Hybrid computational and real data-based positioning of small cells in 5G networks

Coverage and capacity optimization of 5G networks

An analytical study on the coverage, handoff and cost optimization for heterogeneous 5G networks (5G-HetNet) has been written by Ouamri et al. (2020). A path loss model is suggested for both Line-of-Sight (LoS) and Non-Line-of-Sight (NLoS) situations, with different path loss exponents, and the handling of coverage and handoff probability is done by stochastic geometry with values in SINR (Signal to Interference-Plus-Noise Ratio). By the method of Cellular Network Planning (CNP), network investment cost issues are optimized.

Khan, García-Armada & Escudero-Garzás (2020) talks about a heuristic method to predict the employment of 5G base-stations in Spain by obtaining user and traffic data from LTE networks, using the free database of OpenCellid. This dimensioning of a 5G network takes into consideration the various aspects of this technology: heterogeneous architecture, the necessity for a high Quality of Experience (QoE) and smart resource allocation. Their objectives are achieved by separating the highest traffic areas, and then deciding where 5G cells are to be deployed.

Khan et al. (2022), however, improves on their model of 5G planning by assigning a clustering algorithm to the task of deploying 5G base-stations into high data traffic areas. A K-means algorithm has been utilized, using the Elbow heuristic as a benchmark. Not only does it deal with the coverage characteristics of the network, but also demonstrates an extensive study on its capacity dimensioning. The goal of the study is to decrease the network cost, as well as provide a more robust way to interpret the data for network planning decision.

Another article that talks about coverage area optimization for 5G is found in Wang, Lee & Wu (2020). It is mathematically complex, as the goal is to provide numerical solutions for the employment of macro and small cells in 5G. Therefore, different clustering approaches are suggested by the authors, as well as models to prevent noise and interference. Numerical case examples are given for different scenarios and, as it deals with heterogeneous networks, the coverage simulations can be seen with generally one or two macro cells and several small cells around them.

Jia, Ji & Chen (2019) denotes another mathematically complex problem: the intra-clustering organization and interference issues of 5G millimeter wave (mmWave) networks. Given that mmWave has very little coverage, the usage of Pico cells and Femto cells proves to be necessary. This amounts to a large number of cells and nodes within the network that cluster up and results to large amounts of inter and intra-cluster interferences. The article then lists some of the differences between Pico and Femto cell user association, and then derive said UE associations and Laplace transforms of interference via stochastic geometry.

Meanwhile, Bektas et al. (2021) deals with the planning of private 5G networks. In situations which require fast or even temporary network solutions, the authors have provided an unsupervised machine leaning (UML) technique for base-station placement. It takes into consideration the boundaries and environmental variables of the area to be covered, as well as service quality (in this case, a signal greater than −90 dBm). Ray Tracing simulations on a virtual environment emulating the Monaco Grand Prix are made to validate and test the outputs of the UML implementation.

Jain, Lopez-Aguilera & Demirkol (2021), in turn, proposes a framework for 5G capacity optimization based on User Association and Resource Allocation (AURA-5G). It aims to optimize for both maximum sum of throughtputs and best bandwidth allocation within the network. This framework is then applied to be tested in many distinct architectural applications of 4G and 5G, such as Enhanced Mobile Broadband (eMBB) and Massive Machine Type Communication (mMTC). The study demonstrates that this framework improves throughput and other metrics such as latency when compared to baseline user association techniques.

Heterogeneous wireless networks and 5G architecture

As data volume and demands increase with the higher connection data rates of 5G, it is necessary to implement different ways of handling user access to improve the connectivity and capacity of future networks. Hence, 5G implementations aims to provide heterogeneous access and network slicing to meet different data criteria. Here, we shall discourse on some works on the matter, given that it is a fundamental field of study for future applications.

In Haile, Mutafungwa & Hämäläinen (2020), there are sections dedicated on elucidating about ownership models and heterogeneity of architectures in small cell-based 5G-NR networks. Given that the article focuses on the hyperdensification due to growing user demands, it claims that operator-based small cells may not be sufficient to support user data traffic. So, neutral third-parties (governmental or private companies) or even user-deployed small cells may be employed to deal with such problems.

The authors then discourse that a shift to a “service-oriented” paradigm for network planning will become more common eventually, and that 5G network slicing is one of its main features. Network slicing is a way to utilize distinct, end-to-end logical networks taking into consideration different types of user demands whilst still operating within the same physical structures—see Zhang (2019).

One common architecture and user access standard for 5G addressed in many works such as Harper & Sirotkin (2020), Haile, Mutafungwa & Hämäläinen (2020) and Bertenyi et al. (2018) is NG-Radio Access Network (NG-RAN). It is classified as a heterogeneous network as it aims to function with nodes that can provide either 5G New Radio (5G-NR) access or 4G-LTE access (ng-eNB). Its operation can be standalone or non-standalone. Standalone means that only the 5G core architecture (5GC) will be utilized, whilst non-standalone is an integration of NR and LTE-EPC architectures allowing user access for both 4G and 5G. Moreover, network virtualization of 5G base-stations can allow traffic separation into central units (CU), distributed units (DU) and radio units (RU), as well as provide cloud compatibility (C-RAN).

Another technique to improve user access is non-orthogonal multiple access (NOMA), which can be applied to a plethora of situations within 5G networks, such as Femto, Pico and Micro Cell 5G coverage and Networked Flying Platforms (NFPs). According to Lv et al. (2020) and Diamanti et al. (2020), as opposed to orthogonal multiple access (OMA) that allocates resources for each user separately, NOMA utilizes multiplexing in the power domain in order to accommodate multiple users into the same time/frequency resources, promoting a better utilization of available bandwidth, via a method called superposition coding (SC). The management of interference between users is then processed by secure interference cancellation (SIC), a technique that decodes this multiplexed signal at the receiving end.

Given its extremely focused on power allocation and that channel state information (CSI) can be difficult to acquire on this type of technique, optimizations for NOMA network tend to lie under these subjects. Examples are the aforementioned work of Diamanti et al. (2020) that proposes power allocation optimization via reinforcement learning (RL) and contract theory (CT), Xu & Cumanan (2017) that proposes the same problem but considering the statistical CSI on transmitter and using the $\alpha$-fairness method, and Fang et al. (2017) that considers imperfect CSI and aims to improve resource allocation within a multi-carrier NOMA (MC-NOMA) framework.

Kuribayashi et al. (2020) has developed a cell range expansion technique based on a particle swarm algorithm. This work fills both roles of providing coverage planning and expansion while maintaining a HetNet context. The objective is to achieve said small cell coverage expansion by maximizing the number of users whose downlink connection requirements are met. The algorithm then balances the load of small cells via maximum SINR of users, and uses it to model an objective function that aims to compute a cell range Expansion bias (CRE). Conventional PSO and other particle-based approaches have been tested against their proposed algorithm, the PSO-Based CRE Bias (PCB), which has shown to be better for this type of application.

Bioinspired algorithm hybridizations for data clustering

Jensi & Jiji (2015) have accomplished to generate a FPA hybrid with K-means, a similar method to the study presented herein. In the article, they present this technique to test and clusterize eight different datasets, with promising results. It is attested that the hybridization has given better average fitness results for all tests, thus proving it to be more efficient.

However, the hybridization process shown in the article takes the current best solution in the FPA and executes a local search around it using the K-means. This is different from the implementation we have provided, to be further explained in the Methodology section.

Hatamlou (2017) is another article that deals with nature-inspired algorithms, with their proposed technique of utilizing particle swarm optimization (PSO) with the big bang-big crunch (BB-BC). The objective, like in Jensi & Jiji (2015), is to provide an efficient method of data clustering, and both of them use five equal datasets (Iris, Wine, Glass, CMC, Cancer) in order to test their techniques. However, for all the same datasets, the PSO-BB-BC hybrid has shown better average fitness than its FPA-KM counterpart.

Another robust work in the subject is found in Logesh et al. (2020). Based on TripAdvisor travel recommendations data, the authors propose several hybrid solutions. These are based on swarm intelligence algorithms, such as an improved PSO model, Brainstorm Optimization (BSO), Quantum-Behaved Brainstorm Optimization (QBSO) and the Immune Genetic (IG) technique. In most cases, the best optimizer is the novel QBSO-IG hybrid that they have proposed themselves. This study also provides many details on recommendation systems and draws a comparison on many other articles that have dealt with its data clustering optimizations.

Khan, Aftab & Zhang (2019) have proposed a clustering technique for Flying Ad-Hoc Networks (FANETs), named Bio-Inspired Clustering Scheme for FANETs (BICSF), that consist of organizing drones in the air based on their positioning and energy management. The goal is to reduce energy consumption, elevate the air time of the devices and minimize the time that the drones take to organize themselves into clusters. The hybrid is between the glowworm swarm optimization (GSO) and krill herd (KH) algorithms, and it is tested against more commonly found bioinspired ones—Grey Wolf Optimization (GWO) and Ant Colony Optimization (ACO), to specify.

Pitchaimanickam & Murugaboopathi (2020) also proposes a hybrid approach for network clustering, however, with an emphasis on ameliorating battery life and information management in wireless sensor networks. The creation of clusters itself is done by a LEACH-C algorithm, which is then optimized with a conjuction of a PSO and the Firefly Algorithm (FA). HFAPSO, as the hybridization is abbreviated, proves to be considerably more efficient than the separated techniques in a simulation of 100 sensors and an area of (100 × 100) m². Results have shown that not only does it better coordinates the clusters, but it sustains the battery life of sensor nodes for longer.

Cao et al. (2021) demonstrates a heterogeneous Wireless Sensor Network (WSN) coverage area optimization, in which a Chaos-Improved Social Spider Optimization (CSSO) is used to accomplish the task. The solution aims to deploy the sensors by saving as much energy consumption and network cost as possible, as well as reducing coverage redundancy and trying to cover blind spots as best as possible.

Lastly, a study that also uses K-means in a hybrid bioinspired computing setting is conducted by Aswani, Kar & Vigneswara Ilavarasan (2018). It is combined with a Firefly Algorithm, but also with a modified, Lévy flight and chaos-induced version of it (LFA-chaos). This clustering technique is then tested against a Fuzzy C-Means (FCM) algorithm, and its application is to find spam accounts on social media website Twitter. It uses many significant behavioral factors of the user as input to solve this problem, such as its hashtag frequency, number of mentions, tweet count, follower count and many more, up to a total of 13. The Lévy flight applied to this FA is similar to the one used in the FPA exposed in the Methodology section.

Contributions

The main contribution to our study is the utilization of a fairly simple low complexity bioinspired computing and clustering hybrid in order to solve a cell planning (CP) problem by means of small cell coverage and transmitted power optimization.

This study also utilizes real-life data of LTE by the methods of Khan, García-Armada & Escudero-Garzás (2020) in order to provide a life-like user deployment in the city of Belém, Brazil—with suburban to urban characteristics and within the Amazon rainforest biome. The works of Khan, García-Armada & Escudero-Garzás (2020) and Khan et al. (2022) may be evolved in user selection and data mining but still leave a gap to come up with more developed, faster computational methods to solve 5G coverage problems—the former study used a heuristic created by the authors themselves, and the latter used only a clustering algorithm. The hybrid solution present herein is a better and more developed technique overall, therefore it shall produce better coverage optimization results.

Cell planning is considered to be a NP-hard problem, therefore, a solution in real time is unlikely to be found and an exact solution is hard to find in polynomial time, as explained by Weicker et al. (2003) and Taufique et al. (2017). Bioinspired algorithms, however, tend to have low algorithmic complexity but are able to solve highly complex problems—see He et al. (2017). They can be useful tools by providing a metaheurisitc approach that takes less time than other techniques to produce satisfactory network planning results—e.g., the studies of Kaur, Pal & Singh (2020) on the matter. Thus, another contribution of this work is the Running Time analysis conducted to attest that such NP-hard, non real-life problem can be further optimized in a matter of a dozen or couple dozen of minutes only.

As a secondary objective, the work herein can also become a reference for further coverage and capacity planning of 5G networks in suburban and urban environments of the Amazon rainforest region, or even other regions of Brazil, by the usage of real-life user data estimation and dimensioning—which enhances the verisimilitude of simulational results.

Multi-objective cuckoo search

One of the most utilized bioinspired algorithms, the Cuckoo Search algorithm has been created by Yang & Deb (2009). It has been used to optimize multiple applications ever since, mainly in the areas of mathematics and engineering, according to Shehab, Khader & Al-Betar (2017) and Ferreira et al. (2021). The main bioinspired idea behind this method is the computational modeling of the parasitic behavior of cuckoo-type birds, that often lay their eggs inside the nests of other types of birds. This is a natural occurrence in nature, as the host species often does not perceive the cuckoo’s “alien” egg inside the nest, or either choose to ignore it completely or abandon the nest altogether, choosing another place to lay its eggs on. However, if the egg is ignored and left to grow inside the host bird’s nest, the cuckoo hatchling is born, and reaches maturity much faster than the other eggs, pushing the host bird’s eggs outward. Thus, the cuckoo baby bird expels the other eggs from the nest, resulting in a higher food share for it, and becoming well-fed.

Thus, the whole process is based upon three major rules:

1. Each cuckoo lays one egg at a time, and deposits it in a random nest. Each egg is considered a potential solution—metaheuristic

2. The best nests carry the best eggs (solutions), and these will survive the next generations due the parasitic nature of the cuckoo hatchlings—elitism.

3. The number of available nests is constant, and defined by the code developer. The probability of a cuckoo egg being discovered by the host bird is defined as $Pa\in [0,1)$. After this, the bird may choose to discard this egg or abandon its nest—discarding the worst solutions.

The latter rule can also be described by indicating that a probability fraction Pa from the various n nests of host birds are replaced by new nests, presenting randomized solutions.

The movement of cuckoos within the search space is dictated by a stochastic method called Lévy flights, which provides each cuckoo (from a number of i cuckoos) in the code iteration a path to find possible solutions (nests). Equations (2) and (3) represent, mathematically, a Lévy flight and its Lévy distribution, respectively:

(2) $$X_i^{t+1}=X_i^t+\alpha \oplus Levy(\beta )$$

(3) $$Levy(\beta )\cong u=t^{-(\beta +1)};(1<\beta \le 3)$$where i and t are, respectively, the maximum number of cuckoos and the current code iteration. $\alpha$ is a parametric value denoting the step size for the cuckoo flight, that must generally be greater than zero. In this study the value is set to $\alpha =1$, as Yang & Deb (2009) claim that an unitary alpha is sufficient for most optimization cases.

In Eq. (2), the Lévy distribution is associated to the Lévy flight with the product $\oplus$, which means “entrywise multiplication”—a kind of product between two matrices of the same size. In the PSO algorithm, a similar product can be found, however, for the Lévy flight method, the search space can be much better harnessed.

As for Eq. (3), this is about the Lévy distribution. It possesses infinite variance and average values. The variable $\beta$ is the random step length, needed for providing a variable magnitude to the random walk performed by the Lévy flight.

Given that the Lévy distribution presents infinite variance, the search space is virtually limitless, meaning that the length of the flight taken could be very short or incredibly long. However, generally, the new solutions are generated through the Lévy method around the best obtained solution on a given instant, accelerating the process and concentrating the computational effort in a part of the search space. Oftentimes, however, solutions are generated randomly across the space—this is good to prevent the algorithm from getting stuck in local optima, which are not the globally best possible solution.

An illustration showing a Lévy flight with around a hundred steps can be found in Fig. 2.

A pseudocode of the CS algorithm, as well as its multi-objective counterpart, can be found in the article in which it was proposed by Yang & Deb (2009). Below, in Algorithm 1, a transcription of this code is presented.

Multi-objective flower pollination

A solution based on the behavior of natural flower pollinators has been proposed by Yang (2012), the same author of the cuckoo search, which also utilizes the Lévy flight method of optimal space search. Its efficiency, in many single and multi-objective applications is proven to be greater than particle swarm and genetic optimization algorithms. Its multi-objective counterpart has been proposed in 2013, also by Yang, Karamanoglu & He (2013).

As for the stages of the algorithm, four rules are defined thusly:

1. Biotic and cross-pollination is considered as global pollination process with pollen-carrying pollinators performing Lévy flights.

2. Abiotic and self-pollination are considered as local pollination.

3. Flower constancy can be considered as the reproduction probability and is proportional to the similarity of two flowers involved.

4. Local pollination and global pollination is controlled by a switch probability p $\in$ [0, 1]. Due to the physical proximity and other factors such as wind, local pollination can have a significant fraction p in the overall pollination activities.

Two kinds of pollination are considered and simulated: global and local pollination. This assures that the code does not only fall for local solutions, healthily seeking to encounter a global solution to the objectives. For simplicity manners, the algorithm is based on the idea that every plant possesses only one flower and can pollinate also just one other flower at a time, when in true biological terms they can hold a few flowers and millions of pollinating gametes. This is so that a plant/flower/pollinating gamete are all considered to be part of one solution altogether.

Hence, the first rule (global pollination) and third rule (flower constancy) of FPA are mathematically represented as shown in Eq. (4):

(4) $$X_i^{t+1}=X_i^t+L(X_i^t-g\ast )$$where $X_i^t$ is the pollen i or solution vector $X_i$ at iteration t, and g ${}^{\ast }$ is the current best solution among all solutions at the current iteration.

Pollination strength L is dealt via Lévy flight, in which it is a measure of each flight’s step size, as denoted in Eq. (5). Here, the flights symbolize the path of insects and pollinator animals in a given area—in the algorithm, this area is the optimization’s global search space. However, the equation used in this algorithm differs from the one found in cuckoo flights, as it is based on producing Lévy flights via the Mantegna algorithm. This is basically a technique to generate pseudo-random step sizes via normal distributions in order to provide an optimal performance whilst still maintaining the demands of the Lévy distribution.

(5) $$L\approx \frac{\lambda \mathrm{\Gamma }(\lambda )\sin (\pi \lambda /2)}{\pi s^{1+\lambda }},$$in which $\mathrm{\Gamma }(\lambda )$ is the standard, classic gamma function found in Lévy flights, and other probabilistic and complex number applications.

The Mantegna step size algorithm can be explained as the Eq. (6).

(6) $$s=\frac{U}{|V{|}^{\frac{1}{\lambda }}},$$with $s$ being the step size, U being drawn from a Gaussian distribution of variance ${\sigma }^2$ and V also being drawn from a Gaussian distribution but with unitary variance, as can be verified in Yang, Karamanoglu & He (2014). Generally, the lambda is treated as a parametric value and it is safe to assume that it is a constant with a possible value of around $\lambda \in [0.5,1.5]$. When $\lambda =1$, the variance also equals 1, and results are in such case easier to predict.

For the purpose of Rule 2 (local pollination), the flower constancy is mimicked for a limited neighborhood near to the reproductive flower’s position. It is represented as

(7) $$X_i^{t+1}=X_i^t+\epsilon (X_j^t-X_k^t)$$where $X_j^t$ and $X_k^t$ are pollens from different flowers of the same plant species.

The fourth rule is a probabilistic switch between global and local pollination, and the probability p can be parametrically and singularly adjusted to improve optimization performance, depending on the need of the objective function.

All stages of the algorithm are represented in pseudocode form by the recommendations in Yang (2012), which are transcribed in Algorithm 2. Some details previously discussed can be noticed, such as an if/else switch for global and local pollination, which are done by Lévy flights and random selection, respectively.

The K-means algorithm

An unsupervised learning method of computational intelligence, the K-means clustering algorithm has the goal of grouping similar data points. Each of those $k$ amount of clusters, then, possesses a centroid, which is the mean value M of all positions of the data points. Due to being fast and easy to reproduce, it is one of the most popular clustering algorithms. Sinaga & Yang (2020) defines the objective function of K-means according to the following equation (Eq. (8)):

(8) $$J(A,Z)=\sum _{i=1}^n\sum _{k=1}^c{z}_{ik}||x_i-a_k|{|}^2$$in which $x_i$ is a data point i belonging to a dataset $X=\{x_1,x_2,...,x_n\}$ spread over an Euclidean space ${ℝ}^d$ of d-dimensions, $a_k\in A=\{a_1,a_2,...,a_n\}$ is the centroid of the k-th cluster and $z_{ik}$ is a binary variable (either 0 or 1) that signals if the user i belongs to cluster k. Given that the objective function needs to be minimized, the algorithm then proceeds to alter the centroids by also minimizing the Euclidean distance between them and the data points that belong to them, according to Eq. (9):

(9) $$a_k=\frac{{\sum }_{i=1}^n{z}_{ik}{x}_i}{{\sum }_{i=1}^n{z}_{ik}},z_{ik}=\{\begin{array}{cc}1,\hfill & \mathrm{i}\mathrm{f}||{\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{k}}|{|}^2={min}_{(1\le \mathrm{k}\le \mathrm{c})}||{\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{k}}|{|}^2.\hfill \\ 0,\hfill & \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.\hfill \end{array}$$where $||x_i-a_k|{|}^2$ is the Euclidean distance.

The ECC-33 propagation model

A channel path loss model is required to evaluate which users are connected to the network. This propagation model should both belong to a frequency range where the 5G-NR channels are going to be propagated, as well as adapt to the environment in which it shall be applied. Some examples in the literature by Samad, Choi & Choi (2022) and Oladimeji, Kumar & Elmezughi (2022) address models such as the Close-In (CI), Floating Intercept (FI) and Alpha-Beta-Gamma (ABG), along with measurement campaigns to verify their usage in 5G channel modelling characteristics.

However, in this study, an ECC-33 model was chosen, which is adapted to the urban but forested environments of the city of Belém, as proposed by de Carvalho et al. (2021).

ECC-33 is derived from the famous Okumura-Hata model, but taking into consideration the behavior of higher frequencies, thus extending the frequency range. In Mollel & Michael (2014), experimental results have been drawn by comparing different path loss models conducted in Dar es Salaam, Tanzania. The COST-231 and ECC-33 models had the lesser Root Mean Square Error (RMSE), meaning that—for urban, suburban and rural settings—they had the most approximate results to measured signal data.

Therefore, the model used in this work is defined thusly:

(10) $$P{L}_{ECC}=A_{fs}+A_{bm}-G{F}_t-G{F}_r$$in which $A_{fs}$ is the free-space attenuation, $A_{bm}$ is the base median path loss and $G{F}_t$ and $G{F}_r$ are the transmitting antenna gain factor and received antenna gain factor, respectively. The values used for the model correspond to the large city scenario, as all acquired data correspond to an urban region of medium-to-high density. The following Eqs. (11)–(14) further detail the variables of the model:

(11) $$A_{fs}=92.4+20log(d)+20log(f_c)$$

(12) $$A_{bm}=20.41+9.83log(d)+7.89lo{g}_{(f_c)}+9.56(log(f_c){)}^2$$

(13) $$G{F}_t=log(\frac{h_{BS}}{200})(13.958+5.8log(d){)}^2$$

(14) $$G{F}_r=0.759(h_{UE})-1.862$$in which $f_c$, $h_{BS}$ and $h_{UE}$ are the central frequency, the base-station antenna height and the user device’s antenna height, respectively. While $d$ is the distance between a user and a base-station, that in this study is calculated by the haversine distance formula, as it appears on Gade (2010) and Chopde & Nichat (2013):

(15) $$d=2R\arcsin \sqrt{si{n}^2(\frac{{\mu }_2-{\mu }_1}{2})+cos({\mu }_1)cos({\mu }_2)si{n}^2(\frac{{\lambda }_2-{\lambda }_1}{2})}$$where $\lambda$ are the latitudinal positions, $\mu$ are the longitudinal positions, and R is the approximate radius of the Earth ( ${\mathrm{6.371.10}}^6$ m). Furthermore, the received power for an user is represented in Eq. (16):

(16) $$P_r=P_t-P{L}_{ECC}+G_t+G_r,$$in which $G_t,G_r$ are the transmitting (Tx) antenna gain and receiving (Rx) antenna gain.

The hybridization process

In real-life scenarios, not often is the number of clusters to achieve optimal results is known beforehand. For that, there are some heuristic methods that can be utilized to get approximations of how many clusters one might need to satisfy an application. For instance, Khan et al. (2022) have chosen the Elbow heuristic. However, given that one might need a more complex and sophisticated way of deciding how to cluster, and how much to cluster, like in a scenario of network dimensioning and coverage area optimization, these heuristics may not provide optimal solutions. This is why a metaheuristic method, like a bioinspired algorithm, is more suited for this task. The hybridization of a clustering algorithm like K-means and a metaheurisitc BIC means that both of those methods benefit from each other. The K-means provides the base-station coordinates on a given area, and the BIC provides the values of the ideal number of clusters, taking into consideration multiple inputs that will decide how much clustering there must be. Figure 3 demonstrates the whole process of the study. Please notice how the K-means processes a geographical position for cluster centers and the BIC returns it a number of clusters to attempt and see if an optimal result is reached. This process has the goal, in our study, to provide an optimal value of k clusters, as well as a satisfactory transmitted power value.

A pseudocode displaying the structure of the hybridization can be found in Fig. 3. For each iteration of the BIC, the K-means will run alongside it, providing cluster center (or centroid) values that the BIC shall take and calculate the received power for each and every user, one cluster at a time. If the user has a received power of −90 dBm or greater, it will be considered as connected to the network and, thus, covered.

Results for 700 MHz

Firstly, Figs. 4 and 5 display the results achieved for the 700 MHz frequency band for, respectively, the MOCS-KM and MOFPA-KM. This is a band that possesses a fairly low data rate and bandwidth, but since it is one of the auctioned bands for 5G operation in Brazil, it is included here as means of comparison and to test the BIC + K-means hybrids.

The X symbols in the figures displaying the clusters are referent to the position of centroids, and the colored dots correspond to the users.

Here, the algorithm had no problems whatsoever in optimizing for zero fitness, needing only a few iterations to converge. This is because path loss is less intense for lower frequencies, and so a small group of small cells can already provide perfect coverage.

The best solver in this case is MOCS-KM, with a solution of 12 clusters and power of 30 dBm. However, MOFPA-KM comes very close to a zero fitness value as well, with an inferior number of clusters of only 9 and 30 dBm power.

Results for 2.3 GHz

The band that promises to function both in LTE and in 5G is shown next. Figures 6 and 7 are the results for the 2.3 GHz frequency band. It is expected to operate in smaller cities as a means of digital inclusion, providing more bandwidth to 4G-LTE demands, or in urban centers as a complementary traffic alternative to the main band of 5G, which is 3.5 GHz.

Results for this band are already harder to solve than 700 MHz, with the best fitness results coming from MOCS-KM once again. It has achieved $f_{best}=2$, with 46 clusters and a satisfactory transmitted power of 31 dBm. MOFPA-KM has proposed a higher $P_t$ value with less clusters, resulting in $f_{best}=2.4267$, $k=34$, and $P_t=33.3352$ dBm.

Results for 3.5 GHz

Finally, the main and most used frequency band of 5G. The 3.5 GHz spectrum possesses up to 80 MHz of band per operator in Brazil, and is set to expand in highly-urbanized areas, as it can handle greater traffic and provide faster data rates than the other two. Figures 8 and 9 demonstrate the results of the simulation for this frequency.

For this case, the results are basically the same in terms of coverage, but fitness values turn greater because of the choice in both algorithms to utilize more transmitted power instead of increasing the number of clusters too much. In terms of fitness, MOCS-KM marginally wins again, with $f_{best}=4.006$. That being said, MOFPA-KM has converged to a solution with 34 clusters again—the same number from the 2.3 GHz simulation. It has, however, increased the transmitted power to its maximum value of 40 dBm in order to achieve that.

Table of results

In Table 3, results for the optimization processes of MOCS-KM and MOFPA-KM hybrids are shown. For the best generated population (the one which produced $f_{best}$), it records the maximum, medium and best fitness values, as well as the outputs and the running time of the respective simulations.

The parameters and variables displayed in Table 3 are: the optimal number of clusters (k); optimal transmitted power per cluster ( $P_t$, in dBm); the percentage of users suffering outage ( $N_{outage}$, in %); the differential between optimal $P_t$ and the lower bound value of $P_t=30$ dBm ( $P_{diff}$, in dBm); maximum objective function fitness value found within the best generation ( $f_{max}$); average fitness value of all individuals in the best generation ( $f_{avg}$); the fitness value of the best individual, in the best generation ( $f_{best}$); and the total running time of the optimizations (Time, in seconds).

Computational analysis

In order to elucidate the performance of both algorithms, a numerical analysis of their computational complexity has been conducted.

Theoretically, the study of algorithmic complexity of bioinspired computing and clustering hybrids has already been conducted. Kaur, Pal & Singh (2020) denotes that the complexity of both CS-KM and FPA-KM hybrids are set to linear variables in the Big-O formula—see Bae & Bae (2019). These are still valid for multi-objective counterparts of the same techniques. Thus, the equations for algorithmic time complexity, adapted from Kaur, Pal & Singh (2020), are as follows:

(18) $$O_{MOCS-KM}=O(N_i\ast k\ast n_{CS}\ast N_{users})$$

(19) $$O_{MOFPA-KM}=O(N_i\ast k\ast n_{FPA}\ast N_{users})$$in which $N_i$, $k$, $n_{CS}$, $n_{FPA}$ and $N_{users}$ are the number of iterations in the code, the number of clusters, the cuckoo population (for MOCS), the pollen population (for MOFPA), and the number of users to be covered by the network, respectively. The area of coverage, as considered in the original formula, can be replaced into the number of users to be covered by the network. In general terms, the more users a network needs to cover, the greater the area of coverage should be.

Numerical simulations have been performed taking into account the running time of the algorithms. Therefore, a set of 32 data points, 16 per technique, were obtained via simulations in order to measure the linearity of computational complexity. That is because, for each simulation, one of four parameters (iterations, clusters, population and users) is chosen to vary between four values, while all others are kept to fixed values.

Cardoso et al. (2020) is an example of an article that provides a numerical analysis of results. Although it utilized another network planning method (cellular automata, or CA), its methodology, along with Kaur, Pal & Singh (2020), have served as basis for the computational analysis made in this work.

Table 4 shows the variations chosen for each parameter, denoting in italic letters which ones are kept constant—for both MOCS-KM and MOFPA-KM. Lastly, Figures 10 and 11 display the numerical analysis results for MOCS-KM and MOFPA-KM parameters, respectively. These are curve fittings (blue line) taking into account the simulated data points (circles) in MATLAB.

Given that simulational results have shown in Table 3 to operate, in standard parameter values, in around 10 to 15 min, these are satisfactory approximations for the NP-hard problem of cell planning. Furthermore, their performance are linearly scalable, thus easy to predict for a set of parameter values, as seen in Figs. 10 and 11.

Table 5 denotes the running times and coverage capacity (by number of users covered) of data points for MOCS-KM, whilst Table 6 does so for MOFPA-KM, both for the frequency of 3.5 GHz. These two variables are measured in the analysis of Cardoso et al. (2020), thus it is useful to detail them further here as well. The coverage ratio, which is the amount of users connected to the clusters, can be calculated as $N_{connected}$ taken from objective function (Eq. (17)).

In Table 5, it is noticeable that the maximum coverage achieved by MOCS-KM, considering all users at 3.5 GHz, is capped at 97,4%. A likewise behavior is seen for MOFPA-KM, in Table 6, where maximum capacity is capped at 98%. When reducing the number of users, they may become more scattered in the search space, thus resulting in slightly lower coverage in some cases.