# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 2 Materials and Methods
In this case, it was assumed that interference follows a normal distribution, which allows the CDF to be used to evaluate the probability that the signal-to-interference ratio (SIR) is greater than or equal to a desired threshold (\(\gamma\)), thus ensuring reliable communication. Given this assumption, the next step is to leverage the properties of the CDF to systematically evaluate the probability of achieving a signal-to-interference ratio (SIR) above a given threshold. The CDF of a normal distribution evaluates the cumulative probability that a random variable takes a value less than or equal to a given number. Normalizing the values allows the interference to be converted to a standard scale, removing the dependence on the original units and facilitating the use of standard tables or software to calculate the CDF. This approach has several advantages. First, the CDF guarantees an analytical solution to evaluate probabilities in stochastic systems, which is useful in scenarios similar to this where the interference parameters are not deterministic. Second, the use of the CDF allows for directly incorporating statistical characteristics of the communication environment, such as the mean and standard deviation of the interference, providing a more accurate representation of the probability of success [11; 12]. Based on this, the calculation of \(\mu\) as the mean of the interference and \(\sigma\) as half of \(\mu\) is grounded in the premise of a symmetric normal distribution, an approximation valid in LoRa networks with multiple transmitters operating under similar conditions [13; 14]. By substituting the obtained values (\(\mu\) = 1.447 \(\times\) 10\({}^{-10}\) and \(\sigma\) = 7.235 \(\times\) 10\({}^{-11}\)) into the CDF formula, a high value is obtained (CDF (6.72) \(\approx\) 1.0), indicating that the probability of success is close to 100%. This high value of the probability of success is essential to ensure that the LoRa network operates effectively, minimizing data loss and maintaining reliable communication [15; 16]. Furthermore, this CDF-based method can be extended to analyze more complex scenarios, such as networks with higher transmitter density or more variable interference conditions. By optimizing the system based on these probabilities, it is possible to balance network reliability with energy efficiency and other available resources [17; 18]. Substituting \(\mu\) = 1.447 \(\times\) 10\({}^{-10}\) and \(\sigma\) = 7.235 \(\times\) 10\({}^{-11}\), the probability of success is obtained by consulting the cumulative distribution function (CDF (6.72)):  
CDF (6.72) \(\approx\) 1.0  
This results in a success probability of 99.9%. With these results in mind, the next step is to maintain the same success percentage while minimizing power to optimize energy consumption [19; 20]. Considering the same values with a minimum and maximum allowed power limit of \(P_{\text{R,min}}\) = 2 dBm, \(P_{\text{tx,max}}\) = 18 dBm, and a minimum desired success probability \(p\) = 0.9, the power optimization calculation continues. To achieve this optimization while ensuring reliable communication, it is essential to accurately determine the path loss (PL), which quantifies the attenuation of signal power as it propagates from the transmitter to the receiver. Path loss refers to the attenuation of the signal power as it propagates through space from the transmitter to the receiver. It is a critical factor in determining the reliability and efficiency of LoRa communication systems [21; 22]. The Friis transmission equation, is given as follows:  
\[PL=20\text{log}_{10}\bigg{(}\frac{4\pi df}{c}\bigg{)}, \tag{5}\]  
where \(d\) is the distance between the transmitter and the receiver, \(f\) is the carrier frequency, \(c\) is the speed of light, and is used to compute \(PL\). This equation highlights the dependency of path loss on distance and frequency, making it a cornerstone in understanding signal attenuation.