# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 3 Results
These observations underscore the potential of carefully calibrated transmission parameters to maximize network reliability. The data from the table confirm that the model's predicted success probabilities tended to overestimate the actual outcomes, particularly for transmitters 1 and 3. Despite this, the model still enabled the system to achieve high reliability, with most packets being successfully transmitted as expected. This slight overestimation suggests room for refinement in the interference and path loss modeling, especially under varying network densities or environmental conditions. The results also highlight the relationship between success probability and the average RSSI values. Transmitters with lower RSSI values (e.g., transmitters 4 and 5) achieved perfect success rates, emphasizing the robustness of the LoRa modulation scheme even under conditions of moderate signal attenuation. This finding aligns with the theoretical expectations of LoRa technology, where its inherent resistance to interference allows for reliable communication over extended ranges. From a broader perspective, these findings reinforce the importance of optimizing transmission parameters, such as power levels and spreading factors, to adapt to the specific conditions of a network deployment. By tuning these parameters, the system can achieve higher reliability while minimizing energy consumption, an essential consideration for IoT applications. In conclusion, the table demonstrates the effectiveness of the proposed model in maintaining high success probabilities across all tested transmitters. While there is room for further refinement of the model to reduce overestimation, the results validate its applicability for real-world LoRa deployments. Future experiments could expand on these findings by testing in environments with higher node densities, increased interference, or greater distances from the gateway to further evaluate the robustness of the model and the network's scalability. Figure 5 depicts the relationship between transmission power (measured \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Transmitter** & **No. of Data Points (10 min)** & **SP (\%)** & **Avg. RSSI** \\ \hline
1 & 63 & 87.5\% & \(-43\) dBm \\
3 & 63 & 87.5\% & \(-41\) dBm \\
4 & 72 & 100\% & \(-47\) dBm \\
5 & 72 & 100\% & \(-46\) dBm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Transmission results obtained over a 10-min period. in dBm) and the success probability of transmitted packets (expressed as a percentage). This graph provides critical insights into the effect of varying transmission power on the reliability of the communication in the experimental setup described in the article. The trend shown in Figure 5 is a clear positive correlation between transmission power and success probability. At the lowest transmission power of 2 dBm, the success probability is approximately 90%. This relatively low value suggests that at minimal transmission power levels, factors such as path loss, interference, and signal attenuation significantly reduce the likelihood of successful packet delivery. However, as the transmission power increases, the success probability also rises, indicating improved reliability of the communication link. At a transmission power of 6 dBm, the success probability increases to around 94%. This marks a notable improvement, demonstrating that even a moderate increase in power can enhance the robustness of the communication link. The improvement in success probability between 6 dBm and 12 dBm is similarly pronounced, with the success rate reaching approximately 96%. This further supports the hypothesis that higher transmission power mitigates the negative effects of environmental factors and interference on packet delivery.