# Towards Digital-Twin Assisted Software-Defined Quantum Satellite Networks
## 3 System Model  
A QSB can ensure significantly better performance than OFs. However, an efficient QSB requires extremely precise control, and a DT working in synergy with an SDN Controller is considered essential to achieving this objective. A DT designed to manage a quantum constellation effectively must include a reliable model of the entire network, the links, and the main parameters that characterize the involved devices. In the case of a LEO satellite network, where the scenario evolves rapidly due to the high velocity of the satellites, a TG can constitute an excellent instrument for appropriately representing the scenario evolution. Moreover, an efficient temporal path selection algorithm operating on a TG can be a solution to enhance the entanglement rate and fidelity on the designated path. Specifically, a TG can be expressed as \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), with \(\mathcal{V}\) and \(\mathcal{E}\), which are, respectively, the sets of vertices and edges that compose \(\mathcal{G}\)[29]. Every single edge \(e\in\mathcal{E}\) consists of the following parameters \((u,v,t_{start},t_{end},\mathcal{W}_{u,v})\). In particular, \(u,v\in\mathcal{V}\) are the two adjacent nodes, and \(t_{start}\) and \(t_{end}\) are the _start time_ and the _end time_, the latter depicted in Figure 2. In the case of satellite networks, a TG can be derived from the Two-line Element Sets (TLEs) [55], a specific data format that encodes a list of satellites that compose a constellation. In this paper, we propose to apply a method for optimizing the flows between the DT and the satellites that compose the E2E paths. The optimization strategy was derived from that reported in [56] and is expressed as follows:  
\[\min_{f_{uv}}\max_{k\in\mathcal{D}}\text{RTT}_{k}= \sum_{(u,v)\in\mathcal{P}_{k}}\frac{d_{u,v}}{c}+\sum_{(v,u)\in \mathcal{P}_{k}}\frac{d_{v,u}}{c},\quad\forall k\in\mathcal{D} \tag{1}\] \[\text{s. t.}\sum_{v\in\overline{\mathcal{N}}(u)}f_{u,v}-\sum_{v \in\overline{\mathcal{N}}(u)}f_{v,u}= \begin{cases}r_{k}-f_{s,k}\approx r_{k}& \text{if }u=k,\;k\in\mathcal{D}\\ \sum_{k\in\mathcal{D}}f_{s,k}-\sum_{k\in\mathcal{D}}r_{k}\approx-\sum_{k\in \mathcal{D}}r_{k}&\text{if }u=s\\ 0&\text{otherwise}\end{cases} \] \[0\leq f_{uv}\leq\mathcal{C}_{u,v},\quad\forall(u,v)\in\mathcal{E}\] \[f_{uv}=0,\quad\forall(u,v)\
otin\mathcal{E}\]  
where the parameters and the related constraints are defined as follows:  
* \(s\): DT node. * \(\mathcal{D}\subseteq\mathcal{V}\): Set of destination nodes. * \(f_{u,v}\): Flow of data on edge \((u,v)\). * \(d_{u,v}\): Distance on edge \((u,v)\). * \(c\): Speed of light. * \(\mathcal{C}_{u,v}\): Capacity of edge \((u,v)\). * \(r_{k}\): Data rate at destination node \(k\in\mathcal{D}\). * \(P_{k}\): Path from the DT to satellite \(k\). * \(\mathcal{N}(u)\): Set of neighbors of node \(u\) in the network. * \(\text{RTT}_{k}\): RTT from the DT to satellite \(k\). Specifically, the optimization problem in Equation (1) minimizes the maximum RTT between the DT and each satellite, aiming to reduce the RTT for the slowest one. In the proposed architecture, the DT node, denoted as node \(s\), functions as a client in a client server model that performs a request and aggregates responses from multiple servers. This characteristic is reflected by considering the incoming flow, that is, the overall flow of response data to be significantly greater than the outgoing flow, i.e., the data flow of the requests. As a result, the expression for \(u\) equal to \(s\) is defined as negative. Conversely, in the case where \(u\) is equal to \(k\), the result is equivalent to \(r_{k}\), as the flow from the DT to node \(k\) is assumed to be negligible. One of the parameters that a DT node must consider is the attenuation between the links that compose the paths.