# Resource Mapping Allocation Scheme in 6G Satellite Twin Network
## 3 System Model of 6G Satellite Twin Network  
### A Hierarchical Unified Feature Description Method  
As shown in Figure 2, the communication-twin-computing integration network is modeled as a space, which consists of three layers: one layer is the communication domain, one layer is the twin domain, and one layer is the computing domain. The communication behaviors in the communication domain are carried by the hardware resources to realize the performance indexes. The communication behaviors and resource load of the communication domain are regarded as the tasks of the twin domain to realize the performance indexes. The twin domain is regarded as the task of the computing domain, which is carried by the parallel distributed docker containers. It is difficult to uniformly model heterogeneous systems that are both independently composed within the system and interconnected, interacting, and nested with each other. We hope that a unified network model can be constructed by combining system characteristics and resource characteristics. In this paper, as the performance indexes of the communication domain change over time, the twin domain needs to select different functions for simulation implementation. The computation resources required by the communication domain can change accord  
Figure 2: System model. _There are_ \(p_{jmin}^{G}=min(\hat{\mu}+Ccvs^{-1^{T}Ccvs}(I^{G}-1\hat{\mu}))\)_,_ \(p_{jmax}^{G}=max(\hat{\mu}+Ccvs^{-1^{T}Cvs}(u^{G}-1\hat{\mu}))\) _(see Appendix_ A_), and_ \(p_{k}^{d_{n}}\in\left[p_{jmin}^{G},p_{jmax}^{G}\right]\)_. Then,_ \(d_{n}\in G= \begin{pmatrix}p_{1 min}^{G}\leq p_{1}\leq p_{1 max}^{G}\\ \vdots\\ p_{jmin}^{G}\leq p_{j}\leq p_{jmax}^{G}\end{pmatrix} \)_._  
According to Definition 1, we build a simulation space \(\mathrm{D}=\{\mathrm{G},\mathrm{P},\mathrm{Bound},\alpha\}\). \(\mathrm{G}=\left\{\mathrm{G}^{\mathrm{COM}},\mathrm{G}^{\mathrm{S}},\mathrm{G }^{\mathrm{C}}\right\}\) denotes the communication system, the twin system, and the computing system. \(\mathrm{P}^{\mathrm{G}}=\left\{\mathrm{P}_{1}^{G},\cdots,\mathrm{P}_{J}^{G}\right\}\) denotes the performance index in this space: delay, rate, throughput, delay jitter, bandwidth, packet loss rate, etc. The upper and lower bounds of the performance indexes are \(\mathrm{Bound}^{\mathrm{G}}=\left(\mathrm{P}_{1 min}^{G},\cdots,\mathrm{P}_{Jmax }^{G}\right)\), and the range of space \(\mathrm{D}\) can be expressed as \( \begin{pmatrix}\mathrm{P}_{1 min}^{G}\leq p_{1}\leq p_{1 max}^{G}\\ \vdots\\ \mathrm{P}_{jmin}^{G}\leq p_{j}\leq p_{jmax}^{G}\end{pmatrix} \). There is a set of nodes \(d=\{d_{1},\cdots d_{N}\}\) in the simulation space. According to Definition 2, the communication system can be represented as \(G^{\mathrm{COM}}=\left\{F^{COM},R^{COM},P^{COM}\right\}\). \(F^{COM}=\left\{f_{1}^{COM},\cdots,f_{J}^{COM}\right\}\) denotes the set of functions that make up the communication system. \(R^{COM}=\left\{r_{1}^{COM},\cdots,r_{W}^{COM}\right\}\) denotes the resources required to support the operation of the communication system. \(P^{COM}=\left\{p_{1}^{COM},\cdots,p_{Z}^{COM}\right\}\) denotes the performance indexes of the communication system. According to Theorem 1, when the performance index of the node \(p_{d_{n}}^{COM}\) meets the requirements of the communication system, the node \(d_{n}\) can be divided into the communication domain. The twin system can be represented as \(G^{\mathrm{S}}=\left\{F^{S},R^{\mathrm{S}},P^{\mathrm{S}}\right\}\). \(F^{\mathrm{S}}=\left\{f_{1}^{\mathrm{S}},\cdots,f_{J}^{\mathrm{S}}\right\}\) denotes the set of functions that make up the twin system. \(R^{\mathrm{S}}=\left\{r_{1}^{\mathrm{S}},\cdots,r_{W}^{\mathrm{S}}\right\}\) denotes the resources required to run each set of functions, \(r_{j}^{\mathrm{S}}=\left\{\mathrm{cap}f_{j}^{\mathrm{S}},\mathrm{mem}f_{j}^{ \mathrm{S}},\mathrm{bank}f_{j}^{\mathrm{S}}\right\}\). \(P^{\mathrm{S}}=\left\{p_{1}^{\mathrm{S}},\cdots,p_{Z}^{\mathrm{S}}\right\}\) denotes the performance index of the twin system.