## 2 Materials and Methods  
### System Model  
(1) VLC Channel  
In the optical line of sight (LOS) path shown in Figure 1, the VLC channel gain for the indoor \(k\)-th user is assumed by (1), where \(\phi_{k}\) is the angle of irradiation of the light-emitting diode (LED) lights, \(\psi_{k}\) is the angle of incidence, \(\Phi_{1/2}\) is the semi-angle of the LED, \(A\) is the received area of a photo diode, \(\gamma\) is the optical-electrical conversion efficiency and \(T_{f}\) is the gain of the optical filter:  
\[h_{k}=\frac{\left(m+1\right)A\gamma}{2\pi d_{k}^{2}}\cdot\cos^{m}\left(\phi_{ k}\right)\cdot T_{f}\cdot g\left(\psi_{k}\right)\cdot\cos\left(\psi_{k}\right)= \frac{\rho\cdot\left(m+1\right)\cdot L^{m+1}}{\left(r_{k}^{2}+L^{2}\right)^{ \frac{m+3}{2}}}. \tag{1}\]  
Figure 1: Visible light communication (VLC) user scenario. The labels FOV, LED, and QoS are field of vision, light emitting diode, and quality of service, respectively. Let us denote the Lambertian emission order as \(m=\frac{\ln 2}{\ln(\cos\phi_{1/2})}\) and the gain of the optical concentrator as \(g\left(\psi_{k}\right)=\frac{n^{2}}{\sin^{2}(\psi_{FOV})}\) with the refractive index of \(n\) and the field of view (FOV) \(\psi_{FOV}\). In addition, \(\rho=\frac{A\cdot\gamma\cdot T_{f}\cdot g\left(\psi_{k}\right)}{2\pi}\) and \(d_{k}^{2}=r_{k}^{2}+L^{2}\). If illumination scenario parameters are given, the VLC channel gain is determined by the distance between the user and the LED. (2) NOMA GRPA  
In current GRPA studies, both the fairness and efficiency of power resources are considered to follow \(h_{1}<h_{2}<\cdots<h_{k}<\cdots<h_{K}\) with \(a_{1}>a_{2}>\cdots>a_{k}>\cdots>a_{K}\), where \(a_{k}\) is a power allocation factor. If the total signal power for communication denotes \(P_{t}\), the sub-channel power \(\Delta P_{t}^{user_{k}}\) for each user is allocated by (2) and all of \(a_{k}\) are normalized. \[\Delta P_{t}^{user_{k}}=P_{t}\cdot a_{k},\,k<K \tag{2}\] \[SINR_{k}=\frac{\left(h_{k}\cdot a_{k}\cdot P_{t}\right)^{2}}{ \sum\limits_{i=k+1}^{K}\left(h_{k}\cdot a_{i}\cdot P_{t}\right)^{2}+N_{0} \cdot B},\,k<K\] (3) \[C_{k}\left(h_{k},a_{k}\right)=B\cdot\log_{2}\left(1+SINR_{k} \right),\,k<K \tag{4}\]  
Using on-off keying (OOK), the signal to interference plus noise ratio (SINR) and theoretical data rate for the \(k\)-th user can be determined by (3) and (4). If the positions of users are given, the traditional GRPA introduces the static interference components into each user, which means that each user is prioritised by the pre-defined order. Users with poor channel gains are given higher priority with high signal transmitting power, causing the static resource management of QoS requirements by sorted channel gains. \[f_{h_{k}}\left(h\right) = \left|\frac{\partial r^{-1}\left(h\right)}{\partial h}\right| \cdot f_{h}\left(r^{-1}\left(h\right)\right)\overset{\Delta}{=}F_{1}\left(h \right)\cdot F_{2}\left(h\right) \tag{5}\] \[F_{1}\left(h\right) = \frac{\partial r^{-1}\left(h_{k}^{2}\right)}{\partial h_{k}^{2} }=\frac{1}{\left(\frac{\partial h\left(r_{k}\right)}{\partial r_{k}}\right)}\] (6) \[= \frac{1}{\left(\rho\cdot\left(m+1\right)\cdot L^{m+1}\right)^{2} \cdot\left(-\left(m+3\right)\right)\cdot\left(r_{k}^{2}+L^{2}\right)^{-\left( m+3\right)-1}\cdot 2r_{k}}\] \[= \frac{1}{\left(\rho\cdot\left(m+1\right)\cdot L^{m+1}\right)^{2} \cdot\left(-\left(m+3\right)\right)\cdot\left(\left(\frac{\rho\cdot\left(m+1 \right)\cdot L^{m+1}}{h_{k}}\right)^{\frac{2}{m+3}}\right)^{-\left(m+3\right)- 1}\cdot 2r_{k}}\] \[= -\frac{1}{m+3}\cdot\left(\rho\cdot\left(m+1\right)\cdot L^{m+1} \right)^{\frac{2}{m+3}}\cdot\left(h_{k}^{2}\right)^{-\frac{1}{m+3}-1}\cdot \frac{1}{2r_{k}}\] Figure 1:0 (7) Figure 1:1 Figure 1:2  
(3) NOMA-VLC User Outage Probability  
In NOMA-VLC networks, the user outage probability should be considered because of the vulnerability of optical channels [22; 23]. According to the conclusion of (1), we can regard it as and denote \(h_{k}\) as the function of \(h_{k}\left(r_{k}\right)\).