# MEC Server Sleep Strategy for Energy Efficient Operation of an MEC System
## 4 Deep Learning-Inspired Sleep Control Strategy
### Task Arrival Rate Prediction  
The amount of tasks offloaded to an MECS \(i\) is determined by the number of users in \(U_{i}(t)\) and their service preferences. Since the distance that a user can move during a time slot is limited, \(U_{i}(t+1)\) is influenced mainly by \(U_{i}(t)\) and \(U_{j}(t)\)s, where \(j\in\mathcal{M}_{i}\). Therefore, \(\hat{\lambda}_{i}(t+1)\) is affected not only by \(\{\hat{\lambda}_{i}(t),\ldots,\hat{\lambda}_{i}(t-h+1)\}\) but also \(\hat{\lambda}_{j}(t)\). Consequently, to increase the prediction accuracy, the spatio-temporal correlation among \(\{\hat{\lambda}_{i}(t),\ldots,\hat{\lambda}_{i}(t-h+1)\}\) and \(\hat{\lambda}_{j}(t)\) must be exploited. To make use of the spatio-temporal relationship, we model an MEC system as a graph \(G=(V,E)\). The set of nodes \(V\) is the set of MECSs \(\mathcal{M}\). The elements of the edge set \(E\) are the links which reflect the adjacency between MECSs. In other words, the element \((i,j)\) in \(E\) is 1 if \(j\in M_{i}\), otherwise it is zero. Our goal is to predict the amount of tasks imposed on each MECS during the next time slot. Thus, we use the history of the workload generated in each \(S_{i}\) as a node feature. We denote the node feature vector as \(x(t)=\{\tilde{\lambda}_{i}(t):i\in\mathcal{M}\}\). Therefore, our task arrival prediction problem becomes a problem to find a mapping function \(f_{p}\) such that  
\[[\tilde{\lambda}_{1}(t+1),\ldots,\tilde{\lambda}_{|\mathcal{M}|}(t+1)]=f_{p} (G;[x(t),\ldots,x(t-h+1)]), \tag{6}\]  
where \(h\) is the length of the task-load history. Since an MEC system has a graph structure, we adopt a STGCN model to approximate \(f_{p}\) by capturing and exploiting the spatio-temporal dependency among \(\tilde{\lambda}_{i}(t-a)\)s for all \(i\in\mathcal{M}\) and \(a\in\{1,\ldots,h-1\}\). As we can see in Figure 1, STGCN is composed of two consecutive ST-Conv blocks and one output layer. The ST-Conv block is composed of a spatial Graph-Conv module in between two temporal Gated-Conv modules (Figure 2). The Gated-Conv module is composed of a 1-D convolution unit having a width \(K_{t}\) kernel followed by GLU (gated linear unit). We denote the input to the \(k\)-th temporal Gated-Conv module in the \(l\)-th ST-Conv block as \(v_{k,l}=(v_{k,l}(t),\ldots,v_{k,l}(t-h+1)\) where \(k,l\in\{1,2\}\). Then, for each node in \(G\), the 1-D convolution unit performs temporal convolution on \(v_{k,l}\) by exploring \(K_{t}\) neighbors, which reduces the sequence length from \(h\) to \(h-K_{t}+1\). Two linear layers in the GLU take the temporal features extracted by the 1-D convolution unit and produces \(P_{k,l}\) and \(Q_{k,l}\), respectively. Then, GLU outputs \(c_{k,l}\) by computing \(P_{k,l}\odot\sigma(Q_{k,l})\), where \(\odot\) is the element-wise Hadamard product. Therefore, if we denote the temporal convolution kernel for the \(k\)-th temporal Gated-Conv module in the \(l\)-th ST-Conv block as \(\Gamma_{k,l}\) and the temporal gated convolution operator as \(*_{T}\), the temporal gated convolution is expressed as  
\[c_{k,l}=\Gamma_{k,l}*_{T}v_{k,l}=P_{k,l}\odot\sigma(Q_{k,l}), \tag{7}\]  
where \(\sigma(\cdot)\) is a sigmoid function. The spatial Graph-Conv module is designed based on a ChebNet [38], which is a spectral-based GNN. The goal of the spatial Graph-Conv module is to derive the spatial features from the temporal features extracted by the temporal Gated-Conv module. To achieve the goal, the spatial Graph-Conv module performs graph convolution on \(c_{k,l}\). We denote the graph convolution operator as \(*_{g}\) and the graph convolution kernel used by the  
Figure 1: Deep learning-inspired MECS sleep control framework. Figure 2: ST-Conv block pipeline. spatial Graph-Conv module in the \(l\)-th ST-Conv block as \(\Theta_{l}(l\in\{1,2\})\). We also denote the input size and the output size of the feature maps as \(C_{i}\) and \(C_{o}\), respectively.