Introduction

5G cellular networks have garnered significant interest due to their storage capacity and ability to achieve spectrum efficiency1. A more essential analysis of network technology and architecture is required for wireless communication systems to advance past 5G to satisfy the regularly increasing needs for enhanced connectivity, lower latency, and advanced data rates2. Nevertheless, mobile networks, in this 5G period, are required to offer services to devices with various Quality of Service (QoS) requirements and applications kinds3. Moreover, the capability to assist a large number of users with outstanding information rates is regarded as the major factor for the growth of 5G technology4. Furthermore, 5G networks are also anticipated to address the most significant issues in daily lives, like the digital divide, urbanization, and environmental sustainability. In addition to this, 5G has the ability to advance sustainable development goals by allowing increased connectivity, smart infrastructure management, and effective resource utilization5. Besides, the main need in 5G technology is to provide a high data rate to each user without contemplating the duration of network access6. Moreover, Millimeter wave (mmWave) communication is considered a capable solution for 5G networks due to its ability to support gigabit-per-second data rates7. The most essential enabler for a 5G communication system is Massive Multiple-Input Multiple-Output (M-MIMO) since it enhances bandwidth efficiency, spectral utilization, and data throughput. Further, the reliability and performance of mm-wave transmissions in 5G networks are improved by employing MIMO4.

The implementation of MIMO is limited by the number of users. However, it can be effectively served on a shared time-frequency resource by precoding8. During the process of channel estimation in MIMO networks, pilot signals are sent by all the transmitting antennas to assess the channel9. Moreover, MIMO networks are considered as a dependable option for overcoming the capacity and data storage problems by satisfying the requirements of various users. The major concept of Massive-MIMO (M-MIMO) technology is to train the Base Station (BS) with many wireless antennas for serving many customers at the same time by allowing enhancement in spectrum efficiency1. Furthermore, in 5G wireless networks, M-MIMO is developed as a vital technology for excellent enhancement in transmission quality and spectrum efficiency10. The transceiver optimization of MIMO is more difficult than single-antenna networks due to the existence of a large number of antennas in M-MIMO, and data multiplexing. Here, several optimization techniques for user scheduling, power allocation, and user association have been employed in the past because of the multi-objective nature of M-MIMO transceivers1. In 5G networks, M-MIMO is considered a promising approach for enhancing the performance8,11. Besides, implementing MIMO antenna elements in communication systems is regarded as a crucial process in 5G12. M-MIMO systems are crucial in overcoming the physical constraints, like path attenuation and loss associated with high-frequency bands employed in 5G5. However, energy-constrained environments and low channel quality conditions reduce the Multi User (MU)-MIMO systems’ performance13.

The spectrum efficiency and system capacity are increased in MIMO by employing spatial diversity gain through various antennas14. Nevertheless, BS in M-MIMO systems require several progressive resource allocations because of resource options and antenna’s scale15. Nowadays, in environmentally friendly transmission and wireless communication, resource allocation with efficient energy has gained more popularity. Furthermore, the main aim of resource allocation is to enhance users’ Quality of Experience (QoE). Moreover, various measures, such as energy efficiency, data rate, fairness, QoS, and spectrum efficiency are used for managing the problems in resource allocation6. To satisfy the QoS necessities of every user, resources, namely antenna beams, bandwidth, and transmit power are dynamically assigned through effective resource allocation2. Further, optimal resource allocation is crucial for the effectual performance of the system9. In networks, the decisions of resource allocation are usually taken on a millisecond timescale, and thus optimization approaches have to be scalable to attain better outcomes in a short period16. Resource allocation decisions and configuration have to be made systematically based on a forward-looking offline plan and by employing the system’s stochastic profile for a specific time horizon in networks running on hybrid energy. Additionally, due to the constant advancement of DL, neural networks have been utilized in electromagnetism, antennae, and resource allocation. Besides, well-trained networks are exploited for solving resource allocation issues has better performance with less computational complexity than other conventional methods17. Artificial intelligence (AI) is also very beneficial in assigning resources for autonomous control since it can select a suitable control strategy based on the information taken from historical data3.

This paper intends to introduce a new approach called HGGO_XCovNet for resource allocation in MIMO-enabled wireless networks. Initially, a BS with many users is considered, and then resource allocation is performed by considering various objective parameters including SINR, data rate, and power consumption. Here, resource allocation is carried out by utilizing the XCovNet technique, which is tuned using HGGO. Here, the HGGO algorithm is attained by the amalgamation of two models namely GGO and HO.

The following contribution of the proposed antenna will be illustrated.

  1. 1.

    The Hippo Graylag Goose Optimization (HGGO) is a newly proposed metaheuristic algorithm that combines the strengths of Greylag Goose Optimization (GGO) and Hippopotamus Optimization (HO). This hybrid approach enhances exploration and exploitation, leading to faster convergence and better performance.

  2. 2.

    We adapt the Xception Convolutional Neural Network (XCovNet) for resource allocation, which is not previously applied in this context. Its separable convolutions and efficient feature extraction make it ideal for handling MIMO system parameters.

  3. 3.

    Our experiments demonstrate superior results in energy efficiency (74.943 kbits/joule), sum rate (269.93 Mbps), and throughput (551.262 Mbps), outperforming state-of-the-art methods like PD-DQN, WMMSE, and DRL.

The remaining sections are structured as follows: Sect. 2 displays the literature review of various prevailing models for resource allocation in MIMO, Sect. 3 shows the system model of the MIMO system, Sect. 4 explains the HGGO_XCovNet for resource allocation in the MIMO system, Sect. 5 portrays the results of HGGO_XCovNet, and Sect. 6 illustrates the conclusion.

Motivation

In 5G mobile networks, the MIMO system supports various progressive applications with high reliability, low latency, and high data rates. Nevertheless, various conventional approaches in MIMO resource allocation are challenging because of the network complexity and QoS across layers. Furthermore, a new model is devised for resource allocation in MIMO systems using HGGO_XCovNet. Moreover, the challenges faced by existing methods are listed below. The Important acronyms summary are shown in (Table 1).

Table 1 Important acronyms summary.
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Literature review

Purushothaman et al.6 they devised a Multi-objective Sine Cosine algorithm (MOSCA) for resource allocation in 5G Multi-User Massive MIMO. This approach minimized power consumption, maximized energy efficiency, and offered a high data rate. However, this model failed to consider the usage of resource allocation by 5G users to attain less complexity. A Parameterized Deep Q-Leaning Network (PD-DQN) was introduced by Sharma et al.1 for power allocation in massive MIMO. This technique improved energy efficiency and solved the issues in power allocation and user association. Still, this approach did not consider additional simulation outcomes for attaining higher energy efficiency with diverse learning rates. Shaik et al.18 developed a Weighted Minimum Mean-Square-Error (WMMSE)-based Dunklebach algorithm for resource allocation in cell-free massive MIMO. This model offered less complexity and obtained maximum spectrum efficiency. Nevertheless, this approach did not investigate minimum-rate constraints to reduce the maximum energy efficiency. Misso et al.4 presented a max-min fairness algorithm for improving the spectral efficiency in massive MIMO. This algorithm attained low computational complexity and effectively enabled frequency reuse patterns. However, this technique did not examine the Angle of Arrival and AI-based approaches to improve the massive MIMO system’s performance.

Unsupervised Learning-Aided Wideband Scheduling (ULAWS) was formulated by Hsu et al.14 for resource allocation in wideband MU-MIMO systems. This method enhanced the performance gain and obtained a low outage rate under low noise effects and high-rate requirements. However, this approach required more time for execution. Liu et al.17 introduced a Deep Q-Leaning Network (DQN) for resource allocation in MIMO. This approach efficiently measured the user data packet’s time out and attained high throughput and energy efficiency. Still, this model failed to consider resource allocation problems for simulating the bandwidth, antenna number, and power. Minimum Mean Square Error and Riemannian Conjugate Gradient (MMSE-RCG) were devised by Singh et al.13 for resource allocation in MU-MIMO. This approach attained a faster convergence rate, better spectral performance, and effective power resources. However, this model did not explore advanced channel models to capture the complexity of multi-intelligent Reflecting Surface (IRS) systems, which accounted for dynamic channel conditions, and potential interaction among reflected signals from various IRSs. Huang et al.15 developed a Deep Reinforcement Learning (DRL) model for antenna allocation in Massive MIMO. This technique improved the resource allocation in a flexible and adaptable manner and also assessed in realistic traffic scenarios. However, this model failed to include self-adaptation of Machine Learning (ML) approaches and decisions for improving the performance. The Table 2 presents a Comparative Analysis of State-of-the-Art MIMO Resource Allocation Techniques, summarizing their key features and performance metrics.

Table 2 Comparative analysis of state-of-the-Art MIMO resource allocation techniques.
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Challenges

The challenges faced by several prevailing models for resource allocation in MIMO-enabled wireless networks are given as follows.

  • In1, the PD-DQN approach efficiently handled large-scale multi-agent issues in M-MIMO networks. Nevertheless, this method did not mitigate and manage interference, especially in dense network environments.

  • In18, the WMMSE-based Dunklebach algorithm was simple and offered heuristic power allocation policies. However, this approach attained the best fairness, though it resulted in a loss of performance.

  • In14, the ULAWS technique obtained intrinsic user groups with low Co-Channel Interference (CCI) between co-channel users. Still, this approach failed to reduce the latency in resource allocation, which affected the overall experience of the user.

  • In15, the DRL model efficiently decreased the interference between the users by improving the throughput of the overall system. Nevertheless, this model got trapped in local optima, which affected the reliability of resource allocation policies in MIMO.

  • Various traditional methods exploited for resource allocation in MIMO networks were dependent on extensive databases, which were difficult and expensive to implement. Thus, generating a suitable and reliable resource allocation system was vital to optimize the network performance.

System model of MIMO

MIMO involves wireless communication systems with a large number of antennas F at the BS, typically in the order of hundreds or thousands, serving multiple users q simultaneously. Moreover, these antennas influence the spatial diversity and several abilities of MIMO systems, which aim to enhance the reliability, energy efficiency, and spectral efficiency of wireless communication systems. In addition to this, MIMO19 can also be employed in interference management as the BS is equipped with huge antenna elements. Further, MIMO is considered a promising technology, which is constructed with less cost, and the low-power transceivers make the process more secure and robust.

Since BSs are equipped with very large antenna elements compared to the number of user equipment in a given cell region, transmitted signals can be easily shaped to null interference. Assume that a flat fading channel exists among the q users and BS. The behavior of the system can be modeled by considering several parameters, such as SINR, data rate, and power consumption. Here, SINR is utilized for evaluating the quality of a received signal. The received SINR20 for the \({q^{th}}\) user in the \({\hbar ^{th}}\) time slot is written as,

$$P_{{r,q}}^{\hbar }=\frac{{L_{{r,r,q}}^{\hbar }E_{{r,q}}^{\hbar }}}{{\sum\nolimits_{{q^{\prime} \ne q}} {L_{{r,r,q}}^{\hbar }E_{{r,q^{\prime}}}^{\hbar }+\sum\nolimits_{{r^{\prime} \in {S_r}}} {L_{{r^{\prime},r,q}}^{\hbar }\sum\nolimits_{m} {E_{{r^{\prime}m}}^{\hbar }+{\delta ^2}} } } }}$$
(1)

Here, BS (transmitter) is exemplified as r, \(\hbar\) represents the time slot, channel gain is denoted as L, power is symbolized as E, the additional noise power is characterized as \({\delta ^2}\), interference sell set is indicated as \({S_r}\), \(\sum\nolimits_{{r^{\prime} \in {S_r}}} {L_{{r^{\prime},r,q}}^{\hbar }\sum\nolimits_{m} {E{{_{{r^{\prime}m}}^{\hbar }}^2}} }\) and \(\sum\nolimits_{{q^{\prime} \ne q}} {L_{{r,r,q}}^{\hbar }E_{{r,q^{\prime}}}^{\hbar }}\)denotes the inter-cell and intra-cell interference.

Furthermore, the data rate23 is utilized for evaluating the quantity of data transmitted in a network. For user q, the data rate on BS \(\hbar\)is designated by the below expression.

$$W_{{r,q}}^{\hbar }=\varsigma \log \left( {1+P_{{r,q}}^{\hbar }} \right)$$
(2)

where, the data rate is characterized as \(W_{{r,q}}^{\hbar }\), and bandwidth of \({q^{th}}\)subcarrier is epitomized as \(\varsigma\).

Further, power consumption6 refers to the amount of energy utilized per unit time. Here, the power consumed by BS is articulated as the sum of dynamic and static terms, and is expressed as,

$$A_{{r,q}}^{\hbar }={A_{static}}+{A_{dynamic}}$$
(3)

Moreover, the equation of \({A_{dynamic}}\)and \({A_{static}}\)is signified as follows,

$${A_{static}}={{{\chi _r}} \mathord{\left/ {\vphantom {{{\chi _r}} Z}} \right. \kern-0pt} Z}$$
(4)
$${A_{dynamic}}={\xi _r}\sum\limits_{{b=1}}^{s} {A_{r}^{b}}$$
(5)

wherein, the overall amount of subcarriers is embodied as Z, s symbolizes spatial streams in MIMO system, b specifies the subcarrier, the power amplifier is designated as \({A_r}\), \({\chi _r}\) typifies the circuit’s power dissipation. The system model of Massive MIMO21 is illustrated in (Fig. 1). Table 3 outlines the Parameters and Variables for the System Model Equation, providing a comprehensive overview of the key elements used in the model.

Fig. 1
figure 1

System model of MIMO.

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Table 3 Parameters and variables for system model equations.
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Proposed hippo graylag goose optimization with XCovNet (HGGO_XCovNet ) for resource allocation

MIMO is considered a fundamental technology, which is employed to increase spectral efficiency and throughput. Nevertheless, effective resource allocation is challenging because of the channel conditions, high-dimensional nature, and requirements of effectual utilization of resources, like antenna configurations, bandwidth, power, and so on. Therefore, a new approach named HGGO_XCovNet is developed for resource allocation in MIMO. Firstly, the BS with a large number of users is considered. Here, the resource allocation is carried out by considering several objective parameters, such as SINR, data rate, and power consumption. In addition to this, the resource allocation is executed using the XCovNet22 method, which is tuned by utilizing the HGGO algorithm. Further, the HGGO is generated by integrating both methods, like GGO23 and HO24. The general outline of HGGO_XCovNet for resource allocation in MIMO systems is shown in (Fig. 2).

Multi-objective parameter

In MIMO, the multi-objectives play a vital role, and it is employed to achieve the optimum performance. Moreover, wireless communication is enhanced by MIMO, which is done by exploiting many antennas at both the receiver and transmitter to increase the reliability, data rate, and offering spatial diversity. Here, the multi-objective parameters, namely power consumption, data rate, and SINR are considered the key parameters, which are utilized to ensure energy-efficient and high-quality communication to all users. Furthermore, the explanation of these parameters is given as follows.

  1. (i)

    SINR: SINR is employed to measure the quality of the signal, which refers to the proportion of signal strength and the existence of noise in the communication channel. Here, the high SINR value represents better data transmission with high signal quality. Moreover, the equation of SINR is given in expression (1).

  2. (ii)

    Data rate: This parameter is referred to the speed at which the data is transmitted in a communication channel. Moreover, the data rate is determined by using various factors, like bandwidth, SINR, and so on. Further, the expression of the data rate is mentioned above in Eq. (2).

Fig. 2
figure 2

The general outline of HGGO_XCovNet for resource allocation in MIMO systems.

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  1. (iii)

    Power consumption: The amount of energy utilized by the network devices to transmit and receive data is referred to as power consumption. This parameter is complex due to the factors including networks with energy efficiency concerns or battery-powered devices. Here, the power consumption is expressed in Eq. (3).

Moreover, all the multi-objective parameters are combined to attain the parameter\(\ell\), which is represented as \(\ell =\{ P_{{r,q}}^{\hbar },W_{{r,q}}^{\hbar },A_{{r,q}}^{\hbar }\}\).

Energy efficient resource allocation using HGGO_XCovNet

In MIMO, energy efficiency is considered a significant performance measure, especially during the increase in demand for large-scale connectivity, reliability, and high-speed data. Furthermore, many antennas at both receiver and transmitter are used in the MIMO technology, which enables multiplexing and spatial diversity by improving the robustness of wireless communication. Nevertheless, more amount of resources and antennas are required for optimum performance, which leads to high power consumption, and thus energy-efficient resource allocation process is challenging. Therefore, a new model termed HGGO_XCovNet is developed for attaining energy-efficient resource allocation in MIMO. Here, the XCovNet approach22is fine-tuned using the HGGO algorithm, which integrates HO24 and GGO23.

Architecture of XCovNet

The parameter\(\ell\)with dimension\(R \times C \times 3\) is fed as input to XCovNet22, where the value of power consumption, data rate, and SINR is individually calculated by considering R users with C channels at time \((\hbar - 1)\). This network converts the parameters into a single dimension, which is represented as \(R \times C \times 1\), and the output attained from XCovNet is typified as \({f_k}\). Based on this parameter, the SINR, data rate, and power consumption at time \(\hbar\)can be estimated. The description of XCovNet is explained below.

XCovNet is formed based on the principles of Inception architecture. The XCovNet technique is considered as an effective and efficient model by exploiting separable convolutions (Conv), and it is employed without lowering the performance rate. This network is comprised of three blocks, namely Conv block, Depth-wise Separable Conv block, and fully connected layer block. Moreover, the explanation of these blocks is given below.

Conv block

XCovNet utilizes Conv layers, and the layer after the input produces Conv kernels for constructing various feature maps to present the features of input data. During the generation of the feature map, all the Conv kernels are disseminated across every region of input data. The relative outcomes of the feature map are obtained by using several Conv layers. When the feature value is at the position \((v,g)\)of a \({\eta ^{th}}\) feature map of \({w^{th}}\)layer, then the expression for computing the feature map \(M_{{v,g,\eta }}^{w}\)is given below.

$$M_{{v,g,\eta }}^{w}=G * p_{\eta }^{w} * H_{{v,g}}^{w}+\vartheta * p_{\eta }^{w}$$
(6)

Here, the center of the input patch on the layer w at the \({(v,g)^{th}}\)location is denoted as \(H_{{v,g}}^{w}\), the weight vector is exemplified as \(G * p_{\eta }^{w}\), \(M_{{v,g,\eta }}^{w}\)specifies the generated feature maps, bias values of \({\eta ^{th}}\) feature map and \({w^{th}}\)layer is indicated as \(\vartheta * p_{\eta }^{w}\).

Furthermore, the feature maps \(M_{{v,g,\eta }}^{w}\)are generated using the kernels \(G * p_{\eta }^{w}\) that are shared. The weight distribution mechanism includes many benefits, like less complexity. The activation and max pooling layers are exploited in feature maps after the Conv layer. Here, the generalized rectified unit activation function called parametric PreLU is used with a negative slope, and it is modeled as,

$$\beta ({d_v})=\left\{ \begin{gathered} {d_v}\,\,\,\,\,\,,\,\,\,if\,\,{d_v}>0 \hfill \\ {T_v}{d_v}\,\,,\,\,if\,\,\,{d_v} \le 0\, \hfill \\ \end{gathered} \right.$$
(7)
Fig. 3
figure 3

XCovNet architecture.

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where, the negative slope which is regarded as a learnable parameter is implied as \({T_v}\), \(\beta\)characterizes the activation function, and the network layer’s input on the \({v^{th}}\)channel is embodied as \({d_v}\). In order to determine nonlinear features, the activation function is employed, which attains faster convergence with less overfitting risks, whereas to lessen the feature map’s dimension max-pooling layer is utilized.

DepthWise separable convolution block

This block is a kind of Conv method, which works in depth and space. Here, two phases are included for simplifying the Conv operation, they are depth-wise and pointwise Conv. Generally, DL frameworks use these Conv when filters cannot be decomposed into smaller ones. Moreover, the pointwise Conv utilizes a kernel, which repeats over every point.

XCovNet

An input with dimension\(R \times C \times 3\) is passed to the Conv block, and here the features are mined using two Conv layers. Further, every Conv layer involves two separate Conv2D layers, padding, and PreLU activation. The activation is normalized in batches to enhance the convergence, and the spatial dimension is minimized using the max pooling layer. In addition to this, a dropout layer with a 0.2 rate is employed for regularization. Moreover, the dropout layer is used for preventing overfitting problems. After this, the 3D tensor is reformed into a 1D vector using a flatten layer. Additionally, dense is regarded as the final layer, and then the softmax activation function is exploited for converting the class probabilities into outputs. Further, this network is used for capturing the relevant features by maintaining computation efficiency. This XCovNet converts the parameters into a single dimension, which is epitomized as \(R \times C \times 1\), and the outcome obtained from XCovNet is symbolized as \({f_k}\). Thus, the XCovNet determines the ideal channel for the user at the time \(\hbar\)by considering the SINR, data rate, and power consumption at time \((\hbar - 1)\). Figure 3 signifies the XCovNet architecture.

Training XCovNet using HGGO

To enhance the performance of XCovNet for resource allocation in MIMO networks, HGGO is employed, where HGGO is formed by incorporating HO24 and GGO23. HO24 is a stochastic approach, which is motivated by the inherent behaviours of the hippopotamuses, and it is demonstrated as an innovative model in metaheuristic methodology. This algorithm is theoretically determined by employing a trinary-phase method, which integrates evasion methods, self-protective tactics against predators, and updation of location position in ponds or rivers. HO is a flexible technique, which can adapt to several kinds of optimization issues. GGO23,25 is a metaheuristic optimization technique, and it is inspired by the migratory behaviourof greylag geese. Due to the established track record in comparable optimization issues, and typical characteristics, which provide more advantages over alternative optimization models. GGO has the ability to demonstrate encouraging outcomes in various optimization issues. Thus, the combination of HO and GGO algorithms is employed to enhance the performance with less computation time. Further, the mathematical phases of HGGO are explained below.

Population initialization

The HO’s initial solution is created randomly, and the vector for the decision variable is generated by using the below expression.

$${Q_{lu}}={J_u}+\sigma \times ({Y_u} - {J_u})$$
(8)

Here, a random number among \([0,1]\)is typified as \(\sigma\), \({Y_u}\)and \({J_u}\)denotes the upper and lower bound of \({u^{th}}\)decision variable, the location of the\({l^{th}}\)candidate solution is exemplified as \({Q_{lu}}\). Moreover, the matrix form of initialization is given below.

$$Q={\left[ {\begin{array}{*{20}{c}} {{Q_1}} \\ \vdots \\ {{Q_l}} \\ \vdots \\ {{Q_\tau }} \end{array}} \right]_{\tau \times \zeta }}={\left[ {\begin{array}{*{20}{c}} {{Q_{1,1}}}& \cdots &{{Q_{1,u}}}& \cdots &{{Q_{1,\zeta }}} \\ \vdots & \ddots & \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots \\ {{Q_{l,1}}}& \cdots &{{Q_{l,u}}}& \cdots &{{Q_{1,\zeta }}} \\ \vdots & {\mathinner{\mkern2mu\raise1pt\hbox{.}\mkern2mu \raise4pt\hbox{.}\mkern2mu\raise7pt\hbox{.}\mkern1mu}} & \vdots & \ddots & \vdots \\ {{Q_{\tau ,1}}}& \cdots &{{Q_{\tau ,u}}}& \cdots &{{Q_{\tau ,\zeta }}} \end{array}} \right]_{\tau \times \zeta }}$$
(9)

Here, the population size of hippopotamuses in the herd is specified as \(\tau\), amount of decision variables in the issue is characterized as \(\zeta\).

Fitness function

Mean Square Error (MSE) is exploited for assessing the fitness among the output generated from XCovNet and targeted output. Further, the MSE expression is formulated as,

$$MSE=\frac{1}{t}\sum\limits_{{k=1}}^{t} {{{\left( {f_{k}^{ * } - {f_k}} \right)}^2}}$$
(10)

where, \(f_{k}^{ * }\) embodies the targeted output of XCovNet, \({f_k}\) exemplifies the XCovNet outcome, and t indicates the amount of tuning samples.

Location update

The male hippopotamuses are driven by other dominant males in their adult stage. After that, the dominant male continues to fight with another male hippopotamus for establishing its own benefits. Moreover, the location of the male hippopotamus is formulated as,

$$Q_{{lu}}^{z}(\varpi +1)=Q_{{lu}}^{z}(\varpi )+\mu \left[ {B - \gamma * Q_{{lu}}^{z}(\varpi )} \right]$$
(11)
$$Q_{{lu}}^{z}(\varpi +1)=Q_{{lu}}^{z}(\varpi )+\mu * B - \mu * \gamma * Q_{{lu}}^{z}(\varpi )$$
(12)
$$Q_{{lu}}^{z}(\varpi +1)=Q_{{lu}}^{z}(\varpi )\left[ {1 - \mu * \gamma } \right] - \mu * B$$
(13)

where, a random number between \((0,1)\)is epitomized as \(\mu\), dominant hippopotamus’s location is symbolized as B, location of male hippopotamus at \({(\varpi +1)^{th}}\)iteration is embodied as \(Q_{{lu}}^{z}(\varpi +1)\), an integer ranging in \(\left[ {1,2} \right]\)is signified as \(\gamma\), z typifies the male hippopotamus, \(Q_{{lu}}^{z}(\varpi )\)exemplifies the male hippopotamus’s position at \({\varpi ^{th}}\)iteration.

Additionally, the GGO is integrated with HO for combining the strengths of both algorithms by exploiting complementary features for solving complex optimization issues more effectively. From GGO23,

$$Q_{{lu}}^{z}(\varpi +1)={Q^ * }(\varpi ) - N \cdot \left| {U{Q^ * }(\varpi ) - Q_{{lu}}^{z}(\varpi )} \right|$$
(14)
$$Q_{{lu}}^{z}(\varpi +1)={Q^ * }(\varpi ) - N \cdot U{Q^ * }(\varpi )+N \cdot Q_{{lu}}^{z}(\varpi )$$
(15)
$$N \cdot Q_{{lu}}^{z}(\varpi )=Q_{{lu}}^{z}(\varpi +1) - {Q^ * }(\varpi )+N \cdot U{Q^ * }(\varpi )$$
(16)
$$Q_{{lu}}^{z}(\varpi )=\frac{1}{N}\left[ {Q_{{lu}}^{z}(\varpi +1) - {Q^ * }(\varpi )+N \cdot U{Q^ * }(\varpi )} \right]$$
(17)

By substituting the expression (17) in (13), we get

$$Q_{{lu}}^{z}(\varpi +1)=\left[ {\frac{{Q_{{lu}}^{z}(\varpi +1)}}{N} - \frac{{{Q^ * }(\varpi )}}{N}+U{Q^ * }(\varpi )} \right]\left( {1 - \mu * \gamma } \right) - \mu * B$$
(18)
$$Q_{{lu}}^{z}(\varpi +1) - \frac{{Q_{{lu}}^{z}(\varpi +1)}}{N}\left( {1 - \mu * \gamma } \right)=\left[ {U{Q^ * }(\varpi ) - \frac{{{Q^ * }(\varpi )}}{N}} \right]\left( {1 - \mu * \gamma } \right) - \mu * B$$
(19)
$$Q_{{lu}}^{z}(\varpi +1)\left[ {1 - \frac{{\left( {1 - \mu * \gamma } \right)}}{N}} \right]=\left[ {\frac{{N \cdot U{Q^ * }(\varpi ) - {Q^ * }(\varpi )}}{N}} \right]\left( {1 - \mu * \gamma } \right) - \mu * B$$
(20)
$$\frac{{Q_{{lu}}^{z}(\varpi +1)\left[ {N - 1+\mu * \gamma } \right]}}{N}=\frac{{\left[ {N \cdot U{Q^ * }(\varpi ) - {Q^ * }(\varpi )} \right]\left( {1 - \mu * \gamma } \right) - N \cdot \mu * B}}{N}$$
(21)

Moreover, the updated expression of HGGO algorithm is represented as,

$$Q_{{lu}}^{z}(\varpi +1)=\frac{{\left[ {N \cdot U{Q^ * }(\varpi ) - {Q^ * }(\varpi )} \right]\left( {1 - \mu * \gamma } \right) - N \cdot \mu * B}}{{\left[ {N - 1+\mu * \gamma } \right]}}$$
(22)

wherein, the present location of the search agent is indicated as \(Q_{{lu}}^{z}(\varpi )\), location of the prey is characterized as \({Q^ * }(\varpi )\), vectors comprising random values between [0, 1] is implied as N and U. Further, in order to improve the exploration and global search a vector is used, and it is signified as,

$$a=\left\{ \begin{gathered} {\gamma _2} \times \overrightarrow {{y_1}} +(\sim {K_1}) \hfill \\ 2 \times \overrightarrow {{y_2}} - 1 \hfill \\ \,\,\,\,\,\,\,\overrightarrow {{y_3}} \hfill \\ {\gamma _1} \times \overrightarrow {{y_4}} +(\sim {K_2}) \hfill \\ \,\,\,\,\,\,\,\,\,{y_5} \hfill \\ \end{gathered} \right.$$
(23)

where, \({K_1}\)and \({K_2}\)stipulates integer random numbers with values as ‘0’ or ‘1’.

$$X=\exp \left( { - \frac{\varpi }{\upsilon }} \right)$$
(24)

Here, \(\upsilon\) embodies the amount of iteration, and \(\varpi\)indicates the iteration count.

$$Q_{{lu}}^{{FB}}=\left\{ \begin{gathered} Q_{{lu}}^{z}+{a_1}(B - {\gamma _2}z{\kappa _l})\,\,\,,\,\,\,X>0.6 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\rho \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,else \hfill \\ \end{gathered} \right.$$
(25)

where, \(Q_{{lu}}^{{FB}}\)implies the immature or female hippopotamus, \({\gamma _2}\)specifies the integer. Moreover, the equation of immature hippopotamus \(\rho\)is formulated as,

$$\rho =\left\{ \begin{gathered} Q_{{lu}}^{z}+{a_2}(z{\kappa _l} - B)\,\,\,\,,\,\,{y_6}>0.5 \hfill \\ {J_u}+{y_7}({Y_u} - {J_u})\,\,,\,\,else \hfill \\ \end{gathered} \right.$$
(26)

Here, random numbers in the range \((0,1)\)are specified as X and \({y_6}\), random number among \([1,1]\) is exemplified as \({y_5}\), and \({\overrightarrow y _{1,2,3,4}}\)denotes the random vector ranging in \((0,1)\). \(z\kappa\)symbolizes the mean value of some arbitrarily chosen hippopotamus with equal probability. Further, the location of immature or female hippopotamus in the population is described in expression (25) and (26). When X is larger than 0.6, then the immature hippopotamus distances itself from its mother. When \({y_6}\)is greater than 0.5, then the immature hippopotamus moves away from its mother but stays in the herd or else it is away from the population. Here, \({a_1}\)and \({a_2}\)are arbitrarily chosen vectors or numbers from Eq. (23).

In the population, the male hippopotamus’s location and the updated locations of other weak hippopotamuses are designated in expressions (27) and (28).

$${Q_l}=\left\{ \begin{gathered} Q_{l}^{z}\,\,,\,\,\alpha _{l}^{z}<{\alpha _l}\,\, \hfill \\ {Q_l}\,\,\,,\,\,else \hfill \\ \end{gathered} \right.$$
(27)
$${Q_l}=\left\{ \begin{gathered} Q_{l}^{{FB}}\,\,,\,\,\alpha _{l}^{{FB}}<{\alpha _l}\,\, \hfill \\ {Q_l}\,\,\,,\,\,else \hfill \\ \end{gathered} \right.$$
(28)

where, the value of objective function is represented as \({\alpha _l}\).

Hippo defense against predators

In this stage, the weak hippopotamuses move away from their group for some specific reasons and thus this weak hippopotamus is regarded as the aim for large organisms to attack. However, the immature hippopotamus deviates from the herd because of its inherent curiosity and then becomes a target for spotted hyenas, lions, and Nile crocodiles. Moreover, weak hippopotamuses are also vulnerable to being hunted by the predators. In addition to this, a self-protective strategy is exploited by hippopotamuses to turn towards the predator by producing loud sounds to frighten the predator to move away from them. In this phase, the hippopotamus approaches the predator and hence the predator moves away from the hippopotamus, and it is expressed as,

$${I_u}={J_u}+{\overrightarrow y _8}\left( {{Y_u} - {J_u}} \right)$$
(29)
$$B=\left| {{I_u} - {Q_{lu}}} \right|$$
(30)

Here, the predator at \({u^{th}}\)decision variable is epitomized as\({I_u}\), the distance of the \({l^{th}}\) hippopotamus to the predator is represented using expression (30). Meanwhile, the defensive behavior is adopted by the hippopotamus considering \({\alpha _{{I_j}}}\) for protection from predators. Here, \({\overrightarrow y _8}\) exemplifies the random vector among [0, 1]. The risk of predation for hippopotamus is demonstrated when \({\alpha _{{I_j}}}\)is lesser than \({\alpha _l}\), and hence they move fast toward the predator and make them withdraw. Moreover, \({\alpha _{{I_j}}}\)is large, then the invading hippopotamus or predator’s territory is indicated in expression (31). In this situation, the hippopotamus only moves to the limited range, and will not move to the predator, and hence awareness is made between invaders or predators aware of the existence in its territory.

$$Q_{{lu}}^{{HR}}=\left\{ \begin{gathered} {O_e} \oplus {I_u}+\left( {\frac{n}{{x - \varepsilon \cos (2\pi \mu )}}} \right)\cdot \left( {\frac{1}{B}} \right)\,\,,\,\,{\alpha _{{I_u}}}<{\alpha _l}\, \hfill \\ {O_e} \oplus {I_u}+\left( {\frac{n}{{x - \varepsilon \cos (2\pi \mu )}}} \right)\cdot \left( {\frac{1}{{2 \times B+{{\overrightarrow y }_9}}}} \right)\,\,,\,\,{\alpha _{{I_u}}} \geqslant {\alpha _l}\, \hfill \\ \end{gathered} \right.$$
(31)

Here, the change in the predator’s location when attacking hippopotamuses is epitomized as \({O_e}\), uniform random number among \([1,1.5]\)is symbolized as \(\mu\), hippopotamus’s position when facing predators is symbolized as \(Q_{{lu}}^{{HR}}\), random number among \([ - 1,1]\) is implied as x, \({\overrightarrow y _9}\)signifies the random vector of m-dimensionality, \(\varepsilon\)exemplifies the random \([1,1.5]\), n denotes the random \([2,4]\), B epitomizes random \([2,3]\).

Additionally, Levy distribution is employed for assessing a sudden change in the location when the hippopotamus attacks the predator, and it is calculated by using the below expression.

$$e(t)=0.05 \times \frac{{V \times {j_V}}}{{{{\left| t \right|}^{\frac{1}{c}}}}}$$
(32)

wherein, c exemplifies constant, Furthermore, \({j_V}\)is computed using the expression (32).

$${j_V}={\left[ {\frac{{\upsilon (1+t)\sin \left( {\frac{{\pi t}}{2}} \right)}}{{\upsilon \left( {\frac{{1+t}}{2}} \right)t \times {2^{\left( {\frac{{t - 1}}{2}} \right)}}}}} \right]^{\frac{1}{t}}}$$
(33)

Here, V and t indicates the random numbers between \([0,1]\). From expression (34), the replaced hippopotamus’s location is designated when the \(\alpha _{l}^{{HR}}\)is greater than \({\alpha _l}\), or else it returns to its population.

$${Q_l}=\left\{ \begin{gathered} Q_{l}^{{HR}}\,\,\,,\,\,\,\alpha _{l}^{{HR}}<{\alpha _l} \hfill \\ {Q_l}\,\,\,\,\,\,,\,\,\,\alpha _{l}^{{HR}} \ge {\alpha _l} \hfill \\ \end{gathered} \right.$$
(34)
Escape of hippo from predators

When the hippopotamus faces the predators, a defensive action will be taken by the hippopotamus to repel them. Furthermore, a random location is generated by the hippopotamus near the present location, and hence the behaviour is modeled based on the below expressions.

$$J_{u}^{{local}}=\frac{{{J_u}}}{\varpi },\gamma _{u}^{{local}}=\frac{{{\gamma _u}}}{\varpi },\varpi =1,2,...,\upsilon$$
(35)
$$Q_{{lu}}^{{HE}}={Q_{lu}}+{y_{10}}\left[ {J_{u}^{{local}}+{\iota _1}\left( {\gamma _{u}^{{local}} - J_{u}^{{local}}} \right)} \right]$$
(36)

In expression (36), \(Q_{{lu}}^{{HE}}\)is employed for searching the hippopotamus’s position to identify the near safest position \({\iota _1}\)chosen from the expression (37). Moreover, the situation under consideration has strong local search capability, and the expression of \({\iota _1}\)is given beneath.

$${\iota _1}=\left\{ \begin{gathered} 2 \times {\overrightarrow y _{11}} - 1 \hfill \\ \,\,\,\,\,\,{y_{12}} \hfill \\ \,\,\,\,\,\,{y_{13}} \hfill \\ \end{gathered} \right.$$
(37)

Here, random vector \([0,1]\)is typified as \({\overrightarrow y _{11}}\), \({y_{13}}\)represents the random \([0,1]\),\({y_{12}}\)designates the random variable.

Re-evaluation

The fitness function is employed for each iteration for determining the optimal solution, and it is measured by the using the expression (10). Moreover, the feasibility evaluation is designated as,

$${Q_l}=\left\{ \begin{gathered} Q_{l}^{{HE}}\,\,\,,\,\,\,\alpha _{l}^{{HE}}<{\alpha _l} \hfill \\ {Q_l}\,\,\,\,\,\,,\,\,\,\alpha _{l}^{{HE}} \ge {\alpha _l} \hfill \\ \end{gathered} \right.$$
(38)
Termination

The afore-mentioned mathematical steps of HGGO are frequently until the optimum solution is attained. The HGGO-based optimization algorithm for addressing complex optimization problems is presented in Algorithm 1.

Algorithm 1
figure a

HGGO-based optimization algorithm for solving complex optimization problems.

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Subsequently, the HGGO model, which is formed by fusing HO and GGO obtained optimum performance. Further, HGGO successfully determined the optimal hyperparameters of the XCovNet. Hence, the HGGO_XCovNet effectively finds the optimal trade-off in MIMO-enabled wireless network systems.

Result and discussion

The investigational outcomes of HGGO_XCovNet employed for resource allocation are examined by comparing the HGGO_XCovNet with several traditional approaches. Moreover, the analysis of HGGO_XCovNet is illustrated in this section.

Experimental setup

The implementation of HGGO_XCovNet used for resource allocation is done in Python tool with simulation.

Performance measures

Different evaluation measures, including sum rate, throughput, and energy efficiency are utilized for evaluating efficacy of HGGO_XCovNet. In this section, the description of these measures are given as follows.

Sum rate

It is defined as the overall data attained by every communication link or user in the network. Moreover, the sum rate26 in the system with many users is referred to as the total rate achieved by an individual user, and it is signified as,

$$SR=\sum\limits_{q}^{o} {{{W^{\prime}}_q}}$$
(39)
$${W^{\prime}_q}={\log _2}(1+{s^{\prime}_q})$$
(40)

Here, \({s^{\prime}_q}\) implies the SINR ratio, \(SR\)designates the sum rate, and \(\:{W}_{{q}^{{\prime\:}}}^{{\prime\:}}\)represents the sum rate of \({q^{th}}\)user.

Energy efficiency

It is26 frequently defined as the proportion of sum rate to power consumption, and it is designated as,

$$EE=\frac{{SR}}{{{D_{g^{\prime}}}}}$$
(41)

wherein, \(EE\) indicates the energy efficiency and power consumption is characterized as \({D_{g^{\prime}}}\).

Throughput

Throughput in MIMO networks is a critical measure that is used for determining the rate at which data is transmitted.

Table 4 details the Parameters and Variables for the HGGO_XCovNet Equations, offering a clear summary of the essential components utilized in the model.

Table 4 Parameters and variables for HGGO_XCovNet equations.
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Simulation result

The simulation result of HGGO_XCovNet at time 4s is shown in (Fig. 4). Here, The BS sends the signal to three header nodes, and then the header nodes distribute the signal to all the user equipment under it.

Fig. 4
figure 4

Simulation result of HGGO_XCovNet at time 4s.

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Simulation parameter

The simulation parameter used for implementing the newly developed HGGO_XCovNet technique is represented in (Table 5).

Table 5 Simulation parameter of HGGO_XCovNet.
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Performance analysis

The performance investigation of the newly presented HGGO_XCovNet technique for the Rayleigh channel and Rician channel by varying the number of users, and is demonstrated here.

For Rayleigh channel

In Fig. 5, the investigation of HGGO_XCovNet for the Rayleigh channel by changing the number of users is exemplified. The graph among users and throughput is represented in (Fig. 5a). For 300 users, the throughput achieved by HGGO_XCovNet with 10, 20, 30, and 40 iterations is 515.262, 527.932, 533.16, and 539.262 Mbps, respectively.

Fig. 5
figure 5

Performance assessment of HGGO_XCovNet for Rayleigh channel with (a) Throughput. (b) Sum rate. (c) Energy efficiency.

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Figure 5b depicts the examination of HGGO_XCovNet on the basis of the sum rate. When considering 100 users, the value of the sum rate measured by HGGO_XCovNet for 10 iterations is 227.663Mbps, 20 iterations is 233.696Mbps, 30 iterations is 236.33Mbps, and 40 iterations is 242.633Mbps. In Fig. 5c, the energy efficiency-based analysis of HGGO_XCovNet is signified. The energy efficiency attained by HGGO_XCovNet for 200 users with iteration 10, 20, 30, and 40 is 70.141kbits/joule, 72.070kbits/joule, 79.012kbits/joule, and 87.901kbits/joule.

For Rician channel

The investigation of HGGO_XCovNet for the Rician channel by changing the number of users is represented in Fig. 6. In Fig. 6a, the throughput-based analysis of HGGO_XCovNet is signified. The throughput attained by HGGO_XCovNet for 200 users with 10, 20, 30, and 40 iterations is 485.922Mbps, 497.926Mbps, 503.262Mbps, and 515.256Mbps. Figure 6b depicts the examination of HGGO_XCovNet based on the sum rate. When considering 400 users, the value of the sum rate measured by HGGO_XCovNet with 10 iterations is 233.964Mbps, 20 iterations is 242.696Mbps, 30 iterations is 248.964Mbps, and 40 iterations is 254.6Mbps. The graph among users and energy efficiency is represented in (Fig. 6c). For 300 users, the energy efficiency calculated by HGGO_XCovNet with 10, 20, 30, and 40 iterations is 50.942kbits/joule, 59.458kbits/joule, 66.942kbits/joule, and 75.743kbits/joule.

Fig. 6
figure 6

Performance assessment of HGGO_XCovNet for Rician channel with (a) Throughput. (b) Sum rate. (c) Energy efficiency.

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Comparative methods

The newly introduced HGGO_XCovNet approach exploited for resource allocation is contrasted with several prevailing techniques including PD-DQN1, WMMSE-based Dinkelbach algorithm18, ULAWS14, and DRL15.

Comparative analysis

In this section, the comparative examination of newly formulated HGGO_XCovNet approach for Rayleigh channel and Rician channel by altering the number of users is exemplified.

For Rayleigh channel

The estimation of HGGO_XCovNet by considering the Rayleigh channel with various metrics is shown in (Fig. 7). The investigation of HGGO_XCovNet regarding throughput is depicted in (Fig. 7a). For 300 users, the throughput recorded by HGGO_XCovNet is 539.262Mbps, whereas the prevailing models including PD-DQN, WMMSE-based Dinkelbach algorithm, ULAWS, and DRL recorded throughput of 478.59Mbps, 485.262Mbps, 503.262Mbps, and 521.26Mbps. In (Fig. 7b), the sum rate-based valuation of HGGO_XCovNet is exemplified. The traditional methods, like PD-DQN, WMMSE-based Dinkelbach algorithm, ULAWS, and DRL figured a sum rate of 212.696Mbps, 218.297Mbps, 224.628Mbps, and 233.297Mbps for 200 users, while the proposed HGGO_XCovNet obtained a sum rate of 251.697Mbps. Figure 7c delineates the examination of HGGO_XCovNet with energy efficiency. Here, when assuming 400 users, the energy efficiency quantified by HGGO_XCovNet is 74.943kbits/joule, PD-DQN is 31.012kbits/joule, WMMSE-based Dinkelbach algorithm is 39.021kbits/joule, ULAWS is 55.014kbits/joule, and DRL is 63.102kbits/joule.

Fig. 7
figure 7

Comparative assessment of HGGO_XCovNet for Rayleigh channel with (a) Throughput. (b) Sum rate. (c) Energy efficiency.

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For Rician channel

Figure 8 illustrates the estimation of HGGO_XCovNet by considering the Rician channel with various measures. Figure 8a delineates the examination of HGGO_XCovNet with throughput. Here, when assuming 400 users, the throughput quantified by HGGO_XCovNet is 539.261Mbps, PD-DQN is 467.860Mbps, WMMSE-based Dinkelbach algorithm is 485.272Mbps, ULAWS is 497.387Mbps, and DRL is 514.726Mbps. The investigation of HGGO_XCovNet regarding the sum rate is depicted in (Fig. 8b). For 300 users, the sum rate recorded by HGGO_XCovNet is 248.630Mbps, whereas the sum rate attained by the prevailing models including PD-DQN, WMMSE-based Dinkelbach algorithm, ULAWS, and DRL is 209.328Mbps, 218.631Mbps, 221.630Mbps, and 227.636Mbps. In (Fig. 8c), the energy efficiency-based valuation of HGGO_XCovNet is exemplified. The traditional methods, like PD-DQN, WMMSE-based Dinkelbach algorithm, ULAWS, and DRL figured an energy efficiency of 52.631kbits/joule, 59.023kbits/joule, 59.830kbits/joule, and 63.012 kbits/joule for 200 users, while the proposed HGGO_XCovNet obtained an energy efficiency of 79.983kbits/joule.

Fig. 8
figure 8

Performance assessment of HGGO_XCovNet for Rician channel with (a) Throughput. (b) Sum rate. (c) Energy efficiency.

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Comparative discussion

In Table 6, the comparative discussion of HGGO_XCovNet with respect to several measures for Rician channel and Rayleigh channel is illustrated. Here, for Rayleigh channel the newly presented HGGO_XCovNet model attained maximum throughput of 551.262Mbps, sum rate of 269.930Mbps, and energy efficiency of 74.943kbits/joule. Furthermore, the existing methods, like PD-DQN, WMMSE-based Dinkelbach algorithm, ULAWS, and DRL quantified a throughput of 485.206Mbps, 497.922Mbps, 515.922Mbps, and 533.262Mbps, while the sum rate recorded by these techniques are 221.630Mbps, 230.630Mbps, 236.630Mbps, and 242.610Mbps. Moreover, the energy efficiency measured by PD-DQN is 31.012kbits/joule, WMMSE-based Dinkelbach algorithm is 39.021kbits/joule, ULAWS is 55.014kbits/joule, and DRL is 63.102kbits/joule.

Table 6 Comparative discussion of HGGO_XCovNet.
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Hence, it is demonstrated that the newly developed HGGO_XCovNet attained better results than other prevailing models. Besides, HGGO is employed for training and XCovNet is engineered by the combination of HO and GGO. Here, the HO algorithm is robust and obtained high scalability in the optimization process, while GGO reduced energy consumption with a faster convergence rate. Thus, HGGO lead to faster training time for XCovNet by improving the resource allocation in MIMO systems.

Conclusion

MIMO is regarded as a vital technology, which is utilized in modern wireless communication systems. Resource allocation is the process of effectively distributing several limited resources among multiple users to maximize the performance by improving QoS. Here, optimal resource allocation is required to achieve high throughput and energy efficiency. However, various conventional models were employed but they are not effective in handling complex and large-scale scenarios. Therefore, this paper introduces an HGGO_XCovNet for resource allocation. Firstly, a BS with multiple users is considered and the resource allocation is carried out by considering various objective functions, namely SINR, data rate, and power consumption. Moreover, the resource allocation is performed by employing a DL model called XCovNet, where XCovNet is trained using the proposed HGGO model. Further, the HGGO is formulated by the combination of GGO and HO. Furthermore, the HGGO_XCovNet technique measured a maximum energy efficiency of 74.943 kbits/joule, a sum rate of 269.93 Mbps, and a throughput of 551.262 Mbps. Future work will focus on implementing AI models to optimize resource allocation in MIMO systems while reducing energy consumption.