## Appendix A Optimization Problem  
In this section, we describe the ILP problem formalized for the comparison with the heuristic proposed in Section 4. The problem is inherited from [16] and aims to minimize the overall costs while determining in each stationary interval in which NVFI-PoPs to execute the VNFIs and which lightpath channels to be used to connect them. in this new formulation, the contributions of the total costs are twofold: (i) the rented cloud resources; (ii) the bandwidth associated to the lightpath. Another inherited concept is the one of the path channel that aims to satisfy contiguity and continuity constraints in the optical routing problem. As in [16], we evaluate off-line the available set \(C_{\bar{p},q}\) of path channels that are associated to a given physical path \(\bar{p}\) and given number \(q\) of Frequency Slots (FSs). The formalized problem is subject to two additional constraints that concern switch reconfiguration that occurs when a virtual link is mapped on a different lightpath with respect the previous SI. Figure 7: Total cost (**a**) and percentage cost saving (**b**) as a function of the maximum number \(R\) of switch reconfigurations. The offered traffic in the peak SI equals 800 Gbps. The cloud resource cost unbalancing parameter \(\alpha\) is varied from 1 to 3. The USNET network topology is considered. The first one is the following:  
\[\rho^{(h)}_{\bar{e}\bar{e}}\geq\delta_{\bar{e}\rho}(\sum_{q\in[1..Q_{ max}]}\sum_{c\in C_{\bar{p},q}}y^{(h)}_{cc}-\sum_{q\in[1..Q_{max}]}\sum_{c\in C_{ \bar{p},q}}y^{(h-1)\bmod N}_{cc})\] \[\forall e\in L;\forall\bar{e}\in L;\ \forall h\in[0..T-1] \tag{1}\]  
wherein \(\rho^{(h)}_{\bar{e}\bar{e}}\) is a binary variable that represents the reconfiguration of a virtual link \(e\) on the physical link \(\bar{e}\) during the \(h\)-th SI, \(\delta_{\bar{e}\bar{p}}\) is a binary parameter equal to 1 if the physical path \(\bar{p}\) contains the physical link \(\bar{e}\), and \(y^{(h)}_{cc}\) is the binary variable that assumes value 1 if the virtual link \(e\) is mapped on the path channel \(c\) during the \(h\)-th SI. The constraint (1) establishes that a virtual link \(e\in L\) has been moved in the \(h\)-th SI when it has been mapped on \(\bar{e}\in L\) in the \(h\)-th SI but it does not in the \(((h-1)\bmod N)\)-th SI. Therefore this constraint establishes if a switch configuration occurs. The second constraint introduced in the formalization is related to the the maximum number \(R\) of switch reconfigurations to be applied to the elastic optical network during the cycle-stationary period and could be expressed as follow:  
\[\sum_{e\in L}\sum_{c\in L}\sum_{h\in[0..T-1]}\rho^{(h)}_{\bar{e} \bar{e}}\ \leq\ R \tag{2}\]  
As aforementioned, the optimization problem is based on the minimization of an objective function characterizing the total cost \(C_{tot}\), that is:  
\[min\ \ C_{tot}=C_{CR}+C_{BW} \tag{3}\]  
where \(C_{CR}\) and \(C_{BW}\) are the cloud and bandwidth resource cost respectively. If \(T_{s}\) is the duration of a SI, the rented cloud resources cost \(C_{C}R\) could be expressed as follow:  
\[C_{CR} = T_{s}\sum_{h=0}^{T-1}\sum_{\bar{e}\in V_{NP}}c^{core}_{\bar{e} }\sum_{v\in V_{VI}}x^{(h)}_{v\bar{v}}N^{(h)}_{v} \tag{4}\]  
wherein \(c^{core}_{\bar{e}}\) is the cost for renting a core in the NVFI-PoP \(\bar{e}\) of the set \(V^{-}_{NP}\), \(x^{(h)}_{v\bar{v}}\) is a binary variable that assumes value 1 if the VNFI \(v\) is mapped on the NFVI-PoP \(\bar{v}\) during the \(h\)-th SI, and \(N^{(h)}_{v}\) represents the number of cores that need to be allocated for the VNFI \(v\) during the \(h\)-th SI.