## 4 Numerical Results
The predicted values are evaluated by applying the proposed traffic forecasting algorithm and from the knowledge of the real requested VNF-FG bandwidth values from 31 May 2004 to 20 June 2004 [24]. The real traffic values are used for the LSTM training. To reduce the training times we have considered an LSTM network with the following parameters [16]: (i) the number \(N_{nr}\) of neurons equals 8; (ii) the loop-back parameter \(L\) equals 24; (iii) the batch size \(N_{sz}\) equals 24; (iv) the total number \(N_{ep}\) of epochs has been fixed to 20; that is, the LSTM training process is executed 20 times to find the best model to perform forecasting. We assume the cloud and bandwidth resource allocation is performed by executing two algorithms that have been proposed by the authors in a previous paper [8]. It has been shown how these algorithms, referred to as Least Cloud Resource and Bandwidth (LCBC) and Deployment Costs Aware (DCA) [8], perform well in allocating resources for NFV networks and allows for an operation cost optimization when real traffic data are known. We report the cost in Figure 5 when the parameter \(w\), defined as the ratio of the Resource Allocation cost \(C_{RA}\) to the QoS degradation \(C_{QoS}\), is varied from 0.0001 to 1 and the resource allocation is performed in the cases in which the MSE and Asymmetric (ASYM) LSTM prediction techniques are used. Notice that because \(C_{RA}\) is fixed to 0.025 $/Gb, the variation of \(w\) is obtained by varying \(C_{QoS}\) from 250 to 0.025 $/Gb. In particular we study a case of interest in which the QoS degradation cost \(C_{QoS}\) is higher than or equal to the cloud resource allocation cost \(C_{RA}\). The results of Figure 5 have been achieved in the case of prediction step \(h\) equal to 1, 12 and 24. From the results reported in Figure 5 we can make the following remarks:  
* The proposed forecasting solution based on the asymmetric cost function allows for total costs lower than or equal to the one of the MSE-based forecasting solution; the total costs of the two solutions are equal only for \(w=1\), that is, when the over-provisioning and under-provisioning costs are equal; as a matter of example, the total costs of the MSE and ASYM solutions for \(w=0.04\) and \(h=12\) are 134 $ and 96 $ with 28% cost advantage of our proposed asymmetric LSTM prediction solution;
* The better performance in total cost of the asymmetric prediction solution for \(w\) lower than 1 is due to the fact that it reduces the resource under-provisioning periods and consequently, the costs due to the QoS degradation. To justify the results we also report in Figure 6 the predicted VNF-FG bandwidth values from 21 June 2004 to 25 June 2004 [24] for the traffic offered between two access nodes of the Deutsche Telekom network. In particular we compare the real, MSE and ASYM LSTM predicted traffic for values of \(w\) equal to 0.1 and prediction steps equal to 1, 12 and 24 in Figure 6a-c respectively. From these figures we can remark that the MSE predictions are very near to the real time series values but they do not allow us to reach the goal of over-estimating the time series values because of the higher QoS degradation costs; conversely the ASYM LSTM predictions allows for a correct operation mode by overestimating the predicted values. Figure 5: Cost in allocating resources for the NFV network of Figure 4 when \(w\) varies from 0.0001 to 1. The total cost is reported when the allocation algorithms use the MSE and ASYM LSTM predicted SFC bandwidth values and prediction steps \(h\) equal to 1, 12 and 24. Figure 6: Comparison of the Real, Mean Square Error (MSE) and Asymmetric (ASYM) LSTM prediction values for \(w\) = 0.1 and prediction step \(h\) equal to 1 (**a**), 12 (**b**) and 24 (**c**). These results are confirmed in Figure 7 where we report the total cost as a function of the prediction step \(h\) for values \(w\) varying from 0.01 to 1.