: LoRaWAN node network stack.  
Figure 2: LoRaWAN architecture.