## 5 Numerical Results  
We evaluate the effectiveness of the proposed heuristic for the networks of Figures 3-5. The network of Figure 3 is composed by four access nodes, four optical switches and four NFVI-PoPs;  
Figure 2: Two examples of application of the Label Setting algorithm. The initial multi-stage graphs are reported in (**a,c**) for different reconfiguration costs; the procedures for the evluation of the constrained shortest path are reported in (**b,d**) for the multi-stage graphs (**a,c**) respectively. its small size allows us to compare the results of the heuristic to the ones of the ILP problem inherited from [16] with the addition of a constraint on the maximum number of switch reconfigurations. The Deutsche Telekom network of Figure 4 is composed by fourteen access nodes, fourteen optical switches and four NFVI-PoPs located in the nodes of Leipzig, Hannover, Frankfurt and Nuremberg. The USNET network is composed by twenty-four access nodes, twenty-four switches and four NFVI-PoPs connected to the switches S1, S2, S3 and S4. The Deutsche Telekom and USNET network scenarios will allow us to evaluate the impact of the limited number of switch reconfigurations on the cost saving in realistic case studies. The fiber lengths are reported as labels in Figures 3-5. Figure 4: Deutsche Telekom Network. Figure 3: Optical Network Topology with four access nodes, four optical switches and four NFVI-PoPs. The spectrum in each fiber is organized in frequency slot of 6.25 GHz. We assume that adaptive modulation systems are used whose choice determines the maximum path length. In particular the BPSK, QPSK, 8 QAM and 16 QAM modulations are used with path length \(L_{BPSK}\), \(L_{QPSK}\), \(L_{8 QAM}\) and \(L_{16 QAM}\) reported in Table 2.  
The NFVI-PoPs are equipped with four types of VMs executing the Firewall (FW), Intrusion Detection System (IDS), Network Address Translator (NAT) and Proxy service functions. The number \(n^{c}\) of cores to be allocated to the various types of VMs and their maximum processing capacity \(C_{p}\) are reported in Tables 3 and 4 for the NFVI-PoPs located in the networks of Figures 3 and 4 respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & \(\mathbf{C_{p}}\) & \(\mathbf{n^{c}}\) \\ \hline
**FW** & 30 Gbps & 130 \\
**IDS** & 30 Gbps & 350 \\
**NAT** & 30 Gbps & 65 \\
**PROXY** & 30 Gbps & 130 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Maximum processing capacity \(C_{p}\) and allocated number \(n^{c}\) of cores for the VMs implementing FW, IDS, NAT and Proxy in the network scenario of Figure 4.  
Figure 5: USNET Network. \begin{table}
\begin{tabular}{c c} \hline \hline \(L_{BPSK}\) & 3000 Km \\ \(L_{QPSK}\) & 1500 Km \\ \(L_{8 QAM}\) & 750 Km \\ \(L_{16 QAM}\) & 375 Km \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Maximum Path Length for the Binary Phase Shift Keying (BPSK), Quadrature Phase Shift Keying (QPSK), 8 Quadrature Amplitude Modulation (8 QAM) and 16 QAM modulation systems. To obtain the results of the heuristic we have implemented it in a JAVA code. Conversely the results of the optimization problem have been achieved by using the CPLEX tool. First we carry out a comparison between the results of the heuristic and the ILP problem for the small network scenario of Figure 3. We assume that each NFVI-PoP is equipped with a total number of 2000 cores. The core cost is equal to 0.74 $/h, 0.89 $/h, 1.08 $/h and 1.29 $/h for the NFVI-PoP-1, NFVI-PoP-2, NFVI-PoP-3 and NFVI-PoP-4 respectively [28]. The total number of frequency slots in each optical fiber is 50. The bandwidth cost is equal to 0.001 cent/KmGHzhour. We assume a cycle-stationary traffic [16] with two only SIs. The total offered in the peak SI is 200 Gbps and a 80% traffic reduction occurs in the other SI. Four type of SFCs are considered: the first one composed by a FW, the second one is composed by a FW and an IDS, the third one composed by a FW, an IDS and a NAT and the fourth one is composed by a FW, an IDS, a NAT and a PROXY.