## 3 System Model and Problem Formulation
### Cache Performance Model  
The caching performance is primarily determined by the quality of user service, namely, the latency in accessing the content. To this end, it is necessary to establish an overall delay model for cloud-edge caching and to ensure that the requested content by the user is represented as a hit in its local BS (resulting in minimum content transfer delay). \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(\mathcal{N}=\{1,\ldots,n,\ldots,N\}\) & Set of BSs \\ \(\mathcal{U}=\{1,\ldots,u,\ldots,U\}\) & Set of UEs \\ \(\mathcal{F}=\{1,\ldots,f,\ldots,F\}\) & Set of content \\ \(\mathcal{T}=\{1,\ldots,t,\ldots,T\}\) & Epochs of federated learning/caching decision. \\ \(v_{u,n}\),\(v_{u,n}\),\(v_{u,n}\),\(v_{b}\) & Transmission rate between UEs and BSs, BSs and Cloud, BSs and BSs. \\ \(D_{n}\), \(D_{f}\) & The size of the content in \(BS_{n}\), the size of the content \(f\). \\ \(P_{u,f}\) & UE \(u\) preferences for content \(f\). \\ \(\Re_{u}\), \(\Re_{n}\) & Number of requests from UE \(u\), number of \\ \(H_{t,n}\), \(H_{t,system}\) & requests received by \(BS_{n}\). \\ \(E_{t,n}^{cmp}\), \(E_{t,n}^{up}\) & Cache hit efficiency of system in epoch \(t\). \\ \(E_{t,system}\) & The energy consumption respectively, of local \\ \(s_{i}=(s_{i,n}^{r}s_{i,n}^{c})\) &
\begin{tabular}{c} epoch \(t\). \\ The caching state. \\ The caching decision. \\ The Double Deep Q-Network (DDQN) reward. \\ \(\
abla_{w_{i}}\)loss(\(w_{i}\)) \\ \(\beta_{t,n}\) & The weight factor for adaptive iteration times. \\ \(D_{t,n}\) & Amount of data for a single training session in \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key notations and descriptions. Figure 1: Proposed networks topology architecture. Similar to [31], the transmission medium of content between the BS and UE is wireless and is determined by the Shannon formula:  
\[v_{u,n}=B\log(1+\frac{q_{u}g_{u,n}}{\varpi+\sum\limits_{i\in U\backslash\{u\}:u_{ i}=a_{u}}q_{i}g_{i,n}}) \tag{1}\]  
where \(B\) represents the channel bandwidth, \(\varpi\) represents the background noise, \(q_{u}\) represents the transmission power between the \(BS_{n}\) and the UE \(u\), and \(g_{u,n}\) represents the channel gain. \(a_{i}=a_{u}\) represents that UE \(u\) has the content it needs. Caching content raises the issue of content popularity. High content popularity indicates that users request the content more frequently, while low content popularity indicates that users request the content less frequently. Therefore, content popularity can directly affect cache performance. As a result, edge BSs should cache more popular content. In this article, content popularity is defined as a measure of user preference for content f. Similar to [13], \(d_{U-B}\) is defined as the transmission delay between the user and the BS. \[d_{U-B}=\sum\limits_{f\in\mathcal{U}}\sum\limits_{f\in\mathcal{F}}P_{u,f}\frac {D_{f}}{v_{u,n}} \tag{2}\]  
where \(D_{f}\) represents the size of content \(f\), and \(P_{u,f}\) is the UEs' preferences for content \(f\):  
\[P_{u,f}=r_{u,f}/\Re_{u} \tag{3}\]  
\[ \begin{array}{l}subject\ to:\\ f\in\mathcal{F}\\ \sum\limits_{f\in\mathcal{U},f\in\mathcal{F}}P_{u,f}=1\end{array} \]  
\(\Re_{u}\) represents the total number of requests made by UE \(u\), while \(r_{u,f}\) represents the number of requests made by UE \(u\) for content \(f\). Similarly, we can obtain the transmission delay between the cloud and the edge:  
\[d_{C-B}=\sum\limits_{f\in\mathcal{U}}\sum\limits_{f\in\mathcal{F}}P_{u,f}\frac {D_{f}}{v_{a}} \tag{4}\]  
\(v_{a}\) denotes the transmission rate from the cloud to the edge BS, and similarly, by setting \(v_{b}\) as the transmission rate between BSs, we can obtain the transmission delay between BSs:  
\[d_{B-B}=\sum\limits_{f\in\mathcal{U}}\sum\limits_{f\in\mathcal{F}}P_{u,f}\frac {D_{f}}{v_{b}} \tag{5}\]  
The cache hit rate can directly evaluate the effectiveness of caching policies and reflect the performance of caching systems.