# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 5 Conclusions  
A probabilistic analysis of packet loss in LoRa networks was conducted in this research, resulting in significant insights into optimizing network performance in small environments. The probabilistic model developed in this study predicted packet loss rates with high accuracy and optimized transmission power, ensuring efficient communication. Specifically, the model predicted an average interference power of \(I_{\text{average}}=1.447\times 10^{-10}\) mW under the given experimental setup, which closely matched the simulated values and validated the model's reliability. This accurate prediction allowed for a reduction in transmission power while maintaining packet success rates above 85% in scenarios with moderate to high traffic loads, demonstrating the model's effectiveness in balancing energy efficiency and communication reliability. The study highlighted the importance of key parameters, such as transmission power, network density, and environmental conditions, in determining packet success rates. For example, the relationship between transmission power and success probability, as visualized in Figure 5, demonstrated that optimal power levels could significantly reduce energy consumption without compromising data integrity. Furthermore, under low traffic conditions, the model slightly overestimated interference due to a limited number of overlapping packets, identifying a limitation that future work should address by refining the aggregated interference calculations. A thorough understanding of path loss was also shown to be critical for optimizing LoRa networks. Accurate modeling of _RSSI_ and _SIR_ provided valuable insights for designing power-efficient systems that meet reliability requirements. The results emphasized the potential of dynamic power adjustment algorithms, which could respond to real-time fluctuations in path loss, further enhancing scalability and energy efficiency in LoRa networks. Expanding the scope of simulations to include larger-scale deployments and more complex environments, such as urban areas with high interference levels, would further validate the scalability and applicability of the proposed model. In conclusion, the findings of this study provide a robust foundation for optimizing the performance and reliability of LoRa networks. By accurately predicting packet loss rates and optimizing transmission power, the probabilistic model developedhere offers a practical and energy-efficient approach to enhancing communication in dense, low-power wide-area networks.  
Conceptualization, J.L.R.V. and A.G.-B.; methodology, J.L.R.V.; software, J.L.R.V.; validation, J.L.R.V. and A.G.-B.; formal analysis, J.A.D.-P.-F.; investigation, F.R.C.S.; resources, R.F.I.-H.; data curation, U.P.R.; writing--original draft preparation, J.L.R.V.; writing--review and editing, E.Z.-S.; visualization, A.G.-B., J.A.D.-P.-F. and U.P.R.; supervision, A.G.-B.; project administration, J.L.R.V.; funding acquisition, A.G.-B. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.