# Context-Aware Caching Distribution and UAV Deployment: A Game-Theoretic Approach
## 2 System Model and Problem Formulation
### Problem Formulation
\tag{6}\]  
where \(u^{\prime}_{k}\) and \(u^{\prime\prime}_{o}\) denote the UAV that caches file \(k\) and the central UAV, respectively, \(w_{2}\), \(w_{3}\), and \(w_{4}\) are different overheads per file per distance according to different source of files, \(u^{\prime}_{k}\) is the UAV that caches file \(k\), \(c_{u^{\prime}_{k}}\) means the set of files that UAV \(u^{\prime}_{k}\) caches. The explanation of \(I(k,c_{u^{\prime}_{k}},l_{u_{m}},l_{u^{\prime}_{k}},l_{u^{\prime\prime}_{o}})\) can be classified into four parts. The first expression indicates that UAV \(u_{m}\) caches the file that the users require; the second expression indicates that UAV \(u_{m}\) tends to obtain files from its neighbors; the third expression denotes the fact that UAV \(u_{m}\) tends to obtain files from other small UAVs, which are defined as potential neighbors; the truth that UAV \(u_{m}\) will fly back to the big central UAV to cache missing files is reflected in the last expression. Moreover, it should be noted that the value of each part in indicator function is related to the distance between UAV \(u_{m}\) and the destination for the file requested by users, i.e., \(d(l_{u_{m}},l_{u^{\prime}_{k}})\) and \(d(l_{u_{m}},l_{u^{\prime\prime}_{o}})\). Note that the third and last expression indicate that UAV \(u_{m}\) will leave the service location. Therefore, \(w_{3}\) and \(w_{4}\) involve the flight overhead per distance. Although the UAV can obtain the files requested by users, the successful communication probability is still an important factor which may influence the service performance. Suppose that each communication link uses a pre-given channel to transmit data. It can be completed by applying unicast transmission schemes [23] where each UAV serves corresponding users by different channels. Therefore, it means that the communications between UAVs and users will not interfere with each other. Then, for an interference-free scenario, we define the successful communication probability as the probability that the received signal-to-noise ratio (\(SNR\)) by the received user is over a predefined communication threshold, which is as follows:  
\[P_{success}=P\left(SNR>r_{th}\right), \tag{7}\]  
where \(r_{th}\) is the communication threshold, (\(SNR\)) is calculated by:  
\[SNR=\frac{P_{n}g_{u_{m}m}}{N_{0}}, \tag{8}\]  
where \(P_{n}\) is the transmitting power of UAV \(n\), \(g_{u_{m}m}\) is the channel power gain from UAV \(u_{m}\) to user \(m\), \(N_{0}\) is the background noise. Different from terrestrial communication network, the transmission link can be classified into two propagation groups, i.e., _LOS_ and _NLOS_. Motivated by Reference [24], \(g_{u_{m}m}\) for _LOS_ and _NLOS_ links are respectively calculated by:  
\[g_{u_{m}m}=\left\{ \begin{array}{ll}h\cdot d_{u_{m}m}^{-\alpha},&LOS\ link\\ h\cdot\eta\cdot d_{u_{m}m}^{-\alpha},&NLOS\ link.\end{array} \right. \tag{9}\]  
where \(d_{u_{m}m}\) reflects the distance from UAV \(u_{m}\) to user \(m\), \(\alpha\) denotes the path loss exponent, \(\eta\) denotes an additional attenuation factor caused by the _NLOS_ link, \(h\), i.e., \(h\sim\Gamma\left(\gamma,\xi\right)\), represents the Nakagami fading. According to Reference [24], both the service location and the elevation angle \(\theta\) between the UAV and the user influence the probability of _LOS_ link. Thus, the probability of causing _LOS_ links can be given as follows:  
\[P\left(LOS\right)=\frac{1}{1+a\exp\left(-b(\theta-a)\right)}, \tag{10}\]where \(a\) and \(b\) are the parameters related to the environment, \(\theta\) denotes the elevation angle specified as follows:  
\[\theta=\frac{180}{\pi}\times\sin^{-1}(\frac{z_{u_{m}}}{d_{u_{m}m}}), \tag{11}\]  
where \(z_{u_{m}}\) is the height of UAV \(u_{m}\). The probability of causing _NLOS_ link is \(P(NLOS)=1-P\left(LOS\right)\).