Proportional Fairness

The Fairness–Throughput Tradeoff

Max-min fairness gives the strongest guarantee to the weakest user, but as we saw in Section 5.1, this can come at a steep cost in total throughput. Proportional fairness offers a middle ground: it maximizes the geometric mean of the rates, which penalizes extreme imbalance without sacrificing as much aggregate throughput. The result is an allocation where improving any user's rate by a relative amount must come at a larger relative cost to the other users — the Nash bargaining solution from cooperative game theory.

Definition:

Proportional Fairness

An allocation p⋆\mathbf{p}^\star is proportionally fair if it solves

max⁡p≥0  ∑k=1Klog⁡Rk(p)subject to∑k=1Kpk≤Pt.\max_{\mathbf{p} \geq \mathbf{0}} \; \sum_{k=1}^{K} \log R_k(\mathbf{p}) \quad \text{subject to} \quad \sum_{k=1}^{K} p_k \leq P_t.

Equivalently, p⋆\mathbf{p}^\star maximizes the product ∏k=1KRk(p)\prod_{k=1}^{K} R_k(\mathbf{p}), i.e., the geometric mean of the rates.

The name "proportional fairness" comes from the first-order optimality condition: at p⋆\mathbf{p}^\star, any feasible perturbation δp\delta \mathbf{p} satisfies ∑kδRkRk⋆≤0\sum_k \frac{\delta R_k}{R_k^\star} \leq 0. That is, the sum of relative rate changes is non-positive — you cannot improve all users proportionally.

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Proportional Fairness

A resource allocation that maximizes the sum of log-rates ∑klog⁡Rk\sum_k \log R_k. It balances fairness and throughput by ensuring that no feasible reallocation can improve the relative rate of one user without reducing the relative rate of another by a larger amount.

Related: Max-Min Fair Power Control, Nash Bargaining Solution

Definition:

Nash Bargaining Solution

In cooperative game theory, the Nash bargaining solution for KK players with utility functions uku_k and disagreement point d=(d1,…,dK)\mathbf{d} = (d_1, \ldots, d_{K}) is

u⋆=arg⁡max⁡u∈U  ∏k=1K(uk−dk)\mathbf{u}^\star = \arg\max_{\mathbf{u} \in \mathcal{U}} \; \prod_{k=1}^{K} (u_k - d_k)

where U\mathcal{U} is the feasible utility region. When dk=0d_k = 0 and uk=Rku_k = R_k, this coincides with proportional fairness.

The Nash bargaining solution satisfies four axioms: Pareto optimality, symmetry, independence of irrelevant alternatives, and invariance to affine transformations of utility. These axioms uniquely characterize the product-maximization form.

Nash Bargaining Solution

The unique outcome of a cooperative bargaining game that satisfies Pareto optimality, symmetry, scale invariance, and independence of irrelevant alternatives. Equivalent to maximizing the product of utilities above the disagreement point.

Related: Proportional Fairness

Historical Note: Nash's Bargaining Problem and Wireless Resource Allocation

1950–2005

John Nash introduced the bargaining problem in his 1950 paper "The Bargaining Problem," establishing the axiomatic framework for cooperative game theory. Nearly five decades later, Frank Kelly (1998) recognized that proportional fairness in network bandwidth allocation is precisely the Nash bargaining solution with zero disagreement point. This connection was brought into the wireless domain by Tse and Viswanath, who showed that proportional fair scheduling in fading channels exploits multiuser diversity — serving each user when their channel is strong relative to their own average.

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Theorem: Proportional Fairness as a Geometric Program

Under the generic SINR model from Definition 5.1, the proportional fairness problem

max⁡p≥0  ∑k=1Klog⁡ ⁣log⁡2 ⁣(1+pkak∑l≠kplγkl+σ2bk)s.t.∑kpk≤Pt\max_{\mathbf{p} \geq \mathbf{0}} \; \sum_{k=1}^{K} \log\!\log_2\!\left(1 + \frac{p_k a_k}{\sum_{l \neq k} p_l \gamma_{kl} + \sigma^2 b_k}\right) \quad \text{s.t.} \quad \sum_k p_k \leq P_t

is not convex in general, but can be reformulated as a geometric program (GP) through the change of variables qk=1/pkq_k = 1/p_k. In the high-SINR regime where log⁡2(1+SINRk)≈log⁡2(SINRk)\log_2(1 + \text{SINR}_k) \approx \log_2(\text{SINR}_k), the problem reduces to

max⁡p>0  ∑k=1Klog⁡ ⁣(pkak∑l≠kplγkl+σ2bk)s.t.∑kpk≤Pt\max_{\mathbf{p} > \mathbf{0}} \; \sum_{k=1}^{K} \log\!\left(\frac{p_k a_k}{\sum_{l \neq k} p_l \gamma_{kl} + \sigma^2 b_k}\right) \quad \text{s.t.} \quad \sum_k p_k \leq P_t

which is a standard GP in posynomial form and can be solved in polynomial time using interior-point methods.

The high-SINR approximation log⁡(1+x)≈log⁡(x)\log(1 + x) \approx \log(x) removes the constant "1" that breaks the GP structure. The resulting objective is a sum of log-ratios of posynomials, which is the canonical GP form. Even away from high SINR, successive GP approximations converge to the true optimum.

Proof
High-SINR approximation

When SINRk≫1\text{SINR}_k \gg 1 for all kk, we have log⁡(1+SINRk)≈log⁡(SINRk)\log(1 + \text{SINR}_k) \approx \log(\text{SINR}_k). The objective becomes ∑klog⁡(pkak)−∑klog⁡(∑l≠kplγkl+σ2bk)\sum_k \log(p_k a_k) - \sum_k \log(\sum_{l \neq k} p_l \gamma_{kl} + \sigma^2 b_k).

Identify the GP structure

A geometric program maximizes a ratio of posynomials in log form. The numerator pkakp_k a_k is a monomial. The denominator ∑l≠kplγkl+σ2bk\sum_{l \neq k} p_l \gamma_{kl} + \sigma^2 b_k is a posynomial. The power constraint ∑kpk≤Pt\sum_k p_k \leq P_t is a posynomial inequality. This matches the standard GP form.

Solve via interior-point methods

Standard GP solvers (e.g., CVXPY with gp=True) convert the problem to a convex form via the substitution p~k=log⁡pk\tilde{p}_k = \log p_k and solve in O(K3.5)O(K^{3.5}) time using barrier methods. ■\blacksquare

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Example: Proportional Fair vs. Max-Min Fair Allocation

Consider a 3-user system with Nt=100N_t = 100, path losses β1=0.1\beta_{1} = 0.1, β2=0.01\beta_{2} = 0.01, β3=0.001\beta_{3} = 0.001, total power Pt=1P_t = 1, and σ2=1\sigma^2 = 1. Using MRC with perfect CSI, compare the rate vectors under max-min fairness and proportional fairness.

Solution
Max-min fairness result

Under max-min fairness, the algorithm allocates most power to user 3 (worst channel). The resulting rates are approximately R1≈R2≈R3≈0.9R_1 \approx R_2 \approx R_3 \approx 0.9 bits/s/Hz, with sum rate Rsum≈2.7R_{\text{sum}} \approx 2.7 bits/s/Hz.

Proportional fairness result

Under proportional fairness, the allocation is less extreme. User 3 still receives extra power, but users 1 and 2 retain more of their natural advantage. Typical result: R1≈3.1R_1 \approx 3.1, R2≈1.4R_2 \approx 1.4, R3≈0.5R_3 \approx 0.5 bits/s/Hz, with sum rate Rsum≈5.0R_{\text{sum}} \approx 5.0 bits/s/Hz.

Interpretation

Proportional fairness nearly doubles the sum rate compared to max-min fairness, at the cost of the weakest user losing about half its rate (from 0.9 to 0.5). The geometric mean of rates, however, is higher under proportional fairness, reflecting the better throughput-fairness balance.

Rate CDF Under Different Fairness Criteria

Compare the cumulative distribution function (CDF) of per-user rates under max-min fairness, proportional fairness, and sum-rate maximization. The CDF is computed over random user deployments in a cell.

Parameters
8
64

Definition:

α\alpha-Fair Utility

The α\alpha-fairness framework unifies max-min and proportional fairness through a single parameter α≥0\alpha \geq 0:

Uα(R)={∑k=1KRk1−α1−αα≠1∑k=1Klog⁡Rkα=1U_\alpha(\mathbf{R}) = \begin{cases} \sum_{k=1}^{K} \frac{R_k^{1-\alpha}}{1-\alpha} & \alpha \neq 1 \\ \sum_{k=1}^{K} \log R_k & \alpha = 1 \end{cases}

Special cases: α=0\alpha = 0 gives sum-rate maximization, α=1\alpha = 1 gives proportional fairness, and α→∞\alpha \to \infty gives max-min fairness.

The parameter α\alpha controls the "degree of fairness" — higher α\alpha means more weight on the weakest user. System operators can tune α\alpha to match their service-level agreement: mobile broadband typically uses α∈[0.5,1.5]\alpha \in [0.5, 1.5].

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α\alpha-Fairness

A parameterized family of utility functions that interpolates between sum-rate maximization (α=0\alpha = 0), proportional fairness (α=1\alpha = 1), and max-min fairness (α→∞\alpha \to \infty).

Related: Proportional Fairness, Max-Min Fair Power Control

Max-Min vs. Sum-Rate Tradeoff

Explore the Pareto frontier between minimum user rate and total sum rate as the fairness parameter α\alpha varies. Observe how the system trades off aggregate throughput for fairness to the weakest user.

Parameters
4
64
15

Common Mistake: The Double Logarithm Trap in Proportional Fairness

Mistake:

When maximizing ∑klog⁡Rk\sum_k \log R_k where Rk=log⁡2(1+SINRk)R_k = \log_2(1 + \text{SINR}_k), it is tempting to simplify log⁡(log⁡2(1+x))\log(\log_2(1 + x)) into a cleaner form. Some authors mistakenly drop the outer logarithm and maximize ∑kRk\sum_k R_k instead, which is sum-rate maximization — a fundamentally different problem.

Correction:

The correct proportional fairness objective is ∑klog⁡Rk=∑klog⁡log⁡2(1+SINRk)\sum_k \log R_k = \sum_k \log\log_2(1 + \text{SINR}_k), not ∑klog⁡(1+SINRk)\sum_k \log(1 + \text{SINR}_k). The double logarithm is essential: the outer log⁡\log creates the diminishing returns that penalize rate imbalance. Dropping it removes the fairness property entirely.

Comparison of Fairness Criteria

PropertyMax-MinProportionalSum-Rate
Objectivemax⁡min⁡kRk\max \min_k R_kmax⁡∑klog⁡Rk\max \sum_k \log R_kmax⁡∑kRk\max \sum_k R_k
α\alpha-fairnessalphatoinfty\\alpha \\to \\inftyalpha=1\\alpha = 1alpha=0\\alpha = 0
FavorsWeakest userBalanced allocationStrongest user
Sum-rate lossHighModerateNone (by definition)
Cell-edge rateHighestModerateLowest (possibly zero)
Optimization structureQuasi-concave (bisection)GP (interior point)Non-convex (SCA)
Practical useVoice/safety servicesMobile broadbandBest-effort data

Quick Check

Proportional fairness maximizes ∑klog⁡Rk\sum_k \log R_k. What happens to the solution as one user's channel becomes arbitrarily strong?

That user receives all the power

That user's allocated power decreases as its channel improves

The solution converges to equal power allocation

The problem becomes infeasible

Correction:
That user's allocated power decreases as its channel improves

The log⁡\log in the objective creates diminishing returns. As user kk's channel strengthens, log⁡Rk\log R_k grows slowly, so the marginal benefit of additional power decreases. The optimizer redistributes power to weaker users where the marginal log⁡\log-utility is higher.

⚠️Engineering Note

Proportional Fair Scheduling in Practice

In real cellular systems, proportional fairness is typically implemented via the PF scheduler, which in each time slot nn serves the user k⋆(n)=arg⁡max⁡kRkinst(n)Rˉk(n)k^\star(n) = \arg\max_k \frac{R_k^{\text{inst}}(n)}{\bar{R}_k(n)} where RkinstR_k^{\text{inst}} is the instantaneous achievable rate and Rˉk\bar{R}_k is the exponentially weighted average rate. This simple rule converges to the proportionally fair allocation without solving the GP. It also exploits multiuser diversity: users are scheduled when their channels are relatively good, which is more likely with more users.

Practical Constraints
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    PF scheduling requires instantaneous CSI at the scheduler, refreshed every slot (0.5 ms in 5G NR)

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    The averaging window for the denominator is typically 100-200 ms to track slow fading

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    In massive MIMO with MU-MIMO, the scheduler must jointly select users and allocate power

📋 Ref: 3GPP TS 38.214, Section 5.1.2

Key Takeaway

Proportional fairness maximizes the geometric mean of rates — equivalently, the Nash bargaining solution. It provides a principled tradeoff between throughput and fairness, and is the basis of the PF scheduler used in every modern cellular system from 4G LTE onward.

Max-Min FairnessSum-Rate Maximization