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Unifying Performance and Equity via Percentile Optimization and Cell-Free Massive MIMO

Ahmad Khan, Hussein A. Ammar, Raviraj Adve

arXiv:2610.33982v1cs.ITcs.NIcs.SYeess.SPeess.SYmath.IT

TL;DR

Wireless resource allocation seeks better service for the weakest users, while UC-mMIMO is designed around fairness; whether percentile optimization adds value in this architecture is unstudied. The paper formulates percentile power control for UC-mMIMO and finds gains over WMMSE and competing architectures.

  • Problem

    Whether explicit lower-percentile optimization remains valuable in UC-mMIMO, an architecture designed to favor weaker users, had not previously been studied.

    Source: A. Related Work and Research Gap · p. 2

  • Method

    The paper formulates downlink power control to maximize the sum-least-qth-percentile rate in UC-mMIMO and develops a block-concavity-restoring change of variables for its algorithm.

    Source: B. Contributions · p. 2, §VI · p. 13

  • Results

    56–72%: percentile optimization raises the lowest-10th-percentile average rate over WMMSE, while UC-mMIMO leads competing architectures in lower-percentile rates at every tested AP density.

    Source: B. Contributions · p. 2, §VI · p. 13

  • Takeaways & Limitations

    Percentile optimization retains value even in a network with no cell edges, and UC-mMIMO provides the strongest lower-percentile service among the compared architectures.

    Source: §VI · p. 13

    At zero powers, the change of variables has a singular Jacobian, so directional stationarity in transformed variables need not coincide with that in the original powers.

    Source: D. Convergence · p. 8

Abstract

from arXiv · show

Improving the throughput of the weakest users is a central objective of wireless resource allocation, and the objective around which the user-centric cell-free massive multiple-input multiple-output (UC-mMIMO) architecture is built. This paper brings the two together by formulating and solving percentile programs for UC-mMIMO networks. We maximize the sum-least-qth-percentile (SLqP) rate via downlink power control, a utility that targets the user equipments (UEs) in the lowest q percentile of the rate distribution and subsumes max-min-rate and sum-rate maximization as special cases. The problem is non-convex, non-smooth and NP-hard. The cyclic maximization approach of one of the previous percentile optimization series does not carry over: the coherent joint transmission technique introduces cross terms in the powers that destroy the block-concavity on which it relies. We develop a change of variables that restores block-concavity, yielding an algorithm with closed-form auxiliary updates and a concave power update, and we give a self-contained proof of non-decreasing convergence to a directional stationary point. Simulations establish two facts: first, percentile optimization retains its value in an architecture with no cell edges; relative to the weighted minimum mean-square error benchmark, it raises the average rate of the lowest-10th-percentile UEs by 56--72% and Jain's fairness index from below 0.92 to above 0.99. Second, compared directly with massive MIMO and small cells under the same SLqP utility, UC-mMIMO delivers the highest lower-percentile rates at every access point density tested, by up to 62% and 2.7$\times$ respectively, at the same fairness, and its advantage widens with densification.

I. INTRODUCTION

The paper asks whether explicit lower-percentile rate allocation remains useful in user-centric cell-free networks, which are designed to favor weaker users. It combines percentile power control with UC-mMIMO and reports benefits against both WMMSE and alternative architectures.

I. INTRODUCTION

  • Fairness concerns how evenly resources serve UEs, particularly those in the lowest 5% or 10% of conventional-network rate distributions.The paper describes fairness as helping ensure no UE receives service substantially below QoS while others receive more than required.Source: §I · p. 1

A. Related Work and Research Gap

  • Percentile programs target the lowest q percentile of UEs, spanning max-min-rate and sum-rate extremes while balancing service to weaker users against aggregate utility.Optimizing a percentile differs from focusing on only the single worst UE, which can impose a large rate penalty on others.Source: A. Related Work and Research Gap · pp. 1–2, Figure 1 · p. 1
  • The research gap is whether percentile optimization still matters in UC-mMIMO, an architecture designed around weaker UEs and without conventional cell edges.The cited prior percentile-optimization studies address conventional cellular networks.Source: A. Related Work and Research Gap · p. 2

B. Contributions

  • A change of variables restores block-concavity for a cyclic maximization algorithm, with closed-form auxiliary updates and a concave power update; it is non-decreasing and every limit point is directionally stationary.Coherent joint transmission creates cross products that defeat the earlier approach; the SLqP utility is optimized directly, without smoothing, relaxation, or surrogate approximation.Source: B. Contributions · p. 2
  • 56–72%: percentile optimization raises the lowest-10th-percentile average rate over WMMSE and lifts Jain’s fairness index from 0.80–0.92 to above 0.99.A massive MIMO network under percentile optimization delivers about as much to its lowest-percentile UEs as UC-mMIMO under WMMSE, and considerably more at moderate densities.Source: B. Contributions · p. 2
  • UC-mMIMO delivers the highest lower-percentile rates at every tested AP density and cell size, with 8%–62% advantage over massive MIMO and 1.7×–2.7× over small cells at the same fairness.The comparison uses the same SLqP utility, AP and UE locations, and total power across architectures.Source: B. Contributions · p. 2

II. SYSTEM MODEL

The system model considers downlink UC-mMIMO with cooperating AP clusters serving UEs. The central unit allocates per-AP, per-UE transmit powers under AP power budgets, while MRT beamformers are fixed.

II. SYSTEM MODEL

  • The central unit performs resource allocation for APs connected by wired fronthaul, with M antennas per AP and one antenna per UE.The network comprises APs and active UEs indexed by b and k, respectively.Source: §II · p. 3
  • Each UE is served by a neighboring-AP cluster and each AP serves a corresponding UE set; cluster membership is independent of cellular structure.The cluster includes APs whose large-scale fading exceeds a threshold, together with the strongest AP to ensure a UE is served.Source: §II · p. 3
  • Per-AP, per-UE powers are the only design variables, subject to each AP’s total power budget; beamformers are fixed to maximum-ratio transmission.The model assumes perfect channel state information is available to all APs.Source: §II · p. 3
  • Coherent sums over serving clusters enter both desired-signal and interference terms, coupling transmissions across APs serving the same UE.The resulting rate expression includes coherent combination of cluster transmissions.Source: §II · p. 3

III. PROBLEM FORMULATION

The SLqP utility sums the lowest Kq values, with q setting how many entries are targeted. It is concave and continuous but has nondifferentiable boundaries where the selected entries change.

A. Preliminaries

  • SLqP is the sum of the Kq smallest components, where Kq is the smallest count whose fraction of K values reaches percentile q.For K = 6 and q = 100/3, Kq = 2.Source: A. Preliminaries · p. 3
  • The utility equals the minimum sum over all subsets of Kq entries, but enumerating those selection vectors grows combinatorially with K.The variational form characterizes SLqP; it is not the intended computational recipe.Source: A. Preliminaries · pp. 3–4
  • SLqP is concave, continuous, and non-decreasing in each argument, but is non-smooth when the Kqth and (Kq + 1)th smallest entries coincide.Away from those boundaries, its gradient is piecewise constant.Source: A. Preliminaries · p. 4
  • At boundary points, the gradient and Hessian do not exist, ruling out direct steepest-ascent and Newton methods without addressing the non-smoothness.Fig. 3 depicts the creases where the identity of the Kq smallest entries changes.Source: A. Preliminaries · p. 4, Figure 3 · p. 4
  • The utility’s concavity and monotonicity support the composition arguments, surrogate bounds, and epigraph reformulation used later.These constructions require a concave, non-decreasing outer aggregation function.Source: A. Preliminaries · p. 4
  • At q = 100, SLqP is sum utility; at q = 100/K, it is minimum utility, so the family includes sum-rate and max-min-rate objectives.Intermediate q values span the tradeoff between these extremes.Source: A. Preliminaries · p. 4

B. Problem Definition and Analysis

The section formulates downlink power allocation to maximize the SLqP rate in UC-mMIMO and identifies difficulties from percentile utility and coherent joint transmission. It then introduces equivalent transformations for cyclic maximization.

B. Problem Definition and Analysis

  • Power allocation is formulated to maximize the sum of the Kq lowest UE rates, subject to nonnegative powers and per-AP power budgets.The choice of q sets the targeted percentile; q = 10, for example, optimizes 10th-percentile throughput.Source: B. Problem Definition and Analysis · p. 4
  • The problem is non-concave and non-smooth, and smoothing with one constraint per candidate active set introduces intractable non-convex constraints.The non-smoothness occurs where the identity of the Kq worst UEs changes.Source: B. Problem Definition and Analysis · p. 4
  • Coherent transmission couples powers across APs serving the same UE through cross products, which defeat the direct application of the earlier percentile approach.Because SLqP is global, local channel knowledge alone cannot determine which UEs belong to the targeted percentile set; the paper assumes central coordination.Source: B. Problem Definition and Analysis · pp. 4–5

IV. PROPOSED APPROACH

  • The approach converts the problem to an equivalent block-concave form using the QFT, a power-variable change, and an epigraph formulation, while optimizing SLqP directly.The transformations avoid smoothing, relaxation, or replacing the utility with a proxy.Source: §IV · p. 5

A. The Quadratic Fractional Transform

  • The QFT introduces an auxiliary variable per UE to transform each SINR before aggregating the rates, without changing the optimal powers or objective value.The auxiliary objective lower-bounds the original ratio and is tight at its maximizing auxiliary variable.Source: A. The Quadratic Fractional Transform · p. 5

B. Reformulating the SLqP Problem

The QFT introduces per-UE auxiliary variables to transform the SINRs before applying the percentile objective. The transformed problem preserves the original optimal value and powers.

B. Reformulating the SLqP Problem

  • Applying the QFT to each UE’s SINR introduces one auxiliary variable per UE before the transformed rates are aggregated.The auxiliary variables are collected in X.Source: B. Reformulating the SLqP Problem · p. 5
  • The transformed problem is equivalent to the original: the auxiliary variables enlarge the search space but do not alter the optimal objective value or powers.With powers fixed, the auxiliary variables have an optimal choice given by the QFT update.Source: B. Reformulating the SLqP Problem · p. 5
  • Each auxiliary variable is a per-UE weight reflecting the ratio of desired signal amplitude to experienced interference-plus-noise power.The optimal auxiliary value depends on all UEs’ powers, coupling the network-wide objective into the algorithm.Source: B. Reformulating the SLqP Problem · p. 5

1) Where the approach of [12] breaks down:

Coherent joint transmission creates cross terms that break the power-block concavity needed by the earlier approach. Reparameterizing powers as transmit amplitudes restores concavity while preserving the original feasible problem.

1) Where the approach of [12] breaks down:

  • Cross products between powers at distinct APs serving the same UE make the interference term neither convex nor concave, breaking the direct approach of.The cross products arise from coherent joint transmission and are absent when each UE is served by a single node.Source: 1) Where the approach of [12] breaks… · p. 6

2) Restoring concavity in the power block:

  • The bijection θk,b = √pk,b reparameterizes power as transmit amplitude without losing equivalence, and makes the desired-signal term linear in the transformed variables.The interference term becomes the squared modulus of an affine function of the amplitudes.Source: 2) Restoring concavity in the power… · p. 6
  • The transformed per-AP budget remains a convex quadratic constraint, and replacing it with a tighter linear constraint would shrink the feasible set and break equivalence.The tighter constraint applies when an AP serves more than one UE.Source: 2) Restoring concavity in the power… · p. 6
  • With auxiliary variables fixed, the transformed objective is concave in the amplitude variables, enabling a concave power update.The proof uses concavity of the transformed rates and the SLqP utility.Source: 2) Restoring concavity in the power… · p. 6

3) Epigraph form for a disciplined convex solver:

The epigraph reformulation gives a solver-ready concave maximization for fixed auxiliary variables while preserving the SLqP optimum. Its rate constraints handle the percentile structure without enumerating candidate UE selections.

3) Epigraph form for a disciplined convex solver:

  • Exponentiating the rate constraints makes the left side concave in Θ and the right side convex in each slack variable, giving a valid convex restriction.The slacks represent per-UE rate bounds.Source: 3) Epigraph form for a disciplined… · p. 6
  • The slack-variable rate constraints yield a concave maximization whose optimum matches the original problem and can be solved with CVX or subgradients.The rate constraints are tight at an optimum, and the solver handles the SLqP combinatorial structure implicitly rather than enumerating selection vectors.Source: 3) Epigraph form for a disciplined… · p. 6
  • The smoothed reformulation would require searching over active sets, creating the combinatorial explosion that made that approach intractable.Here the active set is revealed by optimizing the concave atom directly.Source: 3) Epigraph form for a disciplined… · p. 7

C. The Proposed Algorithm

The proposed algorithm alternates a closed-form auxiliary update with a concave power update, exploiting block-concavity. Its power allocation also refines serving clusters in response to interference and the lowest-rate users.

C. The Proposed Algorithm

  • The algorithm alternates a closed-form auxiliary update with a power update that maximizes a concave objective for fixed auxiliary variables.The auxiliary update restores equality between the auxiliary and original objectives before powers are reallocated.Source: C. The Proposed Algorithm · p. 7
  • Equal power initialization is feasible, requires no channel knowledge, and favors no UE a priori; iterations stop at tolerance or the cap using true achievable utility.The auxiliary variables are updated before powers, keeping the auxiliary objective tight at the start of each power update.Source: C. The Proposed Algorithm · p. 7
  • Links to UEs above the lowest Kq rates can receive negligible power, implicitly refining serving clusters while penalizing reductions if a UE enters that lowest-rate set.Such links contribute power only through interference to UEs that affect the objective.Source: C. The Proposed Algorithm · p. 7

D. Convergence

The algorithm produces a non-decreasing, convergent objective sequence, and every limit point is directionally stationary in the amplitude variables. The analysis also specifies when that guarantee transfers to the original power variables.

D. Convergence

  • The objective is non-decreasing after each iteration and converges because the per-AP power budget bounds the rates and objective from above.Each power update maximizes the concave subproblem for the current auxiliary variables.Source: D. Convergence · p. 7
  • Directional stationarity is used because the SLqP objective is non-differentiable, and it is stronger than the smooth weighted-sum-rate stationarity guarantees of prior FP and WMMSE work.The power update maximizes a surrogate built from the current iterate.Source: D. Convergence · p. 7
  • Every limit point is directionally stationary for the SLqP problem in amplitude variables; for the NP-hard problem, the authors describe this as the best polynomial-time guarantee.The theorem rules out stalling at an arbitrary point.Source: D. Convergence · p. 8
  • At strictly positive powers, the amplitude-variable guarantee transfers to the original power variables, but the notions need not coincide when powers vanish.The mapping is a diffeomorphism on the open positive orthant and has a singular Jacobian at zero powers.Source: D. Convergence · p. 8

E. Percentile Optimization for the Benchmark Architectures

The benchmark comparison optimizes the same SLqP utility across UC-mMIMO, massive MIMO, and small cells. In the single-node benchmarks, the power-domain problem already has the block-concavity needed for cyclic maximization.

E. Percentile Optimization for the Benchmark Architectures

  • All three architectures optimize the same SLqP utility, isolating architecture in the comparison; both single-node benchmarks reduce to the QFT algorithm of [12].The benchmark reduction also identifies which part of the development is specific to the cell-free setting.Source: E. Percentile Optimization for the… · p. 8
  • Massive MIMO and small cells serve each UE from one node, so their cluster sums reduce to single terms and their rates use per-node power budgets.The serving node is the macro BS or the AP whose Voronoi region contains the UE, respectively.Source: E. Percentile Optimization for the… · p. 8
  • For single-node benchmarks, the auxiliary objective is already concave in powers, so the change of variables is unnecessary and cyclic maximization applies directly.The auxiliary variables have closed-form updates, followed by a concave power program; the final squaring step is omitted.Source: E. Percentile Optimization for the… · p. 8

F. Computational Complexity and Fronthaul Load

The section bounds Algorithm 1’s computational cost and fronthaul traffic, then specifies the deployment and comparison setup used in simulations.

1) Computational complexity:

  • Algorithm 1’s generic cost is O(KNM) + O(I (N + K)^3.5); bounded cluster sizes make N scale linearly with UE count, and the algorithm reaches over 95% of final SLqP rate within five iterations.The power update dominates; its generic interior-point bound is for a sparse concave program, and a solver exploiting sparsity may perform better.Source: 1) Computational complexity · p. 9

2) Fronthaul load:

  • Uplink fronthaul carries KN complex scalars per coherence block rather than O(KBM) raw-channel scalars, while the downlink returns N real power values.Fronthaul is used once per coherence block, not per iteration; decentralized percentile optimization remains open because an AP cannot determine active-set membership from local information alone.Source: 2) Fronthaul load · p. 9

A. Simulation Setup

  • Simulations use 100 UEs in a 1 × 1 km2 area, with 75-AP and 150-AP deployments, wraparound, and q = 10.Serving clusters use a 100 m distance threshold; channel gains include path loss and log-normal shadowing.Source: A. Simulation Setup · p. 9
  • Massive MIMO co-locates each virtual cell’s AP antennas at a macro base station with the combined AP power budget, while small cells use independent Voronoi-region service without cooperation.Both architectures use the same AP and UE locations as UC-mMIMO; allocation comparisons use WMMSE, while architecture comparisons apply the same SLqP utility and average over independent network realizations.Source: A. Simulation Setup · pp. 9–10

B. Convergence

The results show that percentile optimization improves lower-percentile rates across architectures, while UC-mMIMO leads the architecture comparison and gains most from densification. Its lowest-percentile users are distributed across the area, and convergence is monotonic and fast.

B. Convergence

  • The SLqP objective increases monotonically from equal-power initialization, with more than 95% of final rate reached within five iterations in all cases.UC-mMIMO takes roughly ten iterations to come within 1% of its final UDD value, compared with three to five iterations for the single-node benchmarks.Source: B. Convergence · p. 10

C. Value of Percentile Optimization in Cell-Free Networks

  • Percentile optimization raises UC-mMIMO’s lowest-decile average rate over WMMSE by 72% at 75 APs/km2 and 56% at 250 APs/km2, with gains of the same order in the benchmark architectures.In the MDD and UDD, the median percentile-UE rate rises from about 25 to 45 Mbps and from about 50 to 81 Mbps, respectively.Source: C. Value of Percentile Optimization in… · p. 11
  • Under percentile optimization, more than 80% of UC-mMIMO’s UDD percentile UEs fall between roughly 77 and 90 Mbps, as the SLqP objective transfers power to UEs below the Kth order statistic.WMMSE leaves low-percentile rates spread over a wider range; the lowest tail includes UEs with poor channels to every AP in their cluster.Source: C. Value of Percentile Optimization in… · p. 11

D. Comparison of the Network Architectures

  • UC-mMIMO’s network SLqP rate exceeds massive MIMO by 9% in the MDD and 44% in the UDD, and small cells by 65% and 131%, respectively.Median low-percentile UE rates are 45, 40, and 28 Mbps in the MDD and 81, 55, and 36 Mbps in the UDD for UC-mMIMO, massive MIMO, and small cells, respectively.Source: D. Comparison of the Network… · p. 11
  • As density rises from 75 to 250 APs/km2, percentile-UE average rates grow by factors of 2.6 for UC-mMIMO, 1.7 for massive MIMO, and 1.6 for small cells.UC-mMIMO’s advantage over massive MIMO widens from 8% to 62%, and over small cells from 1.7× to 2.7×; added APs enlarge UE clusters and contribute coherently.Source: D. Comparison of the Network… · pp. 11–12
  • With virtual-cell sides up to 800 m, rates remain essentially flat; beyond 800 m, rates fall, while UC-mMIMO stays ahead by 42–54% over massive MIMO and 2.3–3× over small cells.The network shifts from interference-limited to noise-limited beyond 800 m, and UC-mMIMO’s advantage is largest at 2000 m.Source: D. Comparison of the Network… · p. 12
  • In the UDD, lowest-decile UEs occur throughout the area, with rates around 80–90 Mbps for UC-mMIMO, 50–65 Mbps for massive MIMO, and 25–45 Mbps for small cells.The rate dips are isolated rather than spatially contiguous coverage holes.Source: D. Comparison of the Network… · pp. 12–13
  • At a 70 Mbps requirement, roughly one in ten UC-mMIMO lowest-decile UEs are in outage, versus all such UEs in massive MIMO and small cells; the implied UC-mMIMO network outage is about 1%.At 40 Mbps, only a handful of the percentile UEs in UC-mMIMO and massive MIMO are in outage, versus roughly half in small cells.Source: D. Comparison of the Network… · p. 13

E. User Fairness

The section reports how percentile optimization affects fairness across architectures and summarizes the paper’s UC-mMIMO contribution, algorithm, and open problems.

E. User Fairness

  • Percentile optimization yields Jain’s index of 99%–99.8% across architectures at every AP density, while WMMSE ranges from 62.2% to 89.8% at 75 APs/km2.Under WMMSE, UC-mMIMO is less fair than massive MIMO at every density and cell size; under percentile optimization, architectures have indistinguishable fairness despite UC-mMIMO’s higher rates.Source: E. User Fairness · p. 13

VI. CONCLUSION

  • The paper formulates downlink power control to maximize the sum-least-qth-percentile rate in UC-mMIMO, combining lower-percentile targeting with an architecture designed around fairness.The simulations report that percentile optimization improves lowest-10th-percentile rates by 56–72% over WMMSE and raises Jain’s index above 0.99.Source: §VI · p. 13
  • Changing variables to transmit amplitudes restores block-concavity after coherent joint transmission introduces power cross-products, enabling closed-form auxiliary updates and a concave power update.The algorithm is non-decreasing in SLqP rate and converges to a directional stationary point; in single-node architectures it reduces to the algorithm of [12].Source: §VI · p. 13
  • 56–72% higher average rates for the lowest-10th-percentile UEs than WMMSE, with Jain’s fairness index above 0.99; UC-mMIMO also leads lower-percentile rates across tested architectures at the same fairness.Its advantage over massive MIMO grows from 8% to 62% as AP density rises from 75 to 250 APs/km2.Source: §VI · p. 13
  • Decentralized percentile optimization and joint optimization of beamforming and power in cell-free networks remain open problems.Source: §VI · p. 13

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