# Matching of Energy Provisions in Multihop Wireless Infra-Structures
## 5 Energy Differentiation Factor for Energy Provision in WMI Networks
### Differentiated Energy Consumption in a Line Topology  
In this subsection, we demonstrate the method of estimating differentiated energy consumption with an example of a WMI network with line topology. An example of line topology WMI network with 5 BSs is shown in Figure 5. We suppose that BS are deployed in a line and assume that the distance between neighboring BS corresponds to the wireless transmission range, so each BS can only receive or overhear packets from its neighbors. We also suppose that each BS generates a number \(TX_{local}\) of packets for transmission upstream to the external network via the GW. Let each BS be labelled with an identifier (ID) in a sequence corresponding to the topological distance from the GW, so that a BS that has a topological distance of \(i\) hop to the GW, has ID equal to \(i\). With these assumptions, we then introduce the basic relations between \(TX\) and \(TX_{local}\), \(RX\) and \(TX\), as well as \(OH\) and \(TX\), according to the multihop packet relay features in the line-topology WMI. First, the number of \(TX\) packets at each BS(\(i\)) can be written in terms of the amount of \(TX_{local}\) packets as follows.  
\[TX(i)=\sum_{k=i}^{N}TX_{local}(k) \tag{6}\]  
We call this relation the \(TX\)-\(TX_{local}\) relation.  
Second, the number of \(RX\) packets at each BS(\(i\)) can be expressed according to the amount of \(TX\) packets as:  
\[RX(i)=\begin{cases}TX(i+1)&0<i<N\\ 0&i=N\end{cases} \tag{7}\]  
We call this relation the \(TX\)-\(RX\) relation.  
Third, the number of \(OH\) packets at each BS can be described in terms of the amount of \(TX\) packets by:  
\[OH(i)=\begin{cases}TX(i-1)&2<i\leq N\\ 0&i=1\end{cases} \tag{8}\]  
That is, each BS(\(i\)) overhears the packets transmitted at BS(\(i\)-1). We call this relation the \(TX\)-\(OH\) relation.  
In the example of a line topology WMI network that is shown in Figure 5, there are 5 BSs and 5 STAs in the network. Each STA is associated with one of the BS and generates a flow of packets for  
Figure 5: A line-topology WMI example with 5 traffic flows.  
delivery via its BS and the WMI to the external network through the GW. Given the \(TX_{local}\) generation amount in a certain time duration \(D\), we can derive the activity and energy differentiation in the network. To be specfic, we let the generation amount be the same at each BS, let the energy \(E_{Others}\) be equivalent to 6 units, and let the energy consumption be 1 unit per packet in \(TX\), \(RX\) and \(OH\). Note that since we only need the ratios of energy consumption, it is not necessary to specify the absolute units of energy.  
The number of each type of packet operation (\(TX\), \(RX\), \(OH\)), the energy consumption, and the matching energy provision for each BS are illustrated in Table 3. Differentiation factors can be derived based on Equations (4) and (5). The optimal energy provision for BSs is obtained according to Equation (3), We can see that the energy consumption and energy differentiation factor is largest at node BS(2).  
This simple model serves to illustrate the idea of differentiated energy consumption and Matching-EP. In practice, the accuracy of the estimate of energy consumption will depend on how well the model estimates the actual wireless activity. One way of using this simple model in practice is to consider energy consumption factors as average quantities, including estimates of average energy required for multiple-retransmissions per packet, and the energy of transmissions required for network management.  
Finally, we note that the analysis of energy differentiation in the line-topology WMI can be easily extended to WMI with multiple separated line paths, as shown in Figure 6. Given a WMI with multiple separate line paths to the GW, the dependence of energy consumption differentiation on BS hop distance from the GW is similar to that for the simple line topology.