2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

Task Scheduling with Load Balancing for Computational Grid

Using NSGA II with Fuzzy Mutation

Reza Salimi College of Computer Science

Tabari Institute of Higher Education Babol, Iran

rz,sa63@gmaiLcom

Homayun Motameni Department of Computer Engineering Islamic Azad University, sari Branch

Sari, Iran motameni@iausari,ac,ir

Hesam Omranpour Department of Computer Engineering Amirkabir University of Technology

Tehran, Iran h,omranpour@autac,ir

Abstract- The resources management in a grid computing is a

complicated problem. Scheduling algorithms play important role

in the parallel distributed computing systems for scheduling jobs,

and dispatching them to appropriate resources. An efficient task

scheduling algorithm is needed to reduce the total Time and Cost

for job execution and improve the Load balancing between

resources in the grid. In grid computing, load balancing is a

technique to distribute workload fairly across computational

resources, in order to obtain optimal resource utilization with

minimum response time, and avoid overload. Load balancing is a

crucial problem to grid computing. In this paper, we address

scheduling problem of independent tasks in the market-based

grid. In market grids, resource providers can request payment

from users based on the amount of computational resource that

used by them. Beside we consider Makespan and Load balancing.

In this paper, NSGA II with Fuzzy Adaptive Mutation Operator

is used to address independent task assignments problems in

parallel distributed computing systems. Results obtained proved

that our innovative algorithm converges to Pareto-optimal solutions faster and with more quality.

Keywords- Task scheduling; grid computing; Non-dominated

Sorting Genetic Algorithm (//); Adaptive Mutation Operator; fuzzy logic.

I. INTRODUCTION

Computational grid has originated from a new computing environment that has emerged as a main- stream technology for scientific research and cooperation using large-scale computing resources sharing and distributed system integration, In fact, computational resources in grid are geographically distributed computers or clusters, which are aggregated to serve as a single computing resource logically [1]. On the other hand, the goal of load balancing algorithms is essentially to fairly spread the load on computational resources for maximizing their utilization while minimizing the total task execution time [2]. In distributed computational systems, load balancing has important role in reducing response time and avoiding overload. Load balancing is applied in grid computing system, using some scheduling algorithm to ensure that the ratio of performance of entire resource node computing as an equal, therefore by improving the utilization of resources based on

Corresponding author: Reza Salimi E-mail: rz.sa63@gmail.com

978-1-4673-2925-5/12/$31.00 ©2012 IEEE 79

nodes, the overall task completion time can be reduced [3]. Computational grids enabling resource sharing and coordination are now one of the common and acceptable technologies used for solving computational intensive applications rising in scientific and industrial problems. Nevertheless, due to the heterogeneity, dynamicity and autonomy of the grid resources, task scheduling within these systems has become a challenging research area. Therefore, many research works have been done to overcome these challenges by proposing new algorithms and mechanisms. Applying the market model to the grids is a good approach which can easily take the dynamic characteristics of the grid resources into account and simplifY the scheduling problem considering user-centric trends. In this paper, we made use of a multi-objective heuristic genetic algorithm, NSGA II for optimizing three objectives: Cost, Makespan and maximum load balance in scheduling problem in grid computing. Also we implement fuzzy adaptive mutation operator for this algorithm. In the experiments, we have used the standard one-point crossover. Some algorithms are used and then are compared together. The rest of paper is organized as follows. We begin with an overview of related works in section 2. NSGA II and our approach are presented in section 3. Finally, experimental results and conclusion are represented in sections 4 and 5.

II. RELATED WORKS

Previously proposed approaches for scheduling problem in traditional grids are not comprehensive. These approaches [1, 2] mostly consider system and grid factors like Maximum load balance and Makespan of the system as main objective in scheduling, ignoring the interests and requirements of users or [4] considers cost and Makespan as main objective in scheduling, without load balance. Buyya et aL Reference [5] proposed an economics model for Grid resource management and scheduling, using marketing concepts such as commodity market, posted price modeling, bargaining modeling, contract net modeling, auction modeling and other. [6] proposes two models for predicting the completion time of jobs in a grid. The first service model predicts the completion time of a job in a grid that offers only one type of service in grid. The second is

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

multiple services model that estimates the completion time of a job that runs in a service grid which represents multiple types of services. Then, it has developed two algorithms that use the predictive models to schedule jobs in the service grid at both system level and application level, so that in application-level scheduling, using genetic algorithm, the average completion time of jobs is minimized through optimal job allocation on each node. [7] has proposed a job grouping method using Particle Swarm Optimization (PSO) to reduce the communication overhead and consequently reduce the completion time of the processes in computational grid and improve resource utilization. The objective of that paper is to dynamically assemble the individual fme-grained jobs of an application into a group of jobs and then transfer these coarse­grained jobs to the grid resources, so that optimizes the utilization of grid resources and reduces the overall completion time for processing user jobs. In [8] is represented a pure load balancing in computational grid using genetic algorithm without considering Makespan or cost for grid resources. In [9] has been studied the various load balancing strategies based on a tree representation of a grid. This study enables transforming any grid architecture into a unique tree with at most four levels. Using this proposed model, they defmed a hierarchical load balancing strategy so that privileges local load balancing in first. After load balancing is applied at group level, but if load is not balanced at group level, then it will be balanced at region level, fmally if load is not balanced at region level, it will be balanced at grid level. [10] represents a task scheduling GSACA (combination of simulated annealing algorithm and genetic algorithm) optimal algorithm based on resource load balance. GSACA algorithm uses immune clonal algorithm and simulated annealing and quantum algorithm to solve load balancing problem in the computational grid environment. A hierarchical layered architecture for grid computing services in [11] has been offered. It proposed an adaptive two level load balancing algorithm, which attempts to minimize the overall completion time and maximize the system throughput. Use of multi-objectives optimization algorithms such as NSGA II in [12] is observed. It uses NSGA II for optimizing Tasks scheduling problem in heterogeneous distributed computing system considering two objectives, Makespan and flow time.

III. NSGA II AND PROPOSED APPROACH

The NSGA-II algorithm [14] is the first and one of the commonly used evolutionary multi-objective optimization (EMO) algorithms which search solution space to fmd Pareto­optimal solutions in a multi objective optimization problem. NSGA-II uses the elitist principle and an explicit diversity preserving mechanism. In addition to, it emphasizes non­dominated solutions and forms the Pareto front as Pareto­optimal solutions [16]. The NSGA-II algorithm uses two effective strategies including an elite-preserving and an explicit diversity-preserving. NSGA-II uses an explicit diversity­preservation or niching strategy to assign a diversity rank to all the individuals that are in the same non-dominated front and thus have the same non-dominated rank in the population. The members within each non-dominated front that are in the least crowded region in that front are assigned a higher rank. For calculating the density of solutions surrounding a particular solution in the population, a crowding distance metric is used

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that is achieved from the average distance of the two solutions on either side of the solution along each of the objectives. As respects this particular niching strategy does not require any external parameters, so it was chosen for NSGA II. Details can be found elsewhere [13, 14]. Because of the nature of the models of the multi-objective optimization problems, non­dominated sorting genetic algorithm (NSGA) can be used to find the non-dominant optimal solutions. In the absence of any additional information about multi-objective optimization problem, one of these Pareto-optimal solutions cannot be considered as better solution than the others. Superiority and Suitability of one solution over the others depends on several factors including user's choice and problem environment. Therefore, the NSGA II determines a set of dominant solution and so Pareto front is obtained [17]. In this paper, NSGA II with Fuzzy Adaptive Mutation Operator is used to address independent task assignments problems in parallel distributed computing systems. Task scheduling in grid with three approaches is done. First, Tasks scheduling in grid is done with NSGA II, then this problem is investigated with MOPSO. Then, Tasks scheduling with two objective functions, Price and Makespan solved with NSGA II without fuzzy logic and with fuzzy logic. Finally, Tasks scheduling is optimized with three objective functions, Price, Makespan and Load balancing. In NSGA II with fuzzy mutation, inputs for fuzzy function are Variance between Costs of individuals and Gene Variance that expresses difference between values of bits in order to prevent individuals from being equal in population as well as more variety. In next section we solve the problem with proposed approach.

A. Encoding Mechanism In the coding scheme we have developed for our problem,

each solution is encoded as a vector of integers. For a problem with n tasks and m resources, the length of the vector which can be considered as a chromosome is n. As well as, the content of each cell of vector which shows a gene value in chromosome can take a number between 1 and m that representing the resource allocated to that task Fig. 1.

For generating an initial population with k individuals, a random number between 1 and m is assigned to each cell of the vector.

B. Objectives and Fitness Functions Our main objective here is to get task assignments that will

obtain minimum completion time and a well-balanced load across all nodes as well as minimum Price for user. Therefore, our fuzzy NSGA II algorithm is three dimensions optimization. In this problem two objectives Price and Makespan are in conflict with each other naturally so that when Price is reduced then Makespan increases. But Load Balance objective do not conflict with any one.

T, T2 T3 T4 Ts T6 T7 Tg T9 TIO

, , , , • • , , , , 2 5 I 1 I 3 I 3 I 4 I 5 I 1 I 4 I 4

Figure 1. A chromosome in coding Scheme

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

• Minimum Makespan: The fIrst objective function of our algorithm is the Makespan or latest completion time of the task schedule. Makespan (or maxspan) is the longest completion time between all the processors in the system [1, 2]. Consider Ti and Cj denote the size of the task i and processing speed of the resource j, respectively. Then, the execution time of the task i on the resource j can be formulated as follow:

T texe(i, j)=

C' (1) J

For each processor there will be a completion time for tasks which assigned to it. In general, completion time in any processor is calculated as follow:

LkEA Tk t G)- J l:S j :Sm (2) comp""c Cj

Where, Aj is the set of tasks indexes which are assigned to resource j. Now, Makespan is:

Makespan= Max{ teomp""c G)} l:S j :Sm (3)

Therefore, one of goals is to minimize (3), which means that the assigned tasks to resources will be completed in the shortest time.

• Maximum load balance: In distributed systems, load balancing is a technique (usually performed by load balancers) to spread workload between computational resources, in order to achieve optimal resource utilization, throughput, and response time [3]. The load balancing mechanism attempts to distribute the load on each computing node equitably, and maximize the utilization and minimize the total task execution time. In order to get these goals, the load balancing mechanism should be 'fair' in distributing the load across the resources; it implies that the load difference between the "heaviest-loaded" node and the "lightest-loaded" node should be minimized. So the second objective function of our approach is obtaining the maximum load balance. We fIrst specity the average node utilization. Note that high average node utilization ahnost means that the distributed load is well balanced across all nodes in the system [1].

We obtain the average node utilization through dividing the sum of all the nodes' utilization by the total number of nodes [1]. So, we calculate the expected utilization of each node based on the given tasks assignment. We calculate it by dividing the task completion time of each node by the Makespan. Thus, utilization of each node is:

P G)= tmp""c G)

u Makespan l:S j :Sm (4)

We must note that a high value of the average node utilization doesn't always imply a desirable load balance [1], so we calculate average node utilization:

P (L�I PuG))

(5) m

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Then, the algorithm must minimize the mean square deviation of P u (j) to improve Load balance across all nodes. Mean square deviation of Pu (j) is achieved as follow:

(6)

Hence, second objective function for OptlITIIZing is improving load balance across all processors that is gained from minimizing (6).

• Minimum Price: as mentioned, resource providers in market-based grids can request price from users based on the amount of resource that requested by them. Therefore, scheduling algorithms in market-based grid should consider users' willingness to complete their applications in the most economical way possible [4]. So, third objective function is total price that must be minimized. Suppose Wj denotes to unit price for resource j. Therefore, the execution cost of the task i on the resource j can be computed using following equal:

Price(j)=t (j) XW complete J (7)

Then, total cost for scheduling is calculated as follow: m

Total Cost= L Price(j) (8) j�l

Where Total Cost denotes the overall cost resulting from a chromosome in population that representing a scheduling.

C. Fuzzy Mutation Operator In this paper, specialized Fuzzy Mutation operator is

developed and compared with Mutation operator without using fuzzy logic and demonstrated superior to the other. Two functions are used for calculating the Variance. First calculates the variance between genes values in different chromosomes in order to increase the variety and investigate all states for assignment the tasks to resources. This function takes individuals existing in Pareto front as input and selects the better member, then compares genes of other individuals with genes of better member. This function will generate a number in the interval [0, 1]. The inputs of second function are also the Pareto front members. It calculates the variance of fItness average. This is to ensure diversity among the members according to their Cost. This function will generate a number in the interval [0, 1.57]. A fuzzy system is designed which consists of two inputs and one output for inputs. Then these results will be as inputs to fuzzy system. The output of the fuzzy system is used for determining the probability of mutate in population and also probability of mutate of genes in chromosomes which determines the number of genes for mutation in any chromosome. Input membership functions are shown in Fig. 2 and Fig. 3. Fuzzy rules for this operator also can be seen in Table. IV. The membership functions of the output of the fuzzy system are also shown in Fig. 4.

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

IV. EXPERIMENTAL RESULTS

In this section, for the experiments we fIrst optimize tasks scheduling with three algorithms MOPSO, NSGA II and fuzzy NSGA II with two objectives, Price and Makespan, so that we don't take into account the Load Balancing. The algorithms are implemented according to their description in the literature [14, 15]. In this stage we observe that our algorithm is better to other so that, our proposed method converges to Pareto-optimal solutions faster and with more quality. Experimental results show that fuzzy NSGA II creates the Pareto front with all individuals existing in population in less iteration and also created Pareto front has more quality. Furthermore during the experiments, we found that non-dominated sorting genetic algorithms perform better than the multi-objective particle swarm optimization for scheduling problem. The results are shown in the next section.

In second stage, we take into account the Load Balancing. The observations can be seen in the next subsections. In the experiments, we have used the standard one-point crossover. The population size is kept constant. All performance analyses are carried out at different numbers of generations, 50 and 100. Table. I and Table. II are used as the specifIc parameters. In Table. II Mutation probability is the likelihood of mutating a particular solution and Bit mutation probability is the likelihood of mutating each bit of a solution in mutation.

A. Experimenting stage J Since, in the market-based grid environment, two factors

Price and Makespan is very important, therefor we fIrst optimize tasks scheduling problem in terms of these two factors. The optimization is done with three algorithms MOPSO, NSGA II and NSGA II with fuzzy mutation. Tests were performed in 50 and 100 iterations. In any case, tests 10 times were done and there exist the average of obtained results Fig. 5 and Fig. 6.

TABLE I. PROBLEM PARAMETERS Population size: Number of generations: Number of tasks: Max size of tasks: Number of Resources: Max Price for resources: Max processing speed of resources:

200 50, 100

200 50 15

20 units of price 5

TABLE II GENETIC OPERA TORS

� Crossover Mutation Bit mutation Algorithm probability probability probability

NSGA II 0.8 0.2 0.2

Fuzzy NSGA II 0.8 mu (Fig.5) mu (Fig.5)

As can be seen from the fIgures the performance of our proposed method is better. So that in the experiment with 50 iterations, NSGA II with fuzzy mutation on average in iteration 34 total of individuals i.e. 200 members entered into Pareto­optimal Front, while in NSGA II in iteration 50 only 183 members and in MOPSO in iteration 50 only 28 members are entered into Pareto front. Also in the experiment with 100 iterations, NSGA II with fuzzy mutation on average in iteration 34 total of individuals i.e. 200 members entered into Pareto­optimal Front, while in NSGA II in iteration 80, 200 members and in MOPSO in iteration 100 only 38 members are entered into Pareto front. In all experiments, Pareto Front in our proposed method was created faster and with more quality.

B. Experimenting stage 2 In stage 2, we perform the experiments for optimizing tasks

scheduling problem with three objectives; Load balancing, Makespan and Price. Initially distribution of individuals is shown in Fig. 7. Average of obtained results with performing these algorithms in ten times is shown in fIgures 8 and 9. In the analysis for instance, the points A, B and C in Fig. 9, are the best solutions in terms of only one objective. For example point B is the best member of Pareto-optimal front in terms of Makespan while the point C is the best solution in terms of Price. Table. III shows the best solutions in terms of each objective separately. The points D are good solutions in terms of both objectives Makespan and Price but not in terms of Load Balancing. The points A and B are optimal solutions for both Load Balancing and Makespan but not in terms of Price and the points C are good and optimal solutions in terms of both Price and Load Balancing but not in terms of Makespan. Since Price has a higher magnitude over Makespan and Load Balancing so proposed method tries to minimum all three objectives therefor in Table. III is observed that price has been reduced by proposed method more than other algorithms. The points Z are the best in terms of all three objectives. These points almost are in the middle of range of each parameter. Furthermore Pareto­optimal front is created by NSGA II with fuzzy mutation much faster than others. So that in experiment with 50 iterations for fuzzy NSGA II, total of individuals are placed in Pareto­optimal front in 5th iteration (Fig. 8) while in NSGA II all of members are placed in Pareto-optimal front in lIth iteration and in MOPSO only 127 members are placed in Pareto-optimal front in 50th iteration. And in experiment with 100 iterations in fuzzy NSGA II, total of individuals are placed in Pareto­optimal front in 7th iteration (Fig. 9) while in NSGA II all of individuals are placed in Pareto-optimal front in 10th iteration and in MOPSO only 145 members are placed in Pareto-optimal front in 100th iteration. In all of these cases speed of achieving to Pareto-optimal solutions by fuzzy NSGA II is faster.

TABLE III PARETO-OPTIMAL SOLUTIONS FOR DIFFERENT CHOICES

� point A: Best Load Balancing point B: Best Makespan point C: Best Price

Load Makespan Price

Load Makespan Price

Load Makespan Price Methods Balancing Balancing Balancing

Fuzzy NSGA II 0.14 223.75 43733 0.16 192 41821 0.25 8320 10332

NSGA II 0.13 242.7 45653 0.17 187.5 45194 0.23 6328 16512

MOPSO 0.18 418 41619 0.2 312 42873 0.21 834 37698

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If If If If If If If If If If If If If If If

low

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

A is verylow A is verylow A is verylow A is low A is low A is low A is medium A is medium A is medium A is high A is high A is high A is veryhigh A is veryhigh A is veryhigh

me<!

TABLE IV. Fuzzy RULE DATABASE FOR FUZZY MUTATION and B is low then mu is veryhigh and B is medium then mu is high and B is high then mu is medium and B is low then mu is high and B is medium then mu is medium and B is high then mu is medium and B is low then mu is high and B is medium then mu is medium and B is high then mu is low and B is low then mu is medium and B is medium then mu is low and B is high then mu is verylow and B is low then mu is medium and B is medium then mu is medium and B is high then mu is verylow

high �eryhig�

0.5

In addition, when we investigated the mean of the best solutions about NSGA II and NSGA II with fuzzy mutation, we observed that proposed method has less mean of the best solutions and the mean of solutions in terms of all three objectives compared with NSGA II, Fig. l 0, and Fig. 11.

O c==±==�==c=�==��c=���==�� o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2. Input membership function: Gene value variance (A)

10 '·' me" high

05

o c=========�========�==========� o Q5 1.5

Figure 3. Input membership function: Cost variance (B)

me<! high veryhig

0.5

O c==±==�==c=������==�==�� 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0. 26 0.28 0.3

Figure 4. output membership function: mu

o NSGA II with fuzzy Mutation

I> NSGAII

o MOPSO 4 . 5 ........... , ..... .... .. , ........... : ........... : ...... ... .. : ......... .

I> ! ! ! ! : · . . . . ---------"-----------"----------_._---------_._---------_ ._---------· . . . . : : : : : · . . . . · . . . . - - - - - - - - � - - - - - - - - - - - � - ----------! -- - - - - - - - - - � -- ---- -----! ----------

________ � ___________ t ___________ i ___________ i ___________ I _________ _ · . . . .

! . ! ! ! - -, - - - � - - - - - --- - - - � - - - - - - - - - - - ! - - - - - - - - - - - ! - - - - - - - - - - - ! - - - - - - - - - -· . . · . . · . .

1.5 •...••..•.. _ •... :::::::::::········:::: ::: :::::1:::::::::::1::::::: ::: 2000 4000 6000 8000

f1 :Makespan 10000 12000

Figure 5. Obtained Pareto-Optimal Fronts in 50 iterations

83

<i'

X 10· 5�� �-'---'--�'---'--'��O�N�S�G�A�" �= ·�th�fu�Z=Z=Y �M?u t=a �t i o =n'il

4.5 ------!------�-------l-------�------ 6 ����� ... : ...... i" .... -:- ...... , ..... c.,. :7: •• 7: .. 7: .. -'-:,.7. .. 7: .• :7: . • ':c ... :::. ,:::" ",.,::: .. c:-•• c:-• • cc: .C:-•• :-:-• • :-'..1

3.5 - ---! - - - -- - � -------;----- - -� ------ : - - - - - - � -- - - --�-- - ----� - - - - - - � -----

- - - � - - - - - - � - - - - - - + -- - - - -�--- - - - - t-- - - - - � -- - - - - -l-- - ----i--- - - - -� -----

.;..J--. 2.5 --; - - - - - - �--- - -. �--- --- -� - - - - - - � --- --

���....L; ....".�······i······j······i·······,······� ····· 1.5 ..... + ..... . []: ...... -:-=r ..... : � ... i ····�·+····�·:·······df .. . . .

1 - ----- � ----- - � --- - - - + -- ----� - - ---- i - ----- � - - --- -; -------:-- - - -- ,

0.5 O�----;-;, Of;:O:CO ---;;:20;!; 0"'O-----03"'0�00;--4:--;0f;:0 0;;-""50;!; 0"'O-----06"'00�0;--7"'0!:: 00;;--;;:BO;!;0"' 0----;;9c:!OO"' O'-------:,:::-!OOOO

f1 :Makespan Figure 6. Obtained Pareto-Optimal Fronts in 100 iterations

11000

10000

9000

8000

7000

6000

5000 1500

..•

3.'

� .;.r. 2.S

1.' 1 12000

----r' -_

:

-,-

-,-

-

-"-,-

"'"

500

f,:LoadBalance

Figure 7. initially distribution of solutions

Figure 8. Obtained Pareto-Optimal Fronts in 50 iterations

0.4

2012 2nd IEEE International Conference on Parallel, Distributed and Grid Computing

c u:: � In

�

.g 3 Il. ':"i->

x 10� - - .-� . -_ .. , :'"

_ ... . -:. ----

--. . : . . ---- ---: ... .• - ... :..

B

__ : _ ___ __ _ l --A--�--

...... --,"� .. -

o 0.05 0.1

o NSGA II with fuzzy Mutation • -'�._ t> NSGAII : " o MOPSO : l-",>.

-r-'- .... --L " -,� : . '.

f1:loadBalance

0.45

Figure 9. Obtained Pareto-Optimal Fronts in 100 iterations

145 �--�--�--�--�--�--�i=�==?c==��=1l ---- F uzzy N SGA II 140

135

130

125

120

115

1100

-- NSGAII

10 20 30 40 50 60 70 Iteration

80 90 100

Figure 10. Mean of best solutions in terms of all three objectives

200 r.-�-�-�-�-�-�r===::::::::'=::::::!====il -- F uzzy NSGA II

195 --NSGAII

190

� 185 u::

c � 180

175

170

Iteration

Figure II. Mean of solutions in terms of all three objectives

V. CONCLUSION

In this paper NSGA II with fuzzy adaptive mutation is implemented for task scheduling and compared with NSGA II and MOPSO. It is clear that the quality of schedules achieved by proposed method is almost better. In comparing the performance of the algorithms it is seen that NSGA-II with fuzzy mutation maintains a uniform spread of solutions in the obtained non-dominated front. Spread and span of solutions in proposed method is more than other method because fuzzy mutation operator is flexible compared with simple mutation therefor it explores the solution space better and finds solutions in more extensive areas. Fuzzy mutation ensures that mutation do not make solutions impaired. This is why the Pareto front can be created much faster and requires less iteration than

84

others because when the variety between individuals is high then mutation rate is reduced and will prevent solutions from getting impaired. On the other hand when the variety between individuals is low then mutation rate is increased to avoid the recession.

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