## 2 System Model and Problem Formulation
### Channel Model  
Before problem formulation, we first present detailed channel modeling. As the BIOS is deployed at the BS to achieve efficient SIC, \(\mathbf{H}_{\text{TB}}\), \(\mathbf{H}_{\text{TR}}\), and \(\mathbf{H}_{\text{BR}}\) are considered near-field channels without non-line-of-sight (NLoS) paths. Thus, considering the properties of spherical waves, we model the LoS path gain from the \(n\)-th receive antenna of \(\text{BS}_{\text{Tx}}\) to the \(m\)-th element of \(\text{IOS}_{1}\) as follows [37]:  
\[[\mathbf{H}_{\text{TB}}]_{m,n}=\sqrt{\frac{a^{2}G_{\text{TB}}F_{\text{TB}} \big{(}\theta_{m,n}^{r},\phi_{m,n}^{r}\big{)}}{4\pi d_{m,n}^{2}}}\exp\left( \frac{-\text{j}2\pi d_{m,n}}{\lambda}\right), \tag{4}\]  
where \(a\) is the size of scattering elements, \(\lambda\) is the carrier wavelength, and \(d_{m,n}\) is the distance between the \(n\)-th antenna of \(\text{BS}_{\text{Tx}}\) and the \(m\)-th element of \(\text{IOS}_{1}\). \(\theta_{m,n}^{r}\) and \(\phi_{m,n}^{r}\), respectively, denote the elevation and azimuth angle of the received signal at the \(m\)-th element of \(\text{IOS}_{1}\). Moreover, \(G_{\text{TB}}=2\) and \(F_{\text{TB}}(\theta,\phi)=|\cos\theta|\), respectively, denote parameters related to the equivalent antenna gain and the normalized power radiation pattern between \(\text{BS}_{\text{Tx}}\) and \(\text{IOS}_{1}\).  
Similarly, the channel gain of \(\mathbf{H}_{\text{BR}}\), \(\mathbf{H}_{\text{TR}}\), and \(\mathbf{G}\) can be modeled as follows:  
\[[\mathbf{H}_{\text{BR}}]_{m,n}=\sqrt{\frac{a^{2}G_{\text{BR}}F_{\text{BR}} \big{(}\theta_{m,n}^{r},\phi_{m,n}^{r}\big{)}}{4\pi d_{m,n}^{2}}}\exp\left( \frac{-\text{j}2\pi d_{m,n}}{\lambda}\right), \tag{5}\]  
\[[\mathbf{H}_{\text{TR}}]_{m,n}=\sqrt{\frac{a^{2}}{4\pi d_{m,n}^{2}}}\exp\left( \frac{-\text{j}2\pi d_{m,n}}{\lambda}\right), \tag{6}\]  
\[[\mathbf{G}]_{m,n}=\sqrt{\frac{a^{2}G_{\text{II}}F_{\text{II}}\big{(}\theta_{m,n }^{r},\phi_{m,n}\big{)}}{4\pi d_{m,n}^{2}}}\exp\left(\frac{-\text{j}2\pi d_{m,n }}{\lambda}\right), \tag{7}\]  
where \(G_{\text{BR}}=2\), \(G_{\text{II}}=1\), \(F_{\text{BR}}(\theta,\phi)=|\cos\theta|\), and \(F_{\text{II}}(\theta,\phi)=|\cos\theta|^{2}\).  
Considering the NLoS paths, \(\mathbf{h}_{\text{D},k}\) and \(\mathbf{h}_{\text{U},l}\) can be modeled as Rician channels. Specifically, the channel gain between the \(k\)-th DL UE and \(m\)-th element of \(\text{IOS}_{2}\) can be modeled as follows:  
\[[\mathbf{h}_{\text{D},k}]_{m}=\sqrt{\frac{\kappa}{1+\kappa}}h_{m}^{\text{LoS}}+ \sqrt{\frac{1}{1+\kappa}}h_{m}^{\text{NLoS}}, \tag{8}\]\[h_{m}^{\text{LoS}}=\sqrt{\alpha_{0}\left(\frac{d_{m}}{d_{0}}\right)^{-\beta}G_{ \text{UE}}F_{\text{UE}}(\theta_{m}^{\text{t}},\phi_{m}^{\text{t}})}\exp\left( \frac{-\text{j}2\pi d_{m}}{\lambda}\right), \tag{9}\]  
\[h_{m}^{\text{NLoS}}=\sqrt{\alpha_{0}\left(\frac{d_{m}}{d_{0}}\right)^{-\beta}G_ {\text{UE}}F_{\text{UE}}(\theta_{m}^{\text{t}},\phi_{m}^{\text{t}})}h_{m}^{ \text{s}}, \tag{10}\]  
where \(\kappa\) is the Rician factor. \(h_{m}^{\text{LoS}}\) and \(h_{m}^{\text{NLoS}}\), respectively, denote the LoS and NLoS path gain from the \(m\)-th element of IOS\({}_{2}\) to the \(k\)-th DL UE. \(d_{m}\) denotes the distance between the \(m\)-th element of IOS\({}_{2}\) and the \(k\)-th DL UE. \(\alpha_{0}\) denotes the pass loss of the reference distance \(d_{0}\) with the path loss exponent \(\beta\). \(h_{m_{1},h}^{\text{s}}\sim\mathcal{CN}(0,1)\) denotes the cumulative effect of a large number of scattered paths. Moreover, \(G_{\text{UE}}=2\), \(F_{\text{UE}}(\theta,\phi)=|\cos\theta|\). Similarly, \(\mathbf{h}_{\text{U},J}\) can be modeled in the same way.