# Cooperative Caching and Resource Allocation in Integrated Satellite-Terrestrial Networks
## 4 Problem Formulation
### Cooperative Caching and Resource Allocation Problem
#### 4.1.1. Caching Model  
When satellite network \(G(V,E)\) provides edge caching services for users, each satellite can deploy parts of all the contents as the onboard storage resources are limited. Furthermore, the content placed on a satellite can be accessed by different users, where a content sharing approach can be used to decrease the onboard storage resource usage. Therefore, in order to describe the formulated caching problem more clearly, the onboard storage resource usage for users can be viewed in two scenarios of first and non-first content services. In the first content service scenario, content \(w\) on satellite \(v\) is firstly accessed by user \(u\), then the storage resources deploying content \(w\) on satellite \(v\) are viewed as the storage resource usage for the content service of user \(u\). In the non-first content service scenario, content \(w\) on satellite \(v\) is not firstly accessed by user \(u\), then the storage resources deploying content  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Name** & \multicolumn{2}{c}{**Description of Symbols**} \\ \hline  & Satellite–terrestrial Network \\ \hline \(G(V,E)\) & Satellite network with satellites \(V\) and links \(E\). \\ \(dc\) & Cloud center. \\ \(h\) & Number of network hops. \\ \(C_{v}\) & Available storage resources for satellite \(v\). \\ \(v,e\) & The \(v\)-th satellite and the \(e\)-th link. \\ \(B_{e}\) & Available bandwidth resources for link \(e\). \\ \(t_{e}\) & Transmission delay for link \(e\). \\ \(D_{max}^{in}\) & Maximum number of users provided content services for satellite \(v\). \\ \(E_{v}^{in},E_{v}^{out}\) & Sets of input and output degrees for satellite \(v\). \\ \(v_{dc,a}\) & Access satellite of cloud center \(dc\). \\ \(k_{dc}\) & Distance between cloud center \(dc\) and the access satellite. \\ \hline  & Users and Contents \\ \hline \(U,M\) & Set of users and the number of users. \\ \(u,v_{u,a}\) & The \(u\)-th user and the access satellite. \\ \(bw_{u}\) & Bandwidth between user \(u\) and the access satellite. \\ \(g_{u,c}\) & Channel gain of retrieving the content for user \(u\). \\ \(p_{u,n}\) & Noise power of retrieving the content for user \(u\). \\ \(p_{u,tr}\) & Transmission power of retrieving the content for user \(u\). \\ \(k_{u}\) & Distance between user \(u\) and the access satellite. \\ \(c\) & Velocity of data transmission. \\ \(W,Q\) & Set of contents and the number of contents. \\ \(w\) & The \(w\)-th content. \\ \(s_{w},p_{w}\) & Data size and popularity for content \(w\). \\ \(\mu_{u,w}^{max}\) & Maximum service delay of retrieving the content for user \(u\). \\ \hline  & Decision Variables \\ \hline \(x_{u,w}\) & \(x_{u,w}=1\) if user \(u\) retrieves content \(w\). \\ \(y_{w,v}\) & \(y_{w,v}=1\) if content \(w\) is deployed on satellite \(v\). \\ \(z_{u,v}^{w}\) & \(z_{u,v}^{w}=1\) if user \(u\) retrieves content \(w\) on satellite \(v\). \\ \(q_{u,e}^{u}\) & \(q_{u,e}^{u}=1\) if user \(u\) retrieves content \(w\) by link \(e\). \\ \(a_{u,dc}\) & \(a_{u,dc}=1\) if user \(u\) retrieves content \(w\) from cloud center \(dc\). \\ \hline  & Variables \\ \hline \(\phi_{U}\) & Objective function of content services for all users. \\ \(\lambda_{u,w}^{slot}\) & Cost factor of storage resource usage for user \(u\). \\ \(\lambda_{u,w}^{hw}\) & Cost factor of network bandwidth consumption for user \(u\). \\ \(j_{v,v_{u,a}}\) & \(j_{v,v_{u,a}}=1\) indicates satellite \(v\) can be viewed as satellite \(v_{dc,a}\). \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of SYMBOLS.  
\(w\) on satellite \(v\) are viewed as zero for the content service of user \(u\). Therefore, the onboard storage resource usage for the content service of user \(u\) can be indicated as  
\[\varphi_{u,w}^{stor}=\left\{\begin{aligned} & x_{u,w}\cdot(1-a_{u,dc})\cdot \sum\limits_{p\in V}y_{w,p}\cdot z_{u,w}^{w}\cdot s_{uw},\qquad\text{First service},\\ & 0,\qquad\qquad\qquad\qquad\text{Non-first service}.\end{aligned}\right. \tag{1}\]