# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 2 Materials and Methods
Path loss directly impacts the received signal strength (RSSI) and, consequently, the ability to decode packets accurately [23; 24]. In LoRa networks, the \(RSSI\) is given by the following:  
\[RSSI=P_{\text{tx}}-PL, \tag{6}\]where \(P_{\rm tx}\) is the transmission power. For example, in our experimental setup, with \(P_{\rm tx}=14\) dBm, and distances \(d_{1}=2\) m, \(d_{2}=4\) m, \(d_{3}=6\) m, and \(d_{4}=8\) m, the calculated path losses are 37.7 dB, 47.3 dB, 47.2 dB, and 49.7 dB, respectively. This results in \(RSSI\) values of 23.7 dBm, \(-33.3\) dBm, \(-33.2\) dBm, and \(-35.7\) dBm. The average aggregate interference is calculated as before, obtaining:  
\[\approx 3.79\times 10^{-3}\;\rm mW\]  
To optimize transmission powers, the following constraint must be met:  
\[P_{\rm tx,n}\geq PL_{n}\cdot\gamma\cdot[\mu+\sigma\cdot\rm{probit}(0.9)] \tag{7}\]  
where \(\gamma=3.98\), \(\mu=3.79\times 10^{-3}\), \(\sigma=\frac{\mu}{2}=1.895\times 10^{-3}\), and probit \((0.9)\approx 1.28\). This results in optimal transmission powers for each module (in dBm):  
\[Ptx1=6.5,\,Ptx2=12.5,\,Ptx3=16,\,Ptx4=16\]  
Moreover, while power optimization enhances the overall reliability and efficiency of the LoRa system [25; 26], it is also crucial to consider the limitations of certain performance metrics, such as RSSI, particularly in short-range experimental setups. The experimental setup employed in this study involves distances between 2 m and 8 m, which may lead to non-negligible effects from the electronic circuits of the LoRa transceiver. Specifically, the Low Noise Amplifier (LNA) could introduce nonlinearities due to signal saturation, and potential feedback mechanisms within the receiver chain may also affect the measured values. These factors can cause deviations from theoretical path loss models, resulting in RSSI values that might not accurately reflect real-world long-range LoRa deployments. Additionally, in short-range scenarios, multipath effects and internal circuit characteristics of the transceiver (such as automatic gain control or impedance mismatches) can further distort RSSI measurements. While these effects are minimized in our setup by maintaining controlled environmental conditions, their presence should be taken into account when interpreting the results. In addition to these considerations regarding RSSI limitations, the experimental setup utilized a single-channel gateway (GW) to receive transmissions from four LoRa nodes, which introduces additional challenges related to packet reception and interference. In this configuration, all devices operated within the same frequency band, using an ALOHA-based medium access control (MAC) protocol. Given the absence of carrier sense multiple access (CSMA) in LoRa, the transmission scheduling is uncoordinated, which inherently leads to packet collisions, especially when multiple nodes transmit simultaneously. Since the gateway operates on a single channel, it can only decode one packet at a time per spreading factor (SF). If two or more packets arrive at the gateway at the same frequency and SF simultaneously, a collision occurs unless the signals are sufficiently separated in power (typically by at least 6 dB) for capture effect to take place. In addition to packet collisions caused by uncoordinated transmissions, another critical issue in this setup is intra-SF interference. Unlike traditional frequency-division multiple access (FDMA) or time-division multiple access (TDMA) systems, the LoRa chirp spread spectrum (CSS) modulation allows for multiple spreading factors to coexist within the same frequency channel, leveraging their quasi-orthogonality. However, transmissions using the same SF are not orthogonal, and simultaneous packets with identical SF values cause mutual interference, leading to increased packet loss. This effect is exacerbated in scenarios with high transmission density, where multiple nodes compete for access to the gateway.