# On the Evaluation of Complex Networks Designs for an Energy-Efficient IP/WDM Core Network
## 6 MILP Model  
In this section, we study the impact of the implementation of the small-world, random, and SFN physical topologies on the total network power consumption of IP over WDM NSF network. For completeness, we re-introduce the MILP optimization model developed in [Tucker and Dong] for a non-bypass IP over WDM network to evaluate and compare such proposed designs with the current physical topology of the NSF network. The model shall minimize the network power consumption by optimizing the network resources. Optimizing network resources through dynamic approaches by switching off underutilized resources at low traffic demands and grouping, grooming, and consolidating traffic efficiently on fewer resources. The focus in this work is on the core network of the IP over WDM; the access network and aggregation routers are not considered in the evaluation study. Before introducing the model, we list the notations and parameters used in the model as follows (Tables 2 and 3). Figure 8: IP over WDM network architecture [13]. The total network power consumption of the modeled network consists of:  
1. Total power consumption of router ports \[\sum_{x\in N}\sum_{y\in Nm}P_{RP}*\gamma_{xy}\] (4)
2. The power consumption of transponders \[\sum_{x\in N}\sum_{y\in Nm}P_{Transponder}*\gamma_{xy}\] (5)
3. The power consumption of EDFAs \[\sum_{x\in N}\sum_{y\in Nm}P_{EDFA}*F_{xy}*\varphi_{xy}\] (6)
4. The power consumption of optical switches \[\sum_{x\in N}P_{OXC}\] (7)
5. Total power consumption of aggregation ports \[\sum_{i\in N}P_{RP}*\beta_{i}\] (8) \begin{table}
\begin{tabular}{c l} \hline \(N\) & Set of vertices (nodes) \\ \hline \(Nm_{\{i\}}\) & Set of neighbors of node \(i\) \\ \hline \(s\) _and_\(d\) & Denote source and destination pair \\ \(i\) _and_\(j\) & Denote end points of a virtual link in the IP layer \\ \hline \(x\) _and_\(y\) & Denote end points of a physical link in the optical layer \\ \(P_{RP}\) & Denote power consumption of router ports \\ \hline \(P_{Transponder}\) & Denote power consumption of a transponder \\ \hline \(P_{OXC}\) & Denote power consumption of optical switch \\ \hline \(P_{EDFA}\) & Denote power consumption of an EDFA \\ \hline \(P_{Reg}\) & Denote power consumption of a regenerator \\ \hline \(B\) & Denote capacity of a wavelength \\ \hline \(\varphi_{xy}\) & Denote number of EDFAs on physical link \((x,y)\). \(\varphi_{xy}\) can be calculated as: \(\varphi_{xy}=\left(\frac{\delta_{xy}}{d}-1\right)+2\), where \(d\) is the distance \\  & between two neighboring EDFAs and \(\delta_{xy}\) is the distance between \((x,y)\) nodes. \\ \hline \(\tau^{sd}\) & Size of traffic demand from node \(s\) to node \(d\) in \(b/s\) \\ \hline \(\eta\) & Denotes number of wavelengths in each fiber \\ \hline \end{tabular}
\end{table} 
Table 2: Notations and parameters [1]. \begin{table}
\begin{tabular}{c l} \hline \(\tau^{sd}_{ij}\) & The size of traffic flow from node \(s\) to node \(d\) that traverse the virtual link \((i,j)\) \\ \(C_{ij}\) & The number of wavelength channels used in the virtual link \((i,j)\) \\ \hline \(\gamma^{ij}_{xy}\) & The number of wavelength channels used in the virtual link \((i,j)\) \\  & that traverse physical link \((x,y)\) \\ \hline \(\tau_{xy}\) & The number of wavelength channels that propagate physical link \\  & \((x,y)\) \\ \hline \(F_{xy}\) & Number of fiber strands in physical link \((x,y)\) \\ \hline \(\beta_{i}\) & Number of routers’\(i\) aggregation ports \\ \hline \(NPC\) & Total network power consumption \\ \hline \end{tabular}
\end{table} 
Table 3: Variables [1]. The total number of aggregation ports in each node:  
\[\beta_{i}=\frac{1}{B}\sum\limits_{d\in N}\tau^{id}\forall i\in N \tag{9}\]
6. Power consumption of regenerators: \[\sum\limits_{x\in N}\sum\limits_{y\in N_{m}}P_{Reg}*RG_{xy}*\tau_{xy}\] (10) The MILP model is defined as follows: Objective: Minimize NPC The total network power consumption (NPC) is given in (11): \[