# Provision of Energy- and Wavelength-Efficient Traffic
## 2 Scenario and Models of Traffic Grooming in the DSCNs
### Optimization Models  
#### 2.2.1 Minimize the Energy Consumption \begin{tabular}{l l} \(\min\,E^{Total}=\sum_{k}E_{k}^{total}\) \\  & \(=\sum_{k}\,(E_{k}^{sou}+E_{k}^{des}+E_{k}^{\mathsf{int}}+\sum\limits_{n\in Q_ {k}}I_{kn}E_{TX})\) \\ \end{tabular}   
where  
\[E_{k}^{su}=\sum\limits_{d:d\
eq k}\left[\left[ \begin{array}{c} \frac{\sum_{f\in[1,F_{k,d}]}b_{k,d}^{f}}{\Omega_{s}}\\ \Omega_{s}\end{array} \right]\cdot(E_{agg}+\frac{E_{eq}+E_{effa}}{\delta})\] \[\cdot\min(1,\sum\limits_{p}\xi_{f,p}\cdot\sum\limits_{j}\sum \limits_{i\
eq j}Z_{ij}^{f,p,w})\right],\;\forall k\in V,w\in[1,W]\] \[E_{k}^{des}=\sum\limits_{ss\
eq k}\left[\left[ \begin{array}{c} \frac{\sum_{f\in[1,F_{k,d}]}b_{k,d}^{f}}{\Omega_{s}}\\ \end{array} \right]\cdot(E_{agg}+\frac{E_{eq}}{\delta}) \tag{2}\] \[\cdot\min(1,\sum\limits_{p}\xi_{f,p}\cdot\sum\limits_{j}\sum \limits_{i\
eq j}Z_{ij}^{f,p,w})\right],\;\forall k\in V,w\in[1,W]\] \[E_{k}^{\text{int}}=\sum\limits_{s\
eq k}\sum\limits_{d\
eq k} \sum\limits_{j\
eq k}\sum\limits_{i\
eq k}\left( \begin{array}{c}\sum \limits_{m:m\in Q_{k}}\Gamma_{ij}^{sd}\Theta_{kn}^{j}+\sum\limits_{m:m\in Q_ {k}}\Gamma_{ij}^{sd}\Theta_{mk}^{j})\cdot E_{edfa}\end{array} \right.\]  
The minimum energy consumption of satellite \(k\) can be calculated by \(E_{k}^{Total}\) in Equation (1), where \(E_{k}^{su}\), \(E_{k}^{des}\), and \(E_{k}^{\text{int}}\) are the used energy when \(k\) serves as the source, destination, and intermediate nodes, respectively. The term \(I_{kn}E_{TX}\) in (1) represents the energy consumed by ISL establishing and maintaining between satellite \(k\) and \(n\), (i.e., the energy consumption of the transceivers). The term \(\left[\sum_{f\in[1,F_{k,d}]}b_{k,d}^{f}/\Omega_{s}\right]\) in (2) is the number of aggregated connection requests, which originate from \(k\) and terminate at \(d\). It is assumed that the number of ports used for electric/optical conversion is equivalent to that for the EDFA amplifier. The term \(\min(1,\;\sum_{p}\xi_{f,p}\cdot\sum_{j}\sum_{i\
eq j}Z_{ij}^{f,p,w})\) indicates that the aggregated traffic requests are taken as a whole in the process of optical/electric conversion and optical amplification. If the satellite \(k\) serves as the intermediate node, it is responsible for the signal receiving and forwarding, and the number of corresponding occupied ports can be expressed as \(\sum_{m:m\in Q_{k}}\Gamma_{ij}^{sd}\Theta_{mk}^{j}\) and \(\sum_{t:m\in Q_{k}}\Gamma_{ij}^{sd}\Theta_{kt}^{j}\), respectively. Moreover, many constraints must be satisfied in the optimization model. First, we list the port constraints \(C_{1}\)\(\sim\)\(C_{3}\), where \(C_{1}\) means that the occupied ports cannot exceed the maximum number of ports for traffic aggregation, \(C_{2}\) represents that the optical/electric and electric/optical conversions of traffic requests must be completed within the limited \(\Delta_{k}^{oe}\). Apart from ports used for traffic transformation, other ports are responsible for lightpath bypass, which is depicted as \(C_{3}\). \[C_{1} : \sum\limits_{d:d\
eq k}\left[\sum_{f\in[1,F_{k,d}]}b_{k,d}^{f}/ \Omega_{s}\right]\leq\Delta_{k}^{agg}=\
abla_{k}^{e}-\delta\;\Delta_{k}^{oe}\] \[C_{2} : \frac{1}{\delta}(\sum\limits_{d:d\
eq k}\left[\sum_{f\in[1,F_{k, d}]}b_{k,d}^{f}/\Omega_{s}\right]+\sum\limits_{ss\
eq k}\left[\sum_{f\in[1,F_{k, d}]}b_{s,k}^{f}/\Omega_{s}\right])\leq\Delta_{k}^{oe} \tag{3}\] \[C_{3} : \sum\limits_{m:m\in Q_{k}}\Gamma_{ij}^{sd}\Theta_{kn}^{j}+\sum \limits_{m:m\in Q_{k}}\Gamma_{ij}^{sd}\Theta_{mk}^{j}\leq\
abla_{k}^{o}- \Delta_{k}^{oe},\;\;\;\forall s,d,i,j\in V\]  
Second, we list other constraints \(C_{4}\)\(\sim\)\(C_{7}\). Wavelength constraint \(C_{4}\) is utilized to guarantee that the number of wavelengths used in every ISL cannot exceed \(W\). It is assumed that lightpaths in an ISL comprise both the forward and the reverse ones. \(C_{5}\) ensures that each traffic flow can traverse at most one virtual lightpath on any wavelengths, which implies that the traffic flows cannot be split in the process of forwarding.