# Resource Allocation and Offloading Strategy for UAV-Assisted LEO Satellite Edge Computing
## 3 System Model and Problem Description
### Problem Definition  
In terrestrial satellite networks, the management of available computing resources is crucial. One of the critical aspects of resource management is allocating tasks from IoT devices to satellite nodes for processing, where different offloading decisions lead to additional costs, affecting system performance and energy consumption.  
Based on the system model and assumptions discussed above, the primary objective is the minimisation of the balanced totals of system latency and energy consumption by the collaborative optimisation of offloading decisions and resource allocation. The system satisfies the storage requirements, given the available bandwidth and computational resources for all tasks simultaneously, and the problem can be expressed mathematically, as follows.  
\[min\ Z=\zeta T+\eta E \tag{21}\] \[s.t.\ \text{C1}:\sum_{j\in\mathcal{V}}a_{j}\overline{X_{j}}\geq \sum S_{i}(t)\] (22) \[\text{C2}:\sum_{i\in\mathcal{N}}S_{ij}\leq\text{C}_{j}\] (23) \[\text{C3}:N\leq K\] (24) \[\text{C4}:\sum_{i\in\mathcal{N}}f_{ij}\leq f_{j}^{*}\] (25) \[\text{C5}:\sum_{j\in\mathcal{V}}S_{ij}=S_{i}\] (26) \[\text{C6}:\sum_{j\in\mathcal{V}}P_{mj}\leq P_{m}^{*} \tag{27}\]  
where \(\overline{X_{j}}\) denotes the size of the remaining storage resources of \(leq_{j}\), \(f_{ij}\) is the computing resources allocated to \(s_{ij}\) by \(leq_{j}\), \(f_{j}^{*}\) is the satellite's highest CPU frequency \(j\), \(P_{m}^{*}\) is the full uplink transmission power of the UAV, \(C_{j}\) is the total storage space, \(N\) is the number of individual satellite connections, \(K\) is the maximum number of channels, and the size of the task block is allocated by \(S_{ij}\). Where \(\zeta\), \(\eta\in[1,10]\), \(\zeta\) and \(\eta\) are the weights of delay and energy consumption, respectively.  
C1 is that the free storage capacity of the LEO satellite to which the device is connected is not less than the device's task size, \(\text{C2}\) is that the storage space already used by the satellite is not more significant than the total storage space, \(\text{C3}\) is that the number of individual satellite connections does not exceed the maximum number of channels, and \(\text{C4}\) is a constraint on satellite computing resources to ensure that the CPU resources being allocated to IoT device tasks do not overwhelm the total CPU computing resources. \(\text{C5}\) is the sum of the tasks given to different satellites by \(S_{i}\) and the whole task size, and \(\text{C6}\) is the UAV to uplink the transmission power that is not more significant than the maximum UAV power.  
The complexity of the problem is increased by the coupling relation among the optimizing variables. In addition, the proposed optimisation problem is a mixed integer nonlinear problem, though the function and constraints have binary variables. As IoT devices continue to rise, the complexity grows exponentially. To reduce the problem's complexity, decomposing the original problem into sub-problems provides a new solution. It decouples the optimising problem and turns it into two sub-problems: satellite selection, task volume and computational resource allocation.  
Transformation, according to the optimization objective, yields  
\[min\ Z=\zeta T+\eta E=\zeta T_{prop}+\left[\zeta\big{(}T_{tran}+T_{j}\big{)}+ \eta E\right] \tag{28}\]  
The satellite selection needs to satisfy the constraints \(\text{C1}\), \(\text{C2}\) and \(\text{C3}\), and the objective of optimisation is phrased as  
\[min\ Z_{1}=\zeta T_{prop} \tag{29}\]  
The task and computational resource allocation policy must satisfy constraints \(\text{C4}\), \(\text{C5}\) and \(\text{C6}\) with the following optimization objectives.  
\[min\ Z_{2}=\zeta\big{(}T_{Tran}+T_{j}\big{)}+\eta E \tag{30}\]