# Matching of Energy Provisions in Multihop Wireless Infra-Structures
## 5 Energy Differentiation Factor for Energy Provision in WMI Networks
### Zone Model of Differentiated Energy Consumption in 2D Topology  
In this section, we introduce a zone model for estimating differentiated energy consumption in a WMI with a two-dimensional topology. In the zone model, the WMI is partitioned into zones according to the topological distance of BS to the GW, so that the traffic flow can be represented by aggregate packet flows between contiguous zones. The topological distance is the relay hop count. The energy-consuming packet operations in the network include transmitting, receiving and overhearing. We define zone(\(k\)) as the set of all BSs in the network with the number of hops from BS(\(i\)) to the GW equal to \(k\). \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**BS** & \(TX_{local}\) & \(TX\) & \(RX\) & \(OH\) & \(E_{Others}\) & \(E_{\epsilon}\) & _Diff_ & \(E_{0}[i]\) \\ \hline BS(1) & 1 & 5 & 4 & 0 & 6 & 15 & 1.09 & 10.9 \\ BS(2) & 1 & 4 & 3 & 5 & 6 & 18 & 1.30 & 13.0 \\ BS(3) & 1 & 3 & 2 & 4 & 6 & 15 & 1.09 & 10.9 \\ BS(4) & 1 & 2 & 1 & 3 & 6 & 12 & 0.87 & 8.7 \\ BS(5) & 1 & 1 & 0 & 2 & 6 & 9 & 0.65 & 6.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The traffic, energy consumption and differentiation at each BS in a line-topology WMI wth 5 BS, each generating \(TX_{local}=1\) packet. \(E_{0-all}=50\) J. The energy consumption difference in the zone-based model can be analysed in terms of aggregate numbers of packet operations (_TX_, _RX_ and _OH_) in each zone. Figure 7 shows an example of zones in a 2D mesh network, and Table 4 shows the packet operations, energy consumption and differentiation factors in each zone. The total energy consumption in a zone(\(k\)) is given by:  
\[Total_{Z}(k)=E_{Pt}ZTX(k)+E_{Pr}ZRX(k)+E_{Pr}ZOH(k)+Oth(k) \tag{9}\]  
where \(ZTX(k)\), \(ZRX(k)\), \(ZOH(k)\) are the number of _TX_, _RX_ and _OH_ packets, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**zones** & \(\mathbf{N_{Z}}\) & \(\mathbf{ZTX_{local}}\) & \(\mathbf{ZTX}\) & \(\mathbf{ZRX}\) & \(\mathbf{ZOH}\) & \(\mathbf{E_{others}}\) & \(\mathbf{E_{Total-Z}}\) & \(\mathbf{E_{Total-B}}\) & \(\mathbf{Diff_{28S}}\) \\ \hline zone(1) & \(N_{Z}(1)\) & \(\mathit{Sr}_{Z}(1)\) & \(\mathit{ZTX}(1)\) & \(\mathit{ZRX}(1)\) & \(\mathit{ZOH}(1)\) & \(\mathit{Oh}(1)\) & \(\mathit{Total_{Z}(1)}\) & \(\mathit{Total_{B}(1)}\) & \(\mathit{Diff(1)}\) \\ \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– \\ zone(k) & \(N_{Z}(k)\) & \(\mathit{Sr}_{Z}(k)\) & \(\mathit{ZTX}(k)\) & \(\mathit{ZRX}(k)\) & \(\mathit{ZOH}(k)\) & \(\mathit{Oh}(k)\) & \(\mathit{Total_{Z}(k)}\) & \(\mathit{Total_{B}(k)}\) & \(\mathit{Diff(k)}\) \\ \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– & \(\tilde{\ }\)– \\ zone(H) & \(N_{Z}(H)\) & \(\mathit{Sr}_{Z}(H)\) & \(\mathit{ZTX}(H)\) & \(\mathit{ZRX}(H)\) & \(\mathit{ZOH}(H)\) & \(\mathit{Oh}(H)\) & \(\mathit{Total_{Z}(H)}\) & \(\mathit{Total_{B}(H)}\) & \(\mathit{Diff(H)}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The traffic, energy consumption and differentiation in BS zones. Figure 6: Applications of line-topology WMI analysis. Energy consumption analysis of line-topology WMI can be applied to a WMI with multple separated lines paths to the GW. Figure 7: Zone model of network traffic and energy consumption. The average energy consumption at a BS in zone(\(k\)) is given by:  
\[Total_{B}(k)=Total_{Z}(k)/N_{Z}(k) \tag{10}\]  
where \(N_{Z}(k)\) is the number of BS in zone(\(k\)). Consequently, for each BS in zone(\(k\)), the differentiated energy consumption ratio \(Diff(k)\) can be expressed as:  
\[Diff(k)=\frac{Total_{B}(k)}{\sum_{i=1}^{N}Total_{B}(i)/N} \tag{11}\]  
where \(N\) is the total number of BS. Let \(ZTX(k)\), \(ZRX(k)\) and \(ZOH(k)\) represent the number of packet operations \(TX\), \(RX\) and \(OH\) in a zone(\(k\)), respectively.