## 3 Results: Network Performance
### Fairness  
Throughput and fairness are usually in conflict. On one hand, the highest throughput is obtained with a greedy schedule. On the other hand, a fair schedule yields a poor throughput. When all node locations are equally important in terms of data acquisition, transmission fairness [28] is a scheduling objective. In this analysis, fairness means that all nodes transmit the same amount of their own data in the long-term, regardless of their distance to the sink node. Previous works by other authors deal \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{**FIFO**} & \multicolumn{2}{c}{**Round-Robin**} \\ \cline{3-6}
**Size (N)** & \begin{tabular}{c} **Offered Traffic** \\ \(\left[\lambda_{2}\,\lambda_{3}\,\ldots\,\lambda_{N}\right]\) \\ \end{tabular} & \begin{tabular}{c} **W\({}^{ini}\)** \\ \end{tabular} & \begin{tabular}{c} **Delay** \\ **Max. (Av.)** \\ \end{tabular} & \
\begin{tabular}{c} **W\({}^{ini}\)** \\ \end{tabular} &
\begin{tabular}{c} **Delay** \\ **Max. (Av.)** \\ \end{tabular} \\ \hline
4 & \(\left[\begin{array}{ccc}1&8&6\\ \hline \end{array}\right]\) & \(\left[\begin{array}{ccc}1&2\\ \hline \end{array}\right]\) & \(11\) (5.29) & \(11\) (5.29) \\
4 & \(\left[\begin{array}{ccc}1&8&6\\ \hline \end{array}\right]\) & \(\left[\begin{array}{ccc}1&3\\ \hline \end{array}\right]\) & \(11\) (5.1) & \(11\) (5.29) & \(11\) (5.3) & \(11\) (5.1) \\
4 & \(\left[\begin{array}{ccc}1&1&1\\ \hline \end{array}\right]\) & \(\left[\begin{array}{ccc}1&0&1\\ \hline \end{array}\right]\) & \(11\) (4.69) & \(\left[\begin{array}{ccc}0&2&3\\ \hline \end{array}\right]\) & \(11\) (4.69) \\
5 & \(\left[\begin{array}{ccc}1&8&8\\ \hline \end{array}\right]\) & \(\left[\begin{array}{ccc}3&0&1\\ \hline \end{array}\right]\) & \(15\) (7.99) & \(\left[\begin{array}{ccc}3&0&1\\ \hline \end{array}\right]\) & \(14\) (7.99) \\
6 & \(\lambda_{j}=\frac{1}{10}\) (\(j=2,..,6\)) & \(\left[\begin{array}{ccc}0&1&2&4&1\\ \hline \end{array}\right]\) & \(21\) (10.95) & \(\left[\begin{array}{ccc}0&1&1&4&0\\ \hline \end{array}\right]\) & \(21\) (10.16) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum delay of the optimal scheduling (slots) Round-Robin vs. FIFO. Simulation time: 1000 time slots. Figure 11: Average end-to-end delay sweeping \(\mathbf{W}^{ini}\) values for an \(N=6\) nodes network and simulation time of 1000 slots. The two service policies considered (Round-Robin (**a**) and FIFO (**b**)) have similar values. The discretization in the \(y\)-axis is because the transmissions are allowed by the scheduler only at the beginning of a time slot. with fair scheduling in STDMA networks. [PERSON] et al. proposed a scheduling algorithm [29], but they emphasized adaptive scheduling instead of the shortest frame. Concerning UWSNs, [PERSON] and [PERSON] proposed STDMA protocol for ad hoc UWSNs where fairness was considered [30] but not uniformly achieved. [PERSON] et al. also presented an algorithm to find optimal fair scheduling for linear topology in TDMA networks [31], but the duration of the frame (called cycle in their paper) is greater than our frame length (four slots) and dependent of the network size \(N\), i.e., for a \(N=3\) network, their frame length is six slots, and 12 slots in the case of \(N=5\). Besides that, the end-to-end delay is higher than the present case, as we have seen in Section 3.2. In general, a multi-hop TDMA network does not exhibit fair behavior. In our case, the optimal scheduling jointly with the packet generation rate in (14) guarantees a fair operation in the network, maintaining the number of packets delivered to the Gateway from every node.