# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 2 Materials and Methods
Since interference power is a key factor in LoRa network performance, its impact must be analyzed alongside other parameters such as coding rate (CR) and bandwidth (BW), which further  
Figure 1: The experimental setup consists of four transmitters, a gateway, and a server. The transmitters are responsible for collecting and sending data to the gateway. The gateway processes the received data and forwards them to the server for further analysis and storage. influence interference, energy consumption, and transmission efficiency. The coding rate (CR) determines the level of redundancy added to transmitted data for error correction. For example, a CR of 4/5 indicates that 4 bits are data and 1 bit is redundant. Higher CR values enhance robustness against errors in noisy or interference-heavy environments but increase the airtime of packets. This extended airtime raises the probability of overlapping transmissions, directly impacting the aggregated interference power \(I_{\text{average}}\). While a higher CR can mitigate some effects of interference, it does so at the cost of increased resource usage. The bandwidth (BW), on the other hand, affects the data rate and sensitivity of the receiver. A larger BW enables faster data transmission, reducing the airtime and, consequently, the likelihood of packet collisions. However, it also decreases the receiver's sensitivity, potentially requiring higher transmission power to maintain reliable communication. Conversely, a smaller BW improves sensitivity but increases the airtime, which may lead to higher \(I_{\text{average}}\) values due to prolonged exposure to interference. The indirect effects of BW and CR on \(I_{\text{average}}\), airtime, and network performance are significant and are controlled by the experiment presented in this work. In addition to CR and BW, the spreading factor (SF) plays a critical role in determining the airtime of transmitted packets, further influencing the likelihood of interference. Higher SF values result in longer packet airtime, increasing the probability of overlapping transmissions and, consequently, the aggregated interference power \(I_{\text{average}}\). This relationship is particularly important as LoRa networks rely on the orthogonality of different SFs to minimize interference between packets. By incorporating SF into the interference model, this study ensures a more accurate prediction of \(I_{\text{average}}\) under varying network conditions. Building on this interference model, the next step is to compute the SIR using the formula:  
\[\text{SIR}_{n}=\frac{P_{\text{signal}}}{I_{\text{average}}} \tag{1}\]  
In this specific scenario, substituting the calculated values \(P_{\text{signal}}\) = \(2.51\times 10^{-9}\) mW and \(I_{\text{average}}\) = \(1.447\times 10^{-10}\) mW, the SIR is as follows:  
\[\text{SIR}_{n}=\frac{2.51\times 10^{-9}}{1.447\times 10^{-10}}\approx 17.35\; \text{dB} \tag{2}\]  
This detailed computation emphasizes the importance of both accurate path loss modeling and interference estimation, as errors in either can significantly impact the SIR and, consequently, the network's performance and reliability [8]. The probability of success is calculated using the cumulative distribution function (CDF) of a normal distribution, assuming interference follows a normal distribution:  
\[P[\text{SIR}_{n}\geq\gamma]=\text{CDF}\Bigg{(}\frac{10\frac{\text{RSS}_{n}}{10 }-\mu}{\gamma}\Bigg{)} \tag{3}\]  
where  
\[\mu=I_{\text{average}},\ \ \sigma=\frac{\mu}{2} \tag{4}\]  
The choice of the cumulative distribution function (CDF) to calculate the probability of success in this analysis is based on the need to model the statistical nature of interference and the random factors that affect communication in LoRa networks [9]. Interference generated by neighboring transmitters, as well as signal attenuation and other noise, introduce variations in the system that can be represented statistically [10].