# Improving Energy Efficiency in LoRaWAN Networks with Multiple Gateways
## 4 System Model
### Network Utility Model  
In this section, we will present the network utility function and study its convexity. LoRaWAN uses the pure Aloha random access mechanism, where nodes transmit packets without channel sensing. Then, the Aloha mathematical equations can be used to estimate the expected performance of a typical LoRaWAN deployment [24]. We denote by \(p(s)\) the ratio of nodes that use SF \(s\in\Omega=\{7,8,9,10,11,12\}\). \(S_{min}\) refers to the minimum SF 7 and \(S_{max}\) to the maximum SF 12.  
Assuming that the transmit attempts are performed according to a Poisson distribution with a mean arrival rate of \(\lambda\) packets per second, the normalized channel traffic load on each SF \(s\) can be calculated as follows [25]:  
\[\begin{split} G(s)&=\lambda.D(s).ToA(s)\\ &=\lambda.p(s).N_{t}.ToA(s)\end{split} \tag{7}\]  
The probability of having \(k\) transmissions during two packet times is expressed according to the Poisson distribution, as follows [25]:  
\[P(k\text{ transmissions})=\frac{(2G)^{k}\exp{(-2G)}}{k!}. \tag{8}\]  
Then, the probability of a successful frame transmission (\(k=0\)) is equal to \(e^{-2.G(s)}\). The Aloha throughput per frame time can be obtained as the normalized channel load multiplied by the success probability: \(G(s).e^{-2.G(s)}\). Thus, to obtain the throughput (\(\mathcal{S}\)) in  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Description** \\ \hline \(\lambda\) & Packet generation rate \\ \(T\) & Time between two successive transmissions \\ \(ToA(s)\) & Time to transmit a packet using SF \(s\) \\ \(T_{r1}(s)\) & First receive window duration on SF \(s\) \\ \(T_{r2}(s)\) & Second receive window duration on SF \(s\) \\ \(R\_{D1}\) & RECEIVE\_DELAY1 Time from transmitting before opening the first receive window \\ \(R\_{D2}\) & RECEIVE\_DELAY2 Time from transmitting before opening the second receive window \\ \(k\) & Payload size \\ \(d\) & Duty cycle \\ \(N_{l}\) & Number of nodes \\ \(D(s)\) & Number of nodes that use SF \(s\) \\ \(N(s)\) & Number of nodes that use SF \(\geq s\) \\ \(p(s)\) & Ratio of nodes that use SF \(s\) \\ \(l\) & Transmission power level in dBm \\ \(l_{tx}(l)\) & Current consumption in transmit mode at transmission power level \(l\) \\ \(l_{tx}\) & Current consumption in receive mode \\ \(l_{xt}\) & Current consumption in standby mode \\ \(l_{lat}\) & Current consumption in idle mode \\ \(V^{\prime}\) & Node operating voltage \\ \(d_{1}\) & Probability that a node successfully receives a downlink message at \(RX1\) \\ \(d_{2}\) & Probability that a node does not successfully receive a downlink message at \(RX1\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: System model parameters.  
Figure 5: Time diagram for a class A node during a period of time \(T\).  
bits/s on each SF, we divide by the frame time and multiply by the packet length. \(\mathcal{S}\) is expressed as follows [26]:  
\[\begin{split}\mathcal{S}(s)&=\lambda.D(s).b.e^{-2.G(s)} \\ &=\lambda.p(s).N_{l}.b.e^{-2.\lambda.p(s).N_{l}.ToA(s)}\end{split} \tag{9}\]  
where \(b\) represents the number of bits per packet and is equal to the sum of the payload size \(k\) and the preamble length \(n_{\textit{pream}}\).  
We define the utility function of the network as the total throughput (\(\mathcal{R}\)) of the LoRaWAN network in bits/s. As the different SFs are assumed to be orthogonal, \(\mathcal{R}\) can be calculated as the sum of the throughput on each SF:  
\[\mathcal{R}=\sum_{s=S_{\textit{min}}}^{S_{\textit{max}}}\mathcal{S}(s) \tag{10}\]