Symmetrical User Fairness in Asymmetric Indoor Channels: A Max–Min Framework for Joint Discrete RIS Partitioning and Power Allocation in NOMA Systems

Abstract

Reconfigurable intelligent surface (RIS)-assisted non-orthogonal multiple access (NOMA) has emerged as a promising technique to enhance spectral efficiency and coverage in fifth- and sixth-generation wireless networks. However, asymmetric indoor propagation conditions characterized by heterogeneous line-of-sight (LoS) and non-line-of-sight (NLoS) links often degrade user fairness. This paper investigates a downlink RIS-assisted NOMA system under the standardized 3GPP indoor office (InH) channel model to address fairness-oriented design under realistic link-budget constraints. We formulate an optimization problem for max–min fairness that jointly considers discrete RIS element partitioning and NOMA power allocation to achieve a symmetrical allocation of quality of service (QoS). To enable efficient computation, the non-convex problem is transformed into an epigraph form and solved using a low-complexity, bisection-based quasi-convex optimization framework combined with enumeration over RIS partitions. Numerical results demonstrate significant fairness gains; for instance, doubling the RIS array size yields a substantial improvement in the ergodic max–min rate, corresponding to approximately a 66% gain at moderate transmit power levels. Furthermore, by accounting for practical impairments such as imperfect successive interference cancellation (iSIC), imperfect channel state information (iCSI), and RIS implementation losses, the results reveal that fairness-optimal operation consistently prioritizes the far user to overcome severe indoor NLoS attenuation. The proposed framework is also compared with alternating optimization (AO)-based RIS-NOMA, conventional RIS beamforming without partition and RIS-assisted orthogonal multiple access (OMA) schemes. Simulation results confirm that the proposed framework achieves low computational complexity, making it suitable for practical indoor wireless environments.

1. Introduction

Non-orthogonal multiple access (NOMA) has emerged as a promising solution to the rapid growth of higher-data-rate, latency-sensitive applications in fifth-generation (5G) and emerging sixth-generation (6G) networks [1,2]. This multiple-access technique enhances the spectral efficiency of multiple users sharing the same time-frequency resource by employing power-domain multiplexing and successive interference cancellation (SIC). However, the performance of NOMA systems is strongly dependent on channel discrepancies among users, and in indoor environments, severe path loss, blockage, and line-of-sight (LoS) propagation can significantly degrade user fairness, particularly for cell-edge or far users (FUs). Reconfigurable intelligent surfaces (RISs) have recently attracted considerable attention as a cost-effective, energy-efficient solution for improving wireless propagation conditions [3,4,5,6]. By intelligently adjusting the phase shifts in the passive reflecting elements, RIS can reconfigure the wireless environment to enhance signal coverage, improve link reliability, and mitigate the effects of blockage. The integration of RIS with NOMA has emerged as a powerful paradigm to simultaneously improve spectral efficiency and coverage, especially in challenging indoor hotspot scenarios. Despite the promising benefits of RIS-assisted NOMA systems, key practical challenges in resource management and optimization remain unexplored. Since RIS elements are fixed in real-world deployments, it is necessary to partition them among users to maintain user fairness. Apart from RIS element allocation, power allocation in next-generation base stations (gNBs)/access points (APs) also plays an important role in maintaining user fairness. Moreover, in the RIS-assisted NOMA system, the joint design of power allocation and RIS configuration is critical to achieve user fairness, particularly in a heterogeneous propagation environment comprising LoS and non-line-of-sight (NLoS) links. Motivated by these challenges, this paper investigates the indoor RIS-assisted NOMA system under the 3GPP indoor office (InH) channel model.

1.1. Research Gaps

Although extensive research has been conducted on RIS-assisted NOMA systems, several important gaps remain:

• Existing works on RIS partitioning predominantly focus on sum-rate maximization, employ learning-based frameworks with high computational and training overhead, or consider uplink, aerial, or mmWave scenarios.
• Most studies assume idealized channel models and overlook realistic indoor propagation characteristics, including mixed LoS/NLoS links, wall-penetration losses, and physical link-budget constraints. In particular, fairness-oriented optimization for downlink indoor RIS-assisted NOMA systems, under standardized 3GPP InH channel models, has not been adequately addressed. The joint impact of RIS element partitioning and power allocation on max–min user fairness in such environments remains an open research problem.
• Most traditional studies are tested with only two user scenarios and do not account for real-world impairments such as imperfect SIC (iSIC), imperfect channel state information (iCSI) and RIS implementation loss.

1.2. Major Contributions

To address these gaps, this paper offers the following principal contributions:

• Indoor RIS-assisted NOMA system modeling under realistic propagation conditions: We develop a downlink RIS-assisted NOMA framework tailored for indoor hotspot environments by incorporating the 3GPP InH channel model, which accounts for distance-dependent path loss, shadowing, and wall penetration losses. This allows the system to capture the asymmetric propagation conditions typically encountered in indoor wireless deployments.
• Fairness-oriented joint RIS partitioning and power allocation: Unlike most existing RIS-NOMA studies that primarily focus on sum-rate maximization, this work formulates a max–min fairness optimization problem to achieve balanced quality-of-service (QoS) between near users (NUs) and FUs. The proposed approach jointly optimizes discrete RIS element partitioning and NOMA power allocation, enabling symmetric service performance even under heterogeneous channel conditions.
• Low-complexity quasi-convex optimization framework: The original non-convex fairness optimization problem is transformed into an epigraph formulation, which allows the problem to be solved using a bisection-based quasi-convex optimization algorithm combined with RIS partition enumeration. The proposed framework significantly reduces computational complexity while maintaining efficient resource allocation.
• Practical impairment analysis: The framework is extended to analyze the impact of iSIC, iCSI, and RIS implementation losses, providing insights into how residual interference and hardware-induced attenuation create critical performance floors.
• Performance comparison with conventional RIS-assisted schemes: To validate the effectiveness of the proposed framework, the proposed bisection-based quasi-convex optimization method is compared with several benchmark schemes, including alternating optimization (AO)-based RIS–NOMA, RIS beamforming without partitioning, and RIS-assisted orthogonal multiple access (OMA). Simulation results demonstrate that the proposed RIS-partitioned framework achieves superior ergodic max–min fairness compared to conventional RIS beamforming and OMA baselines, while achieving performance comparable to AO-based optimization methods with substantially lower computational complexity.
• System generalization and scalability: To enhance modeling realism, the study incorporates direct AP-user equipment (UE) links, multiple-input multiple-output (MIMO)-integrated configurations, and multi-user support through hybrid NOMA, demonstrating that the proposed bisection-based logic serves as a scalable baseline for distributed multi-RIS architectures.

1.3. Organization

The rest of the paper is organized as follows: Section 2 describes the existing works on RIS partitioning and NOMA power allocation. Section 3 describes the system model, which incorporates the 3GPP InH channel model and practical link budget constraints. The proposed user fairness optimization problem is discussed in Section 4, and Section 5 validates the proposed framework under realistic indoor conditions. Section 6 concludes the work and discusses future research directions.

2. Related Works

Early work on RIS partitioning for NOMA was presented in [7], where the RIS was divided into multiple sub-surfaces, each dedicated to a specific user, to mitigate inter-user interference and improve fairness. The resulting fairness-oriented resource allocation problem was formulated as a non-convex integer program, and low-complexity algorithms were proposed to obtain near-optimal solutions. A dynamic RIS partitioning scheme that jointly addressed RIS identification and beamforming by allocating RIS elements according to heterogeneous performance objectives [8]. Their approach efficiently balanced identification reliability and beamforming gain, resulting in a significant signal-to-noise ratio (SNR) improvement while maintaining accurate RIS identification. Analytical results, supported by simulations, validated the effectiveness of the proposed partitioning strategy. As RIS-assisted systems become increasingly complex, several learning-based approaches have been explored. In [9], RIS partitioning for millimeter-wave (mmWave) MIMO–NOMA systems was investigated using a multi-agent deep reinforcement learning framework to overcome the high channel state information (CSI) acquisition overhead. By jointly optimizing RIS partitioning, phase shifts, beam selection, and power allocation without requiring user CSI, near-optimal sum-rate performance was achieved.

A partition-based RIS-assisted multi-user multiple-input single-output (MISO) symbiotic radio system was proposed to simultaneously support primary communications and Internet of Things (IoT) transmissions by dividing the RIS into dedicated subsurfaces [10]. The authors formulated a joint beamforming and RIS phase-shift optimization problem under rate constraints and developed AO and difference-of-convex methods to address the resulting nonconvexity. Their results demonstrated that RIS partitioning effectively mitigated inter-user interference and significantly outperformed conventional RIS designs that employed all elements for secondary transmissions. Similarly, an artificial intelligence (AI)-driven resource allocation framework for RIS-assisted NOMA in the IoT networks was proposed to jointly optimize power allocation, RIS phase shifts, and energy efficiency [11]. By combining AO, fractional programming, and learning-based approaches using deep learning and reinforcement learning, the authors addressed the nonconvexity and dynamic nature of RIS–NOMA optimization. Numerical results demonstrated notable improvements in system sum-rate and energy efficiency, particularly in large-scale and data-intensive IoT scenarios. A partitioned-RIS-assisted NOMA framework was investigated to maximize the system sum-rate by jointly optimizing RIS phase shifts and user-subsurface matching [12]. The authors formulated a non-convex sum-rate maximization problem and proposed an AO approach combined with a channel-strength-based hierarchical user-pairing strategy. Simulation results demonstrated notable sum-rate gains over conventional RIS-assisted and NOMA benchmark schemes.

A machine learning (ML)-enabled RIS-assisted mmWave NOMA system was investigated, in which the RIS is virtually partitioned into multiple sub-surfaces to serve different user clusters [13]. The authors proposed a three-stage ML framework combining K-means and Gaussian mixture models for user clustering, deep neural networks for RIS partitioning, and deep reinforcement learning for joint beamforming and power allocation. Simulation results demonstrated that the proposed ML-based approach significantly outperformed conventional AO schemes in terms of system sum-rate. An efficient RIS–partition–based multi-beam NOMA transmission scheme was proposed for RIS-assisted mmWave downlink systems [14]. By developing an asymptotic analytical framework for the effective channel gain, the authors derived closed-form solutions for RIS partitioning and decoupled the joint optimization of beamforming, decoding order, and power control. Compared with existing approaches, the proposed method achieved lower computational complexity while maintaining strong performance gains.

Antenna partitioning was investigated in a downlink vehicular NOMA system to enhance the system sum-rate by optimally forming subarrays and associating each subarray with a dedicated NOMA stream [15]. By jointly optimizing antenna partitioning and power-domain multiplexing, the proposed approach achieved higher sum-rate performance while significantly reducing computational complexity compared to conventional schemes. Optimal virtual partitioning of RIS was investigated for uplink grant-free NOMA networks to enhance channel gain disparity among users and eliminate the need for uplink power control [16]. Closed-form solutions were derived under multiple operational regimes, including sufficient QoS, efficient RIS utilization, and max–min fairness regimes, subject to uplink QoS constraints. Simulation results confirmed that RIS partitioning significantly improved system performance while reducing signaling overhead and computational complexity. Aerial RIS partitioning and unmanned aerial vehicle (UAV) deployment were jointly optimized for uplink NOMA-based IoT networks under practical transceiver hardware impairments and iSIC [17]. The authors formulated a max–min rate optimization problem accounting for residual hardware distortions and proposed an efficient solution to enhance user fairness. Simulation results demonstrated that a UAV-mounted, partitioned RIS significantly improved the max–min achievable rate compared to conventional benchmark schemes.

The outage and sum-rate performance in a multi-user environment were enhanced through an intelligent omni-surface (IOS)-enabled hybrid NOMA system [18]. This study examined three user-pairing strategies: near-near and far-far (NN-FF) pairing, odd-even pairing, and near-far (N-F) pairing to support multiple users. Additionally, a low-complexity nonconvex closed-form power-fraction optimization was discussed to maximize the sum-rate. The impact of iSIC on system performance was also analyzed. In [19], the authors studied an optimization framework for a RIS-assisted amplify-and-forward relay network aimed at maximizing the system’s energy efficiency. The problem was solved using an AO approach that combined successive convex approximation (SCA), the Dinkelbach transformation, semidefinite relaxation (SDR), and Karush–Kuhn–Tucker (KKT) conditions to jointly optimize beamforming, RIS phase shifts, and transmit power. In [20], the authors investigated fairness optimization in simultaneously transmitting and reflecting (STAR)-RIS and NOMA-assisted integrated sensing and communication (ISAC) systems. The fairness between communication users and sensing targets was maximized by jointly optimizing the base station transmit beamforming and the STAR-RIS coefficient matrices using a SCA and semidefinite programming (SDP)-based algorithm. In contrast to this work, which focuses on STAR-RIS-assisted ISAC with joint beamforming design, the present study considers RIS-assisted NOMA systems and develops a low-complexity fairness optimization framework based on RIS element partitioning and power allocation under realistic indoor channel conditions. Although several studies have investigated RIS-assisted NOMA systems, most existing works focus on sum-rate maximization or learning-based resource allocation and rarely address fairness-oriented optimization under realistic indoor propagation conditions.

3. System Model

Consider an indoor two-user downlink RIS-assisted NOMA wireless communication system shown in Figure 1. In the proposed system, the direct links from the indoor hotspot AP to NU and FU UEs are assumed to be blocked or weak, hence ignored. A passive RIS with M reflecting elements is deployed to enhance the coverage and blockage effects in indoor environments. Reflecting elements in the RIS produces a controllable phase shift with unit-modulus reflection. The indoor hotspot AP and NOMA users are all equipped with a single antenna. The AP transmits the superimposed coded signal of NOMA users by allocating different power levels to the users while sharing the same time/frequency resources, and it is given by

$$x_{NOMA}=\sqrt{{\alpha }_{NU}{P}_t}{x}_{NU}+\sqrt{{\alpha }_{FU}{P}_t}{x}_{FU}$$

(1)

where, ${\alpha }_{NU}$, and ${\alpha }_{FU}$ are the power fraction coefficients for NU and FU, respectively, satisfying ${\alpha }_{NU}+{\alpha }_{FU}=1$, $0<{\alpha }_{NU},{\alpha }_{FU}<1$ and ${\alpha }_{FU}>{\alpha }_{NU}$. The symbols $x_{NU}$, and $x_{FU}$ are assumed to have unit average energy and $P_t$ denote the total transmit power budget available at AP. In the proposed deployment, the RIS is positioned such that the link between AP and RIS, RIS and NU experiences LoS whereas the link between RIS and FU experiences NLoS. The LoS channel links in the system model are modeled as Rician fading. The channel vector $\mathbf{h}$ between AP and the RIS is given by [21]

$$\mathbf{h}=\sqrt{{\beta }_{AR}\left(d_{AR}\right)}\left(\sqrt{{\displaystyle \frac{K_{AR}}{K_{AR}+1}}}{\mathbf{h}}_{AR}^{LoS}+\sqrt{{\displaystyle \frac{1}{K_{AR}+1}}}{\mathbf{h}}_{AR}^{NLoS}\right)$$

(2)

For the link, AP-RIS with distance $d_{AR}$, the large-scale power gain ${\beta }_{AR}\left(d_{AR}\right)$ is defined as

$${\beta }_{AR}\left(d_{AR}\right)={10}^{-PL\left(d_{AR}\right)/10}$$

(3)

where ${\mathbf{h}}_{AR}^{LoS}$ denotes the deterministic LoS components between $AP$ and RIS, ${\mathbf{h}}_{AR}^{NLoS}\sim \mathcal{CN}(0,\mathbf{I})$ denotes the scattered component and $K_{AR}$ is the Rician factor for the link AP-RIS. The large-scale path loss for the link AP-RIS is modeled as [22]

$$PL\left(d_{AR}\right)={\chi }_{\mathrm{LoS}}+P{L}_{\mathrm{LoS}}\left(d_{AR}\right)+L_{\mathrm{RIS}}-G_t-G_r$$

(4)

where ${\chi }_{LoS}\sim \mathcal{N}(0,{\sigma }_{LoS}^2)$ is the shadowing loss, $L_{RIS}$ is the RIS implementation loss, $G_t$ and $G_r$ are the transmit and receive antenna gains of AP and NOMA UEs respectively. The mean path loss (dB) for carrier frequency $f_c$ (GHz) is given by

$$P{L}_{\mathrm{LoS}}\left(d_{AR}\right)=32.4+17.3{\log }_{10}\left(d_{AR}\right)+20{\log }_{10}\left(f_c\right)$$

(5)

The channel vector ${\mathbf{g}}_{NU}$ between the RIS and NU is given by

$${\mathbf{g}}_{NU}=\sqrt{{\beta }_{RNU}\left(d_{RNU}\right)}\left(\sqrt{\frac{K_{RNU}}{K_{RNU}+1}}{\mathbf{g}}_{RNU}^{\mathrm{LoS}}+\sqrt{\frac{1}{K_{RNU}+1}}{\mathbf{g}}_{RNU}^{\mathrm{NLoS}}\right)$$

(6)

For the link, RIS-NU with distance $d_{RNU}$, the large-scale power gain ${\beta }_{RNU}\left(d_{RNU}\right)$ is defined as

$${\beta }_{RNU}\left(d_{RNU}\right)={10}^{-PL\left(d_{RNU}\right)/10}$$

(7)

where ${\mathbf{g}}_{RNU}^{LoS}$ denotes the deterministic LoS components between RIS and NU. ${\mathbf{g}}_{RNU}^{NLoS}\sim \mathcal{CN}(0,\mathbf{I})$ denotes the scattered component and $K_{RNU}$ is the Rician factor for the link RIS-NU. The large-scale path loss for the link RIS-NU is modeled as

$$PL\left(d_{RNU}\right)=P{L}_{\mathrm{LoS}}\left(d_{RNU}\right)+{\chi }_{\mathrm{LoS}}+L_{\mathrm{RIS}}-G_t-G_r$$

(8)

The mean path loss (dB) for $f_c$ is given by

$$P{L}_{\mathrm{LoS}}\left(d_{RNU}\right)=32.4+17.3{\log }_{10}\left(d_{RNU}\right)+20{\log }_{10}\left(f_c\right)$$

(9)

Since the RIS-FU link is assumed to be NLoS due to blockage, it is modeled as Rayleigh fading. The NLoS channel vector ${\mathbf{g}}_{FU}$ between RIS and FU is modeled as

$${\mathbf{g}}_{FU}=\sqrt{{\beta }_{RFU}}{\mathbf{g}}_{RFU}$$

(10)

where ${\mathbf{g}}_{RFU}\sim \mathcal{CN}(0,I)$, and each element in ${\mathbf{g}}_{RFU}=[g_{RF{U}_1},g_{RF{U}_2}\dots g_{RF{U}_M}]$ is defined as

$$g_{RF{U}_i}={\eta }_i{e}^{j\theta g_{RF{U}_i}}i=1,2,\dots M$$

(11)

The magnitude ${\eta }_i$ is modeled as independent and identically distributed (IID) Rayleigh random variables and ${\theta }_{g_{RF{U}_i}}\sim U(0,2\pi )$. For the link RIS-FU with distance $d_{RFU}$, the large-scale power gain ${\beta }_{RFU}\left(d_{RFU}\right)$ is defined as

$${\beta }_{RFU}\left(d_{RFU}\right)={10}^{-PL\left(d_{RFU}\right)/10}$$

(12)

The large-scale path loss between RIS and FU is modeled as

$$PL\left(d_{RFU}\right)=PL\left(d_{RFU}\right)+{\chi }_{NLoS}+L_{wall}+L_{RIS}-G_t-G_r$$

(13)

where ${\chi }_{NLoS}\sim \mathcal{N}(0,{\sigma }_{NLoS}^2)$, $L_{wall}$ is the wall loss present in the indoor environment [23]. The mean path loss (dB) is given by

$$PL\left(d_{RFU}\right)=32.4+38.3{\log }_{10}\left(d_{RFU}\right)+20{\log }_{10}\left(f_c\right)$$

(14)

The received signal at the NU is given by

$$y_{NU}={\mathbf{g}}_{NU}\mathbf{\Phi }\mathbf{h}{x}_{NOMA}+n_{NU}$$

(15)

where $\mathbf{\Phi }=\mathrm{diag}[e^{j{\phi }_1},e^{j{\phi }_2},\dots e^{j{\phi }_M}]$, ${\phi }_i\in (0,2\pi ),i=1,2\dots M$ and the NU receiver noise is modeled as $n_{NU}\sim \mathcal{CN}(0,{\sigma }_{NU}^2)$.

Using SIC, FU information is decoded at the NU. The signal-to-interference plus noise ratio (SINR) of detecting FU data at NU is given by [24]

$${\gamma }_{FU,NU}={\displaystyle \frac{|{\mathbf{g}}_{NU}{\mathbf{\Phi }\mathbf{h}|}^2{\alpha }_{FU}\rho }{|{\mathbf{g}}_{NU}{\mathbf{\Phi }\mathbf{h}|}^2{\alpha }_{NU}\rho +1}}$$

(16)

where $\rho ={\displaystyle \frac{P_t}{N_o}}$ is the SNR. After successful decoding of FU information, the NU decodes its information, and the corresponding SNR is given by

$${\gamma }_{NU,NU}={\left|{\mathbf{g}}_{NU}\mathbf{\Phi }\mathbf{h}\right|}^2{\alpha }_{NU}\rho$$

(17)

The received signal at the FU is given by

$$y_{FU}={\mathbf{g}}_{FU}\mathbf{\Phi }\mathbf{h}{x}_{NOMA}+n_{FU}$$

(18)

where FU receiver noise is modeled as $n_{FU}\sim \mathcal{CN}(0,{\sigma }_{NU}^2)$. SINR for detecting FU data at the FU is given by

$${\gamma }_{FU,FU}={\displaystyle \frac{|{\mathbf{g}}_{FU}{\mathbf{\Phi }\mathbf{h}|}^2{\alpha }_{FU}\rho }{|{\mathbf{g}}_{FU}{\mathbf{\Phi }\mathbf{h}|}^2{\alpha }_{NU}\rho +1}}$$

(19)

Let ${\mathrm{A}}_{NU}={\mathbf{g}}_{NU}\mathbf{\Phi }\mathbf{h}$, and ${\mathrm{A}}_{FU}={\mathbf{g}}_{FU}\mathbf{\Phi }\mathbf{h}$, the data rate achieved by NU after SIC is given by

$$R_{NU}={\log }_2(1+|{\mathrm{A}}_{NU}{|}^2{\alpha }_{NU}\rho )$$

(20)

The data rate achieved by FU is given by

$$R_{FU}={\log }_2\left(1+{\displaystyle \frac{|{\mathrm{A}}_{FU}{|}^2{\alpha }_{FU}\rho }{|{\mathrm{A}}_{FU}{|}^2{\alpha }_{NU}\rho +1}}\right)$$

(21)

The achievable data rates for NU and FU are derived based on the signal and channel models. The ergodic rate for the NU is given by

$${\overline{R}}_{NU}=\mathbb{E}\left[{\log }_2(1+{\gamma }_{NU,NU})\right]$$

(22)

Similarly, the ergodic rate for the FU is given by

$${\overline{R}}_{FU}=\mathbb{E}\left[{\log }_2(1+{\gamma }_{FU,FU})\right]$$

(23)

In the next section, a max–min optimization problem is formulated by jointly optimizing discrete RIS partitioning and power allocation in a NOMA system. In practical RIS-assisted systems, CSI can be obtained using pilot-based estimation or cascaded channel estimation techniques. The proposed fairness optimization framework remains applicable when estimated CSI is used, although performance may degrade depending on estimation errors.

4. Symmetrical User Fairness Optimization Problem

In RIS-assisted NOMA systems, the performance of NU and FU depends on both power domain multiplexing and shared RIS resources. Even though NOMA provides improvement in spectral efficiency, it lacks rate imbalances between users, especially for FU in the indoor/outdoor environments. Hence, resource allocation among NOMA users plays an important role in RIS-assisted NOMA wireless systems. In this section, the resource allocation in terms of power fraction coefficients and RIS element allocation for NOMA users is focused on improving the user fairness in the NOMA system. The objective is to maximize the minimum ergodic rate among the users while satisfying practical constraints, including total power allocation, RIS element conservation, SIC feasibility at the NU, and minimum QoS requirements for both users. The resulting problem captures the fundamental trade-off between fairness and resource allocation in indoor RIS-assisted NOMA systems and serves as the basis for the proposed optimization framework. Let $M_1$, $M_2$, be the number of RIS elements allocated to NU and FU, respectively, for maintaining user fairness among NOMA users. The mathematical optimization for the symmetrical user fairness of ergodic rate is given by

$$\underset{{\alpha }_{NU},{\alpha }_{FU},M_1,M_2}{\max }\min \left({\overline{R}}_{NU},{\overline{R}}_{FU}\right)$$

(24)

Subject to

$${\alpha }_{NU}+{\alpha }_{FU}=1$$

(25a)

$$\begin{array}{cc}\hfill 0\le {\alpha }_{NU},{\alpha }_{FU}& \le 1\hfill \end{array}$$

(25b)

$$M_1+M_2=M,$$

(25c)

$$M_1,M_2\in \mathbb{Z},\mathbb{Z}\ge 0$$

(25d)

$${\gamma }_{FU,NU}({\alpha }_{FU},M_1)\ge {\gamma }_{FU}^{\mathrm{th}}$$

(25e)

$${\gamma }_{FU,FU}({\alpha }_{FU},M_2)\ge {\gamma }_{FU}^{\mathrm{th}}$$

(25f)

$${\gamma }_{NU,NU}({\alpha }_{NU},M_1)\ge {\gamma }_{\mathrm{NU}}^{\mathrm{th}}$$

(25g)

where ${\gamma }_{FU}^{th}$ is the threshold SINR for successful SIC at NU, ${\gamma }_{NU}^{th}$ is the threshold for maintaining QoS, and constraints (25e), (25f), and (25g) are the SIC feasibility at NU, FU QoS, and NU QoS, respectively.

4.1. Epigraph Form

The ergodic rate of max–min optimization problem formulated in the previous section is non-convex due to the coupled objective function, the integer-valued RIS partition variables, and the non-linear SINR constraints. To facilitate tractable analysis and solution development, the max–min objective is equivalently transformed into an epigraph form by introducing an auxiliary variable t, which represents the minimum achievable rate among the users [25].

$$\underset{{\alpha }_{NU},{\alpha }_{FU}{M}_1,M_2,t}{\max }t$$

(26)

Subject to    

$$\begin{array}{c}\hfill R_{NU}({\alpha }_{NU},M_1)\ge t\end{array}$$

(27a)

$$\begin{array}{c}\hfill R_{FU}({\alpha }_{FU},M_2)\ge t\end{array}$$

(27b)

$$\begin{array}{c}\hfill {\alpha }_{NU}+{\alpha }_{FU}=1\end{array}$$

(27c)

$$\begin{array}{c}\hfill 0\le {\alpha }_{NU},{\alpha }_{FU}\le 1\end{array}$$

(27d)

$$\begin{array}{c}\hfill M_1+M_2=M\end{array}$$

(27e)

$$\begin{array}{c}\hfill M_1,M_2\in \mathbb{Z},\mathbb{Z}\ge 0\end{array}$$

(27f)

$$\begin{array}{c}\hfill {\gamma }_{FU,NU}({\alpha }_{FU},M_1)\ge {\gamma }_{FU}^{\mathrm{th}}\end{array}$$

(27g)

$$\begin{array}{c}\hfill {\gamma }_{FU,FU}({\alpha }_{FU},M_2)\ge {\gamma }_{FU}^{\mathrm{th}}\end{array}$$

(27h)

$$\begin{array}{c}\hfill {\gamma }_{NU,NU}({\alpha }_{NU},M_1)\ge {\gamma }_{NU}^{\mathrm{th}}\end{array}$$

(27i)

The optimization problem (26) remains non-convex due to the integer-valued RIS partition variables $M_1$ and $M_2$, as well as the coupled SINR expressions. To handle the integer nature of the problem, an enumeration-based approach is adopted [7]. Specifically, for each feasible value $M_1\in \{0,1,\dots ,M\}$, set $M_2=M-M_1$ and solve the resulting continuous subproblem. The optimal RIS partition is then obtained by selecting the value $M_1$ that maximizes the objective value.

Furthermore, using the constraint ${\alpha }_{NU}+{\alpha }_{FU}=1$, the number of optimization variables is reduced by defining $\alpha \triangleq {\alpha }_{FU}$, which implies ${\alpha }_{NU}=1-\alpha$. To ensure successful SIC, the feasible range $\alpha$ is restricted to $\alpha \in [0.5,1]$. For a fixed RIS partition, the optimization problem thus reduces to a two-variable feasibility problem $(\alpha ,t)$ for fixed t.

Since the achievable rate is a monotonic function of the SINR, the rate constraints $R_u=lo{g}_2(1+{\gamma }_{u,u})\ge t$, $u\in \{NU,FU\}$, are equivalently transformed into SINR constraints ${\gamma }_{u,u}\ge {\gamma }_t$, where ${\gamma }_t=2^t-1$. Let $a_1\triangleq {\left|A_{NU}\left(M_1\right)\right|}^2\rho$, $a_2\triangleq {\left|A_{FU}\left(M_2\right)\right|}^2\rho$, where $A_{NU}\left(M_1\right)$ is the cascaded channel gain ${\mathbf{g}}_{NU}\mathbf{\Phi }\mathbf{h}$ allocated with $M_1$ elements and $A_{FU}\left(M_2\right)$ is the cascaded channel gain ${\mathbf{g}}_{FU}\mathbf{\Phi }\mathbf{h}$ with $M_2$ elements. The resulting SNR expression for NU after SIC is given by

$${\gamma }_{NU,NU}\left(\alpha \right)=a_1(1-\alpha )$$

(28)

The SINR expression for FU is rewritten as

$${\gamma }_{FU,FU}\left(\alpha \right)={\displaystyle \frac{a_2\alpha }{a_2(1-\alpha )+1}}$$

(29)

The SINR expression for NU based on SIC condition is rewritten as

$${\gamma }_{FU,NU}\left(\alpha \right)={\displaystyle \frac{a_1\alpha }{a_1(1-\alpha )+1}}$$

(30)

The non-convex SINR ratio constraints can be rearranged into affine bounds on the power allocation variable $\alpha$:

(I)

NU rate constraint:

$${\gamma }_{NU,NU}\ge {\gamma }_t\Rightarrow \alpha \le 1-{\displaystyle \frac{{\gamma }_t}{a_1}}$$

(31)

(II)

FU rate constraint:

$${\gamma }_{FU,FU}\ge {\gamma }_t\Rightarrow \alpha \ge {\displaystyle \frac{{\gamma }_t(1+a_2)}{a_2(1+{\gamma }_t)}}$$

(32)

(III)

SIC feasibility constraint:

$${\gamma }_{FU,NU}\ge {\gamma }_{FU}^{\mathrm{th}}\Rightarrow \alpha \ge {\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(a_1+1)}{a_1(1+{\gamma }_{FU}^{\mathrm{th}})}}$$

(33)

(IV)

FU QoS constraint:

$${\gamma }_{FU,FU}\ge {\gamma }_{FU}^{\mathrm{th}}\Rightarrow \alpha \ge {\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(1+a_2)}{a_2(1+{\gamma }_{FU}^{\mathrm{th}})}}$$

(34)

(V)

NU QoS constraint:

$${\gamma }_{NU,NU}\ge {\gamma }_{NU}^{\mathrm{th}}\Rightarrow \alpha \le 1-{\displaystyle \frac{{\gamma }_{NU}^{\mathrm{th}}}{a_1}}$$

(35)

Together with the NOMA power allocation constraint $0.5\le \alpha \le 1$, the feasibility of the continuous subproblem reduces to a box constraint, and the interval is non-empty:

$$L\left(t\right)\le \alpha \le U\left(t\right),$$

(36)

where

$$L\left(t\right)=\max \left\{0.5,{\displaystyle \frac{{\gamma }_t(1+a_2)}{a_2(1+{\gamma }_t)}},{\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(a_1+1)}{a_1(1+{\gamma }_{FU}^{\mathrm{th}})}},{\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(1+a_2)}{a_2(1+{\gamma }_{FU}^{\mathrm{th}})}}\right\}$$

(37)

$$U\left(t\right)=\min \left\{1,1-{\displaystyle \frac{{\gamma }_t}{a_1}},1-{\displaystyle \frac{{\gamma }_{NU}^{\mathrm{th}}}{a_1}}\right\}$$

(38)

Hence, for a fixed RIS partition, the feasible set is convex and the constraints are linear in $\alpha$. This property follows from the monotonic behavior of the SINR expression with respect to the power allocation coefficient $\alpha$. Specifically, the NU SNR ${\gamma }_{NU,NU}=a_1(1-\alpha )$ is strictly decreasing in $\alpha$, whereas FU SINR ${\gamma }_{FU,FU}={\displaystyle \frac{a_2\alpha }{a_2(1-\alpha )+1}}$ and the SIC SINR ${\gamma }_{FU,NU}$ are strictly increasing in $\alpha$. Accordingly, the NU constraint produces the upper bound on $\alpha$, while the FU and SIC constraints produce lower bounds, resulting in a feasible interval $L\left(t\right)\le \alpha \le U\left(t\right)$. Since the achievable rate $R_u$ is a monotonic function of SINR, the feasibility condition is also monotonic in the epigraph variable t. The ergodic max–min fairness problem is therefore addressed using a bisection-based quasi-convex optimization on the epigraph variable t. For each feasible RIS partition, $M_1\in \{0,1,$…,$M\}$ the remaining elements are assigned as $M_2=M-M_1$. The continuous feasibility problem in variables $(\alpha ,t)$ is solved using bisection t as shown in Algorithm 1 [26]. The optimal solution is obtained by comparing the fairness rate across all enumerated RIS partitions and selecting the configuration that maximizes the minimum achievable rate.

4.2. Computational Complexity Analysis

In this paper, complexity is defined as the total number of constant-time feasibility checks required per channel realization to reach a target accuracy ${ϵ}_t$. This is a standard measure for bisection-based algorithms. This algorithm iteratively narrows down the search interval of t by checking the feasibility constraints on $L\left(t\right)$ and $U\left(t\right)$ thereby guaranteeing convergence to the optimal max–min rate with minimum complexity. Specifically, the algorithm enumerates all $M+1$ RIS partitions and applies a bisection search with $I_t=\lceil {\log }_2((t_{\max }-t_{\min })/{\epsilon }_t)\rceil$ iterations for each partition. Since each bisection step involves constant-time feasibility checks, the overall complexity scales as $O\left(M{I}_t\right)$. For ergodic evaluation with $N_{\mathrm{MC}}$ channel realizations, the total complexity becomes $O\left(M{I}_t{N}_{\mathrm{MC}}\right)$, which is polynomial in the RIS size and logarithmic in the target accuracy, while avoiding learning-based or iterative AO methods.

Traditional RIS optimization often relies on SDR or SCA to handle the phase-shift matrix $\mathbf{\Phi }$. These methods typically involve solving a linear matrix inequality of size $M\times M$, which has a well-known worst-case complexity of $O\left(M^3\right)$ to $O\left(M^{4.5}\right)$. In general, the complexity of iterative methods scales exponentially with the number of users/antennas/RIS elements. However, the proposed optimization framework scales $O\left(M\right)$ with the RIS size. This is considerably less than learning-based frameworks that need large training datasets and complex high-dimensional matrix computations. Without the bisection refinement for power allocation ($\alpha$), a full grid search would result in $O\left(M{\displaystyle \frac{1}{{\epsilon }_d}}\right)$, where ${\epsilon }_d$ is the power step size, leading to much higher overhead. Transforming the problem into an epigraph form allows the bisection search to efficiently reduce the search for the optimal power fraction to a logarithmic process.

Table 1. Computational complexity and ergodic max–min rate performance of the proposed bisection-based joint discrete RIS partition for different RIS configurations.

RIS Configuration $\left(\mathit{M}\right)$	Bisection Iterations $\left({\mathit{I}}_{\mathit{t}}\right)$	Total Complexity $(\mathit{M}.{\mathit{I}}_{\mathit{t}})$	Rate at 20 dBm (bps/Hz)	Rate at 30 dBm (bps/Hz)	Rate at 40 dBm (bps/Hz)	Fairness Gain for ${\mathit{P}}_{\mathit{t}}$ = 20 dBm
64	15	960	1.95	4.22	6.94	-
128	15	1920	3.24	5.85	8.59	66.15%
256	15	3840	4.90	7.90	10.20	151.28%

illustrates the total number of operations required per channel realization across different RIS configurations investigated in the study. It is assumed that a bisection tolerance of ${\epsilon }_t={10}^{-3}$, which typically results in $I_t\approx 10$ to 15 iterations for convergence. It is clear that as the RIS array size increases, the computational complexity per realization increases linearly, but it yields substantial spectral efficiency gains. The minimal memory and processing requirements of these constant-time feasibility checks make this approach well suited to real-time indoor 5G/6G hardware.

Algorithm 1 Bisection-based symmetrical user fairness optimization with RIS partition enumeration.

Input parameters: Total number of RIS elements M; SNR scaling factor $\rho$; SINR thresholds ${\gamma }_{FU}^{\mathrm{th}}$ and ${\gamma }_{NU}^{\mathrm{th}}$; bisection tolerance ${\epsilon }_t$; maximum number of bisection iterations $I_{\max }$. Output: Optimal RIS partition $(M_1^{★},M_2^{★})$, optimal FU power allocation ${\alpha }^{★}$, and max–min rate $t^{★}$. 1: Initialize $t^{★}\leftarrow 0$, ${\alpha }^{★}\leftarrow 0.5$ 2: Initialize $M_1^{★}\leftarrow 0$, $M_2^{★}\leftarrow M$ 3: for $M_1=0,1,\dots ,M$  do 4:       $M_2\leftarrow M-M_1$ 5:       Compute effective channel gains $$a_1\leftarrow \rho |A_{NU}\left(M_1\right){|}^2,a_2\leftarrow \rho {\left|A_{FU}\left(M_2\right)\right|}^2$$ 6:       if $a_1\le 0$ or $a_2\le 0$ then 7:           continue 8:       end if 9:       Initialize bisection interval $$t_{\min }\leftarrow 0,t_{\max }\leftarrow {\log }_2\left(1+\max \{a_1,a_2\}\right)$$ 10:       for $i=1$ to $I_{\max }$ do 11:             $t\leftarrow (t_{\min }+t_{\max })/2$ 12:             ${\gamma }_t\leftarrow 2^t-1$ 13:             Compute lower feasibility bound $$L\left(t\right)\leftarrow \max \left\{0.5,{\displaystyle \frac{{\gamma }_t(1+a_2)}{a_2(1+{\gamma }_t)}},{\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(a_1+1)}{a_1(1+{\gamma }_{FU}^{\mathrm{th}})}},{\displaystyle \frac{{\gamma }_{FU}^{\mathrm{th}}(1+a_2)}{a_2(1+{\gamma }_{FU}^{\mathrm{th}})}}\right\}$$ 14:             Compute upper feasibility bound $$U\left(t\right)\leftarrow \min \left\{1,1-{\displaystyle \frac{{\gamma }_t}{a_1}},1-{\displaystyle \frac{{\gamma }_{NU}^{\mathrm{th}}}{a_1}}\right\}$$ 15:             if $L\left(t\right)\le U\left(t\right)$ then 16:                   $t_{\min }\leftarrow t$ 17:             else 18:                   $t_{\max }\leftarrow t$ 19:             end if 20:             if $t_{\max }-t_{\min }\le {\epsilon }_t$ then 21:                   break 22:             end if 23:        end for 24:        $t^{\left(M_1\right)}\leftarrow t_{\min }$ 25:        ${\alpha }^{\left(M_1\right)}\leftarrow L\left(t^{\left(M_1\right)}\right)$ 26:        if $t^{\left(M_1\right)}>t^{★}$ then 27:             $t^{★}\leftarrow t^{\left(M_1\right)}$ 28:             ${\alpha }^{★}\leftarrow {\alpha }^{\left(M_1\right)}$ 29:             $M_1^{★}\leftarrow M_1$ 30:             $M_2^{★}\leftarrow M_2$ 31:        end if 32: end for 33: return  $(M_1^{★},M_2^{★},{\alpha }^{★},t^{★})$

4.3. Outage Analysis

For SIC-based decoding at NU, the outage event occurs when either FU data decode fails or NU data decode fails, and it is given by

$$\begin{array}{c}\hfill P_{out}^{NU}=\mathrm{Pr}(\{{\log }_2(1+{\gamma }_{FU,NU})<R_{FU}^{th}\}\cup \{{\log }_2(1+{\gamma }_{NU,NU})<R_{NU}^{th}\})\end{array}$$

(39)

where $R_{NU}^{th}$ and $R_{FU}^{th}$ are the required data rate demands of NU and FU, respectively. For FU, the outage event occurs when FU data decoding fails, and it is given by

$$P_{out}^{FU}=\mathrm{Pr}[{\log }_2(1+{\gamma }_{FU,FU})<R_{FU}^{th}]$$

(40)

The fairness objective for outage analysis with the same constraints as in problem (26) given by

$$\underset{M_1,\alpha }{\min }\max \left(P_{out}^{NU},P_{out}^{FU}\right)$$

(41)

The optimization problem in (41) is non-convex with respect to the power allocation factor $\alpha$ and $M_1$. However, for a fixed RIS partition $M_1$, the $P_{out}^{NU}$, $P_{out}^{FU}$ typically vary in opposite directions with $\alpha$, enabling a bisection search to balance the two outages within the feasible interval. The overall solution is obtained by repeating the above procedure for all feasible RIS partitions and selecting the configurations that minimize the worst-user outage probability.

4.4. Practical Impairments and Architectural Generalization

This section presents a sensitivity analysis of the ergodic rate concerning iSIC, iCSI, and RIS implementation loss. It also discusses the effects of the direct AP-RIS link and explores extensions to multi-user, multi-RIS, and MIMO scenarios.

4.4.1. Impact of iSIC

In practical scenarios, a perfect SIC would be impractical. Due to iSIC, a residual FU component will still be present in the signal received by NU [18]. Due to iSIC, the SINR of detecting NU data becomes

$${\gamma }_{NU,NU}={\displaystyle \frac{|A_{NU}{|}^2{\alpha }_{NU}\rho }{ϵ|A_{NU}{|}^2{\alpha }_{FU}\rho +1}}$$

(42)

where $ϵ$ is the SIC residue error. The increase in $ϵ$ affects the achievable rate of NU. In traditional NOMA systems, the sum-rate diminishes as the number of users exceeds a certain threshold.

4.4.2. Impact of iCSI

In RIS-assisted NOMA systems, the assumption of perfect CSI is idealized and may not hold in practice due to training limitations, feedback delay, pilot contamination, and estimation errors in the cascaded AP-RIS-user channels. Such imperfections are particularly important in RIS-partitioned architectures, where the effective channel depends on the selected partition and must be re-estimated accordingly. Under iCSI, the received signal model is governed by estimated cascaded channels together with residual estimation uncertainty, which introduces additional interference into the SINR expressions. This degradation affects both the direct decoding performance of the FU and the SIC process at the NU, thereby reducing the achievable ergodic fairness rate and potentially changing the optimal RIS partition and power-allocation strategy. To capture this practical effect, the present work models estimation errors through a channel-scaled iCSI formulation and studies their impact on the ergodic max–min fairness performance over transmit power.

Let the estimated channels at the NU and FU be related to estimation error ${\widehat{A}}_u=A_u+e_u$, $u\in \{NU,FU\}$ modeled as Gaussian $e_u\sim \mathcal{CN}(0,{\sigma }_{e,u}^2)$. Let ${\widehat{a}}_1=\rho {|{\widehat{A}}_{NU}|}^2$ and ${\widehat{a}}_2=\rho {|{\widehat{A}}_{FU}|}^2$, the SINR expression for NU-based on SIC condition is rewritten as

$${\gamma }_{FU,NU}={\displaystyle \frac{{\widehat{a}}_1{\alpha }_{FU}}{{\widehat{a}}_1{\alpha }_{NU}+{\widehat{a}}_1{\sigma }_{e,NU}^2+1}}$$

(43)

The SINR expression for NU after SIC is given by

$${\gamma }_{NU,NU}={\displaystyle \frac{{\widehat{a}}_1{\alpha }_{NU}}{{\widehat{a}}_1{\sigma }_{e,NU}^2+1}}$$

(44)

Now, the SINR expression for FU is given by

$${\gamma }_{FU,FU}={\displaystyle \frac{{\widehat{a}}_2{\alpha }_{FU}}{{\widehat{a}}_2{\alpha }_{NU}+{\widehat{a}}_2{\sigma }_{e,FU}^2+1}}$$

(45)

The lower and upper bounds $\left(L\right(t\left)\right)$ and $\left(U\right(t\left)\right)$ in Algorithm 1 must be adjusted to use these modified SINR thresholds, ensuring the bisection search accounts for the iCSI-induced performance floor.

4.4.3. Impact of Direct AP-RIS Link

In many indoor environments, the direct path between the AP and UE is strongly blocked by structural obstacles such as concrete walls or metallic partitions. The proposed system assumes that the AP is positioned strategically so that the direct link experiences significant shadowing or deep fading, making the RIS-assisted path the main contributor to the received signal. However, for environments where a direct path persists, the received signal at the user $u\in \{FU,NU\}$ can be modified to include the direct channel component $h_{d,u}$,

$$y_u=({\mathbf{g}}_u\mathbf{\Phi }\mathbf{h}+h_{d,u})x_{NOMA}+n_u$$

(46)

Based on the effective channel, the SINR/SNR and rate expressions will be updated accordingly. Even without precise phase alignment in RIS, the performance baseline will improve due to the direct link. This addition enhances the system’s diversity but often makes phase-shift optimization more complex, as the RIS must now align the reflected signal with a potentially fluctuating direct path.

4.4.4. Multi-User Extension

Additionally, the system model is extended for L downlink UEs in indoor environments served by the AP-assisted by RIS and it is shown in Figure 2. This L user power domain NOMA requires multi-layer SIC decoding with an order of $O\left(L\right)$. Consequently, achieving massive connectivity through NOMA is considered ideal in theory but impractical for real-world IoT scenarios. To address this, hybrid NOMA has been proposed, which combines conventional OMA with NOMA [18]. To demonstrate the multi-user capability of the proposed framework, the time division multiple access (TDMA) is integrated with NOMA. The total frame duration is divided into subframes, each of which resembles a two-user NOMA system. Various strategies exist for pairing users within each subframe. In the proposed system model, two commonly used pairing strategies are considered, namely, NN-FF and N-F. In N-F pairing, the UE nearest to the gNB is paired with the UE farthest from the gNB. The next-nearest UE is paired with the next-farthest UE, and so on. In NN-FF pairing, the nearest UE is paired with the next nearest UE and the farthest UE is paired with the next farthest UE and so on. Previous studies have shown that pairing exceeds traditional NOMA with the same number of UEs, and N-F outperforms NN-FF pairing [18]. For L number of UEs, the system model remains the same with ${\displaystyle \frac{2}{L}}$ factor before the rate expressions in (20) to (23). The SINR/SNR constraints are modified to ${\gamma }_t=2^{{\displaystyle \frac{Lt}{2}}}-1$ and the same changes to ${\gamma }_{FU}^{th}$ and ${\gamma }_{NU}^{th}$.

4.4.5. MIMO Extension

The transition from a single-antenna AP to a multi-antenna configuration significantly enhances the capability of the system to suppress multi-user interference and improve link reliability through spatial beamforming. Consider an AP equipped with N transmit antennas serving two users via an RIS consisting of M reflecting elements. Let $\mathbf{H}\in {\mathbb{C}}^{M\times N}$ denote the AP–RIS channel matrix and ${\mathbf{g}}_u\in {\mathbb{C}}^{1\times M}$ represent the RIS–user-u channel vector, where $u\in \{\mathrm{FU},\mathrm{NU}\}$ corresponds to the FU and NU, respectively. The cascaded effective channel between the AP and user u through the RIS can be expressed as

$${\mathbf{q}}_u={\mathbf{g}}_u\mathbf{\Phi }\mathbf{H},$$

(47)

where ${\mathbf{q}}_u\in {\mathbb{C}}^{1\times N}$ represents the equivalent AP-to-user channel vector after RIS reflection. Equivalently, the cascaded channel can be written as

$${\mathbf{q}}_u=\sum _{m=1}^M{g}_u\left(m\right){\varphi }_m{\mathbf{h}}_m,$$

(48)

where ${\mathbf{h}}_m$ denotes the mth row of $\mathbf{H}$. Assuming maximal ratio transmission (MRT), the beamforming vector for user u is aligned with the effective channel and given by

$${\mathbf{w}}_u={\displaystyle \frac{{\mathbf{q}}_u^H}{\parallel {\mathbf{q}}_u\parallel }},u\in \{\mathrm{FU},\mathrm{NU}\}.$$

(49)

The transmitted superposed NOMA signal vector is therefore written as

$$\mathbf{x}=\sqrt{{\alpha }_{FU}{P}_t}{\mathbf{w}}_{FU}{x}_{FU}+\sqrt{{\alpha }_{NU}{P}_t}{\mathbf{w}}_{NU}{x}_{NU},$$

(50)

The received signal at user u can be expressed as

$$y_u={\mathbf{q}}_u\mathbf{x}+n_u,$$

(51)

Substituting the transmit signal into the above equation yields

$$y_u=\sqrt{{\alpha }_{FU}{P}_t}{\mathbf{q}}_u{\mathbf{w}}_{FU}{x}_{FU}+\sqrt{{\alpha }_{NU}{P}_t}{\mathbf{q}}_u{\mathbf{w}}_{NU}{x}_{NU}+n_u.$$

(52)

When the AP is equipped with a large number of antennas and the RIS contains a large number of reflecting elements, the effective cascaded channel exhibits the channel hardening property. In this regime, the random channel gain becomes nearly deterministic, such that [27]

$${\displaystyle \frac{\parallel {\mathbf{q}}_u{\parallel }^2}{\mathbb{E}\left[\parallel {\mathbf{q}}_u{\parallel }^2\right]}}\to 1.$$

(53)

Consequently, the desired beamforming gain under MRT can be approximated as

$$|{\mathbf{q}}_u{\mathbf{w}}_u{|}^2={\parallel {\mathbf{q}}_u\parallel }^2\approx \mathbb{E}\left[\parallel {\mathbf{q}}_u{\parallel }^2\right].$$

(54)

Assuming independent small-scale fading and large-scale fading coefficients ${\beta }_{AR}$ and ${\beta }_{Ru}$ for the AP–RIS and RIS–user-u links, respectively, the deterministic equivalent of the cascaded channel gain becomes [28]

$$\mathbb{E}\left[\parallel {\mathbf{q}}_u{\parallel }^2\right]=NM{\beta }_{AR}{\beta }_{Ru}.$$

(55)

Therefore, for large N and M, the effective beamforming gain of the RIS-assisted MIMO link becomes approximately deterministic, which greatly simplifies the performance analysis and enables tractable evaluation of the ergodic max–min fairness rates. Due to asymptotic orthogonality of user channels in massive MIMO systems, the inter-user beamforming interference satisfies

$$|{\mathbf{q}}_u{\mathbf{w}}_i{|}^2\approx 0,u\ne i.$$

(56)

Therefore, the SINR expressions can be approximated using deterministic equivalents. The FU decodes its own signal by treating the NU signal as interference. The instantaneous SINR is given by

$${\gamma }_{FU,FU}={\displaystyle \frac{{\alpha }_{FU}\rho {\left|{\mathbf{q}}_{FU}{\mathbf{w}}_{FU}\right|}^2}{{\alpha }_{NU}\rho {\left|{\mathbf{q}}_{FU}{\mathbf{w}}_{NU}\right|}^2+1}}.$$

(57)

Applying the channel hardening approximation yields the deterministic-equivalent SINR

$${\gamma }_{FU,FU}^{\mathrm{DE}}\approx {\alpha }_{FU}\rho NM{\beta }_{AR}{\beta }_{RFU}$$

(58)

The NU performs SIC to decode the FU signal and then detects its own signal. After successful SIC, the SNR for detecting the NU signal is

$${\gamma }_{NU,NU}={\alpha }_{NU}\rho {\left|{\mathbf{q}}_{NU}{\mathbf{w}}_{NU}\right|}^2$$

(59)

Using the deterministic equivalent channel gain, the SNR expression is rewritten as

$${\gamma }_{NU,NU}^{\mathrm{DE}}\approx {\alpha }_{NU}\rho NM{\beta }_{AR}{\beta }_{RNU}$$

(60)

These deterministic-equivalent SINR expressions significantly simplify the analytical performance evaluation, as the randomness associated with small-scale fading is effectively averaged out in the large-antenna regime.

4.4.6. Multi-RIS Extension

In this work, we focused on a single-RIS setup to establish a clear mathematical baseline for the joint discrete partitioning and power allocation problem under the standardized 3GPP InH channel model. However, the proposed max–min fairness framework is designed with scalability in mind. The analytical expressions and bisection-based optimization can be extended to multi-RIS scenarios by incorporating a RIS selection. Our single-RIS model serves as a prerequisite for multi-panel indoor architectures. The channel from the AP to l th RIS panel is ${\mathbf{h}}_l\in {\mathrm{C}}^{M\times 1}$. In Algorithm 1, the AP chooses the l-th RIS panel that provides the best LoS path to the user before initiating the $M_1$ partitioning loop. All other steps remain the same.

5. Simulation Results

The RIS configuration parameters and system settings used in the numerical simulations are summarized in

Table 2. Simulation parameters.

Symbol	Parameter	Value
$f_c$	Carrier frequency	3.5 GHz
$d_{AR}$	$AP-RIS$ distance	10 m
$d_{RNU}$	RIS-NU distance	10 m
$d_{RFU}$	RIS-FU distance	30 m
${\sigma }_{LoS}$	LoS shadowing standard deviation	3 dB
${\sigma }_{NLoS}$	NLoS shadowing standard deviation	8 dB
$L_{wall}$	Wall Loss	6 dB
$G_t$,$G_r$	Tx/Rx antenna gain	0 dBi, 0 dBi
$R_{NU}^{th}$	NU data rate demand	1 bps/Hz
$R_{FU}^{th}$	FU data rate demand	0.5 bps/Hz
$N_{MC}$	Number of realizations	$3\times {10}^4$
$\alpha$	FU power fraction	[0.5, 1]
B	Bandwidth	10 MHz
$NF$	Noise Figure	7 dB

, including the carrier frequency, propagation parameters, and Monte-Carlo realizations employed for performance evaluation. RIS hardware characteristics, including element counts, implementation loss, and the heterogeneous fading models applied to the respective sub-surfaces, are summarized in

Table 3. Key RIS configuration and assumptions.

RIS Parameter	Value/Assumption
Total Elements $\left(M\right)$	64, 128, 256
Partitioning Strategy	Joint discrete RIS partitioning (allocation of elements between users $(M_1$ for NU, $M_2$ for FU))
Implementation Loss $\left(L_{RIS}\right)$	0, 2, 4, 6 dB
Reflection Model	Unit-modulus phase shift behavior
NU Channel	Rician fading (AP–RIS and RIS–NU propagation conditions)
FU Channel	Rayleigh fading (RIS–FU propagation under blockage)
RIS Element Spacing	$\lambda /2$
Quantization	Continuous phase shift

. The large-scale fading is modeled using the 3GPP InH channel model as specified in 3GPP TR38.901 [29,30]. Both LoS and NLoS propagation conditions account for distance-dependent path loss and log-normal shadowing. The receiver noise power is modeled according to the thermal noise floor and receiver noise figure. The noise power in dBm is calculated as [31]

$$N_{dBm}=-174+10{\log }_{10}\left(B\right)+NF$$

(61)

where −174 dBm/Hz denotes the thermal noise power spectral density at room temperature, B is the system bandwidth in Hz, $NF$ represents the receiver noise figure in dB.

The variation of the optimal RIS partition $M_1^{*}$ corresponding to the number of RIS elements allocated to the NU-FU as a function of transmit power $P_t$ is shown in Figure 3. It is observed that $M_1^{*}$ increases monotonically with $P_t$ for different RIS sizes, indicating that as the available transmit power increases, a slightly larger portion of RIS resources can be assigned to the NU without compromising user fairness. However, $M_1^{*}$ remains a relatively small fraction of the total RIS elements even at high $P_t$, confirming that the FU subject to NLoS propagation and higher path loss remains the bottleneck in indoor InH scenarios. Moreover, for a larger RIS size ($M=256$), the optimal $M_1^{*}$ is consistently higher than that for $M=128\mathrm{and}M=64$, reflecting the additional spatial degrees of freedom provided by a larger RIS aperture. This result highlights the joint dependence of fairness-optimal RIS partitioning on both transmit power $P_t$ and RIS size M, while reinforcing that the majority of RIS elements should be dedicated to strengthening the FU link.

The variation of the optimal power allocation coefficient ${\alpha }^{*}$, corresponding to the fraction of transmit power allocated to the FU, as a function of the transmit power $P_t$ is shown in Figure 4. It is observed that ${\alpha }^{*}$ remains close to unity across the entire power range for all the RIS sizes and gradually increases with $P_t$. This behavior indicates that the FU consistently remains the bottleneck user under InH conditions due to its NLoS propagation and higher path loss, thereby requiring a dominant share of transmit power to maintain fairness. Moreover, for the larger RIS configuration ($M=256$), the optimal ${\alpha }^{*}$ is higher than that for $M=128\mathrm{and}M=64$ at low and moderate transmit powers, reflecting the system’s ability to further exploit the increased RIS aperture to support the FU. As $P_t$ increases, ${\alpha }^{*}$ approaches unity for all the cases, suggesting that at high SNR the fairness optimization primarily relies on RIS partitioning rather than power redistribution.

The ergodic max–min fairness rate versus transmit power $P_t$ for RIS-assisted NOMA under the 3GPP InH channel model is shown in Figure 5. The ergodic max–min fairness increases linearly with the transmit power $P_t$ for different configurations of RIS sizes. At $P_t=20$ dBm, the fairness rate improves from 1.951 bps/Hz for $M=64$ to 3.243 bps/Hz for $M=128$, corresponding to a 66.1% gain and further increases to approximately 4.9 bps/Hz for $M=256$, yielding an additional gain of 51% over $M=128$. At $P_t=30$ dBm, the ergodic rate increases from 4.22 bps/Hz to 5.85 bps/Hz (38.4% gain) and further to around 7.9 bps/Hz for $M=256$, representing a 35% improvement over $M=128$. Similarly, at $P_t=40$ dBm, the fairness rate improves from 6.94 bps/Hz to 8.59 bps/Hz (23.8% gain) when M is doubled from 64 to 128 and further reaches approximately 10.2 bps/Hz, for $M=256$, corresponding to an additional gain of about 18–19%. These results indicate that increasing the number of RIS elements in a uniform planar array yields substantial absolute fairness-rate gains across all power levels, while the relative gain decreases at high $P_t$ due to the logarithmic rate growth in the high-SNR regime. Moreover, the optimal RIS partition allocates only a small fraction of elements to the NU, and the optimal power allocation $\alpha$ remains close to unity.

The ergodic max–min fairness rate performance of the proposed joint optimization framework for a downlink RIS-assisted NOMA system is compared with the baselines in Figure 6. The joint optimization of RIS elements and power allocation coefficients linearly increases the ergodic max–min rate as the transmit power increases, reaching 8.59 bps/Hz for the $P_t$ of 40 dBm and $M=128$. Using a fixed and equal RIS partition with optimized power fractions leads to a visible performance gap, whereas using a fixed and equal RIS partition with fixed power fractions leads to the lowest performance, significantly lower than the joint optimal case. The equal RIS partition with optimal power allocation [18] achieves a rate of 7.8 bps/Hz, while the fixed-power allocation [2] case achieves 1.6 bps/Hz for $P_t=40$ dBm. The significant gap between this and the optimized versions shows that fixed power allocation cannot overcome the severe blockages and NLoS conditions inherent in InH scenarios. These findings confirm that fairness-optimal operation requires allocating most RIS elements and transmit power to the FU to overcome the severe path loss and blockage typical of indoor NLoS environments. The ergodic max–min fairness rate achieved by the proposed bisection-based user fairness optimization and the AO method [32] as a function of the transmit power $P_t$ is shown in Figure 7 for two RIS sizes, $M=64$ and $M=128$. It is observed that the ergodic max–min rate increases monotonically with the transmit power for both optimization methods and RIS configurations. The proposed bisection-based optimization method consistently outperforms the AO-based benchmark for both RIS sizes. The performance gap is more pronounced in the low- and medium-transmit-power regions, where the proposed method achieves higher fairness rates. This indicates that the proposed algorithm is more effective in balancing the rate allocation between the NU and FU through optimal RIS partitioning and power allocation. As the transmit power becomes large, the performance gap between the proposed method and the AO method slightly reduces. This occurs because, at high SNR, the system performance becomes less sensitive to the exact partitioning and power allocation, and both methods approach similar fairness limits.

The ergodic max–min fairness performance of the proposed RIS-partitioned NOMA scheme is compared with RIS beamforming without partitioning and RIS-assisted OMA [18] in Figure 8. The fairness rate increases with transmit power $P_t$ for all schemes due to the improvement in received signal strength. The proposed RIS-partitioned NOMA consistently achieves the highest fairness rate across the entire transmit power range. This improvement is attributed to the joint optimization of RIS element partitioning and NOMA power allocation, which effectively balances the channel conditions of the NUs and FUs. The RIS beamforming scheme without partitioning performs better than RIS-assisted OMA but remains inferior to the proposed method since it does not explicitly address fairness through RIS resource allocation. For instance, at $P_t=20$ dBm, the proposed scheme achieves about 2 bps/Hz, while RIS beamforming without partitioning and RIS-assisted OMA achieve approximately 1.4 bps/Hz and 1.1 bps/Hz, respectively. At $P_t=40$ dBm, the proposed method reaches nearly 7 bps/Hz, compared to 6.4 bps/Hz and 3.9 bps/Hz for the other two schemes. These results demonstrate the effectiveness of the proposed RIS-partitioned NOMA in improving fairness and spectral efficiency. A comparative discussion of these benchmark methods and their applicability to the proposed problem formulation is provided in

Table 4. Comparison of optimization approaches for RIS-assisted systems.

Method	Optimization Variables	Complexity	Suitability for Discrete RIS Partitioning	Training Requirement	Performance Characteristics
Conventional RIS Beamforming [18]	Continuous phase shifts	Moderate	Not suitable	No	Mainly maximizes received power or sum-rate; fairness is not explicitly addressed
SDR-based Joint Optimization [33]	Continuous variables via relaxation	High (matrix lifting and semidefinite programming)	Limited suitability due to integer relaxation	No	Provides near-optimal solutions but requires Gaussian randomization and high computational cost
Alternating Optimization (AO) [32]	Iterative update of power allocation and RIS partition variables	Moderate–High (iterative convergence required)	Applicable but computationally intensive	No	Achieves performance close to optimal but convergence speed depends on initialization
Deep Reinforcement Learning (DRL) [34]	Policy-based RIS configuration and power control	High (training and inference stages)	Applicable for dynamic environments	Yes	Suitable for time-varying environments but lacks analytical guarantees
Proposed Bisection–Enumeration Method	Discrete RIS partition and fairness parameter	Low–Moderate	Directly suitable	No	Achieves near-optimal ergodic max–min fairness with reduced computational complexity

.

The outage probability at the optimum of min–max fairness versus $P_t$ for RIS-assisted NOMA is shown in Figure 9. The outage probability is evaluated for joint optimal RIS partition $M_1^{*}$ and power allocation ${\alpha }^{*}$ in RIS-assisted downlink NOMA with SIC at the NU, under an InH channel model. For each $P_t$, and every RIS size $(M=64,128,256)$, the system has been optimized to minimize the maximum outage probability at NU and FU. The plotted curves in Figure 6 represent $P_{out}^{NU}(M_1^{*},{\alpha }^{*})$ and $P_{out}^{FU}(M_1^{*}$,${\alpha }^{*})$. The outage curves of the NU and FU overlap across the entire ${\mathrm{P}}_t$ range, confirming that the proposed optimization achieves balanced outage performance between users. It is also observed that increasing the RIS size yields a substantial reduction in outage probability for both NU and FU across the entire transmit power $P_t$. Specifically from $M=64$ to $M=128$ and to further $M=256$, the outage curve shifts leftward, indicating that the same target reliability can be achieved at a lower transmit power $P_t$.

The impact of iSIC on the ergodic max–min fairness performance of the RIS-assisted NOMA system is shown in Figure 10 for $M=64$. When perfect SIC is assumed ($ϵ=0$), the system achieves the highest ergodic max–min rate across the transmit power range and the performance gain increases noticeably at medium and high $P_t$, indicating the effective interference cancellation at the NU fully exploits the NOMA and RIS gains. As the iSIC level increases, the ergodic max–min rate decreases monotonically for all $P_t$. This degradation is primarily due to the residual FU interference at the NU, which limits the achievable NU SINR even at high $P_t$. The performance gap between perfect SIC and iSIC is more pronounced at high $P_t$ where residual interference dominates over the thermal noise.

The impact of iCSI at the NU and FU on the ergodic max–min fairness rate of the RIS-assisted NOMA system as a function of the transmit power $P_t$ is shown in Figure 11. The results show that the fairness rate increases with $P_t$ for all cases, but the presence of channel estimation errors degrades the achievable performance compared to the perfect CSI case. When both estimation errors are zero $({\sigma }_{e,NU}^2=0,{\sigma }_{e,FU}^2=0)$, the system achieves the highest fairness rate across the entire $P_t$ range. As the estimation error variances increase, the ergodic max–min rate decreases due to the additional interference introduced by inaccurate channel estimates. The degradation becomes more noticeable at higher transmit powers, where the impact of residual interference caused by estimation errors becomes more significant. A comparison between asymmetric error cases indicates that the system performance is slightly more sensitive to the estimation error at the NU than at the FU. This is because the NU is responsible for performing SIC before decoding its own signal, making accurate channel estimation more critical at this user. These results demonstrate that iCSI reduces the achievable fairness performance in RIS-assisted NOMA systems, and accurate channel estimation particularly at the NU is essential to fully exploit the benefits of RIS partitioning and NOMA transmission.

The ergodic max–min fairness rate versus transmit power $P_t$ for different RIS implementation losses $L_{RIS}$ is shown in Figure 12. The fairness rate increases with $P_t$ for all cases due to improved received signal power. However, larger RIS loss degrades the performance because it reduces the effective cascaded channel gain. The best performance is achieved when $L_{RIS}=0$ dB, while increasing the loss to 2, 4, and 6 dB progressively lowers the fairness rate. For example, at $P_t=40$ dBm, the rate decreases from about 7.5 bps/Hz for $L_{RIS}=0$ dB to about 5.8 bps/Hz for $L_{RIS}=6$ dB. The findings reveal a steady decline in the achievable fair rate with increasing loss, emphasizing that hardware-related attenuation directly reduces the beamforming gain that RIS can deliver. This indicates that, as power increases, the quality of RIS hardware becomes a key limiting factor for system performance.

The ergodic sum-rate performance of the RIS-assisted TDMA-NOMA system under N-F and NN-FF user pairing strategies, considering without SIC feasibility and with SIC feasibility constraint, is shown in Figure 13. The sum-rate increases rapidly with an increase in transmit power $P_t$ for all user pairings, and larger RIS size enhance the system performance gain by providing passive beamforming. It is observed that the N-F and NN-FF user pairings achieve nearly identical sum-rates across the entire $P_t$ range, indicating that when only sum-rate maximization is considered, user pairing has a marginal impact. When the SIC feasibility constraint is enforced, a noticeable reduction in sum-rate is observed for all RIS sizes and pairing strategies. This degradation arises from the additional requirement that the NU must reliably decode the FU signal prior to SIC, which restricts the feasible power-allocation region. The results confirm that while SIC feasibility constraints impose a tradeoff between reliability and throughput, the proposed RIS-assisted NOMA system remains robust, and increasing the RIS size significantly enhances sum-rate performance under both pairing strategies.

6. Conclusions

This paper investigated a symmetrical, user fairness-oriented resource allocation framework for downlink RIS-assisted NOMA systems operating in realistic indoor environments. By adopting the standardized 3GPP InH channel model, the study explicitly accounted for heterogeneous propagation conditions, including LoS and NLoS links, shadowing, and physical link-budget constraints, which are often overlooked in existing works. To address the inherent user fairness challenges in indoor NOMA deployments, a max–min fairness optimization problem was formulated by jointly considering discrete RIS element partitioning and NOMA power allocation under SIC feasibility and QoS constraints. The resulting non-convex problem was efficiently solved using an epigraph-based transformation combined with a bisection-based quasi-convex optimization approach and RIS partition enumeration. The proposed bisection-based enumeration algorithm achieves globally optimal max–min fairness with polynomial computational complexity, thereby eliminating the need for learning-based schemes or the computationally intensive iterative AO method.

Numerical results demonstrated significant improvements in ergodic max–min fairness across a wide range of transmit powers and RIS sizes. In particular, increasing the RIS size from 64 to 256 elements yielded substantial fairness gains, while the optimal resource allocation consistently assigned a dominant share of RIS elements and transmit power to the FU experiencing NLoS propagation. At the same time, a small but increasing fraction of RIS elements was allocated to the NU as the transmit power increased, revealing an important fairness-driven trade-off between power allocation and spatial resource partitioning. These findings provide valuable design insights for practical indoor RIS-assisted NOMA deployments, confirming that fairness performance is primarily constrained by the FU link under indoor conditions. To ensure practical relevance, the framework incorporates multi-user support via hybrid TDMA-NOMA and accounts for iSIC, and iCSI. The study is further generalized to include multi-antenna configurations and distributed multi-RIS architectures, demonstrating that the proposed bisection-based partitioning logic serves as a scalable foundational baseline for complex, next-generation indoor 6G deployments. Additionally, the current enumeration-based method does not consider sub-millisecond dynamic switching. Future work may extend this framework to dynamic user mobility and hybrid active–passive RIS architectures, as well as explore joint optimization with advanced scheduling strategies and experimental testbed validation.