# First Request First Service Entanglement Routing Scheme for Quantum Networks
## 3 Modeling for Entanglement Distribution Network
### Instantiation  
Here, we give the specific instantiations of common implemented nodes, including entangled photon source, \(2\times 2\) BS, \(2\times 2\) AOS, \(1\times 2\) DWDM, \(1\times 2\) WSS and user, shown in Figure 2.  
The entangled photon source S can be instantiated as  
\[v_{\text{S}}=\left(\left[\infty\right],\left[-3\right]\right), \tag{15}\]  
where the entangled photon source has 1 output port.  
The \(2\times 2\) BS can be instantiated as  
\[v_{\text{BS}}=\left(\begin{bmatrix}\infty&\infty&3&3\\ \infty&\infty&3&3\\ 3&3&\infty&\infty\\ 3&3&\infty&\infty\end{bmatrix},\left[-1\right]\right), \tag{16}\]  
where the pass-through loss among the BS ports is assumed to 3 dB.  
The \(2\times 2\) AOS can be instantiated as  
\[v_{\text{AOS}}=\left(\begin{bmatrix}\infty&1&1&1\\ 1&\infty&1&1\\ 1&1&\infty&1\\ 1&1&1&\infty\end{bmatrix},\left[0\right]\right), \tag{17}\]  
where the pass-through loss among the AOS ports is assumed to 1 dB.  
The instantiation of the \(1\times 2\) DWDM is shown as  
\[v_{\text{DWDM}}=\left(\begin{bmatrix}\infty&0.5&\infty\\ 0.5&\infty&\infty\\ \infty&\infty&\infty\end{bmatrix},\begin{bmatrix}\infty&\infty&0.5\\ \infty&\infty&\infty\\ 0.5&\infty&\infty\end{bmatrix}\right),\left[33,35\right]\right), \tag{18}\]  
where the pass-through loss among the DWDM ports is assumed to 0.5 dB and 2 wavelength channels are supported by the DWDM.  
The instantiation of the \(1\times 2\) WSS is shown as  
\[v_{\text{WSS}}=\left(W_{\text{WSS}},T_{\text{WSS}}\right), \tag{19}\]  
where  
\[W_{\text{WSS}}=\left[\begin{bmatrix}\infty&5&5\\ 5&\infty&5\\ 5&5&\infty\end{bmatrix},\ldots,\begin{bmatrix}\infty&5&5\\ 5&\infty&5\\ 5&5&\infty\end{bmatrix}\right] \tag{20}\]  
and  
\[T_{\text{WSS}}=\left[32,33,35,36\right]. \tag{21}\]  
Figure 2: Schematic diagram of commonly implemented nodes. (**a**) Entangled photon source. (**b**) \(2\times 2\) beam splitter (BS). (**c**) \(2\times 2\) all-pass optical switch (AOS). (**d**) \(1\times 2\) dense wavelength division multiplexer (DWDM), CH33 and CH35 are supported here. (**e**) \(1\times 2\) wavelength selective switch (WSS), CH32, CH33, CH35 and CH36 are supported here. (**f**) User.  
Here, the pass-through loss among the WSS ports is assumed to 5 dB and 4 wavelength channels are supported by the WSS.  
The user can be instantiated as  
\[v_{\text{User}}=\left(\left[\infty\right],\left[-2\right]\right), \tag{22}\]  
where the user has 1 receive port.  
As shown in Figure 3, we take the links between a \(2\times 2\) BS (\(v_{1}\)) and \(2\times 2\) AOS (\(v_{2}\)) as the examples to explain the modeling of edges.  
The edges between \(v_{1}\) and \(v_{2}\) are instantiated as  
\[e_{4,1}^{1,2} =(1,4,2,1,3), \tag{23}\] \[e_{3,2}^{1,2} =(1,3,2,2,5),\] (24) \[e_{1,4}^{2,1} =(2,1,1,4,3),\] (25) \[e_{2,3}^{2,1} =(2,2,1,3,5). \tag{26}\]  
The link loss matrix for connecting \(v_{1}\) and \(v_{2}\) can be described as  
\[C_{1,2}=\begin{bmatrix}\infty&\infty&\infty&\infty&\infty\\ \infty&\infty&\infty&\infty&\infty\\ \infty&5&\infty&\infty&\infty\\ 3&\infty&\infty&\infty&\infty\end{bmatrix}. \tag{27}\]  
\[C_{2,1}=\begin{bmatrix}\infty&\infty&\infty&3\\ \infty&\infty&5&\infty\\ \infty&\infty&\infty&\infty&\infty\\ \infty&\infty&\infty&\infty&\infty\end{bmatrix}. \tag{28}\]