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Energy Efficiency as the Design Objective

The previous sections focused on maximizing SINR and rate. But in dense cell-free deployments, energy efficiency is often the binding constraint. Each AP consumes circuit power (amplifiers, oscillators, ADCs) regardless of the traffic load, and the fronthaul links consume power proportional to their data rate. Adding more APs improves coverage but increases the total power consumption. Similarly, serving more users per AP increases the spectral efficiency but also the per-AP computation and fronthaul power. The question is: what is the optimal user load per AP that maximizes the bits-per-joule efficiency? This section presents the answer, drawing on the CommIT contribution by Göttsch and Caire.

Definition:
Network Energy Efficiency

The network energy efficiency (EE) is defined as the ratio of the sum throughput to the total power consumption:

$\text{EE} = \frac{\sum_{k=1}^{K} R_k}{P_{\text{total}}} \quad [\text{bit/Joule}]$

where $R_k$ is the achievable rate of user $k$ (in bit/s) and the total power is

$P_{\text{total}} = \sum_{m=1}^{M} \left( P_{\text{circuit}} + P_{\text{fh},m} + \frac{1}{\eta} \sum_{k \in \mathcal{D}_m} p_{mk} \right) + P_{\text{CPU}}$

Here $P_{\text{circuit}}$ is the per-AP circuit power (fixed cost of keeping an AP on), $P_{\text{fh},m}$ is the fronthaul power at AP $m$ (proportional to the fronthaul rate), $\eta \in (0,1]$ is the power amplifier efficiency, $p_{mk}$ is the transmit power from AP $m$ to user $k$ , and $P_{\text{CPU}}$ is the CPU processing power.

The key insight is that $P_{\text{circuit}}$ and $P_{\text{fh},m}$ do not vanish as the transmit power decreases. In cell-free networks with many APs, the circuit power can dominate the total power budget.

Definition:
User Load

The user load of the network is the ratio of the number of users to the number of APs:

$\rho = \frac{K}{M}$

In user-centric cell-free massive MIMO, the effective user load at AP $m$ is $|\mathcal{D}_m| / N$ , measuring the number of users served per antenna. The user load determines the interference level: when $\rho$ is small (many APs per user), interference is low and the system is noise-limited. When $\rho$ is large, the system becomes interference-limited and the per-user rate saturates.

Theorem: Energy Efficiency vs. User Load Tradeoff

Under the UatF bound with local MMSE combining and max-min power control, the network energy efficiency as a function of user load $\rho = K/M$ has the following structure:

For small $\rho$ , the sum rate grows linearly with $K$ while the power is dominated by the fixed circuit power $M P_{\text{circuit}}$ . Thus $\text{EE} \approx \frac{\rho \, \bar{R}}{P_{\text{circuit}} + P_{\text{fh},0}}$ where $\bar{R}$ is the average per-user rate, which is approximately constant for small $\rho$ .
For large $\rho$ , the per-user rate decreases due to interference, while the fronthaul power grows. The EE decreases.
There exists an optimal user load $\rho^{\star}$ that maximizes the EE. This optimum depends on $P_{\text{circuit}}$ , the path loss distribution, and the fronthaul power model.

In typical deployments, $\rho^{\star} \in [0.3, 0.8]$ , meaning the optimal number of users is 30–80% of the number of APs.

With too few users, the APs are underutilized — their circuit power is "wasted" on few bits. With too many users, interference degrades the rates faster than the sum rate grows. The optimum balances these two effects.

Proof

Low-load regime

For $\rho \to 0$ , each user sees negligible interference and the per-user rate approaches the interference-free bound $\bar{R}_{\max}$ . The sum rate is $K \bar{R}_{\max} = \rho M \bar{R}_{\max}$ . The power is approximately $M P_{\text{circuit}}$ (fixed). Thus $\text{EE} \approx \rho \bar{R}_{\max} / P_{\text{circuit}}$ , which increases linearly with $\rho$ .

High-load regime

For large $\rho$ , the system is interference-limited. The per-user SINR scales as $O(1/\rho)$ (for fixed $M$ and $N$ ), so $R_k \approx \log_2(1 + c/\rho)$ for some constant $c$ . The sum rate $K R_k \approx \rho M \log_2(1 + c/\rho)$ grows sublinearly. Meanwhile, the fronthaul power $P_{\text{fh}} = \sum_m P_{\text{fh},m}$ grows with $|\mathcal{D}_m|$ , which increases with $\rho$ . Thus the EE decreases.

Existence of optimum

Since $\text{EE}(\rho)$ increases from zero (at $\rho = 0$ ) and eventually decreases (as $\rho \to \infty$ ), by continuity there exists at least one $\rho^{\star}$ maximizing the EE. The exact value depends on the power model parameters and must be computed numerically. $\blacksquare$

🎓CommIT Contribution(2023)

Optimal User Load and Energy Efficiency in User-Centric Cell-Free Networks

F. Göttsch, G. Caire — IEEE Trans. Wireless Communications, vol. 22, no. 12, pp. 9674–9688

Göttsch and Caire address a fundamental system-design question: given a cell-free deployment with $M$ APs, how many users should be admitted to maximize the network energy efficiency? The analysis reveals that:

The optimal user load $\rho^{\star}$ is surprisingly moderate. For typical deployments with $N = 4$ antennas per AP and urban path loss, $\rho^{\star} \approx 0.5$ , meaning each AP should serve about $2N = 8$ users on average (across its cluster).
AP switching saves 30–50% power. When the user load is below $\rho^{\star}$ , turning off excess APs and redistributing users to the remaining active APs improves the EE significantly.
Fronthaul power is the dominant cost. In many deployments, the fronthaul power exceeds the transmit power by an order of magnitude. This makes the choice of cooperation level (L1–L4) an energy efficiency decision, not just a performance one.
Level 3 (LSFD) offers the best energy efficiency. The SINR gain from Level 4 does not compensate for the $N \times$ increase in fronthaul power in most scenarios.

The paper provides closed-form approximations for $\rho^{\star}$ as a function of the power model parameters, enabling system designers to determine the optimal admission policy without extensive simulations.

energy-efficiencyuser-loadcell-freeCommITAP-switchingView Paper →

Energy Efficiency vs. User Load

Visualize the network energy efficiency as a function of the user load $\rho = K/M$ . Observe the optimal user load $\rho^{\star}$ and how it shifts with circuit power, fronthaul power, and the number of antennas per AP.

Parameters

M

(APs)200

Number of access points

N

(antennas/AP)4

Antennas per access point

P_{\\text{circuit}}

(dBm)20

Per-AP circuit power in dBm

P_{\\text{fh}}

(W/Gbps)0.1

Fronthaul power per Gbps

Example: Computing the Optimal User Load

A cell-free network has $M = 100$ APs with $N = 4$ antennas each. The power model parameters are: $P_{\text{circuit}} = 100$ mW per AP, fronthaul power $P_{\text{fh}} = 0.1$ W/Gbps, PA efficiency $\eta = 0.3$ , transmit power per user $p_k = 100$ mW. Under Level 3 (LSFD), the average per-user rate as a function of user load is empirically fitted as $\bar{R}(\rho) \approx 2.5 (1 - 0.8 \rho)$ bit/s/Hz for $\rho \in [0, 1]$ . System bandwidth $B = 20$ MHz. Fronthaul rate per AP is $R_{\text{fh}} \approx 2 |\mathcal{D}_m| B$ bit/s (2 bits per real sample, Nyquist). Find the user load that maximizes EE.

Solution

Express the sum rate

Average cluster size $|\mathcal{D}_m| \approx N_{\text{cl}} \rho$ where $N_{\text{cl}} \approx 20$ (average cluster size per user). Total users: $K = \rho M$ .

Sum throughput: $S(\rho) = \rho M \bar{R}(\rho) B = 100 \rho \times 2.5(1 - 0.8\rho) \times 20 \times 10^6$ $= 5 \times 10^9 \rho (1 - 0.8\rho)$ bit/s.

Express the total power

Circuit power: $M P_{\text{circuit}} = 100 \times 0.1 = 10$ W.

Fronthaul power: $P_{\text{fh,total}} = 0.1 \times M \times 2 |\mathcal{D}_m| B / 10^9$ $\approx 0.1 \times 100 \times 2 \times 20\rho \times 20 \times 10^6 / 10^9 = 0.1 \times 100 \times 800 \rho \times 10^{-3} = 8\rho$ W.

Transmit power: $P_{\text{tx}} = K p_k / \eta = 100\rho \times 0.1 / 0.3 = 33.3\rho$ W.

Total: $P(\rho) = 10 + 8\rho + 33.3\rho = 10 + 41.3\rho$ W.

Optimize

$\text{EE}(\rho) = \frac{5 \times 10^9 \rho(1 - 0.8\rho)}{10 + 41.3\rho}$

Taking the derivative and setting to zero (or evaluating numerically): $\frac{d}{d\rho} \text{EE}(\rho) = 0$ yields $\rho^{\star} \approx 0.42$ .

At $\rho^{\star} = 0.42$ : $K^{\star} = 42$ users, $S(0.42) = 5 \times 10^9 \times 0.42 \times 0.664 = 1.39$ Gbit/s, $P(0.42) = 10 + 17.3 = 27.3$ W, $\text{EE}^{\star} = 1.39 \times 10^9 / 27.3 \approx 51$ Mbit/Joule.

Common Mistake: Ignoring Circuit Power in Energy Efficiency

Mistake:

Computing energy efficiency as $\text{EE} = \sum_k R_k / P_{\text{tx}}$ , counting only the transmit power.

Correction:

In cell-free networks, the circuit power and fronthaul power typically exceed the transmit power by a factor of 5–20. Ignoring these fixed costs leads to the misleading conclusion that more APs always improve EE (because the transmit power per user decreases as $1/M$ ). The total power model must include $P_{\text{circuit}}$ , $P_{\text{fh}}$ , and $P_{\text{CPU}}$ .

⚠️Engineering Note

AP On/Off Switching for Energy Savings

When the traffic load is below the optimal user load $\rho^{\star}$ , the network is wasting circuit power on underutilized APs. AP switching turns off a fraction of the APs and redistributes users to the remaining active APs. The algorithm is:

Compute the current user load $\rho = K / M$ .
If $\rho < \rho^{\star} / 2$ , determine the number of APs to deactivate: $M_{\text{off}} = M - \lceil K / \rho^{\star} \rceil$ .
Select the APs with the lowest aggregate large-scale fading to any active user for deactivation.
Reassign users to the remaining active APs and recompute LSFD weights.

In the Göttsch/Caire analysis, AP switching reduces the total power consumption by 30–50% during low-traffic periods (nighttime, rural areas) with less than 5% rate loss.

Practical Constraints

•
AP activation/deactivation takes 10–100 ms (oscillator settling, synchronization)
•
Frequent switching creates service interruptions; hysteresis of 1–5 minutes recommended
•
Deactivated APs must remain powered for pilot transmission if using TDD reciprocity

Historical Note: The Green Massive MIMO Movement

2010–2023

Energy efficiency became a central metric in wireless research around 2012, driven by the observation that base station energy consumption was growing at 15–20% per year while revenue was flat. The EARTH project (EU FP7, 2010–2012) produced the first comprehensive power models for base stations, showing that the power amplifier accounts for only 30–40% of total consumption — the rest is circuit power, cooling, and backhaul. Bjornson, Hoydis, and Sanguinetti's 2017 monograph extended these models to massive MIMO and showed that massive arrays are inherently energy-efficient: the array gain allows reducing transmit power as $1/M$ while the circuit power grows only linearly. The cell-free paradigm adds a new dimension: now the circuit and fronthaul power of distributed APs must be accounted for, making energy efficiency optimization both more important and more complex.

Quick Check

In a dense cell-free deployment with $M = 500$ APs, $N = 4$ antennas per AP, and $K = 200$ users, which power component typically dominates the total network power consumption?

Transmit power $\sum_k p_k / \eta$

AP circuit power $M P_{\text{circuit}}$

Fronthaul power $\sum_m P_{\text{fh},m}$

CPU processing power $P_{\text{CPU}}$

Correction:

Fronthaul power

\sum_m P_{\text{fh},m}

Correct. In dense cell-free deployments, the fronthaul power dominates because each of the $M = 500$ APs must communicate with the CPU. With fiber fronthaul, the per-link power is 0.5–2 W, giving $250$ – $1000$ W total.

Why This Matters: Energy Efficiency Targets in 5G and Beyond

The ITU-R IMT-2020 framework specifies that 5G should achieve at least $10^7$ bit/Joule network energy efficiency, a $100 \times$ improvement over LTE. Cell-free massive MIMO with LSFD and AP switching can approach this target in dense urban deployments by combining the macro-diversity gain (reducing transmit power) with intelligent AP management (reducing circuit power). The 6G vision targets $10^8$ bit/Joule, which will require further innovations in fronthaul compression and reconfigurable intelligent surfaces (see Book RIS).

User Load

The ratio $\rho = K/M$ of the number of users to the number of APs in a cell-free network. Determines the interference level and the per-AP processing and fronthaul requirements. The optimal user load $\rho^{\star}$ maximizes the network energy efficiency.

Circuit Power

The power consumed by an access point's electronic components (oscillators, ADCs/DACs, mixers, digital processing) regardless of the transmitted signal power. In cell-free networks with many single-antenna or few-antenna APs, the aggregate circuit power can dominate the total network power consumption.

Optimal User Load and Energy Efficiency