research article b3: fuzzy-based data center load optimization in cloud...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 612182 11 pageshttpdxdoiorg1011552013612182

Research ArticleB3 Fuzzy-Based Data Center Load Optimization inCloud Computing

M Jaiganesh and A Vincent Antony Kumar

Department of Information Technology PSNA College of Engineering and Technology Dindigul Tamil Nadu 624 622 India

Correspondence should be addressed to M Jaiganesh jaidevlingamgmailcom

Received 31 December 2012 Revised 24 February 2013 Accepted 25 February 2013

Academic Editor Ming Li

Copyright copy 2013 M Jaiganesh and A V Antony Kumar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Cloud computing started a new era in getting variety of information puddles through various internet connections by anyconnective devices It provides pay and use method for grasping the services by the clients Data center is a sophisticated highdefinition server which runs applications virtually in cloud computing It moves the application services and data to a large datacenter Data center provides more service level which covers maximum of users In order to find the overall load efficiency theutilization service in data center is a definite task Hence we propose a novel method to find the efficiency of the data center in cloudcomputingThegoal is to optimize date center utilization in terms of three big factorsmdashBandwidthMemory andCentral ProcessingUnit (CPU) cycle We constructed a fuzzy expert system model to obtain maximum Data Center Load Efficiency (DCLE) in cloudcomputing environments The advantage of the proposed system lies in DCLE computing While computing it allows regularevaluation of services to any number of clients This approach indicates that the current cloud needs an order of magnitude in datacenter management to be used in next generation computing

1 Introduction

Cloud computing is an evolving paradigm to access assort-ment of data pool via internet by using connective devicessuch as Personal Digital Assistant (PDA) work station andmobile [1ndash4] It is a utility-based computing which has thecapability to deliver services over the internet It provideson-demand access without any human intervention Thestandard deployment object that is used in cloud computingis Virtual Machines (VM) It enhances flexibility and enablesdata center to be dynamic in nature The techniques ofdividing a physical computer into several partly or completelyisolated machines are known as virtualization [5 6] Acollection of data is stored in a centralized pool calledData Center (DC) [7ndash9] Cloud computing is the art ofmanaging tasks and applications by altering the softwareplatform and infrastructure and by organizing third partydata centers known as Cloud Service Providers (CSP) such asYahoo Amazon Google and VMware [2 10] Data centeris deployed as an individual server room which is hostedwithin the organization It runs several applications on a

single server In cloud computing the data center providesmore services which covers maximum numbers of users [11]So cloud service providers are prepared in better toleranceto manage and update the data centers Cloud computingprovides myriad of services [12] Therefore the data center istoo costly to build andmanageThe challenges of data centersare the following

(i) Irrefutable Cost Construction of low cost data centeris unaffordable for a single compound Cloud com-puting built a centralized data center which requiresincreasing cost in servers and storage

(ii) Workload Utilization Cloud computing needs newservers to be installed in data centers Virtualizationhas enabled many applications to run on a singleserver or couple of servers Some key factors ofutilization are storage power cooling response timecapacity and efficiency

(iii) Optimization of Services Numerous data centersapplications provide variety of services So findingoverall load efficiency and utilization of services is

2 Mathematical Problems in Engineering

a complex task associated with data centerDue toenormous applications running on it optimization ofdata center service is a major challenge

The major difficulty in a data center is to deploy that pro-ducers are expected to have better knowledge in monitoringdata centers so that they are able to find the service utilizationissues by managing the data center load configurations [13ndash16] In [17] presented a data center utilization scenario tomonitor and analyze cloud system the utilization of clientspecification bounds such as bandwidth memory and CPUutilization

Fuzzy Logic was introduced by Zadah [18ndash20] It is aproblem-solving system methodology that lends itself tosurvive systems ranging from simple to sophisticated tosurvive It is used in embedded networked distributedsystems Fuzzy set is a common set that has collection ofelements measuring improbability in the set It has varyingdegrees of membership in the set A typical function of acrisp set allocates a value of either 1 or 0 to each individualin the universal set The function can be comprehensive insuch a manner that the values are assigned to the elements ofthe universal set Huge values represent upper degrees of setmembership and it is calledmembership function and the setis identified as fuzzy set The most usually employed range ofvalues ofmembership function is the unit interval [0 1] Eachmembership function plots elements of a given universal set119883 and it is always a crisp set into real numbers in [0 1] Themembership function of fuzzy set 119860 is denoted by 120583

119860 that is

120583119860

119883 rarr [0 1] Each fuzzy set is completely and uniquelydefined by one particular membership function and it mayalso be used as labels of the associated fuzzy sets [21] Eachelement of fuzzy set is mapped to a universal membershipvalue by using function theoretic form [22] It is having anelement in the universal set119883 is amember of fuzzy set119860 andthen this mapping is given by 120583

119860(119883) isin [0 1] where 120583

119860(119883) is

called grade of membershipIn this work extensive use of Fuzzy logic has been

deployed to find the data center load efficiency Here weused crisp value of input as real numbers and in the nextanalysis we intend to go in for Fuzzy Fractal Dimensions[23 24] Data center load efficiency is the key object Here thefuzzy fractal dimension is denoted by the pair of Bandwidth(BW) and Memory of the CPU fields [25 26] Here BW isthe numerical value of the fractal dimension of bandwidthand 119872

0is the membership function of bandwidth namely

the memory and CPU It is mainly because the Memoryand CPU are dependent on the bandwidth The unevennessof the dynamically changing resource requirements and theemerging demand pattern can be compared to the differentgeometric objects [27 28] Hence here we apply fuzzy rulesto differentiate the different patterns and cluster them Fractalgeometry can be used to classify different objects based ontheir roughness [24 26 29] In this case the focus is basedonly on the smoother objects where the limitation of thefractal value is only up to one and if it is closer to one thenthat means maximum utilization of the memory and CPUresources If memory is 119872

1and CPU cycle is 119862

1 then the

data center load efficiency is DC1 Thus based on the input

parameter the output object efficiency is predicted using thesimple fuzzy rules Only disadvantage here is that based onthe parameters the total number of rules increases causingproblem due to dimensionality Based on this model whichhas been created the future values of the demand for CPUandmemory can be predicted leading to accurately assess theefficiency of the data center in the varying situations

11 Background In recent times more attention is shownon the framework of cloud computing and the performanceevaluation Iosup et al [30] have done ldquoperformance anal-ysis of cloud computing servicesrdquo approach for supportingefficiency of cloud computing In their model they analyzethe performance of Many Task Computing (MTC) work-loads They have proposed a comparison on performancecharacteristics and cost models Moreno-Vozmediano et al[31] have deployed a computing cluster on the top of manytask computing applications In this subsequent work clusterloads have been used for resources from different cloudsto construct high availability strategies These are used forproving viability to perform scalability of resources andperformances for large scale cluster infrastructure Dutreilhet al [32] have considered the recent research to constructa data center management framework for atomic resourceallocation in virtual applications They evaluated in twoways namely threshold-based and reinforcement learningmethods to dynamically scale resources

Yazir [33] presented a virtualization tool that providesthis gap by applying ideas from computational geometryIt proved valuable assistance in providing quick and easypreliminary performance analyses Data processing manage-ment is difficult to get as many machines as an applicationneeds The large scale jobs are distributed on differentmachines as parallel running processes The control andcoordination of these processes is complex with time depen-dent Cloud Architectures [34] have solved such difficultiesCloud administrators usually worry about hardware procur-ing (when they run out of capacity) and better infrastructureutilization (when they have excess and idle capacity) Thelower network bandwidth and the inherent lower hardwaredependability force enterprises to reorganize cloud appli-cation architecture [35] From the data center challengesand methodologies the two key questions arise How arethe efficiency of data centers and performance of cloudcomputing calculated What are the key factors to decidethe efficiency of data center in cloud computing This paperanswers these questions The contributions of the paper areoutlined as follows

(1) We compute the DCLE be (120578) for load-based datacenter management in cloud computing (120578) is avaluable perception for cloud service providers tomonitor manage and mitigate the cloud computingservices and justify the ability to hire a single virtualclient or thousands of virtual clients

(2) We construct it as a fuzzy expert systemmodel to findthe DCLE with basic three factors like BandwidthMemory and CPU cycles to validate the steps of our

Mathematical Problems in Engineering 3

model which is possible through tangible implemen-tation and assessment

This paper is organized as follows Section 2 gives theproblem identification Section 3 the deals with problem for-mulation preliminaries and definitions Section 4 presentsfinding of data center load efficiency using fuzzy modelingSection 5 provides the performance analyzes and experimentresults Section 6 gives the conclusion of the paper

2 Problem Identification

The objective of this work is to assess the data center loadefficiency when more number of clients and several requestsare running on the same server The typical web applicationused in cloud computing has the potential capacity con-straints such as bandwidth into the load balancer CPU cycleand memory of the load balancer [36 37] The ability of theload balancer depends upon (i) bandwidth between the loadbalancer with application server [38 39] (ii) CPU cycle andmemory of the application server (iii) bandwidth betweenapplication server and network storage devices (iv) datastorage and Disk IO of database server [40] The followingmajor three factors play a vital role in cloud computing

(1) Bandwidth(2) Memory(3) Central Processing UnitCycle (CPU Cycle)

21 Bandwidth In corporate motto the cloud computingis operationally exhaustive and obviously parallel In anysoftware that runs on entire virtual client it should becommunicative It is not giving operational transaction andbandwidth assurance The cloud service provider [28] canoffer a bandwidth which is found through their networkconnections of data center with internal as well as inpublic internet The data centers can provide consistencyand service delivery efficiently It includes the guaranteedamount of bandwidth that every client should get [41 42]The number of service tends to grow and cloud serviceprovider increases the cloud information rate which alsobrings increase in their bandwidth [43ndash45] Based on HighPerformance Computing (HPC) challenging results existin [44 46] Figure 1 depicts the bandwidth utilization ofHigh Performance Cluster Computing (HPCC) for GoGridcloud computing platforms Here bandwidth is calculatedfor HPCC performance prediction The volume of serviceson the cloud computing keeps on growing and tends tomore bandwidth [24 26 47 48] The bandwidth utilizationand the data center load are directly proportional to eachother that is when the bandwidth utility in cloud increasesthe data center load also increases and vice versa Hencethe bandwidth utilization is considered as one among thebig three factors for providing a good cloud service to thecustomers

22 Memory It is a major difficulty for storage and deliveryof services in cloud computing It is purely depending uponthe application or task used by the client In cloud computing

5432125

80

135

190

245

300

GoGRID services

GoGRID bandwidth performances

Band

wid

th (M

Bps)

Figure 1 GoGRID bandwidth utilization

Amazon disk categories

2000

1610

1220

830

440

50

Disk

capa

city

(GB)

1 2 3 4 5Amazon disk services

Figure 2 Amazon EC2 Instances-Memory utilization

the applications and the files are permanently stored indata center by the access of third party clients and usersAmazonrsquos Simple Storage Service (S3) (eg) In cloud survey[49] Figure 2 shows the memory usage of Amazon EC2platforms m1small to c1xlarge In dynamic nature of datacenters [46] the database management system requires moreamount of memory for processing the services The memoryshould be elastic in nature such that applications are beingperformed Memory is comparatively low while runningSaaS applications So the memory elasticity and memoryvisualization aremanageable see [50 51] In cloud computing


many of CPUrsquos transaction is done in a single data center Somemory is able to tolerate the CPU transactions and serviceperformance calculations Because of this aforementionedfacts The memory is another important factor to constructDCLE

23 Central Processing Unit Cycle (CPU Cycle) Third cloudcomputing needs core of processors present in a single frag-ment and providing high concurrent throughput for serviceswith parallel operation In cloud computing utilization ofCPU is an important factor An input supplied factor toa processorrsquos computing power is its clock speed It is anapproximation to the division of clock speeds that actuallytake place for a given processor design In addition the adventof new processors affects purchase of existing processorsData center applications need large amount of memory notat all having CPUS responsible for processing According tothis situation CPU with efficient performance called workstation is installed In cloud computing the samework stationis termed as data center In the real world memory is limitedand not infiniteThen we only prefer CPU cycle to be the oneof the prime factor to decideDCLEThedatabase applicationsare deployed on mainframe computer or server with hugecapacity In [46] the grid workload archive traces along withCPU utilizationThe cloud computing systemwill need someof 100rsquos CPUrsquos formultiprocessing architectures It starts fromCPU ranges from 64 to 128 We identified that previous threebig factors play a major role in computing of DCLE Wepresent these big three factors to obtain an optimized valueof maximized data center efficiency It is done through a validproblem solving control system using fuzzy modeling

3 Problem Formulation

The proposed model is formulated as knowledge base fuzzyexpert system modeling [52 53] We propose a novelapproach that has been tightening in data center to find thenew perception called Data Center Load Efficiency (DCLE)This factor is predicted in network load configuration regionDCLE is depicted as three important fundamental factorsThe factors are Bandwidth (BW) Memory (MEM) andCPU cycle or Speed (CPU) of data center This knowledgeof finding DCLE is mentioned in terms of fuzzy inferencerules which connect antecedents with consequences A fewdefinitions will be provided to demonstrate this perceptionmodel

31 PreliminariesDefinition 1 (approximate reasoning) Fuzzy set correspond-ing to the linguistic values defined as 119860

1 1198611 We include a

reasoning as multiconditional in the form

Rule 1 IF 119909 is 1198601 THEN 119910 is 119861

1

Rule 2 IF 119909 is 1198602 THEN 119910 is 119861

2

Rule 119899 IF 119909 is 119860119899 THEN 119910 is 119861

119899

Fact 119909 is 119860

Conclusion 119884 119894119904 119861

(1)

Given 119899 If - then Rule rule 1 through 119899 and a fact ldquo119909 is119860rdquoWe conclude that ldquo119910 is 119861rdquo where 119860 119860

119895isin 119891(119909) 119861 119861

119895isin 119891(119910)

for all 119895 isin 119873119899and 119883 119884 are sets of variables of 119909 and 119910

Definition 2 (fuzzy implication) In general fuzzy implica-tion 120597 is defined as the function of the form

120597 [0 1] lowast [0 1] 997888rarr [0 1] (2)

It gives any of possible true values 119886 119887 of given fuzzypropositions 119901 119902 respectively define the true value 120597(119886 119887)

of the conditional proposition called IF Then rules likeldquoIF 119901 then 119902rdquo This is called classical implication of 119901 rarr 119902from the restricted domain 0 1 to the full domain [0 1] oftrue values in fuzzy logic deriving ldquo120597rdquo in classical formulabeing

120597 (119886 119887) = 119886 or 119887 (3)

for all 119886 119887 isin 0 1 We interpret disjunction and negation asa fuzzy union and fuzzy complement and then 120597 in classicallogic is to employ the formula

120597 (119886 119887) = max 119909 isin 0 1 | 119886 and 119909 le 119887 (4)

Moreover equation (4) may also be rewritten due to law ofabsorption of negation in classical logic as either

120597 (119886 119887) = 119886 or (119886 and 119887) (5)

Definition 3 (relation ldquoRrdquo) The fuzzy relation 119877 employed inreasoning is obtained from the given if- then rules in (2) Foreach rule 119895 in (2) we determine a relation 119877

119895by the formula

119877119895(119909 119910) = min [119860

119895 (119909) 119861119895 (119910)] (6)

for all 119909 isin 119883 119910 isin 119884 then 119877 is defined by the unions ofrelations 119877

119895for all rule in Definition 1 gives

119877 = ⋃119895isin119873119899

119877119895 (7)

In this paper consider the problem as disjunctive innature So the interpretation of the rules in disjunctive canbe returned as

1198611015840= ⋃119895isin119873119899

1198601015840sdot 119877119895 (8)

In general 119877119895may be determined by a suitable fuzzy impli-

cation mentioned in Definition 2 as

119877119895(119909 119910) = 120597 (119860

119895 (119909) 119861119895 (119910)) (9)

a general counterpart of (2)


Definition 4 (fuzzy proposition) The proposition is mea-sured in its ranges and true values It depends on the matterof degree So each fuzzy proposition is uttered by a numberin the element interval [0 1] We consider our model asconditional and unqualified propositions Propositions ldquo119875rdquoof this type are expressed by the canonical form

119875 IF 119909 = 119860 THEN119910 = 119861 (10)

where 119909 and 119910 are variables whose values are in set 119883 and119884 respectively Finally 119860 and 119861 are fuzzy sets on 119883 and 119884respectively The propositions may also be viewed as

⟨119909 119910⟩ is 119877 (11)

where 119877 is a fuzzy set on 119883 lowast 119884 that is determined for each119909 isin 119883 119910 isin 119884 by formula

119877 (119909 119910) = 120597 [119860 (119909) 119861 (119910)] (12)

where 120597 denotes a binary operation on [0 1] representing asuitable fuzzy implication

Definition 5 (compositional rule inference) Consider vari-ables 119909 and 119910 that take values from sets119883 and119884 respectivelyand assume that for all 119909 isin 119883 and all 119910 isin 119884 the variables arerelated by a function 119910 = 119891(119909) and 119909 is in a given set 119860 and119910 in a given set 119861 is given by

119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 (13)

Similarly since 119909 isin 119860 we can infer that 119910 isin 119861 where

119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 119909 isin 119860 (14)

Examine that this inference may be expressed equally well interms of characteristics functions119883

119860119883119861119883119877of sets 119860 119861 119877

respectively by the equation

119883119861(119910) = sup

119909isin119883[119883119860 (119909) 119883119861 (119909 119910)] (15)

for all 119910 isin 119884 Let us proceed now one step further and assumethat 119877 is fuzzy relation on119883lowast119884 and 119860 119861 are fuzzy sets on119883

and 119884 respectively Then if 119877 and 119860 are given

119861 = sup119909isin119883

[119860 (119909) 119877 (119909 119910)] (16)

for all 119910 isin 119884 which is the generalization of (7) obtainedby replacing the characteristics functions in (7) with cor-responding membership functions We prefer this equationas generalization called compositional rule of inference tofacilitate approximate reasoning

4 Cloud Data Center Efficiency PredictionUsing Fuzzy Expert System

Fuzzy controller is working as a feedback system by repeatingthe cycles to all and attaining a desired output To establish thefuzzy controller modeling first we have to define the inputand output variables Data center management is progressedby the DCLE (120578) which is calculated among three factors

Table 1 Fuzzy linguistic values and notations

Linguistic variables NotationBandwidth(BW)

Low LMedium MHigh H

Memory (MEM)Small SMedium MLarge LA

CPU Utilization (CPU)Low LMedium MHigh H

Data Center Load Efficiency (DCLE)Minimum MNModerate MDMaximum MX

Bandwidth (BW) Memory (MEM) and CPU Cycles (CPU)In our assumption these three factors are considered as inputvariables and data center load efficiency as output variableThe solution is judged by data center management as controlproblem in nature To define the load efficiency of data centeris a single output variable of cloud environment This systemconsists of three modules

(i) fuzzification and defuzzification(ii) fuzzy inference engine(iii) fuzzy rule base

First observations are done of all input and outputvariables which mention conditions of the data centermanagement control process Then these observations areconverted into appropriate fuzzy set to propose observationuncertainties called fuzzification To define the data centerload efficiency 120578 of a single variable inspite of bandwidthmemory and CPU cycles we consider the combinations ofany two input variables 119889 119889 to be considered as bandwidthCPU cycle or memory By utilizing these values the fuzzycontroller produces a control variable 120578 that is DCLE Lin-guistic variables and their notations are depicted in Table 1

41 Step 1 It is a process of identifying inputoutput variablesand to assign a meaningful linguistic states and their rangesTo prefer exact linguistic states for each variable and posethem by corresponding fuzzy sets these linguistic states areproposed as fuzzy sets (or) fuzzy numbers Consider that theranges of input variables 119889 belongs to [minus119886 119886] 119889 belongs to[minus119887 119887] and the range of output variable 120578 belongs to [minus119888 119888]The linguistic input variables are Bandwidth and MemoryCPUcycle and output variable isDataCenter LoadEfficiency(DCLE) The ranges of the each input variables are havingthree linguistic states as shown in Figures 3 and 4 Also theoutput variable has three linguistic states


1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p

Bandwidth (normalized)

Figure 3 Fuzzy trapezoid view of bandwidth

1

08

06

04

02

001 02 03 04 05 06

CPU cycles (normalized)

Deg

rees

of m

embe

rshi

p

L M H

Figure 4 Fuzzy trapezoid view of CPU cycles

42 Step 2 In this step we introduce a fuzzification functionfor each input variable to propose the associate observationuncertainness To find grades of membership of linguisticvalues of linear variable corresponding to an input numberor fuzzy number it is used to calculate and interpret observa-tions of input variable each expressed as a real number

Consider a fuzzification function of the form

119891119889 [minus119886 119886] 997888rarr 119877 (17)

where 119877 denotes the set of all fuzzy numbers and 119891119889(1199090)

is a fuzzy number chosen by 119891119889as approximation of the

measurement 119889 = 1199090

We introduced trapezoidal shape as membership func-tion to define 119891

119889(1199090) It is showing the two control variables

and their trapezoidal view to represent fuzzy numbers Weillustrate fuzzification by showing the membership functionfor Bandwidth and Memory together with a trapezoid viewof variables depicted in Figure 5

43 Step 3 Fuzzy inference system can be generated asrelevant fuzzy inference rules by fuzzy associated memory

1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip

Input-membership function

Figure 5 Input membership function of bandwidth and memory(Normalized)

MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th

Data center loadefficiency

M Memory

Figure 6 FAM square-rule 1

called FAM square They can be conveniently represented byFigures 6 7 and 8 as a FAM square

In our approach 119889 119889 are inputs 120578 is output variable andthen

IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)

where 119860 119861 119862 are fuzzy numbers chosen from the setof numbers and their linguistic states The possible rulegenerated for each input and output variable is 3 so 32 = 9and totallywe have 36 rules To find the fuzzy rules practicallywe need a set of input-output data of the following

119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX

MXMX Data center loadefficiency


where 119911119896is a attained value of output variable 120578 for given value

119909119896and 119910

119896of the input variable 119889 and 119889 respectively 119870 is an

appropriate index setLet 119860(119909

119896) 119861(119910

119896) 119862(119911

119896) denote the largest membership

grades Then the degree of relevance can be expressed by

1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)

where 1198941 1198942are t-norms

Note A function 119894 [0 1]2 rarr [0 1] such that for all 119886 119887 119889 isin

[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =

119894(119887 119886) 119894(119886 119894(119887 119889)) = 119894(119894(119886 119887) 119889)The function is usually also continuous such that 119894(119886 119886) le

119886 for all 119886 isin [0 1]

44 Step 4 The observation of input variable must beperiodically matched with fuzzy inference rules to makeinference in terms of output variables

We choose composite inference logic mentioned inDefinition 5 to define our variables We convert the given

fuzzy inference rules represented in (18) which are equivalentto simple fuzzy conditional proposition of the form

IF ⟨119889 119889⟩ is 119860 times 119861 THEN 120578 is 119862 (21)

where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)

for all 119909 isin [minus119886 119886] and 119910 isin [minus119887 119887]The output variable DCLE 120578 becomes the problem

of approximate reasoning with composite inference fuzzyproposition mentioned in Definitions 4 and 5 respectivelyThe fuzzy rule base consists of ldquo119899rdquo fuzzy inference valuesthen

Rule 1 IF (119889 119889) is 1198601times 1198611 Then 120578 is 119862

1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)

Conclusion 120578 is 119862

(23)

The symbols 119860119895 119861119895 119862119895(119895 = 1 2 119899) denote fuzzy

sets that represent the linguistic states of variables 119889 119889 120578respectively

The rule is explained in terms of relation 119877119895 which is

mentioned in Definition 2The rules are considered as disjunctive in nature We

derive (17) to conclude that the output variable 120578 is definedby the fuzzy set as

119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)

where 119900119894 is the sup-119894 composition for a t-norm 119894 The choice

of the t-norm is a matter similar to the choice of fuzzy sets forgiven linguistic labels

45 Step 5 The process of computing single fuzzy numberfrom 119862 is called defuzzification The fuzzy output variable isalso a linguistic variable whose values have been assigninggrades of membership In the last step we find a single num-ber compatible with membership function in Data CenterLoad Efficiency (DCLE) called output membership functiondepicted in Figure 9 This number will be the output fromthis final step in defuzzification process There are severalmethods for calculating a single defuzzified number Weused a centroid method to convert the output values ofinference engine as a crisp numbers expressed as fuzzy setWe


1

08

06

04

02

0

07 075 08 085 09Datacenter load efficiency

Maximum Moderate MinimumOutput-membership function

Deg

ree o

f mem

bers

hip

Figure 9 Output membership function DCLE

calculated the output variable with centroid method whichcan be expressed as

119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)

Let 120583119860(119909) be the corresponding grade of membership in the

aggregated membership function and let

(1) 119883min be the minimum 119909 value attain the minimum ofdata center load efficiency 120578

(2) 119883 mod the moderate 119909 value attain the moderate ofdata center load efficiency 120578

(3) 119883max the maximum 119909 value attain the maximum ofdata center load efficiency 120578

119883lowast is defuzzified output as a real number value

5 Performance Analysis

We now asses the performance of the proposed cloud datacenter efficiency using the Fuzzy Expert system model toshow that they are load efficient We will focus on the loadefficiency of the data center in all the factors like bandwidthmemory and CPU Cycles The experiment is conductedusing MATLAB Version 78 with an Intel Core 2 Processorrunning at 186GHz 2048MB of RAM Among the threekey variables namely bandwidth memory and CPU cyclesthe first step of the simulation focuses on fuzzification byconverting them into input membership functions This isperformed using the tool called as membership functioneditor provided in the MATLAB Each variable in the experi-ment is quantified into small medium and large formemorylow medium and high for bandwidth and CPU cycles Theinput variables are segregated because the comparison of the

Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E

DCLE data center load efficiency

Figure 10 Fuzzy 3D view of bandwidth and memory versus DCLE

325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E


Figure 11 Fuzzy 3D view of bandwidth and CPU cycles versusDCLE

variables becomes effective and it helps in providing betterresults The If-Then rules of the experiment are formulatedusing rule editor

We performed our required operation in FIS editor whichhandles the high level issues The membership functioneditor which defines the shapes of all membership functionis associated with each variable and rule editor for editingthe list of rules The surface viewer plots an output surfacemap for the system The input vectors of the fuzzy inferenceengine as calculated by the simple attribute function are0812 0872 and 0884 and the unique output generatedby the Mamdani method is 0959 All the rules have beendepicted as 3D graphs called surface viewer in Figures10 11 and 12 Through Figure 10 we infer that when thebandwidth and memory linearly increase the load efficiencyof the data center increases at the same time when theydecrease it brings down the efficiency of the data centerlinearly In Figure 11 the Bandwidth and the CPU cycles arecompared with the efficiency of the data center load Whenthe bandwidth and the CPU are higher the efficiency of thedata center is also higher and vice versa In Figure 12memoryand CPU cycles are compared with the DCLE The resultsindicate thatwhen thememory and theCPUcycles are higherthe DCLE is also higher and lower in the opposite caseHowever the experiments suggest that our system is moreaccurate in predicting the efficiency of a data center thana human expert Here DCLE is used as a prime factor indetermining the overall system utilization and assessment of


325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E


Figure 12 Fuzzy 3D view of memory and CPU cycles versus DCLE

100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()

Efficiency comparison HPL versus DCLE

HPLDCLE

Figure 13 Efficiency Comparison

the system efficiency The results proved the increase in thenumber of services in the data center leading to increase inthe complexity of the calculation in the DCLE We list thefeatures of our system Figure 13 and also make a comparisonof our scheme with HPL performances (LINPACK Scheme)[44] It was observed that they performed the experimentusing the virtual clusters for GoGrid cloud service providerinstances according to the varying number of nodes andpercentage of efficiencyThe efficiency is varied from60 to 70In this experiment they consider bandwidth memory andprocessing cycles It was observed that when the bandwidthmemory and the CPU Cycles ranges were higher for theinstances this resulted in the increase in efficiency of theGOGrid instances Whereas even when any of the threebig factors were reduced it impacted on the efficiency ofthe HPL system The three big factors have been used tostudy the data center load efficiency and it was observed theattribute values of the three factorswhen increased resulted in

higher efficiency of any cluster or virtual systems It is clearlyevident that the simulation results are 20 percentage higherin comparison to the results offered by HPL systems

6 Conclusion

The most important task in the successful service of theinternet is access through maximum data center load effi-ciency In this paper we examined the load efficiency of datacenter which is essentially needed for the cloud computingsystems This system is designed according to the servicelayers of cloud computing cloud service provider estimatingthe strategy Data center maintains a chart to monitor thebig three factors suggested in this workThe advantage of theproposed system lies in DCLE computingWhile computingit allows regular evaluation of services to any number ofclients This work is extended in the way of providingresource adaptation and trustworthiness of cloud computingenvironment

References

[1] C Modi D Patel B Borisaniya H Patel A Patel and MRajarajan ldquoA survey of intrusion detection techniques in cloudrdquoJournal of Network and Computer Applications vol 36 no 1 pp42ndash57 2013

[2] F M Aymerich G Fenu and S Surcis ldquoAn approach to a cloudcomputing networkrdquo in Proceedings of the 1st InternationalConference on the Applications of Digital Information andWeb Technologies (ICADIWT rsquo08) pp 113ndash118 Ostrava CzechRepublic August 2008

[3] Enterasys Secure NetworksData Center Networking-ManagingVirtualized Environment Enterasys Networks Salem NHUSA 2011

[4] I Foster Y Zhao I Raicu and S Lu ldquoCloud computing and gridcomputing 360-degree comparedrdquo in Proceedings of the GridComputing EnvironmentsWorkshop (GCE rsquo08) pp 1ndash10 AustinTex USA November 2008

[5] M Jaiganesh and A Vincent Antony Kumar ldquoJNLP basedsecure software as a service in cloud computingrdquo Communica-tions in Computer and Information Science vol 283 pp 495ndash504 2012

[6] M Armbrust A Fox R Griffth et al ldquoAbove the clouds aberkeley view of cloud computingrdquo Tech Rep UCBEESCS-2009-28 EECS Department University of California BerkeleyCalif USA 2009

[7] M Beckert B A Ellison and S Krishnapura ldquoIntel it datacenter solutions strategies to improve efficiencyrdquo IT IntelWhite Paper Intel Information Technology Intel CorporationSanta Clara Calif USA 2009 1ndash11

[8] P Stryer ldquoUnderstanding data centers and cloud computingrdquoExpert Reference Series of White Papers Global KnowledgeTraining LLC Cary NC USA 2010 1ndash7

[9] R Buyya ldquoMarket-oriented cloud computing vision hype andreality of delivering computing as the 5th utilityrdquo in Proceedingsof the 9th IEEEACM International Symposium on ClusterComputing and the Grid (CCGRID rsquo09) pp 1ndash13 ShanghaiChina May 2009

[10] J Varia ldquoCloud architecturesrdquo White Paper Amazon Web Ser-vices Amazon Company 2008 1ndash14


[11] L Wang J Zhan W Shi Y Liang and L Yuan ldquoIn cloud doMTC or HTC service providers benefit from the economies ofscalerdquo in Proceedings of the 2nd ACMWorkshop on Many-TaskComputing on Grids and Supercomputers (MTAGS rsquo09) pp 7ndash11November 2009

[12] M Jaiganesh andA Vincent AntonyKumar ldquoACDP predictionof application cloud data center proficiency using fuzzy model-ingrdquo in Proceedings of the International Conference onModelingOptimization and Computing Procedia Engineering vol 38 pp3005ndash3018 Elsevier Publications 2012

[13] B J S Chee and C Franklin Cloud ComputingmdashTechnologiesand Strategies of the Ubiquitous Data Center CRC Press BocaRaton Fla USA 2010

[14] C Cattani S Chen and G Aldashev ldquoInformation and model-ing in complexityrdquo Mathematical Problems in Engineering vol2012 Article ID 868413 3 pages 2012

[15] X JieM LiWZhao and S YChen ldquoBoundmaxima as a trafficfeature under DDOS flood attacksrdquo Mathematical Problems inEngineering vol 2012 Article ID 419319 20 pages 2012

[16] Y Zheng S-Y Chen Y Lin and W-L Wang ldquoBio-inspiredoptimization of sustainable energy systems a reviewrdquo Mathe-matical Problems in Engineering vol 2013 Article ID 354523 12pages 2013

[17] M Kutare G Eisenhauer C Wang K Schwan V Talwarand M Wolf ldquoMonalytics online monitoring and analytics formanaging large scale data centersrdquo in Proceedings of the 7thIEEEACM International Conference on Autonomic Computingand Communications (ICAC rsquo10) pp 141ndash150 Washington DCUSA June 2010

[18] P Lu S Chen and Y Zheng ldquoArtificial intelligence in civilengineeringrdquo Mathematical Problems in Engineering vol 2012Article ID 145974 22 pages 2012

[19] L A Zadeh ldquoFuzzy sets and systemsrdquo International Journal ofGeneral Systems pp 129ndash138 1990

[20] L A Zadeh ldquoOutline of new approach to the analysis ofcomplex systems and decision processesrdquo IEEE Transactions onSystems Man and Cybernetics vol SMC-3 pp 28ndash44 1973

[21] A K Lohani R Kumar and R D Singh ldquoHydrological timeseries modeling a comparison between adaptive neuro-fuzzyneural network and autoregressive techniquesrdquo Journal ofHydrology vol 442-443 pp 23ndash35 2012

[22] E H Mamdani and S Assilian ldquoExperiment in linguisticsynthesis with a fuzzy logic controllerrdquo International Journal ofMan-Machine Studies vol 7 no 1 pp 1ndash13 1975

[23] B B Mandelbrot Gaussian Self-Affinity and Fractals SpringerNew York NY USA 2001

[24] M Li and W Zhao ldquoAbstract description of internet traffic ofgeneralized Cauchy typerdquo Mathematical Problems in Engineer-ing Article ID 821215 18 pages 2012

[25] D Veitch N Hohn and P Abry ldquoMultifractality in TCPIPtraffic the case againstrdquo Computer Networks vol 48 no 3 pp293ndash313 2005

[26] M Li and W Zhao ldquoVisiting power laws in cyber-physicalnetworking systemsrdquo Mathematical Problems in Engineeringvol 2012 Article ID 302786 13 pages 2012

[27] P Abry R Baraniuk P Flandrin R Riedi and D VeitchldquoMultiscale nature of network trafficrdquo IEEE Signal ProcessingMagazine vol 19 no 3 pp 28ndash46 2002

[28] M Li and W Zhao ldquoRepresentation of a stochastic trafficboundrdquo IEEE Transactions on Parallel and Distributed Systemsvol 21 no 9 pp 1368ndash1372 2010

[29] G Terdik and T Gyires ldquoLevy flights and fractal modeling ofinternet trafficrdquo IEEEACM Transactions on Networking vol 17no 1 pp 120ndash129 2009

[30] A Iosup S Ostermann N Yigitbasi R Prodan T Fahringerand D Epema ldquoPerformance analysis of cloud computing ser-vices for many-tasks scientific computingrdquo IEEE Transactionson Parallel and Distributed Systems vol 22 no 6 pp 931ndash9452011

[31] R Moreno-Vozmediano R S Montero and I M LlorenteldquoMulticloud deployment of computing clusters for looselycoupled MTC applicationsrdquo IEEE Transactions on Parallel andDistributed Systems vol 22 no 6 pp 924ndash930 2011

[32] X Dutreilh N Rivierre A Moreau J Malenfant and I TruckldquoFrom data center resource allocation to control theory andbackrdquo in Proceedings of the 3rd IEEE International Conference onCloud Computing (CLOUD rsquo10) pp 410ndash417 Miami Fla USAJuly 2010

[33] Y O Yazir ldquoVisual assessment of cloud resource consolidationmanagers using convex hullsrdquo in Proceedings of the IEEE EighthWorld Congress on Services pp 293ndash300 Honolulu HawaiiUSA June 2012

[34] M Litoiu M Woodside J Wong J Ng and G Iszlai ldquoAbusiness driven cloud optimization architecturerdquo in Proceedingsof the 25th Annual ACM Symposium on Applied Computing(SAC rsquo10) pp 380ndash385 ACM New York NY USAMarch 2010

[35] S W Liao T H Hung D Nguyen H Zhou C Chouand C Tu ldquoPrefetch optimizations on large-scale applicationsvia parameter value predictionrdquo in Proceedings of the 23rdInternational Conference on Supercomputing (ICS rsquo09) pp 519ndash520 Yorktown Heights NY USA June 2009

[36] A Patel M Taghavi K Bakhtiyari and J C Jnior ldquoAn intrusiondetection and prevention system in cloud computing a system-aticrdquo Journal of Network and Computer Applications vol 36 no1 pp 25ndash41 2013

[37] G Reese Cloud Application Architectures Building Applicationsand Infrastructure in the Cloud OrsquoReilly Media PublicationsGravenstein Highway North Sebastopol Calif USA 2009

[38] M Li and W Zhao ldquoAsymptotic identity in min-plus algebra areport on CPNSrdquo Computational and Mathematical Methods inMedicine Article ID 154038 11 pages 2012

[39] M Malekzadeh A A A Ghani and S Subramaniam ldquoAnew security model to prevent denial-of-service attacks andviolation of availability in wireless networksrdquo InternationalJournal of Communication Systems vol 25 no 7 pp 903ndash9252012

[40] The parallel Workloads archive Team ldquoThe parallel WorkloadsArchive logs Januaryrdquo 2009

[41] M Li and W Zhao ldquoA model to partly but reliably distinguishDDOS flood traffic from aggregated onerdquo Mathematical Prob-lems in Engineering vol 2012 Article ID 860569 12 pages 2012

[42] M Mirahmadi and A Shami ldquoTraffic-prediction-assisteddynamic bandwidth assignment for hybrid optical wirelessnetworksrdquo Computer Networks vol 56 no 1 pp 244ndash259 2012

[43] H S Kim and N B Shroff ldquoThe notion of end-to-end capacityand its application to the estimation of end-to-end networkdelaysrdquo Computer Networks vol 48 no 3 pp 475ndash488 2005

[44] The HPCC Team ldquoHPC challenge resultsrdquo 2009[45] H Liu and D Orban ldquoGridBatch cloud computing for large-

scale data-intensive batch applicationsrdquo in Proceedings of the 8thIEEE International Symposium on Cluster Computing and theGrid (CCGRID rsquo08) pp 295ndash305 Lyon France May 2008


[46] A Iosup H Li M Jan et al ldquoThe grid workloads archiverdquoFuture Generation Computer Systems vol 24 no 7 pp 672ndash6862008

[47] A Rizk and M Fidler ldquoNon-asymptotic end-to-end perfor-mance bounds for networks with long range dependent fBmcross trafficrdquoComputerNetworks vol 56 no 1 pp 127ndash141 2012

[48] X Jie M Li andW Zhao SY Chen ldquoBound maxima as a trafficfeature under DDOS flood attacksrdquo Mathematical Problems inEngineering vol 2012 Article ID 419319 20 pages 2012

[49] M Maurer I Brandic and R Sakellariou ldquoSelf-adaptive andresource-efficient SLA enactment for cloud computing infras-tructuresrdquo in Proceedings of the IEEE 5th International Confer-ence on Cloud Computing (CLOUD) pp 368ndash3375 2012


[51] L Youseff M Butrico and D Da Silva ldquoToward a unifiedontology of cloud computingrdquo in Proceedings of the GridComputing Environments Workshop (GCE rsquo08) Austin TexUSA November 2008

[52] M Saleem Khan ldquoFuzzy time control modeling of discreteevent systemsrdquo in Proceedings of the World Congress on Engi-neering andComputer Science International Association of Engi-neers (IAENG rsquo08) pp 683ndash688 2008

[53] W Duch R Adamczak and K Gra bczewski ldquoA new method-ology of extraction optimization and application of crisp andfuzzy logical rulesrdquo IEEE Transactions on Neural Networks vol12 no 2 pp 277ndash306 2001

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


a complex task associated with data centerDue toenormous applications running on it optimization ofdata center service is a major challenge

The major difficulty in a data center is to deploy that pro-ducers are expected to have better knowledge in monitoringdata centers so that they are able to find the service utilizationissues by managing the data center load configurations [13ndash16] In [17] presented a data center utilization scenario tomonitor and analyze cloud system the utilization of clientspecification bounds such as bandwidth memory and CPUutilization

Fuzzy Logic was introduced by Zadah [18ndash20] It is aproblem-solving system methodology that lends itself tosurvive systems ranging from simple to sophisticated tosurvive It is used in embedded networked distributedsystems Fuzzy set is a common set that has collection ofelements measuring improbability in the set It has varyingdegrees of membership in the set A typical function of acrisp set allocates a value of either 1 or 0 to each individualin the universal set The function can be comprehensive insuch a manner that the values are assigned to the elements ofthe universal set Huge values represent upper degrees of setmembership and it is calledmembership function and the setis identified as fuzzy set The most usually employed range ofvalues ofmembership function is the unit interval [0 1] Eachmembership function plots elements of a given universal set119883 and it is always a crisp set into real numbers in [0 1] Themembership function of fuzzy set 119860 is denoted by 120583

119860 that is

120583119860

119883 rarr [0 1] Each fuzzy set is completely and uniquelydefined by one particular membership function and it mayalso be used as labels of the associated fuzzy sets [21] Eachelement of fuzzy set is mapped to a universal membershipvalue by using function theoretic form [22] It is having anelement in the universal set119883 is amember of fuzzy set119860 andthen this mapping is given by 120583

119860(119883) isin [0 1] where 120583

119860(119883) is

called grade of membershipIn this work extensive use of Fuzzy logic has been

deployed to find the data center load efficiency Here weused crisp value of input as real numbers and in the nextanalysis we intend to go in for Fuzzy Fractal Dimensions[23 24] Data center load efficiency is the key object Here thefuzzy fractal dimension is denoted by the pair of Bandwidth(BW) and Memory of the CPU fields [25 26] Here BW isthe numerical value of the fractal dimension of bandwidthand 119872

0is the membership function of bandwidth namely

the memory and CPU It is mainly because the Memoryand CPU are dependent on the bandwidth The unevennessof the dynamically changing resource requirements and theemerging demand pattern can be compared to the differentgeometric objects [27 28] Hence here we apply fuzzy rulesto differentiate the different patterns and cluster them Fractalgeometry can be used to classify different objects based ontheir roughness [24 26 29] In this case the focus is basedonly on the smoother objects where the limitation of thefractal value is only up to one and if it is closer to one thenthat means maximum utilization of the memory and CPUresources If memory is 119872

1and CPU cycle is 119862

1 then the

data center load efficiency is DC1 Thus based on the input

parameter the output object efficiency is predicted using thesimple fuzzy rules Only disadvantage here is that based onthe parameters the total number of rules increases causingproblem due to dimensionality Based on this model whichhas been created the future values of the demand for CPUandmemory can be predicted leading to accurately assess theefficiency of the data center in the varying situations

11 Background In recent times more attention is shownon the framework of cloud computing and the performanceevaluation Iosup et al [30] have done ldquoperformance anal-ysis of cloud computing servicesrdquo approach for supportingefficiency of cloud computing In their model they analyzethe performance of Many Task Computing (MTC) work-loads They have proposed a comparison on performancecharacteristics and cost models Moreno-Vozmediano et al[31] have deployed a computing cluster on the top of manytask computing applications In this subsequent work clusterloads have been used for resources from different cloudsto construct high availability strategies These are used forproving viability to perform scalability of resources andperformances for large scale cluster infrastructure Dutreilhet al [32] have considered the recent research to constructa data center management framework for atomic resourceallocation in virtual applications They evaluated in twoways namely threshold-based and reinforcement learningmethods to dynamically scale resources

Yazir [33] presented a virtualization tool that providesthis gap by applying ideas from computational geometryIt proved valuable assistance in providing quick and easypreliminary performance analyses Data processing manage-ment is difficult to get as many machines as an applicationneeds The large scale jobs are distributed on differentmachines as parallel running processes The control andcoordination of these processes is complex with time depen-dent Cloud Architectures [34] have solved such difficultiesCloud administrators usually worry about hardware procur-ing (when they run out of capacity) and better infrastructureutilization (when they have excess and idle capacity) Thelower network bandwidth and the inherent lower hardwaredependability force enterprises to reorganize cloud appli-cation architecture [35] From the data center challengesand methodologies the two key questions arise How arethe efficiency of data centers and performance of cloudcomputing calculated What are the key factors to decidethe efficiency of data center in cloud computing This paperanswers these questions The contributions of the paper areoutlined as follows

(1) We compute the DCLE be (120578) for load-based datacenter management in cloud computing (120578) is avaluable perception for cloud service providers tomonitor manage and mitigate the cloud computingservices and justify the ability to hire a single virtualclient or thousands of virtual clients

(2) We construct it as a fuzzy expert systemmodel to findthe DCLE with basic three factors like BandwidthMemory and CPU cycles to validate the steps of our









5432125

80

135

190

245

300

GoGRID services


Band

wid

th (M

Bps)



2000

1610

1220

830

440

50

Disk

capa

city

(GB)












Rule 1 IF 119909 is 1198601 THEN 119910 is 119861

1

Rule 2 IF 119909 is 1198602 THEN 119910 is 119861

2

Rule 119899 IF 119909 is 119860119899 THEN 119910 is 119861

119899

Fact 119909 is 119860

Conclusion 119884 119894119904 119861

(1)


119895isin 119891(119909) 119861 119861

119895isin 119891(119910)



120597 [0 1] lowast [0 1] 997888rarr [0 1] (2)



120597 (119886 119887) = 119886 or 119887 (3)


120597 (119886 119887) = max 119909 isin 0 1 | 119886 and 119909 le 119887 (4)


120597 (119886 119887) = 119886 or (119886 and 119887) (5)



119877119895(119909 119910) = min [119860

119895 (119909) 119861119895 (119910)] (6)



119877 = ⋃119895isin119873119899

119877119895 (7)


1198611015840= ⋃119895isin119873119899

1198601015840sdot 119877119895 (8)



119877119895(119909 119910) = 120597 (119860

119895 (119909) 119861119895 (119910)) (9)




119875 IF 119909 = 119860 THEN119910 = 119861 (10)


⟨119909 119910⟩ is 119877 (11)


119877 (119909 119910) = 120597 [119860 (119909) 119861 (119910)] (12)



119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 (13)


119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 119909 isin 119860 (14)


119860119883119861119883119877of sets 119860 119861 119877


119883119861(119910) = sup

119909isin119883[119883119860 (119909) 119883119861 (119909 119910)] (15)



119861 = sup119909isin119883

[119860 (119909) 119877 (119909 119910)] (16)






Low LMedium MHigh H









1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p



1

08

06

04

02

001 02 03 04 05 06


Deg

rees

of m

embe

rshi

p

L M H




119891119889 [minus119886 119886] 997888rarr 119877 (17)



measurement 119889 = 1199090





1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip



MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th


M Memory




IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)


119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















5432125

80

135

190

245

300

GoGRID services


Band

wid

th (M

Bps)



2000

1610

1220

830

440

50

Disk

capa

city

(GB)












Rule 1 IF 119909 is 1198601 THEN 119910 is 119861

1

Rule 2 IF 119909 is 1198602 THEN 119910 is 119861

2

Rule 119899 IF 119909 is 119860119899 THEN 119910 is 119861

119899

Fact 119909 is 119860

Conclusion 119884 119894119904 119861

(1)


119895isin 119891(119909) 119861 119861

119895isin 119891(119910)



120597 [0 1] lowast [0 1] 997888rarr [0 1] (2)



120597 (119886 119887) = 119886 or 119887 (3)


120597 (119886 119887) = max 119909 isin 0 1 | 119886 and 119909 le 119887 (4)


120597 (119886 119887) = 119886 or (119886 and 119887) (5)



119877119895(119909 119910) = min [119860

119895 (119909) 119861119895 (119910)] (6)



119877 = ⋃119895isin119873119899

119877119895 (7)


1198611015840= ⋃119895isin119873119899

1198601015840sdot 119877119895 (8)



119877119895(119909 119910) = 120597 (119860

119895 (119909) 119861119895 (119910)) (9)




119875 IF 119909 = 119860 THEN119910 = 119861 (10)


⟨119909 119910⟩ is 119877 (11)


119877 (119909 119910) = 120597 [119860 (119909) 119861 (119910)] (12)



119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 (13)


119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 119909 isin 119860 (14)


119860119883119861119883119877of sets 119860 119861 119877


119883119861(119910) = sup

119909isin119883[119883119860 (119909) 119883119861 (119909 119910)] (15)



119861 = sup119909isin119883

[119860 (119909) 119877 (119909 119910)] (16)






Low LMedium MHigh H









1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p



1

08

06

04

02

001 02 03 04 05 06


Deg

rees

of m

embe

rshi

p

L M H




119891119889 [minus119886 119886] 997888rarr 119877 (17)



measurement 119889 = 1199090





1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip



MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th


M Memory




IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)


119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















Rule 1 IF 119909 is 1198601 THEN 119910 is 119861

1

Rule 2 IF 119909 is 1198602 THEN 119910 is 119861

2

Rule 119899 IF 119909 is 119860119899 THEN 119910 is 119861

119899

Fact 119909 is 119860

Conclusion 119884 119894119904 119861

(1)


119895isin 119891(119909) 119861 119861

119895isin 119891(119910)



120597 [0 1] lowast [0 1] 997888rarr [0 1] (2)



120597 (119886 119887) = 119886 or 119887 (3)


120597 (119886 119887) = max 119909 isin 0 1 | 119886 and 119909 le 119887 (4)


120597 (119886 119887) = 119886 or (119886 and 119887) (5)



119877119895(119909 119910) = min [119860

119895 (119909) 119861119895 (119910)] (6)



119877 = ⋃119895isin119873119899

119877119895 (7)


1198611015840= ⋃119895isin119873119899

1198601015840sdot 119877119895 (8)



119877119895(119909 119910) = 120597 (119860

119895 (119909) 119861119895 (119910)) (9)




119875 IF 119909 = 119860 THEN119910 = 119861 (10)


⟨119909 119910⟩ is 119877 (11)


119877 (119909 119910) = 120597 [119860 (119909) 119861 (119910)] (12)



119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 (13)


119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 119909 isin 119860 (14)


119860119883119861119883119877of sets 119860 119861 119877


119883119861(119910) = sup

119909isin119883[119883119860 (119909) 119883119861 (119909 119910)] (15)



119861 = sup119909isin119883

[119860 (119909) 119877 (119909 119910)] (16)






Low LMedium MHigh H









1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p



1

08

06

04

02

001 02 03 04 05 06


Deg

rees

of m

embe

rshi

p

L M H




119891119889 [minus119886 119886] 997888rarr 119877 (17)



measurement 119889 = 1199090





1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip



MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th


M Memory




IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)


119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875 IF 119909 = 119860 THEN119910 = 119861 (10)


⟨119909 119910⟩ is 119877 (11)


119877 (119909 119910) = 120597 [119860 (119909) 119861 (119910)] (12)



119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 (13)


119861 = 119910 isin 119884 | ⟨119909 119910⟩ isin 119877 119909 isin 119860 (14)


119860119883119861119883119877of sets 119860 119861 119877


119883119861(119910) = sup

119909isin119883[119883119860 (119909) 119883119861 (119909 119910)] (15)



119861 = sup119909isin119883

[119860 (119909) 119877 (119909 119910)] (16)






Low LMedium MHigh H









1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p



1

08

06

04

02

001 02 03 04 05 06


Deg

rees

of m

embe

rshi

p

L M H




119891119889 [minus119886 119886] 997888rarr 119877 (17)



measurement 119889 = 1199090





1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip



MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th


M Memory




IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)


119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

00201 03 04 05 06 07

L M H

Deg

rees

of m

embe

rshi

p



1

08

06

04

02

001 02 03 04 05 06


Deg

rees

of m

embe

rshi

p

L M H




119891119889 [minus119886 119886] 997888rarr 119877 (17)



measurement 119889 = 1199090





1

05

006 065 07 075 08

085 09Bandwidth

Memory

Low Medium

Medium

high

Small Large

1

05

0065 07 075 08

Deg

ree o

f mem

bers

hip

Deg

ree o

f mem

bers

hip



MD MD MD

MN MD MX

MN MN MD

H

M

L

S LA

Band

wid

th


M Memory




IF 119889 = 119860 119889 = 119861 THEN 120578 = 119862 (18)


119883⟨119909119896 119910119896 119911119896⟩ | 119896 isin 119870 (19)


MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










MD MX MX

MD MD MX

MN MD MD

H

M

L

L H

Band

wid

th

M


CPU cycles


MD

MN MD

MN MDMD

H

M

L

S LAM Memory

CPU

cycle

s

MX




119909119896and 119910



119896) 119861(119910

119896) 119862(119911



1198941[1198942 (119860 (119909

119896) 119861 (119910

119896)) 119862 (119911

119896)] (20)



[0 1] 119894(119886 1) = 119886 119887 le 119889 implies 119894(119886 119887) le (119886 119889) 119894(119886 119887) =


119886 for all 119886 isin [0 1]





where

[119860 times 119861] (119909 119910) = min [119860 (119909) 119861 (119910)] (22)




1


2


119899

Fact (119889 119889) is 119891119889(1199090) times 119891119889(1199100)


(23)






119862 = ⋃119895

[119891119889(1199090) times 119891119889(1199100)] 119900119894119877119895 (24)





1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1

08

06

04

02

0



Deg

ree o

f mem

bers

hip



119909lowast=

int119887

119886

120583119860 (119909) 119909 119889119909

int119887

119886

120583119860 (119909) 119889119909

(25)









Memory

325

215

1

0604

020 0 02 04 06 08 1

Bandwidth

DCL

E



325

215

1

07 06 05 04 03 02 01 0 0 01 02 0305 06

Bandwidth04

CPU cycles

DCL

E






325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










325

215

1

07 06 05 04 03 02 01 0 0 01 02 03 04 06 07Memory

05

CPU cycles

DCL

E



100

90

80

70

60

50

40

30

20

10

0168642

Number of nodes

Effici

ency

()


HPLDCLE




6 Conclusion


References































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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