# Offloading of Atomic Tasks in Satellite Networks:
## 4 Method
### Algorithm Design  
#### 4.2.1 Maximum Task Flow Transmission  
Satellites change with time and space, so the visible time window between satellites is limited, which leads to precious link resources between satellites. If multiple satellites need to offload unquantized missions to high-orbiting satellites, the time window limitation can make it difficult. Therefore, we would need to calculate the maximum amount of data to be transferred on a network consisting of satellites that currently need to offload atomic  
Figure 3: Mission offloading of low-medium orbit satellites and re-offloading process of high-orbit satellites. missions. We transform part of the problem into the problem of calculating the maximum flow in a dynamic network, which can be expressed as follows,  
\[\max \sum_{(u,v)\in E}f(u,v) \tag{8}\] \[s.t.\] \[C1: \sum_{(u,v)\in E}f(u,v)-f(v,u)=0,\ (u\
eq s,t),\] \[C2: \sum_{(u,v)\in E}f(i,v)-f(v,j)=v(f),\ (i=s),\] \[C3: \sum_{(u,v)\in E}f(u,v)-f(v,u)=-v(f),\ (u=t),\]  
As shown in the Formula (8) \(f\) denotes a feasible flow of a link in a satellite network, \(u\) and \(v\) denote the satellite nodes in the network which can establish a time window, \(s\) denotes the only source node with indegree zero, and \(t\) denotes the only sink node with outdegree zero. The Formula (8), which solves a feasible flow in the network to maximize the flow \(v(f)\) on the network, where the constraint \(C1\) represents the conservation of flow, and the so-called conservation of flow refers to the intermediate node no flow is stored. When initially dividing the number of satellite tasks to unload, we can assume that the node does not store the flow in solving the maximum flow, and the task is unloaded after the maximum flow is solved. \(C2\) and \(C3\) indicate the need to meet traffic demand. In addition, the bandwidth constraints of the link need to be satisfied. As shown in Figure 4, it describes the process of calculating the maximum flow on a given instantaneous satellite network. The result of the maximum flow calculation can be used to determine the maximum number of task offloads that can be transmitted on a satellite. As shown in Subfigure (a) represents a small satellite network consisting of low-medium orbiting satellites at \(t\) time. There is a visible time window between the satellites, and each satellite has an atom that needs to be offloaded. During the task offloading process, sources, and sinks are not unique in the network. Subfigure (b) represents the computational process of the maximum flow. First, we create two virtual nodes as unique sources and sinks in the satellite network. We can use the single source and single sink maximum flow calculation method to solve the satellite network. The maximum traffic that can be transmitted on the subgraph Subfigure (c) is the number of missions that can be offloaded from each satellite based on the result of the maximum traffic solution. The missions to be transmitted are then transferred to the specific satellite and the offloaded mission data is transmitted by the satellite to the high-orbiting satellite. Figure 4: Maximum flow calculation method for task offloading quantity allocation of dynamic satellite network. The solving process of the maximum tasks offloaded on each satellite in the satellite network composed of low-medium orbit satellites is shown in the Algorithm 1. It mainly calculates the maximum traffic path of the network composed of all low-medium orbit satellites that receive the atomic task request queue and the maximum allowable transmission traffic. Considering the satellite network has time-varying characteristics, we calculate the task offloading plan in satellite networks over time. First, given a fixed period and a graph composed of satellites and communication links \(G^{{}^{\prime}}\) (lines 1-2).