2479 lines
284 KiB
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2479 lines
284 KiB
Plaintext
Computer Standards & Interfaces 97 (2026) 104106
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Contents lists available at ScienceDirect
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Computer Standards & Interfaces
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journal homepage: www.elsevier.com/locate/csi
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A novel hybrid WOA–GWO algorithm for multi-objective optimization of
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energy efficiency and reliability in heterogeneous computing
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∗
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Karishma , Harendra Kumar
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Department of Mathematics and Statistics, Gurukula Kangri (Deemed to be University) Haridwar, Uttarakhand 249404, India
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ARTICLE INFO ABSTRACT
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Keywords: Heterogeneous computing systems are widely adopted for their capacity to optimize performance and energy
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Energy-efficient scheduling efficiency across diverse computational environments. However, most existing task scheduling techniques
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Heterogeneous computing address either energy reduction or reliability enhancement, rarely achieving both simultaneously. This study
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Hybrid WOA–GWO
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proposes a novel hybrid whale optimization algorithm–grey wolf optimizer (WOA–GWO) integrated with
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Metaheuristics
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dynamic voltage and frequency scaling (DVFS) and an insert-reversed block operation to overcome this
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Reliability optimization
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Sensitivity analysis
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dual challenge. The proposed Hybrid WOA–GWO (HWWO) framework enhances task prioritization using the
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dynamic variant rank heterogeneous earliest-finish-time (DVR-HEFT) approach to ensure efficient processor al-
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location and reduced computation time. The algorithm’s performance was evaluated on real-world constrained
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optimization problems from CEC 2020, as well as Fast Fourier Transform (FFT) and Gaussian Elimination (GE)
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applications. Experimental results demonstrate that HWWO achieves substantial gains in both energy efficiency
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and reliability, reducing total energy consumption by 55% (from 170.52 to 75.67 units) while increasing system
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reliability from 0.8804 to 0.9785 compared to state-of-the-art methods such as SASS, EnMODE, sCMAgES, and
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COLSHADE. The experimental results, implemented on varying tasks and processor counts, further demonstrate
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that the proposed algorithmic approach outperforms existing state-of-the-art and metaheuristic algorithms by
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delivering superior energy efficiency, maximizing reliability, minimizing computation time, reducing schedule
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length ratio (SLR), optimizing the communication-to-computation ratio (CCR), enhancing resource utilization,
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and minimizing sensitivity analysis. These findings confirm that the proposed model effectively bridges the
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existing research gap by providing a robust, energy-aware, and reliability-optimized scheduling framework for
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heterogeneous computing environments.
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1. Introduction as multiprocessor task scheduling is an NP-hard optimization problem,
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delivering a valid solution within a predefined deadline remains a
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1.1. Motivation significant challenge for real-time applications in heterogeneous sys-
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tems [4]. Alternatively, the delicate balance between performance
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In recent years, the exponential growth in data volume and com- and power consumption stands as a pivotal factor in the design of
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putational demands has propelled the development of heterogeneous multiprocessor systems [5]. To attain optimal performance, it is im-
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computing systems. Numerous computing resources are required for perative to implement efficient scheduling of applications across the
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various heterogeneous computing models, such as utility computing, diverse resources within heterogeneous computing systems, comple-
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peer-to-peer, and grid computing. These resources can be allocated mented by efficient runtime support mechanisms [6]. In multiprocessor
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through the network in order to fulfill the needs of carrying out systems, scheduling of tasks involves arranging the sequence of tasks
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high performing tasks [1]. Resource scheduling is a fundamental chal- and facilitating their execution across selected processors to achieve
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lenge in heterogeneous computing, especially as the number of tasks a predetermined goal, such as meeting deadlines, minimizing over-
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and resources increases. Inefficient task allocation can lead to pro- all execution time (makespan), conserving energy, enhancing system
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reliability, among other objectives [7].
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cessor overutilization or underutilization, complicating the scheduling
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Efficiently managing energy consumption is pivotal in the design
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process [2]. Heterogeneous distributed computing has proven highly
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of heterogeneous distributed systems. This is essential as the dissi-
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effective in handling diverse and complex end-user tasks, driven by ad-
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pation of energy directly influences not only the development and
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vancements in network technologies and infrastructure [2,3]. However,
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∗ Corresponding author.
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E-mail addresses: maths.karishma97@gmail.com (Karishma), balyan.kumar@gmail.com (H. Kumar).
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https://doi.org/10.1016/j.csi.2025.104106
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Received 14 February 2025; Received in revised form 14 November 2025; Accepted 28 November 2025
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Available online 7 December 2025
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0920-5489/© 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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operation of the system but also profoundly impacts the individuals HWWO leverages this approach to achieve superior scheduling perfor-
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within the living environment [8]. The rise in energy consumption mance by harnessing the complementary strengths of WOA and GWO
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has emerged as a significant concern, which has a direct impact on while mitigating their individual drawbacks.
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the costs associated with computing services. This consumption typ- The WOA demonstrates superior exploration capabilities through
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ically comprises dynamic energy resulting from switching activities its advanced updating mechanism, employing a randomized search
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and static energy arising from leakage currents [9]. Recognizing the approach to dynamically shift positions and navigate towards optimal
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importance of energy conservation, researchers have explored and solutions. As highlighted by [30], WOA strikes a good balance between
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developed several techniques to address this issue. These include mem- exploration and exploitation, exhibiting notable convergence speed in
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solving optimization problems. However, despite its effectiveness rela-
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ory optimization, DVFS, and resource hibernation. DVFS, also known
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tive to traditional algorithms, WOA can struggle to escape local optima
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as dynamic speed scaling (DSS), dynamic power scaling (DPS), and
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due to its encircling mechanism [31] and may fail to effectively refine
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dynamic frequency scaling (DFS), is particularly noteworthy for its
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the best solutions. Conversely, GWO excels in exploitation through
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potential to save energy [10,11]. This technique facilitates energy-
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strong local search capabilities but suffers from limited diversity in
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efficient scheduling by dynamically adjusting the supply voltage and the early stages, which can hinder global search. To address these
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frequency of a processor while tasks are running, thereby optimizing limitations, we have proposed a hybrid approach that augments WOA
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energy usage [12–14]. The implementation of dynamic voltage scaling with mutation operators and integrates it with GWO to enhance overall
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for energy-efficient optimization presents a noteworthy advancement. scheduling performance. Recognizing that excessive mutation can dis-
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Nevertheless, it is essential to acknowledge a potential drawback: an rupt previously discovered good solutions and impede convergence, we
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elevated risk of transient failures in processors, which could undermine have incorporated an insert-reversed block operation after mutation to
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the reliability of systems [9,15]. Reliability pertains to the probability preserve solution quality and improve the algorithm’s efficiency.
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that a schedule will successfully complete its execution within the This study endeavors to attain an optimized energy-efficient
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defined parameters [16,17]. Higher frequencies typically correspond scheduling algorithm with a maximal systems’ reliability, thus minimiz-
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to both high energy consumption and enhanced reliability, whereas ing the aggregate energy consumption of precedence-constrained tasks
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lower frequencies are associated with decreased energy consumption in parallel applications executed on heterogeneous computing systems.
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and reduced reliability [18]. When an application meets its designated The primary contributions of this research are succinctly outlined as
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reliability objective – referred to as a reliability goal, requirement, follows:
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assurance, or constraint in various studies – it is deemed reliable in • This article proposes innovative hybrid algorithmic approaches
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accordance with functional safety standards. These standards include that combine the WOA and GWO to tackle the intricate problems
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DO-178B for avionics systems, ISO 26262 for automotive systems, and related to energy-efficient tasks scheduling.
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IEC 61508 for a broad spectrum of industrial software systems [8,19]. • This study meticulously designs energy-efficient scheduling algo-
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rithms that leverage the hybrid WOA–GWO to effectively opti-
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1.2. Our contributions mize two key objectives; energy consumption and the reliabil-
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ity of the system. The algorithms ensure compliance with the
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Task scheduling, an NP-hard problem, increases the complexity of deadline constraints of parallel applications.
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voltage adjustment choices in heterogeneous computing systems [20, • The proposed algorithm assigns tasks to suitable processors by
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21]. Balancing energy efficiency and reliability presents a major chal- synergistically integrating the HWWO technique with DVFS tech-
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lenge, as prioritizing one often complicates optimizing the other [18]. nology. Here, the proposed algorithm applies the mutation opera-
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Scheduling algorithms are broadly classified into heuristics and meta- tor in conjunction with the insert-reversed block operation as part
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heuristics. Heuristic methods, which use greedy strategies for optimal of the HWWO technique and helps to mitigate the static energy
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selection [22], are computationally efficient but often fail to perform consumption. Additionally, the DVFS technique has been utilized
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to mitigate the dynamic energy consumption.
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well in complex or large-scale scheduling problems [21,23]. In contrast,
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• Comprehensive experimental evaluations are carried out by com-
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metaheuristic algorithms, inspired by natural processes, offer more reli-
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paring the proposed HWWO algorithm’s performance against sev-
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able results and greater flexibility [24]. Metaheuristics are popular due
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eral well-known algorithms, including the FFT, GE, and four
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to their simplicity, adaptability, independence from derivative-based
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benchmark algorithms from the ’CEC2020 Competition’ — SASS,
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methods, and ability to avoid local optima. Authors [25] developed
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EnMODE, sCMAgES, and COLSHADE. Furthermore, the algorithm
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an optimized gravitational search algorithm (GSA) to enhance feature- is subjected to testing on a set of unimodal benchmark test
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level fusion in multimodal biometric systems. Their work demonstrated functions.
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how metaheuristic optimization can effectively improve system perfor- • The experimental results demonstrate that the proposed algo-
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mance through better parameter tuning and search-space exploration. rithmic approach outperforms existing state-of-the-art and meta-
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Authors [26] developed a hybrid white shark optimizer–support vector heuristic algorithms by delivering superior energy efficiency,
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machine (WSO–SVM) model for gender classification from video data, maximizing reliability, minimizing computation time for tasks
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where the white shark optimizer was used to fine-tune SVM param- assignment, minimizing sensitivity analysis and SLR, CCR, and
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eters, leading to improved accuracy and faster processing compared enhancing resource utilization. These promising results hold true
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to traditional SVM methods. In [27] authors developed two novel across various scale conditions and deadline constraints.
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task scheduling models based on the metaheuristic GWO technique to • Conduct an evaluation of the proposed technique’s computational
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optimize energy consumption while minimizing computational time for complexity during execution and employ the Wilcoxon-signed
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parallel applications. In this article, a novel hybridization (HWWO) of rank statistical test to validate its performance.
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the WOA [28] and GWO [29] is employed to tackle the task scheduling
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The structure of the article is outlined as follows. Section 2 offers
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problem. Hybrid algorithms are designed to integrate the features of an extensive review of relevant literature and related works. Detailed
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various metaheuristic approaches, exploiting their synergy to address explanations of the WOA and GWO techniques are covered in Section 3.
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complex optimization challenges. This fusion not only enhances the In Section 4, pertinent models and problem formulations are explored,
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efficiency and flexibility of the algorithms but also augments their along with essential notations used throughout the study. Section 5 pro-
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overall performance, often surpassing that of traditional metaheuristic vides a thorough description of the proposed model. The development
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algorithms. Furthermore, a wide range of such hybrid algorithms has of simulation experiments to evaluate the proposed model is detailed
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been developed, driving the evolution of new-generation metaheuristics in Section 6. Finally, Section 7 elaborates on the study’s conclusion,
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that effectively balance exploration and exploitation. The proposed discussing limitations and directions for future research.
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2
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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2. Literature review and physical systems. Among these, swarm intelligence (SI) methods
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form a prominent category, drawing on the collective behavior of
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The substantial energy consumption associated with computing living organisms. These algorithms utilize population-based, stochastic,
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systems poses a significant impediment to their rapid advancement. and iterative strategies. Numerous real-world challenges, including
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Therefore, minimizing energy usage while ensuring system reliability drone deployment, image processing, wireless sensor network localiza-
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has become a pressing concern for fostering sustainable computing tion, machine learning optimization, and others, have found effective
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methodologies. Researchers from multiple fields have devoted substan- solutions by employing SI techniques [2,50]. Beyond their diverse
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tial efforts to investigating the intricate challenges associated with task applications, SI algorithms have undergone continuous enhancements
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scheduling techniques that strike a balance between minimizing energy through modifications, hybridizations with other techniques [51], and
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consumption and maximizing system reliability. parallel computing implementations [52]. These efforts aim to further
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In the realm of sustainable computing systems, DVFS technique has optimize their performance and obtain superior solutions across di-
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emerged as a prominent and widely employed method for efficiently verse problem domains. SI algorithms are inspired by the collaborative
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curtailing energy consumption [32]. A study by [33] addressed energy- behaviors observed in various animal and insect communities, where
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aware task assignment for deadline-constrained workflow applications entities interact and respond to their environment collectively. This
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in heterogeneous computing environments. Earlier, [34] explored the- includes animal herds, ant colonies, fish schools, bacterial aggregations,
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oretical models for DVFS and proposed an energy-aware schedul- and bird flocks etc. These algorithms exhibit notable advantages in-
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ing strategy for single-processor platforms. Advanced algorithms like cluding adaptability, user-friendliness, and reliability [53]. Some of the
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enhanced energy-efficient scheduling (EES) [35] and DVFS-enabled recently developed SI algorithms are artificial bee colony (ABC) [54],
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energy-efficient workflow task scheduling (DEWTS) [36] were later ant colony optimization (ACO) [55], cuckoo optimization algorithm
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developed to reduce energy consumption in parallel applications. EES (COA) [56], particle swarm optimization (PSO) [57], horse herd op-
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uses DVFS to slack non-critical tasks while meeting time constraints, timization algorithm [58], krill herd (KH) [59], crow search algorithm
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while DEWTS enhances energy efficiency by selectively turning off (CSA) [60], GWO [29], sailfish optimizer (SFO) [61], and WOA [28]
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processors to minimize static energy consumption. In [37], authors etc.
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proposed the downward energy consumption minimization (DECM) This study implements the WOA algorithm to address energy-
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algorithm that innovatively transfers application deadlines to task-level efficient and reliable task scheduling in heterogeneous computing
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deadlines using deadline-slack and task level concepts, enabling low- environments. The approach enhances WOA by hybridizing it with
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complexity energy minimization. Authors [38] proposed a two-stage GWO, leveraging the strengths of both techniques. WOA, introduced
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solution to enhance the reliability of automotive applications while by Mirjalili and Lewis in 2016 [28], is a notable method in swarm
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satisfying energy and response time constraints. First, it solved response intelligence optimization. Inspired by the remarkable hunting strategies
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time reduction under energy constraint (RREC) via average energy employed by humpback whales in the vast ocean, this algorithm
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pre-allocation. While, the second stage enhanced reliability within the demonstrated competitive or superior performance compared to sev-
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remaining energy–time budgets from the RREC stage. Addressing the eral existing optimization methods [28]. Similar to WOA, GWO is
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challenge of energy-efficient tasks scheduling in cloud environments, a nature-inspired metaheuristic optimization algorithm based on the
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the authors of [39] proposed an algorithm based on DVFS that pri- social behavior and hunting strategies of grey wolves [29]. It is widely
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oritizes tasks by deadline, categorizes physical machines, and assigns used for solving complex optimization problems. WOA, recognized for
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tasks to nearby machines in the same priority class. Researchers in- its unique approach, has been effectively applied to scheduling tasks
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troduced the energy makespan multi-objective optimization algorithm in cloud computing, aiming to enhance system performance within
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for energy-efficient, low-latency workflow scheduling across fog–cloud constrained computing capacities [62]. The researchers proposed an
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resources [40]. The research work of [41,42] aimed to devise an innovative scheduling approach based on WOA that combined multi-
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approach that could reduce the overall execution time for parallel objective optimization with trust awareness. It mapped tasks to virtual
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applications running on high-performance distributed computing en- resources based on priorities, evaluated trust via SLA parameters,
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vironment, while concurrently enforcing adherence to predetermined and enforced deadline constraints for task execution on VMs [63].
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energy consumption thresholds. The study tackled the challenge of scheduling tasks on heterogeneous
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Reliability-aware design algorithms aimed at ensuring reliability multiprocessor systems equipped with DVFS capabilities. The objective
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typically try to reduce certain objectives while also satisfying relia- was to optimize energy consumption while adhering to constraints
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bility requirements. Improving the reliability of parallel applications related to makespan and system reliability. To achieve this, [64]
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frequently results in longer schedules or higher energy usage. Opti- proposed an enhanced variant of the WOA, incorporating opposition-
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mizing both schedule length (or energy consumption) and reliability based learning and an individual selection strategy. In the article [65],
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concurrently poses a classic bi-criteria optimization problem requiring the authors introduced an improved whale algorithm (IWA) to opti-
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the identification of pareto-optimal solutions [43,44]. The researchers mize tasks allocation in multiprocessing systems (MPS), minimizing
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in [45] introduced an approach to reliably assign and schedule tasks energy consumption and makespan. They utilized DVFS and addressed
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on heterogeneous multiprocessor systems, tackling the complexities task scheduling’s NP-hard nature. The article [66] addressed power
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and potential failures associated with critical applications. Researchers consumption in cloud infrastructure, underscoring the necessity for
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in [46,47] tackled the intricate problem of workflow scheduling on energy-efficient algorithms and load balancing techniques. The authors
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heterogeneous computing systems, aiming to achieve high reliability employed various optimization algorithms, such as PSO, COA, and
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while minimizing the unnecessary duplication of resources. Energy WOA, to achieve efficient resource scheduling and mitigate energy
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consumption and reliability are closely intertwined concepts. In [48], consumption.
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researchers explored this relationship and developed a model that The researchers proposed an innovative tasks scheduling algorithm
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linked energy consumption to reliability levels. Their work aimed to leveraging the GWO technique for cloud computing environment [67].
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maximize the reliability of parallel applications executed on uniproces- This GWO-based tasks scheduling (GWOTS) approach aimed to min-
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sor systems while adhering to strict deadlines and energy consumption imize execution costs, reduce energy consumption, and shorten the
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constraints. Authors in [49], proposed power management schemes overall makespan . Researchers developed an advanced multi-objective
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for homogeneous multiprocessors that targeted energy savings while optimization technique inspired by the GWO to address the growing
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upholding specified system reliability levels. computational demands on cloud data centers [68]. Their primary goals
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Metaheuristic techniques are versatile methods inspired by natu- were to maximize the efficient utilization of cloud resources, minimize
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ral phenomena, such as evolutionary adaptation, biological swarms, energy consumption, and reduce the overall execution time, while
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3
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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Table 1
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Performance parameters of various WOA based metaheuristic tasks scheduling techniques.
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S. No. Reference Core technique Environment Considered parameters
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Energy Reliability 𝐶𝑇𝑇 𝐴 Sensitivity Makespan Resource utilization
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1 [72] VWOA Heterogeneous
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2 [73] WOA Heterogeneous
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3 [74] M-WODE Heterogeneous
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4 [75] WOA Heterogeneous
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5 [76] h-DEWOA Heterogeneous
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6 [77] IWC Homogeneous
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7 [24] HGWWO Heterogeneous
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8 [78] HWOA based MBA Homogeneous
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9 [79] HPSWOA Heterogeneous
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10 [80] CWOA Heterogeneous
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11 [81] HWOA Heterogeneous
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12 [82] WHOA Heterogeneous
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13 [83] ANN-WOA Homogeneous
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14 [84] WOA Heterogeneous
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15 Proposed model HWWO Heterogeneous
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ensuring the requested services were delivered effectively. In [69], au- ranging from 30 m in length and weighting up to 180 tons. These
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thors proposed a novel hybrid model that combined PSO and GWO for giants of the sea are classified into various species, including the killer
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workflow scheduling in cloud computing environments. This integrated whale, the sei, the finback, the humpback, the minke, and the awe-
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approach aimed to enhance the overall performance by optimizing total inspiring blue whale. Beyond their sheer size, whales are remarkable
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execution costs and reducing the time required for task completion . for their intelligence and emotional depth, exhibiting complex social
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The article [70] addressed the challenges of tasks allocation and quality behaviors. They are often observed traveling and living in close-knit
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of service (QoS) optimization in cloud–fog computing environment. groups. One of the most fascinating species is the humpback whale,
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The authors proposed a multi-objectives GWO algorithm, implemented
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renowned for its intricate hunting technique known as bubble-net
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within the fog broker system, to minimize delay and energy consump-
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feeding [85]. When it comes to hunting, humpback whales exhibit a
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tion. In article [71], authors addressed tasks scheduling challenges in
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remarkable preference for preying on schools of krill or small fish that
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cloud computing by proposing a GWO-based algorithm. The approach
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congregate near the ocean’s surface. Their intricate hunting strategy
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aimed to efficiently allocate resources and minimize task completion
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times. involves diving to depths of around 12 m and then employing a
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WOA employs simple yet powerful search mechanisms to efficiently fascinating maneuver. The whales release a spiral of bubbles, carefully
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identify optimal solutions. However, like other SI algorithms, WOA can encircling their prey, and then gracefully swim upwards towards the
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face challenges such as getting trapped in local optima and maintaining surface, trapping their quarry within the bubble net. It is notewor-
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population diversity. To address these limitations, numerous WOA thy that this bubble-net feeding technique is a unique behavior that
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variants have been proposed, enhancing the core algorithm through has been observed exclusively in humpback whales. The exceptional
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modifications or hybridization. These improved versions have been hunting prowess of these whales has served as a source of inspiration
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successfully applied to a variety of optimization problems, including for the development of a swarm intelligence algorithm known as the
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task scheduling in distributed computing environment. WOA [28]. Proposed for solving continuous optimization problems,
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Table 1 provides a concise summary of recent studies that utilize the WOA aims to mimic the humpback whales’ remarkable hunting
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WOA-based metaheuristic techniques for task scheduling in distributed strategies. The WOA represents each potential solution as a whale
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systems. The table categorizes these approaches based on key per- searching for the optimal position, guided by the best solution found.
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formance parameters, such as energy consumption, reliability, 𝐶𝑇𝑇 𝐴 ,
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It uses two mechanisms: encircling the prey (exploring promising
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sensitivity, makespan, and resource utilization. These parameters are
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areas) and creating bubble nets (exploiting by trapping targets). As
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chosen for their relevance in evaluating the proposed HWWO method in
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depicted in Fig. 1, humpback whales exhibit a remarkable coordi-
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this article. As evident from Table 1, while many studies consider these
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nated feeding strategy involving the creation of bubble nets to trap
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parameters individually or in limited combinations, none evaluate them
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as comprehensively as we have done in this work. This underscores and capture their prey. The exploration phase searches for potential
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the uniqueness of our approach and its potential to provide a more solution regions, while exploitation focuses on the most viable solutions
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well-rounded assessment of task scheduling optimization. within those areas, balancing exploration and exploitation for efficient
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optimization
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3. Preliminaries
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The following discussion aims to provide a succinct yet comprehen- 3.1.1. A mathematical-based model
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sive understanding of the core concepts driving the WOA and GWO This section initially models the whale behaviors of encircling tar-
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metaheuristics. Additionally, it shall elucidate the distinctive features gets, prey searching, and spiral bubble-net feeding maneuvers mathe-
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of these algorithms and their versatile applicability across a wide matically.
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spectrum of problem spaces.
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3.1.1.1. Encircling prey. The WOA algorithm takes inspiration from the
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3.1. Whale optimization algorithm hunting behavior of humpback whales, which can effectively locate and
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encircle prey. Since the optimal solution’s position is not known ini-
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Whales are majestic creatures that captivate the imagination. Among tially, the algorithm assumes the current best solution is near the global
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the animal kingdom, they hold the distinction of being the largest optimum. The other candidate solutions then attempt to update their
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mammals on earth. An adult whale can reach staggering proportions, positions towards this best solution identified so far. This encircling and
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4
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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Fig. 1. Coordinated feeding strategy employed by humpback whales involving bubble nets.
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localization process is mathematically modeled through Eqs. (1) and 𝜈 signifies a randomly generated scalar quantity constrained
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(2). within the bounds of [−1, 1].
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→ → → → → → →
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𝐷 = | 𝜉 ∗ 𝑍 ∗ (𝑟) − 𝑍(𝑟)| (1) 𝐷′ = |𝑍 ∗ (𝑟) − 𝑍(𝑟)| (6)
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→ → → → → → →
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∗
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𝑍(𝑟 + 1) = |𝑍 (𝑟) − 𝜁 ∗ 𝐷| (2) 𝑍(𝑟 + 1) = 𝐷 ∗ 𝑒 ′ 𝑏𝜈 ∗
|
||
∗ 𝑐𝑜𝑠(2𝜋𝜈) + 𝑍 (𝑟) (7)
|
||
The Eqs. (1) and (2) involve several variables and vectors. The variable
|
||
→ →
|
||
𝑟 denotes the current iteration number. 𝜁 and 𝜉 represent coefficient The humpback whale circles its prey in a tightening spiral pattern while
|
||
→
|
||
hunting. The WOA models this by randomly choosing between the
|
||
vector quantities. The vector 𝑍 ∗ representing the current best solution
|
||
shrinking encirclement or spiral model, each with 50% probability, to
|
||
must be updated during any iteration where a superior solution is
|
||
→ update whale positions during optimization. This stochastic positional
|
||
identified. The vector 𝑍 indicates another position vector being consid-
|
||
update process is given by Eq. (8), wherein 𝑝 denotes a randomly
|
||
ered. The absolute value operation is represented by the ∥ symbol. The
|
||
→ → generated scalar confined within the range [0,1].
|
||
calculation of the coefficient vectors 𝜁 and 𝜉 proceeds in the following
|
||
manner [85]: ⎧ → → →
|
||
→ ⎪ 𝑍 ∗ (𝑟) − 𝜁 ∗ 𝐷 if 𝑝 < 0.5
|
||
→ → → → 𝑍(𝑟 + 1) = ⎨ → (8)
|
||
𝜁 = 2 ∗ 𝜇 ∗ 𝑙1 − 𝜇 (3) ⎪ 𝐷′ ∗ 𝑒𝑏𝜈 ∗ 𝑐𝑜𝑠(2𝜋𝜈) + 𝑍 ∗ (𝑟) if 𝑝 ≥ 0.5
|
||
→ →
|
||
⎩
|
||
𝜉 = 2 ∗ 𝑙2 (4) 3.1.1.3. Exploration phase (searching for prey strategy). This strategy
|
||
→ facilitates the whales in surveying the problem domain to uncover
|
||
In Eq. (3), the parameter 𝜇 linearly decreases from 2 to 0 across all
|
||
unexplored regions and augment the diversity within the population. A
|
||
iterations, encompassing the exploration and exploitation phases, while
|
||
→ randomly selected search agent dictates the positional update for each
|
||
𝑙1 , 𝑙2 are random vectors in the range [0, 1]. The formulation of 𝜇 can →
|
||
be expressed as [2]: individual whale. The parameter 𝜁 enables steering the search agent
|
||
{ } away from an arbitrarily chosen humpback whale. The exploration
|
||
→ 2
|
||
𝜇 =2−𝑟∗ (5) phase, governed by Eq. (10), prevents the premature convergence to
|
||
𝑚𝑎𝑥_ 𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛
|
||
local optima [28].
|
||
3.1.1.2. Exploitation phase (bubble-net attacking method). Two distinct → → →
|
||
methodologies have been proposed aimed at constructing mathematical 𝐷 = |𝜉 ∗ 𝑍 𝑟𝑎𝑛𝑑 − 𝑍(𝑟)| (9)
|
||
→ → →
|
||
models to characterize the bubble-net feeding behavior exhibited by
|
||
𝑍(𝑟 + 1) = 𝑍 𝑟𝑎𝑛𝑑 − 𝜁 ∗ 𝐷 (10)
|
||
humpback whales.
|
||
→
|
||
i Shrinking encircling mechanism: This behavior is modeled by where, 𝑍 𝑟𝑎𝑛𝑑 denotes a randomly selected whale position vector from
|
||
→ the current population within the search space.
|
||
diminishing the convergence parameter 𝜇 in Eq. (3). Moreover,
|
||
→ →
|
||
the oscillation range of 𝜁 contracts linearly via 𝜇, transitioning
|
||
→ 3.2. Grey wolf optimization algorithm
|
||
from 2 to 0 across iterations. Stated differently, 𝜁 represents a
|
||
random value within the interval [−𝜇, 𝜇]. The GWO is a SI technique that emulates the hierarchical leadership
|
||
ii Spiral updating position mechanism: The approach structure and cooperative hunting strategies exhibited by grey wolves
|
||
→
|
||
commences by quantifying the distance between the vector 𝑍 ∗ , in their natural habitats [29]. The GWO algorithm mathematically
|
||
representing the best solution identified thus far, and another formalizes the search, encirclement, and attack behaviors observed in
|
||
→
|
||
whale position vector 𝑍, through Eq. (6). Subsequently, it de- the predatory conduct of grey wolves. It incorporates the hierarchical
|
||
fines the spiral motion pattern originating from the present social structure present within wolf packs as a core concept. Wolves
|
||
location and progressing towards an enhanced solution, as ex- are classified into four distinct hierarchical tiers based on their levels
|
||
pressed through Eq. (7). In these equations, the constant 𝑏 of dominance. The 𝛼, 𝛽, and 𝛿 ranks represent the leaders, presumed
|
||
governs the logarithmic spiral’s geometric characteristics, while to possess superior capabilities that guide the pack. In contrast, the 𝜔
|
||
|
||
5
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 2. Organizational structure and role hierarchy in grey wolf packs.
|
||
|
||
|
||
wolves assume a subordinate role, following the navigation directives 3.2.2. Hunting
|
||
provided by the dominant leaders (see Fig. 2). The grey wolf algorithm mimics the intricate hunting strategies
|
||
GWO employs mathematical models that emulate the intricate hunt- employed by wolf packs in nature. Central to this optimization process
|
||
ing tactics exhibited by grey wolves, including their pursuit, encir- is the 𝛼 wolf, acting as the lead entity guiding the search for the optimal
|
||
clement, and eventual capture of prey, as a framework to guide the solution. Through iterative refinement, the 𝛼 continuously updates and
|
||
optimization process. stores the best solution encountered, replacing it with an improved
|
||
one if found in subsequent iterations. This iterative refinement allows
|
||
3.2.1. Social behavior convergence towards the optimal result. While the 𝛼 takes the lead, the
|
||
The mathematical formulation supposes 𝛼 to be the preeminent 𝛽 and 𝛿 wolves contribute their prowess to the hunt. By mathematically
|
||
solution, embodying the social behavior of the lead wolf. The subse- simulating the hunting behavior, the algorithm assumes that the 𝛼, 𝛽,
|
||
quent solutions, 𝛽 and 𝛿, constitute the second and third-best outcomes, and 𝛿 solutions possess superior knowledge of the potential optimal
|
||
respectively. All remaining solutions are collectively classified as 𝜔. The position. Consequently, the top three candidate solutions are retained.
|
||
hunting process within the GWO algorithm is steered by the triumvirate The remaining search agents, including 𝜔, must update their positions
|
||
of 𝛼, 𝛽, and 𝛿, while the 𝜔 solutions are governed by adherence to this based on the 𝛼 location. In essence, the 𝛼, 𝛽, and 𝛿 predict the optimal
|
||
leading trio. location, while other wolves randomly explore the surrounding areas,
|
||
Encircling the prey: The predatory strategy of grey wolves involves driven by the overarching goal of locating the prey — the global opti-
|
||
meticulously encircling and confining their prey during the hunt. This mum. The positions of the wolves are iteratively updated through the
|
||
critical stage of encirclement is mathematically modeled by the ensuing subsequent mathematical equations that emulate the hunting behavior.
|
||
system of equations: → → → → ⎫
|
||
→ → → → 𝛶𝛼 = | 𝜉 1 ∗ 𝑍 𝛼 − 𝑍| ⎪
|
||
𝛶 = | 𝜉 ∗ 𝑍 𝑝 (𝑟) − 𝑍(𝑟)| (11) → → → → ⎪
|
||
𝛶𝛽 = | 𝜉 2 ∗ 𝑍 𝛽 − 𝑍| ⎬ (13)
|
||
→ → → → → → → → ⎪
|
||
𝑍(𝑟 + 1) = 𝑍 𝑝 (𝑟) − 𝜁 ∗ 𝛶 (12) 𝛶𝛿 = | 𝜉 3 ∗ 𝑍 𝛿 − 𝑍| ⎪
|
||
⎭
|
||
→ → → → →
|
||
In the given context, the variable 𝛶 symbolizes the vector distance 𝑍1 = 𝑍𝛼 − 𝜁 1 ∗ 𝛶 𝛼 (14)
|
||
separating the prey’s location from the wolf’s position. The variable → → → →
|
||
𝑟 denotes the current iteration number, while (𝑟 + 1) signifies the 𝑍2 = 𝑍𝛽 − 𝜁 2 ∗ 𝛶 𝛽 (15)
|
||
→
|
||
→ → → →
|
||
iteration number that follows. The variable 𝑍 𝑝 (𝑟) signifies the position 𝑍3 = 𝑍𝛿 − 𝜁 3 ∗ 𝛶 𝛿 (16)
|
||
→
|
||
of the prey within the optimization process, whereas 𝑍(𝑟) denotes → → →
|
||
→ 𝑍1 + 𝑍2 + 𝑍3
|
||
the position of the wolf. These variables are employed to model the 𝑍(𝑟 + 1) = (17)
|
||
interaction between the prey and the wolf, which is a crucial aspect of 3
|
||
the optimization algorithm being discussed.
|
||
The optimization algorithm iteratively calculates and refines the 3.2.3. Exploitation (attacking prey)
|
||
→ →
|
||
coefficient vectors 𝜁 and 𝜉 through the use of the mathematical ex- The hunting process involves a strategy employed by the grey
|
||
pressions represented by Eqs. (3) and (4), which are provided as wolves to restrict the prey’s mobility, rendering it vulnerable to an
|
||
follows: attack. This approach is implemented by gradually decreasing the value
|
||
→
|
||
→ → → → → → of a parameter 𝜇(decrease from 2 to 0). Concurrently, the value of
|
||
→
|
||
𝜁 = 2 ∗ 𝜇 ∗ 𝑙1 − 𝜇 and 𝜉 = 2 ∗ 𝑙2
|
||
another parameter, 𝜁 , is also reduced in accordance with the value of
|
||
→ →
|
||
where, the parameter 𝜇 linearly decreases from 2 to 0 across all 𝜇, ensuring it remains within the range of [−1, 1]. The grey wolves
|
||
→
|
||
iterations, encompassing the exploration and exploitation phases, while initiate an attack on the prey when the value of 𝜁 falls between −1
|
||
𝑙1 , 𝑙2 are random vectors in the range [0, 1]. and 1.
|
||
|
||
6
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 3. Dynamic positioning of search agents based on parameter interactions.
|
||
|
||
|
||
3.2.4. Exploration (search for prey) Table 2
|
||
The lead wolves, 𝛼, 𝛽, and 𝛿, strategically position themselves in Key symbols for the present work.
|
||
a manner that balances the pursuit of the prey with the readiness to Notation Description
|
||
→
|
||
strike. This dual approach is modeled through a parameter 𝜁 , where 𝐺 Direct acyclic graph (DAG) representing the
|
||
distributed parallel application
|
||
values exceeding 1 represent a diversion from the prey’s immediate
|
||
𝑋 = {𝜏1 , 𝜏2 , … , 𝜏|𝑋 } Set of |𝑋| tasks
|
||
vicinity, yet still within striking range. Another influential factor gov- 𝑌 = {𝑌1 , 𝑌2 , … , 𝑌|𝑌 | } Set of |𝑌 | processors
|
||
→
|
||
erning the exploration process is denoted by 𝜉 , which plays a crucial 𝑐̂𝑖,𝑘 Worst-case response time between the tasks 𝜏1 and 𝜏𝑘
|
||
𝑤̂ 𝑖,𝑙 Worst-case execution time of the task 𝜏𝑖 on the
|
||
role, particularly in scenarios where the algorithm encounters local
|
||
→ processor 𝑌𝑙
|
||
optima. The range of 𝜉 lies between 0 and 2, and its value is determined LB(G) Lower bound of G
|
||
through a mathematical expression, labeled as Eq. (4). DL(G) Deadline of application G
|
||
MS(G) Makespan of application G
|
||
Fig. 3 illustrates a search agent dynamically positioning itself within
|
||
𝐸̂ 𝑠 (𝐺) Static energy consumption of G
|
||
a search space using 𝛼, 𝛽, and 𝛿 parameters. The agent’s final po- 𝐸̂ 𝑑 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ) The dynamic energy consumption of task𝜏𝑖 on
|
||
sition, representing an estimated prey location, is depicted within a processor 𝑌𝑙 with frequency 𝑓𝑙,ℎ
|
||
circle defined by these parameters. Surrounding agents adapt their 𝐸̂ 𝑑 (𝐺) Dynamic energy consumption of G
|
||
positions around this estimated location, introducing randomness and 𝐸̂ 𝑡𝑜𝑡𝑎𝑙 (𝐺) The aggregate energy consumption of G
|
||
𝑅𝑒 (𝐺) Reliability of the application G
|
||
coordinated behavior akin to predators.
|
||
𝑅𝑒 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ) Reliability of task 𝜏𝑖 executed on the 𝑌𝑙 with
|
||
frequency 𝑓𝑙,ℎ
|
||
4. Notations and mathematical modeling 𝑅𝑒(𝑚𝑖𝑛) (𝐺) Minimum reliability value of G
|
||
𝑅𝑒(𝑚𝑎𝑥) (𝐺) Maximum reliability value of G
|
||
𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) Reliability goal of the application G
|
||
4.1. Notations
|
||
𝜆𝑙,ℎ Failure rate of processor 𝑌𝑙 at frequency 𝑓𝑙,ℎ
|
||
𝑅𝑒(𝑚𝑖𝑛) (𝜏𝑖 ) Minimum reliability value of 𝜏𝑖
|
||
The key symbols and their meanings, as employed in the present 𝑅𝑒(𝑚𝑎𝑥) (𝜏𝑖 ) Maximum reliability value of 𝜏𝑖
|
||
work, are summarized in Table 2. 𝑅𝑒 (𝜏𝑖 ) Reliability of task 𝜏𝑖
|
||
𝐸𝑆𝑇 (𝜏𝑘 , 𝑌𝑙 ) Earliest start time of 𝑘th task on processor 𝑌𝑙
|
||
𝐸𝐹 𝑇 (𝜏𝑘 , 𝑌𝑙 ) Earliest finish time of 𝑘th task on processor 𝑌𝑙
|
||
4.2. Application model
|
||
𝐴𝐹 𝑇 (𝜏𝑘 ) Actual finish time of 𝑘th task
|
||
|
||
The directed acyclic graph (DAG) serves as a versatile represen-
|
||
tation widely adopted in academic research for modeling distributed
|
||
parallel applications. In this study, the application is effectively mod- zero-weight dependencies are introduced into the graph to maintain
|
||
eled as a DAG, denoted as 𝐺 = (𝑋, 𝐸, ̂ 𝑊̂ , 𝐶).
|
||
̂ This model encompasses consistency [8]. For |𝑌 | no. of processors, 𝑤̂ 𝑖,𝑙 ∈ 𝑊̂ |𝑋|×|𝑌 | gives the
|
||
a set 𝑋, comprising various computational tasks with distinct worst- 𝑊̂ 𝐶𝐸𝑇 of task 𝜏𝑖 on processor 𝑌𝑙 .
|
||
case execution times (𝑊̂ 𝐶𝐸𝑇 𝑠) on different processors. Furthermore, The expressions LB(G) and DL(G) represent the lower bound and
|
||
the model incorporates a set 𝐸, ̂ representing communication edges
|
||
deadline of G, respectively. It is imperative to maintain a condition
|
||
between these tasks. Each element in 𝐸, ̂ represented as 𝑒̂𝑖,𝑘 , signifies
|
||
wherein the lower bound of an application remains below its associated
|
||
a communication link from task 𝜏𝑖 to 𝜏𝑘 , accompanied by a precedence
|
||
deadline, i.e., 𝐿𝐵(𝐺) ≤ 𝐷𝐿(𝐺) In the course of this work, the concept of
|
||
constraint that mandates task 𝜏𝑘 to commence only upon the com-
|
||
lower bound pertains to the minimal achievable makespan by an appli-
|
||
pletion of task 𝜏𝑖 . Consequently, every 𝑐̂𝑖,𝑘 represents the worst-case
|
||
response time (𝑊̂ 𝐶𝑅𝑇 ) of 𝑒̂𝑖,𝑘 . The set 𝑠𝑢𝑐𝑐(𝜏𝑖 ) represents the immediate cation developed through a conventional scheduling algorithm, where
|
||
successor tasks of 𝜏𝑖 , and 𝑝𝑟𝑒𝑑(𝜏𝑖 ) represents the immediate predecessor each processor is singularly dedicated to the application, operating at
|
||
tasks of 𝜏𝑖 . Tasks lacking predecessors are designated as 𝜏𝑒𝑛𝑡𝑟𝑦 , while its maximum frequency [36,86]. 𝑀𝑆(𝐺) denotes the actual makespan
|
||
those lacking successors are designated as 𝜏𝑒𝑥𝑖𝑡 . In instances where achieved by application G, which signifies the precise conclusion time
|
||
a function includes multiple 𝜏𝑒𝑛𝑡𝑟𝑦 or 𝜏𝑒𝑥𝑖𝑡 tasks, dummy tasks with of 𝜏𝑒𝑥𝑖𝑡 within the corresponding 𝐷𝐴𝐺.
|
||
|
||
7
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 4. Example of a DAG featuring 10 tasks.
|
||
|
||
|
||
Table 3 exhibits a dependency on frequency. The parameter 𝜙 signifies the
|
||
(𝑊̂ 𝐶𝐸𝑇 𝑠) of tasks in Fig. 4. system states and serves as an indicator of whether dynamic power is
|
||
Tasks \Processors 𝑌1 𝑌2 𝑌3 presently being consumed within the system, where 𝜙 = 1 signifies an
|
||
𝜏1 14 16 9 active state, and 𝜙 = 0 represents an inactive condition. The term 𝐶𝑒𝑓
|
||
𝜏2 13 19 18 symbolizes the effective switching capacitance. The exponent m, known
|
||
𝜏3 11 13 19
|
||
𝜏4 13 8 17
|
||
as the dynamic power exponent with a minimum value of 2. Both 𝐶𝑒𝑓
|
||
𝜏5 12 13 10 and m are processor-specific constants.
|
||
𝜏6 13 16 9 Strategically reducing the operating frequency presents an avenue
|
||
𝜏7 7 15 11 to curb frequency-dependent power dissipation. However, prolonged
|
||
𝜏8 5 11 14
|
||
𝜏9 18 12 20
|
||
execution times may ensue, augmenting static power consumption and
|
||
𝜏10 21 7 16 frequency-independent power expenditure. Several studies, including
|
||
those conducted by [8,87], and [88], have established the existence
|
||
of an optimal energy-efficient frequency, denoted as 𝑓𝑒𝑒 , at which the
|
||
system achieves minimal power consumption. This optimal frequency
|
||
Fig. 4 illustrates a DAG-based parallel application as an example [8,
|
||
can be formulated as follows:
|
||
86]. This demonstration comprises 10 tasks processed across three √
|
||
designated processors {𝑌1 , 𝑌2 , 𝑌3 }. In the illustration, the weight of 18 𝑃𝑖𝑛𝑑
|
||
𝑓𝑒𝑒 = 𝑚 (19)
|
||
on the edge 𝑒̂1,2 connecting 𝜏1 and 𝜏2 symbolizes the response time, (𝑚 − 1)𝐶𝑒𝑓
|
||
denoted as 𝑐̂1,2 , if 𝜏1 and 𝜏2 are not allocated to the same processor.
|
||
The data in Table 3 represents the worst-case execution times Under the premise that a processor’s operating frequency can vary
|
||
(𝑊̂ 𝐶𝐸𝑇 𝑠) corresponding to the maximum frequency illustrated in Fig. between a minimum available frequency, 𝑓𝑚𝑖𝑛 , and a maximum fre-
|
||
4. The value of 14 assigned to the intersection of 𝜏1 and 𝑌1 in Table quency, 𝑓𝑚𝑎𝑥 , the optimal energy-efficient frequency for executing a
|
||
3 denotes the (𝑊̂ 𝐶𝐸𝑇 𝑠), symbolized as 𝑤̂ 1,1 = 14. The variations in given task should adhere to the following formulation:
|
||
(𝑊̂ 𝐶𝐸𝑇 𝑠) for an identical task across different processors arise from
|
||
𝑓𝑙𝑜𝑤 = 𝑚𝑎𝑥(𝑓𝑒𝑒 , 𝑓𝑚𝑖𝑛 ) (20)
|
||
the intrinsic diversity of the processors.
|
||
As a result, any of the actual effective frequencies, denoted as 𝑓ℎ ,
|
||
4.3. Power and energy models should reside within the range delineated by 𝑓𝑙𝑜𝑤 ≤ 𝑓ℎ ≤ 𝑓𝑚𝑎𝑥 [8].
|
||
For a system with |𝑌 | heterogeneous processors, each processor
|
||
Considering the nearly linear correlation between voltage and fre-
|
||
requires individual power parameters. Here, the static power set is
|
||
quency, DVFS techniques are employed to scale down these parameters,
|
||
defined as:
|
||
thereby achieving energy conservation. Consistent with the approaches
|
||
adopted in [8,87], the term frequency change is utilized to denote {𝑃1,𝑠 , 𝑃2,𝑠 , … , 𝑃|𝑌 |,𝑠 }
|
||
the simultaneous alteration of both voltage and frequency. For DVFS-
|
||
capable systems, a widely adopted system-level power model, as ex- frequency-independent and dependent dynamic power sets are repre-
|
||
emplified in [8,87], is leveraged. This model expresses the power sented as:
|
||
consumption at a given frequency (f) as follows:
|
||
{𝑃1,𝑖𝑛𝑑 , 𝑃2,𝑖𝑛𝑑 , … , 𝑃|𝑌 |,𝑖𝑛𝑑 } and {𝑃1,𝑑 , 𝑃2,𝑑 , … , 𝑃|𝑌 |,𝑑 }
|
||
𝑃 (𝑓 ) = 𝑃𝑠 + 𝜙(𝑃𝑖𝑛𝑑 + 𝑃𝑑 ) = 𝑃𝑠 + 𝜙(𝑃𝑖𝑛𝑑 + 𝐶𝑒𝑓 𝑓 𝑚 ) (18)
|
||
the effective switching capacitance is defined as:
|
||
Within this power model, 𝑃𝑠 symbolizes the static power component,
|
||
which can be mitigated solely by deactivating the entire system. The {𝐶1,𝑒𝑓 , 𝐶2,𝑒𝑓 , … , 𝐶|𝑌 |,𝑒𝑓 }
|
||
frequency-independent dynamic power is represented by 𝑃𝑖𝑛𝑑 , and this lowest energy-efficient frequency set is represented as:
|
||
component can be eliminated by transitioning the system into a low-
|
||
power sleep state. 𝑃𝑑 denotes the dynamic power component that {𝑓1,𝑙𝑜𝑤 , 𝑓2,𝑙𝑜𝑤 , … , 𝑓|𝑌 |,𝑙𝑜𝑤 }
|
||
|
||
8
|
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
and actual effective frequency set is: To determine the application’s reliability bounds, an evaluation across
|
||
{ all available processors is conducted. The minimum and maximum
|
||
{𝑓1,𝑙𝑜𝑤 , 𝑓1,𝑐 , 𝑓1,𝑑 , … , 𝑓1,𝑚𝑎𝑥 }, {𝑓2,𝑙𝑜𝑤 , 𝑓2,𝑐 , 𝑓2,𝑑 , … , 𝑓2,𝑚𝑎𝑥 }, …
|
||
} reliability values are then derived using the respective equations:
|
||
{𝑓|𝑌 |,𝑙𝑜𝑤 , 𝑓|𝑌 |,𝑐 , 𝑓|𝑌 |,𝑑 , … , 𝑓|𝑌 |,𝑚𝑎𝑥 }
|
||
𝑅𝑒(𝑚𝑖𝑛) (𝜏𝑖 ) = min 𝑅𝑒 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,𝑙𝑜𝑤 ) (30)
|
||
𝑌𝑙 ∈𝑌
|
||
Let 𝐸̂ 𝑠 (𝐺) denote the static energy consumed by active processors ∏
|
||
executing application G. Since inactive processors do not consume 𝑅𝑒(𝑚𝑎𝑥) (𝜏𝑖 ) = 1 − (1 − 𝑅𝑒 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,𝑚𝑎𝑥 )) (31)
|
||
energy, 𝐸̂ 𝑠 (𝐺) is the sum of static energy consumption across all active 𝑌𝑙 ∈𝑌
|
||
processors, calculated as: As per Eq. (29), the reliability of application G is the product of task
|
||
|𝑌 |
|
||
∑ reliabilities. Hence the minimum and maximum reliability values of G
|
||
( )
|
||
𝐸̂ 𝑠 (𝐺) = 𝑃𝑙,𝑠 ∗ 𝑀𝑆(𝐺) (21) can be computed as:
|
||
𝑙=1,𝑌𝑙 is on ∏
|
||
𝑅𝑒(𝑚𝑖𝑛) (𝐺) = (𝜏𝑖 ) (32)
|
||
The dynamic power consumption of task 𝜏𝑖 executing on 𝑌𝑙 at frequency
|
||
𝑒(𝑚𝑖𝑛)
|
||
𝑓𝑙,ℎ is represented by 𝑃𝑑 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ). This can be formulated as: ∏
|
||
𝑅𝑒(𝑚𝑎𝑥) (𝐺) = (𝜏𝑖 ) (33)
|
||
𝑚
|
||
𝑃𝑑 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ) = (𝑃𝑙,𝑖𝑛𝑑 + 𝐶𝑙,𝑒𝑓 ∗ 𝑓𝑙,ℎ𝑙 ) (22) 𝑒(𝑚𝑎𝑥)
|
||
|
||
The application is deemed reliable if its reliability metric satisfies the
|
||
And the dynamic energy consumption of 𝜏𝑖 is calculated as:
|
||
specified reliability goal, denoted as 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) i.e.,
|
||
𝑚 𝑓𝑙,𝑚𝑎𝑥
|
||
𝐸̂ 𝑑 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ) = (𝑃𝑙,𝑖𝑛𝑑 + 𝐶𝑙,𝑒𝑓 ∗ 𝑓𝑙,ℎ𝑙 ) ∗ ∗ 𝑤̂ 𝑖,𝑙 (23)
|
||
𝑓𝑙,ℎ 𝑅𝑒(𝑚𝑖𝑛) (𝐺) ≤ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 𝑅𝑒(𝑚𝑎𝑥) (𝐺) (34)
|
||
|
||
Let 𝐸̂ 𝑑 (𝐺) represent the total dynamic energy consumption of applica-
|
||
tion G, calculated as the sum of dynamic energy consumed by each 4.5. Description of scheduling problem
|
||
task.
|
||
|𝑋|
|
||
∑ The focus of this subsection is on the tasks scheduling problem
|
||
𝐸̂ 𝑑 (𝐺) = 𝐸̂ 𝑑 (𝜏𝑖 , 𝑌𝑎𝑐(𝑖) , 𝑓𝑎𝑐(𝑖),ℎ𝑧(𝑖) ) (24) in distributed computing environments consisting of parallel applica-
|
||
𝑖=1 tions G and heterogeneous processor set Y. Specifically, it addresses a
|
||
where 𝑌𝑎𝑐(𝑖) signifies the processor and 𝑓𝑎𝑐(𝑖),ℎ𝑧(𝑖) represents the fre- scenario where tasks must be executed on a set Y of processors that
|
||
quency at which task 𝜏𝑖 is actively executing. support varying frequency levels. This assignment must simultaneously
|
||
Hence, the aggregated energy consumption of G can be deduced optimize two critical objectives: minimizing overall energy consump-
|
||
using the ensuing formulation: tion and ensuring the application’s reliability metric 𝑅𝑒 (𝐺) meets or
|
||
surpasses a predefined reliability goal, 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺). Ultimately, the goal
|
||
𝐸̂ 𝑡𝑜𝑡𝑎𝑙 (𝐺) = 𝐸̂ 𝑠 (𝐺) + 𝐸̂ 𝑑 (𝐺) (25)
|
||
is to judiciously map tasks to processor-frequency combinations that
|
||
strike an optimal balance between reducing energy usage and boosting
|
||
4.4. Reliability model and reliability goal system reliability for the given application.
|
||
|
||
𝐸̂ 𝑡𝑜𝑡𝑎𝑙 (𝐺) = 𝐸̂ 𝑠 (𝐺) + 𝐸̂ 𝑑 (𝐺)
|
||
The concept of processor reliability probability adhering to a Pois- ∏
|
||
son distribution has been extensively studied and widely accepted subject to: 𝑅𝑒 (𝐺) = 𝑅𝑒 (𝜏𝑖 ) ≥ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺)
|
||
𝜏𝑖 ∈𝑋
|
||
within the relevant literature [8]. The variable 𝜆𝑙 denotes the failure
|
||
rate per unit time of processor 𝑌𝑙 and the reliability of a task 𝜏𝑖 executed
|
||
5. Proposed hybrid approach
|
||
on processor 𝑌𝑙 within its 𝑊̂ 𝐶𝐸𝑇 can be quantified using the following
|
||
mathematical expression:
|
||
This study presents an inventive methodology that seamlessly inte-
|
||
𝑅𝑒 (𝜏𝑖 , 𝑌𝑙 ) = 𝑒−𝜆𝑙 ∗𝑤̂ 𝑖,𝑙 (26) grates the HWWO algorithm and the DVFS technique, addressing the
|
||
crucial need for energy-efficient tasks scheduling strategies that can
|
||
For DVFS-enabled processors, research shows varying failure rates
|
||
adeptly address both static and dynamic energy considerations. The
|
||
across frequencies. Let 𝜆𝑙,𝑚𝑎𝑥 denote the failure rate of processor 𝑌𝑙 at
|
||
dynamic adjustment of processor voltage and frequency, facilitated by
|
||
maximum frequency. Then, the failure rate 𝜆𝑙,ℎ of 𝑌𝑙 at frequency 𝑓𝑙,ℎ
|
||
the DVFS mechanism, enables energy consumption optimization during
|
||
is calculated as:
|
||
task execution. This integrated methodology possesses the potential
|
||
𝑑(𝑓𝑙,𝑚𝑎𝑥 −𝑓𝑙,ℎ )
|
||
to revolutionize tasks scheduling in computational environments by
|
||
𝜆𝑙,ℎ = 𝜆𝑙,𝑚𝑎𝑥 ∗ 10 𝑓𝑙,𝑚𝑎𝑥 −𝑓𝑙,𝑚𝑖𝑛 (27)
|
||
achieving substantial energy savings, enhancing system reliability, and
|
||
Where the constant d indicates the sensitivity of failure rates to voltage concurrently optimizing computational time. The HWWO algorithm
|
||
scaling. employs a mutation operator in conjunction with an insert-reversed
|
||
Subsequently, a correlation is established between task reliability block operation, contributing to the minimization of static energy
|
||
and frequency, as outlined by Eqs. (26) and (27) i.e., the reliability consumption. Complementing this, the DVFS technique is strategically
|
||
of the task 𝜏𝑖 executed on the processor 𝑌𝑙 with the frequency 𝑓𝑙,ℎ is employed to tackle dynamic energy consumption, further augment-
|
||
calculated as follows: ing the energy-efficiency of the proposed solution. The incorporation
|
||
𝑤̂ ∗𝑓
|
||
−𝜆𝑙,ℎ ∗ 𝑖,𝑙 𝑓 𝑙,𝑚𝑎𝑥
|
||
of the WOA and the GWO into the task assignment methodology is
|
||
𝑅𝑒 (𝜏𝑖 , 𝑌𝑙 , 𝑓𝑙,ℎ ) = 𝑒 𝑙,ℎ
|
||
substantiated by their exceptional versatility. These techniques have
|
||
𝑑(𝑓𝑙,𝑚𝑎𝑥 −𝑓𝑙,ℎ ) consistently exhibited superior performance in addressing assignment
|
||
𝑤̂ ∗𝑓
|
||
−𝜆𝑙,𝑚𝑎𝑥 ∗10 𝑓𝑙,𝑚𝑎𝑥 −𝑓𝑙,𝑚𝑖𝑛 ∗ 𝑖,𝑙 𝑓 𝑙,𝑚𝑎𝑥
|
||
=𝑒 𝑙,ℎ (28) issues, demonstrating remarkable convergence characteristics [89].
|
||
In any SI algorithm, achieving a balance between exploitation and
|
||
The overall reliability of an application can be expressed as the product exploration is crucial for its effectiveness in terms of convergence
|
||
of the reliability values associated with each of its constituent tasks. The speed and solution quality. As underscored in [30], the WOA algorithm
|
||
reliability value of the application G is denoted as: showcases remarkable performance regarding convergence speed while
|
||
∏ adeptly balancing exploration and exploitation in addressing optimiza-
|
||
𝑅𝑒 (𝐺) = 𝑅𝑒 (𝜏𝑖 ) (29)
|
||
𝜏𝑖 ∈𝑋 tion issues. Although the WOA demonstrates effectiveness compared
|
||
|
||
9
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
to conventional algorithms, it may encounter challenges such as strug- 1. Determine task weights through the calculation of the average
|
||
gling to evade local optima because of its encircling search mechanism execution time across all processors, mirroring the HEFT algo-
|
||
and insufficient solution enhancement after each iteration. These limi- rithm’s methodology. In the HEFT algorithm, the function 𝑓 (𝑤̂ 𝑖 )
|
||
tations of WOA prompted the proposal of a hybrid approach with GWO. is computed by averaging the execution time of 𝜏𝑖 across each
|
||
GWO excels in exploitation, offering a solution to WOA’s challenges machine, depicted as:
|
||
through two strategies: preserving the best solution per iteration and
|
||
𝑓 (𝑤̂ 𝑖 ) = 𝑎𝑣𝑟(𝑤̂ 𝑌1 , 𝑤̂ 𝑌2 , … , 𝑤̂ 𝑌|𝑌 | ) (36)
|
||
evaluating new solutions with the best one during exploration. If the
|
||
outcome improves upon the best solution, agents’ positions change; oth- 2. Alternatively, it can be derived from the best-case scenario.
|
||
erwise, they remain unchanged. Additionally, to enhance the HWWO
|
||
algorithm, authors proposed integrating mutation operators into WOA 𝑓 (𝑤̂ 𝑖 ) = min(𝑤̂ 𝑌1 , 𝑤̂ 𝑌2 , … , 𝑤̂ 𝑌|𝑌 | ) (37)
|
||
and combining it with GWO.
|
||
Mapping tasks to multiprocessors poses a significant NP-hard chal- 3. Task weighting based on the worst-case scenario.
|
||
lenge. Consequently, the hybrid metaheuristic HWWO scheduling tech- 𝑓 (𝑤̂ 𝑖 ) = max(𝑤̂ 𝑌1 , 𝑤̂ 𝑌2 , … , 𝑤̂ 𝑌|𝑌 | ) (38)
|
||
nique is employed as a strategic solution. This approach unfolds across
|
||
two distinct phases: initial task prioritization, wherein tasks are se-
|
||
Each of these schemes results in a unique task ordering. For instance,
|
||
quenced in descending order of priorities, and subsequent task alloca-
|
||
when applied to the DAG example shown in Fig. 4, they produce differ-
|
||
tion to suitable processors.
|
||
ent upward rank lists for tasks. The most optimal ranking is achieved
|
||
Phase 1 by the first scheme, the HEFT algorithm, with the corresponding values
|
||
for the tasks depicted in Fig. 4 summarized in Table 4.
|
||
5.1. Prioritizing tasks and deadline determination Upon employing the DVR HEFT technique to generate an optimized
|
||
upward rank list for the tasks, the next step involved calculating the
|
||
To calculate the lower bound for tasks scheduling, this study uti- computation time required to allocate each task to the available pro-
|
||
lizes the heterogeneous earliest-finish-time (HEFT) algorithm proposed cessors. This allocation process followed the methodology outlined in
|
||
by [86]. The HEFT algorithm is chosen due to its proven effective- the HEFT algorithm. The task with the highest rank value is identified
|
||
ness in generating high-quality schedules for heterogeneous computing and then assigned to the processor that could complete its execution
|
||
systems, making it a reliable choice for this purpose. Obtaining an at the earliest possible finish time (EFT). To determine the earliest
|
||
exact lower bound is a challenging endeavor, so the scheduling length feasible execution time for a given task 𝜏𝑘 on processor 𝑌𝑙 , the values
|
||
estimated by the HEFT method is adopted as the standard. It is further for the 𝐸𝑆𝑇 (𝜏𝑘 , 𝑌𝑙 ) (earliest start time) and 𝐸𝐹 𝑇 (𝜏𝑘 , 𝑌𝑙 ) are calculated
|
||
assumed that the application’s deadline constraint is not known until recursively as follows:
|
||
after this lower bound reference has been determined. ⎧𝐸𝑆𝑇 (𝜏
|
||
𝑒𝑛𝑡𝑟𝑦,𝑌𝑙 ) = 0;
|
||
The strategic allocation of tasks stands as a pivotal challenge in ⎪
|
||
⎪ ⎧
|
||
the context of (DAG) list scheduling within heterogeneous distributed ⎨ ⎪𝑎𝑣𝑎𝑖𝑙[𝑙], (39)
|
||
systems. In an endeavor to tackle this challenge, the present article ⎪ 𝐸𝑆𝑇 (𝜏 ,
|
||
𝑘 𝑙𝑌 ) = 𝑚𝑎𝑥 ⎨ 𝑚𝑎𝑥 {𝐴𝐹 𝑇 (𝜏 ) + 𝑐̂ } ;
|
||
adopts the dynamic variant rank HEFT algorithm (DVR HEFT), as intro- ⎪ ⎪𝜏𝑖 ∈𝑝𝑟𝑒𝑑(𝜏𝑘 ) 𝑖 𝑖,𝑘
|
||
⎩ ⎩
|
||
duced by [90]. As per the findings elucidated by [90], the DVR HEFT
|
||
and 𝐸𝐹 𝑇 (𝜏𝑘 , 𝑌𝑙 ) = 𝐸𝑆𝑇 (𝜏𝑘 , 𝑌𝑙 ) + 𝑤̂ 𝑘,𝑙 (40)
|
||
algorithm demonstrates enhanced performance over its conventional
|
||
counterpart, HEFT, by yielding superior outcomes while concurrently The term 𝑎𝑣𝑎𝑖𝑙[𝑙] denotes the earliest moment when processor 𝑌𝑙 is
|
||
mitigating time complexity. By utilizing this improved task prioritiza- ready to execute a task, while 𝐴𝐹 𝑇 (𝜏𝑖 ) refers to the actual finish time of
|
||
tion approach, the DVR HEFT algorithm aims to enhance the efficiency task 𝜏𝑖 . If tasks 𝜏𝑖 and 𝜏𝑘 are allocated to the same processor, the vari-
|
||
of scheduling interdependent tasks across heterogeneous computing able 𝑐̂𝑖,𝑘 is assigned a value of zero. The makespan of the application de-
|
||
resources within a distributed system. In its initial phase, akin to other fines the exact completion time of task 𝜏𝑒𝑥𝑖𝑡 , accounting for the schedul-
|
||
static algorithms, the DVR HEFT algorithm undertakes the computation ing of all tasks within a DAG. As previously described, the process to
|
||
of task priorities. Within the DVR HEFT framework, this entails metic- compute the lower bound of application G unfolds as follows:
|
||
ulously establishing task priorities through a comprehensive evaluation
|
||
𝐿𝐵(𝐺) = 𝐿𝐵(𝜏𝑒𝑥𝑖𝑡 ) (41)
|
||
of their upward rank values, following a procedure similar to that of the
|
||
HEFT algorithm. The upward rank, denoted as 𝑅𝑎𝑛𝑘𝑈 (𝜏𝑖 ), for a given Therefore, the relative deadline can be fulfilled. For the illustrated
|
||
task 𝜏𝑖 , is recursively determined through the following equation: example in Fig. 4, the application’s deadline is considered as the sum
|
||
{ } of its lower bound and 20 [86].
|
||
𝑅𝑎𝑛𝑘𝑈 (𝜏𝑖 ) = 𝑓 (𝑤̂ 𝑖 ) + max [𝑎𝑣𝑟(𝑐̂𝑖,𝑘 ) + 𝑅𝑎𝑛𝑘𝑈 (𝜏𝑘 )] (35) The comparative scheduling results for the specified DAG illustrated
|
||
𝜏𝑘 ∈𝑠𝑢𝑐𝑐(𝜏𝑖 )
|
||
in Fig. 4 have been determined using a variety of algorithms, including
|
||
where, the symbol 𝑤̂ 𝑖 represents the 𝑊̂ 𝐶𝐸𝑇 of task 𝜏𝑖 . The task weight HEFT [86], DECM [37], energy-aware processor merging (EPM) [91],
|
||
value, produced by the function 𝑓 (𝑤̂ 𝑖 ), is contingent upon the task’s reliability enhancement under energy and response time constraints
|
||
execution duration on each processor whereas 𝑐̂𝑖,𝑘 signifies the 𝑊̂ 𝐶𝑅𝑇 (REREC) [38], and energy-efficient scheduling with a reliability goal
|
||
between the tasks 𝜏𝑖 and 𝜏𝑘 . (ESRG) [8]. Assessing energy consumption and reliability involves ref-
|
||
In a heterogeneous computing environment, the time required to erencing the power parameters listed in Table 5 for all processors. Each
|
||
execute a task can fluctuate based on the capabilities and performance processor’s energy-efficient frequency, denoted as 𝑓𝑒𝑒 , is calculated
|
||
characteristics of the specific machine handling that task. As a result, in based on Eq. (19), while the maximum frequency, 𝑓𝑚𝑎𝑥 , for each proces-
|
||
such heterogeneous settings, there exist multiple distinct methodologies sor is considered to be 1.0, as indicated in previous studies [8,91]. The
|
||
to calculate the computational weight associated with each node or schedule generated by employing the HEFT algorithm at its maximum
|
||
task. The approach chosen to determine a node’s weight 𝑤̂ 𝑖 could frequency level is graphically presented in Fig. 5. The aggregate energy,
|
||
potentially optimize computation time in certain scenarios, however, denoted as 𝐸̂ 𝑡𝑜𝑡𝑎𝑙 (𝐺), and the overall reliability, represented as 𝑅𝑒 (𝐺),
|
||
it does not guarantee improvements across all cases. As a result, the resulting from the execution of the HEFT algorithm, are quantified as
|
||
researchers in [90] proposed three distinct methods for calculating the 170.52 and 0.880426, respectively [91]. The visual depictions of the
|
||
upward rank values assigned to tasks. These alternative schemes for scheduling outcomes for the other algorithms are illustrated in Figs.
|
||
determining task priorities are described as follows: 6–9. It is worth highlighting that the processors displayed with shading
|
||
|
||
10
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 4
|
||
𝑅𝑎𝑛𝑘𝑈 values for tasks illustrated in Fig. 4.
|
||
Tasks 𝜏1 𝜏3 𝜏2 𝜏4 𝜏5 𝜏6 𝜏7 𝜏8 𝜏9 𝜏10
|
||
𝑅𝑎𝑛𝑘𝑈 133.19 118.19 115.86 114.19 101.53 87.27 70.86 59.86 44.36 14.7
|
||
|
||
|
||
|
||
|
||
Fig. 5. Scheduling result of the DAG shown in Fig. 4 using the HEFT technique.
|
||
|
||
|
||
|
||
|
||
Fig. 6. Scheduling result of the DAG shown in Fig. 4 using the DECM technique.
|
||
|
||
|
||
|
||
|
||
Fig. 7. Scheduling result of the DAG shown in Fig. 4 using the EPM technique.
|
||
|
||
|
||
|
||
|
||
11
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 8. Scheduling result of the DAG shown in Fig. 4 using the REREC technique.
|
||
|
||
|
||
|
||
|
||
Fig. 9. Scheduling result of the DAG shown in Fig. 4 using the ESRG technique.
|
||
|
||
|
||
Table 5 employing the DVR HEFT technique. This method aids in reducing
|
||
Power parameters of processors (𝑌1 , 𝑌2 , 𝑎𝑛𝑑𝑌3 ). the static energy consumed. This section proposes a novel algorithmic
|
||
𝑌𝑙 𝑃𝑙,𝑠 𝐶𝑙,𝑒𝑓 𝑃𝑙,𝑖𝑛𝑑 𝑚𝑙 𝑓𝑙,𝑙𝑜𝑤 𝑓𝑙,𝑚𝑎𝑥 𝜆𝑙,𝑚𝑎𝑥 approach that leverages the DVFS-integrated EES technique. This algo-
|
||
𝑌1 0.3 0.8 0.06 2.9 0.33 1.0 0.0005 rithm incorporates a HWWO model, featuring an insert-reversed block
|
||
𝑌2 0.2 0.7 0.07 2.7 0.29 1.0 0.0002 operation and a mutation operator. The primary goal is to efficiently
|
||
𝑌3 0.1 1.0 0.07 2.4 0.29 1.0 0.0009
|
||
assign tasks to appropriate processors within computational environ-
|
||
ments, thereby achieving substantial reductions in energy consumption
|
||
while simultaneously enhancing the overall reliability of the system.
|
||
in the figures signify their inactive or idle status during the execution The mutation operator’s main role is to diversify the population and
|
||
of the respective schedules. boost HWWO’s global exploration. This study employs the inversion
|
||
The tasks displayed in blue indicate a higher execution frequency. mutation operator to fulfill this function.
|
||
Initially, all processors shown in Fig. 6 and processors 𝑌1 and 𝑌2
|
||
illustrated in Fig. 7 become active. Following that, every processor 5.2.1. Termination criteria
|
||
depicted in Fig. 8 is activated. Afterward, Fig. 9 demonstrates the
|
||
Optimization algorithms must have proper stopping conditions to
|
||
activation of processors 𝑌1 and 𝑌2 specifically.
|
||
ensure they do not run indefinitely. The algorithm iterates until conver-
|
||
The total energy consumption when employing the DECM algorithm
|
||
gence to find the best solution, making it essential to set termination
|
||
(depicted in Fig. 6) is lower than other algorithms, followed by REREC
|
||
criteria to assess the optimization process’s convergence. In this case,
|
||
and ESRG algorithms. Crucially, this technique also enhances the sys-
|
||
the algorithm is designed to carry out 20 optimization iterations ini-
|
||
tem’s reliability compared to these other methods. By analyzing Figs.
|
||
tially, before assessing any convergence criteria. Subsequent to this
|
||
5–9, it becomes evident that the DECM technique delivers superior per-
|
||
initial phase, it examines two criteria:
|
||
formance concerning both energy efficiency and reliability, surpassing
|
||
the EPM, HEFT, REREC, and ESRG algorithms. i The first criterion measures the change in the fitness function
|
||
Phase 2 between the latest optimization iteration 𝑓𝑓 𝑛 (𝑟) and the previous
|
||
iteration 𝑓𝑓 𝑛 (𝑟 − 1). When the relative change is less than a
|
||
5.2. DVFS-integrated hybrid WOA-GWO scheduling model specified tolerance 𝜖𝑓 e.g., 𝜖𝑓 = 0.001 or 0.1%, the stop criterion
|
||
is met. This stop criterion is defined as:
|
||
The section before this one explains the task prioritization pro- 𝑓𝑓 𝑛 (𝑟) − 𝑓𝑓 𝑛 (𝑟 − 1)
|
||
cess, where tasks are arranged in descending order of priorities by < 𝜖𝑓 (42)
|
||
𝑓𝑓 𝑛 (𝑟)
|
||
|
||
12
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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|
||
|
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||
|
||
Fig. 10. Inversion mutation operator.
|
||
|
||
|
||
ii If the above criterion is not satisfied, the optimization process Algorithm 1 Pseudo code for the insert-reversed block operation
|
||
halts after reaching the maximum number of iterations, set here ( )
|
||
|𝑋|
|
||
to 500. 1: temp = random 2, ;
|
||
2
|
||
2: arr positions [temp];
|
||
5.2.2. Scheduling fitness function 3: function replace(search_agent, i)
|
||
HWWO integrates metaheuristic techniques for tasks scheduling, 4: for j = 0 to temp - 1 do
|
||
with the effectiveness contingent upon the crafted fitness function 5: search_agent[j] = switch(i, positions[value - j - 1]);
|
||
determining suitable computing resources. Eq. (43) outlines the formu- 6: end for
|
||
lation of the fitness function for each particle in HWWO. 7: return search_agent;
|
||
[ |𝑌 | |𝑋|
|
||
( )] 8: end function
|
||
∑ 𝑀𝑆(𝐺) ∑ −𝜆𝑙,𝑚𝑎𝑥 ∗
|
||
𝑤̂ 𝑖,𝑙 ∗𝑓𝑙,𝑚𝑎𝑥
|
||
|
||
𝑓𝑓 𝑛(𝜏𝑖 ,𝑌𝑙 ,𝑓𝑙,ℎ ) = 𝑚𝑖𝑛 𝜎1 ∗ 𝑃𝑙,𝑠 ∗ + 𝜎2 ∗ 𝑒 𝑓𝑙,ℎ
|
||
(43) 9: function insert_reversed_block_operation(assignment)
|
||
𝐷𝐿(𝐺)
|
||
𝑙=1 𝑖=1 10: matrix permutations[|𝑋| − 1][|𝑋|];
|
||
Here, 𝜎1 and 𝜎2 signify the optimal weights for energy consumption 11: positions = sorted(random(0, |𝑋| - 1, temp)); ⊳ Select ‘temp‘
|
||
and computation time, subject to 𝜎1 + 𝜎2 = 1. During selection, each number of positions of tasks.
|
||
solution’s objectives receive random weights, encouraging exploration 12: permutations[0] = assignment;
|
||
in various directions. 13: for i = 0 to |𝑋| - 3 do
|
||
14: if i not in positions then
|
||
5.2.3. Inversion mutation 15: permutations[i + 1] = replace(permutations[i], i);
|
||
Inversion mutation [92], an operator in evolutionary algorithms, 16: end if
|
||
randomly selects a segment of genes within a chromosome and reverses 17: end for
|
||
their order. This process fosters genetic diversity by exploring different 18: end function
|
||
regions of the search space, potentially leading to improved solu-
|
||
tions. This can be understood by referring to the illustrative example
|
||
presented in Fig. 10.
|
||
formulating an effective HWWO approach for tackling optimization
|
||
challenges lies in the intricate process of developing an appropriate en-
|
||
5.2.4. Insert-reversed block operation
|
||
coding mechanism to model the constituent elements or search agents
|
||
Excessive mutation can disrupt previously found good solutions and
|
||
within the HWWO framework. For optimization scenarios involving |𝑌 |
|
||
impede the algorithm’s convergence toward optimal or near-optimal
|
||
processors, a viable strategy to model the constituent search particles
|
||
solutions. This occurs because random changes introduced by mutation
|
||
is through the utilization of |𝑌 |-dimensional coordinate vectors, as
|
||
may not always improve the solutions. Consequently, after applying the
|
||
illustrated in the subsequent representation.
|
||
inversion mutation operator, the insert-reversed block operation [93] is
|
||
integrated into the algorithm. This operation inserts a reversed block of → → → →
|
||
task permutation into all (|𝑋| − 1) conceivable positions within a |𝑋|- 𝑋 = (𝑋 1 , 𝑋 2 , … , 𝑋 |𝑌 | )
|
||
dimensional search agent, as outlined in Algorithm 1 and illustrated in
|
||
Fig. 11. Step 2: Evaluating fitness values for each particle in assignment.
|
||
This step calculates the fitness score for each HWWO particle,
|
||
5.2.5. HWWO scheduling algorithm reflecting the task-to-processor assignment. The hybrid WOA–GWO
|
||
The pseudocode in Algorithm 2 outlines the procedure for tasks framework evaluates individual particle fitness using the function de-
|
||
scheduling employing the HWWO technique. A more in-depth eluci- fined in Eq. (43). When a particle’s current fitness exceeds its previous
|
||
dation of these steps is presented in the following paragraphs: best, the global best fitness is updated to the new higher value.
|
||
Step 1: Whale-wolf encoding and position vector initialization.
|
||
Step 3: Updating the position vector.
|
||
The developed technique views tasks as elements within the HWWO
|
||
framework, which systematically refines their designated positions over The proposed HWWO approach updates the position of each whale
|
||
successive iterations, ultimately converging on an optimal task allo- using equations ((2), (8), (10)) under different scenarios, whereas
|
||
cation among the available processors. One of the key obstacles in Eq. (17) facilitates the positional update of the wolf for every iteration,
|
||
|
||
13
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 11. Example demonstrating the insert-reversed block operation.
|
||
|
||
→ → →
|
||
taking into consideration the values of 𝑍 𝛼 , 𝑍 𝛽 , and 𝑍 𝛿 as defined by assignment, carried out between lines 19–21, has a time complexity
|
||
Eqs. (13)–(16). of 𝑂(|𝑌 |). The iterative loop spanning steps 33 to 63 also requires a
|
||
time complexity scaling as 𝑂(|𝑌 | ∗ |𝑋|). As a result, the overall time
|
||
Step 4: Emergence of optimal scheduling. complexity per iteration of the HWWO algorithm is the cumulative sum
|
||
The algorithm models task assignment using whales and wolves. Ap- of these individual complexities, amounting to 𝑂(|𝑋| ∗ |𝑌 |) + 𝑂(|𝑌 |).
|
||
plying mutated and permutated HWWO, particle positions are adjusted To store the assignments of |𝑋| tasks across |𝑌 | processors, a matrix
|
||
for optimal tasks scheduling on processors. With optimal scheduling structure of size |𝑌 | ∗ |𝑋| is required, and an array of length |𝑌 | is
|
||
obtained, the EES algorithm then optimizes voltage and frequency needed to hold the fitness value of each assignment. Consequently,
|
||
distribution among all processors. the overall space complexity for representing and evaluating these
|
||
Step 5: Calculating total energy consumption, and reliability. assignments is 𝑂(|𝑋| ∗ |𝑌 |) + 𝑂(|𝑌 |).
|
||
The aggregate energy consumption of the resulting set of particles Alternatively, the algorithms based on energy considerations, such
|
||
can be determined by utilizing Eq. (25), while their reliability metric as ESRG, EPM, and REREC techniques, exhibit time complexities of
|
||
is ascertained through the application of Eq. (29). 𝑂(|𝑋|2 ∗ |𝑌 |), 𝑂(|𝑋|2 ∗ |𝑌 |3 ), and 𝑂(|𝑋|2 ∗ |𝑌 |), respectively. It is
|
||
The outlined steps should be iteratively executed until the stopping noteworthy that, in terms of time complexity, the proposed algorithm
|
||
criteria or termination condition is met. The following provides a surpasses the performance of the other existing algorithms in this
|
||
comprehensive description of Algorithm 2. domain.
|
||
Fig. 12 visually depicts the scheduling outcomes obtained with the
|
||
proposed HWWO algorithm, and Fig. 13 illustrates the flowchart of 6. Experimental evaluation
|
||
the proposed model. The algorithm effectively reduces total energy
|
||
consumption and enhances reliability. Using Eq. (25), the total energy This study aims to develop an approach that improves system
|
||
consumption 𝐸̂ 𝑡𝑜𝑡𝑎𝑙 (𝐺) is calculated as 75.673 units, with 𝐸̂ 𝑠 (𝐺) being reliability, reduces the overall computation time, and conserves energy
|
||
60.6 units and 𝐸̂ 𝑑 (𝐺) being 15.073 units. Furthermore, to satisfy the usage. The proposed model’s performance in handling tasks scheduling
|
||
reliability goal 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) i.e., 𝑅𝑒(𝑚𝑖𝑛) (𝐺) ≤ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 𝑅𝑒(𝑚𝑎𝑥) (𝐺), the across heterogeneous distributed systems has been rigorously evaluated
|
||
maximum reliability 𝑅𝑒(𝑚𝑎𝑥) (𝐺) and the minimum reliability 𝑅𝑒(𝑚𝑖𝑛) (𝐺) through a comprehensive set of experiments. This section outlines the
|
||
for the DAG are evaluated as 0.99989 and 0.89738 respectively. Here, simulations conducted and the evaluation metrics configured to address
|
||
the reliability goal, 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺), is set at 0.95, and the obtained reliabil- the stated problem using the proposed algorithm. An in-depth analysis
|
||
ity, 𝑅𝑒 (𝐺), stands at 0.978526 > 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺). The improvement is clear of each experiment is provided in the following subsections.
|
||
when contrasted with the energy consumption and reliability results
|
||
shown in Figs. 5–9, representing the HEFT, DECM, EPM, REREC, and 6.1. Experimental metrics
|
||
ESRG algorithms.
|
||
The results for the DAG, as depicted in Fig. 4, clearly highlight the The HWWO algorithm’s efficacy in distributed computing is com-
|
||
superiority of the proposed algorithm in reducing energy consumption prehensively evaluated using multiple performance metrics for an-
|
||
and enhancing system reliability. alytical assessment. These metrics cover total energy consumption,
|
||
system reliability, computation time, resource utilization, SLR, CCR,
|
||
5.2.6. Time and space complexities and sensitivity analysis. This comprehensive assessment aims to provide
|
||
The computational time complexities required by the WOA and the insights into the algorithm’s performance and its potential impact on
|
||
GWO expressed as 𝑂(|𝑋| ∗ |𝑌 |). In the proposed HWWO technique, the distributed computing environment.
|
||
the time taken for steps 14–17, during which the mutation operation With regard to performance evaluation, the proposed algorithm’s
|
||
is performed on each assignment, exhibits an identical complexity of capabilities are benchmarked through two distinct stages. Firstly, a
|
||
𝑂(|𝑋| ∗ |𝑌 |). Moreover, the evaluation of the fitness function for each comparative analysis is conducted between the introduced HWWO
|
||
|
||
14
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 12. Best scheduling result of the DAG shown in Fig. 4 using proposed HWWO technique.
|
||
|
||
|
||
|
||
|
||
Fig. 13. Flowchart of the proposed HWWO model.
|
||
|
||
|
||
technique and several existing state-of-the-art methods, namely HEFT, is carried out across multiple scenarios, generated by varying the
|
||
DECM, EPM, REREC, and ESRG. Additionally, the successful state- number of tasks over different generations, to provide a comprehensive
|
||
of-the-art algorithms from the ‘CEC2020 competition on real-world understanding of the algorithm’s performance under diverse conditions.
|
||
single objective constrained optimization’ – specifically, SASS [94], The assessment of the presented benchmark suite is conducted on a
|
||
sCMAgES [95], EnMODE [96], and COLSHADE [97] – are incorporated personal computer equipped with the Microsoft Windows 11 operating
|
||
as four benchmark algorithms for comparative evaluation in the con- system, featuring an INTEL Core i3 CPU and 8 GB of RAM.
|
||
text of real-world optimization challenges, as outlined in the relevant Stage I
|
||
literature [98]. Furthermore, the proposed methodology is evaluated
|
||
against several metaheuristic approaches, including PSO, GWO, ACO, 6.2. Benchmark analysis with state-of-the-art algorithms
|
||
KH, WOA, DA (dragonfly algorithm) [67], and AHA (artificial hum-
|
||
mingbird algorithm) [67], using various performance metrics. The The subsection delves into an assessment of the proposed algo-
|
||
algorithm’s capabilities are also assessed through a series of benchmark rithm’s effectiveness, drawing comparisons with state-of-the-art meth-
|
||
tests involving unimodal test functions, which are compared against the ods across various performance metrics. The evaluation encompasses
|
||
aforementioned metaheuristic techniques. This comparative analysis three distinct scenarios, wherein the validation process incorporates
|
||
FFT, GE, and constrained optimization challenges from the renowned
|
||
|
||
15
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Algorithm 2 The HWWO tasks scheduling algorithm
|
||
52: assignments[𝑖] = inversion_mutation(assignments[𝑖], 0.2);
|
||
Input:
|
||
53: assignments[𝑖] = insert_reversed_block_operation
|
||
Dataset 𝑊̂ 𝐶𝐸𝑇
|
||
No. of processors, |𝑌 | (assignments[𝑖]);
|
||
No. of tasks, |𝑋| 54: end( for )
|
||
𝑓𝑓 𝑛 (𝑟)−𝑓𝑓 𝑛 (𝑟−1)
|
||
Population size, |𝑌 | 55: if < 𝜖𝑓 (= 0.001) then
|
||
→ 𝑓𝑓 𝑛 (𝑟)
|
||
Control coefficient, 𝜇
|
||
56: 𝑟 = (max_iteration);
|
||
Maximum no. of iterations, 𝑟
|
||
57: else
|
||
Output:
|
||
Global best solution (best tasks assignment) 58: 𝑟 = 𝑟 + 1;
|
||
Begin 59: end if
|
||
1: matrix assignments[|𝑌 |][|𝑋|] = randomly assign |𝑋| tasks in the 60: for 𝑖 = 0 to |𝑌 | − 1 do
|
||
processors prioritizing on the basis of rank; 61: fitness[𝑖] = fitness(assignments[𝑖]);
|
||
2: tasks = order by (rank, decreasing); 62: end for
|
||
3: function fitness(assignment)[ ( )] 63: end while
|
||
𝑤̂ ∗𝑓
|
||
∑|𝑌 | ∑|𝑋| −𝜆𝑙,𝑚𝑎𝑥 ∗ 𝑖,𝑙 𝑓𝑙,ℎ𝑙,𝑚𝑎𝑥 64: best_assignment = assignment with least fitness value;
|
||
4: 𝑓𝑓 𝑛(𝜏𝑖 ,𝑌𝑙 ,𝑓𝑙,ℎ ) = 𝑚𝑖𝑛 𝜎1 ∗ 𝑙=1 𝑃𝑙,𝑠 ∗ 𝑀𝑆(𝐺)
|
||
𝐷𝐿(𝐺)
|
||
+ 𝜎 2 ∗ 𝑖=1
|
||
𝑒 ;
|
||
65: Implement the EES algorithm to optimize voltage and frequency
|
||
5: return 𝑓𝑓 𝑛(𝜏𝑖 ,𝑌𝑙 ,𝑓𝑙,ℎ ) ; distribution among all processors
|
||
6: end function
|
||
7: function inversion_mutation(assignment, mutation_rate)
|
||
8: if 𝑟𝑎𝑛𝑑𝑜𝑚(0, 1) < 𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛_𝑟𝑎𝑡𝑒 then Table 6
|
||
9: 𝑖, 𝑗 = sorted(random(|𝑋|, 2)); Parameter setting for stage I.
|
||
10: assignment[𝑖: 𝑗 + 1] = assignment[𝑖: 𝑗 + 1][:: -1]; Parameter Values
|
||
11: end if
|
||
𝑤̂ 𝑖,𝑙 (𝑚𝑠) [10, 100]
|
||
12: return assignment; 𝑐̂𝑖,𝑘 (𝑚𝑠) [10, 100]
|
||
13: end function 𝑃𝑙,𝑠 [0.1, 0.5]
|
||
14: for 𝑖 = 0 to |𝑌 | − 1 do 𝑃𝑙,𝑖𝑛𝑑 [0.03, 0.07]
|
||
15: assignments[𝑖] = random assignment to processors (tasks); 𝐶𝑙,𝑒𝑓 [0.8, 1.2]
|
||
16: assignments[𝑖] = inversion_mutation(assignments[𝑖], 0.2); 𝑚𝑙 [2.5, 3.0]
|
||
17: end for 𝑓𝑙,𝑚𝑎𝑥 1 GHz
|
||
18: arr fitness_values[𝑌 ]; 𝑙1 and 𝑙2 [0, 1]
|
||
→
|
||
19: for 𝑖 = 0 to |𝑌 | − 1 do 𝜇 [0, 2]
|
||
𝜆𝑙,𝑚𝑎𝑥 [0.0003, 0.0009]
|
||
20: fitness_values[𝑖] = fitness(assignments[𝑖]);
|
||
𝐶𝐶𝑅 0.1, 0.5, 1, 5, 10
|
||
21: end for
|
||
→
|
||
22: 𝑍[0] = alpha assignment;
|
||
→
|
||
23: 𝑍[1] = beta assignment;
|
||
→
|
||
CEC2020 competition. Complementing these scenarios, a diverse array
|
||
24: 𝑍[3] = delta assignment;
|
||
→ → → of experiments centered around tasks scheduling in a multiprocess-
|
||
25: function wolf(𝑀 ′ , 𝜇, 𝜉 , 𝜁 )
|
||
ing environment is conducted. The validation outcomes underscore
|
||
26: matrix wolf_particles[3][|𝑋|];
|
||
the algorithm’s efficacy. It is noteworthy that each algorithm un-
|
||
27: for 𝑖 = 0 to 2 do
|
||
→ → dergoes independent iterations, refining its functions in pursuit of
|
||
28: 𝛶 = | 𝜉 ∗ 𝑍[𝑖] − 𝑀 ′ |;
|
||
→ → → optimal performance. The comprehensive evaluation aims to under-
|
||
29: 𝑤𝑜𝑙𝑓 _𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠[𝑖] = 𝑍[𝑖] − 𝜁 ∗ 𝛶 ;
|
||
stand the algorithm’s capabilities in addressing real-world optimization
|
||
30: end for
|
||
and scheduling challenges. The validation of the algorithm is facili-
|
||
31: return 𝑤𝑜𝑙𝑓 _𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠[0]+𝑤𝑜𝑙𝑓 _𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠[1]+𝑤𝑜𝑙𝑓 _𝑝𝑎𝑟𝑡𝑖𝑐𝑙𝑒𝑠[2]
|
||
;
|
||
3 tated through simulations conducted within a replicated heterogeneous
|
||
32: end function
|
||
33: while 𝑟 < max_iteration do distributed embedded system environment, comprising 95 processors
|
||
→
|
||
( )
|
||
34: 2
|
||
𝜇 = 2 − 𝑟 ∗ 𝑚𝑎𝑥_𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 ; capable of handling tasks of varying complexities. These processors
|
||
35: 𝑏 = random constant; exhibit diverse processing capabilities, with their specifications and
|
||
36: update best assignments on the basis of least fitness value; the corresponding application parameters closely mirroring the details
|
||
37: for 𝑖 = 0 to |𝑌 | − 1 do outlined in Refs. [8,91]. The input data parameters employed in this
|
||
→ →
|
||
38: update 𝑟, 𝜁 , 𝜉 , 𝜈, p; stage are delineated in Table 6, wherein each frequency magnitude un-
|
||
39: if 𝑝 < 0.5 then dergoes discretization and is represented with a precision of 0.01 GHz.
|
||
→ →
|
||
40: 𝐷 = | 𝜉 ∗ 𝑏𝑒𝑠𝑡_𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡 − 𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠[𝑖]|;
|
||
Following the simulations, a comprehensive evaluation is undertaken,
|
||
→ wherein various assessment metrics are computed, including the av-
|
||
41: if | 𝜁 | < 1 then
|
||
→ → erage execution time, as well as the standard deviation and mean
|
||
42: assignments[𝑖]=|𝑏𝑒𝑠𝑡_𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡 − 𝜁 ∗ 𝐷|; of solutions across the iterative process, as described in [98]. This
|
||
43: else
|
||
evaluation replicates real-world heterogeneous distributed embedded
|
||
44: 𝑗 = random(0, |𝑌 |);
|
||
→ → systems, providing insights into the algorithm’s performance.
|
||
45: 𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠[𝑖] = 𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠[𝑗] − 𝜁 ∗ 𝐷;
|
||
46: end if
|
||
47: else 6.2.1. Scenario 1
|
||
→
|
||
48: 𝐷′ = |𝑏𝑒𝑠𝑡_𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡 − 𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠[𝑖]|; This scenario conducts a rigorous assessment of the efficacy of the
|
||
→
|
||
49: 𝑀 ′ = 𝐷′ ∗ 𝑒𝑏𝜈 ∗ 𝑐𝑜𝑠(2𝜋𝜈) + 𝑎𝑠𝑠𝑖𝑔𝑛𝑚𝑒𝑛𝑡𝑠[𝑖]; HWWO-based technique through an examination of 15 real-world op-
|
||
→ → →
|
||
50: assignments[𝑖] = wolf(𝑀 ′ , 𝜇, 𝜁 , 𝜉 ); timization problems. The assessment utilizes four algorithms identified
|
||
51: end if in the ‘CEC2020 competition on real-world single objective constrained
|
||
optimization’, namely the SASS algorithm, the sCMAgES algorithm,
|
||
the EnMODE, and the COLSHADE algorithm. The article leverages
|
||
these algorithms to evaluate the performance of the HWWO-based
|
||
|
||
16
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 7
|
||
15 real-world constrained optimization problems.
|
||
Problem Name D g h
|
||
𝐶𝑜𝑝𝑡1 Optimal power flow (minimization of active power loss) 126 0 116
|
||
𝐶𝑜𝑝𝑡2 Topology optimization 30 30 0
|
||
𝐶𝑜𝑝𝑡3 Process flow sheeting problem 3 3 0
|
||
𝐶𝑜𝑝𝑡4 Gas transmission compressor design (GTCD) 4 1 0
|
||
𝐶𝑜𝑝𝑡5 SOPWM for 3-level inverters 25 24 1
|
||
𝐶𝑜𝑝𝑡6 Optimal power flow (minimization of fuel cost) 126 0 116
|
||
𝐶𝑜𝑝𝑡7 Optimal power flow (minimization of active power loss and fuel cost) 126 0 116
|
||
𝐶𝑜𝑝𝑡8 SOPWM for 5-level inverters 25 24 1
|
||
𝐶𝑜𝑝𝑡9 Pressure vessel design 4 4 0
|
||
𝐶𝑜𝑝𝑡10 Optimal sizing of distributed generation for active power loss minimization 153 0 148
|
||
𝐶𝑜𝑝𝑡11 Wind farm layout problem 30 91 0
|
||
𝐶𝑜𝑝𝑡12 Microgrid power flow (islanded case) 76 0 76
|
||
𝐶𝑜𝑝𝑡13 Optimal setting of droop controller for minimization of active power loss in islanded microgrids 86 0 76
|
||
𝐶𝑜𝑝𝑡14 Microgrid power flow (grid-connected case) 74 0 74
|
||
𝐶𝑜𝑝𝑡15 Optimal setting of droop controller for minimization of reactive power loss in islanded microgrids 86 0 76
|
||
|
||
|
||
|
||
technique on the selected real-world optimization problems. The ar-
|
||
ticle presents a detailed description of the attributes that define the
|
||
real-world problems under investigation, including their dimensions
|
||
and the number of equality and inequality constraints involved. This
|
||
information is comprehensively outlined in Table 7. To ensure a fair
|
||
and consistent comparison, the parameters of the four algorithms are
|
||
maintained at their original settings as documented in the relevant
|
||
literature [94–97]. To ensure an impartial evaluation, the proposed
|
||
algorithm is executed for 500 iterations, adhering to the guidelines
|
||
outlined in [98]. This ensures an equal number of function evaluations
|
||
across methods. Subsequently, a statistical analysis is conducted using
|
||
the Wilcoxon signed-rank test [99] to assess the HWWO’s performance
|
||
relative to the other algorithms under consideration.
|
||
|
||
The simulation outcomes for 15 real-world challenges are show-
|
||
cased in Table 8, which provides comprehensive information regarding
|
||
the best fitness value, mean fitness value, worst fitness value, and the
|
||
standard deviation (St. Dev) of the fitness values.
|
||
|
||
The results displayed in Table 8 demonstrate the proficiency of the
|
||
HWWO algorithm in tackling the majority of these problems, exhibiting
|
||
commendable performance. Notably, some of the solutions obtained
|
||
Fig. 14. DAG of FFT application with 𝜌 = 4.
|
||
by HWWO surpass those achieved by competing algorithms. To facil-
|
||
itate a comprehensive comparison of algorithm performance on the
|
||
proposed benchmark suite, we have adopted the ranking methodology
|
||
outlined in the CEC2020 competition [98]. The evaluation process benchmarks (𝑝 < 0.05). These findings suggest that the HWWO algo-
|
||
assigns weighted scores to the best, mean, and median results obtained rithm provides faster convergence and improved accuracy in solving
|
||
from 25 independent runs of each algorithm to quantify their perfor- optimization problems, which could translate into better outcomes in
|
||
mance, as outlined in [98]. The weighted performance measures (PM) practical applications such as logistics and scheduling optimization.
|
||
are: HWWO (0.321089, rank 1), SASS (0.335719, rank 2), EnMODE
|
||
(0.351856, rank 3), sCMAgES (0.415387, rank 4), and COLSHADE 6.2.2. Scenario 2
|
||
(0.493992, rank 5). Outperforming the others, HWWO demonstrates its In this scenario, the proposed approach rigorously evaluates the
|
||
effectiveness in handling diverse real-world problems with acceptable effectiveness of the HWWO technique through an analysis of the FFT
|
||
performance. algorithm. The visual depiction, illustrated in Fig. 14, showcases a
|
||
parallel implementation of the FFT application [8,86], incorporating
|
||
The results of the Wilcoxon signed-rank test, presented in Table
|
||
a crucial parameter value of 𝜌 = 4. The parameter ′ 𝜌′ governs the
|
||
9, highlight the effectiveness of the proposed HWWO algorithm when
|
||
application’s task count |𝑋| via |𝑋| = (2 ∗ 𝜌 − 1) + 𝜌 ∗ 𝑙𝑜𝑔2 (𝜌), and
|
||
compared to other methods. The table shows the ranks of the HWWO
|
||
is exponentially related to an integer ′ 𝑦′ through 𝜌 = 2𝑦 . As illustrated
|
||
algorithm relative to the second algorithm in terms of best fitness
|
||
in Fig. 14, the application achieves a task count of |𝑋| = 15, which
|
||
values, with 𝑇 + , 𝑇 − , and 𝑇 indicating the statistical results. 𝑇 + repre-
|
||
occurs when 𝜌 is set to 4. The following are three experiments evaluated
|
||
sents the superiority of the HWWO algorithm. The p-values, calculated
|
||
utilizing the FFT application.
|
||
at a 5% significance level, test the null hypothesis that the median
|
||
difference between the algorithms is zero. Additionally, the final row of Experiment 1: This experimental work aims to conduct a comprehen-
|
||
Table 9 summarizes the counts of 𝑇 + and 𝑇 − , as well as the test statistic, sive comparative evaluation of the proposed HWWO technique against
|
||
offering a clear overview of the results. The Wilcoxon signed-rank test is existing algorithms. The primary focus is to assess their respective
|
||
particularly suited for paired comparisons in situations where normality performances concerning total energy consumption and computational
|
||
cannot be assumed, making it a reliable tool for validating performance time requirements within the context of parallel applications involving
|
||
differences in real-world optimization problems. The results indicate FFT. This experimental study employs a rigorous deadline and reliabil-
|
||
significant improvements in the performance of the HWWO algorithm, ity constraints, expressed as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4 and 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) =
|
||
with 𝑇 + accounting for 86.96% and 𝑇 − for 13.03% of the evaluated 0.90 respectively, where the complexity of the parallel application is
|
||
|
||
17
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
Table 8
|
||
Results of 15 real-world constrained optimization problems.
|
||
Problem Performance Optimization Algorithm
|
||
metric SASS COLSHADE EnMODE sCMAgES HWWO
|
||
𝐶𝑜𝑝𝑡1 Best fitness 5.13E+4 7.08E+4 7.10E+4 7.12E+4 5.19E+4
|
||
Mean fitness 5.33E+4 7.08E+4 7.22E+4 7.22E+4 5.37E+4
|
||
Worst fitness 5.51E+4 7.08E+4 7.61E+4 7.71E+4 5.66E+4
|
||
St. Dev 1.07E+2 2.12E−12 1.14E+2 2.14E+2 1.15E+2
|
||
𝐶𝑜𝑝𝑡2 Best fitness 3.06E+4 3.06E+4 3.06E+04 3.06E+04 3.05E+4
|
||
Mean fitness 3.06E+4 3.06E+4 3.08E+04 3.08E+04 3.06E+4
|
||
Worst fitness 3.06E+4 3.07E+4 3.08E+04 3.08E+04 3.06E+4
|
||
St. Dev 3.07E−11 2.51E−7 2.53E+1 2.51E+1 7.46E−1
|
||
𝐶𝑜𝑝𝑡3 Best fitness 3.68E+04 3.67E+4 3.67E+4 3.67E+4 3.63E+4
|
||
Mean fitness 3.68E+04 3.69E+4 3.67E+4 3.67E+4 3.63E+4
|
||
Worst fitness 3.68E+04 3.69E+4 3.67E+4 3.67E+4 3.63E+4
|
||
St. Dev 1.69E−15 7.51E+1 4.53E−15 4.53E−15 1.66E−16
|
||
𝐶𝑜𝑝𝑡4 Best fitness 5.29E+05 5.46E+05 5.32E+05 5.45E+05 5.26E+05
|
||
Mean fitness 5.29E+05 5.46E+05 5.35E+05 5.45E+05 5.28E+05
|
||
Worst fitness 5.31E+05 5.48E+05 5.36E+05 5.48E+05 5.28E+05
|
||
St. Dev 7.13E−4 2.36E−4 1.66E+2 3.31E−04 1.16E−1
|
||
𝐶𝑜𝑝𝑡5 Best fitness 1.60E+6 1.67E+06 1.34E+06 1.34E+6 1.34E+06
|
||
Mean fitness 1.67E+6 1.67E+06 1.34E+06 1.38E+6 1.34E+06
|
||
Worst fitness 1.67E+6 1.67E+06 1.36E+06 1.38E+6 1.36E+06
|
||
St. Dev 2.06E+4 1.05E−09 3.66E−7 5.14E+2 3.66E−7
|
||
𝐶𝑜𝑝𝑡6 Best fitness 6.85E+07 6.91E+7 6.85E+07 6.85E+07 6.85E+07
|
||
Mean fitness 6.87E+07 6.91E+7 6.87E+07 6.89E+07 6.87E+07
|
||
Worst fitness 6.89E+07 6.93E+7 6.89E+07 6.89E+07 6.89E+07
|
||
St. Dev 2.27E+04 6.61E−04 2.27E+04 5.42E+04 2.27E+04
|
||
𝐶𝑜𝑝𝑡7 Best fitness 5.73E+6 5.76E+6 5.73E+6 5.77E+6 5.76E+6
|
||
Mean fitness 5.74E+6 5.79E+6 5.74E+6 5.79E+6 5.77E+6
|
||
Worst fitness 5.76E+6 5.79E+6 5.76E+6 5.79E+6 5.77E+6
|
||
St. Dev 1.05E+4 6.05E+4 1.05E+4 3.08E+4 5.05E+3
|
||
𝐶𝑜𝑝𝑡8 Best fitness 9.93E−4 9.97E−04 8.92E−4 8.92E−4 8.92E−4
|
||
Mean fitness 9.95E−4 9.97E−04 8.92E−4 8.92E−4 8.95E−4
|
||
Worst fitness 9.96E−4 9.97E−04 8.96E−4 8.93E−4 8.95E−4
|
||
St. Dev 4.34E−6 5.41E−06 6.63E−4 4.31E−6 4.31E−6
|
||
𝐶𝑜𝑝𝑡9 Best fitness 1.89E+2 4.08E+03 4.08E+03 4.18E+03 1.89E+2
|
||
Mean fitness 1.91E+2 4.25E+03 4.31E+03 4.26E+03 1.91E+2
|
||
Worst fitness 1.99E+2 4.37E+03 4.37E+03 4.31E+03 1.99E+2
|
||
St. Dev 2.80E−1 8.85E+01 5.85E+01 1.55E+01 2.80E−1
|
||
𝐶𝑜𝑝𝑡10 Best fitness 6.16E−02 6.74E−02 6.26E−02 3.26E−02 3.26E−02
|
||
Mean fitness 6.96E−02 7.85E−02 7.59E−02 6.96E−02 6.96E−02
|
||
Worst fitness 7.92E−02 9.04E−02 9.23E−02 7.32E−02 7.32E−02
|
||
St. Dev 5.28E−02 5.26E−02 4.49E−05 4.28E−05 4.28E−05
|
||
𝐶𝑜𝑝𝑡11 Best fitness 3.06E+4 3.06E+4 3.11E+4 3.02E+4 2.94E+4
|
||
Mean fitness 3.08E+4 3.09E+4 3.11E+4 3.07E+4 2.94E+4
|
||
Worst fitness 3.11E+4 3.13E+4 3.13E+4 3.07E+4 2.94E+4
|
||
St. Dev 7.12E+1 2.61E+1 4.64E−4 4.64E+1 0
|
||
𝐶𝑜𝑝𝑡12 Best fitness 1.67E+1 1.67E+1 1.68E+1 1.72E+1 1.70E+1
|
||
Mean fitness 1.67E+1 1.68E+1 1.68E+1 1.74E+1 1.73E+1
|
||
Worst fitness 1.69E+1 1.69E+1 1.71E+1 1.74E+1 1.73E+1
|
||
St. Dev 2.03E−1 2.03E−1 3.08E−1 4.13E+1 2.03E−1
|
||
𝐶𝑜𝑝𝑡13 Best fitness 5.75E+2 5.71E+2 5.24E+2 5.57E+2 5.24E+2
|
||
Mean fitness 5.78E+2 5.79E+2 5.29E+2 5.59E+2 5.29E+2
|
||
Worst fitness 5.79E+2 5.79E+2 5.33E+2 5.61E+2 5.33E+2
|
||
St. Dev 2.51E+1 4.43E+1 1.61E+1 7.01E+1 1.61E+1
|
||
𝐶𝑜𝑝𝑡14 Best fitness 3.55E+2 3.62E+2 3.60E+2 3.55E+2 3.53E+2
|
||
Mean fitness 3.74E+2 3.78E+2 4.45E+2 3.61E+2 3.61E+2
|
||
Worst fitness 3.79E+2 3.79E+2 4.71E+2 3.66E+2 3.63E+2
|
||
St. Dev 8.01E+2 4.23E+2 7.32E+2 3.39E+1 3.37E+1
|
||
𝐶𝑜𝑝𝑡15 Best fitness 1.93E+5 1.89E+5 1.91E+5 1.89E+5 1.89E+5
|
||
Mean fitness 1.95E+5 1.91E+5 1.96E+5 1.89E+5 1.92E+5
|
||
Worst fitness 1.99E+5 1.97E+5 1.96E+5 1.89E+5 1.97E+5
|
||
St. Dev 1.32E+3 4.15E+3 5.80E+1 2.97E−11 3.35E+1
|
||
|
||
|
||
|
||
intrinsically linked to the quantity of constituent tasks it comprises. The the HWWO algorithm in comparison to other existing algorithms. As
|
||
study deliberately varies |𝑋| from 95 (smaller scenarios) to 2559 (larger the task count rises, both the DECM and REREC algorithms yield
|
||
scenarios), while concurrently investigating the effects of 𝜌 ranging comparable performance levels. Notably, up to a task value of 511, the
|
||
from 16 to 256. ESRG algorithm stands out for its lower energy consumption compared
|
||
to EPM. Beyond this threshold, EPM gradually refines its outcomes,
|
||
Tables 10 and 11 present the outcomes from using FFT applications
|
||
albeit at the expense of higher energy usage in contrast to DECM
|
||
with varying 𝜌 values. In all experiments, HEFT (without DVFS tech-
|
||
and REREC. The best outcomes, highlighted in bold text, are further
|
||
nique) consistently consumes more energy. In Table 10, the parameter
|
||
illustrated in Fig. 15, which visually represents the data from Table 10,
|
||
|𝑋| demonstrates a spectrum of values ranging from 95 to 2559. This
|
||
offering a comprehensive comparative analysis.
|
||
underscores the superior energy consumption outcomes achieved by
|
||
|
||
18
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 9
|
||
Results of Wilcoxon signed-rank test of Table 8.
|
||
Problem HWWO vs SASS HWWO vs COLSHADE HWWO vs EnMODE HWWO vs sCMAgES
|
||
Rank Rank Rank Rank
|
||
𝐶𝑜𝑝𝑡1 7 4 3 4
|
||
𝐶𝑜𝑝𝑡2 2 1 1 1.5
|
||
𝐶𝑜𝑝𝑡3 6 10 9 9
|
||
𝐶𝑜𝑝𝑡4 4.5 5 10 3
|
||
𝐶𝑜𝑝𝑡5 10 7 – –
|
||
𝐶𝑜𝑝𝑡6 – 12 – –
|
||
𝐶𝑜𝑝𝑡7 4.5 – 6.5 1.5
|
||
𝐶𝑜𝑝𝑡8 1 2 – –
|
||
𝐶𝑜𝑝𝑡9 – 9 8 8
|
||
𝐶𝑜𝑝𝑡10 3 8 6.5 –
|
||
𝐶𝑜𝑝𝑡11 8 3 2 10
|
||
𝐶𝑜𝑝𝑡12 11 6 4.5 5.5
|
||
𝐶𝑜𝑝𝑡13 13 11 7
|
||
𝐶𝑜𝑝𝑡14 9 13 11 5.5
|
||
𝐶𝑜𝑝𝑡15 12 – 4.5 –
|
||
p-value 0.0116 0.0452 0.0370 0.0352
|
||
𝑇 + = Sum of positive number ranks 68.5 85 55 55
|
||
𝑇 − = Sum of negative number ranks 22.5 6 11 0
|
||
𝑇 = min(𝑇 + , 𝑇 − ) 22.5 6 11 0
|
||
|
||
|
||
Table 10
|
||
Energy consumption analysis for FFT parallel applications across task configurations.
|
||
|𝑋| Performance Algorithm
|
||
metric HEFT DECM EPM REREC ESRG HWWO
|
||
95 Best 1.48E+4 2.73E+3 6.83E+3 3.07E+3 6.81E+3 1.68E+3
|
||
Mean 1.53E+4 2.80E+3 6.90E+3 3.26E+3 6.89E+3 1.72E+3
|
||
Worst 1.58E+4 2.83E+3 6.94E+3 3.33E+3 6.94E+3 1.83E+3
|
||
St. Dev 3.17E+3 5.78E+1 7.31E+1 4.13E+1 7.23E+1 3.14E+1
|
||
223 Best 2.53E+4 7.13E+3 8.43E+3 7.44E+3 8.39E+3 4.11E+3
|
||
Mean 2.61E+4 7.21E+3 8.47E+3 7.49E+3 8.47E+3 4.18E+3
|
||
Worst 2.67E+4 7.33E+3 8.56E+3 7.57E+3 8.54E+3 4.23E+3
|
||
St. Dev 6.20E+1 1.28E+1 6.16E+1 2.69E+2 3.80E+3 1.02E+1
|
||
511 Best 3.64E+4 1.32E+4 2.18E+4 1.27E+4 2.06E+4 6.58E+3
|
||
Mean 3.72E+4 1.32E+4 2.27E+4 1.35E+4 2.17E+4 6.61E+3
|
||
Worst 3.72E+4 1.32E+4 2.37E+4 1.46E+4 2.29E+4 6.77E+3
|
||
St. Dev 1.28E+1 3.47E−7 5.34E+1 7.28E+2 4.21E+2 5.54E+1
|
||
1151 Best 7.34E+4 3.26E+4 4.73E+4 3.26E+4 4.79E+4 9.75E+3
|
||
Mean 7.41E+4 3.35E+4 4.81E+4 3.37E+4 4.88E+4 9.79E+3
|
||
Worst 7.49E+4 3.43E+4 4.83E+4 3.43E+4 4.97E+4 9.79E+3
|
||
St. Dev 4.13E+4 2.13E+1 7.34E+2 5.81E+3 7.03E+2 2.34E+1
|
||
2559 Best 9.35E+4 6.17E+4 6.44E+4 6.21E+4 6.71E+4 4.73E+4
|
||
Mean 9.43E+4 6.28E+4 6.51E+4 6.37E+4 6.82E+4 4.86E+4
|
||
Worst 9.51E+4 6.39E+4 6.59E+4 6.47E+4 6.88E+4 4.93E+4
|
||
St. Dev 5.82E+4 3.72E+2 1.80E+2 4.63E+2 1.80E+2 7.37E+2
|
||
|
||
|
||
|
||
Within Table 11, it is notable that the EPM algorithm requires the specified reliability goals compared to other existing methods.
|
||
significantly more 𝐶𝑇𝑇 𝐴 . However, as the number of tasks increases, Contrastingly, the HEFT, EPM, and ESRG algorithms exhibit an inability
|
||
ESRG surpasses the other three algorithms in producing higher energy to fulfill the reliability constraints in the majority of scenarios. As
|
||
values. The 𝐶𝑇𝑇 𝐴 of the newly proposed HWWO algorithm is projected the reliability objective escalates from 0.91 to 0.95, HEFT, EPM, and
|
||
to occupy between 31.20% and 35.61% of the computational time ESRG manage to comply with the requirements, but struggle beyond
|
||
required by the DECM and REREC algorithms. Regarding performance that range. Conversely, DECM, REREC, and HWWO successfully meet
|
||
metrics, across a spectrum of values for |𝑋| from 95 to 2559, the 𝐶𝑇𝑇 𝐴 the reliability constraint within the range of 0.91 to 0.98, although
|
||
of the HWWO closely mirrors that of the DECM algorithm, consistently none of the algorithms can fulfill the rigorous 0.99 requirement. It
|
||
surpassing it. is noteworthy that if the upper bound for the reliability objective is
|
||
established at an excessively elevated level, the maximum attainable
|
||
Experiment 2: The study evaluates the reliability metrics and total reliability values for partial tasks may fall short of this upper bound in
|
||
energy consumption of an extensive FFT application under varying practical implementation scenarios.
|
||
reliability constraints. The experimental configuration involves 1151 Data presented in Table 12 has been shown graphically in Fig. 17.
|
||
tasks, with 𝜌 = 128. Additionally, the reliability goal, 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺), is The table evaluates the energy consumption profiles of FFT applications
|
||
systematically varied from 0.91 to 0.99 in increments of 0.01, enabling when subjected to varying reliability criteria. For reliability thresholds
|
||
an assessment of the corresponding effects on reliability performance up to 0.98, the techniques DECM, REREC, and HWWO exhibit superior
|
||
and energy utilization. energy consumption performance in comparison to HEFT, EPM, and
|
||
The graphical representation in Fig. 16 illustrates the actual relia- ESRG. The algorithms HEFT, EPM, and ESRG are capable of producing
|
||
bility values attained by the large-scale FFT application when subjected energy outcomes only up to a reliability criterion of 0.95, as they fail
|
||
to varying reliability criteria. Among the techniques evaluated, the to meet the reliability constraints beyond this point, as evidenced by
|
||
HWWO algorithm demonstrates superior performance in accomplishing the findings illustrated in Fig. 16. Until the 0.98 reliability threshold,
|
||
|
||
19
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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|
||
|
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|
||
|
||
Fig. 15. Graphical representation of Table 10.
|
||
|
||
|
||
|
||
|
||
Table 11
|
||
𝐶𝑇𝑇 𝐴 values of FFT applications across diverse task quantities.
|
||
|𝑋| Performance Algorithm
|
||
metric DECM EPM REREC ESRG HWWO
|
||
95 Best 1.5E+1 2.57E+2 1.9E+1 2.31E+2 1.2E+1
|
||
Mean 2.62E+1 2.73E+2 2.6E+1 2.66E+2 2.2E+1
|
||
Worst 2.80E+1 3.27E+2 2.91E+1 3.25E+2 2.6E+1
|
||
St. Dev 3.15E+1 1.19E+2 5.4E+1 2.4E+2 4.8E+0
|
||
223 Best 2.1E+1 5.04E+2 3.9E+1 4.6E+2 2.1E+1
|
||
Mean 2.7E+1 5.73E+2 4.3E+1 4.8E+2 2.7E+1
|
||
Worst 3.8E+1 5.91E+2 4.3E+1 5.3E+2 3.1E+1
|
||
St. Dev 3.42E−1 4.21E+2 2.06E+0 3.07E+2 3.42E+0
|
||
511 Best 5.6E+1 3.5E+3 6.3E+1 3.2E+3 4.7E+1
|
||
Mean 5.81E+1 3.6E+3 6.8E+1 3.4E+3 4.7E+1
|
||
Worst 6.04E+1 3.6E+3 7.3E+1 3.5E+3 4.92E+1
|
||
St. Dev 3.66E+1 3.01E−1 7.2E+1 6.7E+0 8.9E−1
|
||
1151 Best 2.62E+2 4.16E+3 3.31E+2 4.73E+3 2.49E+2
|
||
Mean 2.74E+2 4.16E+3 3.31E+2 4.73E+3 2.51E+2
|
||
Worst 2.74E+2 4.47E+3 3.47E+2 4.73E+3 2.51E+2
|
||
St. Dev 3.31E−1 1.71E+3 1.08E−1 3.26E−1 3.31E−1
|
||
2559 Best 4.7E+2 8.33E+3 5.4E+2 8.72E+3 3.48E+2
|
||
Mean 4.8E+2 8.42E+3 5.6E+2 8.74E+3 3.71E+2
|
||
Worst 4.8E+2 8.46E+3 5.6E+2 8.83E+3 3.71E+2
|
||
St. Dev 1.07E+1 3.72E+1 1.07E+1 6.01E+2 7.06E+2
|
||
|
||
|
||
|
||
DECM, REREC, and HWWO successfully fulfill the reliability constraints larger-scale scenarios), while simultaneously exploring the impacts of
|
||
in the majority of scenarios. Notably, among these three techniques, 𝜌 ranging from 16 to 256.
|
||
the HWWO algorithm demonstrates more favorable results by further The scheduling length ratio (SLR) is a widely adopted metric em-
|
||
optimizing energy consumption through an expanded exploration of ployed for evaluating and contrasting various scheduling algorithms. It
|
||
processor and frequency combination possibilities. HWWO surpasses is quantified as the ratio of the makespan to the cumulative sum of the
|
||
DECM and REREC in energy savings, reducing consumption by 33% minimum execution times of all tasks residing on the critical path of the
|
||
and 36% on average, correspondingly. However, it is pertinent to note DAG [86]. This can be expressed through the following mathematical
|
||
that none of the algorithms evaluated can achieve the stringent 0.99 formulation:
|
||
reliability requirement. 𝑀𝑆(𝐺)
|
||
𝑆𝐿𝑅 = ∑ (44)
|
||
𝜏𝑖 ∈𝐶𝑃𝑀𝐼𝑁 𝑚𝑖𝑛𝑌𝑙 ∈𝑌 (𝑤
|
||
̂ 𝑖,𝑙 )
|
||
Experiment 3: The current experiment examines the SLR and CCR
|
||
metrics for a comprehensive FFT application, considering variations The data presented in Table 13 and visually represented in Fig.
|
||
in 𝜌. This experimental approach incorporates a stringent deadline 18 demonstrates the average performance of various tasks scheduling
|
||
requirement, expressed as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4, where the com- algorithms in terms of the SLR metric. The proposed HWWO algorithm
|
||
plexity of the parallel application is inherently tied to the number of exhibited the lowest SLR values across all experiments, outperforming
|
||
constituent tasks it encompasses. The study systematically alters |𝑋| the other techniques evaluated. Concerning SLR, HWWO established
|
||
from 95 (representing smaller-scale scenarios) to 2559 (representing itself as the superior approach. Across all task sizes, the HEFT algorithm
|
||
|
||
20
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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|
||
|
||
|
||
|
||
Fig. 16. Graphical representation of actual reliability values under varying reliability constraints (in case of FFT).
|
||
|
||
|
||
Table 12
|
||
Energy consumption assessment for FFT applications with varying reliability criteria.
|
||
Reliability Performance Algorithm
|
||
goal metric HEFT DECM EPM REREC ESRG HWWO
|
||
0.91 Best 4.44E+4 2.36E+4 3.73E+4 2.46E+4 3.53E+4 1.25E+4
|
||
Mean 4.51E+4 2.45E+4 3.81E+4 2.47E+4 3.71E+4 1.29E+4
|
||
Worst 4.58E+4 2.47E+4 3.83E+4 2.53E+4 3.83E+4 1.32E+4
|
||
St. Dev 4.23E+1 4.13E−2 5.24E+2 5.81E−2 6.34E+2 5.24E−4
|
||
0.92 Best 4.74E+4 2.43E+4 3.89E+4 2.54E+4 3.83E+4 1.71E+4
|
||
Mean 4.81E+4 2.47E+4 3.91E+4 2.54E+4 3.86E+4 1.78E+4
|
||
Worst 4.92E+4 2.47E+4 3.91E+4 2.54E+4 3.86E+4 1.78E+4
|
||
St. Dev 7.32E+4 6.28E−11 7.04E−4 6.28E−11 5.34E−4 4.02E−7
|
||
0.93 Best 5.64E+4 3.71E+4 4.28E+4 3.71E+4 4.06E+4 2.71E+4
|
||
Mean 5.72E+4 3.78E+4 4.33E+4 3.85E+4 4.37E+4 2.88E+4
|
||
Worst 5.82E+4 3.88E+4 4.47E+4 3.92E+4 4.49E+4 2.94E+4
|
||
St. Dev 3.28E+4 7.12E+4 2.02E+4 5.08E+4 2.02E+4 2.02E+4
|
||
0.94 Best 6.74E+4 3.87E+4 4.89E+4 3.81E+4 4.82E+4 2.87E+4
|
||
Mean 6.74E+4 3.93E+4 4.93E+4 3.95E+4 4.89E+4 2.97E+4
|
||
Worst 6.74E+4 3.99E+4 4.97E+4 3.99E+4 4.97E+4 2.99E+4
|
||
St. Dev 3.28E+4 1.12E+4 7.02E+4 1.12E+4 5.52E+4 6.42E+4
|
||
0.95 Best 8.45E+4 6.17E+4 6.84E+4 6.37E+4 6.71E+4 4.24E+4
|
||
Mean 8.48E+4 6.28E+4 6.88E+4 6.37E+4 6.82E+4 4.46E+4
|
||
Worst 8.51E+4 6.39E+4 6.95E+4 6.37E+4 6.88E+4 4.53E+4
|
||
St. Dev 5.82E+4 3.72E+2 1.80E+4 4.63E−8 8.75E+4 7.37E+4
|
||
0.96 Best – 6.66E+4 – 6.71E+4 – 4.51E+4
|
||
Mean – 6.78E+4 – 6.87E+4 – 4.51E+4
|
||
Worst – 6.78E+4 – 6.91E+4 – 4.63E+4
|
||
St. Dev – 3.72E+4 – 4.63E−4 – 1.17E−7
|
||
0.97 Best – 7.05E+4 – 7.21E+4 – 6.19E+4
|
||
Mean – 7.08E+4 – 7.21E+4 – 6.19E+4
|
||
Worst – 7.09E+4 – 7.21E+4 – 6.19E+4
|
||
St. Dev – 3.13E+3 – 3.33E−11 – 3.33E−11
|
||
0.98 Best – 8.29E+4 – 8.41E+4 – 7.39E+4
|
||
Mean – 8.31E+4 – 8.48E+4 – 7.57E+4
|
||
Worst – 8.34E+4 – 8.48E+4 – 7.61E+4
|
||
St. Dev – 6.37E+2 – 1.17E+2 – 4.52E+3
|
||
0.99 Best – – – – – –
|
||
Mean – – – – – –
|
||
Worst – – – – – –
|
||
St. Dev – – – – – –
|
||
|
||
|
||
|
||
consistently generated the poorest schedules, trailing behind EPM and size. The average SLR performance of HWWO across all generated
|
||
ESRG. Initially, EPM underperformed compared to ESRG, but as the graphs exceeded that of the DECM algorithm by 15% and the REREC
|
||
number of tasks increased, its performance surpassed that of ESRG. algorithm by 20.98%.
|
||
Notably, in scenarios where every path within the DAG constituted a
|
||
The communication to computation ratio (CCR) is a metric that
|
||
critical path, the DECM and REREC algorithms achieved comparable
|
||
quantifies the relative significance of communication overhead by di-
|
||
and superior results to HEFT, EPM, and ESRG, regardless of the input
|
||
viding the cumulative communication times across all edges by the
|
||
|
||
21
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
Table 13
|
||
Average SLR for all algorithms with respect to various tasks (in case of FFT).
|
||
|𝑋| Average SLR
|
||
HEFT DECM EPM REREC ESRG HWWO
|
||
95 5.1E+1 3.4E+1 4.2E+1 3.7E+1 4.0E+1 2.4E+1
|
||
223 6.2E+1 4.6E+1 4.9E+1 4.6E+1 4.9E+1 3.8E+1
|
||
511 8.8E+1 5.2E+1 6.5E+1 5.8E+1 6.2E+1 4.7E+1
|
||
1151 1.1E+2 7.8E+1 8.8E+1 8.5E+1 9.2E+1 6.9E+1
|
||
2559 1.3E+2 8.3E+1 9.2E+1 8.9E+1 9.6E+1 7.6E+1
|
||
|
||
|
||
|
||
|
||
Fig. 17. Graphical representation of Table 12.
|
||
|
||
|
||
|
||
|
||
Fig. 18. Graphical representation of Table 13.
|
||
|
||
|
||
total execution times across all nodes in a DAG. Fig. 19 illustrates the DECM and REREC are comparable across different CCR values. How-
|
||
average SLR performance of various algorithms as a function of the ever, HWWO emerged as the top-performing algorithm, yielding the
|
||
CCR. When considering CCR values, HEFT consistently exhibited the best SLR outcomes for all CCR values considered. Notably, the average
|
||
poorest SLR results, surpassed by both EPM and ESRG, while DECM, SLR performance of HWWO across all generated graphs surpassed that
|
||
REREC, and HWWO demonstrated their ability to generate superior of DECM by 14.16% and REREC by 19.91%.
|
||
schedules compared to these algorithms. The schedules produced by
|
||
|
||
|
||
22
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 19. Graphical representation of SLR values with respect to CCR (in case of FFT).
|
||
|
||
|
||
6.2.3. Scenario 3
|
||
To rigorously assess the proposed technique’s effectiveness, the
|
||
study conducts a comprehensive evaluation of the HWWO algorithm’s
|
||
performance through an analysis of the gaussian elimination (GE)
|
||
problem. The visual representation in Fig. 20 depicts a parallel im-
|
||
plementation of the GE application, incorporating a critical parameter
|
||
value of 𝜌 = 5, as described in [8,86]. Notably, the total number of
|
||
tasks, denoted as |𝑋|, is dynamically determined by the expression
|
||
2
|
||
|𝑋| = 𝜌 +𝜌−2
|
||
2
|
||
. Specifically, when 𝜌 = 5, the resulting task count is
|
||
|𝑋| = 14, as illustrated in Fig. 20. This scenario highlights the intricate
|
||
interplay between the parameters 𝜌 and |𝑋| in the context of parallel
|
||
computing applications. The following three experiments are conducted
|
||
to evaluate the performance of the proposed approach using the GE
|
||
application as a benchmark.
|
||
|
||
Experiment 4: The overarching goal of this experimental endeavor is to
|
||
conduct a meticulous comparative assessment of the proposed HWWO
|
||
technique against existing algorithms. The primary goal is placed on
|
||
evaluating their respective performances concerning total energy con-
|
||
sumption and computational time requirements within the realm of
|
||
parallel applications involving GE. Underpinning this experimental
|
||
study is a stringent deadline and reliability constraints, formulated
|
||
as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4 and 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) = 0.90 respectively, where
|
||
the complexity of the parallel application is intrinsically linked to the
|
||
quantity of constituent tasks it comprises. To comprehensively analyze
|
||
the techniques’ behavior, the study methodically varies the task count Fig. 20. DAG of GE application with 𝜌 = 5.
|
||
|𝑋| over a substantial range, spanning from 90 tasks (representing
|
||
smaller-scale scenarios) to 2555 tasks (larger-scale scenarios), while
|
||
concurrently investigating the effects of 𝜌 ranging from 13 to 71. in Fig. 21, providing a visual comparative analysis of the data from
|
||
Table 14.
|
||
Tables 14 and 15 present the outcomes from using GE applications
|
||
with varying 𝜌 values. In all experiments, HEFT (without DVFS tech-
|
||
nique) consistently consumes more energy. Table 14 shows that the
|
||
parameter |𝑋| has a wide range of values from 90 to 2555. This high- Table 15 shows that the EPM algorithm requires significantly higher
|
||
lights the superior energy efficiency of the HWWO algorithm compared 𝐶𝑇𝑇 𝐴 . However, as the number of tasks increases, ESRG outperforms
|
||
to other existing algorithms. As the number of tasks increases, both the other three algorithms in terms of higher energy values. The
|
||
the DECM and REREC algorithms exhibit similar performance levels. proposed HWWO algorithm’s 𝐶𝑇𝑇 𝐴 is projected to be 33.4% to 37.39%
|
||
Notably, the ESRG algorithm outperforms EPM in terms of lower energy of the computational time required by DECM and REREC algorithms.
|
||
consumption up to 495 tasks. Beyond that, EPM gradually improves its In terms of performance metrics, for |𝑋| values ranging from 90 to
|
||
results but at the cost of higher energy usage compared to DECM and 2555, the HWWO algorithm’s 𝐶𝑇𝑇 𝐴 closely follows and consistently
|
||
REREC. The best outcomes, highlighted in bold, are further illustrated outperforms the DECM algorithm.
|
||
|
||
23
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 14
|
||
Energy consumption analysis for GE parallel applications across task configurations.
|
||
|𝑋| Performance Algorithm
|
||
metric HEFT DECM EPM REREC ESRG HWWO
|
||
90 Best 3.18E+4 3.36E+3 7.88E+3 3.77E+3 7.81E+3 2.78E+3
|
||
Mean 3.33E+4 3.43E+3 7.94E+3 3.86E+3 7.89E+3 2.81E+3
|
||
Worst 3.38E+4 3.43E+3 7.94E+3 3.93E+3 7.94E+3 2.88E+3
|
||
St. Dev 5.04E+1 1.42E+0 3.31E+1 4.13E+1 2.13E+1 5.07E+1
|
||
230 Best 4.73E+4 7.93E+3 8.63E+3 7.93E+3 8.49E+3 4.71E+3
|
||
Mean 4.81E+4 7.93E+3 8.77E+3 7.93E+3 8.53E+3 4.79E+3
|
||
Worst 4.81E+4 7.93E+3 8.78E+3 7.93E+3 8.59E+3 4.83E+3
|
||
St. Dev 6.20E+4 2.32E−7 3.26E+1 2.32E−7 3.80E+1 2.32E+1
|
||
495 Best 6.27E+4 2.62E+4 3.08E+4 2.67E+4 3.05E+4 9.68E+3
|
||
Mean 6.27E+4 2.62E+4 3.16E+4 2.73E+4 3.11E+4 9.68E+3
|
||
Worst 6.27E+4 2.62E+4 3.16E+4 2.83E+4 3.16E+4 9.72E+3
|
||
St. Dev 8.08E−4 3.47E−7 2.34E+2 1.28E+3 4.44E+1 5.54E−3
|
||
1127 Best 9.44E+4 6.05E+4 7.03E+4 6.35E+4 7.59E+4 4.25E+4
|
||
Mean 9.44E+4 6.05E+4 7.13E+4 6.35E+4 7.62E+4 4.25E+4
|
||
Worst 9.49E+4 6.11E+4 7.13E+4 6.41E+4 7.67E+4 4.25E+4
|
||
St. Dev 1.13E+2 1.25E+4 2.14E+4 1.15E+4 5.03E+4 7.34E−9
|
||
2555 Best 9.75E+4 7.65E+4 9.44E+4 7.97E+4 9.54E+4 7.13E+4
|
||
Mean 9.82E+4 7.67E+4 9.51E+4 7.97E+4 9.54E+4 7.17E+4
|
||
Worst 9.88E+4 7.67E+4 9.59E+4 7.97E+4 9.59E+4 7.17E+4
|
||
St. Dev 1.02E+3 2.37E−1 4.80E+4 5.87E−7 3.80E+1 2.37E+2
|
||
|
||
|
||
|
||
|
||
Fig. 21. Graphical representation of Table 14.
|
||
|
||
|
||
reliability constraint from 0.91 to 0.98, but none of the algorithms can
|
||
meet the stringent 0.99 requirement.
|
||
Experiment 5: This experiment evaluates the reliability metrics and
|
||
total energy consumption of an extensive GE application under varying Fig. 23 visually represents the data from Table 16, showing the
|
||
reliability constraints. The experimental configuration involves 1127 energy consumption of GE applications under different reliability con-
|
||
tasks, with 𝜌 = 47. Additionally, the reliability goal, 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺), is straints. Up to 0.98 reliability, DECM, REREC, and HWWO are more
|
||
systematically varied from 0.91 to 0.99 in increments of 0.01, enabling energy-efficient than HEFT, EPM, and ESRG. While HEFT, EPM, and
|
||
an assessment of the corresponding effects on reliability performance ESRG fail to meet reliability constraints beyond 0.95, as evident from
|
||
and energy utilization. Fig. 22, DECM, REREC, and HWWO successfully fulfill the reliability
|
||
The graphical representation in Fig. 22 illustrates the actual reliabil- requirements up to 0.98 in most cases. Among them, HWWO achieves
|
||
ity values attained by the GE application when subjected to varying reli- better energy savings by exploring more processor and frequency com-
|
||
ability criteria. Among the techniques evaluated, the HWWO algorithm binations, reducing consumption by 33% and 37% compared to DECM
|
||
demonstrates superior performance in accomplishing the specified reli- and REREC, respectively. However, none of the algorithms meet the
|
||
ability goals compared to other existing methods. In contrast, the HEFT, stringent 0.99 reliability requirement.
|
||
EPM, and ESRG algorithms struggle to meet the reliability constraints
|
||
in most scenarios. While they can comply when 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) is between Experiment 6: The current experiment examines the SLR and CCR
|
||
0.91 and 0.95, their performance deteriorates beyond that range. On metrics for a comprehensive GE application, considering variations
|
||
the other hand, DECM, REREC, and HWWO successfully satisfy the in 𝜌. This experimental approach incorporates a stringent deadline
|
||
|
||
24
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 15
|
||
𝐶𝑇𝑇 𝐴 values of GE applications across diverse task quantities.
|
||
|𝑋| Performance Algorithm
|
||
metric DECM EPM REREC ESRG HWWO
|
||
90 Best 3.5E+1 4.31E+2 3.5E+1 3.81E+2 2.6E+1
|
||
Mean 3.6E+1 4.43E+2 3.6E+1 3.91E+2 2.6E+1
|
||
Worst 3.8E+1 4.43E+2 3.8E+1 3.95E+2 2.6E+1
|
||
St. Dev 7.5E−1 5.6E+0 7.5E−1 5.8E+0 0
|
||
230 Best 5.1E+1 7.14E+2 5.1E+1 6.84E+2 3.9E+1
|
||
Mean 5.9E+1 7.23E+2 6.7E+1 6.93E+2 4.4E+1
|
||
Worst 6.8E+1 7.41E+2 6.7E+1 6.94E+2 4.9E+1
|
||
St. Dev 6.4E+0 1.21E+2 7.4E+0 3.20E−1 4.02E−1
|
||
495 Best 6.4E+1 9.5E+3 7.1E+1 8.5E+3 6.4E+1
|
||
Mean 6.9E+1 1.6E+4 8.3E+1 1.3E+4 6.9E+1
|
||
Worst 7.2E+1 2.2E+4 9.2E+1 2.7E+4 7.2E+1
|
||
St. Dev 3.3E+0 5.10E+3 8.3E+1 2.09E+2 3.3E+0
|
||
1127 Best 7.7E+2 8.19E+3 7.9E+2 8.73E+3 7.49E+2
|
||
Mean 7.88E+2 8.25E+3 8.15E+2 8.83E+3 7.67E+2
|
||
Worst 7.94E+2 8.37E+3 8.7E+2 8.87E+3 7.77E+2
|
||
St. Dev 4.31E+1 2.71E+3 7.08E+1 5.08E+0 1.1E+1
|
||
2555 Best 8.57E+2 9.03E+3 8.77E+2 9.12E+3 8.57E+2
|
||
Mean 8.61E+2 9.22E+3 8.81E+2 9.28E+3 8.61E+2
|
||
Worst 8.69E+2 9.36E+3 8.91E+2 9.39E+3 8.69E+2
|
||
St. Dev 4.9E−2 3.32E+1 6.07E+1 6.17E+1 4.9E+1
|
||
|
||
|
||
|
||
|
||
Fig. 22. Graphical representation of actual reliability values under varying reliability constraints (in case of GE).
|
||
|
||
|
||
|
||
|
||
constraint, formulated as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4, where the com- results, while DECM, REREC, and HWWO generated superior schedules
|
||
plexity of the parallel application is inherently tied to the number of compared to HEFT, EPM, and ESRG. DECM and REREC produced com-
|
||
constituent tasks it encompasses. The study systematically alters |𝑋| parable schedules across different CCR values. However, HWWO out-
|
||
from 90 (representing smaller-scale scenarios) to 2555 (representing performed all others, yielding the best SLR outcomes for all considered
|
||
larger-scale scenarios), while simultaneously exploring the impacts of CCR values.
|
||
𝜌 ranging from 13 to 71.
|
||
Stage II
|
||
Table 17 and Fig. 24 demonstrate the average SLR performance
|
||
of various tasks scheduling algorithms. The proposed HWWO algo- 6.3. Benchmark analysis with metaheuristic algorithms
|
||
rithm exhibited the lowest SLR values, outperforming others. HWWO
|
||
emerged as the superior approach in terms of SLR. Across all task sizes,
|
||
In this stage the proposed algorithm HWWO performance is eval-
|
||
HEFT consistently generated the poorest schedules, trailing EPM and
|
||
uated across three scenarios and compared with several metaheuristic
|
||
ESRG. Initially, EPM underperformed compared to ESRG but surpassed
|
||
methods using various metrics. The evaluation considers different task
|
||
it as task count increased. In critical DAG path scenarios, DECM and
|
||
and processor configurations, as well as benchmark tests involving
|
||
REREC outperformed HEFT, EPM, and ESRG, regardless of input size.
|
||
unimodal functions. Additionally, experiments are conducted for tasks
|
||
Fig. 25 shows the average SLR performance of various algorithms scheduling in a multiprocessing environment, with input parameters
|
||
as a function of CCR. HEFT consistently exhibited the poorest SLR described in Table 18. After simulations, a comprehensive assessment
|
||
|
||
25
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 16
|
||
Energy consumption assessment for GE applications with varying reliability criteria.
|
||
Reliability Performance Algorithm
|
||
goal metric HEFT DECM EPM REREC ESRG HWWO
|
||
0.91 Best 4.54E+4 3.45E+4 3.83E+4 3.53E+4 3.73E+4 2.55E+4
|
||
Mean 4.54E+4 3.45E+4 3.83E+4 3.53E+4 3.73E+4 2.55E+4
|
||
Worst 4.58E+4 3.45E+4 3.83E+4 3.53E+4 3.79E+4 2.55E+4
|
||
St. Dev 4.23E+2 4.13E−4 5.24E−4 1.13E−4 6.34E−2 5.24E−8
|
||
0.92 Best 4.74E+4 3.77E+4 4.18E+4 3.96E+4 4.16E+4 2.88E+4
|
||
Mean 4.81E+4 3.78E+4 4.19E+4 3.98E+4 4.19E+4 2.88E+4
|
||
Worst 4.92E+4 3.78E+4 4.19E+4 3.98E+4 4.19E+4 2.88E+4
|
||
St. Dev 7.40E+2 4.27E−1 7.04E+1 9.42E+1 2.04E+2 7.02E−7
|
||
0.93 Best 5.74E+4 3.81E+4 4.28E+4 4.18E+4 4.28E+4 2.91E+4
|
||
Mean 5.74E+4 3.83E+4 4.33E+4 4.20E+4 4.33E+4 2.94E+4
|
||
Worst 5.74E+4 3.86E+4 4.47E+4 4.26E+4 4.47E+4 2.95E+4
|
||
St. Dev 3.28E−4 4.40E+3 8.02E+2 3.30E+3 8.02E+2 1.69E+2
|
||
0.94 Best 6.64E+4 3.97E+4 4.93E+4 4.51E+4 4.97E+4 3.97E+4
|
||
Mean 6.69E+4 3.97E+4 4.93E+4 4.65E+4 4.98E+4 3.97E+4
|
||
Worst 6.74E+4 3.99E+4 4.93E+4 4.77E+4 4.98E+4 3.99E+4
|
||
St. Dev 4.08E+2 9.42E+1 7.12E+2 3.12E+2 4.13E+2 9.42E+1
|
||
0.95 Best 8.45E+4 6.52E+4 6.84E+4 6.63E+4 6.84E+4 4.44E+4
|
||
Mean 8.48E+4 6.52E+4 6.84E+4 6.67E+4 6.84E+4 4.46E+4
|
||
Worst 8.51E+4 6.59E+4 6.85E+4 6.67E+4 6.85E+4 4.53E+4
|
||
St. Dev 5.02E+3 3.72E−4 4.07E+3 4.63E+3 4.07E+3 5.17E+2
|
||
0.96 Best – 6.76E+4 – 6.90E+4 – 4.86E+4
|
||
Mean – 6.78E+4 – 6.90E+4 – 4.86E+4
|
||
Worst – 6.78E+4 – 6.90E+4 – 4.86E+4
|
||
St. Dev – 3.72E−2 – 4.63E−5 – 4.06E−13
|
||
0.97 Best – 7.15E+4 – 7.25E+4 – 6.55E+4
|
||
Mean – 7.18E+4 – 7.25E+4 – 6.55E+4
|
||
Worst – 7.18E+4 – 7.25E+4 – 6.55E+4
|
||
St. Dev – 4.03E+2 – 5.33E−11 – 4.22E−11
|
||
0.98 Best – 9.37E+4 – 9.41E+4 – 7.31E+4
|
||
Mean – 9.37E+4 – 9.41E+4 – 7.31E+4
|
||
Worst – 9.37E+4 – 9.48E+4 – 7.37E+4
|
||
St. Dev – 4.37E−9 – 1.17E+2 – 4.52E−4
|
||
0.99 Best – – – – – –
|
||
Mean – – – – – –
|
||
Worst – – – – – –
|
||
St. Dev – – – – – –
|
||
|
||
|
||
|
||
|
||
Fig. 23. Graphical representation of Table 16.
|
||
|
||
|
||
is carried out, calculating metrics such as average execution time, the algorithm’s effectiveness in terms of energy consumption, system
|
||
standard deviation, and mean across iterations. The results highlight reliability, resource utilization, and sensitivity analysis.
|
||
|
||
|
||
26
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 17
|
||
Average SLR for all algorithms with respect to various tasks (in case of GE).
|
||
|𝑋| Average SLR
|
||
HEFT DECM EPM REREC ESRG HWWO
|
||
90 6.3E+1 3.4E+1 4.4E+1 3.9E+1 4.0E+1 3.4E+1
|
||
230 7.2E+1 4.7E+1 5.8E+1 4.9E+1 5.4E+1 3.8E+1
|
||
495 8.9E+1 5.8E+1 6.6E+1 6.1E+1 6.2E+1 4.9E+1
|
||
1127 1.2E+2 7.8E+1 9.3E+1 8.5E+1 9.8E+1 7.2E+1
|
||
2555 1.3E+2 8.7E+1 9.5E+1 8.9E+1 9.8E+1 7.9E+1
|
||
|
||
|
||
|
||
|
||
Fig. 24. Graphical representation of Table 17.
|
||
|
||
|
||
|
||
|
||
Fig. 25. Graphical representation of SLR values with respect to CCR (in case of GE).
|
||
|
||
|
||
6.3.1. Scenario 1 of processors constant. The detailed findings and analysis are presented
|
||
subsequently.
|
||
The study aims to thoroughly evaluate the effectiveness of the pro-
|
||
posed HWWO-based approach by testing it with different numbers of Tasks range: 100–1000
|
||
tasks. Seven well-known metaheuristic algorithms – PSO, ACO, KH, DA, Processor count: 100
|
||
AHA, GWO, and WOA – are employed alongside the HWWO algorithm.
|
||
These algorithms are utilized to assess the performance of the HWWO For an impartial and consistent evaluation, the parameter settings
|
||
technique when varying the number of tasks while keeping the number of the seven algorithms remained unchanged from their default values
|
||
|
||
27
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
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||
|
||
|
||
|
||
|
||
Fig. 26. Graphical representation of Table 19.
|
||
|
||
|
||
applications that are randomly generated via a DAG genera-
|
||
Table 18 tor [100], where the deadline and reliability requirements for
|
||
Parameter setting for stage II. completing each application are calculated as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗
|
||
Algorithm Parameter 1.4 and 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) = 0.90 respectively.
|
||
PSO [57] Inertia weight = 0.4-0.9 A Table 19 is presented that compares the energy consump-
|
||
Cognitive component = 1.50 tion results from the proposed algorithm against seven other
|
||
Social component= 2 metaheuristic algorithms. This allows a thorough analysis and
|
||
ACO [55] Evaporation rate = 0.2
|
||
Weight of pheromone on decision = 0.5
|
||
side-by-side comparison of the energy efficiencies across these
|
||
Weight of heuristic information on decision = 0.5 different solution techniques when dealing with the randomized
|
||
Q = 2 parallel application workloads with the specified deadlines.
|
||
KH [59] Foraging motion = 0.2 The data presented in Table 19 indicates that when evaluating
|
||
Induced motion = 0.006
|
||
performance metrics, the ACO algorithm consistently exhibits
|
||
Inertia weight = 0.5
|
||
DA [67] Separation weight = 0.12 higher energy consumption compared to the seven other algo-
|
||
Alignment weight = 0.12 rithms under consideration. In the initial stages, the AHA and DA
|
||
Cohesion weight = 0.75 algorithms deliver superior and comparable results to KH and
|
||
Food factor = 1 PSO, respectively. However, as the problem size scales up, the
|
||
Enemy factor =1
|
||
AHA [67] Inertia weight = 0.5
|
||
KH and PSO algorithms outperform AHA and DA, demonstrat-
|
||
Local search probability = 0.5 ing more efficient outcomes. Among the remaining algorithms,
|
||
→
|
||
GWO [29] 𝜇 = [0, 2] the GWO technique demonstrates its strength by outperforming
|
||
𝑙1 , 𝑙2 = [0,1] the WOA method while attaining comparable energy efficiency.
|
||
→
|
||
𝜁 = [−1, 1] Strikingly, the newly proposed HWWO algorithm surpasses all
|
||
WOA [28] 𝜈 = [−1, 1] the other contenders, exhibiting substantially lower energy con-
|
||
→
|
||
𝜇 = [0, 2] sumption levels. The HWWO algorithm establishes itself as the
|
||
𝑙1 , 𝑙2 = [0,1]
|
||
leading performer in terms of optimizing energy consumption
|
||
Proposed HWWO 𝑤̂ 𝑖,𝑙 (𝑚𝑠) = [10, 100]
|
||
𝑐̂𝑖,𝑘 (𝑚𝑠) = [10, 100] across the diverse set of test scenarios explored in this evalu-
|
||
𝑃𝑙,𝑠 = [0.1, 0.5] ation. The graphical representation in Fig. 26 depicts energy
|
||
𝑃𝑙,𝑖𝑛𝑑 = [0.03, 0.07] consumption levels across different sets of tasks. A clear pattern
|
||
𝐶𝑙,𝑒𝑓 = [0.8, 1.2] emerges: higher energy usage as more tasks is added. However,
|
||
𝑚𝑙 = [2.5, 3.0]
|
||
𝑓𝑙,𝑚𝑎𝑥 = 1 GHz
|
||
the proposed algorithm’s energy consumption values are notice-
|
||
𝜆𝑙,𝑚𝑎𝑥 = [0.0003, 0.0009] ably lower than existing methods, indicating superior efficiency.
|
||
For 100 processors, the HWWO algorithm minimizes energy con-
|
||
sumption by 18%–24% less than GWO and WOA respectively.
|
||
This substantial reduction highlights the proposed approach’s
|
||
as presented in Table 18. The findings accentuated the algorithms’ energy-saving advantages, especially with increasing tasks.
|
||
proficiency in several key areas, including energy efficiency, system ii System reliability
|
||
reliability, resource utilization, and sensitivity analysis across a range This part evaluates the proposed algorithm’s effectiveness by ex-
|
||
of input variations. amining reliability across varying task combinations. It analyzes
|
||
reliability metrics for different task counts, targeting a reliability
|
||
i Energy consumption goal of 𝑅𝑒(𝑚𝑖𝑛) (𝐺) ≤ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 𝑅𝑒(𝑚𝑎𝑥) (𝐺) or 0.88371 ≤
|
||
The goal of this evaluation is to assess the proposed algorithm’s 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 0.98577 at 0.90. The evaluation utilizes randomly
|
||
performance by analyzing its energy consumption for different generated parallel applications with deadlines calculated as
|
||
combinations of tasks. The assessment utilizes a set of parallel
|
||
|
||
28
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 27. Reliability outcomes for metaheuristic algorithms with varying task numbers.
|
||
|
||
|
||
𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4. Fig. 27 presents a comparative analysis the experimental findings validates that the proposed HWWO
|
||
of reliability results obtained from the proposed algorithm and approach facilitates more efficient resource utilization than the
|
||
seven other metaheuristic algorithms. This comparison enables a existing metaheuristic frameworks.
|
||
thorough assessment of reliability performance across these solu- iv Sensitivity analysis
|
||
tion techniques when handling randomized parallel application In this subsection, the performance of the proposed method is
|
||
workloads with specified deadlines. evaluated through sensitivity analysis. Sensitivity analysis is a
|
||
Fig. 27 highlights the performance metrics, indicating that the technique employed to ascertain the extent to which the output
|
||
ACO algorithm consistently exhibits lower system reliability of a model is influenced by variations in the input parameters.
|
||
compared to the seven other algorithms evaluated. Initially, It helps to identify the inputs that have the most significant
|
||
AHA and DA algorithms demonstrate superior and comparable influence on the output and assess the model’s robustness to
|
||
reliability results to KH and PSO respectively, but as the problem variations in these inputs.
|
||
size increases, the latter two outperform the former, exhibiting The assessment encompasses a variety of randomly generated
|
||
more efficient outcomes. Among the remaining algorithms, the parallel applications, where the application deadline is deter-
|
||
GWO technique outperforms the WOA method, while the newly mined by the formula 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4. In this evaluation,
|
||
proposed HWWO algorithm surpasses all other contenders, ex- the sensitivity of the proposed HWWO model is investigated
|
||
hibiting substantially higher reliability levels in most cases and concerning the completion time of tasks scheduling. The overall
|
||
maximizing system reliability by 8%–9% more than GWO and completion time for HWWO and other existing metaheuristic
|
||
WOA for 100 processors. approaches is presented in Table 20 with varying task quantities,
|
||
iii Resource utilization for which a sensitivity analysis is conducted. The optimiza-
|
||
This meticulously designed study aims to evaluate the effec- tion problem is addressed using a One-at-a-Time (OAT) based
|
||
tiveness of the proposed algorithm by thoroughly examining method, where the proposed technique’s performance is assessed
|
||
resource utilization across various task combinations. The exam- through sensitivity analysis.
|
||
ination of resource utilization [78,79] mainly focuses on com- From Table 20, it can be concluded that the proposed HWWO
|
||
putation time and compares it with other existing models. The technique gave superior outcomes compared to other techniques.
|
||
assessment encompasses a variety of randomly generated par- It decreased computation time by 46.07%, 47.59%, 43.81%,
|
||
allel applications, where the application deadline is determined 46.11%, 41.84%, 28.85%, and 30.27% in comparison to PSO,
|
||
by the formula 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4. To offer an in-depth and ACO, KH, DA, AHA, GWO, and WOA respectively. To further an-
|
||
thorough comparative analysis of resource utilization, Fig. 28 (in alyze the performance, the average sensitivity of each algorithm
|
||
%) has been carefully crafted. This figure showcases the optimal is calculated using the OAT technique, as detailed in Table 21.
|
||
results achieved by eight distinct metaheuristic algorithms. The table analysis shows the proposed HWWO model has the
|
||
Optimal resource utilization is a critical factor in achieving lowest average sensitivity ratio (0.19), indicating least sensitivity
|
||
profitability for heterogeneous computing systems. Higher re- to task number changes among the algorithms. This lower sen-
|
||
source utilization directly corresponds to increased profits for sitivity suggests HWWO is more robust and reliable for varying
|
||
service providers. The figure presents a comparative analysis workloads, making it preferable in environments with fluctuat-
|
||
of resource utilization between the proposed HWWO approach ing task quantities.
|
||
and established metaheuristic frameworks. The results indicate
|
||
that the HWWO algorithm demonstrates superior performance,
|
||
substantially enhancing resource utilization by 42%, 62%, 22%, 6.3.2. Scenario 2
|
||
42.48%, 21.86%, 11.83%, and 14% in comparison to PSO, ACO, This part focuses on an in-depth evaluation of the proposed HWWO-
|
||
KH, DA, AHA, GWO, and WOA respectively, across a range based approach by experimenting with different processor counts. The
|
||
of computational tasks. The empirical evidence derived from HWWO algorithm is tested alongside seven well-established meta-
|
||
heuristic algorithms: PSO, ACO, KH, DA, AHA, GWO, and WOA. These
|
||
|
||
29
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 19
|
||
Analysis of energy consumptions for varying tasks.
|
||
|𝑋| Performance Algorithm
|
||
metric PSO ACO KH DA AHA GWO WOA HWWO
|
||
100 Best 6.08E+3 6.32E+3 5.25E+3 5.82E+3 5.25E+3 3.57E+3 3.67E+3 3.52E+3
|
||
Mean 6.13E+3 6.37E+3 5.25E+3 5.84E+3 5.25E+3 3.57E+3 3.67E+3 3.52E+3
|
||
Worst 6.13E+3 6.37E+3 5.29E+3 5.84E+3 5.29E+3 3.57E+3 3.68E+3 3.52E+3
|
||
St. Dev 5.30E+1 7.23E+2 5.55E−3 4.15E+1 5.55E−3 1.13E−13 4.61E−4 1.13E−13
|
||
200 Best 6.51E+3 6.74E+3 5.83E+3 6.31E+3 5.69E+3 3.92E+3 3.88E+3 3.88E+3
|
||
Mean 6.51E+3 6.79E+3 5.85E+3 6.31E+3 5.69E+3 3.93E+3 3.89E+3 3.89E+3
|
||
Worst 6.51E+3 6.79E+3 5.87E+3 6.35E+3 5.69E+3 3.95E+3 3.89E+3 3.89E+3
|
||
St. Dev 2.07E−4 7.03E+1 3.33E+1 7.09E+1 3.08E−7 5.12E+1 1.02E+1 1.02E+1
|
||
300 Best 6.97E+3 7.58E+3 6.42E+3 6.93E+3 6.31E+3 4.87E+3 4.64E+3 4.58E+3
|
||
Mean 6.99E+3 7.58E+3 6.49E+3 6.95E+3 6.33E+3 4.87E+3 4.66E+3 4.59E+3
|
||
Worst 6.99E+3 7.58E+3 6.49E+3 6.95E+3 6.38E+3 5.95E+3 4.69E+3 4.61E+3
|
||
St. Dev 7.35E+1 2.43E−6 3.11E+2 7.61E+1 1.01E+1 4.02E−1 3.12E+0 7.17E+1
|
||
400 Best 7.48E+3 7.98E+3 6.89E+3 7.23E+3 6.77E+3 6.05E+3 6.05E+3 6.05E+3
|
||
Mean 7.48E+3 7.99E+3 6.91E+3 7.28E+3 6.77E+3 6.05E+3 6.05E+3 6.05E+3
|
||
Worst 7.48E+3 7.99E+3 6.96E+3 7.28E+3 6.77E+3 6.05E+3 6.05E+3 6.05E+3
|
||
St. Dev 7.15E−4 1.85E−1 1.13E+2 4.25E−4 1.33E−4 3.02E−8 3.02E−8 3.02E−8
|
||
500 Best 8.36E+3 9.39E+3 7.49E+3 8.16E+3 7.89E+3 6.76E+3 6.66E+3 6.57E+3
|
||
Mean 8.53E+3 9.46E+3 7.49E+3 8.23E+3 7.89E+3 6.81E+3 6.68E+3 6.65E+3
|
||
Worst 8.67E+3 9.51E+3 7.49E+3 8.27E+3 7.92E+3 6.81E+3 6.68E+3 6.65E+3
|
||
St. Dev 1.81E+1 1.33E+1 2.73E−6 1.44E+0 4.53E+1 4.11E+3 5.00E+1 9.01E+3
|
||
600 Best 1.31E+4 1.63E+4 9.78E+3 1.53E+4 9.97E+3 8.77E+3 9.04E+3 8.52E+3
|
||
Mean 1.33E+4 1.67E+4 9.78E+3 1.55E+4 9.99E+3 8.79E+3 9.14E+3 8.52E+3
|
||
Worst 1.37E+4 1.67E+4 9.81E+3 1.59E+4 9.99E+3 8.93E+3 9.14E+3 8.52E+3
|
||
St. Dev 6.01E−1 4.35E−2 4.47E+0 2.33E−4 3.43E−2 9.11E−1 7.12E+1 6.16E−12
|
||
700 Best 1.83E+4 2.44E+4 1.41E+4 2.34E+4 1.83E+4 9.89E+3 1.03E+4 9.46E+3
|
||
Mean 1.83E+4 2.54E+4 1.49E+4 2.39E+4 1.83E+4 9.89E+3 1.11E+4 9.47E+3
|
||
Worst 1.83E+4 2.57E+4 1.58E+4 2.43E+4 1.83E+4 9.93E+3 1.22E+4 9.49E+3
|
||
St. Dev 3.01E−8 6.63E+4 2.65E+2 2.73E+4 3.01E−8 6.66E−2 3.01E+2 3.22E−6
|
||
800 Best 3.15E+4 3.51E+4 2.41E+4 3.35E+4 2.76E+4 1.52E+4 1.94E+4 1.06E+4
|
||
Mean 3.35E+4 3.51E+4 2.53E+4 3.35E+4 2.78E+4 1.64E+4 1.94E+4 1.06E+4
|
||
Worst 3.35E+4 3.51E+4 2.58E+4 3.37E+4 2.78E+4 1.64E+4 1.94E+4 1.06E+4
|
||
St. Dev 2.32E+4 2.32E−7 4.08E+2 2.32E+1 8.22E+1 3.07E+4 8.17E−8 3.07E−8
|
||
900 Best 4.15E+4 4.22E+4 3.95E+4 4.19E+4 4.05E+4 3.51E+4 3.91E+4 2.55E+4
|
||
Mean 4.15E+4 4.22E+4 3.95E+4 4.22E+4 4.11E+4 3.55E+4 3.95E+4 2.55E+4
|
||
Worst 4.15E+4 4.27E+4 3.95E+4 4.29E+4 4.11E+4 3.55E+4 3.95E+4 2.55E+4
|
||
St. Dev 1.81E−6 6.21E+0 5.15E−12 1.81E+2 1.81E+1 8.25E+0 5.15E−2 5.15E−12
|
||
1000 Best 4.71E+4 5.28E+4 4.13E+4 5.08E+4 4.60E+4 3.81E+4 4.13E+4 3.25E+4
|
||
Mean 4.73E+4 5.32E+4 4.23E+4 5.13E+4 4.60E+4 3.85E+4 4.23E+4 3.41E+4
|
||
Worst 4.73E+4 5.32E+4 4.23E+4 5.13E+4 4.60E+4 3.85E+4 4.23E+4 3.47E+4
|
||
St. Dev 2.01E+1 6.08E+1 1.51E+3 8.08E+4 3.01E−7 4.44E−2 1.51E+3 6.61E+4
|
||
|
||
|
||
|
||
|
||
Table 20
|
||
Comparative analysis of task completion times across various metaheuristic techniques under varying tasks.
|
||
|𝑋| Algorithm
|
||
PSO ACO KH DA AHA GWO WOA HWWO
|
||
100 1.58E+2 1.58E+2 1.56E+2 1.56E+2 1.35E+2 9.57E+1 1.03E+2 7.62E+1
|
||
200 2.01E+2 2.01E+2 1.88E+2 1.98E+2 1.78E+2 1.68E+2 1.22E+2 8.98E+1
|
||
300 2.67E+2 2.58E+2 2.42E+2 2.51E+2 2.36E+2 2.18E+2 1.88E+2 1.07E+2
|
||
400 4.49E+2 4.68E+2 4.17E+2 4.38E+2 4.17E+2 2.75E+2 2.54E+2 1.59E+2
|
||
500 4.96E+2 4.96E+2 4.85E+2 5.09E+2 4.98E+2 3.12E+2 3.37E+2 2.32E+2
|
||
600 5.31E+2 5.53E+2 5.08E+2 5.63E+2 5.37E+2 3.78E+2 3.95E+2 2.98E+2
|
||
700 5.83E+2 5.96E+2 5.51E+2 5.93E+2 5.83E+2 4.39E+2 4.39E+2 3.55E+2
|
||
800 6.15E+2 6.58E+2 6.01E+2 6.55E+2 6.26E+2 5.07E+2 5.07E+2 3.83E+2
|
||
900 6.61E+2 7.12E+2 6.66E+2 7.09E+2 6.66E+2 5.61E+2 5.71E+2 4.02E+2
|
||
1000 7.11E+2 7.55E+2 6.76E+2 7.38E+2 6.88E+2 6.04E+2 6.04E+2 4.59E+2
|
||
|
||
|
||
Table 21
|
||
The average sensitivity for each algorithm of Table 20.
|
||
Algorithm PSO ACO KH DA AHA GWO WOA HWWO
|
||
Avg sensitivity 0.37 0.42 0.39 0.38 0.34 0.27 0.29 0.19
|
||
|
||
|
||
|
||
algorithms are employed to examine the performance of the HWWO The findings accentuated the algorithms’ proficiency in several key
|
||
method under varying numbers of processors, while maintaining a con- areas, including energy efficiency, system reliability, resource utiliza-
|
||
stant number of tasks. The detailed findings and analysis are presented tion, and sensitivity analysis across a range of input variations.
|
||
subsequently.
|
||
i Energy consumption
|
||
Processors range: 100–1000 The goal of this evaluation is to assess the proposed algorithm’s
|
||
Task count: 1000 performance by analyzing its energy consumption for different
|
||
|
||
30
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 28. Comparative analysis graph of resource utilization for metaheuristic techniques with varying tasks.
|
||
|
||
|
||
|
||
|
||
Fig. 29. Energy consumption for metaheuristic algorithms with respect to varying processors.
|
||
|
||
|
||
combinations of processors. The assessment utilizes a set of mechanism, enabling the HWWO algorithm to outperform oth-
|
||
parallel applications that are randomly generated via a DAG ers and find better solutions. When tested with 1000 tasks,
|
||
generator [100], where the deadline and reliability requirements the HWWO algorithm reduced energy consumption by 13.46-
|
||
for completing each application are calculated as 𝐷𝐿(𝐺) = 23.81% compared to GWO and WOA, respectively. This sub-
|
||
𝐿𝐵(𝐺) ∗ 1.4 and 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) = 0.90 respectively. stantial reduction underscores the energy-saving advantages of
|
||
The energy consumption results displayed in Fig. 29 compare the proposed approach, especially as the number of processors
|
||
the proposed algorithm against other metaheuristic algorithms. increases.
|
||
The figure reveals that the ACO algorithm consistently consumes ii System reliability
|
||
more energy than the other algorithms evaluated. Among these This part evaluates the proposed algorithm’s effectiveness by
|
||
algorithms, the GWO technique outperforms the WOA method examining reliability across varying processors combinations. It
|
||
while achieving similar energy efficiency as the AHA algorithm. analyzes reliability metrics for different task counts, targeting
|
||
Notably, the proposed algorithm exhibits significantly lower a reliability goal of 𝑅𝑒(𝑚𝑖𝑛) (𝐺) ≤ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 𝑅𝑒(𝑚𝑎𝑥) (𝐺) or
|
||
energy consumption than existing methods, indicating superior 0.88371 ≤ 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) ≤ 0.98577 at 0.90. The evaluation uti-
|
||
efficiency. This is due to the proposed algorithm defining a cir- lizes randomly generated parallel applications with deadlines
|
||
cular neighborhood around solutions based on its encirclement calculated as 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4.
|
||
|
||
|
||
|
||
31
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
||
Fig. 30. Reliability outcomes for metaheuristic algorithms with varying processor numbers.
|
||
|
||
|
||
Fig. 30 highlights the performance metrics, indicating that the to 47.94% compared to various metaheuristic techniques. Ad-
|
||
ACO algorithm consistently exhibits lower system reliability ditionally, Table 23 provides the average sensitivity of each
|
||
compared to the seven other algorithms evaluated. Among the algorithm, determined through the OAT technique, for further
|
||
algorithms, the GWO technique outperforms the WOA method, performance analysis.
|
||
while the newly proposed HWWO algorithm surpasses all other The table analysis unveils that the proposed HWWO model
|
||
contenders, exhibiting substantially higher reliability levels in demonstrates the minimal average sensitivity ratio (0.22), indi-
|
||
most cases and maximizing system reliability by 5%–8% more cating its superior resistance to fluctuations in processor avail-
|
||
than GWO and WOA for 1000 tasks. ability compared to other algorithms. This lower sensitivity
|
||
iii Resource utilization makes HWWO more robust and reliable for different work-
|
||
This part intends to assess the efficacy of the proposed algo- loads, making it ideal for environments with varying processor
|
||
rithmic approach by conducting a comprehensive analysis of re- numbers.
|
||
source utilization across different processor configurations. The
|
||
assessment encompasses a variety of randomly generated par-
|
||
allel applications, where the application deadline is determined 6.3.3. Scenario 3
|
||
by the formula 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4. To offer an in-depth and To validate the proposed HWWO algorithm’s efficacy against es-
|
||
thorough comparative analysis of resource utilization, Fig. 31 (in tablished metaheuristic optimization techniques, this section employs
|
||
%) has been carefully crafted. This figure showcases the optimal a set of unimodal test functions. These benchmark functions, sourced
|
||
results achieved by eight distinct metaheuristic algorithms. from [28] and tabulated in Table 24, assess the algorithm’s exploita-
|
||
The results from figure indicate that the HWWO algorithm tion capabilities and overall optimization performance. Ensuring a fair
|
||
demonstrates superior performance, substantially enhancing re- comparison, all tests utilize a population size of 30, with a maximum
|
||
source utilization by 26.92%, 29.79%, 19.34%, 16.48%, 16.86%, of 15,000 function evaluations across 500 iterations. Each algorithm
|
||
9.83%, and 25.15% in comparison to PSO, ACO, KH, DA, AHA, is executed 30 times independently on these functions. The evaluation
|
||
GWO, and WOA respectively, across a range of computational metrics, including mean, standard deviation, best and worst fitness
|
||
processors. The empirical evidence derived from the experi- values from the independent runs, are then computed and presented
|
||
mental findings validates that the proposed HWWO approach in Table 25.
|
||
facilitates more efficient resource utilization than the existing An analysis of Table 25, which presents the results for unimodal
|
||
metaheuristic frameworks. functions, clearly demonstrates the superior exploitation capability of
|
||
iv Sensitivity analysis the proposed HWWO algorithm. This is evident from the fact that the
|
||
In this subsection, the effectiveness of the proposed method HWWO algorithm achieves the best mean fitness values in the majority
|
||
is assessed via sensitivity analysis in relation to different pro- of cases, as indicated by the bold entries. In contrast, the existing algo-
|
||
cessor counts. The assessment encompasses a variety of ran- rithms being evaluated display comparatively inferior performance.
|
||
domly generated parallel applications, where the application In evaluating the algorithms’ performance based on the highest
|
||
deadline is determined by the formula 𝐷𝐿(𝐺) = 𝐿𝐵(𝐺) ∗ 1.4. In fitness scores across 30 runs, it is observed that the HWWO algorithm
|
||
this evaluation, the sensitivity of the HWWO model concerning outperformed others, securing the highest number of best fitness scores
|
||
tasks scheduling completion times is explored. Table 22 displays (5/7). In comparison, WOA and GWO attained fewer best fitness scores
|
||
the overall completion times for HWWO and other existing (1/7 each), while all other algorithms do not achieve the best fitness
|
||
metaheuristic approaches across varying processors, for which in any of the runs. These results suggest that the HWWO algorithm
|
||
a sensitivity analysis is performed. demonstrates greater consistency and reliability in attaining optimal
|
||
Table 22 shows that the HWWO technique yielded better re- fitness values compared to others.
|
||
sults than other methods, reducing computation time by 30.68% The statistical analysis using the Wilcoxon signed-rank test is pre-
|
||
sented in Table 26, which evaluates the performance of the HWWO
|
||
|
||
32
|
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Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 22
|
||
Comparative analysis of task completion times across various metaheuristic techniques under varying processors.
|
||
|𝑌 | Algorithm
|
||
PSO ACO KH DA AHA GWO WOA HWWO
|
||
100 1.28E+2 1.28E+2 1.26E+2 9.56E+1 8.62E+1 8.02E+1 1.07E+2 6.82E+1
|
||
200 2.12E+2 2.21E+2 1.88E+2 1.88E+2 1.73E+2 1.64E+2 1.85E+2 8.88E+1
|
||
300 2.67E+2 2.71E+2 2.71E+2 2.51E+2 2.38E+2 2.38E+2 2.65E+2 1.17E+2
|
||
400 4.89E+2 4.98E+2 4.72E+2 4.48E+2 4.37E+2 2.84E+2 4.59E+2 1.54E+2
|
||
500 4.98E+2 5.21E+2 4.85E+2 5.19E+2 4.78E+2 3.62E+2 5.19E+2 2.22E+2
|
||
600 5.71E+2 5.73E+2 5.64E+2 5.62E+2 5.17E+2 3.73E+2 5.64E+2 2.98E+2
|
||
700 6.13E+2 6.26E+2 5.95E+2 5.89E+2 5.73E+2 4.49E+2 5.93E+2 3.75E+2
|
||
800 6.65E+2 6.68E+2 6.61E+2 6.46E+2 6.46E+2 5.17E+2 6.55E+2 3.86E+2
|
||
900 7.41E+2 7.52E+2 7.36E+2 7.15E+2 6.76E+2 5.61E+2 7.27E+2 4.32E+2
|
||
1000 7.53E+2 7.75E+2 7.46E+2 7.43E+2 6.91E+2 6.12E+2 7.43E+2 4.54E+2
|
||
|
||
|
||
Table 23
|
||
The average sensitivity for each algorithm of Table 22.
|
||
Algorithm PSO ACO KH DA AHA GWO WOA HWWO
|
||
Avg sensitivity 0.41 0.45 0.42 0.39 0.33 0.29 0.37 0.22
|
||
|
||
|
||
|
||
|
||
Fig. 31. Comparative analysis graph of resource utilization for metaheuristic techniques with varying processors.
|
||
|
||
|
||
Table 24
|
||
Description of unimodal benchmark functions.
|
||
Function Dimensions Range 𝑓𝑚𝑖𝑛
|
||
∑𝑛
|
||
𝐹1 = 𝑖=1 𝑥2𝑖 30 [−100, 100] 0
|
||
∑𝑛 ∏𝑛
|
||
𝐹2 = 𝑖=1 |𝑥𝑖 | + 𝑖=1 |𝑥𝑖 | 30 [−10, 10] 0
|
||
∑𝑛 (∑𝑖 )2
|
||
𝐹3 = 𝑖=1 𝑥
|
||
𝑗=1 𝑗
|
||
30 [−100, 100] 0
|
||
𝐹4 = 𝑚𝑎𝑥𝑖 {|𝑥𝑖 |, 1 ≤ 𝑖 ≤ 𝑛} 30 [−100, 100] 0
|
||
∑𝑛−1 [ ]
|
||
𝐹5 = 𝑖=1 100(𝑥𝑖+1 − 𝑥2𝑖 )2 + (𝑥𝑖 − 1)2 30 [−30, 30] 0
|
||
∑𝑛
|
||
𝐹6 = 𝑖=1 ([𝑥𝑖 + 0.5])2 30 [−100, 100] 0
|
||
∑𝑛
|
||
𝐹7 = 𝑖=1 𝑖𝑥4𝑖 + 𝑟𝑎𝑛𝑑𝑜𝑚[0, 1) 30 [−1.28, 1.28] 0
|
||
|
||
|
||
|
||
algorithm based on unimodal benchmark functions. This table shows that the proposed algorithm achieves superior performance in solving
|
||
the rank of the HWWO algorithm in comparison to the second algo- unimodal benchmark problems, demonstrating faster convergence and
|
||
rithm, focusing on the best fitness values. 𝑇 + represents the superiority greater accuracy compared to existing methods.
|
||
of the proposed HWWO technique. P-values, calculated at a 5% signifi-
|
||
Finally, the convergence behavior of the proposed HWWO algo-
|
||
cance level, test the null hypothesis that the median difference between
|
||
rithm is depicted through convergence curves compared with other
|
||
the algorithms is zero. The final row of Table 26 consolidates the counts
|
||
algorithms in Fig. 32. The convergence rate, which evaluates an al-
|
||
of 𝑇 + and 𝑇 − , along with the test statistic, offering a clear summary of
|
||
gorithm’s efficiency in reaching the optimal solution, is analyzed by
|
||
the results. The analysis reveals significant improvements in the HWWO
|
||
comparing HWWO’s performance with existing metaheuristic tech-
|
||
algorithm’s performance, with 𝑇 + accounting for 93.87% and 𝑇 − for
|
||
niques. In the graph, the 𝑥-axis represents the number of iterations,
|
||
6.12% of the evaluated benchmarks (𝑝 < 0.05). These results suggest
|
||
while the 𝑦-axis shows the average fitness values computed over 1000
|
||
|
||
33
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
Table 25
|
||
Outcomes of using unimodal functions.
|
||
F Measure Optimization algorithm
|
||
PSO ACO KH DA AHA GWO WOA HWWO
|
||
𝐹1 Best 1.709E−13 1.471E−10 1.660E−19 1.835E−13 3.003E−49 3.010E−79 5.763E−60 3.755E−94
|
||
Mean 8.644E−11 1.121E−09 6.660E−17 4.992E−12 9.206E−48 2.013E−67 1.847E−57 2.401E−87
|
||
Worst 2.483E−10 6.170E−09 5.873E−16 3.849E−11 2.162E−47 8.057E−67 9.399E−57 8.552E−87
|
||
St. Dev 1.889E−01 7.223E−04 1.861E−01 3.046E−03 1.418E−47 3.413E−72 2.149E−03 4.027E−72
|
||
𝐹2 Best 5.069E−12 8.035E−06 1.803E−34 1.609E−09 7.112E−29 4.925E−55 2.757E−65 2.283E−80
|
||
Mean 9.183E−11 1.290E−03 6.257E−33 1.325E−08 1.116E−28 2.350E−52 9.432E−45 5.662E−60
|
||
Worst 2.272E−10 5.056E−03 1.230E−32 3.626E−08 1.931E−28 9.217E−52 3.775E−44 2.220E−59
|
||
St. Dev 9.594E−11 2.511E−03 6.834E−33 1.611E−08 5.517E−27 3.559E−52 1.088E−44 1.106E−59
|
||
𝐹3 Best 5.077E−07 5.783E−02 2.497E−06 3.273E−26 3.156E−05 3.10E−128 6.837E−03 6.389E−28
|
||
Mean 3.377E−01 1.123E+00 8.221E−05 1.984E−24 9.296E−03 2.683E−82 1.808E−02 2.711E−27
|
||
Worst 1.350E+00 3.310E+00 1.607E−04 7.912E−24 3.513E−02 1.073E−81 4.471E−02 1.080E−26
|
||
St. Dev 6.753E−01 1.481E+00 8.191E−05 3.952E−24 1.723E−02 5.367E−82 1.784E−02 5.398E−27
|
||
𝐹4 Best 7.275E−04 1.655E−03 5.300E−17 3.102E−05 1.969E−54 2.551E−21 1.388E−62 2.277E−78
|
||
Mean 1.654E−02 5.793E−02 8.969E−16 2.358E−04 2.149E−49 1.030E−19 2.346E−43 2.971E−53
|
||
Worst 4.365E−02 1.838E−01 1.704E−15 4.706E−04 8.596E−49 3.632E−19 9.346E−43 1.188E−52
|
||
St. Dev 1.975E−02 8.624E−02 8.320E−16 2.214E−04 4.297E−49 1.741E−19 4.667E−43 5.942E−53
|
||
𝐹5 Best 7.65461E 7.68388E 9.70569E 3.126E−01 6.15363E 4.291E−02 1.18142E 3.113E−04
|
||
Mean 7.99255E 8.42474E 5.91556E+02 2.924E+00 6.68606E 7.121E−01 4.65548E 1.267E−03
|
||
Worst 8.42156E 9.31820E 1.20253E+03 5.389E+00 7.23273E 1.596E+00 6.10117E 2.696E−03
|
||
St. Dev 3.32511E−01 7.36628E−01 5.00585E+02 2.924E+00 6.07412E−1 7.068E−01 2.32924E 1.013E−03
|
||
𝐹6 Best 3.543E−01 7.685E−05 3.620E−06 9.22444E−10 2.039E−03 2.907E−16 6.490E−20 1.549E−12
|
||
Mean 6.382E−01 6.237E−03 4.380E−06 1.23821E−09 4.113E−03 7.377E−14 3.630E−16 7.981E−11
|
||
Worst 8.362E−01 1.772E−02 4.993E−06 1.62791E−09 7.156E−03 2.903E−13 1.324E−15 1.891E−10
|
||
St. Dev 2.061E−01 1.444E−13 5.766E−07 2.97023E−10 2.229E−03 8.229E−03 6.436E−16 9.242E−11
|
||
𝐹7 Best 3.457E−03 6.312E−03 8.926E−04 9.48224E−04 2.857E−04 3.366E−05 1.03E+4 6.443E−06
|
||
Mean 9.097E−03 2.395E−02 1.791E−03 1.97178E−03 6.290E−04 2.899E−04 1.11E+4 2.188E−04
|
||
Worst 2.001E−02 5.483E−02 2.728E−03 3.58221E−03 8.085E−04 8.911E−04 1.22E+4 5.489E−04
|
||
St. Dev 7.412E−03 2.124E−02 7.973E−04 1.24604E−03 2.388E−04 4.095E−04 3.01E+2 2.247E−04
|
||
|
||
|
||
|
||
Table 26
|
||
Results of Wilcoxon signed-rank test of Table 25.
|
||
Problem HWWO vs PSO HWWO vs ACO HWWO vs KH HWWO vs DA HWWO vs AHA HWWO vs GWO HWWO vs WOA
|
||
Rank Rank Rank Rank Rank Rank Rank
|
||
𝐹1 1 1 2 2 2 1 3
|
||
𝐹2 2 2 1 4 3 2 1
|
||
𝐹3 3 6 4 1 4 3 5
|
||
𝐹4 4 4 3 5 1 4 2
|
||
𝐹5 7 7 7 7 7 7 6
|
||
𝐹6 6 3 5 3 6 5 4
|
||
𝐹7 5 5 6 6 5 6 7
|
||
p-value 0.0355 0.0013 0.0281 0.0176 0.0262 0.0474 0.0087
|
||
𝑇 + = Sum of positive number ranks 28 28 28 28 28 20 24
|
||
𝑇 − = Sum of negative number ranks 0 0 0 0 0 8 4
|
||
𝑇 = min(𝑇 + , 𝑇 − ) 0 0 0 0 0 8 4
|
||
|
||
|
||
|
||
|
||
tasks using 100 processors. For clarity, the graph illustrates the average reflecting its well-balanced integration of exploration and exploitation
|
||
fitness values from 10 independent runs, each evaluated over 500 for enhanced optimization performance.
|
||
iterations. As shown in Fig. 32, the HWWO algorithm demonstrates
|
||
rapid convergence toward optimal solutions, outperforming other algo-
|
||
7. Conclusions
|
||
rithms. This superior performance stems from HWWO’s hybrid design,
|
||
which integrates the strengths of WOA and GWO. By combining these
|
||
To address the challenge of tasks scheduling in a heterogeneous
|
||
techniques, HWWO effectively overcomes the limitations of premature
|
||
distributed computing environment, this research proposes a hybrid
|
||
and slow convergence inherent in WOA. The figure underscores the lim-
|
||
meta-heuristic technique called HWWO, which amalgamates the WOA
|
||
itations of PSO and ACO algorithms, primarily their weak exploitation
|
||
and the GWO. The paper presents a reliability-based energy-efficient
|
||
capabilities. Despite iterating extensively through the solution space,
|
||
scheduling model designed to reduce energy consumption and enhance
|
||
these algorithms often fail to reach the optimal solution due to an
|
||
the reliability of applications running on heterogeneous computing
|
||
imbalance between exploration and exploitation phases. Similarly, DA
|
||
platforms, all while adhering to strict deadline requirements. The ap-
|
||
and KH exhibit strong exploratory abilities in the initial stages but
|
||
plications are elegantly modeled using DAGs. The article proposes a
|
||
frequently become trapped in local optima, preventing convergence
|
||
novel scheduling algorithm that combines the WOA and the GWO with
|
||
to the optimal solution. In contrast, WOA initially outperforms GWO
|
||
DVFS capabilities, along with an insert-reversed block operation. This
|
||
in generating promising solutions, but GWO surpasses WOA in later
|
||
hybrid approach aims to minimize both static and dynamic energy
|
||
iterations by refining the search process. HWWO, however, demon-
|
||
consumption. The article presents a refined technique to simultaneously
|
||
strates steady improvement in fitness value with increasing iterations,
|
||
tackle the challenges of tasks scheduling on appropriate processors
|
||
while considering multiple objectives. The proposed method seeks to
|
||
|
||
34
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||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
|
||
|
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Fig. 32. Comparison of convergence curves of HWWO and literature algorithms.
|
||
|
||
|
||
optimize overall energy consumption, computational time, and system HWWO consistently demonstrated the minimal average sensi-
|
||
reliability concurrently, offering a comprehensive solution to address tivity ratio and rapid convergence towards optimal solutions,
|
||
these critical factors. Extensive experiments highlight the proposed outperforming existing metaheuristic algorithms.
|
||
model’s effectiveness in considerably reducing energy consumption and vii A Wilcoxon-signed rank test is utilized to statistically evaluate
|
||
processing time, increasing system reliability, and maintaining low the effectiveness of the results.
|
||
complexities. The key contributions based on experimental findings are: viii The run time and space complexities for the proposed method
|
||
are calculated, both equating to 𝑂(|𝑌 | ∗ |𝑋|) + 𝑂(|𝑌 |). Notably,
|
||
i The article introduces a hybrid scheduling mechanism, termed in terms of complexities, the proposed algorithm demonstrates
|
||
HWWO, that integrates SI techniques, specifically WOA and the superior performance compared to other existing algorithms in
|
||
GWO, to tackle real-world applications effectively. this domain.
|
||
ii The WOA algorithm exhibits rapid convergence and strikes a
|
||
balance between exploration and exploitation when solving op-
|
||
timization problems. However, its encircling search mechanism 7.1. Limitations and future work
|
||
can occasionally lead it to converge prematurely on local optima.
|
||
To mitigate this issue, a hybrid approach has been devised
|
||
by incorporating the GWO, synergizing the capabilities of both In this article, a hybrid model is developed to solve the reliability-
|
||
optimization techniques. based energy-efficient tasks scheduling problem with multiple objec-
|
||
iii Extensive evaluations are conducted on real-world FFT and tives. The model successfully reduces total energy consumption com-
|
||
GE applications to compare the proposed model’s performance pared to existing methods, although it fails to reduce static energy
|
||
against various state-of-the-art methods. consumption individually. As shown in Tables 12 and 16, the proposed
|
||
iv The proposed algorithm is rigorously evaluated on real-world HWWO approach exhibits superior energy consumption performance
|
||
single-objective constrained optimization problems from the for reliability thresholds up to 0.98. However, beyond this threshold,
|
||
CEC 2020 competition. Comprehensive comparisons are con- the method fails to meet reliability constraints due to the absence of
|
||
ducted against the competition’s state-of-the-art algorithms, in- fault-tolerance mechanisms, indicating that 𝑅𝑒(𝑔𝑜𝑎𝑙) (𝐺) cannot always
|
||
cluding SASS, EnMODE, sCMAgES, and COLSHADE. Addition- be satisfied. Addressing these limitations, future research will focus
|
||
ally, the algorithm’s performance is assessed on a set of uni- on integrating fault-tolerance mechanisms into the hybrid model more
|
||
modal benchmark test functions and compared to established efficiently. This integration could involve implementing error detection
|
||
metaheuristic approaches. and correction techniques and robust optimization strategies to ensure
|
||
v The experiments reveal the proposed algorithm’s superiority continuous and accurate tasks scheduling, even in the presence of
|
||
over existing state-of-the-art and metaheuristic methods. It ex- faults.
|
||
cels in energy efficiency, reliability maximization, computation The method is effective within computing environments where
|
||
time and SLR minimization, CCR optimization, and resource uti- processors are fully connected. To address its limitations, enhancing the
|
||
lization enhancement across diverse scale conditions and dead- framework involves refining scheduling algorithms and assessing them
|
||
line constraints. across various workflows such as LIGO, SIPHT, and molecular dynamic
|
||
vi The effectiveness and scalability of the proposed HWWO method code. Furthermore, the proposed framework’s versatility allows for
|
||
are assessed through sensitivity analysis and implementation on potential extensions to diverse computing system environments such
|
||
varying tasks and processor counts. The outcomes revealed that as grid computing, cloud computing, and cluster computing.
|
||
|
||
35
|
||
Karishma and H. Kumar Computer Standards & Interfaces 97 (2026) 104106
|
||
|
||
|
||
CRediT authorship contribution statement [13] H. Xu, R. Li, C. Pan, K. Li, Minimizing energy consumption with reliability goal
|
||
on heterogeneous embedded systems, J. Parallel Distrib. Comput. 127 (2019)
|
||
44–57, http://dx.doi.org/10.1016/j.jpdc.2019.01.006.
|
||
Karishma: Writing – original draft, Validation, Software, Resources,
|
||
[14] L. Zhang, M. Ai, K. Liu, J. Chen, K. Li, Reliability enhancement strategies for
|
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Methodology, Investigation, Data curation, Conceptualization. Haren- workflow scheduling under energy consumption constraints in clouds, IEEE
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dra Kumar: Validation, Supervision, Methodology, Investigation, For- Trans. Sustain. Comput. 9 (2) (2024) 155–169, http://dx.doi.org/10.1109/
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mal analysis, Conceptualization. TSUSC.2023.3314759.
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[15] L. Zhang, K. Li, K. Li, Y. Xu, Joint optimization of energy efficiency and system
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reliability for precedence constrained tasks in heterogeneous systems, Int. J.
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Funding Electr. Power Energy Syst. 78 (2016) 499–512, http://dx.doi.org/10.1016/j.
|
||
ijepes.2015.11.102.
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The authors declare that no funds, grants, or other support were [16] G. Xie, H. Peng, Z. Li, J. Song, Y. Xie, R. Li, K. Li, Reliability enhancement
|
||
toward functional safety goal assurance in energy-aware automotive cyber-
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received during the preparation of this manuscript.
|
||
physical systems, IEEE Trans. Ind. Informatics 14 (12) (2018) 5447–5462,
|
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http://dx.doi.org/10.1109/TII.2018.2854762.
|
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Ethics approval and consent to participate [17] L. Ye, Y. Xia, S. Tao, C. Yan, R. Gao, Y. Zhan, Reliability-aware and energy-
|
||
efficient workflow scheduling in IaaS clouds, IEEE Trans. Autom. Sci. Eng. 20
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(3) (2023) 2156–2169, http://dx.doi.org/10.1109/TASE.2022.3195958.
|
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This article does not contain any studies with human participants
|
||
[18] X. Xiao, G. Xie, C. Xu, C. Fan, R. Li, K. Li, Maximizing reliability of energy con-
|
||
or animals performed by any authors. strained parallel applications on heterogeneous distributed systems, J. Comput.
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Sci. 26 (2018) 344–353, http://dx.doi.org/10.1016/j.jocs.2017.05.002.
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Declaration of competing interest [19] G. Xie, Y. Chen, Y. Liu, Y. Wei, R. Li, K. Li, Resource consumption cost
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minimization of reliable parallel applications on heterogeneous embedded
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systems, IEEE Trans. Ind. Informatics 13 (4) (2016) 1629–1640, http://dx.doi.
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The authors declare that they have no known competing finan- org/10.1109/TII.2016.2641473.
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cial interests or personal relationships that could have appeared to [20] H. Djigal, J. Feng, J. Lu, J. Ge, IPPTS: An efficient algorithm for scientific
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influence the work reported in this paper. workflow scheduling in heterogeneous computing systems, IEEE Trans. Parallel
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Data availability [21] Z. Deng, Z. Yan, H. Huang, H. Shen, Energy-aware task scheduling on
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heterogeneous computing systems with time constraint, IEEE Access 8 (2020)
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Data will be made available on request. 23936–23950, http://dx.doi.org/10.1109/ACCESS.2020.2970166.
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[22] Z. Quan, Z.-J. Wang, T. Ye, S. Guo, Task scheduling for energy consumption
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constrained parallel applications on heterogeneous computing systems, IEEE
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