# Application and Performance Evaluation of Resource Pool Architecture in Satellite Edge Computing
## 3 Proposed Architecture
### Satellite Node Selection
#### 3.3.1 Average Task Response Time  
When satellite \(i\) migrates some tasks to satellite \(j\) for execution, the following three steps need to be completed in the whole process: sending the task to satellite \(j\), satellite \(j\) executes the task, and then returns the task result to satellite \(i\). Under the resource pool, the response time of task \(T_{i,j}\) is composed of five parts: task sending waiting time \(T_{i,j}^{w,off}\), task transmission time \(T_{i,j}^{trans}\), task waiting time in the execution queue \(T_{i,j}^{w,que}\), task execution time \(T_{i,j}^{sat}\), and result return time \(T_{i,j}^{w,back}\). Some tasks may not return results. In this case, the time and energy consumption of returning results are zero. Local processing does not require data transmission. Among them, the satellite transmits data through the inter-satellite link, and the task transmission of the inter-satellite link includes link establishment delay and transmission delay. The chain establishment delay can be divided into tracking  
Figure 3: Process of satellite resource pool service regional tasks.  
delay, synchronization delay, and handshake delay. Taking the common microwave intersatellite link as an example, the two sides of satellite communication need to communicate. Firstly, the link must be built under conditions which ensure the visibility of the satellite. As shown in Figure 4 below, the visible condition of the satellite means that the connection between the two satellites does not pass through the earth, that is \(\Delta h\succ 0\).  
\[\varphi=\arccos(\frac{(r_{1}\bullet r_{2})}{|r_{1}|\bullet|r_{2}|}) \tag{3}\]  
\[\Delta h=h-R_{e}=\frac{|r_{1}|\bullet|r_{2}|\bullet sin\varphi}{|r_{2}-r_{1}|} \tag{4}\]  
In the process of establishing the link between two satellites, the antenna must be aligned first. However, since the satellite ephemeris is known, future satellite links can be predicted, so the tracking delay can generally be regarded as a stable value. The synchronization delay is related to the coding mode of the inter-satellite link, generally QPSK, which can also be regarded as a stable value. The handshake delay is mainly caused by propagation. Assuming that the distance between two satellites is \(|r_{2}-r_{1}|\) and the handshake is \(m\) times during chain establishment, \(c\) is the speed of light, and the handshake delay is:  
\[h_{\rm hand}=m\frac{|r_{2}-r_{1}|}{c} \tag{5}\]  
Assuming that the inter-satellite link transmission rate is \(R_{i,j}^{send}\), the expression of \(T_{i,j}^{trans}\) is obtained:  
\[T_{i,j}^{trans}=\frac{d_{j}^{send}}{R_{i,j}}+t_{hand} \tag{6}\]  
Expression for \(T_{i,j}^{sat}\) is:  
\[T_{i,j}^{sat}=\frac{rc_{j}}{C_{j}^{(s)}} \tag{7}\]  
During the visible window time between the satellite and the ground, the satellite cluster network topology constituting the resource pool can be regarded as unchanged [24]. At this time, the task sending waiting time and result return waiting time are only related to the message sending queue, which will be counted by the task queue in subsequent experiments. To sum up, the response time \(T_{i,j}\) of the task migrating to satellite \(j\) is:  
\[T_{i,j}=T_{i,j}^{w,off}+T_{i,j}^{trans}+T_{i,j}^{w,que}+T_{i,j}^{ sat}+T_{i,j}^{w,back}\] \[=T_{i,j}^{w,off}+T_{i,j}^{w,que}+\frac{d_{j}^{rand}}{R_{i,j}^{wind} }+\pi^{hop}m\frac{|r_{2}-r_{1}|}{c}+\frac{rc_{j}}{c_{i}^{(s)}}+\frac{d_{j}^{ receive}}{R_{i,j}^{rece}} \tag{8}\]  
Figure 4: Satellite visibility relationship.  
In particular, when the task is processed locally, data transmission is not required, so the response time of the task \(T_{i,i}\) only consists of two parts: \(T_{i,i}^{un\_que}\), the waiting time of the task in the execution queue, and \(T_{i,i}^{local}\), the execution time of the task. Furthermore,\(T_{i,i}^{wait}\) equals to the \(T_{i,i}^{un\_que}\) for the local task processing. To sum up, the average response time of tasks performed locally on the satellite is:  
\[T_{i,i}=T_{i,i}^{un\_que}+T_{i,i}^{local}=T_{i,i}^{wait}+\frac{rc_{i}}{C_{i}^{ (s)}} \tag{9}\]