# Energy efficiency in O-RAN through sleep modes

Jean-Baptiste Monteil, Salah-Eddine Elayoubi, Alexis Aravanis

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Jean-Baptiste Monteil, Salah-Eddine Elayoubi, Alexis Aravanis. Energy efficiency in O-RAN through sleep modes. 2025. hal-04878197

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# Energy efficiency in O-RAN through sleep modes

Jean-Baptiste Monteil*, Salah-Eddine Elayoubi* Alexis Aravanis*

*CentraleSupélec, Université Paris-Saclay, CNRS, L2S, Gif-sur-Yvette, France

***Abstract***—Ensuring energy efficiency is an important requirement for mobile networks and sleep mode is an essential enabler for it. With the rise of open-RAN, the disaggregation of network functions at different locations has enabled to put the baseband processing functions into sleep modes whenever possible. In this work, we aim to reduce the energy consumption of the radio access network by leveraging advanced sleep modes, namely micro, light, and deep, with different energy profiles. The main difficulty when implementing these sleep modes on small time scales originates from the non-negligible transition times. We model the problem of the processing units activation/deactivation that host the DUs as an MILP (Mixed Integer Linear Program). We offer an LP-MIP (Linear Program - Mixed Integer Program) version, with binary and continuous variables. We propose exact solutions for the two formulated problems and an additional heuristic with lower complexity.

***Index Terms***—Energy efficiency, integer programming, problem transformation.

## I. INTRODUCTION

**Motivation.** The energy consumption of the network predominantly takes place in the RAN (radio access network): it counts for 73% of the energy consumption [1]. The technical report from 3GPP-release 18 [2] proposes a new energy consumption model for the Base Station (BS) with three types of sleep modes (micro, light and deep), each with specific transition time and energy. In this paper, we aim to apply these sleep modes in an Open-RAN (O-RAN) architecture framework that allows to disaggregate the network functions and place them at different locations, namely the Radio Unit (RU), the Distributed Unit (DU), and the Central Unit (CU) [3]. From O-RAN stems the opportunity to deactivate the baseband components whenever possible to reduce the energy consumption while avoiding traffic outage.

**Related Work.** Several works addressed sleep mode in the context of traditional RAN, where the network functions (radio and baseband processing) are co-located at the BS and processed by dedicated hardware. [4] is the first paper to take account of different sleep modes, which have different power values and transition times. The authors in [5] opt for a Q-learning approach limited to regular time decisions and validate their solution against very low load settings. Similarly to [5], [6] considers multi-level sleep modes for small BSs within a cell and uses Q-learning by finding the state-action pairs first, then applying the trained solution online. [7] leverages multi-step ahead traffic prediction with a neural network module to solve the allocation of power and bandwidth, and selects with a local search the optimal subset of active BSs. [8] enables to defer the awaking decision of a sleeping BS, by setting timers for the sleep modes. The authors explore the trade-off between delay cost and energy savings. In the same line, [9] guarantees an average reactivation delay while maximizing energy saving,

through closed-form formulation based on the probability of request arrival during deactivation transient states and sleep states for all modes.

In the framework of virtualized RAN, the authors in [10] propose a joint approach for network slicing and VNF placement to ensure maximum delay for URLLC users and minimum throughput for eMBB users. [11] exposes O-RAN principles and investigates the challenges of its implementation in 6G networks. [12] surveys the works that address the radio and transport networks energy consumption. [13] models the RU placement and UE-RU association problem as a NP-hard integer linear problem and proposes to solve it based on Lagrangian relaxation. The solution consists of a sub-gradient method on the Lagrange multipliers to reduce the optimality gap. The CU-DU placement problem connects the installed RUs with access-edge servers that host the DUs and CUs. Works on O-RAN have achieved energy efficiency with the dynamic placement of network functions over Virtual Machines (VMs) located at the DU and the CU [14], [15]. [16] addresses the energy consumption problem in O-RAN settings with delay guarantees. The authors focus on RB (Resource Block) selection and DU deactivation but do not account for transition costs in time and energy. [17] models the placement of the functional split at Core Network nodes as a Markov Decision Process and solves it with a distributed reinforcement learning approach.

**Contributions.** In our work, we consider a mixed-time scale decision model in the context of an O-RAN architecture applied to the activation/deactivation of servers that host the DUs. The controller can apply instantaneous micro sleep decisions and delayed light/deep sleep decisions at the DU/baseband processing level.

We model the light/deep sleep decision at the DU level as an MILP that accounts for the transition times and costs. First, we propose to solve the problem with a basic heuristic. Secondly, we transform the problem into an LP-MIP problem, with continuous and binary variables to reach optimality. The continuous approach provides significant energy gains compared to the binary methods. We propose to add micro sleep decisions during the frame on top of light/deep sleep decision and demonstrate additional gains with the mixed-time scale solutions.

Our numerical results show that the LP-MIP with continuous switching times achieves the best performance, but with a high complexity cost. However, if micro sleep mode with a relatively high efficiency is implemented on top of the MILP for constituting a mixed time scale algorithm, our simple heuristic achieves comparable energy gains with linear complexity.
## II. SYSTEM MODEL

### A. RAN model

We consider an O-RAN architecture as defined by the O-RAN alliance framework: the RUs host the RF (Radio Frequency) and PHY layers with components like RF transceivers and power amplifiers, and DUs host intermediate layers like MAC and RLC. We simply refer to radio functions (located at the RU) and processing functions (located at the DU) in the sequel. We assume the distributed RUs ensure non-overlapping coverage of the cell of interest. Each RU is paired to a virtual server/DU instance. The DUs belong to the same set of access edge servers and we assume they inter-connect without delay. In this paper, we consider the baseband processing functions located at the DU. We do not consider processing at the core network and restrain the system model to the RAN.

### B. Sleep Modes and Energy Model

3GPP release 18 [2] introduced a new energy consumption for the BS. The controller can select between different sleep modes and determine accordingly the power state of the radio carrier components. [2] enumerates four different power states: the active state, the micro sleep state, the light sleep state, and the deep sleep state. The latter two states offer bigger energy savings but present the drawback of a longer transition. The total transition time to switch from active mode to light sleep (deep sleep resp.) is 6 milliseconds (50 milliseconds resp.). The total transition time encompasses the time needed for ramping down (from active to sleep) and ramping up (from sleep to active). In Table I, we display the transition times to ramp down and up for each sleep mode: e.g. $T_s = 25$ ms and $T_w = 25$ ms for the deep sleep mode. The energy to transition is proportional to the time it takes: e.g. the transition energy to switch to deep sleep is 90 per slot times 25 ms. The energy values are expressed in terms of relative power to the deepest sleep mode (annotated in bold in Table I) times the duration (in ms). Once the sleeping decision is taken, the resources become unavailable instantaneously, but the energy consumption remains equal to $E_{s,1} > 0$ during a period of $T_s$ ms, and then goes down to a positive value $E_{s,2} < E_{s,1}$. The values of $E_{s,1}$, $E_{s,2}$ and $T_s$ are different according to the selected sleep mode (see Table I). Note that there is neither transition time nor transition energy between active and micro sleep. Therefore, micro sleeps decisions can be made every Transmission Time Interval (TTI) (equal to 1 ms).

We consider that the controller can set the network functions into the three kinds of sleep aforementioned. Once the processing units at the DU are effectively in sleep mode (light or sleep), the process of awakening can start; it lasts for $T_w$ ms during which the energy consumption is equal to $E_w > E_{s,2}$.

While, in a non-overlapping coverage scenario, the controller can switch the radio functions at the RU off at most for 1 ms, so to not lose coverage of the cell, there is no limit to the sleep time of baseband functions at the DUs, provided that the QoS requirements of the different UEs are met.

### C. Mixed-time scale decision model

In Figs. 1, 2, and 3, we display the energy consumption of two sleep modes against the effective sleep time of the deeper

<table><caption>TABLE I: Table of transitions (energy and time)</caption><thead><tr><th></th><th>Deep</th><th>Light</th><th>Micro</th></tr></thead><tbody><tr><td>energy to ramp-down $E_{s,1}$</td><td>90*25 (in total)</td><td>90*3 (in total)</td><td>0 (in total)</td></tr><tr><td>energy to maintain $E_{s,2}$</td><td>1 (every ms)</td><td>25 (every ms)</td><td>55 (every ms)</td></tr><tr><td>time to ramp-down $T_s$</td><td>25 ms</td><td>3 ms</td><td>0 ms</td></tr><tr><td>time to ramp-up $T_w$</td><td>25 ms</td><td>3 ms</td><td>0 ms</td></tr><tr><td>energy to ramp-up $E_w$</td><td>90*25 (in total)</td><td>90*3 (in total)</td><td>0 (in total)</td></tr></tbody></table>

<figure><img src="image_2.png" /><figcaption>Fig. 1: light/micro</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Fig. 2: deep/micro</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 3: deep/light</figcaption></figure>

mode (deep deeper than light deeper than micro). The effective sleep time starts after $T_s$ ms, which is the time to ramp down to the selected sleep mode. From Table I, $T_s$ is 0, 3, and 25 ms respectively for micro, light, and deep sleep. If the effective sleep time is less than 7 ms, the cheapest mode is micro. The deep mode is optimal only from 120 ms of effective sleep time. Thus it requires no less than $T_s + 120 + T_w = 170$ ms without transmission before considering deep sleep a viable option. The light sleep mode is less stringent as it yields the best savings between 13 and 170 ms without transmission. In the following, our objective is to devise the optimal sleep mode decisions at mixed time scales knowing these basic tradeoffs.

### III. PERIODIC MIXED-TIME SCALE SLEEP DECISIONS

We start by the basic case where a DU can be put in light or deep sleep mode on a periodic basis, i.e. the decision is binary (0 or 1) and is applied for a whole time period. For light sleep, the time period may correspond to a radio frame of 10 ms, while in deep sleep, a longer sleep period should be considered as the transition time is larger than a time frame.

#### A. Light/deep sleep decisions

The controller takes the decision whether to put the servers/DUs in light/deep sleep mode or not. The goal is to minimize the total energy consumption of the system over one frame, while ensuring that the total workload $L$ over all interconnected DUs is timely served.

$$ (\mathbb{P}): \quad \min_{x} \sum_{i=1}^{n} TP_{a}x_{i}y_{i} + TP_{s}(1-x_{i})(1-y_{i}) + T_{\text{off}}P_{\text{off}}(1-x_{i})y_{i} + (T-T_{\text{off}})P_{s}(1-x_{i})y_{i} + T_{\text{on}}P_{\text{on}}(1-y_{i})x_{i} + (T-T_{\text{on}})P_{a}(1-y_{i})x_{i} \quad (1) $$

$$ \text{s.t.} \quad x_i = \{0, 1\}, \quad \forall i \in \{1, \dots, n\} \quad (2) $$

$$ y_i = \{0, 1\}, \quad \forall i \in \{1, \dots, n\} \quad (3) $$

$$ \sum_{i=1}^{n} TC_{i}x_{i}y_{i} + (T - T_{\text{on}})C_{i}(1 - y_{i})x_{i} \geq L, \quad (4) $$
**Algorithm 1: Heuristic**

Input: $L, C, T, T_{on}, n, \mathbf{x}_0$

1 $\mathbf{x} = \mathbf{x}_0$

2 Sort $C$ items in non-decreasing order

3 if $TC \cdot (\mathbf{x} \cdot \mathbf{x}_0) + (T - T_{on})C \cdot ((1 - \mathbf{x}_0) \cdot \mathbf{x}) < L$ then

4 while
5 $TC \cdot (\mathbf{x} \cdot \mathbf{x}_0) + (T - T_{on})C \cdot ((1 - \mathbf{x}_0) \cdot \mathbf{x}) < L$ do
6 $\sum_{i=1}^{n} \mathbf{x}[i] = \sum_{i=1}^{n} \mathbf{x}[i] + 1$
7 Output: $\mathbf{x}$

6 else

7 while
8 $\mathbf{x}_{\text{memory}} = \mathbf{x}$;
9 $\sum_{i=1}^{n} \mathbf{x}[i] = \sum_{i=1}^{n} \mathbf{x}[i] - 1$
**Output:** $\mathbf{x}_{\text{memory}}$

In problem (P), the objective (1) to minimize is the energy consumption of the current frame. We enumerate the variables in our formulation:

* $x_i$ represents the decision of the controller concerning the DU $i$: $x_i = 1$ means DU $i$ is now active, while $x_i = 0$ means DU $i$ is now asleep. The vector $\mathbf{x}$ contains the controller's decisions.

* $\{y_i\}_{i=1}^n$ stores the controller's decisions taken in the previous frame. This variable is important as the controller must account for the transitions from sleep to active or from active to sleep, when it decides whether to switch on/off the DUs.

* $P_a$ is the power consumption per time unit while active.

* $P_s$ is the power consumption per time unit while in sleep mode.

* $P_{off}$ is the power consumption per time unit during the transition from active to sleep mode.

* $P_{on}$ is the power consumption per time unit during the transition from sleep to active mode.

* $T$ is the total time within one period.

* $T_{off}$ is the transition time from active to sleep mode.

* $T_{on}$ is the transition time from sleep to active mode.

* $C_i$ is the capacity of the active DU $i$ for one slot.

* $C = [C_1, \dots, C_n]^\top$.

* $L$ is the total workload over one frame.

## B. Heuristic

Problem (P) has a high complexity, as will be seen in the complexity analysis of next section. We then propose a sub-optimal heuristic with enhanced time complexity. The heuristic proposes to check the constraint (4) with the current combination of active and inactive DUs $\mathbf{x}$. If the constraint is violated, then we add one active DU and check again (4); we repeat the process until we find the first combination of active DUs that respects (4). Reversely, if the constraint is respected, then we remove one active DU from the current combination $\mathbf{x}$ and check again (4); we repeat the process until we violate the constraint. We keep the last in date combination $\mathbf{x}$ that respected the constraint (4). This solution has polynomial

**Algorithm 2: Micro sleep procedure**

Input: $L, C, T, P_a, P_s, P_s^\mu, P_{on}, P_{off}, T_{on}, T_{off}, n, \mathbf{x}, \mathbf{y}$

1 Compute waste

$$W = TC \cdot \mathbf{x} \cdot \mathbf{y} + (T - T_{on})C \cdot (1 - \mathbf{y}) \cdot \mathbf{x} - L$$

2 Put $\lfloor \frac{W}{\min(C)} \rfloor$ slots in micro sleep

3 Compute savings $S = \lfloor \frac{W}{\min(C)} \rfloor (P_a - P_s^\mu)$

4 Compute new consumed energy value

$$ 
\begin{aligned}
E &= TP_a \mathbf{x}^\top \mathbf{y} + TP_s (1 - \mathbf{x})^\top (1 - \mathbf{y}) + T_{off} P_{off} (1 - \mathbf{x})^\top \mathbf{y} \\
&\quad + (T - T_{off}) P_s (1 - \mathbf{x})^\top \mathbf{y} + T_{on} P_{on} (1 - \mathbf{y})^\top \mathbf{x} \\
&\quad + (T - T_{on}) P_a (1 - \mathbf{y})^\top \mathbf{x} - S
\end{aligned}
$$

5 Compute new margin/waste

$$W = W - L - \min(C) \lfloor \frac{W}{\min(C)} \rfloor$$

complexity and gives fairly good results, as we demonstrate in the sequel.

## C. Micro sleep decisions

During the frame, the controller has the opportunity to leverage the micro sleep mode so to make additional savings. The active DUs provide a certain amount of capacity that possibly overshoots the workload $L$. This leads to the presence of a margin $W = TC \cdot \mathbf{x} \cdot \mathbf{y} + (T - T_{on})C \cdot (1 - \mathbf{y}) \cdot \mathbf{x} - L$. The controller can then decide to put some slots of the frame into micro sleep mode, at most $\lfloor W/\min(C) \rfloor$, as each slot provides $\min(C)$. This provides additional savings of $S = \lfloor W/\min(C) \rfloor (P_a - P_s^\mu)$. We detail the micro-sleep procedure in algorithm 2. We expect our heuristic solution to benefit from the micro sleep, as the binary decision most time overshoots the workload and the micro sleep procedure can spend the margin to derive additional savings.

## IV. FLEXIBLE SLEEP MODE DECISION

While the periodic sleep mode decision is easier to implement by the controller, it does not fully exploit the potential of energy savings. We extend in this section the scheme to a more flexible sleep mode start.

### A. From discrete to continuous

We change in this subsection the formulation of (P) into a continuous program, where the new decision variable $x_i$ is the time to shut down an active server or the time to activate a sleeping server within the frame. We expect this approach to show better results as the degree of freedom is higher.

For DUs that were active at the end of the previous period ($y_i = 1$), we define the energy consumed by the DU as a function of $x_i$, time of deactivation:

$$ 
\begin{aligned}
E(x_i|y_i = 1) &= x_i P_a + T_{off} P_{off} + (T - T_{off} - x_i) P_s \\
&= (P_a - P_s)x_i + T_{off} P_{off} + (T - T_{off}) P_s. \quad (5)
\end{aligned}
$$

$$ 
\begin{aligned}
E(x_i|y_i = 1) &= x_i P_a + (T - x_i) P_{off} + 0P_s \\
&= x_i(P_a - P_{off}) + TP_{off}. \quad (6)
\end{aligned}
$$
<figure><img src="image_5.png" /><figcaption>(a) Initially active DU</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(b) Initially asleep DU</figcaption></figure>

<caption>Fig. 4: Piece-wise affine functions</caption>

$$E(x_i|y_i = 1) = TP_a. \tag{7}$$

We remark that the function $E(x_i|y_i = 1)$ defined on $[0, T]$ is concave, as shown on Fig. 4-a.

For the DU $i$ such as $y_i = 0$ (the power state of the DU is off), we define the energy consumed by the DU as a function of $x_i$, time of activation:

$$ 
\begin{aligned}
& \text{for } x_i \in [0, T - T_{\text{on}}), \\
& E(x_i|y_i = 0) = P_s x_i + T_{\text{on}} P_{\text{on}} + (T - T_{\text{on}} - x_i) P_a \\
& \qquad = (P_s - P_a) x_i + T_{\text{on}} P_{\text{on}} + (T - T_{\text{on}}) P_a. \quad (8)
\end{aligned}
$$

for $x_i \in (T - T_{\text{on}}, T)$,

$$ 
\begin{aligned}
& \text{for } x_i \in (T - T_{\text{on}}, T), \\
& E(x_i|y_i = 0) = P_s x_i + (T - x_i) P_{\text{on}} + 0 P_a \\
& \qquad = (P_s - P_{\text{on}}) x_i + T P_{\text{on}}. \quad (9)
\end{aligned}
$$

for $x_i \ge T$,

$$E(x_i|y_i = 0) = TP_s. \tag{10}$$

We remark that the function $E(x_i|y_i = 0)$ defined on $[0, T]$ is convex, as shown on Fig. 4-b.

We formulate the following optimization problem:

$$ (\mathcal{L}) : \min_{x} \sum_{i \in \mathcal{A}} E(x_i|y_i = 1) + \sum_{i \in \mathcal{S}} E(x_i|y_i = 0) \tag{11} $$

$$ \text{s.t.} \quad \sum_{i \in \mathcal{A}} C_i x_i + \sum_{i \in \mathcal{S}} C_i \max(0, T - T_{\text{on}} - x_i) \ge L \tag{12} $$

$$ \forall i, \quad x_i \in [0, T] \tag{13} $$

We define $\mathcal{A}$ the set of initially active DUs, and $\mathcal{S}$ the set of initially asleep DUs.

As both functions $E(x|y = 1)$ and $E(x|y = 0)$ are piece-wise affine functions, they can be expressed as $\min_{j=1,2} a_j x + b_i$ and $\max_{j=1,2} c_j x + d_j$, respectively. The reader can refer to Fig. 4 to find the values of $a_1, a_2, b_1, b_2$, etc.

## B. Problem linearization

We attempt to linearize the problem $(\mathcal{L})$. As the function $E(x|y = 1)$ is concave, the epigraph method does not apply and we must introduce a special ordered set of type 2 or binary variables to account whether $x_i \in [0, T - T_{\text{off}})$ or $x_i \in [T - T_{\text{off}}, T]$. One solution is to restrict the possible values of $\{x_i\}_{i \in \mathcal{A}}$ to the subset $[0, T - T_{\text{off}}]$, where $E(x|y = 1)$ is affine hence convex. The linearization of $\max(0, T - T_{\text{on}} - x_i)$ also requires the introduction of binary variables to differentiate the cases $x_i \in [0, T - T_{\text{on}})$ or $x_i \in [T - T_{\text{on}}, T]$. Similarly, we linearize $\max(0, T - T_{\text{on}} - x_i)$ to $T - T_{\text{on}} - x_i$ by restricting

possible values of $\{x_i\}_{i \in \mathcal{S}}$ to the subset $[0, T - T_{\text{on}}]$. The problem $(\mathcal{L})$ becomes:

$$ (\mathcal{M}) : \min_{x,t} \sum_{i \in \mathcal{A}} t_i + \sum_{i \in \mathcal{S}} t_i \tag{14} $$

$$ \text{s.t.} \quad a_1 x_i + b_1 \le t_i, \quad \forall i \in \mathcal{A} \tag{15} $$

$$ a_2 x_i + b_2 \le t_i, \quad \forall i \in \mathcal{A} \tag{16} $$

$$ c_1 x_i + d_1 \le t_i, \quad \forall i \in \mathcal{S} \tag{17} $$

$$ c_2 x_i + d_2 \le t_i, \quad \forall i \in \mathcal{S} \tag{18} $$

$$ x_i \ge 0 \quad \text{and} \quad x_i \le T - T_{\text{off}}, \quad \forall i \in \mathcal{A} \tag{19} $$

$$ x_i \ge 0 \quad \text{and} \quad x_i \le T - T_{\text{on}}, \quad \forall i \in \mathcal{S} \tag{20} $$

$$ \sum_{i \in \mathcal{A}} C_i x_i + \sum_{i \in \mathcal{S}} C_i (T - T_{\text{on}} - x_i) \ge L \tag{21} $$

The problem $(\mathcal{M})$ is LP but sub-optimal. Indeed, the resolution of $(\mathcal{M})$ forces to switch mode for all DUs: the once active DUs all switch to inactive DUs by the end of the frame, while the inactive DUs all wake up. This leads to high transition costs and yields to sub-optimal performance.

To avoid this downfall, we update the sets $\mathcal{A}$ and $\mathcal{S}$ with the rules:

$$ \text{if } x_i = T - T_{\text{off}}, \quad \text{the DU } i \text{ stays active,} \tag{22} $$

$$ \text{if } x_i = T - T_{\text{on}}, \quad \text{the DU } i \text{ stays asleep.} \tag{23} $$

The obtained problem is no longer LP due to the constraints (32) and (33):

$$ (\mathcal{M}') : \min_{x,t} \sum_{i \in \mathcal{A}} t_i + \sum_{i \in \mathcal{S}} t_i \tag{24} $$

$$ \text{s.t.} \quad a_1 x_i + b_1 \le t_i, \quad \forall i \in \mathcal{A} \tag{25} $$

$$ a_2 x_i + b_2 \le t_i, \quad \forall i \in \mathcal{A} \tag{26} $$

$$ c_1 x_i + d_1 \le t_i, \quad \forall i \in \mathcal{S} \tag{27} $$

$$ c_2 x_i + d_2 \le t_i, \quad \forall i \in \mathcal{S} \tag{28} $$

$$ x_i \ge 0 \quad \text{and} \quad x_i \le T - T_{\text{off}}, \quad \forall i \in \mathcal{A} \tag{29} $$

$$ x_i \ge 0 \quad \text{and} \quad x_i \le T - T_{\text{on}}, \quad \forall i \in \mathcal{S} \tag{30} $$

$$ \sum_{i \in \mathcal{A}} C_i f(x_i) + \sum_{i \in \mathcal{S}} C_i (T - T_{\text{on}} - x_i) \ge L \tag{31} $$

$$ f(x_i) = x_i \quad \text{if} \quad x_i < T - T_{\text{off}} \tag{32} $$

$$ f(x_i) = T \quad \text{if} \quad x_i = T - T_{\text{off}} \tag{33} $$

The linearization of $(\mathcal{M}')$ involves the introduction of $|\mathcal{A}|$ new binary variables to differentiate the cases $x_i \in [0, T - T_{\text{off}})$ and $x_i = T - T_{\text{off}}$.

## C. Complexity analysis

We use the CBC MILP solver to address the problem $(\mathbb{P})$. Its branch and bound method consists of fixing one decision variable $x_i = 1$ (or $x_i = 0$), and then solve the linear subproblems with $n-1$ boolean variables. In the worst case, the algorithm ends up solving $2^n$ linear problems. The bounding technique indeed solves the linear relaxation of each subproblem. In practice, the pruning discards lots of branches of the search tree and leads to faster execution time. Nevertheless, the time complexity of $(\mathbb{P})$ is $O(2^n)$.
The time complexity of the heuristic is $O(n)$, given that the while loop terminates when the condition is no longer valid or when the sum of the $n$-dimensional vector $\boldsymbol{x}$ reaches value $n$ or $0$.

As of the continuous problem, the time complexity of $(\mathbb{M}')$ is $O(2^{|\mathcal{A}|}P(n))$. The $|\mathcal{A}|$ binary variables differentiate the cases $x_i \in [0, T - T_{\text{off}})$ and $x_i = T - T_{\text{off}}$ for the active DUs. The solution must explore the $2^{|\mathcal{A}|}$ possibilities, and for each one solve the continuous linear problem with a simplex method in polynomial time.

## V. SIMULATION RESULTS

Our simulation settings consider a 1 km square cell with $n$ non-overlapping RUs ensuring full coverage. The DUs are inter-connected without delay, hence they can process as a whole the given total workload.

Let $L$ (in Mbps) be the total workload to be processed during the period. Without loss of generality, we consider in our numerical results identical DUs (with common capacity $C$ in Mbps/slot). We define the traffic load $\alpha \in [0, 1]$ as the ratio between the offered workload and the total processing capacity $\alpha$, where $n$ is the number of available DUs and $T$ is the frame duration (in slots). Note that we only consider values of $\alpha = \frac{L}{nCT}$ as, if the workload is greater than the total available capacity, no energy gains can be derived and all DUs must stay active.

### A. Energy consumption

In Fig. 5, we consider two different results against the total traffic expressed in terms of the percentage $\alpha$ of the total available capacity, namely the energy consumed, and the margin of unused resources after the algorithm is applied (in Mbps). We look at a system of $n = 5$ DUs, with a frame length of $T = 10$ ms. The capacity of each DU is $C = 10$ Mbps/ms. The solutions MILP, heuristic, LP-MIP are respectively the exact solutions to problem (P), the heuristic sub-optimal solution of (P) using Algorithm 2, and the solution of the flexible continuous time decision problem $(\mathbb{M}')$. The LP-MIP solution gives the best results and provides substantial energy gains compared to the other methods. It benefits from its possibility to activate or deactivate the DU at any time during the frame, while the two other methods force the decision at the beginning of the frame. As a result, we observe the consumed energy augments in a stair fashion. The heuristic method provides fairly good results given its very low time execution. The LP-MIP solution has the lowest margin, which means the usage of capacity is efficient and waste of active resources is small.

### B. Relative energy gains

In Fig. 6a, we consider the relative energy gains of each method compared to the worst performing method being the heuristic. We look at how the system size (in terms of total number of available DUs) can affect those gains. We simulate 100 runs with different random values of $\alpha$ and display the average gains for each $n$ varying from 1 to 20. The frame duration is $T = 10$ ms. The transition times $T_{\text{off}}$ and $T_{\text{on}}$ are both equal to 3. The capacity of one DU is $C = 10$ Mbps/ms.

<figure><img src="image_7.png" /><figcaption>Fig. 5: $n = 5$ DUs, $T = 10$, $T_{\text{off}} = T_{\text{on}} = 3$, $C = 10$.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6: Relative energy gains.</figcaption></figure>

We expect the continuous approach to derive smaller relative gains as the total number of DUs increases. This is due to the convergence of the binary methods' energy profiles to the continuous approach's. As $n$ grows, it smoothens the stair type shape observed in Fig. 5 (in orange and blue) that converges to the linear shaped function (in green) of the continuous method. We observe that the relative energy gains of the LP-MIP solution are huge for small systems (less than 5 DUs): LP-MIP provides 31.8% gains for a simple system of $n = 2$ DUs, and 13.0% gains for $n = 5$ DUs. As the system size increases, the relative gains derived from LP-MIP diminish (down to 3.78% for a large system of $n = 20$ DUs), while the execution time grows exponentially due to the LP-MIP time complexity of $O(2^{|\mathcal{A}|}P(n))$. This makes the LP-MIP method an interesting solution for small systems, while the con of high complexity overcomes the fewer gains for large systems. Although gains are smaller, they remain substantial for large systems.

In Fig. 6b, we display the relative energy gains against the frame duration. We simulate 100 runs for each value of $T$ that ranges from the minimum value from which light sleep becomes interesting over micro sleep (that is $T = 10$ ms) to $T = 50$ ms. We consider a medium size system of $n = 10$ DUs, transition times $T_{\text{off}} = T_{\text{on}} = 3$ and capacity $C = 10$ Mbps/ms. We notice significant increase of gains from 7.30% to 9.59% of the LP-MIP solution, while the MILP method provides smaller gains with larger frame duration. The relative gains increase is even bigger with smaller systems: for $n = 2$ DUs, the relative gains go from 31.8% at $T = 10$ to 39.0% at $T = 50$. Note that we can enlarge the frame duration only if the traffic type allows it (e.g. type eMBB usually afford a delay around 10 ms).

### C. Mixed-Time Scale gains

In this part, we investigate the additional gains the controller can derive by leveraging micro sleep decisions. In Fig. 7a, we observe that the heuristic method benefits the most from the micro sleep, to the point heuristic-micro outperforms MILP-micro and converges to the optimal performance of LP-MIP
<table><caption>TABLE II: $n = 5$ DUs, $T = 10$, $T_{\text{off}} = T_{\text{on}} = 3$, $C = 10$.</caption><thead><tr><th>Margin/Waste (Mbps)</th><th>w/o micro</th><th>w/ micro</th></tr></thead><tbody><tr><td>MILP</td><td>25</td><td>2.8</td></tr><tr><td>heuristic</td><td>46</td><td>2.8</td></tr><tr><td>LP/MIP</td><td><b>5.2</b></td><td><b>1.4</b></td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>Fig. 7: $C = 10$, $P_{\text{active}} = 280$, $P_{\text{micro}} = 55$, $P_{\text{on}} = P_{\text{off}} = 90$.</figcaption></figure>

and LP-MIP-micro. The gains of LP-MIP-micro relative to LP-MIP are very small (see the zoom in Fig. 7a). As can be seen from Table II, the margin of LP-MIP is already low, while the two other solutions have a high margin that can be exploited. We display average values of the margin over 100 different traffic loads ($\alpha$ varies from 0.01 to 0.99) in Table II. In the first column, we display the average value without micro mode, and with micro mode in the second column.

To mitigate the promising results of the heuristic-micro solution, we need to make its sensitivity analysis against the parameter $P_{\mu}/P_{s}$ that is the ratio between the energy consumptions in micro and light sleep modes. We display the results in Fig. 8a. We select the values $P_{s} = 25$, $P_{a} = 280$ and we vary $P_{\mu}$ from 55 (cf. Table I) to 280. We notice that both MILP-micro and heuristic-micro are sensitive to the increase of $P_{\mu}$ and, are equivalent to MILP and heuristic respectively, when $P_{\mu} = P_{a} = 280$. Therefore, the micro sleep procedure only works in a system where the value of $P_{\mu}$ is sufficiently low. According to [2] settings ($P_{\mu} = 55$), the heuristic-micro becomes a very valid solution. In Fig. 8b, we show the performance of the micro mode solutions against the frame length $T$. As $T$ increases, the relative gains of the micro solutions increase as well. As the margin $W$ grows with $T$, the micro solutions can benefit more and derive bigger gains.

## VI. CONCLUSIONS

In this paper, we propose to reduce the energy consumption at the baseband processing part of the RAN by leveraging advanced sleep modes. We formulate the activation/deactivation of DUs as an optimization problem, accounting for the activa-

<figure><img src="image_10.png" /><figcaption>Fig. 8: Sensitivity analysis.</figcaption></figure>

tion/deactivation delay of the sleep mode. We first formulate a simple problem where decision is taken at the beginning of the time frame and formulate it as a MILP with an associated low complexity heuristic. We then modeled the more advanced scheme where the switching time is continuous and formulated an LP-MIP problem. These algorithms can be combined with micro sleep mode in order to reduce the wasted resources and the resulting mixed time scale decision algorithm significantly reduces the energy consumed by the system.

## ACKNOWLEDGMENTS

This research work was supported by the sustainable 6G chair, held by CentraleSupélec and funded by Orange.

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