Greedy scheduling of tasks with time constraints for energy-efficient cloud-computing data centers
- Ziqian Dong^{1}Email author,
- Ning Liu^{2} and
- Roberto Rojas-Cessa^{3}
https://doi.org/10.1186/s13677-015-0031-y
© Dong et al.; licensee Springer. 2015
Received: 22 October 2014
Accepted: 23 February 2015
Published: 18 March 2015
Abstract
In this paper, we introduce a model of task scheduling for a cloud-computing data center to analyze energy-efficient task scheduling. We formulate the assignments of tasks to servers as an integer-programming problem with the objective of minimizing the energy consumed by the servers of the data center. We prove that the use of a greedy task scheduler bounds the constraint service time whilst minimizing the number of active servers. As a practical approach, we propose the most-efficient-server-first task-scheduling scheme to minimize energy consumption of servers in a data center. Most-efficient-server-first schedules tasks to a minimum number of servers while keeping the data-center response time within a maximum constraint. We also prove the stability of most-efficient-server-first scheme for tasks with exponentially distributed, independent, and identically distributed arrivals. Simulation results show that the server energy consumption of the proposed most-efficient-server-first scheduling scheme is 70 times lower than that of a random-based task-scheduling scheme.
Keywords
Cloud computing Energy efficiency Data center Greedy algorithm Integer programmingIntroduction
Cloud computing has risen as a new computing paradigm that brings unparalleled flexibility and access to shared and scalable computing resources. The increasing demand for data processing and storage in this digital world is leading a significant growth of data centers, the size of which has grown from 1000’s to a few hundred thousands servers [1].
Cloud-computing data centers offer information technology (IT) resources as services. The hardware systems (servers, data center network systems, storage, etc.) and software systems (operating systems, management software, etc.) represent the resources the data center provides as Infrastructure as a Service (IaaS) and Platform as a Service (PaaS), respectively. Applications, such as web search, social networking, computation, etc., offered by cloud-computing data centers are hosted as Software as a Service (SaaS) [2]. These applications run on virtualized IT resources, namely, virtual machines (VMs), provided by IaaS and PaaS. Based on the request, the cloud service providers provision resources such as different types of VMs to the requests.
Energy consumption of a data center constitutes a major operation cost [3-5]. The energy consumed by these large-scale data centers has reached billions of Kilowatt-hours per year and is expected to continue to grow [6]. The increasing energy demand could become a hurdle to data center scalability, let alone the carbon footprint they would leave [3-5]. An Emerson report estimates that the servers of a data center account for 52% of the total consumed energy, while the cooling systems account for 38%, and other miscellaneous supporting systems, such as power distribution, account for the remaining 10% [5]. These three different sub-systems of a data center may be optimized for energy efficiency.
In this paper, we target the reduction of energy expenditure of the servers of a data center and address this issue by bounding the number of active servers for workloads that require a constrained response time. We model the energy consumption of a data center and analyze the trade-off between the response time and the number of active servers as an integer-programming optimization problem. Optimization of resource allocation in large-scale data centers is non-trivial and non-scalable.
As a practical solution, we propose the most-efficient-server-first (MESF) task-scheduling scheme to minimize the energy consumption while keeping the response within a constrained time. Here, a task is a request for a job of the contracted application that may require a defined amount of resources and the creation of a VM to support the application. The job may be data transmission (uploading and downloading), data processing, software access and execution, or storage functions. Each task, as the corresponding VM, is then assigned to one of the available servers. In turn, the task is performed and the result (or a completion notice) is returned to the user. Furthermore, because tasks are assigned to servers as soon as they arrive, queues build up on some of the servers. Therefore, we analyze the queueing delay of tasks for the proposed MESF scheduling scheme and prove that the scheduling scheme is weakly stable under independent and identically distributed (i.i.d.) task arrivals that follow an exponential distribution. By simulation, we show the impact of MESF on the energy consumed by a data center, and compare it to that of a data center that assigns tasks to servers randomly [7]. Our simulation results show that MESF may reduce the data center energy consumption 70 times that consumed by the scheme that assigns tasks randomly.
The remainder of this paper is organized as follows. We discuss the related work in the next section. The Data center model section presents the model of the cloud-computing data center adopted in this paper. Task scheduling and energy consumption section introduces the proposed energy consumption model for a cloud-computing data center. The most-efficient server first scheme section introduces the MESF task-scheduling scheme. The Stability analysis of MESF section presents the stability analysis of the proposed scheduling scheme. The Simulation results section presents our simulation results of energy consumption and task response time for the proposed MESF and random task-scheduling schemes. The Conclusions section presents our conclusions.
Related work
The servers of a data center account for the largest amount of energy consumed by the data center [5]. Recent works focus on schemes aiming at reducing energy consumption by servers through efficient job scheduling, resource allocation optimization, and virtual machine consolidation [8-12]. A conservative allocation of resources and jobs in data centers may lead to powering ON a large number of servers, contributing to a large amount of consumed energy [13]. Energy-aware job allocation schemes may be used towards receding the energy consumed by servers [14]. In such a scheme, the traffic distribution and link states of a data-center network are considered for deciding to which servers jobs are allocated. The objective of these schemes is to consolidate network traffic and server load to reduce the fraction of active network equipment, set link speeds to match traffic demand, and turn off non-critical servers.
Cloud-computing data centers may use VMs to consolidate workloads to reduce the number of active servers [9-12,15-19]. To ensure that the service level agreement (SLA) is met, cloud-computing data centers may set up upper limits for resource utilization while placing VMs. This may lead to poor utilization of resources due to the dynamics of data center workloads. A scheme that uses dynamic thresholds was proposed considering the following policies for placing VMs: minimization of VM migration, balancing of resource utilization and SLA violation, and random selection [9]. This scheme achieves energy efficiency by allowing a level of SLA violations. Another approach introduces a power management scheme that implements multiple feedback controllers at the levels of racks, servers, and VMs to improve data center energy efficiency [11]. Although reducing the number of active servers in a data center may improve energy efficiency, over-consolidation may jeopardize the quality of the service (QoS) provided by the data center. Maintaining QoS whilst increasing energy efficiency is critical for the economical sustainability of a data center [10,15-17]. Minimizing the number of virtual-machine migrations may be employed to maintain service guarantees and to reduce the energy consumption of cloud-computing data centers [16]. In a more direct approach, jobs may be mapped to the existing computing resources to ensure that the satisfaction of the required QoS of different applications [20]. However, these methods require global knowledge of the state of the data center and, in turn, a fast central controller to perform timely decisions for the dynamic data center networks.
Data center model
Data center network
The resources in the cloud-computing data center are shared among a large number of tenants through the data center network. Each tenant may run multiple applications in the data center, thus requesting a large amount of resources. Therefore, the number of servers (and VMs) leased by each tenant is large. Resource provisioning in cloud data centers is a complex process that requires matching of a large number of requests with a large amount of software and hardware while satisfying the SLA. In this paper, we focus on resource provisioning at the SaaS level to study the effect of task scheduling schemes on the energy efficiency of a data center. In this paper, we consider the creation and allocation of VMs in cloud data centers as a part of the task, requiring a well-defined amount of resources such as CPU, memory, storage, etc. from the servers.
Data center workload
Data centers have different types of servers with each type dedicated to handle a specific type of tasks. The processing time and computing resources for different types of tasks may also be different. In 2011, Google released the first set of one of its cluster workload traces to the public, which provides data from a 12K-machine data-center cell recorded over about a month, in May 2011 [26]. The data have enabled studies on trace analysis to characterize data center workload [27,28]. These studies show that the data center workload is highly dynamic and heterogeneous. The workload includes both bursty jobs that demand quick response times and long-running jobs with intensive computing resource requirements. In this paper, we provide a perspective on evaluating the energy consumption of cloud-computing data centers by considering various task deadlines, resource requirements, and server energy profiles.
Task scheduling and energy consumption
A data center is required to handle a large number of tasks demanding different computational resources, e.g. CPU, memory, and communications. Under this variety, servers may provide different response times and consume different levels of energy for different types of tasks. In this paper, we focus on the study of efficient task scheduling to minimize the energy consumption of a data center by reducing the number of active servers.
Optimization of energy consumption by minimizing the number of active servers
Terminology and definitions
We consider a data center with M servers, each denoted as S _{ j }, where 1≤j≤M. The data center can process V types of tasks. A task of type i processed at server j is associated with a deadline or a maximum response time, denoted as B _{ i,j }, where 1≤i≤V,1≤j≤M. Task deadlines may be required by users or self-imposed by the data center [29]. Here, we assume that time is slotted with fixed duration. Considering a constant service rate at the servers, the response time increases proportionally to the number of tasks waiting in the server. Therefore, we use the response time and number of tasks in servers interchangeably in this paper. The processing time for a type-i task at S _{ j } is denoted as μ _{ i,j }. The number of tasks arriving to the data center is denoted as N, and the number of type i tasks is n _{ i }, where
1≤i≤V and \(N= \sum \limits _{i=1}^{V} n_{i}\).
The schedule matrix, X _{ j }, is an \(V \times \sum \limits _{i=1}^{V} x_{i,j}\), where \(\sum \limits _{i=1}^{V} x_{i,j}\) is the total number of tasks assigned to S _{ j }.
Terminology definition
Terminology | Definition |
---|---|
V | Total number of task types |
N | Number of task arrivals |
n _{ i } | Number of type-i task arrivals, where 0≤i≤V |
M | Number of active servers in data center |
S _{ j } | Server j, where 1≤j≤M |
B _{ i,j } | Capacity of S _{ j } to store type-i tasks |
x _{ i,j } | Number of task i assigned to S _{ j } |
μ _{ i,j } | Average processing time of type-i task by S _{ j } |
τ _{ i,j } | Average queueing delay of type-i tasks on S _{ j } |
w _{ i,j } | Queue occupancy of type-i tasks at S _{ j } |
T _{ w } | Average task response time |
X _{ j } | Schedule in which tasks are processed by S _{ j } |
ω | Weight vector |
P _{ i,j } | Power consumed by S _{ j } to process a type-i task |
E _{ i,j } | Energy consumed by S _{ j } to process a type-i task |
E | Total server energy consumption in a data center |
In this paper, we aim to find an optimum task-scheduling scheme to minimize task response time and energy consumed by the data center servers.
Optimization problem
Here, the number of tasks assigned to a server is such that the server ensures that all tasks can be processed within the task deadline constraint. We set the capacity of a server queue at S _{ j } to store type i tasks equal to this deadline or B _{ i,j }. Considering the queueing delay, w _{ i,j }, the number of task arrivals, x _{ i,j }, needs to be no larger than B _{ i,j }−w _{ i,j }.
Analysis of homogeneous tasks
In the remainder of this section, we analyze the assignment of a single type of tasks (V=1) and estimate a bound of the number of servers required to comply with the maximum response time. This can be considered under the assumption that other task types can be decomposed into a linear combination of a unitary task of a basic task type.
with \(\sum \limits _{j=1}^{M}x_{j}=n\).
Here, for a given number of n task arrivals with a Poisson distribution, the number of servers M is bounded by the task deadlines B _{ j } and service rate μ _{ j } at each server. The tightness of the bound is in function of the maximum allowable time to service the task, B _{ j } as (15) shows. For example, when B _{ j }=1, the number of servers required to complete n tasks in one time slot is \(2n- \mu _{j}\sum \limits _{j=1}^{M} \frac {1}{\mu _{j}}\).
The most-efficient server first scheme
In a data center with heterogeneous servers, the servers with the highest computing capacity, which is defined as the maximum number of tasks a sever can process in parallel, are the most preferred servers in the assignment of tasks. This server may provide a lower energy expenditure per processed task (or bytes). In this case, the optimization problem can be interpreted as a greedy-assignment scheme. For this, it is considered that the central scheduler sorts the servers based on their energy efficiency, and assigns tasks to the most energy-efficient servers first and it then continues to allocate tasks to the second most efficient servers on the list, and so on, until no task remains or else, servers’ queues are full.
For a data center with a single server type, MESF assigns a number of tasks to each active server, until the saturation point (where the server performance decays significantly, or the task queueing delay is approaching its delay constraint) is reached. The greedy scheduling scheme is described Algorithm 1.
The central scheduler maintains a sorted list of non-saturated and active servers with their energy profiles. The servers are sorted according to their energy profiles where the most energy-efficient servers are placed on the top of the list. Upon receiving task requests, the scheduler assigns tasks to the servers from the sorted list from top to bottom. The servers receive task assignments and their energy profile is updated. Once the most energy-efficient servers are saturated, they are removed from the list until they become unsaturated.
Discussion on the complexity of the scheduling scheme
The complexity of the proposed MESF scheduling scheme is mostly that of the complexity of sorting servers by their energy efficiency. Without knowledge of the server energy profile, the complexity of the algorithm is based on the sorting complexity of the servers energy profiles, which is O(m ^{2}) to sort m servers [31]. However, the power profile for cloud-computing data center servers are available. The energy profile sorting can be done prior to the server’s activation for function. Therefore, the complexity of the MESF task scheduling scheme is reduced to insertion of a sorted list, which has a time complexity of O(m l o g(m)) [31].
Stability analysis of MESF
In this section, we study the stability of the proposed scheduling scheme and present the conditions to ensure system stability. We first study the condition that makes the system unstable, and in this way, to prove a necessary condition to ensure system stability. We use queueing theory to study the queue length as t approaches infinity. If the queue length diverges (increases indefinitely) and approaches infinity as t approaches infinity, the system is, therefore, unstable.
Here, x _{ i,j }(t) is an independent and identically distributed (i.i.d.) random variable of task type i assigned to server j at time slot t.
where R(t) is the matrix of packet arrivals at time slot t, L(t) is the matrix of the tasks serviced at time slot t.
Here, the task-service matrix, L(k), is defined by the proposed task-scheduling scheme and the response deadline, \(q=\frac {\Delta t}{\tau _{i,j}}\). The scheduler allocates a task based on the minimum of the task deadline and the server queue length. If the task deadline is smaller than the waiting time in the server queue, the task will be placed to a server that meets the deadline constraint.
Evaluation of stability of the proposed task scheduling scheme
Simulation results
In this section, we present the performance evaluation of the proposed MESF task-scheduling scheme. We modeled a data center with a central scheduler and a number of servers in Matlab to evaluate the energy consumption through computing simulation. We simulated the proposed greedy algorithm under homogenous (V=1) and exponentially-distributed task arrivals, with a mean of n=1000 tasks and random server profiles. We measured the average task-response time and total energy consumed with respect to the number of servers available to handle the tasks.
We also modeled and simulated a data center using a random-based task-scheduling scheme [32] to compare the performance of the random-based and the MESF schemes. The random-based task-scheduling scheme assigns tasks to servers on a random basis and without constraints, except for available queue at each server, for task allocation or server selection. We simulated both schemes using 20 task types (V=20) and exponentially distributed task arrivals. Here, we consider that the different types of tasks can be decomposed into a linear combination of a unit task type. Tasks of the same type have the same response time constraint, which is equivalent to the queueing capacity for each type. To simplify the comparison, we set the same response time constraint for all task types in these experiments. We evaluated 1000 experiments (one experiment is a task allocation trial with a duration of sufficient events to allow the distribution of the task to complete) for each scheduling scheme.
Performance comparison of MESF- and random-scheduling schemes
Task scheduling | Average task | Total task | Total energy |
---|---|---|---|
scheme | response time | response time | consumption |
(time units) | (time units) | (energy units) | |
MESF | 472.6 | 4.83E+05 | 2.597E+04 |
Random | 123.9 | 1.33E+05 | 1.989E+06 |
Conclusions
In this paper, we formulated the task assignment for a data center as an integer programming optimization problem and proved the average task response time is bounded with an optimized number of active servers. We proposed a greedy task-scheduling scheme, the most-efficient-server-first scheduling, to reduce energy consumption of data center servers. The proposed MESF scheduling scheme schedules tasks to the most energy-efficient servers of a data center. This scheme minimizes the average task response time and, at the same time, minimizes the server-related energy expenditure. We showed that the system using MESF is weakly stable under i.i.d. task arrivals with an exponential distribution. We evaluated and compared the performance of the proposed scheme with that of a random-based task-scheduling scheme using Matlab simulation. Our simulation results show that a data center using the proposed MESF task-scheduling scheme saves on average over 70 times that of a data center using a random-based task-scheduling scheme. The proposed scheme saves energy at the cost of longer task response times, albeit within the maximum constraint.
Declarations
Acknowledgments
The authors would like to thank the anonymous reviewers for their insightful comments and suggestions on improving this paper.
Authors’ Affiliations
References
- Miller RNew Numbers: Who has the most web servers?http://www.datacenterknowledge.com/archives/2013/07/15/new-numbers-who-has-the-most-web-servers/.
- Calheiros RN, Ranjan R, Buyya R (2011) Virtual machine provisioning based on analytical performance and qos in cloud computing environments In: Parallel Processing (ICPP), 2011 International Conference On, 295–304.. IEEE, Taipei, Taiwan.View ArticleGoogle Scholar
- Patel CD, Shah AJ (2005) Cost Model for Planning, Development and Operation of a Data Center. http://www.hpl.hp.com/techreports/2005/HPL-2005-107R1.pdf.
- Baliga J, Ayre RWA, Hinton K, Tucker RS (2011) Green cloud computing: balancing energy in processing, storage, and transport In: Proceedings of the IEEE, vol. 99, 149–167.Google Scholar
- Energy Logic: Reducing Data Center Energy Consumption by Creating Savings that Cascade Across Systems. A White Paper from Experts in Business-Critical Continuity. http://www.emersonnetworkpower.com/documentation/en-us/latest-thinking/edc/documents/white paper/ energylogicreducingdatacenterenergyconsumption.pdf
- Brown R, Masanet E, Nordman B, Tschudi B, Shehabi A, Stanley J, et al (2008) Report to congress on server and data center energy efficiency: Public law 109-431. National Laboratory, Lawrence, Berkeley. https://escholarship.org/uc/item/74g2r0vg#page-1.Google Scholar
- Berral JL, Goiri Í, Nou R, Julià F, Guitart J, Gavaldà R, Torres J (2010) Towards energy-aware scheduling in data centers using machine learning In: Proceedings of the 1st, International Conference on energy-Efficient Computing and Networking, 215–224.. ACM, New York, NY, USA.View ArticleGoogle Scholar
- Bohra AEH, Chaudhary V (2010) Vmeter: Power modelling for virtualized clouds In: Parallel Distributed Processing, Workshops and Phd Forum (IPDPSW), 2010 IEEE International Symposium On, 1–8.. IEEE, Atlanta, GA, USA.View ArticleGoogle Scholar
- Goiri I, Julia F, Nou R, Berral JL, Guitart J, Torres J (2010) Energy-aware scheduling in virtualized datacenters In: Cluster Computing (CLUSTER), 2010 IEEE International Conference On, 58–67.. IEEE, Heraklion, Crete, Greece.View ArticleGoogle Scholar
- Beloglazov A, Buyya R (2010) Energy efficient resource management in virtualized cloud data centers In: Proceedings of the 2010 10th IEEE/ACM International Conference on Cluster, Cloud and Grid Computing, 826–831.. IEEE Computer Society, Washington, DC, USA.View ArticleGoogle Scholar
- Raghavendra R, Ranganathan P, Talwar V, Wang Z, Zhu X (2008) No “power” struggles: coordinated multi-level power management for the data center In: Proceedings of the 13th International Conference on Architectural Support for Programming Languages and Operating Systems. ASPLOS XIII, 48–59.. ACM, New York, NY, USA.View ArticleGoogle Scholar
- Zhang Q, Cheng L, Boutaba R (2010) Cloud computing: state-of-the-art and research challenges. J Internet Serv Appl 1(1): 7–18.View ArticleGoogle Scholar
- Glanz J (2012) Power, Pollution and the, Internet. The New York Times, Vol. 22. http://www.nytimes.com/2012/09/23/technology/data-centers-waste-vast-amounts-of-energy-belying-industry-image.html.
- Mahadevan P, Sharma P, Banerjee S, Ranganathan P (2009) Energy aware network operations In: INFOCOM Workshops 2009, 1–6.. IEEE, Rio de Janeiro, Brazil.View ArticleGoogle Scholar
- Buyya R, Beloglazov A, Abawajy J (2010) Energy-efficient management of data center resources for cloud computing: A vision, architectural elements, and open challenges In: arXiv preprint arXiv:1006.0308. http://arxiv.org/abs/1006.0308.
- Beloglazov A, Abawajy J, Buyya R (2010) Energy-aware resource allocation heuristics for efficient management of data centers for cloud computing. Future Generation Comput Syst 28(5): 755–768.View ArticleGoogle Scholar
- Xu Y, Rojas-Cessa R, Grebel H (2012) Allocation of discrete energy on a cloud-computing datacenter using a digital power grid In: Green Computing and Communications (GreenCom), 2012 IEEE International Conference On, 615–618.. IEEE, Besancon, France.View ArticleGoogle Scholar
- Beloglazov A, Buyya R (2010) Energy efficient allocation of virtual machines in cloud data centers In: Cluster, Cloud and Grid Computing (CCGrid), 2010 10th IEEE/ACM International Conference On, 577–578.. IEEE.Google Scholar
- Rojas-Cessa R, Pessima S, Tian T (2012) Experimental evaluation of energy savings of virtual machines in the implementation of cloud computing In: Proc. IEEE WOCC 2012, 1–6.. IEEE, Kaohsiung, Taiwan.Google Scholar
- Papagianni C, Leivadeas A, Papavassiliou S, Maglaris V, Cervello-Pastor C, Monje A (2013) On the optimal allocation of virtual resources in cloud computing networks. Comput IEEE Trans 62(6): 1060–1071.View ArticleMathSciNetGoogle Scholar
- Al-Fares M, Loukissas A, Vahdat A (2008) A scalable, commodity data center network architecture In: ACM SIGCOMM Computer Communication Review, vol. 38, 63–74.. ACM, New York, NY, USA.Google Scholar
- Greenberg A, Hamilton JR, Jain N, Kandula S, Kim C, Lahiri P, Maltz DA, Patel P, Sengupta S (2009) Vl2: a scalable and flexible data center network In: ACM SIGCOMM Computer Communication Review, vol. 39, 51–62.. ACM, New York, NY, USA.Google Scholar
- Niranjan Mysore R, Pamboris A, Farrington N, Huang N, Miri P, Radhakrishnan S, Subramanya V, Vahdat A (2009) Portland: a scalable fault-tolerant layer 2 data center network fabric In: ACM SIGCOMM Computer Communication Review, vol. 39, 39–50.. ACM, New York, NY, USA.Google Scholar
- Dong Z, Rojas-Cessa R, Oki E (2011) Memory-memory-memory clos-network packet switches with in-sequence service In: High Performance Switching and Routing (HPSR), 2011 IEEE 12th International Conference On, 121–125.. IEEE, Cartagena, Spain.View ArticleGoogle Scholar
- Dong Z, Rojas-Cessa R (2012) MCS: buffered Clos-network switch with in-sequence packet forwarding In: Sarnoff Symposium (SARNOFF), 2012 35th IEEE, 1–6.. IEEE, Newark, NJ, USA.View ArticleGoogle Scholar
- Reiss C, Wilkes J, Hellerstein JL (2011) Google cluster-usage traces: format + schema. Technical report, Google, Inc, Mountain View, CA, USA. http://code.google.com/p/googleclusterdata/wiki/TraceVersion2.Google Scholar
- Reiss C, Tumanov A, Ganger GR, Katz RH, Kozuch MA (2012) Heterogeneity and dynamicity of clouds at scale: Google trace analysis In: ACM Symposium on Cloud Computing (SoCC).. ACM, San Jose, CA, USA. http://www.pdl.cmu.edu/PDL-FTP/CloudComputing/googletrace-socc2012.pdf.Google Scholar
- Liu Z, Cho S (2012) Characterizing machines and workloads on a Google cluster In: 8th International, Workshop on Scheduling and Resource Management for Parallel and Distributed Systems (SRMPDS’12).. IEEE, Pittsburgh, PA, USA.Google Scholar
- DeCandia G, Hastorun D, Jampani M, Kakulapati G, Lakshman A, Pilchin A, Sivasubramanian S, Vosshall P, Vogels W (2007) Dynamo: amazon’s highly available key-value store In: ACM SIGOPS Operating Systems Review, vol. 41, 205–220.. ACM, New York, NY, USA.Google Scholar
- Tam ASW, Xi K, Chao HJ (2011) Use of devolved controllers in data center networks In: Computer Communications Workshops (INFOCOM WKSHPS), 2011 IEEE Conference On, 596–601.. IEEE, Shanghai, China.View ArticleGoogle Scholar
- Cormen TH, Leiserson CE, Riverst RL, Stein C (2001) Introduction to Algorithms. 2nd edn. The MIT Press, Cambridge, Massachusetts.MATHGoogle Scholar
- Khan SU, Ahmad I (2006) Non-cooperative, semi-cooperative, and cooperative games-based grid resource allocation In: Parallel and Distributed Processing Symposium, 2006. IPDPS 2006. 20th International, 10.. IEEE, Rhodes Island, Greece.Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.