# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 2 Materials and Methods  
The work of [PERSON] et al. [1] considers a system with one or more gateways and multiple end devices. The model assumes that all packets have equal airtimes, communication is line-of-sight, and only uplink communication is used, without centralized scheduling or acknowledgments. Using that model, this study proposes to analyze the calculation of the probability of packet success in a LoRa network composed of four transmitters and one gateway. It seeks to determine the optimal transmission powers and the necessary parameters to ensure a high probability of success based on the model of [PERSON] et al. [1]. Figure 1 presents the experimental setup proposed for the measurements. In this scenario, the following key parameter values are maintained: a transmission power \(P_{\text{tx}}=14\) dBm for all transmitters, a transmission interval 1000 ms (\(\lambda=1\) msg/s), distances between transmitters and the gateway for each one with \(d_{1}=2\) m, \(d_{2}=4\) m, \(d_{3}=6\) m, \(d_{4}=8\) m. These parameters were used to test a small-space scenario under normal working conditions. These parameters not only define the experimental setup, but also play a crucial role in determining the received signal strength (RSSI\({}_{n}\)) and the impact of interference in LoRa networks. The received signal strength (RSSI\({}_{n}\)) in LoRa networks is determined as the difference between the transmission power (\(P_{\text{tx}}\)) and the path loss (\(PL\)), yielding \(-86\) dBm in this scenario. Interference, a critical factor in network performance, occurs when multiple devices transmit simultaneously within the same frequency band, with its strength influenced by the transmission power of interfering devices, the path loss during signal propagation, and the temporal overlap between interfering packets and the packet of interest [6]. To quantify this, the total interference power is averaged over the airtime of the packet of interest, resulting in the average aggregate interference (\(I_{\text{average}}\)), a key metric in dense deployments that represents the cumulative interference experienced by a receiver due to overlapping transmissions from other devices. Accurate modeling of \(I_{\text{average}}\) is crucial for predicting packet loss and optimizing network performance, and its calculation involves summing the received powers from interfering nodes, converted to milliwatts. For example, assuming \(P_{\text{tx,i}}=14\) dBm for interfering nodes and using path losses based on distance, the total interference power in this setup is \(1.447\times 10^{-10}\) mW. To assess communication quality under such interference, the signal-to-interference ratio (SIR) is calculated, representing the ratio of received signal power to total interference power at a receiver, with higher SIR values indicating reduced error rates and more reliable transmissions. The SIR is determined by first calculating the received signal power, derived from the transmitted power adjusted for path loss; for instance, a transmitter with \(P_{\text{tx}}=14\) dBm and a path loss of \(PL=100\) dB result in RSSI\({}_{n}=-86\) dBm, which is then converted to milliwatts for SIR computations. By integrating these metrics, a comprehensive understanding of the interplay between interference, network density, and communication quality is achieved, enabling the design of robust and efficient LoRa networks. The interference power \(I_{\text{average}}\) is next computed by summing the contributions from all other transmitters [7]. Each transmitter's interference power at the receiver is calculated using its transmitted power and the path loss specific to its distance from the receiver. These individual powers are converted from decibels to milliwatts. Once summed, the total interference \(I_{\text{average}}\) is obtained, as shown in this study to be \(1.447\times 10^{-10}\) mW.