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Fairness for Non-Orthogonal Multiple Access in 5G Systems
Stelios Timotheou, Ioannis Krikidis
TL;DR
The paper addresses fair power allocation in downlink NOMA, where power-domain multiplexing and SIC can produce unequal user rates. It develops low-complexity algorithms for instantaneous and average CSI, obtaining optimal solutions and substantially better fairness than TDMA in the considered configurations.
Problem
NOMA power allocation must address unequal user rates, motivating fairness optimization under instantaneous and average CSI.
Method
The paper formulates fairness-based power allocation for data rate under full CSI and outage probability under average CSI, then solves the non-convex problems with low-complexity bisection-based algorithms.
Results
NOMA achieves approximately an order of magnitude better fairness performance than TDMA in the considered configurations.
Takeaways & Limitations
Appropriate power allocation enables NOMA to meet high fairness requirements in the studied downlink settings.
Abstract
from arXiv · showhide
In non-orthogonal multiple access (NOMA) downlink, multiple data flows are superimposed in the power domain and user decoding is based on successive interference cancellation. NOMA's performance highly depends on the power split among the data flows and the associated power allocation (PA) problem. In this letter, we study NOMA from a fairness standpoint and we investigate PA techniques that ensure fairness for the downlink users under i) instantaneous channel state information (CSI) at the transmitter, and ii) average CSI. Although the formulated problems are non-convex, we have developed low-complexity polynomial algorithms that yield the optimal solution in both cases considered.
I. INTRODUCTION
The paper motivates fairness-aware power allocation for downlink NOMA, where power-domain multiplexing and SIC can produce unequal user rates. It studies optimal allocation under instantaneous and average CSI using low-complexity algorithms.
- NOMA combines superposition coding with successive interference cancellation to achieve high spectral efficiency in 5G downlink access.
- Power-domain multiplexing and SIC can yield unequal user rates, making fairness critical in constrained scenarios.
- Prior work analyzed fixed-power NOMA performance and cooperative fairness mechanisms that consume dedicated channel resources.
- The paper investigates power allocation for fairness with user rates adapted to instantaneous CSI and fixed target rates under average CSI.
- Despite non-convex formulations, bisection-based iterative algorithms provide globally optimal solutions with closed or semi-closed-form subproblems.
- Results show that appropriately allocated NOMA power significantly improves the worst-user performance over conventional multiple-access approaches.
II. SYSTEM MODEL
The system is a single-cell NOMA downlink with one single-antenna base station serving saturated single-antenna users over block Rayleigh fading. Users are ordered by instantaneous channel strength, and noise is modeled as normalized AWGN.
- The model contains one base station and N single-antenna users in a saturated single-cell downlink.
- The base station has total available transmit power P and always has data queued for every user.
- All links experience independent, identically distributed block Rayleigh fading with additive white Gaussian noise.
- Channels remain constant within a slot and change independently between slots.
- Users are indexed by increasing channel strength, so U1 has the weakest instantaneous channel and UN the strongest.
A. NOMA scheme
NOMA serves all users across the full bandwidth by superimposing their data flows in the power domain. Successive interference cancellation lets stronger-channel users decode weaker users’ flows before decoding their own.
- The base station transmits a linear superposition of N data flows across the entire bandwidth.
- A fraction β_i of the total transmit power is allocated to data flow i.
- User multiplexing is performed in the power domain rather than by separating users into orthogonal resources.
- Each receiver applies successive interference cancellation and can perfectly decode signals intended for weaker users.
- Because of channel ordering, user i can decode flow m when m ≤ i, with R_m,m ≤ R_i,m.
B. Outage probability performance for NOMA
Without instantaneous channel feedback, NOMA assigns every flow a fixed target spectral efficiency and evaluates fairness through outage probability. Under i.i.d. Rayleigh fading, the outage expression is derived in closed form using ordered-channel statistics and the binomial theorem.
- Without instantaneous channel feedback, every data flow uses a target spectral efficiency r0, and outage probability measures performance.
- An outage occurs when user i cannot decode its own flow or any weaker-user flow m < i.
- The i-th user’s outage probability is derived using order statistics for the channel gains.
- The expression uses λ = 1/σ^2, ˆζ_i = max{ζ_1, ζ_2, …, ζ_i}, and δ_i,k = λ(N − i + 1 + k).
- For i.i.d. Rayleigh fading, the analysis gives a closed-form expression, unlike the different channel distribution considered in prior work.
III. FAIRNESS FOR NOMA SYSTEMS
The section frames NOMA fairness around two criteria: instantaneous CSI and average CSI. These criteria address flexible management of users’ achievable rates.
- NOMA supports two fairness criteria for systems with instantaneous and average CSI, respectively.
A. Max-Min fairness with instantaneous CSI
With instantaneous CSI, fairness is formulated as max-min optimization of users’ achievable rates. The non-convex problem is transformed into linear-program feasibility checks and solved optimally with a low-complexity algorithm.
- Instantaneous CSI enables rate allocation according to channel conditions, using max-min fairness to maximize the minimum user rate.
- Problem (3) is quasi-concave, allowing its solution through a sequence of linear programs bounded by bisection.
- At the optimal LP solution, every power fraction βi is positive and all users’ achievable rates are equal.
- The optimal power allocation is computed successively from the strongest to the weakest channel because each βi depends only on stronger-channel allocations.
- The closed-form solution gives every user rate equal to t and has computational complexity O(N).
B. Min-Max fairness with average CSI
With average CSI, fairness is addressed by optimizing users’ outage probabilities under fixed target spectral efficiencies. Despite non-convexity, channel-order transformations and decomposed one-dimensional subproblems yield an optimal solution.
- Average-CSI optimization uses channel-distribution and channel-order knowledge to optimize outage probability for all NOMA users.
- The average-CSI problem is non-convex because exponential terms are neither convex nor concave, motivating elimination of auxiliary variables.
- At the optimum, each target spectral efficiency is satisfied with equality and the auxiliary thresholds match the ordered channel gains.
- After transformation, the problem decomposes into N one-dimensional subproblems whose active monotonic constraints determine the optimal auxiliary variables.
- Each subproblem can be solved by Newton’s or bisection method in O(Nlog(ε)), with total complexity O(N^2log(ε)) or O(Nlog(ε)) in parallel.
IV. NUMERICAL RESULTS
The proposed NOMA power-allocation methods improve fairness across instantaneous- and average-CSI settings, outperforming TDMA and fixed NOMA in the considered configurations. Fairness gains increase with user count, while higher target spectral efficiency worsens outage probability.
- 1000 randomly generated problems were solved across different parameter configurations to evaluate the developed algorithms.
- Increasing transmit power or reducing the number of users improves the achievable maximum fairness rate.
- The fairness-rate gain from N = 10 to N = 5 is significantly higher than from N = 20 to N = 10, while gains diminish as power increases.The diminishing improvement is attributed to the fairness data rate being logarithmic in power.
- NOMA significantly outperforms TDMA in fairness rate, with its advantage increasing almost linearly as the number of users increases.NOMA also has lower computational complexity because TDMA requires a sequence of convex programs.
- Under average CSI, NOMA outperforms TDMA by an order of magnitude in outage probability and fixed NOMA by at least five times in all considered cases.The comparison uses N = 5 and target spectral efficiencies r0 = 0.05 BPCU and r0 = 0.50 BPCU; higher target efficiency produces worse outage probability.
V. CONCLUSIONS
The paper shows that appropriate power allocation enables high fairness in NOMA downlink systems under full and average CSI. Its low-complexity algorithms provide optimal solutions, and simulations show approximately order-of-magnitude fairness gains over TDMA in the considered configurations.
- The study optimizes NOMA power allocation for fairness using data rate under full CSI and outage probability under average CSI.
- Despite non-convex formulations, the developed simple low-complexity algorithms provide optimal solutions.
- NOMA achieves fairness performance approximately an order of magnitude better than TDMA in the considered configurations.
- The results indicate that appropriate power allocation can support high fairness requirements in NOMA downlink systems.