# Application and Performance Evaluation of Resource Pool Architecture in Satellite Edge Computing
## 3 Proposed Architecture
### Pool Capacity Planning  
Too many satellites participating in the resource pool will result in a waste of resources, while too few will affect the service quality of the task. The participation of satellites over long distances in building resource pools will also bring greater network delay. Thus, in order to meet certain task objectives, it is necessary to have an appropriate number of satellites, which brings the problem of capacity planning of the satellite resource pool. Assume that there are \(k\) satellites participating in the target area service. Each satellite is a server, and the satellite task execution queue is represented by the M/M/1 queuing system, that is, the time interval of task generation obeys the exponential distribution, and the task execution time obeys the arbitrary probability distribution. It is assumed that the rate at which the ground station generates tasks is \(\lambda\), that is, the time interval of task generation is an independent, identically distributed random variable, subject to exponential distribution. The service rate of each satellite is \(\mu\), which represents the number of tasks completed by the satellite in unit time. The task requirements of the target area determine the number of satellites to be involved. Given the target area and task type, we can estimate the number of satellites needed for service. We can then solve the number of satellites based on the square root configuration rule [20]. Considering that the goal is to keep the average response time of the system to less than \(\alpha\), and the minimum number of servers required to achieve such quality of service is \(k_{\alpha}^{*}\), it is obtained from the square root configuration rule:  
\[k_{\alpha}^{*}\approx\rho+\varepsilon\sqrt{\rho} \tag{1}\]  
The \(\rho\) is defined by \(\rho=\frac{\lambda}{\mu}\). The \(\varepsilon\) is given by (2), and \(\Phi(\varepsilon)\) represents the standard normal cumulative distribution function and its probability density function. \[\frac{\varepsilon\Phi(\varepsilon)}{\phi(\varepsilon)}=\frac{1-\alpha}{\alpha} \tag{2}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Definition** \\ \hline \(N\) & the number of satellite nodes \\ \(M\) & the number of satellite resource types \\ \(K\) & the number of satellite tasks \\ \(k_{\alpha}^{*}\) & the minimum number of satellites in the pool \\ \(T_{sat;}\) & the task set of satellite \(i\) \\ \(R_{i}^{k}\) & the \(i\) resource requirement of the task \(k\) \\ \(r_{C_{i}}^{k}\) & the computing requirement of the task \(k\) in satellite \(i\) \\ \(r_{i}^{k}\) & the storage requirement of task \(k\) in satellite \(i\) \\ \(d_{i}^{semi},d_{i}^{receive}\) & the input and output file sizes describing task \(i\) \\ \(T_{i;j}^{no,off}\) & task sending waiting time from satellite \(i\) to \(j\) \\ \(T_{i;j}^{trans}\) & task transmission time from satellite \(i\) to \(j\) \\ \(T_{i;j}^{no,que}\) & task waiting time in the execution queue from satellite \(i\) to \(j\) \\ \(T_{i;j}^{sat}\) & task execution time from satellite \(i\) to \(j\) \\ \(T_{i;j}^{no,back}\) & result return time from satellite \(i\) to \(j\) \\ \(T_{i;j}^{trans}\) & the average transmission energy consumption of the task from satellite \(i\) to \(j\) \\ \(E_{i;j}^{sat}\) & the average execution energy consumption of the task from satellite \(i\) to \(j\) \\ \(C_{i}^{(s)}\) & the number of CPU cycles of satellite \(i\) \\ \(v_{m}\) & The resource utilization of satellite resource \(m\) \\ \(v\) & The average utilization rate of m resources \\ \(d\) & The measurement method of resource load balance \\ \(R_{S_{i}}^{*}\) & * = {cpu,mem,sen}, the * resources of the satellite \(i\), respectively, \\ \(Link_{R_{i}}^{S_{i}}\) & the connection link relationship between the satellite \(i\) and the satellite \(j\) during the time period \(n_{k}\) \\ \(Sync_{n_{k}}^{S_{i}}\) & the relationship between the two satellites \(i\) and \(j\) in the time period \(n_{k}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The notations are used in this paper.