# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 3 Results
At the highest transmission power level of 16 dBm, the success probability in LoRa-based networks reaches near-optimal performance, approaching 98%, demonstrating the reliability of the LoRa modulation scheme under elevated power levels and the model's ability to optimize communication parameters for maximum efficiency. However, the diminishing returns observed between 12 dBm and 16 dBm suggest a threshold beyond which further increases in transmission power provide minimal improvements in success probability. This insight underscores the importance of balancing transmission power with other critical factors such as energy consumption and network scalability. While higher transmission power enhances success probability, it also increases energy usage, which is a significant consideration for battery-powered IoT applications. The ability to achieve a success probability of 94-96% at moderate power levels (e.g., 6-12 dBm) highlights an optimal range where reliability and energy efficiency are balanced. This trend aligns with theoretical models of wireless communication, where received signal strength is proportional to transmission power under constant conditions, but it also reveals the influence of environmental factors and interference, as evidenced by the sub-optimal success probability at lower power levels. These findings emphasize the critical role of transmission power selection in LoRa-based networks, particularly for ensuring reliable communication over long distances or in challenging environments with high interference levels. In addition to transmission power, other key parameters such as the spreading factor (SF), bandwidth (BW), and Time on Air (ToA) also play a crucial role in determining the reliability and  
Figure 5: Transmission power in relation to success probability. efficiency of LoRa networks. Figures 6 and 7 illustrate the relationship between the probability of packet loss and key LoRa network parameters, specifically the spreading factor (SF), bandwidth (BW), and Time on Air (ToA). These parameters play a critical role in determining the reliability and efficiency of LoRa communications, particularly in dense network deployments where interference and packet collisions are more likely to occur. In Figure 6, the probability of packet loss is plotted against the spreading factor (SF) for different bandwidth (BW) values. The results show that as the SF increases, the Time on Air (ToA) also increases significantly. This is because a higher SF results in longer symbol durations, which extends the transmission time of each packet. As a consequence, packets are more exposed to potential interference from other transmissions, leading to a higher probability of packet loss. For example, at SF = 12 and BW = 125 kHz, the probability of packet loss is notably higher compared to SF = 7 under the same BW. This trend is consistent across all BW values, although the impact is more pronounced at narrower bandwidths (e.g., 125 kHz) due to the longer ToA. On the other hand, Figure 7 demonstrates how the bandwidth (BW) influences the probability of packet loss. A wider BW reduces the ToA, as data can be transmitted at a faster rate. This reduction in transmission time decreases the likelihood of packet collisions, thereby lowering the probability of packet loss. For instance, at BW = 500 kHz, the probability of packet loss is significantly lower compared to BW = 125 kHz, especially for higher SF values. However, it is important to note that while a wider BW improves transmission efficiency, it also reduces the receiver's sensitivity. This trade-off may require higher transmission power to maintain reliable communication over longer distances or in environments with higher noise levels. Additionally, the impact of the packet arrival rate (\(\lambda\)) on network performance is evident in these results. As \(\lambda\) increases, the network experiences higher traffic loads, leading to more frequent packet collisions and a higher probability of packet loss.