## 3 Problem Formulation
### Mobility Model  
In the multi-UAV Mobile Edge Computing (MEC) system, we consider a scenario comprising \(N\) mobile devices (MDs) and \(M\) UAVs equipped with edge servers. Each MD has generated a set of computational tasks that exceed its local processing capabilities. To address this, MDs offload a portion of their tasks to UAV edge servers based on a predefined offloading ratio. Additionally, UAVs optimize their positions by selecting appropriate MDs and adjusting their locations to minimize the distance to MDs, thereby leveraging line-of-sight (LoS) channels to achieve enhanced channel gain and transmission rates.  
The edge computing system, consisting of multiple MDs and UAVs, employs Time Division Multiple Access (TDMA) for communication between UAVs and MDs, while Frequency Division Multiple Access (FDMA) is used for communication among UAVs. The service period \(T\) is divided into \(n\) time slots, each with a duration of \(\tau\), during which only one MD occupies the channel to communicate with a UAV. At time slot \(t\), the MD in the system is denoted as \(l\in\{1,2,\ldots,N\}\), with its coordinates represented as \(z_{l}^{md}(t)=[x_{l}^{md}(t),y_{l}^{md}(t),0]^{\top}\in R^{3\times 1}\). Similarly, the UAV is denoted as \(u\in\{1,2,\ldots,M\}\), with its coordinates represented as \(z_{u}(t)=[x_{u}(t),y_{u}(t),H]^{\top}\in R^{3\times 1}\), where \(H\) is the UAV's altitude.  
Given the stochastic mobility pattern where mobile devices (MDs) can randomly transition between spatial points within a given time interval, we employ the Gauss-Markov random mobility model to characterize this movement behavior. Specifically, the position of an MD at time slot \(t+1\) is updated according to  
\[x_{l}^{md}(t+1) =x_{l}^{md}(t)+v_{l}^{md}(t)cos(\theta_{l}^{md}(t))\tau \tag{1}\] \[y_{l}^{md}(t+1) =y_{l}^{md}(t)+v_{l}^{md}(t)sin(\theta_{u}(t))\tau\] (2) \[z_{l}^{md}(t+1) =[x_{l}^{md}(t+1),y_{l}^{md}(t+1),H]^{\top} \tag{3}\]  
In Equations (2) and (3), \(v_{l}^{md}(t)\) and \(\theta_{l}^{md}(t)\) represent the moving speed and direction of the mobile device UAV \(l\) in the time slot \(t\), which can be expressed as \[v_{l}^{md}(t)=c_{1}v_{l}^{md}(t-1)+c_{2}\bar{v}+\sqrt{1-c_{1}^{2}\phi_{l}} \tag{4}\]  
\[\theta_{l}^{md}(t)=d_{1}\theta_{l}^{md}(t-1)+d_{2}\bar{\theta}+\sqrt{1-d_{1}^{2} \psi_{l}} \tag{5}\]  
where \(\bar{v}\) and \(\bar{\theta}\) represent the moving speed and direction of the mobile device UAV \(l\) in the time slot \(t\), and parameters \(c_{1}\), \(c_{2}\), \(d_{1}\), and \(d_{2}\) are used for evaluating the movement consistency of mobile devices UAV \(l\) within continuous time intervals. Meanwhile, \(\phi_{l}\) and \(\psi_{l}\) are two random variables that follow an independent Gaussian distribution, i.e., \(\phi_{l}\sim(\xi_{v_{l}},\xi_{v_{l}}^{2})\) and \(\psi_{l}\sim(\xi_{\theta_{l}},\xi_{\theta_{l}}^{2})\). These two distributions represent the randomness of the movement of the two devices.  
For the multi-UAV MEC system, UAV \(u\) initiates its movement at the beginning of time slot \(t\) and updates its position by the start of time slot \(t+1\) as follows:  
\[x_{u}(t+1)=x_{u}(t)+v_{u}(t)cos(\theta_{u}(t))\tau_{fly} \tag{6}\]  
\[y_{u}(t+1)=y_{u}(t)+v_{u}(t)sin(\theta_{u}(t))\tau_{fly} \tag{7}\]  
\[z_{u}(t+1)=[x_{u}(t+1),y_{u}(t+1),H]^{\top} \tag{8}\]  
In Equations (6) and (7), \(v_{u}(t)\) and \(\theta_{u}(t)\) represent the velocity and direction of UAV \(u\) at time slot \(t\), respectively, and \(\tau_{fly}\) denotes the flight duration of the UAV.  
This model ensures efficient task offloading and resource allocation by dynamically optimizing UAV trajectories and communication protocols, thereby enhancing the overall performance of the MEC system.