Energy-efficient virtual machine placement in distributed cloud using NSGA-III algorithm

Abstract

Cloud computing is the most widely adapted computing model to process scientific workloads in remote servers accessed through the internet. In the IaaS cloud, the virtual machine (VM) is the execution unit that processes the user workloads. Virtualization enables the execution of multiple virtual machines (VMs) on a single physical machine (PM). Virtual machine placement (VMP) strategically assigns VMs to suitable physical devices within a data center. From the cloud provider's perspective, the virtual machine must be placed optimally to reduce resource wastage to aid economic revenue and develop green data centres. Cloud providers need an efficient methodology to minimize resource wastage, power consumption, and network transmission delay. This paper uses NSGA-III, a multi-objective evolutionary algorithm, to simultaneously reduce the mentioned objectives to obtain a non-dominated solution. The performance metrics (Overall Nondominated Vector Generation and Spacing) of the proposed NSGA-III algorithm is compared with other multi-objective algorithms, namely VEGA, MOGA, SPEA, and NSGA-II. It is observed that the proposed algorithm performs 7% better that the existing algorithm in terms of ONVG and 12% better results in terms of spacing. ANOVA and DMRT statistical tests are used to cross-validate the results.

Introduction

Cloud computing is a model for outsourcing an organization's computing power to a rented infrastructure. Cloud computing is possible because of emerging service-oriented architecture, sophisticated servers, and software-defined networking technologies. The physical machine can host multiple operating systems with the help of a hypervisor software module installed in physical devices [1, 2]. Virtualization significantly reduces resource wastage instead of using an entire machine hosted with a single operating system. Resource wastage is the unused CPU and RAM after placing the virtual machine in the respective physical machine (Residual). In simple terms, we can express ${R}_{w}={R}_{a}-{R}_{u}$, where ${R}_{w}$ denotes the resource waste, ${R}_{a}$ denotes the available resource in the physical machine, ${R}_{u}$ denotes the resource consumed or utilized by the number of virtual machines hosted in the physical machine [3].

The networking infrastructure is isolated using SDN and assigned to individual virtual machines for communication. SOA is used to expose the virtualized data center to the end-users over the internet. Cloud supports elasticity, service on demand, and the pay-as-you-go model. Cloud provides three fundamental service models to the end-user: IaaS, PaaS, and SaaS. Many other prefabricated services like databases and Hadoop are also in existence. For creating a virtual machine, the user needs to specify the operating system, Memory, CPU cores, and Storage [4]. The preconfigured operating system is a machine image stored in the SAN network that can be executed directly on the virtualized hardware without installation. The machine image is an operating system deployment file compatible with the hypervisor software. The CPU and RAM are partitioned from the physical server and assigned to run the virtual machines. Virtual machine favors the data center with consolidation, migration, and load balancing. When two or more physical devices are underutilized, the virtual machine can be migrated to a single physical machine to save resources. The unused servers can be put to hibernate mode, to consume minimal energy.

The challenge for efficiently utilising a data center lies in using the underlying data center resources. As per the Gartner report [5], the physical machine consumes 60% of data center power, and the remaining 40% is consumed by networking, cooling, and storage infrastructure. It is crucial to efficiently utilize the data center resources by hosting an appropriate virtual machine to the server. Reducing resource utilization will significantly reduce the expense of a data center. Another vital aspect is placing a virtual machine in a data center with less latency [6]. The data centers are distributed in various geographical locations. When a VM is placed in an area having more latency, it suffers from a performance bottleneck. Consider a virtual machine configured to host a database server in a location with more significant latency. Even though the workload is hosted in sophisticated servers with more excellent configurations, it will only help retrieve the data. The delivery of the information solely depends on network bandwidth and latency. As the latency increases, the user will experience a delay in content delivery in both get and put requests.

The main objectives of this paper are summarized as follows:

• To optimize the placement of virtual machines (VMs) in a data center to minimize resource wastage, power consumption, and network transmission delay.

• To develop an efficient methodology for cloud providers to achieve economic revenue and contribute to developing green data centers.

• To propose using NSGA-III, a multi-objective evolutionary algorithm, to simultaneously reduce resource wastage, power consumption, and network transmission delay.

• To compare the performance metrics (Overall Nondominated Vector Generation and Spacing) of the proposed NSGA-III algorithm with other existing multi-objective algorithms, namely VEGA, MOGA, SPEA, and NSGA-II.

• To validate the results of the proposed algorithm using ANOVA and DMRT statistical tests to ensure the reliability and accuracy of the findings.

The motivation behind designing an efficient algorithm to place virtual machines in appropriate servers is to address resource wastage and power consumption in data centers. Currently, data centers consume approximately 2% of the total electricity generated by nations. This significant energy consumption needs substantial efforts to generate electricity, leading to environmental impacts and resource depletion. With the rapid growth of businesses adopting cloud platforms for their operations, data center electricity consumption is projected to increase to 95% in the coming years. This surge in demand makes it primary to find solutions to reduce electricity consumption in data centers, given its crucial role in meeting the escalating digital needs. By developing practical VM placement algorithms, we can optimize resource utilization, distribute workloads efficiently, and minimize energy consumption in data centers. This proactive approach towards energy efficiency aligns with the urgent need to mitigate environmental impact and promote sustainable computing practices. As cloud computing becomes an integral part of modern business operations, the quest to reduce electricity consumption becomes paramount, and an efficient VM placement algorithm emerges as the need of the hour.

The paper is structured as follows: “Literature survey” section provides a comprehensive literature review of existing algorithms; “Virtual machine placement objective formulation” section presents the formulation of the VMP objective; “Proposed methodology” section introduces the NSGA-III algorithm; “Experimental setup” section describes the experimental setup; “Performance evaluation and discussions” section offers the performance evaluations using ANOVA and DMRT; and finally, the paper concludes with a summary of findings.

Literature survey

Building an energy-efficient data center is a crucial concern for any cloud provider. Server virtualization technologies give the flexibility to host multiple operating systems with a partitioned resource called a virtual machine in the same physical machine [3]. It has improved the utilization of cloud servers to a great extent. The challenges are replaced, and the issues are now related to the placement of virtual machines in the cloud server to increase its utilization even further. Thus, objectives emerged to place VM to PM considering criteria like maximizing resource utilization of servers and networking devices, minimizing power consumption, maximizing economic revenue, etc. Consumption or Power Consumption means the amount of electricity the physical machine consumes [7]. A heuristic algorithm like bin packing [8] and linear programming-based formulation [9] is used to achieve better results in problems on a smaller scale. Many novel stochastic algorithms are proposed to achieve maximum benefits from the large-scale data centre. A bio-inspired and evolutionary algorithm is extensively applied out of many stochastic algorithms, and the literature is presented in this section.

Swarm intelligence

Swarm intelligence (SI) is a technique that mimics the natural behaviour of a species to find a food source or a mate. Many researchers used swarm intelligence algorithms to solve virtual placement problems [10, 11]. Ant exhibits their intelligence in finding the food source, whereas the firefly exhibits intelligence in finding a mate. In swarm intelligence, randomly, each agent works until it finds a solution then the information is communicated with the remaining individuals. The remaining individuals will tune themselves to achieve a better solution. The global solution is the individual that dominates all the remaining individuals. Every swarm intelligence algorithm works based on two factors called exploration and exploitation [12]. Exploration is searching for a solution in the overall solution space, and exploitation is searching within the best-known solution space. The solution space is defined using the objective function. For many of the problems, there might be more than a single objective function that either needs to be minimized or maximized. Minimizing an objective function may have a negative impact on other objective functions. When an algorithm is constructively optimized, two or more objective functions are called a multi-objective optimization algorithm [13].

In [14] proposed a multi-objective ant colony algorithm to minimize power consumption (η₁) and maximize the revenue of communication (η₂). The movement of an ant to a food source is mapped to the VM to be placed in PM. The favorability of placing VM_i to PM_j is based on the pheromone trails η(i, j). The multi-objective problem solution is converted to scalar quantity using the weighted sum approach $\upeta \left(\mathrm{i},\mathrm{j}\right)={\upeta }_{1}\left(\mathrm{i},\mathrm{j}\right)+{\upeta }_{2}(\mathrm{i},\mathrm{j})$. In [15], proposed a modified ACA called Order Exchange and Migration ACS to minimize the number of active servers favours energy-efficient data centres. The proposed algorithm is compared with ACS and shows significant performance improvement with a single objective function. The algorithm also focuses on ordering and migrating overloaded and underloaded server loads. The congested server's VM configurations are sorted, and the VM utilizing higher resources is swapped with an underutilized server called load balancing. A load-balancing operation is a network-intensive task once the virtual machine is placed into a physical machine. In [16] proposed work, ant colony-based power-aware and performance-guaranteed methodology (PPVMP) is used to optimize the data centre power consumption and improve VM performance in a physical machine. In [4] proposed Energy Efficient Knee point driven Evolutionary Algorithm (EEKnEA) uses the evolutionary algorithm framework with a modified selection strategy called KnEA where the highest fit Pareto optimal solutions are considered along with knee points for the next generation. The algorithm uses a single-point crossover technique. The chromosomes are checked for feasibility during each population generation, and infeasible chromosomes are subjected to solution repair. In this work, the author addressed the objectives: the energy consumption of servers, the energy consumption of inter-VM communication, Resource Utilization, and Robustness.

Kuppusamy et al. [17] proposed a reinforced social spider optimization to handle job scheduling in a fog-cloud environment. The author performed extensive experimentation using the FogSim simulator to generate the dataset. In addition, they achieved the minimized cost function by considering the CPU processing time and allocated memory resources. Huanlai Xing et al. [18] proposed an ACO algorithm to address the virtual machine placement problem by considering energy consumption and network bandwidth. The proposed algorithm enables the information exchange that inherits the indirect information exchange among the ants in ACO.

Genetic algorithm

Genetic algorithm is inspired by the evolutions of living beings based on the concepts of Darwin's theory of evolution [19]. The genetic algorithm works based on three techniques – selection, crossover, and mutation. Selection is the process of finding the best individual from the entire population. The selected individual's chromosomes are exchanged in varying proportions to form offspring. The mutation is used to achieve something newer from the population. Mutation is a process of voluntarily changing chromosomes to generate unique offspring. Then the offspring are subjected to the fitness function or objective function. If the offspring is a valid chromosome, it survives to the next generation of the population; else, they discard the chromosome. If the progeny survives with the most significant fitness value, then the offspring is likely to be selected in the next mating pool. A better solution can be achieved by iterating the process [20].

The author in [21] proposed a novel hybrid genetic and PSO algorithm (HGAPSO) to optimize power consumption, resource wastage, and SLA violation. Genetic algorithm concepts of crossover and mutation are used to find globally optimal solutions, whereas PSO is used to achieve faster convergence. Roulette wheel selection, single-point crossover with shuffling mutation operator is used in the GA phase. This work converts an ordered encoding chromosome into a binary encoding method to apply PSO. In [22], they considered the NSGA-II algorithm to optimize computing resources and network bandwidth. In [23], a modified genetic algorithm with the fuzzy model optimises the computing resource and thermal efficiency.

In [24], the authors propose a secure and self-adaptive resource allocation framework integrated with an enhanced spider monkey optimization algorithm. The proposed framework addresses workload imbalance and performance degradation issues while meeting deadline constraints. Experimental results demonstrate its superiority over state-of-the-art approaches like PSO, GSA, ABC, and IMMLB in terms of time, cost, load balancing, energy consumption, and task rejection ratio. In [25], the author addresses the challenges of cloud-fog computing. IoT systems generate vast amounts of data that need to be processed. Instant response tasks are sent to fog nodes for low delay but high end-user energy consumption, while complex tasks go to cloud data centres for extensive computation. To address these challenges, the author proposes the MGWO algorithm, which reduces fog brokers' QoS delay and energy consumption. The proposed algorithm is verified in simulations against state-of-the-art algorithms. In [26], the authors introduce the ARPS framework for efficient multi-objective scheduling of cloud services to meet end-user's QoS requirements. The framework optimizes execution time and costs simultaneously using the spider monkey optimization algorithm. Extensive simulation analysis with Cloudsim demonstrates its superiority over four existing mechanisms regarding processing time, cost, and energy consumption.

In [27], the authors focus on microservices and the challenges of meeting end-user demands in cloud computing while adhering to SLA constraints. Using the Fine-tuned Sunflower Whale Optimization Algorithm (FSWOA), the proposed QoS-aware resource allocation model optimizes microservice deployment for improved efficiency and resource utilization. Experimental results show that the proposed approach outperforms baseline methods (SFWOA, GA, PSO, and ACO) with reductions in time, memory consumption, CPU consumption, and service cost by up to 4.26%, 11.29%, 17.07%, and 24.22% respectively. In [28], the authors develop a task-processing framework for cloud computing that selects optimal resources at runtime using a modified PSO algorithm. The proposed algorithm addresses conflicting objectives, optimizing multiple parameters simultaneously, such as time, cost, throughput, and task acceptance ratio. Experimental results using Cloudsim demonstrate its significant superiority over baseline heuristic and meta-heuristic methods. In [29], the authors address the resource provisioning and scheduling challenges in cloud computing due to resource heterogeneity and dispersion. To mitigate environmental concerns caused by increased data centres for high computational demand, the authors propose an efficient meta-heuristic technique using a modified transfer function for binary particle swarm optimization (BPSO).

Comparison on state of art multi-objective optimization algorithms

This section aims to compare some of the leading multi-objective optimization algorithms comprehensively. The difference between the algorithm working and its performance in attaining optimal solutions varies between the algorithm and the problem. By analyzing each algorithm's strengths, weaknesses, and performance metrics, we aim to identify their suitability for specific problem types and offer insights into their practical applications. The algorithms under review include NSGA-II, MOEA/D, NSGA-III, Genetic Algorithm, Particle Swarm Algorithm and Ant Colony Algorithm. The metrics considered for comparison are explained below, and Table 1 compares the mentioned algorithms.

Table 1 Comparison of state of art multi-objective optimization algorithms

Exploration

The degree to which the algorithm explores the unvisited search space. The mechanisms implemented in the algorithm, such as mutation, pheromone updates and reference points, favour exploration in the algorithms.

Exploitation

The degree to which an algorithm focuses on improving the existing solution is called exploration. As the iteration increases, the monotonic decay or reduction in crossover probability favours exploitation. Exploitation is searching for better solutions that are closer to the existing solution.

Convergence

Convergence measures how quickly an algorithm finds the optimal or Pareto optimal solution for a given problem, given a fixed number of iterations. To generalize the ranks of the algorithms listed in the Table 1, we used the Rosenbrock function and ranked them according to their convergence speed.

Computation complexity

Computational complexity defines the runtime of an algorithm. Optimization problems are generally NP-hard, meaning they are computationally intractable and require exponential time. Polynomial time complexity is the most desirable for optimization problems, as the algorithm will run in a reasonable amount of time. The higher the complexity, the longer the runtime will be.

Extract from the literature

The above literature shows that many leading researchers are applying bio-inspired swarm optimization or genetic algorithms to improve the various efficiency aspects of cloud resources. In specific, ACO is widely used in bio-inspired algorithms. The cloud servers are both time and space components, allowing the cloud provider to overcommit their cloud resources. Our research considers that the servers are only space-shared components, and over-committing server resources are not considered.

Virtual machine placement objective formulation

Minimize resource wastage

A cloud data centre may have any number of physical machines. A physical machine has resources in terms of CPU and RAM. In the cloud environment, the storage is given to a physical machine in terms of a Storage Area Network and can be dynamically increased to any volume. Our research considers only the CPU and memory in the optimization objective calculation. About the illustration above, assume we have two physical machines, namely $PM=\{{PM}_{1}, {PM}_{2}\}$ in a data centre with an available resource capacity of 90% each. The remaining 10% of the CPU and Memory is reserved for running the operating system and the hypervisor software. Three VM requests are, namely $VM=\{{VM}_{1}, {VM}_{2},{VM}_{3}\}$ and each VM need a different resource for execution. ${VM}_{1}$ requires 20% of CPU and 20% of RAM for execution, ${VM}_{2}$ requires 60% CPU and 40% of RAM, ${VM}_{3}$ requires 30% CPU and 30% of RAM.

Figure 1 depicts the possible way to schedule the virtual machine. The ${PM}_{1}$ holds $\{{VM}_{1}, {VM}_{2}\}$ and ${VM}_{3}$ is placed in ${PM}_{2}$ because the ${PM}_{1}$ don't have enough resources. The wastage is highlighted in red color. The total resource wastage is the sum of wastage in $\{{PM}_{1}, {PM}_{2}\}$. The above process can be mathematically represented using the below Eq. 1.

$$Min\sum\nolimits_{i=1}^MW_i=\sum\nolimits_{i=1}^M\left[y_i\times\frac{\left|(\theta_{P\;i}-\sum_{j=1}^N(x_{i,\;j.}\;R_{p,\;j}))-(\theta_{M\;i}-\sum_{j=1}^N(x_{i,\;j.\;}R_{M,\;j}))\right|+\varepsilon}{\sum_{j=1}^N(x_{i,\;j.\;}R_{p,\;j}))+\sum_{j=1}^N(x_{i,\;j.\;}R_{M,\;j}))}\right]$$

(1)

$$x_{i,j}=\left\{\begin{array}{l}1\;if\;{VM}_{j\;}is\;allocated\;to\;{PM}_i\\0\;otherwise\end{array}\right.$$

(2)

$$y_i=\left\{\begin{array}{l}1\;if\;{PM}_{i\;}is\;used\\0\;otherwise\end{array}\right.$$

(3)

where ${R}_{M,j}$, ${R}_{p,j}$ is CPU and memory demand of each Virtual Machine, ${\theta }_{M i}$, ${\theta }_{P i}$ Each PM's upper limit value is usually set to 90%, where the remaining 10% is used to run hypervisors and server monitoring modules.

Fig. 1

Illustration on resource wastage objective calculation

Minimize power consumption

Considering the $\{{VM}_{1}, {VM}_{2}\}$ placed in ${PM}_{1}$ and ${VM}_{3}$ is placed in ${PM}_{2}$ the calculation is carried out in this section. In the literature [16], correlation indicates a linear relationship between CPU utilization and power consumption, where an increase in CPU utilization corresponds to a proportional increase in power consumption. The reference value is also incorporated from the literature [16]. When the CPU is idle, not placed with any VM, the power consumption is observed to be 162W. When the PM is thoroughly utilized, the power consumption is 215W. Hence the power consumption of a physical machine ranges from a lower limit of 162W to a higher limit of 215W. In Fig. 2, ${PM}_{1}$ is hosted with $\left\{{VM}_{1}, {VM}_{2}\right\}$ has a total CPU utilization of 80%. 80% of 53W is consumed in addition to 162W. The power consumption of ${PM}_{1}$ is said to $162W+(80\%*53W)$ equals 204.4W.

Fig. 2

Illustration on power consumption objective calculation

Likewise, the power consumption of all physical machines is calculated to find the total power consumption of the cloud data center. It is mathematically represented as in Eq. 4.

$$Min\sum\nolimits_{i=1}^MP_i=\sum\nolimits_{i=1}^M\left[y_i\times\left(\left(P_i^{active}-P_i^{idle}\right)\times\sum\nolimits_{j=1}^N\left(x_{i,\;j.}\;R_{p,\;j}\right)+P_i^{idle})\right)\right]$$

(4)

${P}_{i}^{idle}$ denotes the power consumption of a physical server without any virtual machine hosted in it, ${P}_{i}^{active}$ is the power consumption of a physical machine at its maximum hosted load.

Minimize propagation time

$$\begin{array}{cc}Min\sum\nolimits_{n=1}^N{PDelay}_{i,j}^n&\forall i\;\epsilon\;M,\forall j\;\varepsilon\;N\end{array}$$

(5)

where $PDelay$ denotes the latency of a virtual machine ${VM}_{i}$ to be placed in a physical machine ${PM}_{j}$.

Proposed methodology

NSGA-III [30] is much like the working model of the NSGA-II algorithm [22]. The NSGA II algorithm proposed by Deb follows the pattern of the Pareto-based approach extensively [31]. During the initialization phase, a problem-specific initial population is generated. Then the population is evaluated using the objective function. Now, the population will have their fitness values. This existing population is called a parent population. This parent population is chosen in random or probabilistic-based approaches to generate the children's population. These parent population pairs are subjected to crossover and mutation operators to produce several individuals in the children population. Until this step, this algorithm follows the identical framework of NSGA II. The NSGA III steps are majorly divided into two. They are non-dominated sorting and calculation of crowding distance.

The parent and children populations are merged to find the Pareto front in non-dominated sorting. For example, consider there are five parent individuals $P=\{{p}_{1}, {p}_{2}, {p}_{3}, {p}_{4}, {p}_{5}\}$ and five children $C=\{{c}_{1}, {c}_{2}, {c}_{3}, {c}_{4}, {c}_{5}\}$ both are combined to form $P\cup C=\{{p}_{1}, {p}_{2}, {p}_{3}, {p}_{4}, {p}_{5}, {c}_{1}, {c}_{2}, {c}_{3}, {c}_{4}, {c}_{5}\}$. Now for the combined population, domination is applied to find the Pareto front. The combined population size is usually 2P. Consider three solutions, ${F}^{1}=\{{p}_{1}, {c}_{2},{c}_{3}\}$ are identified as the first non-dominated solution front. To find the next non-dominated solution front, the F¹ solution is removed $\left(P\cup C\right)-{F}^{1}=\{ {p}_{2}, {p}_{3}, {p}_{4}, {p}_{5}, {c}_{1}, {c}_{4}, {c}_{5}\}$ from the combined population. Once again, the non-domination process is repeated until all the individuals are fitted into the front F^m, where m denotes the number of fronts. The non-dominated sorting method is to find the multiple Pareto fronts in the given objective, which is depicted in Fig. 1.

The second part of the algorithm focuses on the density estimation of the solutions called crowding distance. The combined population values are initially sorted according to the individual objective function values to find the distance to the surrounding solutions. The figure depicts the crowding distance calculation for point c₂. To its proximity, three solutions exist {c₃, c₄, p₁} and the distance is calculated for the point c₂. For the solutions in the front's extreme boundary, the distance is assigned to infinity. The overall crowding distance is calculated by summing the distance of each solution. The solution with a smaller crowding distance implicitly represents several solutions in its proximity. The solutions for the next iteration are selected based on two conditions; the solutions in the lower front are preferred over those in the higher Pareto front [32]. If both solutions are from the same front, then the minimal crowding distance solution is chosen to generate the next parent population of size P. (Fig. 3)

Fig. 3

Crowding distance calculation in comparison with the Pareto front-based solution calculation

A selection operator based on the reference points is proposed to maintain the diversity in a population [33]. It is adapted with the help of detailed or well-spread reference points in the solution space [34]. The working model of NSGA-III and its computation of reference points are discussed. A series of distributed reference points of $G$ dimensions are also generated. $p$ is a number that is used for division, and it is generated by the end-user. To deploy the reference points over the normalized hyperplane with the interception of one for each axis is given by Das and Dennis [35]. The total number of reference points $(H)$ generation can be done by

$$H=\left(\begin{array}{c}G+p-1\\ p\end{array}\right)$$

(6)

These reference points are distributed uniformly to maintain diversity among the population. The direction of the reference points is denoted as a ray that starts from the origin and passes through the reference points, as given in Fig. 4.

Fig. 4

The rays from origin to reference points in a two-objective solution space

NSGA III algorithm balance between exploration and exploitation using the process called Niching. Niching ensures that the algorithm does not converge to the local optimum. Niching works based on the principle of fitness sharing. If a solution $I$ in the Pareto optimal front is close to another solution j (d_ij ≤ R), then the fitness value of all the solutions is shared among them using the equation below. The critical parameter that decides the fitness sharing is the radius R. Sh(d_ij) Eq. 7 represents the sharing value between the solution pair, such that we have n solution. The sharing values are added as shown in Eq. 8. The final fitness values for all solutions closer to each are calculated as given in Eq. 9.

$$Sh\left({d}_{ij}\right)=\left\{\begin{array}{c}1-\frac{{d}_{ij}}{R}\\ 0\end{array}\right. \begin{array}{c}{d}_{ij}\le R\\ otherwise\end{array}$$

(7)

$${nc}_{i}=Sh\left({d}_{i1}\right)+Sh\left({d}_{i2}\right)+\dots +Sh\left({d}_{in}\right)$$

(8)

$${f}^{\prime}=\frac{f}{{nc}_{i}}$$

(9)

Algorithm 1. NSGA III algorithm for virtual machine placement

After the initialization of the population, recombination, mutation, and crossover procedures [34], the size of the merged population depends on mutation and crossover percentage; for elite individual preservation, a non-dominated sorting model is used. Each level of individuals is sorted by crowding distance in NSGA-II. It has been replaced in NSGA-III with reference direction-based niching. Before this operation, the objectives are normalized with the below formulae. The equation updates the ideal point Z* during each iteration. The NSGA II-based tchebychef scalarization method is replaced with the achievement scalarization function (ASF), as mentioned in Eq. 8, to convert objective values from vector to scalar values [36].

$${f}_{g}^{\prime}\left(x\right)={f}_{g}\left(x\right)-{z}_{g}^{min}$$

(10)

$$ASF\left(x,w\right)=\frac{{zmax}_{g=1}^{G}{f}_{g}^{\prime}\left(x\right)}{{w}_{g}}$$

(11)

$${f}_{g}^{n}\left(x\right)=\frac{{f}_{g}^{\prime}\left(x\right)}{{a}_{g}-{z}_{g}^{min}}$$

(12)

where ASF is defined as the extreme points in each objective axis, ${z}_{g}^{min}$ is the ideal value ${f}_{g}^{n}\left(x\right)$ is the normalized objective function value. The pseudocode of the proposed model is given in Algorithm 1.

Experimental setup

Generating virtual machine's CPU and RAM dataset is generated statistically based on the algorithm proposed in [8, 14]. The experiment is carried out with three sets of reference values $\overline{{R }_{CPU}}$ and $\overline{{R }_{RAM}}$ (25%, 35% and 45%) respectively. When the reference value is 25% the $\overline{{R }_{CPU}}$ and $\overline{{R }_{RAM}}$ generated values are approximately ranging from 0 to 50%. The correlation values are calculated after generating the datasets to show variations in datasets. The negative correlation denotes a memory-intensive virtual machine workload, and the positive correlation denotes CPU-intensive virtual machine. The expected correlation values are achieved by varying the probability values $P$ to (0, 25, 50, 25, 100), giving the correlation values of (-0.7, -0.3, 0.04, 0.2, 0.74). Figure 5 represents the distribution of the resources of the virtual machine for $\overline{{R }_{CPU}} =25\%$, $\overline{{R }_{RAM}} =25\%$, $P=0$ (Left) and $\overline{{R }_{CPU}} =45\%$, $\overline{{R }_{RAM}} =45\%$, $P=0.5$ (right). Based on the combination of values of $P$(0, 25, 50, 25, 100), $\overline{{R }_{CPU}} \overline{{R }_{RAM}}$(25%, 35%,45%) and ${N}_{VM}$(100,200) thirty datasets are generated, and the corresponding correlation values are given in the column corr in Tables 1, 2 and 3.

Fig. 5

Distribution of statistically generated datasets with different $\overline{{R }_{CPU}}$, $\overline{{R }_{RAM}}$ and $P$ values

Table 2 Experimental results of $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25% for 100 and 200 VM instances

Table 3 Experimental results of $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35% for 100 and 200 VM instances

To calculate the power consumption of the individual physical machine, the processor utilization is calculated based on$\sum_{j=1}^{N}({x}_{i,j . }{R}_{p,j})$. An experiment performed in [16, 37] concluded that the CPU and power utilization form a linear relationship among them. When CPU utilization increases, power consumption also increases. The power consumption of a physical server is calculated based on two parameters. ${P}_{idle}$ and${P}_{active}$. ${P}_{idle}$ indicates the amount of power consumed when the server is not hosted with any virtual machine. ${P}_{active}$ denotes the amount of power consumption for complete physical machine utilization. Based on the experiment in [14] two values are taken where ${P}_{idle}=162W$ and${P}_{active}=215W$. If the CPU utilization of the physical server is 50%, then the power utilization of the server is 188.5W.

To validate the superiority of the proposed algorithm, it is essential to utilize a statistically generated dataset that exhibits variations, thereby showcasing its performance under diverse conditions. By altering the $\overline{{R }_{CPU}}$, $\overline{{R }_{RAM}}$ and $P$ values, different datasets can be generated. In the above figure, $\overline{{R }_{CPU}} =25\%$, $\overline{{R }_{RAM}} =25\%$ and $P=0$ so that the generated request values will fall in the range > 0% and < 50%. The correlation of the dataset is calculated by generating dataset returns -0.7453 (strong negative correlation). The next figure where we use $\overline{{R }_{CPU}} =45\%$, $\overline{{R }_{RAM}} =45\%$ and $P=0.5$ the CPU and RAM values ranges from > 0% and < 90%. The values $\overline{{R }_{CPU}}$, $\overline{{R }_{RAM}}$ and $P$ are used to generate diversified datasets for experimentation.

Hosting a virtual machine in a remote server across regions will incur network transmission delay. Placing a VM at a nearer data center will have less network latency compared farther data centre. For latency, 18 regions of AWS data centers are considered. The latency is estimated using the TCPPing utility configured in the running instances in 18 regions. An average latency measured every 5 min for 15 days is considered in our research work.

Performance evaluation and discussions

To evaluate the proposed algorithm with experimented vales, two metrics specific to multi-objective algorithms are spacing [38] and Overall Non-Dominated Vector Generation (ONGV). ONGV [39] denotes the average number non dominated solutions stored in the external achieve during each iteration of the algorithm. ONGV indicates the number of better-performing solutions found during each iteration. Spacing represents the coarseness of the resolution towards the minimal objective values. If all the non-dominated solutions are closer, then the spacing value is minimal, indicating that the algorithm controls solution generation. If the spacing value is more, we can conclude that the solution is random, and the algorithm has no control over solution generation.

$$ONGV=\frac{Total\;number\;of\;non\;dominated\;vector\;generated}{Total\;number\;of\;iterations}$$

(13)

$$Sp= \sqrt{\frac{1}{\left|PF\right|-1} \sum\nolimits_{i=1}^{\left|PF\right|}{(\overline{d }-{d}_{i})}^{2}}$$

(14)

The results in Tables 2, 3 and 4 are interpreted using the DMRT (Duncan Multiple Range Test) and ANOVA (Analysis of Variance), the statistical tools to show the significance between the listed algorithms. The tests are being performed for the performance indicators ONVG and SP for all three instances of RAM and CPU, respectively.

Table 4 Experimental results of $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45% for 100 and 200 VM instances

ANOVA

The dataset is generated statistically, and the algorithm uses the guided random approach. ANOVA and Duncan Multiplier Range test are used as statistical tests to measure the algorithm's significance. ANOVA test cannot isolate the best performing algorithm; instead, ANOVA tests if there exists a significant difference in the algorithm's performance. Using the ANOVA test, we can ensure that there is a substantial difference between the algorithm but cannot isolate better-performing algorithms. We conducted the post hoc analysis using the Duncan range test to separate the best-performing algorithm. For example, in Table 6, the Duncan multiplier range test categorises the algorithm into three homogeneous groups (three columns). The first homogeneous group denotes none of the other algorithms are performed as equivalent to the NSGA III algorithm. In the same table, SPEA, MOGA and VEGA are performing equally.

ANOVA test shows the significance between the groups or algorithms in the given sample values. In this paper, the ANOVA test is used to identify whether the results of the algorithms show significance among them. The ANOVA test was conducted at a significance level of 95%. Suppose the Sig.value is lower than the critical value (α = 0.05). In that case, the null hypothesis (H_o) should be rejected, and the alternate hypothesis (H₁) should be accepted, indicating a significant difference among the given group of values. It allows for applying post-hoc tests, with Duncan's Multiple Range Test used in this case. Conversely, if the sig.value exceeds 0.05, the null hypothesis (H_o) should be accepted, and post-hoc tests cannot be conducted.

Duncan’s multiple range test

The Duncan Multiple Range Test (DMRT) is a statistical method used to compare multiple sets' mean values. It utilizes studentized range statistics to establish numerical boundaries for classifying significant and non-significant differences between any two or more groups. The DMRT ranks the sets in ascending or descending order according to the user's preference.

Tables 5 and 6 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 2 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%.

Table 5 Result analysis of 100 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 6 Result analysis of 100 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 5 presents the statistical analysis of ANOVA, while Table 6 showcases the results of DMRT. The tabulated results are interpreted from Table 2 of ONVG of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%. The results from Table 5 indicate that the Sig.value is below the critical importance of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis, DMRT NSGA-III ranks top among the existing algorithms. Three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 7 and 8 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 2 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%.

Table 7 Result analysis of 200 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 8 Result analysis of 200 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 7 presents the statistical analysis of ANOVA, while Table 8 showcases the results of DMRT. The tabulated results are interpreted from Table 2 of ONVG of 200 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%. The results from Table 7 indicate that the Sig.value is below the critical importance of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis, DMRT NSGA-III ranks top among the existing algorithms. Three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 9 and 10 shows the ANOVA and DMRT tests of SP results tabulated in Table 2 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%.

Table 9 Result analysis of 100 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 10 Result analysis of 100 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 9 presents the statistical analysis of ANOVA, while Table 10 showcases the results of DMRT. The tabulated results are interpreted from Table 2 of SP of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%. The results from Table 9 indicate that the Sig.value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis, DMRT NSGA-III ranks top among the existing algorithms. Four homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 11 and 12 shows the ANOVA and DMRT tests of SP results tabulated in Table 2 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%.

Table 11 Result analysis of 200 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 12 Result analysis of 200 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 25%)

Table 11 presents the statistical analysis of ANOVA, while Table 12 showcases the results of DMRT. The tabulated results are interpreted from Table 2 of SP of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 25%. The results from Table 11 indicate that the Sig.value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 12, NSGA-III ranks top among the existing algorithms, four homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 13 and 14 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 3 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%.

Table 13 Result analysis of 100 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 14 Result analysis of 100 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 13 presents the statistical analysis of ANOVA, while Table 14 showcases the results of DMRT. The tabulated results are interpreted from Table 3 of ONVG of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%. The results from Table 13 indicate that the Sig.value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 14, NSGA-III ranks top among the existing algorithms, and there are three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 15 and 16 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 3 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%.

Table 15 Result analysis of 200 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 16 Result analysis of 200 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 15 presents the statistical analysis of ANOVA, while Table 16 showcases the results of DMRT. The tabulated results are interpreted from Table 3 of ONVG of 200 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%. The results from Table 15 indicate that the Sig.Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis, DMRT in Table 16 NSGA-III ranks top among the existing algorithms, three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group.

Tables 17 and 18 shows the ANOVA and DMRT tests of SP results tabulated in Table 3 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%.

Table 17 Result analysis of 100 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 18 Result analysis of 100 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 17 presents the statistical analysis of ANOVA, while Table 18 showcases the results of DMRT. The tabulated results are interpreted from Table 3 of SP of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%. The results from Table 17 indicate that the Sig.Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 18, NSGA-III ranks top among the existing algorithms. Four homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 19 and 20 shows the ANOVA and DMRT tests of SP results tabulated in Table 3 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%.

Table 19 Result analysis of 200 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 20 Result analysis of 200 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 35%)

Table 19 presents the statistical analysis of ANOVA, while Table 20 showcases the results of DMRT. The tabulated results are interpreted from Table 3 of SP of 200 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 35%. The results from Table 19 indicate that the Sig.Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 20, NSGA-III ranks top among the existing algorithms. Four homogenous groups are formed among the algorithms NSGA-III shows its significance by creating a standalone group among the others.

Tables 21 and 22 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 4 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%.

Table 21 Result analysis of 100 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 22 Result analysis of 100 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 21 presents the statistical analysis of ANOVA, while Table 22 showcases the results of DMRT. The tabulated results are interpreted from Table 4 of ONVG of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%. The results from Table 21 indicate that the Sig. value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 22, NSGA-III ranks top among the existing algorithms, three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group.

Tables 23 and 24 shows the ANOVA and DMRT tests of ONVG results tabulated in Table 4 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%.

Table 23 Result analysis of 200 VM instances using ANOVA for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 24 Result analysis of 200 VM instances using DMRT for ONVG ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 23 presents the statistical analysis of ANOVA, while Table 24 showcases the results of DMRT. The tabulated results are interpreted from Table 4 of ONVG of 200 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%. The results from Table 23 indicate that the Sig. Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 24, NSGA-III ranks top among the existing algorithms. Three homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 25 and 26 shows the ANOVA and DMRT tests of SP results tabulated in Table 4 for 100 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%.

Table 25 Result analysis of 100 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 26 Result analysis of 100 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 25 presents the statistical analysis of ANOVA, while Table 26 showcases the results of DMRT. The tabulated results are interpreted from Table 4 of SP of 100 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%. The results from Table 25 indicate that the Sig. Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT in Table 26, NSGA-III ranks top among the existing algorithms. Four homogenous groups are formed among the algorithms, and NSGA-III shows its significance by creating a standalone group among the others.

Tables 27 and 28 shows the ANOVA and DMRT tests of SP results tabulated in Table 4 for 200 instances of VM with $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%.

Table 27 Result analysis of 200 VM instances using ANOVA for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 28 Result analysis of 200 VM instances using DMRT for SP ($\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$ = 45%)

Table 27 presents the statistical analysis of ANOVA, while Table 28 showcases the results of DMRT. The tabulated results are interpreted from Table 4 of SP of 200 VM instances for $\overline{{R }_{CPU}}$ = $\overline{{R }_{RAM}}$  = 45%. The results from Table 27 indicate that the Sig. Value is below the critical value of 0.05, leading to the rejection of the Null Hypothesis (${H}_{0})$ and acceptance of the Alternate Hypothesis $({H}_{1})$. On the post-hoc analysis DMRT on Table 28, NSGA-III ranks top among the existing algorithms. Four homogenous groups are formed among the algorithms NSGA-III shows its significance by creating a standalone group among the others.

The proposed algorithm has been tested with different VM instances, from biased towards CPU (-ve correlation coefficient) to biased RAM (+ ve Correlation coefficient). The state-of-the-art algorithms are also compared on the same generated instances. For comparing the performances of the proposed system with existing algorithms in identifying VMP solutions with multiple objectives, two performance metrics were considered: Spacing (SP) and Overall Non-Dominated Vector Generation (ONGV). Due to the "tchebychef scalarization method" in NSGA-III, the Pareto optimal front solutions are identified. It is observed that the proposed algorithm performs 7% better that the existing algorithm in terms of ONVG and 12% better results in terms of spacing. ANOVA and DMRT statistical tests are used to cross-validate the results. Thus, the NSGA-III algorithm outperforms all existing algorithms in the SP and ONVG performance indicators.

Conclusion

In this paper, the NSGA-III algorithm is implemented to optimize three contradicting objectives: resource wastage, power consumption and network propagation time. The problem is formulated as a multi-objective optimization problem, and a discretized NSGA-III algorithm is implemented to find the best-performing solution in all three objectives. The results are compared with other multi-objective optimization algorithms, namely VEGA, MOGA, SPEA, and NSGA-II, regarding ONVG and Spacing performance metrics. Since the algorithm is guided randomly, we executed 30 independent runs, and the resultant values were statistically tested using ANOVA and DMRT. The statistical test shows that the significance lies among the MOEA's on VM placement, stating that the NSGA-III outperforms all the existing algorithms in all aspects. Comparing NSGA-II and SPEA, both algorithms are in the same homogenous group in many DMRT tests. Hence these two algorithms perform equally to each other. Comparing SPEA to MOGA, like NSGA-II, MOGA shares a similarly homogenous group. VEGA in-performs in all the aspects of the VM placement problem. Considering the average values of 30 independent runs, NSGA-III achieved 7% better than the existing algorithm regarding ONVG and 12% better results in terms of spacing.