## 1 Introduction  
Network Function Virtualization [1] technology allows the implementation of software middleboxes located in data centers, referred to as Network Function Virtual Infrastructure-Point of Presence (NFVI-PoP), and running on virtual machines. In these last few years the problem of resource reconfiguration in NFV environments has been widely investigated [2; 3; 4; 5]. The studies focused on reactive techniques based on which the network is reconfigured as soon as traffic changes occur [7; 8; 9; 10]. Everyone agrees that reactive techniques are ineffective in relation to the high variability of traffic and the high time of cloud resource allocation. For this reason proactive reconfiguration techniques have been proposed where the traffic or the amount of resources needed is predicted [11]. Traffic and cloud resource prediction methodologies have been recently used in Network Function Virtualization environments for cloud and bandwidth resource allocation purposes. Both traditional and innovative prediction methodologies [12] have been proposed. For instance, Long Short Term Memory-based prediction techniques have been shown to be very effective in allocating resources. All of these techniques are based on the minimization of symmetric cost functions as the Mean Square Error (MSE) that equally weighs positive and negative prediction errors. However, the error sign can differently impact the cost increase due to prediction errors. For instance, when the Quality of Service degradation cost due to traffic loss is prevalent with respect to the cloud resource allocation cost, an algorithm is preferable that overestimates the offered traffic; conversely, the traffic underestimation is preferable in the opposite case when the cloud allocation cost is lower than the QoS degradation one.  
To our best knowledge, only [PERSON] et al. [13] proposes a solution for mobile network resource orchestration in which the different values of the over-provisioning and under-provisioning costs are taken into account. DeepCog is proposed, a framework for resource allocation to slicing in a 5G mobile environment. It is based on a deep learning technique in which the cost function attributes a rising cost as the amount of over-allocated resources increases and a constant penalty, which is independent of the lost traffic amount, when a QoS degradation occurs.  
In this paper we propose a prediction technique, which, aware of the fact that traffic cannot be accurately predicted, tries to overestimate or underestimate traffic in relation to the values of over-provisioning and under-provisioning costs. This objective is achieved by minimizing an asymmetric cost function characterized by a parameter that takes into account the over-provisioning and under-provisioning costs. The principle of the proposed solution can be applied to any prediction technique and in this work it is applied to one that predicts traffic with a Long Short Term Memory (LSTM) prediction methodology.  
The main contributions of the manuscript are the following:  
* The study and the investigation of a prediction-based resource allocation algorithm for NFV network environments;
* The study and the investigation of an LSTM-based traffic prediction algorithm with an asymmetric loss function that optimally predicts the traffic values according to the over-provisioning and under-provisioning costs;
* A performance comparison of the proposed solution to other ones proposed in literature and based on predictions with minimization of symmetric loss functions.  
The paper is organized as follows. We describe the state-of-the-art, the problem statement and the prediction-based reconfiguration algorithm in Section 2. The traffic forecasting technique based on asymmetric traffic function is illustrated in Section 3. The numerical results, reported in Section 4, show the effectiveness of the proposed technique with respect to MSE-based traditional forecasting techniques in an NFV network environment.