# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 3 Results  
For experimentation, the ESP32 development board was utilized as the central component due to its versatility, ease of communication with peripheral modules, affordability, and low power consumption. The ESP32, a widely popular microcontroller, integrates both Wi-Fi and Bluetooth capabilities, making it an ideal choice for IoT applications where connectivity is crucial [27, 28]. Its dual-core Tensilica Xtensa LX6 processor, running at a clock speed of up to 240 MHz, provides sufficient computational power to handle real-time data processing and communication tasks [29, 30]. Furthermore, the ESP32 ultra-low power consumption in deep-sleep mode, typically around 10 uA, ensures energy efficiency, which is a critical requirement in scenarios involving battery-powered LoRa devices [31, 32]. In this experimental setup, the ESP32 development board was utilized as the central component, and to facilitate long-range communication, the Reyax RYLR998 (Taipei City, Taiwan)LoRa module was employed as the transmitter (Figure 2 shows the transmitter used). This module operates on the LoRa modulation technique, known for its robust long-range communication capabilities and superior resistance to interference [33; 34]. The RYLR998 supports frequencies ranging from 868 MHz to 915 MHz, making it compatible with unlicensed ISM bands in various regions [35; 36]. It boasts a transmission range of up to 15 km in optimal conditions, making it suitable for applications in smart cities, agricultural monitoring, and remote sensing [37; 38]. Additionally, the module features an easy-to-use UART interface, simplifying integration with microcontrollers such as the ESP32. Its built-in commands for setting transmission power, spreading factor, and other parameters enable flexible configuration, essential for adapting to varying network conditions [39]. The experimental setup also included a single-channel LoRa gateway, configured to operate at a frequency of 915 MHz to ensure compatibility with the Reyax RYLR998 module. This gateway was used to facilitate communication between the transmitter and the network, enabling the evaluation of packet loss and interference under controlled conditions. The choice of the Reyax RYLR998 module, instead of utilizing the LoRa functionality integrated into certain ESP32 boards (e.g., Heltec or TIGO models), was motivated by several factors. Firstly, the RYLR998 module provides a simplified AT command interface, which streamlines configuration and facilitates direct control over critical parameters such as transmission power and spreading factor. Secondly, using an external module offers greater flexibility for expanding the experimental setup, as it allows for interoperability with other microcontroller platforms beyond the ESP32. Additionally, this approach ensures consistency and reproducibility in experiments, as the RYLR998 module is widely adopted and provides standardized hardware performance. These considerations were critical for accurately validating the probabilistic model proposed in this study and ensuring its applicability to a broad range of LoRa network scenarios. The experimental setup was designed to utilize relatively short distances between the transmitter and the gateway, which may appear counterintuitive given that the long-range (LoRa) technology is being evaluated. However, this choice was intentional to ensure precise control over the environmental and interference factors affecting the communication. Testing under short-distance conditions allows for a more detailed and accurate analysis of key parameters such as packet loss, signal-to-interference ratio (SIR), and aggregated interference power (\(I_{\text{average}}\)). These controlled results can then be extrapolated to long-distance scenarios, as the LoRa signal propagation characteristics, governed by the path loss model and environmental conditions, remain consistent regardless of the physical distance.