# Verification of a Probabilistic Model and Optimization in Long-Range Networks
###### Abstract  
This paper presents a comprehensive probabilistic analysis of packet loss in long-range (LoRa) networks, a vital aspect of low-power, wide-area communication systems increasingly employed in IoT applications. The proposed model integrates multiple critical factors, including packet arrival rates, transmission power levels, and the distance between transmitting nodes and the gateway. By incorporating these variables into a unified probabilistic framework, the model not only predicts packet loss and interference patterns but also provides insights into optimizing network parameters. Specifically, it focuses on determining the optimal transmission power required to balance energy efficiency and communication reliability. A distinctive feature of the analysis is its ability to adapt dynamically to varying network conditions, ensuring sustained performance even in environments with high node density or fluctuating traffic loads. The study also explores the interplay between transmission power and interference, demonstrating how careful calibration of power settings can significantly reduce packet collisions while conserving energy resources. The proposed framework not only advances theoretical understanding, but also offers actionable guidelines for network designers seeking to achieve high performance in resource-constrained environments.  
p ARTIC 2025 15 1873  
1 Facultad de Ciencias, Universidad Autonoma de San Luis Potosi, San Luis Potosi 78295, Mexico; [EMAIL_ADDRESS] (F.R.C.S.); [EMAIL_ADDRESS] (R.F.H.-H.); [EMAIL_ADDRESS] (U.P.R.)  
1 Facultad de Ciencias, Universidad Autonoma de San Luis Potosi, San Luis Potosi 78295, Mexico; [EMAIL_ADDRESS] (F.R.C.S.); [EMAIL_ADDRESS] (U.P.R.); [EMAIL_ADDRESS] (U.P.R.)  
2 Facultad de Ingenieria, Universidad Panamericana, Alvaro del Portillo 49, Zapopan 45010, Mexico; [EMAIL_ADDRESS] 2  
###### Abstract  
This paper presents a comprehensive probabilistic analysis of packet loss in long-range (LoRa) networks, a vital aspect of low-power, wide-area communication systems increasingly employed in IoT applications. The proposed model integrates multiple critical factors, including packet arrival rates, transmission power levels, and the distance between transmitting nodes and the gateway. By incorporating these variables into a unified probabilistic framework, the model not only predicts packet loss and interference patterns but also provides insights into optimizing network parameters. Specifically, it focuses on determining the optimal transmission power required to balance energy efficiency and communication reliability. A distinctive feature of the analysis is its ability to adapt dynamically to varying network conditions, ensuring sustained performance even in environments with high node density or fluctuating traffic loads. The study also explores the interplay between transmission power and interference, demonstrating how careful calibration of power settings can significantly reduce packet collisions while conserving energy resources. The proposed framework not only advances theoretical understanding, but also offers actionable guidelines for network designers seeking to achieve high performance in resource-constrained environments.  
Keywords:packet loss optimization; RSSI; IoT; probabilistic analysis; LoRa networks; transmission power; interference patterns; energy efficiency; IoT applications +
Footnote †: journal: Applied sciences