## 4 Numerical Results  
We will evaluate the effectiveness of the asymmetric cost function-based LSTM forecasting model in predicting the requested VNF-FG bandwidth when both the cloud resource allocation and QoS degradation costs are considered. The LSTM forecasting technique will be applied in a real scenario to evaluate the operation cost of an NFV network and compare it to the one achieved when an MSE traditional forecasting technique is applied. We provide some results in the case of the Deutsche Telekom (DT) network reported in Figure 4. The network is composed of 14 switches and 24 links. The main input parameters, their description and the values range are reported in Table 4.  
Figure 3: Definition of the asymmetric loss function. Figure 2: Long Short Term Memory (LSTM) Prediction Framework. The cloud resources are placed in \(N_{NP}=|\bar{V}_{NP}|=4\) NFVI-PoPs located in the cities of Hannover, Leipzig, Frankfurt and Nuremberg. Each NFVI-PoP is equipped with \(N_{\sigma}=48\) cores. The core costs of the NFVI-PoPs are randomly chosen among the values \(c_{N\Pi}^{core}=c^{i}C_{0}\) (\(i=0,\cdots,N_{NP}-1\)) [8,23] where \(C_{0}\) is a normalization cost and the parameter \(\varsigma\) characterizes the unbalancing of the core costs in the different NFVI-PoPs. In particular the cost unbalancing is high, as the parameter \(\varsigma\) is higher. For \(\varsigma=1\) we achieve equal costs and the balanced case. We will carry out the analysis when the average core cost \(c_{av}^{core}\) is fixed to 1 $/h. The knowledge of \(c_{av}^{core}\) leads to the normalization value \(C_{0}=N_{NP}*c_{av}^{core}\frac{1-\varsigma}{1-\varsigma^{N_{NP}}}\). Next we provide the results in the case of \(\varsigma=1.4\).  
We assume the link bandwidth \(B_{L}\) equals 30 Gbps.  
We consider four SFs: Firewall (FW), Intrusion Detection System (IDS), Network Address Translator (NAT) and Proxy. VNFIs can be instantiated in the NFVI-PoPs to support these SFs. They are supported by software modules characterized by the maximum processing capacities 900, 600, 900 and 600 Mbps with the number of allocated cores equal to respectively 4, 8, 2 and 4. We consider linear VNF-FG of the same type and composed by one FW, one IDS, one NAT and one Proxy.  
The choice of the core costs and processing capacities leads to a cloud resource allocation cost \(C_{RA}=0.025\) $/Gb for the VNF-FG considered. \begin{table}
\begin{tabular}{l l l} \hline \hline \(B_{L}\) & Link Bandwidth & 30 Gbps \\ \(B_{NP}\) & Number of NFVI-PoPs & 4 \\ \(N_{\sigma}\) & Number of cores in any NFVI-PoP & 48 \\ \(c_{av}^{core}\) & Average Core Cost & 1 $/h \\ \(\varsigma\) & Core cost unbalancing factor & 1.4 \\ \(T_{s}\) & Time duration of a TI & 1 h \\ \(C_{RA}\) & Cloud Resource Allocation cost ($ to & 0.025 $/Gbit \\  & be paid for one traffic Gbit) \\ \(C_{QoS}\) & QoS degradation cost ($ to be paid for & 0.00025–2.5 $/Gbit \\  & one lost traffic Gbit) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameters. Figure 4: Deutsche Telekom network topology. The distances are expressed in km. We assume that one VNF-FG is established for each tuple of access nodes reported in Figure 4. As VNF-FG bandwidth values, we consider the real traffic values measured at hourly intervals and reported in [24]. These values are used to forecast the future traffic values according to the procedure illustrated in Section 3. We evaluate the operation cost of the NFV network reported in Figure 4 when the resource allocation is based on predicted rather than real VNF-FG bandwidth values. In particular we have evaluated the cost for the period from 21 June 2004 to 25 June 2004 [24] by predicting the VNF-FG bandwidth values requested between all of the tuples of access nodes of Figure 4. Because real traffic is not available for the Deutsche Telecom network, we have used the ones available in [24] for other networks with similar size.