# Context-Aware Caching Distribution and UAV Deployment: A Game-Theoretic Approach
## 2 System Model and Problem Formulation
### System Model  
We consider a UAV-assisted networks with \(N\) cache-enabled UAVs, i.e., \(\mathcal{N}=\left\{1,2,...,N\right\}\), and \(M\) users, i.e., \(\mathcal{M}=\left\{1,2,...,M\right\}\). Denote the set of system files as \(f=\left\{f_{1},f_{2},...,f_{\left|f\right|}\right\}\), where \(\left|f\right|\) denotes the number of system files. Suppose that all users request files from \(f\) according to a special file popularity distribution, which is as follows [20; 21]:  
\[P(f_{i})=\frac{{{x_{i}}^{\theta}}}{\sum\limits_{j=1}^{\left|f\right|}{{x_{j}} ^{\theta}}}, \tag{1}\]  
where \(P(f_{i})\) denotes the probability that the user requests the \(i\)-th file, and \(x_{i}\) is \(x_{i}=1/f_{i}\). \(\theta\) is the parameter of Zipf distribution.  
Meanwhile, we denote user \(m\)'s requested files as \(V_{m}=\left(v_{1},v_{2},...,v_{\left|V_{m}\right|}\right)\), where \(\left|V_{m}\right|\) is the number of files requested by user \(m\). \(v_{i}=1\) denotes that user \(m\) requestes the \(i\)-th file while \(v_{i}=0\) indicates that user \(m\) does not requeste the \(i\)-th file.  
An illustration of UAV-assisted communication networks is shown in Figure 1. There are three small UAVs, a big central UAV and some users waiting to be served. As depicted in Figure 1, the service process of UAVs consists of four parts: firstly, the small UAVs use some prior information provided by the central UAV and make their own caching contents decisions individually; Secondly, small UAVs fly to the service locations with carried caching contents. Then, UAVs start to serve the users under the assumption that the caching contents can be shared among the reachable neighboring UAVs by one hop. Next, if some users' demands are still not met, the service UAV will fly to some UAVs with caching contents that the dedicated user wants, to cooperate. Meanwhile, the following file requests of corresponding users will be rejected until the service UAV comes back. Finally, the service UAV will return to the central UAV to get files if other UAVs cannot provide such services.  
Motivated by Reference [22], we denote reachable neighboring UAVs of UAV \(n\) by one hop as \(J_{n}\), i.e.,  
\[J_{n}=\left\{i\in N:D(n,i)\leq D_{n}\right\}, \tag{2}\]  
where \(D(n,i)\) denotes the distance between UAV \(n\) and \(i\). \(D_{n}\) represents the communication distance thresholds for UAVs.  
Figure 1: An illustration of the emergency communications network (1). The small unmanned aerial vehicles (UAVs) obtain the caching contents by the given decision from the central UAV and fly to the destination with cached contents. (2). UAVs cooperate with reachable neighboring UAVs by one hop. (3). UAV will fly to some UAVs with caching contents that the dedicated user wants. (4). UAV will return to the central UAV to get files if other UAVs cannot provide such services.).  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Explanation** \\ \hline \(M\) & Number of ground users \\ \(N\) & Number of unmanned aerial vehicles (UAVs) \\ \(f\) & Set of system files \\ \(f_{i}\) & \(i\)-th requested file \\ \(P_{f_{i}}\) & Probability of requesting the \(i\)-th file \\ \(\gamma\) & Parameter of a Zipf Distribution \\ \(r_{m}\) & Overhead of user \(m\) to obtain files \\ \(Q_{m}\) & Set of user \(m\)’s interested files \\ \(a_{n}\) & Action of UAV \(n\) \\ \(l_{n}\) & Location of UAV \(n\) \\ \(c_{n}\) & Caching contents of UAV \(n\) \\ \(D_{n}\) & Communication thresholds between UAVs \\ \(D\left(i,j\right)\) & Physical distance from UAV \(i\) to \(j\) \\ \(l_{n}\) & Neighbor UAVs by one hop \\ \(C_{mk}\) & Overhead that User \(m\) gets file \(k\) \\ \(l_{m}\) & UAV that is serving for User \(m\) \\ \({u^{\prime}}_{k}\) & UAV that caches file \(k\) \\ \(w_{1}\) & Overhead per file per distance from UAV \(u_{m}\) to user \(m\) \\ \(P_{success,m}\) & Successful communication probability from UAV \(n\) to user \(m\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key variables used in this paper.