Performance Analysis of Workflow Scheduling Algorithm in Cloud Computing Environment using Priority Attribute

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Performance Analysis of Workflow Scheduling Algorithm in Cloud Computing Environment using Priority Attribute

Nidhi Rajak*

, Diwakar Shukla

1,2Department of Computer Science and Applications Dr.Harisingh Gour Central University, Sagar(M.P.), India [email protected]¹, [email protected]²

Abstract

Workflow scheduling is represented by a well known graph that is Directed Acyclic Graph (DAG) and the scheduling problem is also known as NP-complete. There are number of heuristics algorithms has been developed for workflow scheduling and its primary objective to minimize the overall execution time that is scheduling length of the tasks of a given DAG. This paper presents new algorithm for workflow scheduling in cloud computing environment using priority attribute. The priority attribute is computed by using adjacency matrix which consists of the communication time between the tasks. If there is a direct path between the tasks then assign its communication time as matrix element otherwise zero in the matrix.

Then after ,find maximum column element allocated into max array. This algorithm is worked in two phases such first phase compute max array and sort the max array as per their attribute value. Second phase, to remove entry task(s) from priority queue and allocate entry task as duplicate into all available virtual machine. To allocate non entry if satisfied precedence constraint otherwise insert at end of the queue. The propose algorithm is compared with well known four heuristic algorithms such as Heterogeneous Earliest Finish Time(HEFT), Critical-Path-on-a-Processor(CPOP), As Late As Possible(ALAP), and Performance Effective Task Scheduling (PETS) algorithms. The performance analysis of the algorithms has been done based on well known metrics such as speedup, efficiency, scheduling length ratio, cost and resource utilization. The proposed algorithm gives good results in respect of performance metrics as compared to four heuristic algorithms.

Keywords: DAG, Scheduling Length, Cloud Computing, Speedup, Efficiency , Critical Path.

1. Introduction

Cloud computing is one of recent topic of research in computer science and it optimized to use of technology. Sometime, cloud computing is also called as Internet based computing due to it reduces the cost of resources and fast speed of Internet. These are the following areas where cloud computing have widely used such as advance scientific computing, medical science, DNA computing and computational physics etc. This computing environment follows the principle “Pay-Per-Use basis”[1].

The heterogeneous computing environment[2] which includes Cluster, Grid, and Cloud etc widely used in massive where complex application computing. Complex application computing is process of assigning multiple tasks onto available resources. In another term, it is called as Scheduling which comprise of multiple tasks and these tasks are allocated to various machines. Scheduling is also known as workflow scheduling and it is defined as the process of assigning the tasks to computing resources [3]. The primary objective of task scheduling method is to find optimal scheduling length and it is generally considered as NP-complete problem[4]. There are basically three major components [5] of any task scheduling which are as follows: performance of the machines, tasks mapping and order of execution of the tasks. Deterministic and non-deterministic scheduling are the two basic classification of task scheduling. Deterministic task scheduling is a scheduling method which has information about the number of tasks, number of virtual machines and precedence constraint among the tasks are known in advanced. It is also called as compile time scheduling whereas Non-deterministic task scheduling is scheduling method does not have all information in advanced. It is also called as run time scheduling.

Vol. 28, No. 16, (2019), pp. 1810 – 1831

.

Generally, an application program is represented by a Directed Acyclic Graph(DAG) where node denote tasks and the arcs denote the dependencies between the tasks of a given DAG. Figure 1 shows the mapping of the tasks among the virtual machines. Here, firstly application program is decomposed into the number of subtasks, these subtasks are represented by DAG and finally allocate it onto the virtual machines

Application Program

Subtask

Subtask Subtask

Subtask Subtask

Subtask

Virtual Machine (VM)

Virtual Machine(VM) Virtual Machine(VM) Subtask

Subtask Subtask

Subtask Subtask

Subtask

Task Decomposition

Task Scheduling

Figure 1. Mapping of tasks onto Virtual Machines

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Any task scheduling follow three basic steps for scheduling of the tasks onto the available virtual machines such as find the priority of tasks of a given DAG using priority attribute, sort the tasks as per priority value and allocate the tasks among the available virtual machines as per EST and EFT. This paper present new task scheduling algorithm in cloud computing environment which is based on priority attribute. The priority attribute compute using adjacency matrix of the graph.

The proposed algorithm is worked on two phases: First Phase : find the adjacency matrix [6]of the given DAG using communication time between the tasks, allocate the task priority value as per column maximum . Second Phase: Sort the tasks in non deceasing order and Allocate the tasks into a Priority Queue(PQ). Use PQ and precedence constraint for allocation of the tasks onto available virtual machines. Here , entry task(s) will be allocated as duplicate to all cloud servers which minimize scheduling length. The performance of the new algorithm gives better results as compared to heuristics algorithms such as HEFT, CPOP, ALAP and PETS. The comparison study is based on performance metrics such Scheduling length, Speedup, Efficiency, Scheduling Length Ratio (SLR), Resource Utilizations and Cost.

The paper is organized as follows : Section 1 briefly explain the cloud computing, basis of scheduling and its objective. Problem Statement, Basic terminologies and priority attribute will be discussed in section 2, Section 3 proposed algorithm and explain it with various DAG models.

Performance analysis will be done in section 4 and finally come to conclusion part in section 5.

2. Workflow Scheduling Model

This model consists of Application Model, Platform Model, Basic Scheduling Terminologies, Performance Metrics and Objective function.

2.1.Application Model

Suppose an application program is denoted by Directed Acyclic Graph(DAG).

G=(T,E,T_C) where T={ T_1,

T

,…T

} finite set of n tasks of DAG, E is set of edges between the tasks, T

and j are two different tasks. Generally a DAG model having an entry task or more than

T5

T6

T3

T4

T2

T1

4

3

4 3

5 2 5

5

T_entry

TC

Figure 2. A DAG Model with six tasks [7]

T_exit

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one entry tasks and an exit task. An example of application model with six tasks is depicted in figure 2.

2.2.Platform Model

A Platform model is represented by Cloud Server CS={CS1,CS2,…,CSm} with m number of cloud servers where CS1= CS2 = CS3 = …..CSm={VM1, VM2, VM3,…, VMp} consists p number of VM and each cloud server may consists one or more virtual machines which are interconnected through very high speed network. The communication time(CT) between two virtual machines is considered negligible if they belongs to same cloud server otherwise it will be included. Here, also considered that no preemption and no any interrupt during execution of the tasks on virtual machine. Figure 3 shown mapping of tasks and virtual machines.

2.3. Basic Scheduling Terminologies

Following are the basic terminologies which will be used in workflow scheduling in proposed algorithm and to performance metrics and table 1 consists the notations used in this paper.

I. Estimated Computation Time(ECT)[8]

𝐸𝐶𝑇_𝑖𝑗 =

𝐸𝐶𝑇₁₁ 𝐸𝐶𝑇₁₂⋯ 𝐸𝐶𝑇_1𝑛

𝐸𝐶𝑇₂₁ 𝐸𝐶𝑇22 … . . 𝐸𝐶𝑇_2𝑛

𝐸𝐶𝑇_{𝑚 1} 𝐸𝐶𝑇_{𝑚 2} 𝐸𝐶𝑇_𝑚𝑛

(1) Where ECTij is estimated computation time of task Ti on resource VMj. II. Average ECT (AVG)[9] of a task Ti is defined as

𝐴𝑉𝐺_𝑖 = ^{𝑇𝑜𝑡𝑎𝑙 𝑉𝑀}_{𝑗 =1} 𝐸𝐶𝑇_𝑖,𝑗

𝑇𝑜𝑡𝑎𝑙 𝑉𝑀 (2) III. Critical Path(CP)[10,11] of a DAG is defined as follows:

𝐶𝑃 = max

𝑝𝑎𝑡 𝑕𝜖𝐷𝐴𝐺 𝑙𝑒𝑛𝑔𝑡𝑕 𝑝𝑎𝑡𝑕 3

CS1 ={VM1, VM2, VM3,…, VMp}

CS2 ={VM1, VM2, VM3,…, VMp}

CSm ={VM1, VM2, VM3,…, VMp} Scheduler

T₁,T₂,T₃,…Tn

Task Queue

Figure 3. Mapping of Tasks onto VMs

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𝑙𝑒𝑛𝑔𝑡𝑕 𝑝𝑎𝑡𝑕 = 𝐴𝑉𝐺 𝑇_𝑖 + 𝐶𝑜𝑚𝑚. 𝑇𝑖𝑚𝑒(𝑇_𝑖, 𝑇_𝑗)

𝑒𝜖𝐸 𝑇𝑖𝜖𝑇

4 IV. Earliest Start Time EST[12] is defined as follows:

𝐸𝑆𝑇 𝑇_𝑖, 𝑉𝑀_𝑗

= 0 𝑖𝑓𝑇_𝑖 ∈ 𝑇𝑒𝑛𝑡𝑟𝑦

𝑇_𝑗∈𝑝𝑟𝑒𝑑 𝑇max _𝑖 {𝐸𝐹𝑇( 𝑇_𝑗, 𝑉𝑀_𝑗) + 𝑀𝐸𝑇 𝑇_𝑖 + 𝐶𝑜𝑚𝑚. 𝑇𝑖𝑚𝑒(𝑇_𝑖, 𝑇_𝑗)} 𝑜𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒 (5) V. Minimum Execution Time MET[13] is defined as follows:

𝑀𝐸𝑇 𝑇𝑖 = 𝑚𝑖𝑛. 𝐸𝐶𝑇 𝑇_𝑖, 𝑉𝑀𝑚 (6) VI. Earliest Finished Time EFT[13] is defined as follows:

𝐸𝐹𝑇 𝑇_𝑖, 𝑉𝑀_𝑗 = 𝐸𝐶𝑇𝑖𝑗 + 𝐸𝑆𝑇 𝑇_𝑖, 𝑉𝑀_𝑗 (7)

VII. Static Level SL[11,12]: It is similar to B-level attribute but exclude the communication time between the tasks. It can be defined as follows:

𝑆𝐿 𝑇𝑖 = 𝐴𝑉𝐺_𝑖+ max

𝑇𝑗𝜖𝑆𝑢𝑐𝑐 (𝑇𝑖) 𝑆𝐿 𝑇_𝑗 8 Table 1. Notations Used

Notation Meaning DAG Directed Acyclic Graph

T Set of finite tasks n Total n number of tasks p Total p number of cloud servers m Total m number of virtual machines

r Total r number of rate for m VM TC(Ti,Tj) Communication time between Ti and Tj.

Pred(t_i) Predecessor of task t_i VMm Virtual Machine

CS Cloud Server

ECT Estimated Computation Time EST Earliest Start Time

EFT Earliest Finished Time MET Minimum Execution Time

M_S Scheduling Length AM Adjacency Matrix

PQ Priority Queue

2.4. Performance Metrics

The comparison between proposed algorithm and heuristic algorithm are based on performance metrics which are as follows: Scheduling Length, Scheduling Length Ratio, Speedup and Efficiency, Resource Utilization and Cost. The details of the comparison metrics as follows:

I. Scheduling Length [13]: It is an important comparison metrics because it gives the overall execution time of the tasks of given DAG. Formally, Scheduling length is defined as the exit task of given DAG on available virtual machine will take minimum time. i.e.

𝑆𝑐𝑕𝑒𝑑𝑢𝑙𝑖𝑛𝑔𝐿𝑒𝑛𝑔𝑡𝑕 = 𝑀𝑖𝑛. 𝐸𝐹𝑇 𝑇𝑒𝑥𝑖𝑡, 𝑉𝑀 (9)

II. Scheduling Length Ratio (SLR)[14,15] : It is the ratio of Scheduling Length and the sum of minimum ECT of Critical Path (CPmin) task in given DAG i.e.

𝑆𝐿𝑅 =𝑆𝑐𝑕𝑒𝑑𝑢𝑙𝑖𝑛𝑔 𝐿𝑒𝑛𝑔𝑡 𝑕 𝑜𝑓 𝐴𝑙𝑔𝑜𝑟𝑖𝑡 𝑕𝑚 𝑀𝑖𝑛 (𝐸𝐶𝑇_𝑖,𝑗)

𝑇𝑗 ∈𝐶𝑃𝑚𝑖𝑛 (10 )

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III. Speedup[15]: It is defined as the ratio of sum of ECT of all the tasks of given DAG on a VM which will take minimum and Scheduling Length. i.e.

𝑆𝑝𝑒𝑒𝑑𝑢𝑝 = 𝑀𝑖𝑛. ^𝑚_{𝑗 =1}𝐸𝐶𝑇𝑖, 𝑗

𝑆𝑐𝑕𝑒𝑑𝑢𝑙𝑖𝑛𝑔𝐿𝑒𝑛𝑔𝑡𝑕 (11) where m is number of VMs.

IV. Efficiency[15]: It is defined as the ratio of Speedup and total number of VMs. i.e.

𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝑆𝑝𝑒𝑒𝑑𝑢𝑝

𝑚 × 100 (12)

V. Resource Utilization[16]: It is important metric for cloud computing which has primary objective is to maximize the resource utilization. It is calculated by Average Resource Utilization(ARU).

𝐴𝑅𝑈 = ^𝑚_𝑖=1𝑇(𝑉𝑀_𝑖)

𝑆𝑐𝑕𝑒𝑑𝑢𝑙𝑖𝑛𝑔 𝐿𝑒𝑛𝑔𝑡𝑕 × 𝑚 (13)

Where 𝑇(𝑉𝑀_𝑖) is Time taken by Virtual Machine i to finish all tasks of given DAG.

VI. Cost[17]: This metric is defined by following

𝐶𝑜𝑠𝑡 = 𝐸_𝑖𝑗 × 𝐶(𝑉𝑀_𝑗) (14)

Where Eij is the execution time of the task i on VMj and C(VMj) is the cost of VMj per unit time.

This cost metric is also depended on the user’s requirement and shown in the following table 2[12].

Table 2. Cost Rate for Virtual Machines

Virtual Machine(VM) VM1 VM2 VM3 … VMm

Cost / Unit Time Rate1 Rate2 Rate3 … Rater

2.5. Objective function

The objective of the workflow scheduling algorithm is to minimize overall execution time i.e.

scheduling length or makespan Ms. Formally, An objective function of scheduling algorithm can be defined as an exit task(Te) of a given DAG should be allocate to the available VM and it takes minimum beginning time and maximum finishing of execution time of Te. i.e.

𝑆𝑐𝑕𝑒𝑑𝑢𝑙𝑖𝑛𝑔 𝐿𝑒𝑛𝑔𝑡𝑕 = 𝑀𝑖𝑛[𝑀𝑎𝑥 𝐸𝐹𝑇 𝑇_𝑒, 𝑉𝑀 ] (15

) 3. Proposed Method

The proposed algorithm is based on priority attribute which is found using adjacency matrix [1].

An adjacency matrix for DAG can be defined as the matrix of order T×T (n number of tasks in a given DAG) and the elements of the matrix would be the communication time(TC) between the tasks. The details of proposed algorithm is given in Table 3.

The elements of the adjacency matrix M can be computed as follows:

𝑴_𝒊,𝒋= 𝑻𝒄 𝑻_𝒊, 𝑻_𝒋 𝒊𝒇 𝒅𝒊𝒓𝒆𝒄𝒕 𝒄𝒐𝒎𝒎𝒖𝒏𝒊𝒄𝒂𝒕𝒊𝒐𝒏 𝒍𝒊𝒏𝒌 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝒕𝒘𝒐 𝒕𝒂𝒔𝒌𝒔 𝑻_𝒊 𝒂𝒏𝒅 𝑻_𝒋

𝟎 𝑶𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆 (16) Where 𝑀_𝑖,𝑗: the adjacency matrix elements at i^th row and j^th column of T× T matrix.

Vol. 28, No. 16, (2019), pp. 1810 – 1831 𝑇𝑐 𝑇𝑖, 𝑇𝑗 : Communication Time between two tasks Ti and T_j of a given DAG.

Table 3: Proposed Algorithm 1 Read DAG with finite n number of tasks

2 Compute Adjacency Matrix AM elements using For(i=1 to n^th tasks)

For(j=1 to n^th tasks)

𝑀_𝑖,𝑗= 𝑇𝑐 𝑇_𝑖, 𝑇_𝑗 𝑖𝑓 𝑡𝑕𝑒𝑟𝑒 𝑖𝑠 𝑑𝑖𝑟𝑒𝑐𝑡 𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑜𝑛 𝑙𝑖𝑛𝑘 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑤𝑜 𝑡𝑎𝑠𝑘𝑠 𝑇_𝑖 𝑎𝑛𝑑 𝑇_𝑗 0 𝑂𝑡𝑕𝑒𝑟𝑤𝑖𝑠𝑒 AM[i][j]= 𝑀_𝑖,𝑗

3 Find MaxArray Mx[n] of size n which contains Maximum value of each task of DAG from AM using Column wise.

Max=0

For(j=1 to n^th tasks) {

For(i=1 to n^th tasks) { If(Max<A[i][j]) Max=A[i][j]

} Mx[j]=Max }

4 Sorting of Mx elements in increasing order.

5 Mx Arrary Elements allocated to Priority Queue(PQ)

6 While(PQ!=Empty)

{

Remove a task T from PQ If T is an entry task then

Allocate to all available VM as duplicate task.

If T is a non entry task and satisfied Precedence Constraint(PC) then Allocate T onto available VM as per EST and EFT.

If T is a non entry task and not satisfied Precedence Constraint(PC) then Insert T into rear part of PQ

}

7 Find Scheduling Length i.e.

𝑺𝒄𝒉𝒆𝒅𝒖𝒍𝒊𝒏𝒈 𝑳𝒆𝒏𝒈𝒕𝒉 = 𝑴𝒊𝒏[𝑴𝒂𝒙 𝑬𝑭𝑻 𝑻_𝒆, 𝑽𝑴 ]

8 Stop

4. Performance Analysis

This section evaluate the results of proposed algorithm with four heuristics algorithms such as HEFT[18], CPOP[18], ALAP[5] and PETS[19] algorithms. Consider three DAG models with 10, 15 and 26 tasks with their ECT and VM cost Tables are taken into account for the performance analysis of proposed algorithm and heuristics algorithms. The performance analysis is also being done with the help of some well known scheduling metrics such as Scheduling Length, Scheduling Length Ratio, Speedup and Efficiency, Resource Utilization and Cost

.

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Case1: DAG1 Model with Ten Tasks

Table 4. ECT[12] Matrix for DAG₁ Model

T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀

CS₁ VM₁ 14 13 11 13 12 13 7 5 18 21

CS₁ VM₂ 16 19 13 8 13 16 15 11 12 7

CS₂ VM₃ 9 18 19 17 10 9 11 14 20 16

This DAG1 model[12] having ten tasks, single entry and also take into account two cloud servers CS1 and CS2. CS1 consists two virtual machines VM1 and VM2 and CS2 consists one virtual machine VM3. Figure 4 shown DAG with ten tasks and table 4 shown ECT of given DAG. The cost of VM rate for DAG1

model is taken arbitrary in the table 5. As per the proposed algorithm need to find adjacency matrix AM , Max Array and allocation onto the Priority Queue PQ for given DAG1 which is depicted in the figure 5

. The proposed algorithm gives 59 units of scheduling length and comparison with heuristic algorithms is shown in table 6.

Results of proposed and heuristic algorithms with different metrics shown in figure 7-12.

PQ tasks are removed one by one from front end , if it is satisfied PC , then allocate to VM as per computing of EST and EFT of the removed task. Otherwise , removed task should be inserted

T1

T4

T3 T6

T2

T9

T8

T7

T10

T5 18

12 9 11

14

17 13

11

16 15

23

19

27

13

23

Figure 4. DAG₁Model with ten tasks

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into PQ from rear end of the queue. Therefore, the complete mapping of the ten tasks in the available virtual machines as per proposed algorithm is shown in the Figure 6.

CS₁

VM₁ 0~14

T₁

14~25 T₃

25~38 T₂

43~48 T₈

VM₂ 14~23

T4

25~40 T7

40~52 T9

52~59 T₁₀

CS2 VM3 0~9

T₁

9~19 T₅

19~28 T₆

Figure 6.Gantt. Chart for Proposed Algorithm for DAG1 Model and 59 units Scheduling Length.

Table 6: Comparison of Scheduling Algorithms Based on Metrics for DAG₁ Model Scheduling

Algorithms

Performance Metrics for DAG1 Model

Scheduling Length

Speedup Efficiency(%) SLR Resource Utilization(%)

Cost/Unit Time

Proposed 59 2.15 71.67 1.44 76.27 212.13

HEFT 73 1.74 58.00 1.78 77.12 258.65

CPOP 86 1.48 49.22 2.09 77.51 301.12

ALAP 73 1.74 58.00 1.78 77.16 258.65

PETS 70 1.81 60.33 1.71 80.47 265.39

Tasks T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀

T₁ 0 18 12 9 11 14 0 0 0 0

T₂ 0 0 0 0 0 0 0 19 16 0

T₃ 0 0 0 0 0 0 23 0 0 0

T₄ 0 0 0 0 0 0 0 27 23 0

T5 0 0 0 0 0 0 0 0 13 0

T₆ 0 0 0 0 0 0 0 15 0 0

T₇ 0 0 0 0 0 0 0 0 0 17

T₈ 0 0 0 0 0 0 0 0 0 11

T9 0 0 0 0 0 0 0 0 0 13

T₁₀ 0 0 0 0 0 0 0 0 0 0

T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀

0 18 12 9 11 14 23 27 23 17

T1 T4 T5 T3 T6 T10 T2 T7 T9 T8

0 9 11 12 14 17 18 23 23 27

Table 5: VM rate for DAG₁ Model

Virtual Machine(VM) VM₁ VM₂ VM₃

Cost / Unit Time 1.71 1.63 1.21

Max Array Mx[10]

Priority Queue (PQ)

Front Rear

Figure 5. Find AM, Mx and PQ for DAG₁ Model

AM

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Figure 7. Scheduling Length for DAG₁ Model

Figure 8. Speedup for DAG₁ Model

Figure 9. Efficiency for DAG₁ Model 0

20 40 60 80 100

Proposed HEFT CPOP ALAP PETS

Scheduling Length

Scheduling Algorithms

0 0.5 1 1.5 2 2.5

Proposed HEFT CPOP ALAP PETS

Speedup

Scheduling Algorithms

0 20 40 60 80

Proposed HEFT CPOP ALAP PETS

Efficiency(%)

Scheduling Algorithms

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Figure 10. SLR for DAG1 Model

Figure 11. Resource Utilization for DAG1 Model

Figure 12. Cost/Unit time for VM for DAG₁ Model

Case 2: DAG₂ with fifteen Tasks

This DAG2 Model[13] having fifteen tasks, multiples entry as shown in the figure 13 and two cloud servers CS1 and CS2. CS1 consists two virtual machines VM1 and VM2 and CS2 consists two virtual machines VM3 and VM4. Table 8 shown ECT and Cost of VM is shown in the table 7 of given DAG2.

0 0.5 1 1.5 2 2.5

Proposed HEFT CPOP ALAP PETS

SLR

Scheduling Algorithms

72 74 76 78 80 82

Proposed HEFT CPOP ALAP PETS

Resoucre Utilization(%)

Scheduling Algorithms

0 50 100 150 200 250 300 350

Proposed HEFT CPOP ALAP PETS

Cost/Unit Time

Scheduling Algorithms

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Similarly, As per the proposed algorithm need to find adjacency matrix AM , Max Array and allocation onto the Priority Queue PQ for given DAG2which is depicted in the figure 14.

Table 7. VM rate for DAG2 Model

PQ tasks are removed one by one from front end , if it is satisfied PC , then allocate to processor as per computing of EST and EFT of the removed task. Otherwise , removed task should be inserted into PQ from rear end of the queue. Therefore, the complete mapping of the ten tasks in the available virtual machines as per proposed algorithm is shown in the figure 14.

Table 8. ECT[13] Matrix for DAG2 Model

t1 t2 t3 t4 t5 t6 t7 t8 t9 t10 t11 t12 t13 t14 t15

CS1 VM1 17 14 19 13 19 13 15 19 13 19 13 15 18 20 11

CS1 VM2 14 17 17 20 20 18 15 20 17 15 22 21 17 18 18

CS2 VM3 13 14 16 13 21 13 13 13 13 16 14 22 16 13 21

CS2 VM4 22 16 12 14 15 18 14 18 19 13 12 14 14 16 17

Virtual Machine(VM) VM₁ VM₂ VM₃ VM₄

Cost / Unit Time 1.72 1.52 1.69 1.65

T₁ T2 T₃

T₄ T₅ T6 T7 T₈

T₉

T₁₀

T₁₁

T₁₅ T₁₄

T₁₂ T₁₃

22

23 7 15

8

12

\\

13 16

6 11

\\

14

\\

17

\\

18

\\

17 16 \\

\\

19

\\

7

16

\\

21

\\

11 9 13

\\

13 15

19

\\

Figure 13 .DAG2Model with fifteen tasks

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Figure 14. Find AM, Mx and PQ for DAG₂

Figure 15.Gantt. Chart for Proposed Algorithm for DAG₂ Model and 144 units Scheduling Length.

Table 9: Comparison of Scheduling Algorithms Based on Metrics for DAG₂ Model Scheduling

Algorithms

Performance Metrics for DAG2 Model

Scheduling Length

Speedup Efficiency(%) SLR Resource Utilization(%)

Cost/Unit Time

Proposed 144 1.60 40.00 1.62 60.06 575.03

HEFT 152 1.52 38.00 1.71 69.73 702.75

CPOP 164 1.41 35.25 1.84 64.93 694.92

ALAP 155 1.49 37.25 1.74 72.74 751.53

PETS 152 1.52 38.00 1.71 65.13 666.43

Tasks T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀ T₁₁ T₁₂ T₁₃ T₁₄ T₁₅

T₁ 0 0 0 13 16 0 0 0 0 0 22 0 0 0 0

T₂ 0 0 0 15 0 11 14 0 0 0 0 23 0 0 0

T₃ 0 0 0 0 0 17 18 19 0 0 0 0 7 0 0

T₄ 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0

T₅ 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0

T₆ 0 0 0 0 0 0 0 0 16 0 0 0 0 0 0

T7 0 0 0 0 0 0 0 0 17 0 0 0 0 0 0

T₈ 0 0 0 0 0 0 0 0 19 0 0 0 0 0 0

T₉ 0 0 0 0 0 0 0 0 0 7 0 0 0 0 0

T₁₀ 0 0 0 0 0 0 0 0 0 0 13 16 9 0 0

T₁₁ 0 0 0 0 0 0 0 0 0 0 0 0 0 13 0

T₁₂ 0 0 0 0 0 0 0 0 0 0 0 0 0 21 0

T₁₃ 0 0 0 0 0 0 0 0 0 0 0 0 0 15 0

T₁₄ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11

T₁₅ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

T₁ T₂ T₃ T₄ T₅ T₆ T₇ T₈ T₉ T₁₀ T₁₁ T₁₂ T₁₃ T₁₄ T₁₅

0 0 0 15 16 17 18 19 19 7 22 23 9 21 11

T₁ T₂ T₃ T₁₀ T₁₃ T₁₅ T₄ T₅ T₆ T₇ T₈ T₉ T₁₄ T₁₁ T₁₂

0 0 0 7 9 11 15 16 17 18 19 19 21 22 23

CS1 VM1 0~14 T2

14~27 T4

27~31

*

31~44 T6

VM2 0~14

T1

14~31 T3

CS2 VM3 0~13 T1

13~34 T5

34~47 T8

47~60

*

60~73 T9

73~86

*

86~100 T11

100~114

*

114~127 T14

VM4 0~16

T2

16~28 T3

28~42 T7

42~73

*

73~86 T10

86~100 T13

100~114 T12

114~127

*

127~144 T15

Max Array Mx[15]

Priority Queue (PQ)

Front Rear

AM

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Figure 16. Scheduling Length for DAG₂ Model

Figure 17. Speedup for DAG₂ Model

Figure 18. Efficiency for DAG₂ Model 130

135 140 145 150 155 160 165 170

Proposed HEFT CPOP ALAP PETS

Scheduling Length

Scheduling Algorithms

1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65

Proposed HEFT CPOP ALAP PETS

Speedup

Scheduling Algorithms

32 34 36 38 40 42

Proposed HEFT CPOP ALAP PETS

Efficiency(%)

Scheduling Algorithms

Vol. 28, No. 16, (2019), pp. 1810 – 1831

Figure 19. SLR for DAG₂ Model

Figure 20. Resource Utilization for DAG₂ Model

Figure 21. Cost/Unit Time for VM for DAG₂ Model

Case 3: DAG₃ Model with twenty six tasks

This model [14] having twenty six tasks, single entry task and communication time is added between 5 to 25 . Also ,considered two cloud servers CS1 and CS2. CS1consists two resources VM1 and VM2 and CS2

consists two resourcesVM3 and VM4. Figure 22 shown DAG with twenty six tasks, table 10 and table 11 shown ECT and Cost matrix of given DAG3 which are arbitrary value.

1.5 1.551.6 1.651.7 1.751.8 1.851.9

Proposed HEFT CPOP ALAP PETS

SLR

Scheduling Algorithms

0 20 40 60 80

Proposed HEFT CPOP ALAP PETS

Resoucre Utilization(%)

Scheduling Algorithms

0 200 400 600 800

Proposed HEFT CPOP ALAP PETS

Cost/Unit Time

Scheduling Algorithms

Vol. 28, No. 16, (2019), pp. 1810 – 1831

T0

T9

T5

T3

T2 T4

T1

T10

T7

T8

T6 T11

T18

T17

T14

T15

T13

T16

T12

T23

T20

T19

T22 T24

T21

T25 20

0

16 0 23

0 ²⁰0

9

13 9

16 15

8 12 16

15

11 10

15

12 13

13 17 19

10 10

16 11 14 18

17 19

17 12

15 19

18 14

15 12

10 11

Figure 22.DAG₃Model with twenty six tasks