Chapter Summary

Chapter Summary

Key Points

• 1.

  Max-min fairness maximizes the worst user's rate via bisection on a target SINR. Each feasibility check is a linear system solvable via Perron–Frobenius theory, and the bisection converges geometrically to the global optimum in $O(\log(1/\epsilon))$ iterations.

• 2.

  Proportional fairness maximizes $\sum_k \log R_k$ — the Nash bargaining solution. It offers a principled throughput-fairness tradeoff and, in the high-SINR regime, reduces to a geometric program solvable in polynomial time.

• 3.

  Sum-rate maximization is NP-hard in general. The WMMSE algorithm exploits the rate-MSE duality to decompose the non-convex problem into alternating convex steps, converging to a stationary point. It is the standard tool for sum-rate optimization.

• 4.

  The $\alpha$-fairness family unifies all three criteria: $\alpha = 0$ (sum-rate), $\alpha = 1$ (proportional fair), $\alpha \to \infty$ (max-min). The parameter $\alpha$ provides a continuous knob between throughput and fairness.

• 5.

  Fractional power control $p_k \propto \beta_{k}^\alpha$ is the practical workhorse, implemented in 5G NR with exponent $\alpha \in \{0, 0.4, \ldots, 1.0\}$. The optimal exponent for sum rate is $\alpha \approx 0.5$; for cell-edge fairness, $\alpha \approx 0.7$–$0.8$.

• 6.

  The BHS framework provides achievable rate expressions under imperfect CSI that depend only on large-scale fading and estimation quality. Power control reduces to the same algorithmic structures as the perfect-CSI case but with modified SINR parameters.