# Cooperative Caching and Resource Allocation in Integrated Satellite-Terrestrial Networks
## 4 Problem Formulation
### Cooperative Caching and Resource Allocation Problem
#### 4.1.4 Problem Description  
When the cost factors of storage resource usage and network bandwidth consumption for the content service of user \(u\) are indicated as \(\lambda_{u,u}^{Stor}\) and \(\lambda_{u,u}^{bu}\), the objective function for all users can be given as  
\[\phi_{U}=\frac{\sum\limits_{u\in U}{(\lambda_{u,w}^{Stor}\cdot\varphi_{u,w}^{ stor}+\lambda_{u,w}^{bu}\cdot\varphi_{u,w}^{bu})}}{\sum\limits_{u\in U}{x_{u,w}}} \cdot\frac{M}{\sum\limits_{u\in U}{x_{u,w}}}. \tag{10}\]  
There are multiple physical constraints to be considered for cooperative caching and resource allocation. When the satellite network provides content services for users, user \(u\) can retrieve the required content \(w\) from one and only satellite, which can be given as  
\[\sum\limits_{v\in V}{y_{u,v}\cdot z_{u,v}^{w}}={x_{u,w}\cdot(1-a_{u,dc})}. \tag{11}\]  
It is guaranteed that the required storage resources of deploying contents should not be greater than the current available onboard resources, which can be indicated as  
\[\sum\limits_{w\in W}{y_{u,v}\cdot s_{w}}\leq C_{v},vs.\in V. \tag{12}\]  
It is also guaranteed that the required network bandwidth resources for each link can not be more than the current available bandwidth resources. The network bandwidth resource constraint can be indicated as  
\[\sum\limits_{u\in U}{x_{u,w}\cdot q_{u,e}^{w}\cdot b_{u,w}}\leq B_{e},e\in E. \tag{13}\]  
It is also guaranteed that the content service delay for each user should not be more than the maximum acceptable service delay, which can be indicated as:  
\[t_{u,w}\leq t_{u,w}^{max},u\in U,w\in W. \tag{14}\]  
Furthermore, the number of users provided content services for each satellite should not be more than the maximum value \(D_{v}^{max}\), which can be given as  
\[\sum\limits_{u\in U}{x_{u,w}\cdot(1-a_{u,dc})\cdot y_{u,v}\cdot z_{u,v}^{w}} \leq D_{v}^{max},vs.\in V. \tag{15}\]  
Moreover, when content service are provided by satellite network \(G(V,E)\) for one user, a path will be used to route the data. In that case, the number of input and output degrees for each satellite can not be greater than 1. Therefore, the constraint of input and output degrees can be indicated as  
\[\left\{ \begin{array}{l}\sum\limits_{e\in E_{v}^{in}}q_{u,e}^{w}\leq x_{u,w},v\in V,u\in U,w\in W,\\ \sum\limits_{e\in E_{v}^{out}}q_{u,e}^{w}\leq x_{u,w},v\in V,u\in U,w\in W. \end{array} \right. \tag{16}\]  
It is also guaranteed that the number of input and output degrees can usually be equal for each satellite. Note that a satellite providing content service for one user is viewed as the source, and the access satellite is viewed as the destination. Therefore, the number of output degrees for the destination, as well as the number of input degrees for the source, is viewed as 0. The binary auxiliary variable \(j_{v,v_{dc,a}}=1\) indicates that the satellite \(v\) can beviewed as the access satellite \(v_{dc,a}\) of cloud center \(dc\), otherwise \(j_{v,v_{dc,a}}=0\), and then, the path selection constraint for user \(u\) and content \(w\) can be given as  
\[\left\{ \begin{array}{l}\tau_{v}=z_{u,v}^{w}\cdot(1-a_{u,dc})+j_{v,v_{dc,a}} \cdot a_{u,dc},v\
eq v_{u,a},\\ \tau_{v}=(z_{u,v}^{w}-1)\cdot(1-a_{u,dc})+(j_{v,v_{dc,a}}-1)\cdot a_{u,dc},v_ {s.}=v_{u,a},\end{array} \right.