Introduction

The rise in high-speed devices is driving the need for greater network capacity. Ericsson’s 2022 report predicts mobile data traffic will more than double by 2027, with mobile video traffic growing by nearly 30% annually1. Academics and industry experts agree that current advancements in communication technology are inadequate to meet the future surge in data demands. As a result, there is increasing acknowledgment of the necessity for a fundamental shift in the design and implementation of future HetNet2,3,4,5,6.

The expectations for 5G are extensive, including increased capacity, high data rates, low latency, improved energy efficiency, robust security, seamless connectivity, and support for a diverse array of applications. Figure 1 details some key enabling technologies identified by researchers to address the key requirements of HetNets. In recent years, researchers have highlighted five key potential technologies named the 'Big Five' for achieving the objectives of 5G networks7.

Fig. 1
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5G HetNet demands.

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To meet the catastrophic traffic demands of 5G, cell densification or network densification emerges as the easiest, logical, and straightforward approach, which increases the quantity of deployed network equipment and brings base stations (BSs) closer to mobile users. Out of the five key enabling technologies, HetNets and m-MIMO stand out as the two main types of network densification technologies with significant potential. Both technologies promise effective solutions for achieving high coverage gains and data throughput. However, while m-MIMO has limitations such as high infrastructure cost and complex network analysis, HetNets emerged as the key enabler technology for attaining energy-efficient cellular networks, as identified in many research studies3,8,9,10.

In addition, energy efficiency is a critical issue in the development of 6G networks, with numerous studies emphasizing its significant impact. Researchers are investigating how enhanced energy efficiency can improve network performance, user experience, and resource management, making it a focal point in the advancement of 6G technology. For instance, several studies have examined the integration of energy-efficient technologies and their potential to reduce the overall energy consumption of 6G networks11,12,13. This research highlights the importance of optimizing energy usage to achieve sustainable and efficient network operations. Furthermore, advanced AI-driven solutions are being explored to dynamically manage energy resources, ensuring that 6G networks can meet the growing demands of users while minimizing their environmental footprint.

Table 1 highlights the potential of HetNets as a cost-efficient approach to achieving the main goals of 5G networks. In contrast, m-MIMO falls short compared to HetNets and faces challenges related to coverage and implementation costs. They also deal with issues such as intricate network analysis, channel estimation, pilot contamination, and high deployment expenses14,15,16,17. Therefore, the expected outcomes of m-MIMO are fundamentally limited due to the above factors. Thus, HetNets are regarded as the most promising and practical solution for meeting the core requirements of 5G HetNets18,19.

Table 1 Comparison of HetNets and m-MIMO technologies.
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Literature review

It has been now globally acknowledged that the energy-related issue will have a greater impact in the coming years on wireless communication networks. There are different techniques to minimize the consumption of energy in cellular networks. Broadly, these approaches can be divided into five different categories listed in Fig. 2. Each technique has its own benefits and limitations. Upgrading hardware equipment e.g. power amplifiers, BS site relocation, and reselection, can provide maximum energy saving but requires costly solutions for hardware replacements17.

Fig. 2
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Energy efficiency approaches in cellular network design.

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Optimization techniques for radio transmission, such as cooperative relays, cognitive radios, signal resource allocation, and channel coding schemes, are generally more cost-effective. However, these methods come with higher error risks due to uncertainties and involve trade-offs between energy efficiency and system performance27.

Deployment and planning of advanced infrastructure like integrated macro and small cells will introduce new challenges to the radio interface. Adoption of renewable resources like solar, hydro, and wind power It will entail a high replacement cost but can serve as a long-term solution, and for current BS there are limited gains28. Lastly, sleep mode techniques, such as turning on or off underutilized base stations, offer simpler implementations and cost-effective solutions for improving energy efficiency29,30,31,32,33,34.

The pros and cons of each technique are summarized in Table 2. Among the various EE techniques, sleep mode schemes emerge as the most practical choice for cellular networks. These schemes are cost-effective and potentially viable for enhancing the energy efficiency of current HetNets, as they do not necessitate hardware replacements. However, before activating sleep mode in specific BSs, it is crucial for these BSs to coordinate with neighboring active BSs and relinquish their channel resources. The active BSs must then extend their coverage and data rate services to users who would otherwise be served by the sleeping BSs.

Table 2 Summary of base station energy-saving techniques.
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The BS Controller typically manages the exchange of traffic information between BSs. However, transitioning a BS to sleep mode can result in increased outage probability and call-blocking events, which ultimately deteriorates the QoS experienced by users23.

Figure 3, represents the existing BS sleep-enabling techniques in wireless cellular networks MUA is a critical attribute in cellular networks because it is used to determine the method through which each user will be connected to the BSs.

Fig. 3
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Sleep activation methods.

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In 3GPP standards, the concept of Self-Organizing Networks (SON) was introduced and later implemented in WiMAX LTE and other 4G standards. SON streamlines network operations and adjustments without the need for manual intervention. This capability ensures more efficient and adaptive network performance35.

Cell breathing, also known as cell zooming, is a concept similar to SON but offers greater flexibility. It involves dynamically adjusting cell sizes based on data traffic conditions by modifying the base station’s (BS) antenna transmission power, tilt angle, or height. From an implementation perspective, cell breathing is simpler than switching BSs on or off. Nevertheless, deploying small cells is currently regarded as the most recent and effective method for enhancing energy efficiency in wireless cellular networks19,36,37,38,39. Table 2 represents the summary of BS sleep mode techniques in a cellular network.

In HetNets, research has predominantly concentrated on enhancing data throughput, coverage, and resource efficiency, often overlooking EE. The energy performance of HetNets remains underexplored. When small cells in HetNets enter sleep mode, two main types of sleep schemes are used: static/fixed and dynamic. fixed schemes involve a predetermined number of state changes within a specific timeframe. In contrast, dynamic schemes allow base stations to adjust their states as frequently as necessary, depending on factors such as traffic and channel conditions34,40,41.

The dynamic sleep mode scheme generally outperforms fixed schemes but involves greater computational complexity due to the increased frequency of switching operations. This dynamic approach is further divided into CS and DS.

In Centralized Schemes, a central entity maintains comprehensive knowledge of all BSs within the network, allowing for more effective network optimization and decision-making based on the entire network's status. This centralized control can lead to significant performance improvements. However, it also introduces substantial complexity in managing and processing information from all BSs. Distributed Schemes, by contrast, distribute control and decision-making across multiple BSs. This method simplifies implementation and management as each BS operates based on local information and interactions with neighboring BSs. Despite the ease of implementation, Distributed Schemes often face performance trade-offs because they lack the global network oversight that Centralized Schemes provide. Without a central point of control, optimizing network-wide operations can be less effective. Additionally, incorporating sleep mode in small cells can reduce overall energy consumption compared to a network that relies exclusively on macro cells. By enabling small cells to enter sleep mode during periods of low traffic, the network can lower power usage and improve energy efficiency, offering a significant advantage over the constant operation of macro cells alone42,43. However, sleep mode also presents major challenges that have a direct effect on key QoS like outage rates, coverage, and throughput. To address these challenges, one effective approach involves using random spatial models. These models can integrate stochastic geometry and distributed small-cell sleep methods to more accurately simulate the complex realities of HetNet environments. Figure 4 illustrates the multidimensional issues related to the EE of HetNets.

Fig. 4
figure 4

EE multidimensional issues.

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Author in44 has investigated the algorithm for energy-efficient resource management for split spectrum using the Lagrange Dual Decomposition method for various traffic hours. However, they have adopted the use of static data traffic profiles under normalized conditions45. Investigates energy efficiency by adopting activity-aware sleeping modes strategies are proposed. The strategic sleep scheme was based on the activity of macrocell users. Numerical results confirm that dynamic sleep improves network coverage and EE, the authors conclude that EE will saturate as a number of the small cell reaches a threshold level. As stated, the introduction of sleep mode strategies in small BSs will greatly improve overall network performance by reducing power consumption, but it also brings some trade-offs between QoS and EC. A cluster-based approach that outperforms the other strategies is addressed by42. Based on the network model of46, energy efficiency can also be improved2. However, it should be noticed that the sum data rate maximization of all users may result in unfair allocated of data throughput allocation, which is indicated by the simulated results of Fig. 3 in46, which clearly indicates that the load on small cells is heavier as compared to a macro cell which eventually results in congested small cells. Table 3 summarized the advantages and disadvantages of fixed and dynamic sleep mode schemes.

Table 3 Comparison of small cell BS sleep schemes.
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Consequently, only a select group of users located at the center of the macro cell may experience high data rates, whereas others could encounter service outages. However, this method does not contribute to improving the network's energy efficiency and disregards the adverse effects it may have on QoS parameters. Specifically, while the approach may address certain operational aspects, it does not optimize power consumption, leading to continued high energy use. Additionally, it overlooks how the strategy can negatively impact QoS factors such as latency, reliability, and overall user experience, which are crucial for maintaining effective and satisfactory network performance2.

In HetNets, EE presents a complex array of challenges, particularly concerning the implementation of sleep mode strategies. These challenges manifest across multiple dimensions. First and foremost are deployment issues, encompassing considerations such as network topology, traditional modeling techniques, associated metrics, and the coordination of user equipment (UE). Traditional models often fail to adequately capture the intricacies of HetNet deployments, leading to oversimplified representations, assumptions of homogenous node distributions, and unrealistic scenarios. These models often rely on SNR-based approaches and pre-defined traffic profiles, neglecting the nuanced impact on QoS parameters. To address these challenges, an analytical framework is essential.

One approach is the utilization of random spatial models, which can incorporate stochastic geometric and distributed small-cell sleep methodologies to better simulate the complex realities of HetNet environments. Figure 4, represent the multidimensional issues related to EE of HetNet.

State of the art sleep mode strategies

Upgrading hardware components can enhance network efficiency by incorporating newer, more energy-efficient technology, though this often involves high costs and the complexities associated with installation and integration. Radio transmission optimization schemes aim to improve performance by optimizing radio resources to reduce energy usage. However, these schemes can be complex and require significant updates to existing systems. Sleep mode techniques are designed to save energy by reducing power consumption during periods of low activity. Despite their effectiveness, they may introduce latency and impact network reliability due to frequent transitions between active and sleep states. MUA enhances resource allocation and coverage by efficiently distributing users across available cells, but managing these associations can be complex, especially with dynamic user mobility.

SON automates network configuration and adapts to changes, reducing the need for manual intervention and optimizing performance. Nevertheless, the complexity of SON algorithms and increased computational and management overhead present significant challenges. In Small Cells HetNets settings, static or fixed schemes offer simplicity and stable performance but lack flexibility and may lead to inefficiencies if not optimally designed. Dynamic schemes, including both centralized and distributed control strategies, provide greater adaptability. Among these, Random Sleep Mode (RSM) and Load Adaptive Sleep Mode (LAS) are state-of-the-art techniques in HetNets36,37,39,47. RSM involves turning small BSs on or off based on random probabilistic models such as Bernoulli, uniform, Poisson, and exponential processes. While this approach is straightforward, it has limitations, including inadequate adaptability to unpredictable conditions and potential negative impacts on overall network energy efficiency. LAS addresses energy consumption by turning off less-utilized small cell BSs in response to non-uniform traffic load distributions. LAS methods, which include adaptive algorithms based on complete or incomplete Channel State Information (CSI), aim to minimize time-varying energy consumption due to static data traffic profiles45. Despite their advantages, LAS strategies often assume periodic traffic profiles and may not fully account for real-world variability. Authors in48 investigate the energy efficiency of flooding protocols in multi-hop VANETs. They find that restrictive flooding is more energy-efficient than plain flooding while maintaining similar reliability levels. Authors in49 use Lagrangian duality theory to achieve optimal power and rate control, ensuring interference to cellular users remains below a predefined threshold while maintaining proportional fairness. This study provides valuable insights into energy-efficient interference management in wireless network networks. Authors in50, propose a distance-sensitive distributed repulsive sleeping approach to enhance dependable coverage in heterogeneous cellular networks.

However, Distributed Sleep Strategies offer a more nuanced approach by incorporating realistic network models that account for random spatial impacts, user density variations, and diverse traffic conditions. It provides low computational complexity solutions for energy-efficient HetNets under QoS constraints. It optimizes the trade-off between average throughput and energy consumption while considering QoS constraints such as coverage probability and average wake-up time delay51.

Figure 5 illustrates the state-of-the-art power-off techniques used in HetNets, focusing on various strategies for optimizing energy consumption by turning off small base stations (BSs). The figure highlights the different approaches, including fixed and dynamic schemes, as well as centralized, distributed, and hybrid methods.

Fig. 5
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State-of-the-art sleep schemes in HetNets.

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Proposed dynamical DOSS model

Comparing centralized with distributed schemes, distributed schemes offer superior ease of implementation and manageability. They distribute the decision-making process among the BSs within each cluster of size \({N}_{c}\), reducing the need for extensive central coordination. This local decision-making capability simplifies the management of the network, particularly in large-scale deployments. However, this decentralized approach can come with trade-offs in terms of performance. Distributed schemes may not achieve the same level of network-wide optimization as centralized schemes because each BS operates based on local information rather than having a comprehensive view of the entire network52. As a result, while DS can respond more quickly to local changes and reduce computational burdens, they might miss the global optimization potential that CS can provide.

On the other hand, centralized schemes have the advantage of possessing complete knowledge of all BSs in the entire network. This comprehensive view allows for more holistic and potentially more effective network optimization. Centralized algorithms can make more informed decisions that consider the overall network state, leading to higher performance in terms of resource allocation and energy efficiency. However, this advantage comes at the cost of higher computational complexity and processing time. The central controller must process a vast amount of data and perform complex computations, which can be time-consuming and resource-intensive.

The proposed distributed approach attempts to balance these considerations by dividing the network into manageable clusters, allowing for localized optimization while still maintaining an overarching strategy. The basic building block of the D-DOSS scheme is illustrated in Fig. 6, which shows the division of the network into clusters and the strategic algorithm employed to optimize energy efficiency and performance within each cluster. This method leverages the benefits of distributed decision-making to simplify management and reduce computational complexity, while also aiming to achieve a level of performance that approaches the effectiveness of centralized schemes.

Fig. 6
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Proposed scheme implementation and objectives.

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If \({N}_{c}=1\), the CS and DS both will be identical. In a K-tier network which contains \(N\) total numbers of users. At any point X in VT, within an area other f \(A\), the probability of a BS to serve a user for three different cases is given as,

Case (i): \(N=n\),

$$\text{P}\left(\text{N}=\text{n}\right)={\int }_{0}^{\infty }P\left[N=n|X=A\right]{f}_{x}\left(A\right)dA\cong \frac{{\lambda }_{u}^{n} {(\rho {\lambda }_{BS})}^{K}\Gamma (n=\rho )}{\Gamma \left(\rho \right)n!({\lambda }_{u}+\rho {\lambda }_{BS}{)}^{n+\rho }}$$
(1)

Equation 1, calculates the probability that a base station (BS) serves exactly \(\text{n}\) users. This is done by integrating the conditional probability \(P\left[N=n|X=A\right]\), which represents the likelihood that the BS serves \(\text{n}\) users given that it is within the area \(A\) of the Voronoi tessellation (VT). The integral is taken over all possible values of the area \(A\) and is weighted by the probability density function \({f}_{x}\left(A\right)dA\), which describes how likely it is for a given area \(A\) to occur within the network. Essentially, this integration provides the overall probability that a BS serves \(n\) users, considering all possible configurations of the network.

In this approximation, \({\lambda }_{u}^{n}\) represents the user intensity raised to the power of \(n\), reflecting the likelihood of \(n\) users being present. The term \({(\rho {\lambda }_{BS})}^{K}\) involves the constant \(\rho \) and the base station density \({\lambda }_{BS}\) raised to the power of \(K\), indicating the density of BSs and their impact on serving users. The gamma function term \(\Gamma \left(n=\rho =3.6\right)\) adjusts for the distribution of users within the cells, while \(\Gamma \left(\rho \right)n!\) and \(({\lambda }_{u}+\rho {\lambda }_{BS}{)}^{n+\rho }\) provide normalization and scaling factors to ensure the probability calculation is accurate. This approximation helps simplify the calculation while capturing the essential aspects of the network's user and BS distribution.

Table 4 represent the proposed algorithm for the D-DOSS mechanism. This algorithm strategically balances local decision-making with network-wide optimization, leveraging the benefits of both distributed and centralized schemes to achieve energy-efficient management of BS states in large-scale deployments.

Table 4 Proposed distributed DOSS model algorithm.
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For the general case where a BS serves a small number of users, specifically when \({\text{N}} \le {\text{n}}\) the probability can be approximated as follows. In such scenarios, where the number of users \({\text{N}}\) is less than or equal to nnn, the calculation considers a limited number of users per BS53. This situation is addressed by using the integral form provided earlier and applies the same principles but within the context of fewer users.or the general case when a BS has few or less number of users i-e (\(N \le n)\),

$$\text{P}\left(\text{N}\le \text{n}\right)=\sum_{i=1}^{K}\sum_{j=1}^{N}\frac{ {{{\lambda }_{{u}_{j} }}^{K}(\rho {\lambda }_{{BS}_{i}})}^{\rho }\Gamma (i+\rho )}{\Gamma \left(\rho \right)K!({\lambda }_{{u}_{j}}+\rho {\lambda }_{{BS}_{i}}{)}^{i+\rho }}$$
(2)

In case when N = 0 or N = 1 i-e when BS has at-least single user or it has no user can be represented as,

$$\left(\text{N}=\text{O}\right)= \sum_{i=1}^{K} {\left(1+\frac{{\lambda }_{u}/{\lambda }_{{BS}_{i}}}{\rho } \right)}^{-\rho }$$
(3)
$$\text{P}\left(\text{N}=1\right)= \frac{{\lambda }_{u}}{{\lambda }_{{BS}_{i}}}* \sum_{i=1}^{K} {\left(1+\frac{{\lambda }_{u}/{\lambda }_{{BS}_{i}}}{\rho } \right)}^{-(\rho +1)}$$
(4)

The probability that a BS is active and serves at least one user can be expressed as:

$${p}_{a}=1-\left[P\left(N=0\right)\right]=1-\sum_{i=1}^{K} {\left(1+\frac{{\lambda }_{u}/{\lambda }_{{BS}_{i}}}{\rho } \right)}^{-\rho }$$
(5)

The equations above clearly show that the likelihood of a base station remaining in active mode is influenced by the ratio of user density to base station density which is \(\vartheta ={\lambda }_{u}/{\lambda }_{{BS}_{i}}\) , where \({\lambda }_{{BS}_{i}}\ll {\lambda }_{u}\), then \({p}_{a}\approx 1\text{ and }P\left(N=0\right)\approx 0\) and represents a scenario where the base station density is sufficiently high to ensure that there is at least one user connected to the BS.

The probability that a base station transitions into sleep mode, denoted as \({p}_{s}\), is:

$${p}_{s }=1-{p}_{a}$$
(6)

When comparing the performance of the proposed scheme with RSM and LAS for both CS and DS, it's notable that RSM models the sleep mode as a Bernoulli trial. Each BS remains active with probability \(q\) and switches off with probability \(1-q\), independently of other BSs. Thus, for RSM, the power consumption54 is:

$${P}_{RSM}= {\lambda }_{sc}\left({P}_{Fixed}+\gamma .{P}_{tx}\right)*q+{\lambda }_{sc}*{P}_{RSM}*(1-q)$$
(7)

In periods of low data traffic, deactivating RF and front-end components can lead to a power savings of about 40%, as these components typically account for a significant portion of overall power usage.

Working principle of proposed scheme

In the proposed method, a distributed algorithm is implemented strategically to enhance network efficiency. The entire network is divided into regions consisting of Nc equal-sized squares, each representing a distinct cluster area. This division into clusters is crucial for managing the network effectively. The boundaries of these clusters play a significant role in determining which BS belongs to which cluster. This classification is essential for the purpose of optimizing sleep modes across the network. By ensuring that each BS is correctly assigned to a cluster, the algorithm can effectively manage the power usage of the network, putting certain BSs into sleep mode when they are not needed, thus saving energy.

An important aspect of this clustering approach is that it does not affect the user association (UA). Users will always connect to their nearest BS, ensuring that their connectivity and service quality are maintained. This is a critical consideration, as any optimization in network power usage should not come at the cost of user experience. The algorithm takes into account the aggregate throughput demands from users within each cluster to determine the sleep mode of the network. By analyzing the total data demand in each cluster, the algorithm can make informed decisions about which BSs can be put into sleep mode without compromising the network's performance. This ensures that the network operates efficiently, balancing energy savings with user demand.

Overall, the strategic implementation of this distributed algorithm in the D-DOSS framework allows for an optimized network where power consumption is reduced through intelligent clustering and sleep mode management, while user connectivity remains unaffected.

For the \({i}_{th}\) cluster, the probability of a small cell that to remain in active state \({(q}_{{a}_{i}})\) is,

$${{q}_{a}}_{i}=1-\left(1+{\lambda}_{{u}_{i}}/{\lambda}_{{BS}_{i}}\right)^{-3.5}$$
(8)

In the entire network, few small cell BSs of each cluster are switched to sleep state depending upon its sleep mode. The proposed approach considers the distributed clusters based approach and prioritizes BSs with the least activity in the network. In each cluster, a fraction of BSs will remain in an active state whereas others will turned into sleep mode with the probability of \({q}_{s}={1-q}_{a}.\)

Generally, any given percentage of BSs may be switched off strategically by prioritizing BSs with the least activity. A random choice is made, if any two BSs have the same activity levels and only one is to be switched off. In the D-DOSS scheme, we have defined two regions of interest.

  1. a)

    \({q}_{a}\le {q}_{s}\) (both stand by and sleep modes will be affected)

  2. b)

    \({q}_{a}>{q}_{s}\) (only sleep mode will be affected)

The sleep modes attributed and no. of clusters in both regions are defined on the basis of activity level which is given in Table 5, these attributes are derived from42.

Table 5 D-DOSS attributes.
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Problem formulation

In this research, the objective is to enhance the energy efficiency of a HetNet through the Distributed DOSS approach. The optimization problem is structured with constraints including delay, coverage probability, and Signal-to-Interference-plus-Noise Ratio (SINR) thresholds. The primary aim is to optimize the EE by adjusting the operational states of base stations—specifically, their active, standby, sleep, and switch-off states.

Energy efficiency optimization

The energy efficiency is determined by taking the reciprocal of the adjusted power consumption, which integrates the power used across various base station states (active, standby, and sleep) and the interference from neighboring base stations. The formula also includes a component to account for propagation losses and coverage areas, weighted by the power levels and densities of the base stations. Additionally, a logarithmic term is incorporated to reflect the impact of the minimum required SINR on overall energy efficiency. The goal is to maximize this energy efficiency while satisfying all performance constraints, including delay, coverage probability, and SINR thresholds

$${EE}_{D-Doss({q}_{ON},{q}_{stand\,by},{q}_{sleep},{q}_{sw off})}=\left[\frac{1}{{{\lambda }_{f}(q}_{on}+0.5{q}_{stand\,by}+0.15{q}_{sleep})\frac{{\mathcalligra{p}}_{f}}{{\mathcal{P}}_{T}}+\sum_{i=2}^{K}\left(\frac{{{\lambda }_{i}\mathcalligra{p}}_{i}}{{\mathcal{P}}_{T}}\right)}\right]*\left[\left.\frac{ {\lambda }_{f}{q}_{ON}{\text{P}}_{f}^{2/\alpha } \mathcal{A}( \alpha ,{\beta }_{i},{\beta }_{min})+\sum_{k=2}^{K}{\lambda }_{k}{\text{P}}_{k}^{2/\alpha } \mathcal{A}( \alpha ,{\beta }_{k},{\beta }_{min})}{{\lambda }_{f}{q}_{ON}{\text{P}}_{f}^{-2/\alpha }{{\beta }_{f}}^{-2/\alpha }+\sum_{k=1}^{K}{\lambda }_{k}{\text{P}}_{k}^{2/\alpha } {{\beta }_{k}}^{-2/\alpha })}+\text{log}(1+{\beta }_{\text{min})}\right]\right.$$
(9)

Subject to

$${0\le q}_{on}+{q}_{stand\,by}+{q}_{sleep} \le 1$$
(10)
$${q}_{on}+0.5{q}_{stand\,by}+30{q}_{sleep} \le {delay}_{min}$$
(11)
$${P}_{{C}_{ON}}=\frac{{\lambda }_{f}{q}_{ON}{\text{P}}_{f}^{\frac{2}{\alpha }}{\beta }_{f}^{-\frac{2}{\alpha }}+ \sum_{j=2}^{K}{\lambda }_{i}{\text{P}}_{i}^{\frac{2}{\alpha }}{\beta }_{i}^{-\frac{2}{\alpha }}}{{\lambda }_{f}{\text{P}}_{f}^{\frac{2}{\alpha }}+ \sum_{j=2}^{K}{\lambda }_{i}{\text{P}}_{i}^{\frac{2}{\alpha }}}\frac{1}{2\pi \text{csc}\left(\frac{2\pi }{\alpha }\right){\alpha }^{-1}} >{\beta }_{Th}\left\{\begin{array}{l}{\upbeta }_{{f}_{average}}<{\beta }_{Th}\\ {\upbeta }_{macro}>1\end{array}\right.$$
(12)

Here the variables of interest are \({q}_{on},{q}_{stand by}, \text{and }{q}_{sleep}\) and the optimal solution would be the maximized EE while QoS requirement is satisfied.

In a centralized scheme, the energy consumption equation takes into account the centralized control of base stations and their operational states. In this context, the central controller optimizes the energy consumption across the network based on the operational states of all base stations.

Let

  • \({P}_{on}\) be the power consumption of a base station in active mode.

  • \({P}_{\left(stand by\right)}\) is the power consumption of a base station in standby mode.

  • \({P}_{sleep}\) be the power consumption of a base station in sleep mode.

  • \(N\) be the total number of base stations in the network.

The total energy consumption \({E}_{{total}_{Clustered}}\) of the network in a centralized scheme can be expressed as:

$${E}_{{total}_{Clustered}}={\sum }_{i=1}^{M}\left[{q}_{{on}_{i}}* {P}_{on}+{q}_{{standby}_{\left(i\right)}}*{P}_{standby}+{q}_{{sleep}_{\left(i\right)}}*{P}_{sleep}\right]$$
(13)

The summation of all \(N\) base stations provide the total energy consumption of the network. In a centralized scheme, this calculation is managed by a central controller that optimizes the base stations' operational states to minimize total energy consumption while meeting network performance requirements.

Average achievable throughput

In multi-tier Heterogeneous, the average achievable data rate by a randomly selected user in open access mode, when the user is in the coverage region is55,

$$\overline{R }=\text{log}(1+{\beta }_{\text{min})}+\frac{\sum_{i=1}^{K}{\lambda }_{i}{\text{P}}_{i}^{2/\alpha } \mathcal{A}( \alpha ,{\beta }_{i},{\beta }_{min})}{\sum_{i=1}^{K}{(\lambda }_{i}{\text{P}}_{i}^{2/\alpha } {{\beta }_{i}}^{-2/\alpha })}$$
(14)

where \(\mathcal{A}\left( \alpha ,{\beta }_{i},{\beta }_{min}\right)={\int }_{{\beta }_{min}}^{\infty }\frac{{\text{max}({\beta }_{i},x)}^{-2/}\alpha }{1+x}x\), and \({\beta }_{min}=\text{min}\left\{{\beta }_{1},{\beta }_{2},{\beta }_{3}\dots {\beta }_{K}\right\}.\)

Equation 14 represent the average achievable data rate in an interference-limited environment i.e. \({\sigma }^{2}=0\).

By assuming the same value of SINR \(\beta \) for all tiers it will become

$$\overline{R }=\mathcal{A}( \alpha ,\beta ,\beta +\text{log}\left(1+ \beta \right))$$
(15)

Equation 15 indicates that the average achievable data rate is also unaffected by base station density, similar to the coverage probability when the SINR remains consistent across different tiers.

Network illustration and sleep management

Figure 7, depicts the network layout and how sleep modes are managed. When a mobile user moves into the coverage area of a small cell base station and begins transmission, the integrated low-power sniffer within the cell identifies an increase in the uplink (UL) channel's received power. This detection occurs because the small cell picks up a stronger signal from the mobile user compared to the signals it was previously receiving from other base stations.

Fig. 7
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Network Illustration and sleep management.

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The LP sniffer is responsible for continuously monitoring the received signal strength (RSS) from all mobile users within the small cell’s coverage area. It tracks changes in signal strength and assesses the current data requirements of these users. By keeping a real-time record of RSS, the sniffer ensures that the small cell maintains a connection with users who experience the best signal quality. As part of its operation, the small cell sends aggregated data on user requirements to the core network. This communication allows the core network to evaluate the overall data demand and adjust network resources accordingly. The small cell remains operational and active until the requirement of all users associated with it remains above a predefined threshold. Once the average data rate requirement for these users falls below this threshold, indicating that network demand is low, the small cell may transition into a power-saving state or become inactive, depending on the network’s energy management strategies. This dynamic adjustment helps optimize network efficiency by conserving energy when the demand is low, while still providing adequate service when needed.

Results and analysis

HetNets will have an entirely different network topology than the traditional homogenous network as represented in Fig. 8.

Fig. 8
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Traditional network topology (left side) versus real HetNet topology (right side).

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In traditional cellular networks, which typically use only macro base stations (BSs), the coverage areas are clearly defined based on a uniform placement model. Each macro BS covers a distinct geographic area, resulting in a predictable and consistent network topology, as shown in Fig. 8 (left side).

In contrast, HetNets involve a more complex design that combines macro cells with an overlay of small cells. In HetNets, base stations are placed randomly according to various metrics, such as coverage needs or high data rate demands. This random placement of BSs requires modeling with random processes, reflecting a network topology that is fundamentally different from the traditional macro-only network, as depicted in Fig. 8 (right side).

HetNets are characterized by their use of multiple wireless channels, which share both frequency and spatial resources. Each channel has its own set of transmitters and receivers and can vary in complexity. For example, p2P channels represent the simplest form, where a single transmitter communicates directly with a single receiver, such as in D2D scenarios. Broadcasting channels, such as the downlink channel, involve one transmitter sending information to multiple receivers simultaneously. Conversely, multiple access channels, like the uplink channel, handle communication from multiple users to a single receiver, such as a base station. This variety in channel types introduces distinct challenges for network management and optimization. The heterogeneous nature of HetNets, characterized by their varied channel configurations and randomly distributed base stations, introduces distinct challenges for network design and performance management. This complexity contrasts sharply with the more straightforward traditional macro-only networks, as depicted in Fig. 9. It illustrates a typical multi-tier HetNet setup, where the first tier comprises high-power macro base stations, and the second tier consists of small-cell femto base stations. In this model, both base stations and users are distributed according to a PPP.

Fig. 9
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Multi-tier heterogenous cellular network.

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Simulation assumptions

A two-tier system model based on stochastic geometry is considered. Table 6, summarizes the distribution, density, power consumption, SINR thresholds, path loss models, rate requirements, and minimum distance constraints for the macro and femto BS. The analysis and simulated results in this research are based on several assumptions aimed at providing a simple and tractable analysis. All assumptions are validated and compared with those from existing studies that use similar assumptions, with relevant references provided. The simulation parameters utilized in this research are detailed in Table 6. Additionally, the homogeneous hexagonal network is a special case of HetNet with a single tier only; hence, the network can be modeled accordingly.

Table 6 Simulations parameters.
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Decision making time comparison of DS and CS approach

Figure 10, illustrates a comparison of average decision-making times between centralized and distributed base station approaches across varying numbers of base stations. It shows that with an increasing number of base stations, the average decision-making time in the centralized approach tends to increase. This increase suggests that the centralized system becomes less efficient with larger networks due to the added complexity and overhead of processing and aggregating data from multiple base stations. In contrast, the distributed approach maintains relatively lower and more stable decision-making times, demonstrating better scalability and efficiency.

Fig. 10
figure 10

Average decision-making times in centralized versus distributed systems.

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For a multi-tier HetNet, the distributed approach generally proves superior. It enhances coverage probability by enabling localized decision-making, which adapts more swiftly to changes and maintains effective coverage across multiple tiers. The distributed system also reduces blockage probability by allowing base stations to make real-time adjustments to local conditions, thereby improving network reliability. In terms of energy efficiency, the distributed approach outperforms the centralized system. It can dynamically adjust power levels based on local traffic, leading to significant energy savings. Additionally, energy utilization efficiency is better in a distributed system as it can more precisely control energy use, adapting to immediate needs and optimizing resource allocation. Overall, while the centralized approach may simplify network management, the distributed approach offers greater adaptability, efficiency, and effectiveness in a complex multi-tier HetNet environment.

Sleep strategies typically involve putting the entire base station or some of its components into sleep mode, usually turning off those components that are underutilized29. In this perspective, Fig. 11, represents the 24-h data traffic profile41, I indicates that during certain times, BS energy can be saved by turning off some or all of the BS components when user data traffic requirements are low.

Fig. 11
figure 11

Daily data traffic profile19.

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Before enabling sleep mode in particular BSs, it is essential for these BSs to coordinate with neighboring active BSs and hand over their channel resources. The active BSs must then be able to extend their coverage and maintain data rate services for users within the regions of the sleeping BSs. This coordination and traffic information exchange are typically managed by the BS Controller. However, transitioning a BS to sleep mode can result in an increased outage, thereby decreasing the coverage probability.

Figure 10, provides an in-depth comparison of the average decision-making times for centralized versus distributed base station approaches as the number of base stations changes. The data reveal that while the centralized system's decision-making time increases with the number of base stations, indicating growing inefficiencies, the distributed approach remains more stable and efficient, even in larger networks. Complementing this, Fig. 11 underscores the potential for significant energy savings through sleep mode strategies, which can optimize power consumption during periods of low network utilization, albeit with some trade-offs in coverage probability and outage rates.

Moving forward, our analysis will rigorously evaluate the performance of both centralized and distributed systems under critical Quality of Service (QoS) constraints. We will assess their effectiveness in terms of coverage probability, blockage, energy utilization efficiency, and data rate. This comprehensive comparison aims to provide actionable insights into which approach offers superior performance and adaptability in a complex, multi-tier HetNet environment.

Probability of coverage

Figure 12 illustrates the comparison of coverage probability between centralized and distributed base station approaches across a range of 0–4000 base stations or users. The graph reveals that the centralized approach, depicted with dashed lines, shows a decline in coverage probability as the number of base stations or users increases. This decline is indicative of the centralized system's struggle to efficiently manage the growing network complexity and user demand. In contrast, the distributed approach, shown with dotted lines, maintains a higher and more stable coverage probability regardless of the user density. This stability highlights the distributed system's effectiveness in managing local network conditions and user demands through decentralized control. The graph underscores that the distributed approach offers better scalability and efficiency, providing more reliable coverage in large and complex network environments compared to the centralized approach.

Fig. 12
figure 12

Comparison of centralized and distributed approaches with varying number of users.

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Energy utilization efficiency (EUE)

Energy Utilization Efficiency (EUE) is an essential performance metric for wireless networks. It is determined by dividing the energy conserved by all small cells during sleep mode by the energy used in active mode and then multiplying by the number of users served during peak hours. The EUE can be represented as follows:

$${\varvec{E}}{\varvec{U}}{\varvec{E}}=\frac{{{\varvec{E}}}_{{\varvec{saved}\,\varvec{in}\,\varvec{sleep}\,\varvec{mode}}}\times {\varvec{N}}}{{{\varvec{E}}}_{{\varvec{c}}{\varvec{o}}{\varvec{n}}{\varvec{s}}{\varvec{u}}{\varvec{m}}{\varvec{e}}{\varvec{d}}\,{\varvec{i}}{\varvec{n}}\,{\varvec{a}}{\varvec{c}}{\varvec{t}}{\varvec{i}}{\varvec{v}}{\varvec{e}}\,{\varvec{m}}{\varvec{o}}{\varvec{d}}{\varvec{e}}}}$$
(16)

where \({{\varvec{E}}}_{{\varvec{saved}}}\) is the total energy saved by all small cells in sleep mode. \({\varvec{N}}\) is the no. of users served during high-traffic periods. \({{\varvec{E}}}_{{\varvec{A}}{\varvec{c}}{\varvec{t}}{\varvec{i}}{\varvec{v}}{\varvec{e}}}\) is the total energy consumed by the small cells in active mode.

This formula quantifies how efficiently energy is utilized in the network by considering both energy savings during idle periods and the energy consumed when the network is operational. Figure 13, provides a detailed comparison of energy utilization across different base station strategies, with a particular focus on the proposed Dynamically DOSS strategy versus other sleep strategies under varying data traffic conditions. The graph shows that the dynamic DOSS strategies significantly outperform other approaches in terms of energy efficiency, especially when dealing with high data traffic conditions. This is because the Dynamically DOSS strategy dynamically adjusts the activity status (ON/OFF) of small cell base stations based on real-time traffic variations. When traffic is low, picocells enter sleep mode, reducing energy consumption. In contrast, the LAS strategies rely on static data traffic profiles that do not account for daily fluctuations in traffic conditions. These static profiles lead to inefficient network performance as they keep small base stations active even when actual traffic is low. This results in higher energy consumption as unnecessary picocells remain on, consuming power despite the low user load. The centralized approach, which can include strategies like core network-controlled sleep modes, provides an intermediate solution. Centralized strategies typically involve the core network managing sleep modes based on broader traffic patterns and load forecasts. While this approach improves energy efficiency compared to always-on modes, it still may not match the efficiency of the distributed DOSS strategies. This is because centralized approaches, though more adaptive than static LAS strategies, may not fully utilize the potential of real-time traffic variations. They often rely on pre-determined patterns and may not adjust as quickly or accurately as distributed DOSS strategies.

Fig. 13
figure 13

Energy utilization of network.

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Overall, the Fig. 13, highlights that the distributed DOSS strategies, by incorporating real-time traffic adjustments, deliver superior energy utilization compared to both static LAS strategies and centralized approaches. This dynamic adaptability not only conserves energy but also enhances the network's capacity to serve a larger number of users effectively, leading to a significant improvement in EUE.

Success probability

Figure 14, presents a detailed analysis of the probability that a random user at the center of BS, will successfully receive service, relative to the parameter \(q_{ON}\). In the graph, the impact of different base station sleep modes is evaluated, including centralized strategies, dynamic sleep modes, and a full-on mode scenario. Centralized strategies, represented in the graph, involve a core network that manages sleep modes based on broader traffic patterns. While this approach is more energy-efficient than keeping all base stations active, it can be less adaptive compared to dynamic strategies. The centralized approach's success probability, shown by black lines, indicates that users are more likely to stay within coverage compared to scenarios where all base stations are fully operational.

Fig. 14
figure 14

Probability of success.

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The Distributed DOSS strategies, depicted by cyan lines, showcase an advanced method where base stations dynamically adjust their activity status (ON/OFF) based on real-time traffic conditions. This dynamic adjustment significantly enhances energy utilization compared to static centralized methods. As the number of mobile users increases, the energy efficiency of the Distributed DOSS strategies improves notably. This is due to their ability to adaptively manage small cell sleep modes, thereby reducing energy consumption during periods of low traffic while maintaining high coverage probability during peak times. In contrast, the LAS strategies, illustrated with varying lambda values, rely on fixed traffic profiles to determine sleep modes. These strategies, which use static profiles, are less responsive to actual traffic variations. Although LAS strategies can save energy by deactivating some base stations, they may lead to lower coverage probability when actual traffic deviates from the static profile. The plot reveals that as lambda increases indicating more femtocells in operation, the coverage probability improves, but the overall energy efficiency may not reach the levels achieved with Distributed DOSS strategies.

The simulated results in Fig. 14, highlight that when all small cell base stations are in switch-off mode, the probability of a user receiving service is at its lowest, around 10%. However, with random sleep mode selection, this probability increases significantly, ranging from 30 to 40%. This increase underscores the effectiveness of dynamic sleep mode strategies in maintaining higher coverage probability compared to static approaches, making them a more efficient solution for balancing energy savings and user service quality.

In a load-aware approach, the likelihood of service for a randomly chosen mobile user can be elevated to 90%. Nonetheless, the simulated data clearly show that Disctributed DOSS excels beyond all other methods by delivering the highest probability of success. This suggests that Dynamically DOSS offers superior user experience and better energy efficiency.

Data throughput

Figure 15, illustrates that in a network with no sleep strategy, both the average user rate and the total network rate remain constant because the number of active base stations does not change. This configuration establishes an upper limit on the maximum number of users that can be accommodated by the BSs. In comparison, the RSM strategy, which randomly selects BSs to deactivate, results in the poorest performance. This randomness leads to lower average user rates and total network rates because the deactivation of BSs is not optimized based on actual traffic demands. On the other hand, both the LAS strategy and the distributed DOSS approach demonstrate improved performance. LAS, despite its reliance on fixed traffic profiles, still outperforms RSM by providing more stable average user rates and total network rates. Distributed DOSS, which adapts the activity status of BSs based on real-time traffic conditions, offers the best performance. This approach ensures optimal energy utilization while maintaining high average user rates and total network rates, surpassing both RSM and LAS strategies.

Fig. 15
figure 15

Average achievable per user throughput.

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In addition to this, CS and DS are considered. Centralized Strategies generally manage sleep modes based on aggregated traffic data, which can provide a balanced performance but may lack the adaptability of distributed DOSS.

Conclusion

This research has made significant strides in addressing the critical challenge of optimizing EE within HetNets to meet the increasing demands for high-speed data services. The rapid growth in high-speed devices and the projected surge in mobile data traffic highlight the urgent need for innovative strategies to enhance network performance while conserving energy. This study has demonstrated that the D-DOSS strategy effectively achieves these goals. By strategically forming small clusters throughout the network and evaluating key QoS metrics, such as coverage probability, energy utilization efficiency, success probability, and data throughput, D-DOSS effectively balances energy savings with network performance. While Centralized Schemes generally offer robust performance in terms of EE and management, they have limitations compared to distributed approaches. CS typically involves a core network managing sleep modes based on overall traffic patterns, ensuring high coverage and EE but lacking the adaptability of dynamic distributed methods. CS is often easier to implement but less responsive to real-time traffic variations. In contrast, proposed Distributed Sleep strategies provide a more flexible and adaptive solution. Although DS approaches come with trade-offs compared to CS, D-DOSS stands out for its ease of implementation and adaptability. By dynamically adjusting the activity status of base stations based on real-time traffic conditions, D-DOSS outperforms static approaches, including CS, in maintaining high EE while providing reliable coverage.

Overall, while CS can deliver consistent performance, the flexibility and adaptability of D-DOSS make it a superior choice for modern HetNets. The findings emphasize the importance of incorporating dynamic adaptation capabilities to optimize EE and network performance. Future research should focus on validating D-DOSS through real-world deployments and field trials and exploring further enhancements to its dynamic adaptation and optimization algorithms to maximize performance under varying network conditions.