# Hardware Mechanism for Energy Saving in WiFi Access Points
## 2 The Mechanism for Energy Saving in WiFi AP
### Mathematical Model Analysis  
We modelled the theoretical conditions that must be accomplished to obtain energy saving comparing the power consumption of the system in Figure 1 between using our mechanism and not using our mechanism. We considered the transition between charge and discharge (and vice versa) periods takes a negligible amount of power consumption and for that reason we do not model it. That transition also does not affect the performance of AP transmitting WiFi beacons. Where \(T>0\) is an amount of time during the test of energy saving, \(V\) is the constant and continuous voltage supplied by \(S\) and \(I\) the corresponding current. Figure 2 shows our system without our mechanism and the function of _Aadt_ component (we include it only for uniformity in the discussion). During \(T\) the power consumption (\(P\)) of the AP is:  
\[P=\int_{0}^{T}V_{a}(t)\;I_{a}(t)\;dt=\int_{0}^{T}V(t)\;I(t)\;dt \tag{1}\]  
For analysing the power consumption (\(P\)) with our mechanism we differentiate the charge and discharge periods. Figure 3 shows the name of voltages and currents and the adaptation of them realized in Aadt and Badt components in the Charge period. Where, \(w_{v}\) is a charger voltage divider factor (Badt), which depends on the internal components and the voltage value required to charge the battery, and \(w_{c}\) is a current divider factor of the charger (Badt), which depends on the internal components and the value of the current needed to charge the battery. Figure 1: (a) Graphic schema of the wireless fidelity access point (WiFi AP) and main components of our novel mechanism for energy saving. (b) Processing model: chained alternating charge and discharge periods. Figure 2: Graphic and formal model of our system without our mechanism. If the Charge period lasts \(T_{c}\) units of time, then the power consumption in the AP will be: \(\int_{0}^{T_{c}}V_{a}(t)\ I_{a}(t)\ dt\) and the power consumed by \(B\) will be: \(\int_{0}^{T_{c}}V_{c}(t)\ I_{c}(t)\ dt\). Adding those powers, considering the adaptation of Aadt and Badt and ignoring the power consumed in the \(PR\) (we consider it is negligible), and supposing the \(V\) is enough to supply the AP and charge the B, we will obtain:  
\[ \begin{array}{ll}P^{\prime}&=\int_{0}^{T_{c}}V_{a}(t)\ I_{a}(t)\ dt+\ \int_{0}^{T_{c}}V_{c}(t)\ I_{c}(t)\ dt\\ &=\int_{0}^{T_{c}}V(t)\ (I(t)-\frac{I_{c}(t)}{w_{c}})\ dt+\ \int_{0}^{T_{c}}\frac{V(t)}{w_{c}}\ I_{c}(t)\ dt\\ &=\int_{0}^{T_{c}}V(t)\ I(t)\ dt-\frac{1}{w_{c}}\ \int_{0}^{T_{c}}V(t)\ I_{c}(t)\ dt+\frac{1}{w_{c}}\ \int_{0}^{T_{c}}V(t)\ I_{c}(t)\ dt\\ &=\int_{0}^{T_{c}}V(t)\ I(t)\ dt+(\frac{1}{w_{c}}-\frac{1}{w_{c}})\ \int_{0}^{T_{c}}V(t)\ I_{c}(t)\ dt\end{array}  \tag{2}\]  
Figure 4 shows the name of voltages and currents and the adaptation of them realized in Aadt (Badt does not transform its inputs) component in the discharge period. Where \(w_{d}\) is a voltage divider with reference to the Aadt and \(w_{b}\) is a multiplication factor with reference to the Aadt. If the discharge period lasts \(T_{d}\), then the power consumption in the AP will be:  
\[P^{\prime}=\int_{0}^{T_{d}}V_{a}(t)\ I_{a}(t)\ dt=\int_{0}^{T_{d}}\frac{V(t)} {w_{d}}\ I_{d}(t)\ w_{b}dt=\ \frac{w_{b}}{w_{d}}\int_{0}^{T_{d}}V(t)\ I_{d}(t)\ dt \tag{3}\]  
\(\bullet\) Optimizing \(T_{c}\) and \(T_{d}\): Let us now consider \(T=T_{c}+T_{d}\), that is, we only consider one charge and one discharge period for optimizing \(T_{c}\) and \(T_{d}\) in order to obtain the energy saving. We argue that, in general, energy saving needs that \(T_{c}\) be minimized and \(T_{d}\) be maximized obtaining the appropriated values of coefficients named with letter \(w\) above. The \(T_{c}\) can be minimized optimizing \(w_{v}\), \(w_{c}\) and maximizing \(I_{c}\) in \(P^{\prime}\) maintaining the AP working correctly. The \(T_{d}\) can be maximized optimizing \(w_{d}\), \(w_{b}\) and minimizing \(I_{d}\) such as the AP that could work with a minimum of capability.