Gaussian Mixture Models vs. Fuzzy Rule-Based Systems for Adaptive ...

Gaussian Mixture Models vs. Fuzzy Rule-Based Systems for Adaptive Meta-scheduling in Grid/Cloud Computing R.P. Prado1, J. Braun2 , J. Krettek2, F. Hoffmann2, S. García-Galán1, J.E. Muñoz Expósito1, and T. Bertram2 1

2

Telecommunication Engineering Department in University of Jaén, Alfonso X el Sabio, 28 Linares, Jaén. Spain {rperez,sgalan,jemunoz}@ujaen.es Control System Engineering, University Dortmund, D-44221 Dortmund, Germany {jan.braun,johannes.krettek,frank.hoffmann, torsten.bertram}@tu-dortmund.de

Abstract. Adaptive scheduling strategies are about considering the state of computational grids to obtain efficient and reliable schedules and to prevent the system performance deterioration. In this work, emerging adaptive strategies in grid computing, namely Fuzzy Rule-Based Systems (FRBS) -based strategies and a new adaptive scheduling approach, gaussian scheduling founded on Gaussian Mixture Models (GMMs) are compared. Both types of strategies focus on modeling the state of resources and select the most convenient site of the grid at every scheduling step given the current conditions. FRBSs provide a fuzzy characterization of the grid state and the inference of a suitability index based on their own knowledge given in the form of fuzzy IF-THEN rules. Besides, a GMM can be trained to model a complex probability density distribution indicating the suitability of every site in the grid to be the target of the schedule with the current conditions of its resources. This way the GMM scheduler assigns a probability to every state of the site where a higher probability is associated to a higher suitability of selection. Simulations based on real grid facilities are conducted to test the FRBS and GMM-based models and results are analyzed in terms of accuracy and convergence behaviour of their associated learning processes.

1 Introduction Cloud computing is a new paradigm of distributed computing that provides companies with services such as the execution of applications (SoS), platforms for the development of applications (PaaS) and/or an infrastructure (IaaS) with high computing and storing capacity [7]. The final aim of these services consists of the improvement of productivity that companies can reach by externalizing their computational needs and hiring cloud services. On the other hand, a computational grid is a collection of autonomous, heterogeneous and geographically distributed J. Casillas et al. (Eds.): Management Intelligent Systems, AISC 171, pp. 295–304. c Springer-Verlag Berlin Heidelberg 2012 springerlink.com

296

R.P. Prado et al.

computing resources that cooperate and share capacities to achieve a common goal [3]. Hence, an improvement in the features of a grid system represents an improvement in the features of a cloud system that uses it as underlying infrastructure. A major challenge of these computing networks is determined by their efficient cooperation and coordination or scheduling which is a NP-complete problem. Adaptive scheduling strategies propose to consider both current and future resources conditions to avoid and prevent the grid system performance degradation [11]. In this regard, schedule-based strategies base their decisions on a “known” current state of the grid system to allow a more precise schedule of jobs and satisfy diverse QoS (Quality of Service) specifications [6]. The “known” current state makes reference to available resources domain capabilities, computational demands, number of jobs, etc. A key aspect is how to design a scheduling strategy able to simultaneously integrate and combine the diverse criteria describing the current conditions of the grid and to provide with a mechanism that founded on this state, can select the most suitable resource or site for the actual schedule. In this work, emerging adaptive scheduling strategies based on FRBSs, fuzzy rule-based schedulers [8, 9] and a novel strategy derived from Gaussian Mixture Models (GMMs) [2, 12] are compared. Fuzzy Rule-Based Systems (FRBSs) [1] are expert systems derived from fuzzy logic and rule-based systems recently proposed as an alternative for the development of scheduling systems in grid computing on the basis of their adaptability to environments dynamism and capability to cope with uncertainty in systems information [8, 9]. Essentially, fuzzy rule-based schedulers suggest a fuzzy characterization of the states of sites and they associate a suitability index to every site according to their own knowledge of the system given in the form of IF-THEN rules. In this regard, given the dependence of these schedulers with their associated knowledge, learning strategies must be considered in their design such as Michigan and Pittsburgh approaches [1]. On the other hand, a scheduling strategy based on the consideration of probability density functions to model the state of the diverse resources domains making up the grid is presented. Specifically, the application of GMMs [12] is suggested for meta-scheduling in grid computing to model the state of resources and decide the most suitable resource selection at each stage. A GMM is a statistical model that uses a weighted sum of probability density functions of multiple Gaussian distributions to represent the distribution of a vector in the probability space [2, 12]. A GMM is trained to model a meta-scheduler that bases its decisions on a density distribution associated to the different resource domains (RDs) or sites and the state of these resources. Subject to a state of the grid, a probability of selection for the next schedule is obtained for every RD where the site showing a higher probability is selected as the most suitable target domain. Two approaches based on these models with evolutionary learning are proposed for meta-scheduling in grid computing and results are analyzed in terms of accuracy and convergence behaviour to fuzzy rule-based meta-schedulers with Michigan and Pittsburgh approaches as learning strategies. The rest of the paper is summarized as follows. Sections 2 and 3 introduce an overview of fuzzy rule-based schedulers and present the novel adaptive approach,

Adaptive Meta-scheduling in Grid/Cloud Computing with GMM vs. FRBS

297

respectively. In Section 4, the simulation results are analyzed for both types of schedulers and finally, Section 5 outlines the main conclusions.

2 Fuzzy Rule-Based Schedulers As introduced in the previous section, the state of resources must be considered in the scheduling strategies in order to provide QoS. In this sense, one of the biggest problems is given by the fact that the grids, unlike other classical distributed systems, are characterized by the dynamism and uncertainty in the state of theirs resources. Thus, a state-based scheduling that does not consider any uncertainty or imprecision in the handled information can make decisions founded on conditions that do not correspond with reality. The consideration of imprecision-tolerant techniques can be very beneficial in environments subject to uncertainty such as computational grids. FRBS-based schedulers have recently emerged as alternative scheduling systems to provide an efficient characterization of the state of sites in order to decide the suitability of the selection of the different sites in the schedule [8, 9]. Specifically, this decision is given by the application of expert knowledge in the form of IF-THEN rules that can be automatically optimized with diverse learning procedures in FRBSs based on evolutionary strategies such as Pittsburgh and Michigan approaches [1]. The structure of rules of a fuzzy rule-based scheduler generally follows the classical Mamdani model [1]: Ri = IF x1 is A1g and/or . . . xn is Ank T HEN y is Bz with w

(1)

where x j denotes the component j of the antecedent, y indicates the consequent, and/or represents the possible connectives for the antecedents and w is the weight associated to the rule. Also, A jk indicates the set k of the l possible fuzzy sets allowed for the component j of the antecedent and Bz represents the set z of the t possible fuzzy sets for the consequent. Fig. 1 shows the general structure of the fuzzy rule-based meta-scheduler within the grid environment. As illustrated, the general structure of the fuzzy meta-scheduler follows the classical schema of Mamdani fuzzy logic systems [1]. There exist three main components: the fuzzification system, the inference system and defuzzification system. The joint operation of these systems made up the reasoning strategy of the meta-scheduler. Specifically, as a result of the operation of the meta-scheduler, a RD selector factor yo is obtained that shows the level of suitability for the selection of the RD under analysis to be selected in the current schedule. This operation is repeated for every available RD in the grid.

3 Gaussian Mixture Model Schedulers Also, in this work, a new scheduling scheduling strategy is proposed to model the state of the diverse RDs integrated in the grid. Specifically, the selection of the most convenient site for a given schedule is to be done in regard of a suitability probability

298

R.P. Prado et al. >ŽĐĂů^ĐŚĞĚƵůĞƌ Ϯ ZϮ >ŽĐĂů^ĐŚĞĚƵůĞƌϭ

>ŽĐĂů^ĐŚĞĚƵůĞƌũ

Zϭ

ƌϮ͕ϭ

ƌϮ͕Ŭ ƌũ͕Ŭ

ƌϭ͕ϭ ƌϭ͕Ϯ

ƌϭ͕Ŭ

ZƐƚĂƚĞ ǆ

ƌũ͕Ϯ

ƌũ͕ϭ

'ƌŝĚ^ǇƐƚĞŵ

ƌϭ͕ϯ

Zũ

&ƵǌǌǇ ZƵůĞͲĂƐĞĚ DĞƚĂͲƐĐŚĞĚƵůĞƌ '^ d ^

^ĞůĞĐƚŽƌ͗DĂǆ^ƵŝƚĂďŝůŝƚǇ ŝŶĚĞǆ

Zh> ^

ǆс΂&ĞĂƚƵƌĞϭ͕&ĞĂƚƵƌĞϮ΃͕ ǆ

&ĞĂƚƵƌĞϮ

BAJO 1

MEDIO

D egreeof m em bership

0.4

0.2

Zϭ ZϮ

0 0

0.1

0.2

0.3

BAJO 1

0.4 0.5 0.6 MEDIO RECURSOS

0.7

0.8

0.9

1

ALTO

͙

0.8 D e g re eo fm e m b e rsh ip

&h/&/d/KE

/E&ZE

ALTO

0.8

0.6

0.6

0.4

MUYBAJO 1

BAJO

MEDIO

ALTO

^ƵŝƚĂďŝůŝƚǇŝŶĚĞǆ

MUYALTO

0.8 Deg ree of m em bers hip

&h/&/d/KE

&ĞĂƚƵƌĞϭ

0.6

0.4

0.2

0.2

0 0

0.1

0.2

0.3

0.4 0.5 0.6 RECURSOS

0.7

0.8

0.9

1

ZE

0 0

0.1

0.2

0.3

0.4

0.5 0.6 SALIDA

0.7

0.8

0.9

1

Fig. 1 General structure of a grid meta-scheduler based on FRBSs in a 2-dimensional space

which is calculated based on the state of the site and a weighted sum of probability density functions given by a Gaussian structure or GMM making up the scheduler decision core. A GMM is a statistical model that uses a weighted sum of probability density functions of multiple Gaussian distributions to depict the distribution of a vector in the probability space [2, 12]. A complete GMM is parametrized by mixture weights or priors αi , mean vectors μ i , and the covariance matrices Σi from all the M mixture components and can be denoted as:

λ = {αi , μ i , Σi ; i = 1, 2 . . . , M}

(2)

where x is a N dimensional feature vector to be modeled by GMM, Pi (x), i = 1 . . . M, are the probability density functions of x generated from the ith component of GMM which is denoted by λi and can be given as: Pi (x) = Pi (x | λi ) =

1 −1 (x − exp − μ ) Σ (x − μ ) i i i 2 (2αi )N/2 | Σi |1/2 1

(3)

A weighted sum of probability density functions of all the M mixture components is used to compute the probability that x belonged to model λ . M

P(x | λ ) = ∑ αi Pi (x)

(4)

i=1

where αi , i = 1 . . . M are the mixture weights and satisfy the constraint that 0 ≤ α ≤ 1, ∑M i=1 αi = 1. The suggested scheduling strategy is based on the consideration


299

of a GMM to model the scheduler decision module. Initially, the state of every RD participating in a given schedule is described through a finite set of features provided by the Grid Information System (GIS) as in the case of fuzzy rule-based meta-schedulers. To be precise, the consideration of a N dimensional feature vector x, representing the state of grid RDs, is to be modeled by GMM. I.e., x = Feature1, j , Feature1, j , . . . , FeatureN, j for every RD j. Hence, at every scheduling step, the system retrieves the state of each RD j of the grid x, given by a set of features, and it computes the probability that x belongs to model λ , P(x | λ ) for every site, considering a weighted sum of probability density functions as presented in Eq. 3. The RD obtaining the highest probability, is selected for the schedule and the process is repeated through the whole scheduling process. Fig. 2 represents an example of the scheduling GMMbased strategy with the RDs state described by two features. On the other hand, in order to obtain efficient GMM structures for the scheduling process, they are subject to an evolutionary learning. As regards the configuration of the scheduling system based on GMM, two different proposals are suggested. >ŽĐĂů^ĐŚĞĚƵůĞƌ Ϯ ZϮ >ŽĐĂů^ĐŚĞĚƵůĞƌϭ Zϭ

>ŽĐĂů^ĐŚĞĚƵůĞƌũ ƌϮ͕ϭ

ƌϮ͕Ŭ ƌũ͕Ŭ

ƌϭ͕ϭ ƌϭ͕Ϯ

ƌϭ͕Ŭ

'ƌŝĚ^ǇƐƚĞŵ

ƌϭ͕ϯ

ZƐƚĂƚĞ ǆ

ƌũ͕ϭ

ƌũ͕Ϯ

Zũ

'ĂƵƐƐŝĂŶ DŝǆƚƵƌĞDŽĚĞů ͲďĂƐĞĚ DĞƚĂͲƐĐŚĞĚƵůĞƌ W;ǆͿ

ǆс΂&ĞĂƚƵƌĞϭ͕ &ĞĂƚƵƌĞϮ΃͕ сϭ͕EсϮ͕Dсϭ͕

^ĞůĞĐƚŽƌ͗DĂǆW;ǆͿ

ǆ &ĞĂƚƵƌĞϭ

W;ǆͿ

&ĞĂƚƵƌĞϮ

&ĞĂƚƵƌĞϮ

&ĞĂƚƵƌĞϭ

Fig. 2 Meta-Scheduling Gaussian structure in a 2-dimensional space for grid computing

3.1 Use of Simple Gaussian (GS) for the Whole Grid The scheduling structure can be defined as a single N-dimensional Gaussian, with parameters λ , λ = { μ , Σ } and probability P(x | λ ) = P(x). In this approach, the

300

R.P. Prado et al.

number of optimization parameters can be analyzed as follows. Considering Σ a diagonal matrix, the number of optimization parameters is #λ = # {μ , Σ } = # {N, N} = 2 ∗ N

3.2 Use of Multiple Gaussian-Structure (GMM) for the Whole Grid Another approach is to consider a scheduling structure defined as a D-GMM: D Ndimensional Gausssians make up the structure with λi , λi = {αi , μ i , Σi ; i = 1, 2 . . . , D} and probability Pi (x | λi ) = ∑D i=1 αi Pi (x). In this second approach the number of optimization parameters can be determined as follows. Considering Σi as diagonal matrices and the same prior for all the component, the number of optimization parameters are #λ = # { μ i , Σi ; i = 1, 2 . . . , D}= # {N ∗ D, N ∗ D} = 2 ∗ N ∗ D. As introduced above, the GMM parameters must be optimized in the learning of the GMM scheduler.

4 Simulations Results and Discussion Several tests have been conducted to evaluate the GMM and fuzzy rule-based schedulers. Specifically, the scheduler is tested through simulations with Alea software [5] in its 2.1 version. Alea is a grid scheduling toolkit for simulation based on Java GridSim software that allows the application of grid scenarios and traces from real world. In our tests, the grid environment is based on Czech National Grid Infrastructure Metacentrum project [10]. The grid network is made up of 14 Metacentrum clusters with 806 heterogeneous central process units (i.e. Opteron and Xeon) and speed (i.e. 1,500-3,200 MHz) allocated within 210 machines under Linux with random access memory in the range 1,005,000 to 131,182,840 KB. Further, the configuration of the queues of sites machines, maintenance and reservation behaviour and workload characterization are retrieved from traces of Metacentrum facilities available at [4]. According to previous works in the area [8], the following features could be selected for the feature vector x describing the state of each RD in the grid: • Number of free processing elements (FPE): Number of free processing elements within a participating resource domain, RD j .

• Previous tardiness (PT): Sum of tardiness of all finished jobs in resource domain RD j . • Resource makespan (RM): Current makespan or finalization time of the last considered job in the RD j .

• Resource tardiness (RT): Current tardiness of jobs assigned to the RD j . • Previous score in deadline evaluation (PS): Exceeding deadline time of already finished jobs in the RD j .

• Resource score or number of delayed jobs (RS): Number of non delayed jobs so far in the RD j .

• Resources in execution (RE): Number of resources executing jobs within the RD j currently.


301

I.e., x = FPE j , PT j , RM j , RT j , PS j , RS j , RE j , for every RD j. Besides, the minimization of the latest job finalization time or makespan is pursued in this stage as scheduling goal as it is a general indicator of the grid productivity [11]. Two different approaches are considered for the GMM scheduling strategy. On the one hand, the proposal is evaluated using a scheduling structure is defined as a single 7-dimensional gausssian, i.e., N = 7, with parameters λ = {μ , Σ } with probability P(x | λ ) = P(x). The number of optimization parameters in this approach, considering Σ a diagonal matrix, is: #λ = # {μ , Σ } = # {7, 7} = 14. On the other hand, the scheduling structure is defined as a 3-GMM: three 7-dimensional Gaussians makes up the structure with N = 7 and λi = {αi , μ i , Σi ; i = 1, 2 . . . , 3} and probability 3

Pi (x | λi ) = ∑ αi Pi (x)

(5)

i=1

with αi = 1/3; i = 1, 2 . . . , 3., i.e., same probability for all the Gaussians. The number of optimization parameters, considering Σi as diagonal matrices, the number of optimization parameters is: #λ = # {μ i , Σi ; i = 1, 2 . . . , 3} = # {21, 21} = 42. The Gaussians GS (1-GMM) and GMM models are evolved for 100 generations where 14 and 42 parameters, respectively, corresponding to mean and covariances matrices components are adjusted to find an optimum configuration, i.e., mean and covariance matrices are obtained though an evolutionary process to optimize the performance of the meta-schedulers in terms of makespan. Specifically, a genetic evolutionary process is considered. Furthermore, 30 runs are conducted for each GMM structure and results are compared to those of a fuzzy meta-schedulers based on Pittsburgh and Michigan approaches using the same description for the grid state, number of generations and Gaussian fuzzy sets for both the input and output. Table 1 Simulation results for GS scheduler. Simulation results for 30 simulations in grid Metacentrum. Training Fitness makespan (s). Parameter/Strategy

Michigan Fuzzy scheduler

Pittsburgh Fuzzy scheduler

Max

1,847,375.2

1,684,235.5

Min

1,654,505.4

1,625,058.2

Average

1,773,266.8

1,667,586.2

Standard Deviation

54,210.0

19,757.8

Confidence Interval (95%) 1,755,929.68, 1,790,604.09 1,660,515.96, 1,674,656.43

Table 1 presents the learning results for the meta-scheduler based on a 7dimensional GS and 7-dimensional GMM. Specifically, it shows the mean result achieved by the strategies (Average) with the associated standard deviation (Standard Deviation) and 95% confidence interval (Confidence Interval-95%) and the best result (Min) and worst result (Max). As illustrated, the fuzzy meta-scheduler based on Pittsburgh evolution outperforms the GS and 3-GMM schedulers in terms of final makespan by 2.91% and 2.43% on average, respectively, what it is translated in a shorter time to perform the whole schedule. In addition, the best result

302

R.P. Prado et al.

Table 2 Simulation results for GS scheduler. Simulation results for 30 simulations in grid Metacentrum. Training Fitness makespan (s). Parameter/Strategy

GS-scheduler

GMM-scheduler

Max

1,781,326.0

1,781,326.0

Min

1,644,060.1

1,639,568.4

Average

1,717,557.7

1,709,183.5

Standard Deviation

51,057.0

50,924.8

Confidence Interval (95%) 1,698,492.80, 1,736,622.79 1,690,167.89, 1,728,199.20

(i.e., minimum makespan, Min) and worst result (i.e., maximum makespan, Max) achieved by the GS are 1.16% and 5.45% greater than the results obtained with the Pittsburgh fuzzy scheduler, respectively. Also, the best result (i.e., Min) and worst result (i.e., Max) achieved by the 3-GMM are 0.89% and 5.45% greater than the obtained with the Pittsburgh fuzzy scheduler. However, it can be observed that the GS and 3-GMM schedulers outperform the fuzzy meta-scheduler based on Michigan evolution in terms of final makespan by 3.14% and 3.61% on average, respectively. Furthermore, the best result (i.e., Min) and worst result (i.e., Max) achieved by the GS are 0.63% and 3.58% lower than the ones obtained with the Michigan fuzzy scheduler. On the other hand, the best result (i.e., Min) and worst result (i.e., Max) achieved by the 3-GMM are 0.90% and 3.58% more reduced than the results obtained with the Michigan fuzzy scheduler. Further, the standard deviation is also significantly higher with the following impact in the confidence interval. Finally, it is shown that the 3-GMM scheduler outperforms the GS model by 0.49% on average and that the best result (i.e., Min) is improved by 0.28%. 6

1.95

x 10

GS GMM Genetic-Michigan Genetic-Pittsburgh

1.9

Makespan (s)

1.85

1.8

1.75

1.7

1.65

0

10

20

30

40

50 60 Generation

70

80

Fig. 3 Convergence of the GS scheduler in the training process

90

100


303

Also, the convergence of the GS, 3-GMM, Pittsburgh and Michigan schedulers in the learning process is presented in Fig. 3. As illustrated, the genetic Pittsburgh scheduler achieves the higher accuracy meanwhile the genetic Michigan scheduler presents the lower convergence speed and accuracy what shows the dependence of the fuzzy expert meta-schedulers with the learning strategy to achieve an efficient schedule. On the other hand, the GMM-based schedulers provide an intermediate solution. The GS and 3-GMM approaches follow a similar convergence behaviour. However, as discussed above, the 3-GMM achieves a higher accuracy. It is to be noted that the 3-GMM approach presents a higher number of optimization parameters than the GS model what allows a major flexibility for the systems but also a greater complexity in the simultaneous fixing. From these results it can be derived that the GMM schedulers could be competitive approaches in the design of state-based scheduling strategies for grid computing founded on the definition of probability density functions. However, fuzzy rule-based schedulers offer a greater scalability and accuracy and speed in their learning process in most configurations.

5 Conclusions and Future Work The developments in the technical field related to Grid/Cloud Computing are increasingly becoming interesting and relevant for companies nowadays. Taking advantage of inactive time of computers to make a virtual supercomputer will have important effects on the organizations: reduction of infrastructure, less investments in hardware and software, the possibility to send surplus processing cycles, less risk with hardware problems, highest productivity of existing computers, etc. Thus, Grid/Cloud technologies have a growing impact over the governance structures and the self production support of companies. In this sense, improving the efficiency and scalability in the workload/resource scheduling is critical. In this paper, a comparison of FRBS schedulers for Grid/Cloud systems based on Michigan and Pittsburgh approaches and a new scheduling strategy based on GMM for grid computing has been presented. On the one hand, FRBSs models are flexible and they are more and more being adopted as scheduling systems for scheduling jobs in large-scale distributed networks based on the fuzzy characterization of the grid sites state to obtain an efficient schedule within an uncertain environment. On the other hand, the GMM scheduling strategy is founded on the association of density probability functions to every site in the grid which can provide a probability of suitability of selection at every scheduling stage on the basis of the characterization of sites state. To be precise, two different approaches based on GMM are proposed and the learning of the GMM scheduling structures are addressed through evolutionary processes. Simulation results based on existing grid infrastructure have shown that novel proposed approach, a Gaussian structure for all the sites, can be a compete alternative to some fuzzy state-based meta-scheduler systems such as FRBS Michigan schedulers although it is not able to outperform fuzzy Pittsburgh-schedulers in terms of accuracy and convergence behaviour. Also, fuzzy schedulers are more scalable. In future works, the definition of independent GMM structures for every involved site

304

R.P. Prado et al.

in the grid will be analyzed to increase the flexibility. In addition, a principal components analysis (PCA) will be introduced to achieve a reduction in system search space dimension and thus to accelerate the learning of the Gaussian schedulers. Acknowledgements. This work has been financially supported by the Spanish Government (Research Project P07-TIC-02713).

References 1. Cordón, O., Herrera, F., Hoffmann, F., Magdalena, L.: Genetic fuzzy systems: Evolutionary tuning and learning of fuzzy knowledge bases. World Scientific Pub. Co. Inc. (2001) 2. Duda, R.O., Hart, P.E., Stork, D.G.: Pattern Classification, 2nd edn. Wiley Interscience (2000) 3. Foster, I., Kesselman, C.: The Grid 2: Blueprint for a New Computing Infrastructure. Morgan Kaufmann Publishers Inc., San Francisco (2003) 4. C.N.G. Infrastructure: Metacentrum data sets meta (2009), http://www.fi.muni.cz/~xklusac/index.php?page= 5. Klusáˇcek, D., Matyska, L., Rudová, H.: Alea – Grid Scheduling Simulation Environment. In: Wyrzykowski, R., Dongarra, J., Karczewski, K., Wasniewski, J. (eds.) PPAM 2007. LNCS, vol. 4967, pp. 1029–1038. Springer, Heidelberg (2008) 6. Klusacek, D., Rudova, H.: Improving QoS in computational Grids through schedulebased approach. In: Scheduling and Planning Applications Workshop at the Eighteenth International Conference on Automated Planning and Scheduling (ICAPS 2008), Sydney, Australia (2008) 7. Mohammed, A.B., Altmann, J., Hwang, J.: Cloud computing value chains: Understanding businesses and value creation in the cloud. In: Neumann, D., Baker, M., Altmann, J., Rana, O. (eds.) Economic Models and Algorithms for Distributed Systems, Autonomic Systems, pp. 187–208. Birkhäuser Basel (2010) 8. Prado, R.P., García-Galán, S., Expósito, J.E.M., Yuste, A.J., Bruque, S.: Learning of Fuzzy Rule-Based Meta-schedulers for Grid Computing with Differential Evolution. In: Hüllermeier, E., Kruse, R., Hoffmann, F. (eds.) IPMU 2010. CCIS, vol. 80, pp. 751–760. Springer, Heidelberg (2010) 9. Prado, R., García-Galán, S., Yuste, A., Muñoz Expósito, J.: Genetic fuzzy rule-based scheduling system for grid computing in virtual organizations. Soft Computing - A Fusion of Foundations, Methodologies and Applications, 1–17 (2010) 10. Šustr, Z., Sitera, J., Mulaˇc, M., Ruda, M., Antoš, D., Hejtmánek, L., Holub, P., Salvet, Z., Matyska, L.: MetaCentrum, the Czech Virtualized NGI (2009) 11. Xhafa, F., Abraham, A.: Computational models and heuristic methods for grid scheduling problems. Future Generation Computer Systems 26(4), 608–621 (2010) 12. Yu, G., Sun, J., Li, C.: Machine performance assessment using gaussian mixture model (gmm). In: 2nd International Symposium on Systems and Control in Aerospace and Astronautics, ISSCAA 2008, pp. 1–6 (2008), doi:10.1109/ISSCAA.2008.4776183