BS Sleeping Index in Ultra-Dense Networks

• BS Sleeping Index is a metric that quantifies network redundancy by combining UE-transferability (k_i) and backup resource counts (l_i) to ensure service continuity during BS dormancy.
• It supports coordinated load sharing via a greedy selection algorithm that prioritizes BSs with optimal residual capacity, safeguarding SINR thresholds for all UEs.
• The index enhances energy efficiency and convergence speed in ultra-dense networks by dynamically recalculating potential deactivations while maintaining quality of service.

The term "BS Sleeping Index" refers to a quantitative metric introduced in the context of ultra-dense networks (UDNs), specifically within energy-saving strategies for small-cell base stations (s-BSs) in 5G and beyond. The index is constructed to prioritize candidate BSs for transition into a dormant state, balancing energy efficiency (EE) gains against the requirement for seamless service continuity to user equipment (UE). The notion of BS Sleeping Index is formalized and operationalized in the context of coordinated load-sharing and multi-objective optimization in UDNs (Ren et al., 20 Jan 2026).

1. Formal Definition and Mathematical Structure

The BS Sleeping Index for the $i$-th small-cell BS is defined by

$\beta_{i} = k_{i} \times l_{i}$

where:

• $k_{i} \in \{0, 1\}$ is the UE-transferability indicator.
• $$l_{i} = \sum_{j \in \mathcal{U}_{i}} |\mathcal{B}_{i,j}|$$ quantifies the total backup-BS resource.

Here, $\mathcal{U}_i$ is the set of all UEs currently served by BS $i$, and $\mathcal{B}_{i,j}$ for user $j \in \mathcal{U}_i$ is the set of active backup BSs offering signal-to-interference-plus-noise ratio (SINR) above threshold $\theta_{\mathrm{th}}$: $$\mathcal{B}_{i,j} = \{ n \neq i \mid s_n = 1, \mathrm{SINR}_{n,j} \ge \theta_{\mathrm{th}} \}$$ The indicator $k_i$ is set to 1 if every UE has at least one backup BS ($|\mathcal{B}_{i,j}| \ge 1$ $\forall j$), otherwise 0.

2. Rationale and Design Principles

The product form $\beta_i = k_i l_i$ embeds key operational constraints and optimization criteria:

• Service continuity is strictly enforced by $k_i$. If any UE lacks a backup BS satisfying the SINR threshold, $k_i$ becomes zero, immediately disqualifying BS $i$ from sleeping consideration.
• Redundancy quantification is captured by $l_i$, which accumulates across UEs the number of candidate handover BSs. Higher values signify greater neighborhood flexibility for traffic offloading and thus increased dormancy suitability.

This construction ensures that only BSs whose entire load is safely re-routable (as per network topology and SINR constraints) contribute to the pool of sleep candidates, with higher $l_i$ further prioritizing deactivation order under otherwise similar conditions.

3. Algorithmic Role and Decision Process

The BS Sleeping Index is central to the iterative process of coordinated BS deactivation. The operational workflow proceeds as follows:

1. For each active BS $i$ ($s_i = 1$), $\beta_i$ is computed according to the above formula.
2. Only BSs with $\beta_i > 0$ are eligible for sleeping consideration.
3. At each iteration, the algorithm selects

$\text{BS}^* = \arg\max_{i : s_i = 1} \beta_i$

as the highest-priority sleep candidate.

1. A local load-sharing feasibility check is performed prior to switching off BS$^*$. If takeover BSs can accommodate the full load and maintain SINR above threshold for all UEs, BS$^*$ is set to sleep; otherwise, the next-highest candidate is considered.

This closed-loop procedure is repeated until no further BSs have positive $\beta$ or until load-sharing constraints are violated for all remaining candidates.

4. Stepwise Computation and Pseudocode Workflow

The computational realization may be captured in the following structured pseudocode (see (Ren et al., 20 Jan 2026) for authoritative implementation):

for i in N_active: U_i = {j for j in M if A[i, j] == 1} k_i = 1 l_i = 0 for j in U_i: B_ij = {n for n in N_active if n != i and SINR[n, j] >= theta_th} if len(B_ij) == 0: k_i = 0 l_i += len(B_ij) beta_i = k_i * l_i if max(beta_i) == 0: break i_star = argmax(beta_i) success = AttemptLoadShare(i_star) if success: s[i_star] = 0 update A accordingly else: exclude i_star from further consideration

The AttemptLoadShare subroutine ensures that load migration never overloads neighbor BSs and all UEs maintain required communication quality.

5. Integration with Load-Sharing Takeover and Convergence

The BS Sleeping Index is the linchpin of the larger optimization framework targeting minimizing the number of active BSs and maximizing EE. Its greedy selection mechanism ensures that, at each step, the most redundant and easily-offloadable BS is deactivated first. Feasibility is continuously enforced through dynamic load evaluations and SINR checks. As the active set evolves, both $\beta_i$ and BS loads are recalculated at every iteration, maintaining adaptive responsiveness to network topology and load distribution.

Empirical results demonstrate that, compared to baseline schemes, this approach significantly improves algorithmic convergence speed, more optimally reduces active BS count, and enhances EE, substantiating the strategic role of $\beta_i$ in modern UDN energy management (Ren et al., 20 Jan 2026).

6. Comparison to Sleep Indices in Other Domains

The term "sleeping index" in the context of BS management should be distinguished from similarly named indices in human sleep physiology, notably the Sleep Depth Index (SDI) introduced for polysomnographic signal annotation. SDI is a continuous scalar in $[0,1]$ representing sleep depth and is learned via an end-to-end deep learning framework with pairwise ranking loss, targeting neurophysiological time series rather than network resource allocation (Zhou et al., 2024). Both indices operationalize a notion of "readiness for dormancy" but in fundamentally disparate domains.

7. Significance in Ultra-Dense Network Optimization

The introduction of the BS Sleeping Index marks a substantive advance in the coordinated management of small-cell resources. By quantitatively encoding network redundancy and UE handover feasibility, $\beta_i$ enables scalable, responsive, and service-aware BS deactivation. This contributes to resolving key challenges intrinsic to ultra-dense deployments, such as load imbalance, identification of redundant infrastructure, and joint optimization of energy efficiency with quality of service. Its formalization and practical validation underscore its utility as a core analytical and operational tool in the ongoing evolution of self-organizing energy-efficient wireless systems (Ren et al., 20 Jan 2026).