An efficient Algorithm for Task Allocation in Homogenous computer communication network

7/31/2019 An efficient Algorithm for Task Allocation in Homogenous computer communication network

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An efficient Algorithm for Task Allocation in HomogenousComputer Communication NetworkSunil Sharma1, K. R. Singh2 and P. K. Yadav3

1Ph.d Scholar, Jodhpur National University, E-mail: [email protected]

2Meerut College, Meerut (U.P.)

3CBRI, Roorkee

ABSTRACT

Computer Communication Network (CCN) is of current interest due to the advancement of

microprocessors technology and computers networks. It consists of multiple computing nodes

that communicate with each other by message passing mechanism. The advancement the new

technologies in communication and information lead to the development of the CCN. The task

allocation is an essential step for performance evaluation of CCN. In a CCN, a task is allocated

to a processor in such a way that extensive Inter Task Communication (ITC) is avoided. The

present deals the problem of m tasks and n homogenous processors (where m>>n).

1. INTRODUCTIONThe rapid development in microprocessor technology and the availability of interconnecting

networks have made the computer communicating system feasible and popular. With the help

of computer communicating system, it is possible to utilize a remote computer facility or a

database that does not exist in the local computer. This also facilitates parallel execution of

programs which results in increased computational speed. Computer communicating systems

find wide applications in industrial, scientific and commercial environments. Many factors have

considerable influence on the performance of computer communication systems. Besides

hardware aspects such as speed of processors, memories, interconnection networks, the

software design aspects also have a bearing on the performance of the system. These include

the method of splitting the application software into modules or tasks (task partitioning) and

the method used to allocate these modules or tasks to processors (task allocation). The amount

of parallelism inherent in the application problem and the size of tasks executed on each

processor also influence the working of the system. The major questions to be investigated are

task partitioning and task allocation strategies which influence the distributed software

properties like inter task communication and potential for parallelism. Task partitioning is an

earlier design step than task allocation; its effectiveness depends considerably on allocation. If

these steps are not properly implemented and an increase the number of processors in the

system may resulted in a decrease of the total throughput of the system [1]. Meanwhile, the

traditional notions of best-effort and real-time processing have fractured into a spectrum of

processing classes with different timeliness requirements [2-5]. Many systems are hard and


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2. ASSIGNMENT PROBLEMThe specific problem being addressed to is as follows: Given application program that consists

of mcommunicating Tasks, T = {t1 ,t2,.tm}, and a homogenous distributed system with n

processors, P = {p1 ,p2 ,.pn}, where it is assumed that m>>n. An allocation of tasks to

processors is defined by a function Aalloc from the set T of tasks to the set P of processors such

that:

Aalloc:TP ,where Aalloc(i)=j if task ti is assigned to processor pj, 1im,1jn.

For an assignment, the task set (TSj) of a processor can now be defined as the set of tasks

allocated to the processor [25]

TSj= {i: Aalloc(i) =j, j=1, 2n}

The following cost functions have been used while designing the mathematical model.

2.1. Tasks Size (TS): A task is a sequential program, which performs some predefined

action and possibly communicates with other tasks in a system. The task size ts i (1 i

m) of each task depends on the length of tasks and generally counted in bytes.

2.2. Processors Execution Rate (PER): The execution rate erj (1 j n ) of each

processor is the speed (bytes/ second) of the processor at which they execute the tasks.

2.3. Execution Cost (EC): The EC eij (1 i m , 1 j n) of each task ti depends on the

capability of the processor pj to which it is assigned and the work to be performed by

each task. The exaction time of each processor for a allocation are calculated using the

following equation as:

(1)

where TSj= {i: Aalloc(i) =j, j=1, 2n}

2.4. Inter Tasks Communication Cost (ITCT): The ITCC cik of the interacting tasks ti and

tk is incurred due to the data units exchanged between them during the process of

execution. The inter-task communication times for each processor of a given allocation

are calculated as:


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(2)

2.5 Optimal Processing Cost (OPC) : The OPC is a function of the heaviest aggregate

computation to be performed by each processor. The OPC for a given assignment Aallocis

defined conservatively (assuming that computation cannot be overlapped with

communication) as shown below:

(3)

2.6 Assumptions

Several assumptions have been made to keep the algorithm reasonable in size while designing

the algorithm. The program is assumed to be the collection of m tasks, which are to be

executed on a set of n processors that have same processing capabilities. It is assumed that

the size (in bytes) of each tasks and execution rate (in bytes/ second) for each processor is

known. The size of each task has been defined in column matrix TSM= [ts i] ,where tsi represent

the size of i-th task. The service rate of each processor is taken in the form of row matrix

PSRM=[erj] , where erj denotes the service rate for j-th processor. Each task in the task set may

communicate with zero or more other tasks in the set. Two communicating tasks incur an

inter-processor communication cost when they are assigned to two distinct processors. The

inter-processor communication cost is taken in the form of a symmetric matrix named as inter

task communication time matrix ITCTM= [cij] order m, where cij represents the

communication cost between task ti and tj. Further cij=cji. Whenever a group of tasks is

assigned to the same processor, the ITCT between them is zero. The communication system of

the processors is collision free, thus no messages are lost and all messages are sent in a finite

amount of time .We assume a contention free communication for the processors. A processor

can simultaneously execute a task and communicate with another processor. The overhead

incurred by this is negligible, so for all practical purposes we will consider it as zero.

3. Mathematical Model

The allocation of program tasks is to be carried out so that each task is assigned to a processor

whose capabilities are most appropriate for the tasks, and the inter-processor cost is

minimized. The present model passes through the following steps:


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3.1 Calculation of execution cost: Using task size and execution rate, the execution

time eij of each task ti is calculated and taken in the form of execution time matrix,

ETM=[eij] of order m x n.

Where eij= tsi*erj (1 i m , 1 j n)

3.2 Cluster making and assignment of the clusters to the processors: Initially

we assume that each program tasks form a distinct cluster Ci{ti}. Assigned

tasks to the processor are stored in a linear array Tass={} and the tasks which

are not assigned yet are stored in a linear array Tnon-ass={}. Initially linear

array Tass={} is empty and Tnon-ass={} contains all m tasks. Tasks are

clustered based on their communication requirement. Highly communicating

tasks are clustered together to reduce communication delays. Usually number

of tasks clusters should be equal to the number of processor so that one to onemapping may result. These clusters will be fixed throughout their execution.

Since we have n number of processors in DPS, therefore we will make n

number of tasks clusters. Cluster making and their assignment follows the

following steps:

3.2 (a) Initial assignments of n tasks: In this step we assign n

number of tasks to n number of processors as:

(i) Using ITCTM= [cij] calculateTCi=j=1nci,j for i=1 to m.(ii) Arrange the tasks in ascending order of their TCi and store in a linear

array TCL={}.(iii) Select starting n tasks from TCL= {}.(iv) Assign these n tasks to n processors at which their execution timeeij is

minimum.(v) Modify Tass={} by adding these tasks in Tass={}.(vi) Modify Tnon-ass={} by removing these tasks from Tnon-ass={}.(vii) Modify ETM=[eij] and ITCTM= [cij] by deleting these tasks and get

METM=[eij] and MITCTM= [cij].

3.2 (b) Cluster making of remaining (m-n) tasks and their assignment: To

make the balanced load on each processor we make the restriction on each cluster. A

cluster may contain up to (m-n)n maximum number of tasks and average load on the

system may be up to i=1mei,j/n with 20% variation. Upper diagonal values of MITCTM

(,) are stored in descending order in a three dimension array MAXCT (,) whose first

column represent the first task (say rth task) ,second column represent the second


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task (say s-th task) and third column represent their communication time (crs ) .

Initially each task is treated like a cluster C i{ti } for i=1 to m. Store these clusters in a

linear array CLS= {Ci, 1 i m }. Select the first tasks pair say ( tr , ts) (say tr Cr and

ts Cs ) from MAXCT (, ,). If the sum of number of tasks for clusters Cr and Cs is less

than or equal to (m-n)n and total load for the clusters Cr and Cs , than fuse theclusters Cr with Cs otherwise select the next tasks pair from MAXCT (, ,). Modify CLS= {}

by replacing the cluster Cr as Cr Cr Cs={tr , ts} and deleting the cluster Cs . Modify

the MAXCT (, ,) by deleting this tasks pair (tr , ts) . Modify METM (, ) and MITCTM (,)

as:

a. Modify the METM(,) by adding s-th row into r-th .b. Reduce the communication time between tr and ts to zero.c. Add the communication time cs j to cr j for all j.d. Delete task ts from METM (,) and MITCTM (,).

The above procedure is repeated till we do not get number get number of tasks clusters

equal to number of processors.

3.3. Assignment of the Clusters: After making the n number of tasks clusters we get a

modified METM (,) of order n whose rows are corresponding to clusters and assign these

clusters to that each processor.

4. Results & Discussions:

To justify the application and usefulness of the present method an example of a DPS is

considered consisting of a set of n=3 processors P = {p1, p2, p3} connected by an arbitrary

network and a set of m=9 executable tasks T = {t1, t2, t3, t4, t5, t6, t7, t8, t9} which may be

portion of an executable code or a data file. The size of each task and processors execution rate

have been taken in the form of matrices TSM (,) and PSRM (,) respectively. The Inter tasks

communication time between the tasks has been taken in the form of ITCTM (,) of order m.

Input of the model:

Number of processors in the system (n) = 3

Number of tasks to be executed (m) = 9

p1 p2 p3PSRM(,)

=0.0789 0.078

90.0789


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t1 t2 t3 t

4

t5 t6 t7 t8 t9

t1 0 8 10 4 0 3 4 0 0

t2 8 0 7 0 0 0 0 3 0

t3 10

7 0 1 0 0 0 0 0

t4 4 0 1 0 6 0 0 8 0

ITCC(,)=

t5 0 0 0 6 0 0 0 12 0

t6 3 0 0 0 0 0 0 0 12

t7 4 0 0 0 0 0 0 3 10

t8 0 3 0 8 12 0 3 0 5

t9 0 0 0 0 0 12

10 5 0

t1

3444

4

t2

76899

t3

21000

t4

23010

TSM

(,)= t5

22

211

t6

44422

t7

24532

t8 56


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123

t9

66542

Multiplying TSM (,) matrix with PSTM (,), we get execution time matrix as:

TSM (,) * PSTM (,) =ECM (,)

p1 p2 p3t1 2717.63 2717.63 2717.6

3t2 6067.33 6067.33 6067.3

3

t3 1656.90 1656.90 1656.90

t4 1815.49 1815.49 1815.49

ECM(,)= t5 1752.45 1752.45 1752.45

t6 3504.90 3504.90 3504.90

t7 1035.58 1035.58 1035.58

t8 4428.11 4428.11 4428.11

t9 5250.16 5250.16 5250.16


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Applying the step 3.2 (a), we get the following output:

Tass = {t2,t6,t7}

Tnon_ass = {t1,t3,t4,t5,t8,t9}.

Tasks t2 is assigned to processor p3.Tasks t6 is assigned to processor p1.

Tasks t7 is assigned to processor p2.

p1 p2 p

3

t

1

2717.63

2717.63

2717.63

t

3

1656.90

1656.90

1656.90

t

4

1815.49

1815.49

1815.4

9MECM(,)= t

5

1752.45

1752.45

1752.45

t

8

4428.11

4428.11

4428.

11

t

9

5250.16

5250.16

5250.16


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t1 t3 t4 t5 t8 t9t1 0 10 4 0 0 0

t3 10 0 1 0 0 0

t4 4 1 0 6 8 0

MI

TCC(,)=

t5 0 0 6 0 12 0

t8 0 0 8 12 0 5

t9 0 0 0 0 5 0

Applying the step 3.2 (b), we get the following output:

Maximum number of tasks in a cluster = (m-n)/n = (9-3)/3=2.

Cluster1= { t5 ,t8}.

Cluster2= { t1,t3}.

Cluster3= { t9, t4}

Assign these clusters to each processor as we have assumed the processors are homogeneous.

The results are shown in the following table.

Cluster Processor

Cluster1= { t5 ,t8} p1

Cluster2= { t1,t3} P2

Cluster3= { t9 ,t4} P3

The results of the present algorithm are shown in the following table.

EC ITCC TSCProcessor-1 9685.45 40.00 9725.45Processor-2 5410.11 36.00 5446.11Processor-3 13132.98 64.00 13196.98

Average 9409.51 46.67 9456.18

The above result shows that the response time of the system is 13196.98 units. Figure -1 show

the throughput and services rate of the processors are ideally linked. Form the figure it is

concluded that both are directly proportionate.

Figure-1 Service rare and throughput of the processors

The model discussed in this paper have been coded in Mat lab and tested on HP-dual core

processor workstation considering the random sets of input data and found satisfactory. More


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than 50 problems have been tested and the results are compared with the problem solved by

Sing et al [26]. The following figure 2 shows the comparisons of the results computed by both

the algorithms.

Figure 2 Results Comparison

From the figure it is concluded that the present algorithm give better results in 9% cases, 78%

cases the results are similar with 10% variations and 11% results are worse. The algorithm

reported by [26] gives only 2% better results and 20% worse results. It is concluded that the

algorithm reported in this paper gives better results in most of the cases.

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An efficient Algorithm for Task Allocation in Homogenous computer communication network