# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 4 Discussion  
This study provides critical insights into the optimization and behavior of LoRa networks in dense deployment scenarios, focusing on packet loss, aggregate interference, and the signal-to-interference ratio (SIR). The key findings demonstrate that the proposed probabilistic models accurately predict packet loss rates and aggregated interference under moderate to high traffic loads. These results reinforce the importance of interference modeling in LoRa networks, particularly in dense environments where overlapping transmissions lead to frequent packet collisions. By integrating probabilistic interference models into the optimization of transmission power, this work advances existing methodologies. The optimization framework not only reduces energy consumption but also maintains network reliability, aligning with the findings of [PERSON] et al. [6], who emphasized the importance of adaptive power control in LoRa systems. Our approach, however, extends prior research by treating interference as a stochastic process and incorporating cumulative distribution functions (CDFs), providing a more nuanced understanding compared to the deterministic models used in earlier studies, such as those by [PERSON] et al. [5]. The implications of these findings extend beyond theoretical modeling. The use of ESP32 and Reyax RYLR998 modules in controlled experiments validated the practical applicability of the models, bridging the gap between theoretical advancements and real-world implementations. This study's methodologies can be applied to other low-power wide-area networks (LPWANs) operating in unlicensed bands, offering potential applications in smart cities, precision agriculture, and environmental monitoring. Additionally, the integration of real-time monitoring through an SQL-connected dashboard demonstrates the feasibility of developing self-adaptive networks capable of dynamically adjusting to environmental and traffic conditions, enhancing resilience and performance in variable scenarios. Despite these promising contributions, certain limitations must be acknowledged. The experimental setup assumed a small number of transmitters and idealized environmental conditions, which may not fully reflect the complexities of large-scale, urban deployments. Furthermore, the optimization framework did not account for adaptive spreading factors or dynamic bandwidth allocation, which are critical for optimizing network performance under diverse conditions. Future research should address these limitations by exploring larger-scale deployments in complex environments, such as urban areas with high interference levels. Expanding the optimization framework to include adaptive spreading factors and dynamic bandwidth allocation could lead to more comprehensive performance improvements. Additionally, the integration of machine learning techniques could enhance the probabilistic models,enabling more accurate predictions of interference patterns and packet loss rates. Exploring multi-gateway systems is another promising direction, as such systems could improve network reliability and coverage by distributing the communication load, thereby reducing packet loss and improving energy efficiency. Another avenue for future research involves evaluating the proposed probabilistic model in the context of LoRaWAN, which offers additional mechanisms for improving packet delivery and network performance. LoRaWAN introduces features such as the Adaptive Data Rate (ADR), acknowledgments, and duty cycle regulations, which can mitigate packet loss in dense environments. For instance, the ADR dynamically adjusts transmission parameters like spreading factor and transmission power, potentially complementing the optimization framework proposed in this study. Furthermore, the LoRaWAN support for multi-gateway systems and downlink communication could enhance network scalability and reliability by reducing the impact of interference and optimizing resource allocation.