## 2 System Model and Problem Statement
### Power Consumption Model and Problem Formulation  
Unless otherwise stated, we consistently use the following indices ranging over the respective sets: \(j\in\mathcal{J}\), \(a\in\mathcal{A}\), \(q\in\mathcal{Q}\), \(y\in\mathcal{V}_{c}\), \(\tilde{y}\in\mathcal{V}_{c}\cup\mathcal{J}\), \(f\in\mathcal{F}\), \(q\in\mathcal{Q}\), \(u,v,w\in\mathcal{V}\), \(m,n\in\mathcal{V}_{sw}\). We introduce the decision variables \(x_{j,a}\) as the Boolean indicator of whether UE \(j\) attaches to BS \(a\), and \(\phi_{y,f,a}\) as the Boolean indicator of whether VNF \(f\) requested by service \(q\) is deployed on node \(\tilde{y}\). Furthermore, \(\theta_{e}^{e^{\prime},q}\) (or its alias \(\theta_{u,\tilde{y}}^{e^{\prime},q}\)) is the Boolean indicator variable of whether the directed **physical** link \(e=(u,v)\in\mathcal{E}\) belongs to the **physical** path in \(\mathcal{G}\) onto which the **virtual** edge \(e^{\prime}\in\mathcal{E}^{(\rho_{k})}\) is mapped for SFC \(\rho_{q}\). Finally, \(N_{f,\tilde{y}}\) is the number of **instances** of VNF \(f\) deployed on node \(\tilde{y}\). Towards superior energy-saving performance, we employ resources, devices, and links **only** when needed. Specifically, collocated CPU resources at node \(y\) are employed only when \(y\)**actually** runs deployed VNFs, as captured by the Boolean indicator variable \(\xi_{y}\). Hence, the power consumed by CPU processing at node \(y\) is given by  
\[P_{y}^{(CPU)}=P_{y}^{(CPU)}\xi_{y}+\left(P_{y}^{(CPU,m)}-P_{y}^{(CPU,j)}\right) \cdot U_{y}, \tag{1}\]  
where \(P_{y}^{(CPU,m)}\), \(P_{y}^{(CPU,j)}\) are the maximum and idle power of the CPU deployed at \(y\) and \(U_{y}\triangleq\sum_{f\in\mathcal{F}}\frac{N_{f,g}u^{m}}{\xi_{\tilde{y}}}\) is the CPU load factor at \(y\)[15]. Similarly, for a fiber link \((n,m)\in\mathcal{E}_{fi}\), the Boolean variable \(z_{n,m}\) indicates whether the link **actually** carries traffic, in **either** link direction, for any request. To examine whether link \((n,m)\in\mathcal{E}_{fi}\) carries any traffic in the **specific** direction from \(n\) to \(m\), we introduce the Boolean variable \(w_{n,m}\), which implies that it must hold  
\[ \begin{split} z_{m,n}&\geq w_{m,n}\\ z_{m,n}&\geq w_{n,m}\end{split} \quad\forall\;(n,m )\in\mathcal{E}_{fi}. \tag{2}\]  
We denote with \(\psi_{n}\) the Boolean variable of whether the switch of node \(n\in\mathcal{V}_{sw}\) is **actually** used. There exist certain consistency relations between these variables as shown in (3) below, where \(C_{1}>0\) is a sufficiently large constant. \[ \begin{split}&\sum_{q\in\mathcal{Q}}\sum_{e^{\prime}\in\mathcal{E} ^{(\rho_{k})}}\theta_{e}^{e^{\prime},q}\leq C_{1}w_{e},\;\forall\;e\in\mathcal{ E}_{fi},\\ &\sum_{e\in\mathcal{E}_{fi}:h(e)=n}w_{e}+\sum_{e\in\mathcal{E}_{fi }:l(e)=n}w_{e}\leq C_{1}\psi_{n},\;\forall\;n\in\mathcal{V}_{sw}.\end{split}  \tag{3}\]The above relations capture the fact that a node is active (i.e., \(\psi_{n}=1\)) only if it has active incident links carrying traffic and, similarly, a link is active only if it is carrying traffic for one of the requested SFCs. The total power consumed by the switch in node \(n\) is  
\[P_{n}^{(sw)}=P_{iitle}^{(sw)}\psi_{n}+P_{port}\sum_{m\in\mathcal{V}_{sw}:(n,m) \in\mathcal{E}_{fi}}z_{n,m}, \tag{4}\]  
where \(P_{iitle}^{(sw)}\) denotes the switch idle power and the second term accounts for the **active** fiber links of the switch, with \(P_{port}\) being the power consumed by each active port [15].