# Beacon Success Rate versus Gateway Density in Sub-GHz Sensor Networks
## 2 System Model
### The Co-channel Interference Model  
In this section, the co-channel interference model is presented. The packets can potentially collide due to the shared medium access with a finite number of channels and finite number of transmission slots. Collisions occur when transmission attempts occur on the same channel and overlap in time. Within the same WAN network, this occurs when the same transmission slot and same LBT sub-slot are selected. When both of these conditions are met, the same channel is selected, and LBT returns channel clear during LBT for both transmissions, leading to both devices starting TX. Although collision avoidance is in place for the ENs within the coverage of a given GW, collisions on a beacon can still happen from neighboring devices. This is illustrated in Figure 3, where the beacon of GW\({}_{2}\) is interfered by the UL packet from EN\({}_{2}\) which is connected to GW\({}_{1}\). Co-channel collisions can happen over a portion of the desired packet. Let \(\Delta T_{i}\) represent this portion of time and let \(T\) represent the duration of the desired packet. Let \(r_{i}\)\(\frac{\text{det}}{T}\) represent the interferer overlap ratio for the interferer \(i\). Let \(P_{d}\) and \(P_{i}\) represent the Received Signal Strength (RSS) in mW from the desired transmitter and the interferer \(i\), respectively. These definitions are illustrated in Figure 4. In this illustration, the time boundaries of the intended packet are denoted by a green rectangle, and the time boundaries of the interfering packet are indicated by a red rectangle. The y-axis corresponds to the RSS of each packet upon arrival at the receiver. Let \(\frac{C}{T}\) represent the carrier to interference ratio (in dB) when a co-channel interferer is present. With these definitions, the carrier to interference ratio can be derived as:  
\[\frac{C}{I}=10\times log_{10}\{\frac{P_{d}}{\Sigma_{i}P_{i}\times\frac{\Delta T _{i}}{T}}\}=10\times log_{10}\{\frac{P_{d}}{\Sigma_{i}P_{i}\times r_{i}}\}\ [\text{dB}] \tag{1}\]  
In this formula, the overlapping interference power from all interferers is aggregated to determine the overall carrier-to-interference ratio. Figure 3: The channel access mechanism employing LBT. The PSR is the probability at which packets can be received successfully. The PSR can therefore be related to the Packet Error Rate (PER) by the following formula:  
\[PSR\equiv 1-PER \tag{2}\]  
The PSR depends on various factors such as:  
1. The distance, \(d\), between the receiver and the transmitter, which determines the RSS at the receiver,
2. The receiver sensitivity for the selected MCS,
3. The carrier to interference ratio (C/I) and number of interfering events. The collision and interference model in the simulator considers all these parameters before a packet can be marked as successfully received or dropped. A packet drop can be caused by unfavorable conditions such as large interference or path-loss. This is illustrated in Figure 5. The term \(rx_{sens. }\) represents the receiver sensitivity in dBm. For the packets to be considered as successfully received, the RSS of the desired signal at distance \(d\) from the transmitter needs to be greater than the receiver sensitivity, i.e., \(RSS(d)>rx_{sens.}\). Furthermore, once this condition is satisfied, the C/I needs to be above a threshold in order for the packet to be received successfully. This threshold depends on the selected MCS in the communication link. Let \(\gamma_{th}\) represent this threshold in dB. The higher-bitrate MCSs require a larger threshold and larger sensitivity. These thresholds will be presented in detail in Section 3, assuming that GFSK modulation is used for the MCS.