Ferkans — Interactive Telecom Tutor

Fairness as the Design Objective

Spectral efficiency comparisons based on average or sum rates can be misleading: a system where 90% of users get 10 bits/s/Hz but 10% get 0.1 bits/s/Hz has a high average rate but unacceptable service for one in ten users. The cell-free massive MIMO literature uses the 95%-likely per-user rate as the primary performance metric precisely because it captures what matters in practice: the experience of the worst-served users. This section formalizes this metric and shows how to optimize it.

Definition:
95%-Likely Per-User Rate

Let $R_k$ denote the achievable rate of user $k$ for a given random deployment of users over the coverage area $\mathcal{A}$ . The CDF of the per-user rate is

$F_R(r) = \Pr\!\left[R_k \leq r\right] = \frac{1}{K} \sum_{k=1}^{K} \mathbf{1}(R_k \leq r)$

averaged over random user locations. The 95%-likely per-user rate is

$R_{95\%} = F_R^{-1}(0.05) = \sup\{r : F_R(r) \leq 0.05\}$

i.e., the rate that 95% of users achieve or exceed. This is the primary fairness metric in cell-free massive MIMO: a system with high $R_{95\%}$ provides good service to (almost) everyone.

The complementary metric, the 5th percentile of the rate CDF, is standard in 3GPP system-level evaluations and is used by operators to define "coverage" — a cell is "covered" if $R_{95\%}$ exceeds a minimum threshold.

Theorem: Max-Min Fair Power Control

The max-min fair power control problem for cell-free massive MIMO downlink is

$\max_{\{\eta_{lk}\}} \; \min_{k = 1, \ldots, K} \; \text{SE}_k^{\text{dl}}(\{\eta_{lk}\})$

subject to per-AP power constraints $\sum_{k=1}^{K} \eta_{lk} \gamma_{lk} N \leq {P_t}_{l}$ for all $l = 1, \ldots, L$ .

This is a quasi-convex optimization problem that can be solved via bisection: for a target $\text{SE}^* = t$ , check feasibility of

$\text{SINR}_k^{\text{dl}}(\{\eta_{lk}\}) \geq 2^{t \cdot \tau_c / (\tau_c - \tau_p)} - 1, \quad \forall k$

which is a second-order cone program (SOCP) for each fixed $t$ . The bisection over $t$ converges in $O(\log(1/\epsilon))$ iterations.

Max-min fairness maximizes the rate of the worst-off user. The bisection approach converts a non-convex objective (max-min of a fractional function) into a sequence of convex feasibility problems. The SOCP structure arises because the SINR constraint, after squaring the coherent combining term, becomes a second-order cone constraint.

Proof

Quasi-convexity

For fixed $t$ , the set $\{\eta_{lk} : \text{SE}_k \geq t, \; \forall k\}$ is convex (intersection of SINR constraints, each of which is a generalized inequality). This makes the max-min problem quasi-convex.

SOCP formulation

The SINR constraint $\text{SINR}_k \geq \gamma^*$ with the cell-free SINR structure can be written as a second-order cone constraint: $\|\mathbf{u}_k\|_2 \leq \sum_l \sqrt{\eta_{lk}} \gamma_{lk} / \sqrt{\gamma^*}$ where $\mathbf{u}_k$ stacks the interference and noise terms.

Bisection

Binary search over $t \in [0, t_{\max}]$ : if the SOCP is feasible at $t$ , increase $t$ ; otherwise decrease it. Convergence to $\epsilon$ -optimality in $\lceil \log_2(t_{\max}/\epsilon) \rceil$ SOCP solves. $\blacksquare$

,

Max-Min Fair Power Control via Bisection

Complexity:

O(\log(1/\epsilon))

SOCP solves, each of complexity

O((L \cdot K)^{3.5})

Input: Large-scale fading

\{\beta_{lk}\}

, estimation quality

\{\gamma_{lk}\}

,

per-AP power

\{{P_t}_{l}\}

, tolerance

\epsilon > 0

Output: Power control coefficients

\{\eta_{lk}^*\}

, max-min SE

t^*

1. Set

t_{\text{low}} \leftarrow 0

,

t_{\text{high}} \leftarrow \log_2(1 + L \cdot N \cdot \text{SNR})

2. while

t_{\text{high}} - t_{\text{low}} > \epsilon

do

3.

\quad t \leftarrow (t_{\text{low}} + t_{\text{high}}) / 2

4.

\quad \gamma^* \leftarrow 2^{t \cdot \tau_c / (\tau_c - \tau_p)} - 1

5.

\quad

Solve SOCP: find

\{\eta_{lk}\}

s.t.

\text{SINR}_k \geq \gamma^*

\forall k

,

\sum_k \eta_{lk} \gamma_{lk} N \leq {P_t}_{l}

\forall l

6.

\quad

if feasible then

t_{\text{low}} \leftarrow t

, store

\{\eta_{lk}^*\} \leftarrow \{\eta_{lk}\}

7.

\quad

else

t_{\text{high}} \leftarrow t

8. end while

9. return

\{\eta_{lk}^*\}

,

t^* \leftarrow t_{\text{low}}

For large systems ( $L \cdot K > 10^3$ ), the SOCP can be replaced by a fixed-point iteration based on the standard interference function framework, reducing per-iteration cost to $O(L \cdot K)$ .

,

95%-Likely Rate vs Number of Users

Compare the 95%-likely per-user rate across architectures as the number of users increases. Observe how cell-free maintains fairness even under heavy load, while small cells and co-located systems degrade rapidly.

Parameters

L

(APs)64

K_{\max}

(max users)40

SNR (dB)10

Power control

Example: Quantifying the CDF Improvement

From the SINR CDF comparison (see interactive plot in Section 15.2), extract the following metrics for cell-free vs small cells with $L = 64$ , $K = 20$ , SNR = 10 dB: (a) the 5th-percentile SINR improvement, (b) the median SINR change, (c) the 95th-percentile SINR change.

Solution

5th percentile (cell-edge)

Cell-free: $\text{SINR}_{5\%} \approx 5$ dB. Small cells: $\text{SINR}_{5\%} \approx -5$ dB. Improvement: $\sim 10$ dB. This is the most dramatic gain — cell-free virtually eliminates the cell-edge problem.

Median

Cell-free: $\text{SINR}_{50\%} \approx 12$ dB. Small cells: $\text{SINR}_{50\%} \approx 8$ dB. Improvement: $\sim 4$ dB. Median users benefit but less dramatically.

95th percentile (cell-center)

Cell-free: $\text{SINR}_{95\%} \approx 18$ dB. Small cells: $\text{SINR}_{95\%} \approx 20$ dB. Cell-free is actually $\sim 2$ dB worse for the best users, because power is redistributed from cell-center to cell-edge users via max-min fair power control. This is the price of fairness.

Key Takeaway

The 95%-likely per-user rate is the defining performance metric for cell-free massive MIMO. Under max-min fair power control, cell-free achieves 5-10 times higher 95%-likely rate than small cells with the same total antenna count, by converting cell-edge interference into useful signal through coherent combining from distributed APs.

Definition:
Proportional Fairness

An alternative to max-min fairness, proportional fairness maximizes the sum of log-rates:

$\max_{\{\eta_{lk}\}} \; \sum_{k=1}^{K} \log(R_k)$

subject to the same per-AP power constraints. This balances efficiency and fairness: it allocates more resources to users with poor channels (diminishing marginal utility of $\log$ ) but does not sacrifice as much peak rate as max-min fairness. The solution satisfies the Nash bargaining axioms and is the utility function used in most 4G/5G schedulers.

In practice, proportional fairness often achieves 50-70% of the max-min 95%-likely rate while providing 20-40% higher sum-rate. The choice between max-min and proportional fairness depends on the operator's service-level agreement.

Quick Check

Under max-min fair power control in cell-free massive MIMO, what happens to the sum-rate compared to equal power allocation?

Sum-rate always increases because fairness helps everyone

Sum-rate may decrease, but the min-rate increases

Sum-rate is unchanged because total power is the same

Sum-rate always decreases to exactly half

Correction:

Sum-rate may decrease, but the min-rate increases

Max-min fairness explicitly maximizes the worst user's rate, often at the expense of reducing the best users' rates. The sum-rate may decrease by 10-30%, but the 5th-percentile rate can increase by 5-10 times.

Common Mistake: Using Average Rate Instead of 5th-Percentile Rate

Mistake:

Evaluating cell-free massive MIMO by average per-user rate and concluding it provides only modest gains over small cells (e.g., 2 dB average SINR improvement).

Correction:

The average rate hides the dramatic improvement at the cell edge. The 5th-percentile rate — the metric that matters for coverage — shows 10+ dB improvement. Always report the full CDF or at minimum the 5th, 50th, and 95th percentiles when comparing architectures.

Historical Note: Max-Min Fairness: From Networking to Wireless

1992-2017

Max-min fairness originated in wireline networking (Bertsekas and Gallager, 1992), where it was used for bandwidth allocation in packet-switched networks. The concept was adapted to wireless resource allocation by Rashid-Farrokhi, Liu, and Tassiulas (1998), who showed that max-min SINR balancing in CDMA networks can be solved via eigenvalue methods. The application to cell-free massive MIMO by Ngo et al. (2017) leveraged the particular SINR structure to obtain efficient SOCP formulations. The max-min criterion has since become the standard benchmark for fairness in cell-free systems, though proportional fairness and $\alpha$ -fairness generalizations are increasingly studied.

Example: Max-Min Power Control for a 3-AP, 2-User System

Consider $L = 3$ APs and $K = 2$ users with channel estimation qualities $\gamma_{lk}$ given by the matrix $\boldsymbol{\Gamma} = \begin{bmatrix} 0.8 & 0.2 \\ 0.3 & 0.7 \\ 0.5 & 0.5 \end{bmatrix}$ (rows = APs, columns = users). All APs have equal power $P_t = 1$ . Ignoring noise, find the max-min fair power allocation $\eta_{lk}$ .

Solution

SINR expressions

With $N = 1$ , $\text{SINR}_1 = \frac{(\sum_l \sqrt{\eta_{l1}} \gamma_{l1})^2}{\sum_l \eta_{l2} \gamma_{l1} \gamma_{l2}}$ and $\text{SINR}_2 = \frac{(\sum_l \sqrt{\eta_{l2}} \gamma_{l2})^2}{\sum_l \eta_{l1} \gamma_{l2} \gamma_{l1}}$ .

Equal-rate condition

Max-min fairness requires $\text{SINR}_1 = \text{SINR}_2 = \gamma^*$ at optimality. By symmetry of the problem (user 1 and user 2 have "mirror" channel qualities), setting $\eta_{l1} = \eta_{l2}$ for all $l$ equalizes the rates.

Power constraint

The per-AP constraint is $\eta_{l1} \gamma_{l1} + \eta_{l2} \gamma_{l2} \leq 1$ . With equal power allocation, $\eta_l (\gamma_{l1} + \gamma_{l2}) \leq 1$ , giving $\eta_l \leq 1/(\gamma_{l1} + \gamma_{l2})$ : $\eta_1 \leq 1$ , $\eta_2 \leq 1$ , $\eta_3 \leq 1$ . Setting all to equality maximizes the SINR.

Resulting SINR

$\text{SINR}_1 = \frac{(0.8 + 0.3 + 0.5)^2}{0.2 \cdot 0.8 + 0.7 \cdot 0.3 + 0.5 \cdot 0.5} = \frac{2.56}{0.62} \approx 4.13$ (6.2 dB). Both users achieve the same rate: $R = \log_2(1 + 4.13) \approx 2.36$ bits/s/Hz.

Why This Matters: Coverage Probability in 5G Network Planning

The 95%-likely rate directly maps to the coverage probability used in 5G NR network planning. 3GPP defines a cell as providing adequate coverage if the 5th-percentile user throughput exceeds a minimum threshold (e.g., 100 Mbps for eMBB). Cell-free massive MIMO achieves this threshold at significantly lower AP density than conventional small cells, because the coherent combining gain shifts the entire rate CDF to the right — especially the critical lower tail.

Max-Min Fairness

A resource allocation policy that maximizes the minimum rate across all users. Achieves the most egalitarian outcome: no user's rate can be improved without reducing another user's rate below the current minimum. Solved via bisection over SOCP feasibility problems.

Cumulative Distribution Function (CDF)

$F_R(r) = \Pr[R \leq r]$ , the probability that a randomly located user achieves rate at most $r$ . In cell-free evaluations, the CDF is computed empirically over many random user drops and captures both cell-center and cell-edge performance.

Coverage Uniformity as the Ultimate Goal

The vision of cell-free massive MIMO can be summarized in one sentence: every user should experience the same quality of service regardless of location. The 95%-likely rate approaching the median rate is the quantitative signature of this vision. In a perfectly uniform system, the CDF would be a step function — all users achieve exactly the same rate. While this ideal is unachievable (users at different distances from APs will always have different path losses), cell-free with max-min power control comes remarkably close: the ratio of the 95th percentile to the 5th percentile rate is typically 3-5x, compared to 50-100x in conventional cellular networks.

Fairness and Coverage