# Proactive Service Caching in a MEC System by Using Spatio-Temporal Correlation among MEC Servers
## 3 System Model  
We consider a MEC system composed of \(M\) MECSs and a remote server hosting a controller. We denote the set of MECSs in the system as \(\mathcal{M}=\{1,\dots,M\}\). We depict the system model in Figure 2, which also shows the overall process flow. We assume that each MECS \(m\in\mathcal{M}\) is colocated with a base station, and a server is connected to all the MECSs through a backhaul network. Periodically, the controller collects the service popularity information from each MECS and makes a caching decision for all \(m\in\mathcal{M}\). We denote the set of services that a MEC system provides as \(\mathcal{K}\) and \(K=|\mathcal{K}|\), where \(|\mathcal{K}|\) is the cardinarity of a set \(\mathcal{K}\). Each user asks for a service by offloading its tasks to the nearest MECS by using a wireless link between the user and the base station colocated with the MECS. If an user \(u\) offloads its task to a MECS \(m\) and the task is processed by \(m\), the MECS \(m\) is called the serving MECS of a user \(u\). We consider a discrete time controller by dividing time into slots of length \(\tau\). Figure 1: The inner structure of a ConvLSTM cell at time step \(t\). The operator \(*\) represents the convolutional operation, and \(\circ\) denotes the Hardmard product. In addition, \(\sigma\) represents the sigmoid function, and _tanh_ represents the hyperbolic tangent function. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Meaning** \\ \hline \(X_{t}\) & Input at time step \(t\). \\ \(i_{t}\) & Output of an input gate at time step \(t\). \\ \(f_{t}\) & Output of a forget gate at time step \(t\). \\ \(o_{t}\) & Output of an output gate at time step \(t\). \\ \(c_{t}\) & Cell state at time step \(t\). \\ \(h_{t}\) & Hidden state of a cell at time step \(t\). \\ \(W_{xi}\) & Model parameters used for convolution of \(X_{t}\) in an input gate. \\ \(W_{hi}\) & Model parameters used for convolution of \(h_{t-1}\) in an input gate. \\ \(b_{i}\) & Bias in an input gate. \\ \(W_{ci}\) & Model parameters used for element-wise product with \(c_{t-1}\) in an input gate. \\ \(W_{xf}\) & Model parameters used for convolution of \(X_{t}\) in a forget gate. \\ \(W_{hf}\) & Model parameters used for convolution of \(h_{t-1}\) in a forget gate. \\ \(b_{f}\) & Bias in a forget gate. \\ \(W_{cf}\) & Model parameters used for element-wise product with \(c_{t-1}\) in a forget gate. \\ \(W_{xc}\) & Model parameters used for convolution of \(X_{t}\) when making \(c_{t}\). \\ \(W_{hc}\) & Model parameters used for convolution of \(h_{t-1}\) when making \(c_{t}\). \\ \(b_{c}\) & Bias used when making \(c_{t}\). \\ \(W_{x0}\) & Model parameters used for convolution of \(X_{t}\) in an output gate. \\ \(W_{ho}\) & Model parameters used for convolution of \(h_{t-1}\) in an output gate. \\ \(b_{i}\) & Bias in an output gate. \\ \(W_{ci}\) & Model parameters used for element-wise product with \(c_{t}\) in an output gate. \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Notations used in the inner structure of a ConvLSTM cell at time step \(t\). A server can provide all the services in \(\mathcal{K}\) while a MECS \(m\) can serve a subset of \(\mathcal{K}\) depending on its service cache size \(\eta_{m}\) and a caching decision made by a controller. We introduce a variable \(a_{m,k}(t)\) to represent the service cache state of a MECS \(m\) during a time slot \(t\). Specifically, \(a_{m,k}(t)=1\) indicates that a MECS \(m\) contains the service \(k\) in its service cache during a time slot \(t\). Otherwise, \(a_{m,k}(t)=0\). Then, the service cache state of a MECS \(m\) during a time slot \(t\) is represented by the following service cache vector. \[a_{m}(t)=\{a_{m,k}(t)|k\in\mathcal{K}\}. \tag{2}\]  
Accordingly, the service cache state of a MEC system during a time slot \(t\) is represented by a \(M\times K\) matrix  
\[A_{M\times K}= \begin{bmatrix}a_{1}(t)\\ \dots\\ a_{m}(t)\\ \dots\\ a_{M}(t)\\ \end{bmatrix} =