Ferkans — Interactive Telecom Tutor

Why Power Control Matters

In Chapter 4 we derived achievable rate expressions for massive MIMO under MRC, ZF, and MMSE combining — but we assumed all users transmit at the same power. This is rarely a good idea. Users close to the base station enjoy strong channels while cell-edge users suffer from severe path loss. Without power control, the cell-edge users are drowned by interference from stronger users, and the system operates with extreme rate imbalance.

Power control is the art of choosing the transmit powers $p_1, \ldots, p_{K}$ to achieve a desired notion of fairness or efficiency. This chapter develops the three canonical objectives — max-min fairness, proportional fairness, and sum-rate maximization — and shows how each leads to a different optimization structure.

Power Control

The process of selecting the transmit power $p_k$ for each user $k$ subject to a total or per-user power budget, in order to optimize a system-level objective such as fairness or throughput.

Definition:
Generic SINR Expression

Consider a massive MIMO uplink with $N_t$ antennas at the base station and $K$ single-antenna users. Under the UatF bound from Chapter 4, the effective SINR of user $k$ takes the form

$\text{SINR}_k(\mathbf{p}) = \frac{p_k \, a_k}{\sum_{l \neq k} p_l \, \gamma_{kl} + \sigma^2 \, b_k}$

where $a_k > 0$ is the desired signal strength (depending on the combining scheme and channel statistics), $\gamma_{kl} \geq 0$ is the interference coupling coefficient from user $l$ to user $k$ , and $b_k > 0$ captures the noise enhancement factor. The achievable rate is $R_k(\mathbf{p}) = \log_2(1 + \text{SINR}_k(\mathbf{p}))$ .

The specific values of $a_k$ , $\gamma_{kl}$ , and $b_k$ depend on the combining scheme (MRC, ZF, MMSE) and channel estimation quality. The results in this chapter apply to any combining scheme that yields an SINR of this form.

Definition:
Max-Min Fair Power Control

The max-min fair power control problem seeks the power allocation that maximizes the minimum rate among all users:

$\max_{\mathbf{p} \geq \mathbf{0}} \; \min_{k = 1, \ldots, K} \; R_k(\mathbf{p}) \quad \text{subject to} \quad \sum_{k=1}^{K} p_k \leq P_t.$

Equivalently, since $\log_2(1 + \cdot)$ is monotonically increasing, we can work with the SINR directly:

$\max_{\mathbf{p} \geq \mathbf{0}} \; \min_{k} \; \text{SINR}_k(\mathbf{p}) \quad \text{subject to} \quad \sum_{k=1}^{K} p_k \leq P_t.$

Max-min fairness is the strongest form of fairness: it guarantees that no user can be improved without degrading the worst-off user. This makes it the default choice in many cellular systems.

Max-Min Fairness

A resource allocation policy that maximizes the minimum utility (rate) among all users. The resulting allocation is Pareto-optimal within the constraint that no user receives less than the worst-off user.

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

1987–1995

The concept of max-min fairness originated in wired networking, where Bertsekas and Gallager (1987) used it for bandwidth allocation in data networks. In the wireless context, the additional coupling through interference makes the problem substantially harder — increasing one user's power raises interference for all others. The connection between max-min SINR problems and Perron–Frobenius theory was established by Zander (1992) and Yates (1995), creating the mathematical framework that underpins all modern power control algorithms.

Theorem: Max-Min Fairness via Bisection

The max-min fair SINR problem can be solved by bisection on the target SINR $t \geq 0$ . For a given $t$ , the feasibility problem is:

$\text{Find } \mathbf{p} \geq \mathbf{0} \text{ such that } \text{SINR}_k(\mathbf{p}) \geq t \; \forall k, \quad \sum_{k=1}^{K} p_k \leq P_t.$

This is equivalent to the linear system

$p_k \, a_k \geq t \left( \sum_{l \neq k} p_l \, \gamma_{kl} + \sigma^2 \, b_k \right), \quad k = 1, \ldots, K$

which can be rewritten as $(\mathbf{I} - t \, \mathbf{D}^{-1} \mathbf{F}) \, \mathbf{p} \geq t \, \mathbf{D}^{-1} \boldsymbol{\sigma}$ where $\mathbf{D} = \text{diag}(a_1, \ldots, a_{K})$ , $[\mathbf{F}]_{kl} = \gamma_{kl}$ for $k \neq l$ and $[\mathbf{F}]_{kk} = 0$ , and $\boldsymbol{\sigma} = \sigma^2[b_1, \ldots, b_{K}]^\mathsf{T}$ .

The feasibility problem has a solution if and only if $t < t^\star$ , where $t^\star = 1/\rho(\mathbf{D}^{-1}\mathbf{F})$ and $\rho(\cdot)$ denotes the spectral radius. Bisection on $t \in [0, t_{\max}]$ converges to the optimal max-min SINR in $O(\log(t_{\max}/\epsilon))$ iterations.

The key insight is that checking "can all users achieve SINR $\geq t$ ?" is a linear feasibility problem. We binary-search on $t$ : if it is feasible, increase $t$ ; if not, decrease it. Each feasibility check is cheap (solve a linear system), so the overall algorithm is efficient.

Proof

Reformulate SINR constraints as linear inequalities

The constraint $\text{SINR}_k \geq t$ reads $p_k a_k \geq t(\sum_{l \neq k} p_l \gamma_{kl} + \sigma^2 b_k)$ . Rearranging: $p_k a_k - t \sum_{l \neq k} p_l \gamma_{kl} \geq t \sigma^2 b_k$ . In matrix form: $(\mathbf{D} - t\mathbf{F})\mathbf{p} \geq t\boldsymbol{\sigma}$ , or equivalently $(\mathbf{I} - t\mathbf{D}^{-1}\mathbf{F})\mathbf{p} \geq t\mathbf{D}^{-1}\boldsymbol{\sigma}$ .

Apply Perron-Frobenius theory

The matrix $\mathbf{D}^{-1}\mathbf{F}$ is nonnegative (all entries $\geq 0$ ). By the Perron–Frobenius theorem, $(\mathbf{I} - t\mathbf{D}^{-1}\mathbf{F})$ is invertible with a nonneg-ative inverse if and only if $t < 1/\rho(\mathbf{D}^{-1}\mathbf{F})$ . When this holds, the unique minimum-power solution is $\mathbf{p}^\star(t) = t(\mathbf{I} - t\mathbf{D}^{-1}\mathbf{F})^{-1}\mathbf{D}^{-1}\boldsymbol{\sigma}$ .

Bisect on the target SINR

Since the feasibility region is $t \in [0, t^\star)$ and we also need $\sum_k p_k^\star(t) \leq P_t$ , define $f(t) = \mathbf{1}^\mathsf{T} \mathbf{p}^\star(t)$ . This function is continuous and strictly increasing in $t$ (more SINR requires more power). Standard bisection on $f(t) \leq P_t$ finds the optimal $t$ to precision $\epsilon$ in $\lceil \log_2(t^\star / \epsilon) \rceil$ iterations. $\blacksquare$

,

Bisection Algorithm for Max-Min Fair Power Control

Complexity:

O(I \cdot K^{3})

where

I = \lceil \log_2(t^\star/\epsilon) \rceil

bisection iterations

Input: Coupling matrix

\mathbf{F}

, signal strengths

\{a_k\}

,

noise terms

\{b_k\}

, power budget

P_t

, tolerance

\epsilon > 0

Output: Max-min fair power vector

\mathbf{p}^\star

1. Set

t_{\text{lo}} \leftarrow 0

,

t_{\text{hi}} \leftarrow 1/\rho(\mathbf{D}^{-1}\mathbf{F})

2. while

t_{\text{hi}} - t_{\text{lo}} > \epsilon

do

3.

\quad t \leftarrow (t_{\text{lo}} + t_{\text{hi}}) / 2

4.

\quad \mathbf{p}(t) \leftarrow t (\mathbf{I} - t\mathbf{D}^{-1}\mathbf{F})^{-1}\mathbf{D}^{-1}\boldsymbol{\sigma}

5.

\quad

if

\sum_k p_k(t) \leq P_t

then

6.

\quad\quad t_{\text{lo}} \leftarrow t

7.

\quad

else

8.

\quad\quad t_{\text{hi}} \leftarrow t

9.

\quad

end if

10. end while

11.

\mathbf{p}^\star \leftarrow \mathbf{p}(t_{\text{lo}})

Each iteration requires solving a $K \times K$ linear system, which costs $O(K^{3})$ . For massive MIMO with $K \leq 40$ , this is negligible. The bisection converges geometrically — typically 20–30 iterations suffice for machine precision.

Example: Max-Min Fairness with Two Users

Consider a massive MIMO uplink with $N_t = 100$ antennas, $K = 2$ users, and MRC combining. User 1 has path loss $\beta_{1} = 0.1$ (close to the BS) and user 2 has $\beta_{2} = 0.001$ (cell-edge). With equal power $p_1 = p_2 = 0.5P_t$ , user 1 achieves $R_1 \approx 5.2$ bits/s/Hz while user 2 achieves $R_2 \approx 0.8$ bits/s/Hz. Find the max-min fair power allocation.

Solution

Set up the SINR expressions

Under MRC combining with i.i.d. Rayleigh fading and MMSE channel estimation, $a_k = N_t \beta_{k}^{2} / (\beta_{k} + \sigma^2/p_k^{\text{pilot}})$ and $\gamma_{kl} = \beta_{l}$ . For simplicity, assume perfect CSI so that $a_k = N_t \beta_{k}$ and $b_k = 1$ . Then $\text{SINR}_k = \frac{p_k N_t \beta_{k}}{p_{3-k} \beta_{3-k} + \sigma^2}$ .

Apply the equal-SINR condition

Max-min fairness requires $\text{SINR}_1 = \text{SINR}_2 = t^\star$ . Setting these equal: $\frac{p_1 \cdot 100 \cdot 0.1}{p_2 \cdot 0.001 + \sigma^2} = \frac{p_2 \cdot 100 \cdot 0.001}{p_1 \cdot 0.1 + \sigma^2}$ . With $\sigma^2 = 1$ and $p_1 + p_2 = P_t = 1$ : the cell-edge user receives most of the power budget.

Numerical result

Solving numerically: $p_1^\star \approx 0.01$ , $p_2^\star \approx 0.99$ . The resulting rates are approximately $R_1 \approx R_2 \approx 1.5$ bits/s/Hz. The cell-center user sacrifices rate (from 5.2 to 1.5) so that the cell-edge user improves (from 0.8 to 1.5). This is the price of max-min fairness.

Bisection Convergence for Max-Min Fair Power Control

Watch the bisection algorithm converge to the optimal max-min SINR as we sweep the target $t$ from 0 to the spectral radius bound. Adjust the number of users and the path loss disparity to see how fairness interacts with system parameters.

Parameters

K

(users)4

M

(antennas)64

Path loss spread (dB)15

Range between strongest and weakest user

Max bisection iterations20

Common Mistake: Max-Min Fairness Can Waste Resources

Mistake:

A common misconception is that max-min fairness is always the best policy. In fact, max-min fairness can dramatically reduce total throughput because it forces the system to spend most of its power budget on the weakest user.

Correction:

If one user has 20 dB worse path loss than the others, max-min fairness may allocate 99% of the power to that user — reducing the total sum rate by an order of magnitude compared to what the other users could achieve. The choice of fairness criterion depends on the system operator's priorities. When some rate imbalance is acceptable, proportional fairness (Section 5.2) offers a better throughput-fairness tradeoff.

Quick Check

In the max-min fair power control problem, what is the key structural property that enables the bisection algorithm?

The objective function is concave in the power vector

The SINR constraints become linear inequalities for a fixed target SINR

The problem can be decomposed into independent per-user subproblems

The power constraint is a box constraint

Correction:

The SINR constraints become linear inequalities for a fixed target SINR

For a fixed target $t$ , the constraint $\text{SINR}_k \geq t$ is linear in the power vector $\mathbf{p}$ . This makes each feasibility check a linear program, and the optimal $t$ can be found by bisection since the feasible set shrinks monotonically as $t$ increases.

Key Takeaway

Max-min fairness in massive MIMO reduces to a sequence of linear feasibility checks via bisection. Each check asks: "can all $K$ users simultaneously achieve SINR $\geq t$ ?" The Perron–Frobenius theorem guarantees a unique answer, and the bisection converges geometrically.

Why This Matters: Max-Min Fairness in 5G NR Scheduling

In 5G NR, the scheduler at the gNB (base station) performs power control every slot (0.5 ms at 30 kHz SCS). The "round-robin" scheduling mode approximates max-min fairness by cycling through users and adjusting their power to equalize throughput. The bisection algorithm studied here is too slow for per-slot optimization with hundreds of users, which motivates the heuristic approaches in Section 5.4.

See full treatment in Practical Power Control Algorithms

Max-Min Fairness

Why Power Control Matters

Power Control

Definition: Generic SINR Expression

Definition: Max-Min Fair Power Control

Max-Min Fairness

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

Theorem: Max-Min Fairness via Bisection

Reformulate SINR constraints as linear inequalities

Apply Perron-Frobenius theory

Bisect on the target SINR

Bisection Algorithm for Max-Min Fair Power Control

Example: Max-Min Fairness with Two Users

Set up the SINR expressions

Apply the equal-SINR condition

Numerical result

Bisection Convergence for Max-Min Fair Power Control

Parameters

Common Mistake: Max-Min Fairness Can Waste Resources

Quick Check

Key Takeaway

Why This Matters: Max-Min Fairness in 5G NR Scheduling

Definition:
Generic SINR Expression

Definition:
Max-Min Fair Power Control