## 4 The Up/Down Mechanism of WiFi MAPs
### Algorithm Specification
 \tag{4}\]

Now we will define a measure for evaluating the degree of interference redress of our mechanism. Let \(\kappa_{m,n}(t)\) be the interference experienced by every MAP at instant \(t\). If we define \(\rho_{m,n}^{Z}\) as the matrix of active nodes configuring the route with maximum interference level able to communicate the given Z, then Equation (5) determines its corresponding interference redress level \(R^{Z}(t)\) reached when the set of current active nodes is \(A_{m,n}(t)\). It should be noted that, unlike Equation (4), in this case the minimum level is always set to 0. Otherwise, if every node was equally interfered, the factor \(R^{Z}(t)\) would return the maximum value (\(R^{Z}(t)\ =\ 1\)) which would indicate an optimum interference redress, while we would actually like to alert about any interference, even if it is the minimum. This is a different criterion respect to power saving factor \(S^{Z}(t)\) because, in this latter, there will always exist an extra average power consumed \(E_{m,n}\ >\ 0\); and our target, in this case, is that \(S^{Z}(t)\ =\ 0\) indicates the route for Z which minimizes the overall extra average power consumed:

\[R^{Z}(t)\ =\ \frac{\sum_{m=1}^{M}\sum_{n=1}^{N}\bigl{\{}\kappa_{m,n}(t)\ \times\ \rho_{m,n}^{Z}\bigr{\}}\ -\ \sum_{m=1}^{M}\sum_{n=1}^{N}\bigl{\{}\kappa_{m,n}(t)\ \times\ A_{m,n}(t)\bigr{\}}}{\sum_{m=1}^{M}\sum_{n=1}^{N}\bigl{\{} \kappa_{m,n}(t)\ \times\ \rho_{m,n}^{Z}\bigr{\}}} \tag{5}\]

Therefore, we consider that the desirable condition for our PWMNS is characterized by \(R^{Z}(t)\ \approx\ 1\), but keeping \(S^{Z}(t)\ \approx\ 1\); which implies a high degree of interference redress, but preserving a power saving close to the optimum. ### Simulation Results for Power Saving and Interference Redress Factors

In order to clarify the meaning of these measures we will use a straightforward example consisting of a regular 3 \(\times\) 3 mesh which is illustrated in Figure 7. In this example we will assume as condition that the nodes are separated the maximum coverage distance. As we showed in our work in [26], this condition implies that the routes can only follow the city block geometry, i.e., every node can only establish direct routes with its left, right, up or down neighbours. In Figure 7, every node is labelled with its identifier (a number from 1 to 9 inserted in a red box with yellow background), its \(E_{m,n}\) (a value on cyan background over the identifier) and its \(\kappa_{m,n}\) (a value on purple background down the identifier). The routes established among the nodes are indicated by a red line, so that in Figure 7 all the possible direct routes are established. Following we will study the power saving and interference redress possibilities which result when we establish node 1 as source and node 9 as sink. Figure 8 shows some of the routes which give: (a) one of the maximum power saving cases, concretely S(t) = 1, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}=0.5\), R(t) = 0.1, and \(\sum_{m=1}^{M}\sum_{m=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}=0.7\); (b) one of the intermediate cases, corresponding to S(t) = 0.5, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\times A_{m,n}(t)\}=1.3\), R(t) = 0, and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times A_{m,n}(t)\}=0.8\); and (c) one of the routes with the minimum power saving, specifically S(t) = 0, \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{E_{m,n}\ \times\ A_{m,n}(t)\}=2\), R(t) = 0, and \(\sum_{m=1}^{M}\sum_{n=1}^{N}\{\kappa_{m,n}\ \times\ A_{m,n}(t)\}=0.8\). In all cases more than one route gave the same level of S(t).