# Context-Aware Caching Distribution and UAV Deployment: A Game-Theoretic Approach
## 2 System Model and Problem Formulation
### Problem Formulation  
Note that the performance of UAVs is affected by its caching contents as well as the service locations, i.e., where to obtain the files requested by users and the successful communication probability from UAVs to their dedicated users through the influence of propagation groups. Therefore, we need to balance the tradeoff between the transmission overhead and successful communication probability. Consider \(a_{n}=(l_{n},\epsilon_{n})\) as the action of UAV \(n\). Note that \(l_{n}\) reflects the service location of UAV \(n\), which is actually a set of space coordinates, i.e., \((x_{n},y_{n},z_{n})\). Meanwhile, \(c_{n}=(c_{f_{1}},c_{f_{2}},...,c_{f_{|c_{n}|}})\) indicates the caching action of UAV n, where \(c_{f_{i}}\in f,\forall 1\leq i\leq|c_{n}|\) and \(|c_{n}|\) are the number of caching contents carried by UAV \(n\), which is the cache space of UAVs. We define the transmission overhead by considering both caching contents and service location, as follows:  
\[t_{m}(a)=\sum_{i=1}^{|V_{m}|}p(m,i)C(m,k,a), \tag{3}\]  
where \(|V_{m}|\) is the number of files requested by user \(m\), \(p(m,i)\) denotes the probability that user \(m\) requests \(f_{i}\). Denote \(C(m,k,a)\) as the transmission overhead that user \(m\) obtains \(f_{i}\) by the joint action, i.e., \(a=(a_{1},a_{2},...,a_{N})\), of UAVs under the consideration of different sources of files which is given by:  
\[ \begin{split} C(m,k,a)=I_{1}(l_{u_{m}},l_{m})+I_{2}(k,c_{u^{ \prime}_{k}},l_{u_{m}},l_{u^{\prime}_{k}},l_{u^{\prime}_{k}}),\\ l_{u_{m}}\in(l_{1},l_{2},...,l_{N})\,,c_{u_{m}}\in(c_{1},c_{2},...,c_ {N})\,,u_{m}\
eq u^{\prime}_{k},\end{split}  \tag{4}\]  
where \(l_{1}(l_{u_{m}},l_{m})\) and \(l_{2}(k,c_{u^{\prime}_{k}},l_{u_{m}},l_{u^{\prime}_{k}},l_{u^{\prime}_{k}})\) denote the function of the transmission overhead respectively, which is defined as follows:  
\[I_{1}(l_{u_{m}},l_{m})=w_{1}d(u_{m},m), \tag{5}\]  
where \(u_{m}\) and \(m\) represent the UAV and its dedicated user, \(l_{u_{m}}\) and \(l_{m}\) denote the location of UAV \(u_{m}\) and user \(m\) respectively, \(w_{1}\) is the overhead per file per distance, \(d(u_{m},m)\) denotes the distance from \(u_{m}\) to \(m\), which is related to \(l_{u_{m}}\) and \(l_{m}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Variables** & **Explanation** \\ \hline \(r_{th}\) & Communication threshold \\ \(g_{u_{m}m}\) & Channel power gain from UAV \(u_{m}\) to user \(m\) \\ \(h\) & Nakagami fading \\ \(d(l_{u_{m}},l_{m})\) & Distance from UAV \(u_{m}\) to user \(m\) \\ \(c_{u_{m}}\) & Set of files that UAV \(u^{\prime}_{k}\) caches \\ \(l\) & Indicator function \\ \(p_{n}\) & Transmit power of UAV \(n\) \\ \(\alpha\) & Path loss exponent \\ \(N_{0}\) & Background noise \\ \(\eta\) & Additional attenuation factor due to the non-line-of-sight (_NLOS_) link \\ \(h\) & Nakagami fading \\ \(\gamma\) & Shape parameter \\ \(\xi\) & Scale parameter \\ \(\theta\) & Elevation angle \\ \(z_{u_{m}}\) & Height of UAV \(u_{m}\) \\ \(S_{n}\) & Set of ground users that UAV \(n\) serves \\ \(A_{n}\) & Strategy space of player \(n\) \\ \(U_{n}\) & Utility function of player \(n\) \\ \(R\) & Overhead of all ground users \\ \(\Phi\) & Potential function \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._\[I(k,c_{u^{\prime}_{k}},l_{u_{m}},l_{u^{\prime}_{k}},l_{u^{\prime}_{o}})=\left\{ \begin{array}{ccc}0,&if&k\in c_{u_{m}},\\ w_{2}d(l_{u_{m}},l_{u^{\prime}_{k}}),&if&k\
otin c_{u_{m}},\exists\ k\in c_{u^{ \prime}_{k}}\exists\ u^{\prime}_{k}\in J_{n},\\ w_{3}d(l_{u_{m}},l_{u^{\prime}_{k}}),&if&k\
otin c_{u_{m}},\exists\ k\in c_{u^{ \prime}_{k}}\exists\ u^{\prime}_{k}\
otin J_{n},\\ w_{4}d(l_{u_{m}},l_{u^{\prime}_{o}}),&if&k\
otin c_{u_{m}},\exists\ k\
otin \underset{n\
eq m,n\in\mathcal{N}}{\cup}c_{n}.\end{array} \right.