## 4 The Proposed Use Case: Data Model and Scalability Analysis
### Scalability Analysis  
Some simple analyses can be carried out to estimate the scalability of the proposed LPWAN infrastructure, by computing the maximum value of devices that can be managed by a single LoRaWAN  
Figure 6: Examples of data collected by using the described LoRaWAN architecture on 26 November 2019 between 7:00 and 17:00 UTC.  
base station. Referring to the data exchange procedure described above, from 9 B to 37 B messages are transmitted by field devices to supervisory entities every 15 min. Similarly, during downlink transmissions, 8 B to 36 B messages are exchanged every 3 h. If the additional 13 B required by the LoRaWAN headers and trailers are taken into account, the maximum uplink message is 50 B long, whereas the maximum downlink message is 49 B long. Alternatively, if we consider a more realistic scenario with a mixed set of simple sensors and complex devices, the average uplink message is 39 B long, and the average downlink message is 35 B long. Finally, if we consider the sum of the maximum uplink and downlink messages, including the LoRaWAN headers and trailers, about 649 B are exchanged every 3 h, leading to a maximum data throughput of 0.06 B/s, which means that the duty-cycle limitation of LoRaWAN is always satisfied.  
In the following, the theoretical capacity of a single LoRaWAN base station is computed for the two presented scenarios, i.e., by considering that each installed device operates with the maximum expected payload (both at uplink and downlink), and by considering a mixed set of simple sensors and complex devices, each of them operating with the expected average uplink and downlink payload. In both the cases, the theoretical capacity of a single LoRaWAN base station is computed by assuming that every 3 h each cell must be able to manage up to twelve uplink messages with a length from 39 B to 50 B each (corresponding to the two considered scenarios), and one downlink message (with a length from 35 B to 49 B), per each served node, including the LoRaWAN headers and trailers. In this study, three different channels have been taken into account by varying the Spreading Factor (SF) from 7 to 12. The computed capacity of each LoRa base station is reported in Tables 7 and 8, for the worst-case scenario (i.e., corresponding to the maximum expected payload) and for a realistic scenario (i.e., corresponding to the average expected payload), respectively.  
Under perfect synchronization, the number of modes is given by \(N=T/\sum T_{\text{OA}}\), where \(T\) is the considered refresh period (i.e., 3 h), and \(\sum T_{\text{OA}}\) is sum of the message over-the-air time duration per each node over \(T\). If we consider that the actual throughput of ALOHA access is about 18% of the synchronized scenario [46], the maximum number of devices that can be managed by a single LoRaWAN base station is on the order of 3295 and 3919 devices per channel, for the worst-case and the realistic scenarios, respectively, which correspond to about 9885 and 11,757 devices per cell, when the three compulsory channels are adopted.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Spreading Factor** & \(\sum T_{\text{OA}}\) **(ms)** & **Cell Capacity per Channel \({}^{*}\)** & **Cell Capacity per Channel \({}^{**}\)** \\ \hline
7 & 1263 & 8551 & 1539 \\
8 & 2270 & 4758 & 856 \\
9 & 4253 & 2539 & 457 \\
10 & 7973 & 1354 & 243 \\
11 & 14,881 & 725 & 130 \\
12 & 27,632 & 390 & 70 \\ \hline \hline \end{tabular}  
* under perfect synchronization, ** pure ALOHA access.  
\end{table}
Table 7: Capacity of a single LoRaWAN base station (reported as number of allowed modes per channel) in the worst-case scenario, corresponding to twelve consecutive uplink messages of 50 B each, and one downlink message of 49 B, with CR = 4/5 and BW = 125 kHz. CR: Coding Rate, BW: Bandwidth, \(\sum T_{\text{OA}}\): Sum of the Time-on-Air per each node over the considered refresh period \(T\).