# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 1 Introduction  
The rapid growth of the Internet of Things (IoT) has generated an urgent need for scalable and energy-efficient communication technologies. Among emerging solutions, long-range networks (LoRa) have gained prominence as a leading technology for low-power wide-area networks (LPWANs) [1, 2]. LoRa networks excel in scenarios requiring long-distance communication, minimal power consumption, and the ability to support a high density of connected devices. As a result, they are increasingly adopted in various applications, including smart cities, environmental monitoring, industry, communications, and precision agriculture [3, 4]. Despite their advantages, LoRa networks face challenges in maintaining reliable data transmission, especially in dynamic and densely populated environments. Packet loss, a critical performance metric, can significantly affect the functionality of LoRa-dependent applications. Factors such as interference, signal attenuation,network congestion, and collisions contribute to packet loss, emphasizing the need for a detailed analysis of data transmission [5]. Previous studies, such as those by [PERSON] et al. [5] and [PERSON] et al. [6], have provided foundational insights into interference modeling and adaptive power control. However, most of these works rely on deterministic or simplistic models that fail to capture the stochastic nature of interference and environmental dynamics. This study builds upon existing literature by introducing a probabilistic framework that integrates multiple critical factors, including packet arrival rates, transmission power levels, and the distance between nodes and gateways. Unlike prior approaches, our model dynamically adapts to varying network conditions and considers the interplay between transmission power and interference. This allows for a more nuanced understanding of packet loss and provides actionable insights into optimizing network parameters. A distinctive feature of this research is its focus on balancing energy efficiency and communication reliability through optimal transmission power settings. By employing probabilistic models, the analysis quantifies the probability of packet loss under different scenarios, addressing gaps in previous studies and offering practical guidelines for improving network performance. Furthermore, this study validates its theoretical findings through a combination of modeling and experimental evaluations, setting it apart from purely theoretical analyses. Beyond the core analysis of packet loss and interference, this study also explores additional factors that significantly impact LoRa network performance. The role of the ALOHA protocol in managing retransmissions and its effects on network efficiency are examined, highlighting the differences between Pure ALOHA and Slotted ALOHA in reducing collisions. Additionally, the impact of key LoRa parameters, such as the spreading factor (SF) and bandwidth (BW), on packet loss probability is analyzed, demonstrating how adjustments in these settings can improve network reliability. Another crucial aspect addressed in this work is the energy consumption of LoRa devices. By evaluating different transmission power levels and network conditions, we provide insights into optimizing energy efficiency while maintaining robust communication. Experimental results with varying traffic loads and transmission distances further illustrate the interplay between power settings, network density, and energy efficiency. The findings of this research have significant implications for the design and implementation of LoRa networks, particularly in resource-constrained environments. By identifying the key factors contributing to packet loss and proposing strategies to mitigate them, this study advances the state of the art and provides valuable tools for network designers aiming to achieve high reliability and energy efficiency.