## 4 The Up/Down Mechanism of WiFi MAPs
### Algorithm Specification
Figure 9 illustrates some of the results for the interference redress R(t) study: (a) one of the routes with the maximum redress, concretely R(t) = 1, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}=0\), S(t) = 0.3, and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}=1.5\); (b) one with an intermediate value, corresponding to R(t) = 0.4, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}=0.5\), S(t) = 0.1, and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}=1.8\); and (c) one of the routes with the minimum redress factor, specifically R(t) = 0, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}=0.8\), S(t) = 0, and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}=2\). Though routes shown in Figures 8 and 9 are different, there are other routes which share their same S(t) and R(t) level. Thus Figures 8c and 9c are both valid for S(t) = 0 (minimum power saving) and R(t) = 0 (minimum interference redress factor). On the other hand, we can also appreciate the usefulness of our factors in adjusting to a unitary range the highly-uneven original range of the corresponding measures \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}\) and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}\). This is better illustrated in Table 4, where we show the data generated by all possible routes. Figure 7: Regular 3 \(\times\) 3 mesh with city block routes. One of the issues facing our proposal is the simultaneous reachability of our both targets: power saving and interference redress. In Table 4 we can observe that those targets are not always incompatible, so that in the 3 rd entry (route 1-2-5-8-9) a high value has been reached for both factors: S(t) = 0.7 and R(t) = 0.8. In order to verify how frequently we can reach both targets, we made a new simulation in a \(3\times 3\) mesh where we set \(E_{m,n}~{}=~{}0.1\) W for all nodes, and generated all the possible combinations of binary interferences \(\kappa_{m,n}\). It must be clear that we have only considered binary levels of interference, i.e., \(\kappa_{m,n}~{}=~{}\{0|1\}\), where 0 means not interfered and 1 interfered. On the other hand, though \(E_{m,n}\) is fixed to a constant value for every node, extra average power varies according to the length of the considered route. Figure 10 shows the resulting histogram for R(t) vs. S(t) considering all the possible routes in the described conditions. As we can observe the highest peak is placed at {R(t),S(t)} = {0,0}, which corresponds to the longest routes which in addition crosses the most interfered nodes. We can also appreciate that the combination {R(t),S(t)} = {1,1} seldom occurs (but it does). However we can highlight a significant combination for 0.5 < R(t) < 1 and 0.5 < S(t) < 1, which represents favourable cases for both power saving and interference redress, simultaneously. We should not forget that \(E_{m,n}\) is fixed to a constant value for every node, so the S(t) values are inversely proportional to routes length. In fact, Figure 10 shows three clearly visible groups (0, 0.5 and 1) for S(t), which correspond to the three groups of route length showed in Table 4 (5, 7 and 9 nodes). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Route** & **R(t)** & \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{D_{m,n}\cdot A_{m,n}(t)\}\) & **S(t)** & \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\cdot A_{m,n}(t)\}\) \\ \hline
1-4-5-8-9 & 1 & 0 & 0.3 & 1.5 \\
1-4-7-8-9 & 1 & 0 & 0.3 & 1.6 \\
1-2-5-8-9 & 0.8 & 0.2 & 0.7 & 0.9 \\
1-2-5-4-7-8-9 & 0.8 & 0.2 & 0.1 & 1.8 \\
1-4-5-6-9 & 0.4 & 0.5 & 0.6 & 1.1 \\
1-4-7-8-5-6-9 & 0.4 & 0.5 & 0.1 & 1.8 \\
1-2-5-6-9 & 0.1 & 0.7 & 1 & 0.5 \\
1-2-3-6-9 & 0 & 0.8 & 1 & 0.5 \\
1-2-3-6-5-8-9 & 0 & 0.8 & 0.6 & 1.1 \\
1-4-5-2-3-6-9 & 0 & 0.8 & 0.5 & 1.3 \\
1-2-3-6-5-4-7-8-9 & 0 & 0.8 & 0 & 2 \\
1-4-7-8-5-2-3-6-9 & 0 & 0.8 & 0 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation results for a \(3\times 3\) PWMNS. It seems obvious that interference impact is higher in central nodes of PWMNS.