Proportional Fairness Scheduling

The scheduler selects user $k^*$ at time $t$ according to the rule:

$$k^* = \arg\max_k \frac{R_k(t)}{T_k(t)}$$

Where:

• $R_k(t)$: The instantaneous supportable data rate for user $k$ based on current channel quality.
• $T_k(t)$: The exponentially weighted moving average throughput for user $k$.

This metric inherently favors users with momentarily high channel quality relative to their own historical average, preventing starvation.

PF scheduling sits between two extreme strategies:

• Max C/I (Maximum Rate): Schedules the user with the absolute best channel conditions. Maximizes cell throughput but starves cell-edge users.
• Round Robin (RR): Allocates equal airtime to all users. Maximizes fairness but drastically reduces total spectral efficiency.

PF provides a Pareto-efficient compromise, achieving a significant fraction of Max C/I throughput while guaranteeing that no user is permanently denied service.

The average throughput $T_k(t)$ is updated recursively using a forgetting factor $\alpha$:

$$T_k(t+1) = (1-\alpha)T_k(t) + \alpha \cdot R_k(t) \cdot \mathbb{1}_k$$

Where $\mathbb{1}_k$ is 1 if user $k$ was scheduled, and 0 otherwise.

• A large $\alpha$ (e.g., 0.01) gives more weight to recent transmissions, making the scheduler more responsive to changing conditions.
• A small $\alpha$ (e.g., 0.001) smooths over longer time windows, enforcing stricter long-term fairness.

PF scheduling exploits multi-user diversity—the statistical independence of fading channels across users in a cell.

In a cell with many independent users, there is a high probability that at least one user will have a strong channel at any given moment. PF capitalizes on this by scheduling users near their constructive fading peaks, increasing total system capacity without requiring additional spectrum or power.

This gain scales logarithmically with the number of active users.

PF scheduling is a foundational component in multiple wireless standards:

• LTE/4G: Implemented as a common downlink scheduling policy in the MAC layer.
• 5G NR: Extended to handle massive MIMO beamforming and wideband carrier aggregation, where the metric is evaluated per resource block group.
• Qualcomm's EV-DO (HDR): The original commercial implementation that proved the viability of opportunistic scheduling.

In 5G, the algorithm is often enhanced with QoS class identifiers (QCI/5QI) to add delay-aware weighting.

Standard PF scheduling has known weaknesses in specific scenarios:

• Delay Sensitivity: Pure PF is throughput-optimal but delay-blind. Real-time applications (VoLTE, URLLC) require Modified Largest Weighted Delay First (M-LWDF) extensions.
• Minimum Rate Guarantees: PF provides relative fairness, not absolute guarantees. For strict QoS, it is combined with token bucket mechanisms.
• Heterogeneous Traffic: Users with drastically different file sizes can experience unfairness. Score-based schedulers add offset terms to the PF metric to compensate.

A stable state in a non-cooperative game where no individual player can gain an advantage by unilaterally changing their strategy. In spectrum sharing, proportional fairness algorithms often converge to a Nash bargaining solution, which is a specific type of Nash Equilibrium that maximizes the product of user utilities rather than raw sum throughput.

• Models the outcome of competitive spectrum sharing scenarios
• The logarithmic utility function used in proportional fairness directly maps to Nash bargaining axioms
• Ensures no cognitive radio has an incentive to deviate from its allocated schedule

A sealed-bid, combinatorial auction mechanism that incentivizes truthful bidding by charging winners the marginal harm their presence causes to other bidders. Proportional fairness scheduling shares the same philosophical foundation: both mechanisms optimize for social welfare while internalizing the externality one user imposes on others.

• Used for efficient spectrum license allocation in dynamic markets
• The VCG payment rule mathematically mirrors the fairness penalty in proportional scheduling
• Prevents strategic manipulation of resource requests

A mathematical framework for solving coordination problems where multiple agents, each with local constraints, must agree on a globally optimal assignment of variables. Proportional fairness can be implemented as a weighted constraint satisfaction problem where each link's data rate constraint is balanced against the global fairness objective.

• Applied to distributed channel selection in ad-hoc networks
• Max-Sum algorithm variants often approximate proportional fairness in decentralized systems
• Enables coordination without a central Spectrum Access System (SAS)

A machine learning paradigm where multiple autonomous agents learn optimal policies through interaction and feedback within a shared environment. Modern implementations of proportional fairness scheduling use decentralized MARL where each cognitive radio learns a transmission policy that maximizes its own reward while the global reward function encodes the sum of log-rates.

• Used for decentralized spectrum allocation and interference management
• Credit assignment mechanisms ensure agents learn cooperative rather than greedy behaviors
• Enables real-time adaptation to non-stationary electromagnetic environments

A set of predefined, non-cooperative rules and behavioral protocols for cognitive radios to autonomously manage access and mitigate interference without explicit real-time negotiation. Proportional fairness can be hard-coded into etiquette rules by mandating that devices must yield a channel when their continued use would disproportionately degrade a neighbor's throughput relative to their own gain.

• Enables polite spectrum access without centralized coordination
• Rules often include duty cycle limits and maximum transmit power backoff
• Complements Listen-Before-Talk (LBT) mechanisms in unlicensed bands

A deep learning model that represents wireless networks as graphs, where nodes are transceivers and edges are interference links. GNNs can learn to approximate the proportional fair scheduling solution in milliseconds by predicting the optimal user subset and power allocation directly from the interference graph topology, bypassing traditional iterative optimization.

• Learns and predicts complex interference patterns for optimized resource allocation
• Message passing between nodes captures the mutual interference constraints
• Enables near-optimal scheduling with microsecond latency for 5G NR and beyond