Efﬁcient Task Scheduling in Cloud Computing using an Improved

Particle Swarm Optimization Algorithm

Guang Peng and Katinka Wolter

Department of Mathematics and Computer Science, Free University of Berlin, Takustr. 9, Berlin, Germany

Keywords:

Task Scheduling, Cloud Computing, Scientiﬁc Workﬂow, Particle Swarm Optimization, Multi-objective

Optimization, Gaussian Mutation.

Abstract:

An improved multi-objective discrete particle swarm optimization (IMODPSO) algorithm is proposed to solve

the task scheduling and resource allocation problem for scientiﬁc workﬂows in cloud computing. First, we

use a strategy to limit the velocity of particles and adopt a discrete position updating equation to solve the

multi-objective time and cost optimization model. Second, we adopt a Gaussian mutation operation to update

the personal best position and the external archive, which can retain the diversity and convergence accuracy

of Pareto optimal solutions. Finally, the computational complexity of IMODPSO is compared with three

other state-of-the-art algorithms. We validate the computational speed, the number of solutions found and the

generational distance of IMODPSO and ﬁnd that the new algorithm outperforms the three other algorithms

with respect to all three metrics.

1 INTRODUCTION

Task scheduling and resource allocation is one ac-

tive research area in cloud computing. Cloud service

providers offer three different categories of service

Infrastructure as a Service (IaaS), platform as a Ser-

vice (PaaS), and Software as a Service (SaaS) (Mas-

dari et al., 2017). Since clients want to best utilize

the computing and memory resources offered by IaaS,

more and more complicated applications are applying

full or partial ofﬂoading to the Cloud. The users are

charged based on a pay per use model.

Workﬂow descriptions have been frequently used

to model scientiﬁc problems in areas such as as-

tronomy, biochemistry and physics (Rodriguez and

Buyya, 2014). These scientiﬁc workﬂows usually

have high computational complexity. So, cloud com-

puting provides a high-performance computing envi-

ronment for executing scientiﬁc workﬂows. Differ-

ent cloud resources are located in different regions

and have different capabilities for executing applica-

tions. According to the workﬂow and quality of ser-

vice (QoS) requirements, the task scheduling and re-

source allocation for scientiﬁc workﬂow needs to as-

sign different tasks to different cloud resources.

It is known that the task scheduling and resource

allocation problem is a NP-hard combination opti-

mization problem (Zhan et al., 2015). Rezvani (Rez-

vani et al., 2014) and Ergu (Ergu et al., 2013) used

a traditional integer linear programming (ILP) al-

gorithm and an analytic hierarchy process (AHP)

method to solve the resource allocation problems,

respectively. Traditional mathematical methods are

often used to solve simple models and they have

low efﬁciency for solving complex models. Due

to their strong heuristic search ability, different evo-

lutionary algorithms have been commonly used to

solve the different task scheduling and resource allo-

cation problems. Verma (Verma and Kaushal, 2014)

adopted the heuristic genetic algorithm (HGA) which

uses a chromosome to represent a schedule to solve

the task scheduling and resource allocation problem

with a deadline constraint. Singh (Singh and Kalra,

2014) and Hamad (Hamad and Omara, 2016) pro-

vided an improved genetic algorithm (GA) with a

Max-Min and tournament selection approach to solve

the task scheduling problems in cloud computing, re-

spectively. A GA often has slower convergence speed

and relatively needs more time to search for the opti-

mal solution.

Huang (Huang et al., 2013) presented a work-

ﬂow scheduling model based on the weight aggrega-

tion of cost and makespan and used a heuristic parti-

cle swarm optimization (PSO) algorithm to obtain a

schedule satisfying various QoS requirements. Cao

(Cao et al., 2014) modelled security threats by means

Peng, G. and Wolter, K.

Efﬁcient Task Scheduling in Cloud Computing using an Improved Particle Swarm Optimization Algorithm.

DOI: 10.5220/0007674400580067

In Proceedings of the 9th International Conference on Cloud Computing and Services Science (CLOSER 2019), pages 58-67

ISBN: 978-989-758-365-0

of a workﬂow scheduling model and used a discrete

PSO based on the feasible solution adjustment strate-

gies to solve the model. PSO has a faster search but

is also easily caught in local optima. Liu (Liu et al.,

2011) used an ant colony optimization (ACO) algo-

rithm to solve the service ﬂow scheduling with var-

ious QoS requirements in cloud computing. Li (Li

et al., 2011) used a load balancing ant colony opti-

mization (LBACO) algorithm to solve the cloud task

scheduling problem. The single ACO has low ef-

ﬁciency and is easily trapped in a local minimum.

Some researchers combined these algorithms to im-

prove performance. Gan (Gan et al., 2010) proposed

a hybrid genetic simulated annealing algorithm for

task scheduling in cloud computing. Liu (Liu et al.,

2014) used an integrated ACO with a GA to solve

a task scheduling model with multi QoS constraints

in cloud computing. Ju (Ju et al., 2014) presented a

hybrid task scheduling algorithm based on PSO and

ACO for cloud computing. These hybrid algorithms

can improve the search ability to a certain extent, but

increase the complexity of the algorithm at the sane

time.

The different evolutionary algorithms above are

mainly used to solve the task scheduling and re-

source allocation problems with only one objective

function. However, the task scheduling and resource

allocation problems in cloud computing are nor-

mally multi-objective optimization problems which

need to satisfy different QoS requirements. Soﬁa

(Soﬁa and GaneshKumar, 2018) adopted a nondom-

inated sorting genetic algorithm (NSGA-II) to solve

a multi-objective task scheduling model minimizing

energy consumption and makespan. NSGA-II may

need more time to sort the nondominated solutions.

Ramezani (Ramezani et al., 2013) designed a multi-

objective PSO (MOPSO) algorithm for optimizing

task scheduling to minimize task execution time, task

transfer time and task execution cost. MOPSO sorts

the archive members based on the QoS function with

weight aggregation for time and cost, where it is dif-

ﬁcult to set the weight. Tsai (Tsai et al., 2013) used

an improved differential evolutionary algorithm to op-

timize a multi-objective time and cost model of task

scheduling and resource allocation in a cloud comput-

ing environment, but the model did not consider data

interdependency between the cloud customer tasks.

In this paper, considering the multi-objective op-

timization properties of workﬂow scheduling, we

present a multi-objective cost and time model con-

sidering data interdependency for a workﬂow appli-

cation in cloud computing. Since Durillo (Durillo

et al., 2009) and Nebro (Nebro et al., 2009) analyzed

the performance of six representative MOPSO algo-

rithms and concluded that all of them were unable to

solve some multi-modal problems satisfactorily. Ad-

ditionally, they studied the issue and found that the ve-

locity of the algorithms can become too high, which

can result the swarm explosion. Here, our motiva-

tion is twofold. First, we use a velocity constriction

mechanism to improve the search ability as well as

the convergence speed. Also, the Gaussian mutation

operation is applied to enhance the diversity. Then we

can obtain an efﬁcient IMODPSO algorithm to solve

the proposed multi-objective cost and time model.

Our second goal is to compare the performance of

IMODPSO with three state-of-the-art multi-objective

optimization algorithms, i.e. NSGA-II (Deb et al.,

2002), MOEA/D (Zhang and Li, 2007) and MOEAD-

DE (Li and Zhang, 2009), which are the representa-

tive algorithms based on the Pareto ranking method

and decomposition approach. Especially, the decom-

position approach is a potential method in the current

multi-objective optimization research. The main con-

tribution of this paper is that the proposed IMODPSO

algorithm can achieve better accuracy and diversity

schedules satisfying different QoS requirements at a

lower cost than the competitors.

The rest of this paper is organized as follows: Sec-

tion 2 gives a brief introduction of multi-objective op-

timization. Section 3 presents the structure of the sci-

entiﬁc workﬂow application and the multi-objective

time and cost optimization model. The proposed

IMODPSO algorithm is illustrated in Section 4. Sec-

tion 5 discusses the simulations and results. Conclu-

sions and future directions will be highlighted in Sec-

tion 6.

2 MULTI-OBJECTIVE

OPTIMIZATION

Many real-world engineering optimization problems

often involve the simultaneous satisfaction of multiple

functions which are usually in conﬂict, that means the

improvement of one objective value can result in the

degradation of other objectives. Instead of ﬁnding a

single best one solution in single-objective optimiza-

tion, multi-objective optimization problems have a set

of trade-off solutions to satisfy all conﬂicting objec-

tives. A multi-objective optimization problem can be

deﬁned as follows (Reyes-Sierra et al., 2006):

min y = F (x) = [ f

(x), f

(x), ··· , f

(x)]

s.t. g

(x) ≤ 0, i = 1, 2, ··· , p

(x) = 0, j = 1, 2, · · · , q

(1)

Where x = (x

, x

, ··· , x

) ∈ D is a n-dimension

decision vector in a decision space D; y =

Efﬁcient Task Scheduling in Cloud Computing using an Improved Particle Swarm Optimization Algorithm

( f

, f

, ··· , f

) ∈ Y is a m-dimension objective vec-

tor in an objective space Y . g

(x) ≤ 0 (i = 1, 2, ··· , p)

refers to i-th inequality constraint and h

(x) ≤

0 ( j = 1, 2, ··· , q) refers to j-th equality constraint. In

the following, four important deﬁnitions for the multi-

objective optimization problems are given.

Deﬁnition 1 (Pareto Dominance).

A decision vector x



, x

, ··· , x



∈ D is said

to dominate a decision vector x



, x

, ··· , x



∈

D, denoted by x

≺ x

, if and only if







≤ f





, ∀i ∈

{

1, 2, ··· , m

}





< f





, ∃ j ∈

{

1, 2, ··· , m

}

(2)

Deﬁnition 2 (Pareto Optimal Solution).

A solution vector x



, x

, ··· , x



is called a

Pareto optimal solution, if and only if ¬∃x

: x

≺ x

Deﬁnition 3 (Pareto Optimal Solutions Set).

The set of Pareto optimal solutions is deﬁned as





¬∃x

≺ x



Deﬁnition 4 (Pareto Front).

The Pareto optimal solutions set in the objec-

tive space is called Pareto front, denoted by PF =

{

F (x) = ( f

(x), f

(x), ··· , f

(x))

x ∈ P

}

3 PROBLEM DESCRIPTION

In this section we ﬁrst present an example for the type

of problems we address in this paper. Then we intro-

duce our formally deﬁned time and cost model.

3.1 Application and Schedule

Generation

We illustrate how the applications we consider can be

mapped onto resources by discussing an example.

The workﬂow application can be represented as a

directed acyclic graph G = (T, E), where the set of

vertices T = (T 1, T 2, ··· , T N) denotes N application

tasks and an edge E(Ti, T j) denotes the data depen-

dency between task Ti and T j. Ti is said to be the par-

ent task of T j if T j needs to receive the results from

Ti. Thus, T j cannot be executed until all of its par-

ent tasks are ﬁnished. Fig. 1 shows a directed acyclic

graph of a workﬂow application with nine tasks.

Task 1

Task 2

Task 3

Task 4

Task 5

Task 6

Task 7

Task 8 Task 9

Out

Out_13

Out

Out_25

Out_36

Out_47

Out

Out_68

Out

Out_89 Out

Figure 1: Graph workﬂow application with nine tasks.

In a cloud computing environment, resources are

located in different regions and have different pro-

cessing capacity for the workﬂow application tasks.

Assuming that the processing capacity in terms of

ﬂoating point operations per second (FLOPS) (Ro-

driguez and Buyya, 2014) is known we can calcu-

late the execution time of each application task. The

different resources need some time to receive and

preprocess each application task. The time needed

to transmit data for further processing from a par-

ent task to a child task, where the next processing

step takes place, is associated as data transfer time

to the child task. So, the total time of each task in-

cludes the receive time, the execution time and the

transfer time. The receive time and the transfer time

are proportional to the data size and inversely propor-

tional to the bandwidth of the network. The schedul-

ing of the workﬂow application tasks means to prop-

erly assign the different tasks to different resources.

One of the goals when computing a good schedule

of the tasks is to minimize the total time (Makespan)

needed to complete the workﬂow application. Assum-

ing that there are three different resources, an example

of a schedule generated for a workﬂow application is

shown in Fig. 2.

T1_Totoal

T2_Total

T3_Total

T4_Total

T5_Total

T6_Total

T7_Total

T8_Total T9_Total

Makespan

Time

Resource

Figure 2: Example of a schedule generated for a workﬂow.

3.2 Multi-objective Time and Cost

Model

To generalize the discussion, let us assume that

there are M tasks of one workﬂow application and N

different resources. Let T

Rece

, T

Exe

, T

Trans

, and T

wait

be the receive time, execution time, transfer time

and waiting time, respectively. T (i)

Rece

is computed

by the data size of task i divided by the network

bandwidth. T(i)

Exe

is computed by the data size of

task i divided by the processing ability of assigned

resource. T (i)

Trans

is computed by the transmission

data size produced by parent tasks of task i divided by

the network bandwidth. Let T (i) Total represent the

completion time of task i. T

Res j

denotes the total time

to execute all the tasks in resource j. And Makespan

is the maximum time to complete the whole work-

ﬂow application. T (i) Total, T

Res j

and Makespan

CLOSER 2019 - 9th International Conference on Cloud Computing and Services Science

are shown as follows:

T (i) Total = T (i)

Rece

+ T (i)

Exe

+ T (i)

Trans

(3)

Res j

∑

T (i) Total +

∑

wait j

(4)

Makespan = Max (T

Res 1

, ···T

Res j

, ··· , T

Res N

) (5)

Let C

Rent

be the cost per unit time while us-

ing a cloud resource. C

Rent Rece

(i), C

Rent Exe

(i), and

Rent Trans

(i) represent the receive cost, execution

cost and transfer cost of task i, respectively. C

Total

is the overall cost of completing the workﬂow appli-

cation, which is deﬁned as follows:

Total

∑

i=1

Rent Exe

(i) +

∑

i=1

Rent Rece

(i)

∑

i=1

Rent Trans

(i)

(6)







Rent Rece

(i) = T (i)

Rece

×C

Rent

Rent Exe

(i) = T (i)

Exe

×C

Rent

Rent Trans

(i) = T (i)

Trans

×C

Rent

(7)

Based on the above two objective functions, the

multi-objective optimization model of scheduling the

workﬂow application is constructed. It aims to ﬁnd

a schedule to minimize the total cost and makespan

simultaneously.

4 PROPOSED IMODPSO

APPROACH

4.1 Particle Position Encoding

The scheduling of an workﬂow application is a NP

complete problem. When using a particle swarm op-

timization algorithm to solve the model we ﬁrst adopt

an integer encoding. Each particle position represents

one schedule, and each position variable represents

the task assigned to the corresponding resource. So,

the length of encoding is equal to the number of tasks

of the workﬂow application. Considering the work-

ﬂow application with nine tasks and three different

resources, Table 1 shows an example of an encoded

schedule [3, 1, 2, 2, 1, 2, 1, 2, 3], which means that task

1 is assigned to resource 3, task 2 is assigned to re-

source 1 etc.

Table 1: Particle position encoding.

Dimension 1 2 3 4 5 6 7 8 9

Encoding 3 1 2 2 1 2 1 2 3

4.2 Updating the Velocity and Position

In PSO, each particle position represents a po-

tential solution of the model (Kennedy and

Eberhart, 1995). Each particle has a position

(t) = [X

i,1

(t), X

i,2

(t), ··· , X

i,D

(t)] and a velocity

(t) = [V

i,1

(t),V

i,2

(t), ··· ,V

i,D

(t)] in a D-dimensional

search space. According to the personal best

position Pbest

(t) = [P

i,1

(t), P

i,2

(t), ··· , P

i,D

(t)]

and global best particle position Gbest (t) =

(t), G

(t), ··· , G

(t)], the particle can move

to a better position while ﬁnally approaching the

optimal solution. The velocity and position updating

equations are calculated as follows:

i j

(t + 1) = wV

i j

(t) +C

(Pbest

i j

(t) − X

i j

(t))

(Gbest

(t) − X

i j

(t))

(8)

i j

(t + 1) = X

i j

(t) +V

i j

(t + 1) (9)

Where i and j represent the j-th dimension of the

i-th particle, t is the index of the current iteration, w is

the inertia weight, C

and C

are the acceleration co-

efﬁcients, r

and r

are uniformly distributed in [0, 1].

In order to control the particle’s velocity, we

applied a speed constriction coefﬁcient χ obtained

from the constriction factor developed from Clerc and

Kennedy (Clerc and Kennedy, 2002), which can im-

prove the convergence ability of the algorithm.

χ=

2−ϕ−

−4ϕ

(10)

ϕ=



i f C

> 4

0 i f C

≤ 4

(11)

The new velocity constrained updating equation

and the velocity boundary constraint equation are re-

vised as follows:

i j

(t + 1) = χ •V

i j

(t + 1) (12)

i j

(t + 1) =







delta

i f V

i j

(t + 1) > delta

−delta

i f V

i j

(t + 1) ≤ −delta

i j

(t + 1) otherwise

(13)

delta

U pper limit

− Lower limit

(14)

Where U pper limit

and Lower limit

are the up-

per and lower boundaries of the j-th variable in each

particle.

To make the PSO suitable for solving a discrete

scheduling model, the position updating equation is

revised as follows:

i j

(t + 1) =



i j

(t) +V

i j

(t + 1)



(15)

The sign

indicates that if the new position is be-

yond the lower or upper boundary of the position, then

Efﬁcient Task Scheduling in Cloud Computing using an Improved Particle Swarm Optimization Algorithm

the new position should be the lower or upper bound-

ary. Otherwise, the updating position needs to adopt

the round operation to guarantee that positions are al-

ways integer numbers. With regard to the discrete

scheduling optimization problem, the global best par-

ticle can be selected from the nondominated solutions

stored in external archive randomly for each particle

during each iteration.

4.3 The Gaussian Mutation

Different from only one solution in the single opti-

mization problem, there exist some nondominated so-

lutions which are called Pareto optimal solutions in

multi-objective optimization problems. During the

iterations of the multi-objective particle swarm op-

timization algorithm, an external archive is used to

store the nondominated solutions. Furthermore, the

personal best position is updated by comparing the

Pareto dominance relationship with the current parti-

cle. To retain the diversity of solutions, a Gaussian

mutation operation is utilized for updating the per-

sonal best position. This can improve convergence

properties of the algorithm as it avoids that the algo-

rithm is stuck in a local optimum from which a Gaus-

sian mutation can direct it away. Moreover, the muta-

tion can enhance the diversity of the solutions.

The main idea of the Gaussian mutation operation

is to use the Pareto dominance relationship to com-

pare the current particle and Pbest. If Pbest is dom-

inated by the current particle, then Pbest is updated;

otherwise, the Gaussian mutation is adopted to disturb

the current particle to obtain a new particle which is

used to re-compare the Pareto dominance relationship

with Pbest. If the new particle dominates Pbest, then

it is used to replace Pbest; otherwise, Pbest is pre-

served. However, if they are nondominated, one of

them is selected randomly.

The Gaussian mutation operation is deﬁned as fol-

lows. The mutation rate P

is changed by the itera-

tions.

i j





i j

+ delta

× randn



, i f rand <= P

i j

, i f rand > P

(16)

= 1 −

MaxIt

(17)

Where delta

is calculated according to Eq. 14.

rand is a random uniformly distributed value in [0, 1],

randn is a random Gaussian distributed value. t is the

index of the current iteration, MaxIt is the maximum

number of iterations. The operation (

) guarantees

the position of particles to be integer encoded.

4.4 Updating the External Archive

After using the Gaussian mutation operation to up-

date the personal best solution, the obtained new par-

ticles are applied to update the external archive. The

Pareto dominance relationship is used to compare the

new particle and each solution in the external archive.

Then the new nondominated particles are added to the

external archive and the dominated solutions are elim-

inated from the external archive. Due to the diversity

of new particles created by Gaussian mutation, the

nondominated solutions in the external archive can be

updated with better convergence and diversity during

the iterations. The algorithm of updating the external

archive is shown as Algorithm 1.

Algorithm 1: Updating external archive.

Input: L solutions in archive, new N particles created by mutation

Out: The updated external archive

1. for i=1 to N do

2. for j=1 to L do

3. Compare dominance between particle i and solution j;

4. if Particle i is dominated by solution j then

5. Ignore the particle i;

6. elseif Solution j is dominated by particle i then

7. Delete solution j and add particle i into the external archive;

8. else

9. Add the particle i into the external archive;

10. endif

11. endfor

12. endfor

Based on the above improved operations, the pro-

cedure of the IMODPSO algorithm is shown as Algo-

rithm 2.

Algorithm 2: IMODPSO.

Input: T application tasks, R resources, initialize N particles

Out: The Pareto front

1. for t=1 to MaxIt do

2. for j=1 to N do

3. Select leader for each particle;

4. Update velocity (Eq. 12);

5. Update position (Eq. 15);

6. Evaluate the particle;

7. Update Pbest by Gaussian mutation operation;

8. Update the external archive according to Algorithm 1;

9. endfor

10. if archive size > max archive then

11. Maintain the external archive by crowding distance method;

12. endif

13. endfor

4.5 Comparison of Computational

Complexity with other

State-of-the-Art Algorithms

To analyze the computational complexity of the

proposed IMODPSO algorithm, we compare the

computational complexity of IMODPSO with three

CLOSER 2019 - 9th International Conference on Cloud Computing and Services Science

other state-of-the-art algorithms, i.e. NSGA-II (Deb

et al., 2002), MOEA/D (Zhang and Li, 2007), and

MOEA/D-DE (Li and Zhang, 2009). NSGA-II is

known to be well-suited for fast nondominated sort-

ing. MOEA/D and MOEA/D-DE adopt a decomposi-

tion method using different evolutionary mechanisms.

The population size in all experiments is N, the

number of objectives of the multi-objective optimiza-

tion model is M. The archive size is set to be L.

The number of weight vectors in the neighborhood

of MOEA/D and MOEA/D-DE is T . The compu-

tational complexity for updating the external archive

is O(MNL), so the overall complexity of IMODPSO

is O(MNL). The computational complexity for up-

dating the neighbor population is O(MNT ), so the

overall complexity of MOEA/D and MOEA/D-DE

is O(MNT ). The computational complexity for fast

nondominated sorting is O(MN

), so the overall com-

plexity of NSGA-II is O(MN

). Normally, when

optimizing a multi-objective discrete problem, L is

smaller than N and close to T . Therefore, the pro-

posed IMODPSO algorithm has generally a lower

computational complexity for solving discrete opti-

mization problem.

5 SIMULATIONS AND RESULTS

In this section we ﬁrst describe the experiment setup

and then discuss the results obtained from running the

experiments.

5.1 The Experiment Scenario

To test the performance of the different algorithms

for solving the multi-objective optimization model,

an experiment scenario based on the workﬂow appli-

cation in Section 2 is created (Rodriguez and Buyya,

2014). The matrix of receive time, execution time and

transfer time are set to (a), (b) and (c), respectively.

The unit of time is one hour. The rental cost on the

available three resources is as shown in Table 2. The

above parameters are an example, but the experiments

demonstrate what we theoretically showed above.







0.6 0.3 0.5

1 0.4 0.3

1.2 0.8 1.4

0.7 0.5 1.3

0.9 0.6 0.9

0.4 0.2 1

1.2 0.7 3

0.8 0.5 4

1.5 1.1 2.6







(a)







1.4 0.7 3.5

3 2.6 5.7

8.8 5.2 13.6

6.3 3.5 10.7

7.1 3.4 9.1

2.6 1.8 6

10.8 6.3 15

8.2 4.5 16

10.5 6.9 16.4







(b)







0 9 9 9 0 0 0 0 0

0 0 0 0 5 0 0 0 0

0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 7 0 0

0 0 0 0 0 0 0 2 0

0 0 0 0 0 0 0 3 0

0 0 0 0 0 0 0 2 0

0 0 0 0 0 0 0 0 10

0 0 0 0 0 0 0 0 0







(c)

Table 2: Rental cost on available resource.

Resource no. Rent cost (USD/per hour)

R1 6.2

R2 11.0

R3 2.5

5.2 Experiment Results

To prove the efﬁciency and effectiveness of the pro-

posed IMODPSO algorithm, we have used NSGA-

II, MOEA/D, and MOEA/D-DE to perform the same

experiments for comparison. IMODPSO, NSGA-II,

MOEA/D, and MOEA/D-DE have been implemented

in Matlab and are run on Intel Core i5-6300U 2.4GHz

CPU with 4GB RAM. The parameter settings in our

experiments are as follows.

The population size in IMODPSO, NSGA-II,

MOEA/D, and MOEA/D-DE is set to be 100. All

the algorithms run for 200 generations. The external

archive size is 100. NSGA-II and MOEA/D adopted

the simulated binary crossover (SBX) operator and

polynomial mutation for reproducing an offspring.

The crossover probability is set to be 1 and muta-

tion probability is set to be reciprocal of the number

of decision variables. MOEA/D-DE adopted the DE

operator and polynomial mutation for updating the

neighbor population. The mutation and crossover pa-

rameters of DE operator are set to be 0.5 and 1, re-

spectively. The polynomial mutation probability of

MOEA/D-DE is set to be reciprocal of the number of

decision variables. Both the neighborhood sizes of

MOEA/D and MOEA/D-DE are set to be 20.

The Pareto fronts produced by all four algorithms

are shown in Fig. 3. It can be seen that the results

Efﬁcient Task Scheduling in Cloud Computing using an Improved Particle Swarm Optimization Algorithm

obtained by IMODPSO are better than the other three

algorithms. Not only the convergence of IMODPSO

is the best, but also the diversity of the Pareto so-

lutions. MOEA/D-DE and NSGA-II can get better

Pareto solutions than MOEA/D. Both MOEA/D-DE

and NSGA-II have their own advantages. MOEA/D-

DE may converge better, and NSGA-II can have more

diversity in the solutions. The speciﬁc objective func-

tions of the Pareto solutions are listed in Table 3,

Makespan is denoted by M

, and C

Total

is denoted by

. From Table 3, IMODPSO seems to have stronger

search ability and get the most number of Pareto opti-

mal solutions. MOEA/D and MOEA/D-DE can only

search for fewer quantities of Pareto optimal solu-

tions.

30 40 50 60 70 80 90 100 110 120

Makespan

250

300

350

400

450

500

550

Total

IMODPSO

MOEA/D-DE

MOEA/D

NSGA-II

Figure 3: The comparing Pareto fronts.

Table 3: The different Pareto optimal solutions obtained by

four algorithms.

IMODPSO MOEA/D-DE MOEA/D NSGA-II

37 515.2 38 455.5 48 499 38 455.5

38 455.5 40 440 56 493.7 40 440

40 440 61 424.5 58 466.3 55 438.5

55 438.5 65 403.3 73 454.2 61 424.5

58 422.1 67 400.5 79 450.8 65 403.3

62 421.9 68 391 81 405.6 67 400.5

65 403.3 75 383 85 374.9 68 391

67 400.5 82 382.5 90 368.8 75 383

68 391 86 381 99 360.3 86 381

75 383 89 374 89 374

76 377 91 371.5 91 371.5

85 374.9 97 355.3 98 363.5

89 374 107 292.3 102 362.5

90 368.8 111 277.5 103 356.5

91 348.5 107 300.5

103 338.1 111 277.5

107 292.3

111 277.5

In order to verify the efﬁciency and reliability of

the proposed algorithm, we executed each algorithm

30 times independently and used three performance

metrics to evaluate the different algorithms. The Gen-

erational Distance (GD) (Van Veldhuizen, 1999) has

been used to evaluate convergence, the Number of

Pareto Solutions (NPS) is our metric for diversity and

the Computation Time (CT) for computation speed.

All are calculated to compare the performance of

these four algorithms. GD and CT are the smaller the

better, and NPS is the opposite. The unit of CT is one

second.

The box diagrams of different performance met-

rics obtained by four algorithms are shown in Fig. 4,

Fig. 5, and Fig. 6. From Fig. 4, we can see that

GD obtained by IMODPSO is lower than the gen-

erational distance of the other three algorithms. In

addition, IMODPSO can ﬁnd the Pareto optimal solu-

tions which represent the true Pareto front in nearly

half of the cases. The convergence of MOEA/D-

DE algorithm is relatively better than that of NSGA-

II for solving the discrete scheduling problem. The

worst convergence is MOEA/D. The NPS perfor-

mance metric in Fig. 5 demonstrates that the diversity

of IMODPSO is the best among the four algorithms.

Besides, the stability of Pareto solutions obtained by

IMODPSO is very high. Compared with MOEA/D-

DE, NSGA-II has higher diversity in the solutions.

Moreover, MOEA/D has a large ﬂuctuation in the di-

versity of the solution.

As for the computation time in Fig. 6, IMODPSO

has the shortest search time and is hence much bet-

ter than the other three algorithms. This can be ex-

plained by the fact that MOEA/D-DE and MOEA/D

need more time to update the neighbor population,

and NSGA-II needs more time to perform the fast

nondominated sorting approach. Furthermore, the

different computational complexity also veriﬁes the

computational speed of these four algorithms.

IMODPSO MOEA/D-DE MOEA/D NSGA-II

-2

Figure 4: The boxplot of Generational Distance (small bet-

ter).

CLOSER 2019 - 9th International Conference on Cloud Computing and Services Science

IMODPSO MOEA/D-DE MOEA/D NSGA-II

NPS

Figure 5: The boxplot of Number of Pareto Solutions (large

better).

IMODPSO MOEA/D-DE MOEA/D NSGA-II

Figure 6: The boxplot of Computation Time (small better).

5.3 Analysis of Speciﬁc Schedules

In a speciﬁc scenario, the sum of the receive time and

execution time for each task on each of the available

resource is ﬁxed. The transfer time is an important

factor, that needs to be taken into consideration se-

riously. The best schedule will minimize the total

cost and the Makespan. IMODPSO can obtain differ-

ent nondominated solutions which represent different

optimal schedules at a time. We choose three typi-

cal nondominated solutions from IMODPSO and an-

alyze them speciﬁcally. The Makespan of the differ-

ent schedules are one minimal, an intermediate and a

maximal one, respectively. Table 4, Table 5 and Ta-

ble 6 show the three different speciﬁc schedules and

Fig. 7, Fig. 8 and Fig. 9 show the three different Gantt

Charts. The ﬁrst schedule does not assign any task to

resource 3 because the execution time in resource 3 is

relatively big compared with others. The third sched-

ule assigns all tasks to resource 3 because the rental

cost of resource 3 is the cheapest and the transfer time

can be ignored. Due to the different nondominated

solutions representing the different Makespan and to-

tal cost, the decision maker can decide for a suitable

schedule according to the speciﬁc requirements.

Table 4: The ﬁrst schedule (Makespan=37, C

Total

=515.2).

Task 1 2 3 4 5 6 7 8 9

Resource 2 1 2 2 1 2 2 2 2

9+4

2+5

Makespan

Time

Resource

Figure 7: The ﬁrst Gantt Chart (Makespan=37,

Total

=515.2).

Table 5: The second schedule (Makespan=65,

Total

=403.3).

Task 1 2 3 4 5 6 7 8 9

Resource 2 2 2 2 3 1 2 3 3

5+10

1+3

3+2+20

Makespan

Time

Resource

Figure 8: The second Gantt Chart (Makespan=65,

Total

=403.3).

Table 6: The third schedule (Makespan=111,

Total

=277.5).

Task 1 2 3 4 5 6 7 8 9

Resource 3 3 3 3 3 3 3 3 3

Makespan

Time

Resource

Figure 9: The third Gantt Chart (Makespan=111,

Total

=277.5).

Efﬁcient Task Scheduling in Cloud Computing using an Improved Particle Swarm Optimization Algorithm

6 CONCLUSIONS

This paper has presented an improved multi-objective

discrete particle swarm optimization algorithm to

solve the task scheduling and resource allocation

problem for the scientiﬁc workﬂows in cloud com-

puting. It makes the three main contributions: a) the

velocity constriction strategy is applied to improve the

search ability of the algorithm in the discrete space; b)

the Gaussian mutation operation is adopted to boost

the diversity of the nondominated solutions in the ex-

ternal archive; c) the different performance metrics

of IMODPSO are compared with three other state-

of-the-art algorithms to validate the efﬁciency of pro-

posed algorithm. Experiments have shown that the

IMODPSO algorithm can obtain the Pareto optimal

solutions with good convergence and diversity using

less computation time. It is proved to be a stable

and efﬁcient algorithm for solving the multi-objective

discrete task scheduling and resource allocation prob-

lem.

Future research will focus on optimizing large-

scale task scheduling and resource allocation problem

in the more practical engineering environment.

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