# Verification of a Probabilistic Model and Optimization in Long-Range Networks
## 3 Results
The distances considered ranged from 500 m to 5 km, simulating ideal line-of-sight (LoS) conditions. Practical experiments were performed in an open-field environment to validate the simulation results. The following distances were tested: 500 m, 1 km, 2 km, and 5 km. Both simulated and practical results are summarized in Table 6. The practical results were obtained using the Reyax RYLR998 module configured with a spreading factor (SF) of 12, a bandwidth (BW) of 125 kHz, and a transmission power of 14 dBm. These settings were chosen to maximize communication range while maintaining reliable performance under real-world conditions. The results show that LoRa maintains high performance in long-range applications. At distances between 500 m and 2 km, the RSSI remains significantly higher than the threshold of \(-120\) dBm, ensuring a success probability of 100%. Even at extended distances of 5 km, although the RSSI decreases to \(-91.65\) dBm, the success probability remains high at 83.38%. In the practical experiment, the RSSI also remains significantly higher than the sensitivity threshold as expected in the simulation. Even at extended distances of 5 km, although the RSSI decreases to approximately \(-91.65\) dBm, the success probability remains relatively high, reaching 78.20% in practical tests. This slight reduction in success probability at longer distances can be attributed to real-world factors such as multipath fading, environmental noise, and minor obstructions. These results confirm that the probabilistic model developed is not only applicable in short-range scenarios but also in long-range environments where \begin{table}
\begin{tabular}{c c c} \hline
**Bandwidth (kHz)** & **Airtime (ms)** & **Energy per Packet (mJ)** \\ \hline
125 & 50 & 0.75 \\
250 & 25 & 0.38 \\
500 & 12.5 & 0.19 \\ \hline \end{tabular}
\end{table} 
Table 4: Energy consumption for different bandwidth settings. the impact of path loss and RSSI is critical and reinforces the LoRa utility in applications such as precision agriculture and environmental monitoring. Building upon the insights gained from the results, it is important to consider how communication protocols, such as ALOHA, play a role in the performance of LoRa networks under varying conditions. ALOHA is a random access protocol used in LoRa networks to manage retransmissions in the absence of centralized coordination. The two main variants of ALOHA are Pure ALOHA, where nodes transmit at any time without checking the channel, which can lead to potential collisions, and Slotted ALOHA, where time is divided into slots, and transmissions occur only at the start of a slot, thereby reducing collisions. The probability of a successful transmission in these systems is given by the following:  
\[P_{success}=e^{-G} \tag{8}\]  
where \(G\) is the average number of transmission attempts per time slot. The maximum channel efficiency is as follows:  
\[\eta_{max}=\frac{1}{2e} \tag{9}\]  
indicating that Slotted ALOHA performs better under higher loads. The results show that LoRa maintains high performance in long-range applications, as in an effective range from 500 m to 2 km, the RSSI is significantly higher than the threshold, ensuring a success probability of 100%. At extended distances of up to 5 km, although the \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & \multicolumn{2}{c}{**Total Energy**} & \multicolumn{1}{c}{**Average Current (mA)**} \\ \hline Low Traffic (1 pkt/10 min) & 432 & 0.05 \\ Medium Traffic (1 pkt/5 min) & 864 & 0.10 \\ High Traffic (1 pkt/1 min) & 4320 & 0.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Energy consumption in real-world scenarios. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Traffic Load (G)** & **Success Probability—Pure ALOHA** & **Success Probability—Slotted ALOHA** \\ \hline
0.1 & 0.0819 & 0.0905 \\
0.3 & 0.1619 & 0.2231 \\
0.5 & 0.1839 & 0.3033 \\
1.0 & 0.1353 & 0.3679 \\
1.5 & 0.0672 & 0.3247 \\
2.0 & 0.0366 & 0.2707 \\ \hline \hline \end{tabular}
\end{table}