presentation on new approach to acquire optimum source and...

International Journal of Mathematics and Computer Sciences (IJMCS) ISSN: 2305-7661 Vol.24 December 2013 www.scholarism.net

596

Presentation on new approach to acquire optimum source and creation of good

equilibrium in the services of grid computing networks used to sames thoory.

1Shina hekmat,

2Leila ajam

*

1 Department of computer, Amol branch, Islamic Azad University, Amol, [email protected] 2Department of computer, Aliabadkatoul branch, Islamic Azad University, Aliabadkatoul, Iran , [email protected]

Abstract

To expansion and dynamism grid function (tasks). The control of this sources to be important. This essay

presented to one of the issues Qos sources , that, services applicants want to applicote sources, in a grid

network network to solve the problem of parallel processing to least computation costs and the value of

service, dependon the number computing calculation. To solve problem allocation sources use to game theory

and practicable , proximatly disallow recommends with 2 stages. First all participant first in dependently , the

ssues solved , with out consideration some allocation sources. One method programming is binary selection

that it solves to independent optimization. The second method is a evolutional mechanism that some

strategies disallows, former optimium different attenders minimized and the shortages change of their effects.

This algorithm is in the all.cational evolution mechanism simplify and suitable . If the game altocation

sources has available disallows, then Nash quuilibrium will show.

Key words : grid comoutings calculation game theory source allocation servicing , quality of services

*Corresponding author : Leila ajam ( [email protected])

mailto:[email protected]

mailto:[email protected]


597

1- Introduction

Grid networks computing, in the giant computing calculation and supply to datas , for reason cooperation

source computing in the global level to be important. This system divided physical impediment , in single end

separation systems. Group administration of source has a single identity the computing calculation power and

it supplies data storage facilities for users. The slopes of these users can spread in the internal computer to

aeone computers allocation in the wide world. These tasks are based on the grid systems. To grow number

and dynamism , source observer are important. In this essay is about one of the is sue source allocation Qos,

that , applicants want to solve computing problems, to use , rid networks, and value of per knot in network

depend on computing colcutions. Use to computing platforms is for Qos enforcement and condensed

computing calculations. In this essay refered to issues about parallel task allocation in the unconnected

machines in Internet. Timing figure is one of the based disturbance in applied programs based on gird system.

The most important points that, this essay have, consist of

1) respite (extension of time) proposal and algorithm for optimization of time and cost to dependent tasks

scheduling, to consideration relation between them.

2) a project to one evolutional mechanism that binary strategies change the best solve different former sides

to minimize of effected them.

3) equilibrium Nash representation in the source allocation game to special arrangements.

Algorithm explanation

1-2-To explain of integral is a cost measurement, we explain a suitable integral. It had better that source

execute the cost weight total and all time.

For example a attender consider to value of the time equall to 100 units money , and so , we have

Wm:Wt=1/wm+W1=1

The attender (contestant) of tasks can determine to respite , and max cost shows to Mo independent

allocation algorithm involved one task , and its consist on some secondary sub task.

Let us suppose that , (b) is a allocation Matrix to (k) line in lieu of secondary task and (m) is a column in lieu of

source to this reasoning, we have this integral:

.wijt̂.iP̂WZmin t

x

1t

m

1jm

Turnaround (1)

Where

Turn around

a

1L

a

1I

1n1.tij.bijmaztl (2)

And

It = q.1l,tij.bijMaxnl.tij.bij.max j.ij,i (3)


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(q) is a time relation and it is apracticable equation 3 , the time value end of relation Ln to end of L relation.

nL shows percent value of compelementary task in the two relations. In per L , value of 1it the time relation Ln

depend on (i) secondary task with Ri source. The value of b (ij) appoint to depend a secondary task i to Rj.

Consept of equation 6 is a n1 compete in the secondary task. That Ln relation accepted. The reason , use of

the first Max , that is all task connection doesn't prepare and this relation doesn’t accept.

1ijj,i t.bijMax Consuming time is to Ln relation. Max, for the reason , the network quality is

different, and in consist on the slowest network, that , it appointed the time connect. In the fifth equation

complementary time and the number of k have secondary task figure.

1

qn1 Equation , for the reason the end of relation

1L

1L1

q n1 left to every task.

2-2- Compulsory Situation

To every pair i,j. lik, for before example, appointed the number of tij. To the partish allocation time.

Determine of 1a:1,0a ijij

If the (i) secondary task allocated to j source. Of course the task allocated to a machine, that

M

1jij 1a ,

and the compulsory situation explain in the equations 4 to 8.

m,..1jandk..liallfor1or0bij (4)

T tarn around ≤ to (5)

.Mobij.tij.Pjm

1j

k

1i

(6)

.k..li,allfor,1bijk

1i

(7)

.m..lj,allfor,1bijk

1i

(8)

Sixth and fifth equations show the time respite and they are compulsory budget. Seventh equation show

that every task should depend on one and only one processor.


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Eighth equation show that every base processor processes one of this task. In fact relations can't calculate

before every task to similar percentage.

3- The problem simplify and solve

To compound of 2,3 equations , we have

q

1Ij,ij.iaroundturn tij.bijmaxtij.bijmaxT

a large part of tasks in the grid networks to timing calculate didn't show. Because more of time is to

relation.

When the data transferring is small , we should connivanced of difference between networks. In the

Maxij{bij.tij1} transfer to T1. then , we have this equation :

q

1i1j.iaroundturn ttij.bijmaxT

above equation substitute to 11. we can say, this transferring doesn't change. We have a limitation space

and we don't say any reason.

q

1l1ijaroundturn m,..1iallfor,ttij.bijMaxT

To this differences that (bij) should be Binary and compulsory term to these models a are linear. In order

to, the model is a direct Model and it converted to classic programming and we can use many Models.

In fact , some tasks have other needs like reliable abilities, to this situation all of sources don't suit in the

sponge. If a source isn't suited to one duty, tij number is a large number and it is a (T0) . in this situation

algorithm can a voide to attribute. In fact, algortim can show tasks to the Qos needs programming and

schedule ti.


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4- Example of the solve problem

Let us suppose that , for every three tasks execat independent allocation algorithm, and this result .

a1=(0,0,0,1,1) , a2(0,0,1,1,1),a3=(1,1,1,0,1).

The matrix allocation is

10111

11100

11000

a

The cost of these tasks are :

0459.0)a(um063/0)a(u,0500.0)a(u 331122

But, they don't acquire a real costs. Because the source have more than one secondary task. To

consideration real performance of matrix. We have:

Tij =

2/704/65.34

4/862/700

63/6000

Expense Matrix eij

8/408/42/44

6/54/54/500

63/6000


601

Real costs of three tasks are

i133

22

11

uuthatclearmakeTo0400/0)a(u

3.04/0)a(u

0469/0)a(u

Strategies of a3 = (1,1,1,0,1) , a1 = (0,0,0,1,1).

a2 = (0,0,1,1,1) are the first tasks strategie they are a changeable game.

Strategies of every task to acquire more benefit between attenders. In the next part optimization of

valuable algorithms to secarite application every task.

5- Changeable optimization

In this game, no body doesn't know that before every task , the first strategies select , and what is the

general allocation ?

To connivance of other issues. We select allocation strategy. Possible result of first optimumn disallows

consist on to terms : Multidexing and non-Multiplexing. Let us suppose that before strategies are customary

date and knowldgements. When some source are more than one, the result of first time, the next time game

begin change of the strategy should show, one of essential state. New strategy of every task should graw

benefit of all tasks, and or other task connected to its profit. Change processing is a imposition of secondary

task , it changes and this change should specify to all attendants.

To this part, we explain to conspetion of equilibrium to change of mechanism. The consept of equilibrium

Nash, is a anolgue that desing final allocation by the change of some allocation strategies , to solve optimum

first different attendants. We have 2 terms :

1-5- The non-Multiplexing

Explain , a game for source allocation show to (n) task and (m) source. every task consist on Multi

secondary similar tasks. If Matrix (nm) to (a) is ,a1aijxi)1aij(aij , then a is

a Multiplexing allocation. Proposition 1: If (a) one of the non-Multiplexing that vector ai=(ai1,a12,…,qim) is a

optimumn disallow and (Si) exclusive is a i[1..n].

So (ai) is a optimization task of (Si) to i[1...n] and (a) is a exclusive equilibrium of this game.


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Providing = for the reason

1i

ijn

ijaij )1a()1a.( we don't have allocation for multi

allocation sources all strategies first allocation solve aren't competition. So the former allocation solve every

task, is a final allocation solve. Any allocation doesn't hold true to )a(u)a(uin]n..1[i.a iiiii .

2-5. Multiplexing

Let us suppose ai=(ai1,…,aip…aiq,…aim) is first optimization allocation that it consists on independent

optimizational. So imipii a...a,...1aa is a changeable allocation that show and there are

iqaaiq and ipaaip . The cost of ]n...1[itoequall)a(u)a(u iiii . This reason

explain in fourth part. Explain1. allocation multiplexing , is a source allocation game to (n) task and (m) is a

source. every task consist on similar secondary tasks multiplexing.

If allocation [Matrix a] is 1a()1a(:],n...1[i ijm

1iijj ) So (a) is a one allocation ,

Multiplexing. According explain 2 , the term depend on one source or some secondary task. That it should

show in a allocation with some allocation. There are 2 point that 1) multiplexing doesn’t affect on the time of

task's enforcement 2) multiplexing reduce a time of task's enforcement.

1-2-5- state 1

Reson 2. Multiplexing allocation of (a) that vector is a ai = (ai2,…aim), exclusive disallow to (si) is i *1…n+ ,

to the one source Multiplexing Ru.

If every av *1…m+

1)aiv =1 2)

n

1.I iv 1a 3)tiv = maxj = *1,…,m+ ti j, i,e,tiu < tiV

(Ai) is a best strategy of task for si that i [1…n+ and (a) is a unique equilibrium to play. Logic the task of

application interval show that

1) the time of task cost is simaxtit {tij} because uV ,tiv=maxj=*1,…,m+ tij

2) the cost of eijj doesn't change. So this application of (si) task hasn't any effect. T.consider reason 1

we see there isn't change. The result similar to non Multiplexing allocation.


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2-2-5- state 2

The allocation of Multiplexing reduce the application of task , that it is one of the longest time to complete. For

a task , completion when minimize that one task inverted to some secondary tasks.

Theory 3. there isn't exact explanation to show characteristics of shortage and their affect on one allocation,

that a secondary task refer to before allocation. Our purpose is to minimize shortage affects per stage

Algorithm 1. SRAT minimization

Input : Matrix a , task IDi, Pi

Out put : q

{ For all j [1.m]^ j≠ p

Compute srat ralloc (i,p,j)

If min (srat ralloc (i,p,j}<0

Minj {srat ralloc (i,p,j}

q = else

q = -1

return q

}

Explanation 3. If a task shift a secondary task, so the value of SRAT is More than before cost application. For

example , let us suppose (ai) is before allocation si task that tip=Maxj*1,…m+ ,tij+. If secondary task is

changeable.Rp convert Rq and .aaaand0a iqipipip

Let us suppose relloc (i,p,q) show this allocation. Then SRAT interval explain to ui (ai)-ui ( ia ).So our

purpose of know q , enforcement to allocation of si task. Min qe (1,…m). when we find out number minimum

SRAT in a play allocation competitional when si is strategy on task, it effected on the other tasks – it possible

that more than task depend on a secondary similar task., and the time to complete of this secondary tasks, it is

similar value of minimum secondary issues. In this situation, to consept allocation, we inverted affected

shortage and we explain them in the fourth theory presentation 4.

If a task transfer of one source to a secondary task it can shift SRRT, the Value is that total applications

reduce a final case of all tasks.

Let us suppose a=(a1,…a1,…am)T is allocation matrix, the ai vector is a allocation task si. That need to transfer

and shift of a secondary task. If T

m11 )a,...a,...,a(a became an allocation Matrix that has before of

an allocation, in fact ai to a'I change and other case are un changeable. So , the value of SRRT, in this allocation

is

aai aia

iiii )a(u)a(u


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Algorithm 2.SRRT minimization

Input : matrix a, Resource j

Output : task IPj

MinGlobal (a,j)

{

Get multiplexing task set mts in resource Rj

For all ask k Mts

{

q = min single (a, k,j) if q≠-1 {

rallock (k,j,a); // suppose reallocate

if uk (ak)-uk(a'k)<0

add k to negative srat task set nsts: }};

If nsts is null i = -1 ; else i = mink {srrtt} for all k nsts Return;

}

When one more than srat task is negative, every task select that has Minimum SRRT , To acquire. When the

task appoint to again allocation the number's of SRRT and SRAT are minimized. and it avoide of simplicity and

accuracy.

according Algorithm 2, optimization that , we explain to get final disallow that it designs to source

allocation game. To consider respite of complet time , we explain execute of total time and connected time

that execute of time changed to task source allocation you suppose, the connected time, is a fixed value that it

doesn't alternate to alternation strategy of allocation source. therfor , we suppose that the Ti respite time

practical and Mi is Max Cost that it Can be gave to si.

For all tasks , To respite time and Max cost in the compulsory term optimization are alternative. Let us

suppose, (a) is a best final optimum of play, so tasks budgets cause to alternate optimization processing.

Perhaps a doesn't become allocation. If si task is in lieu of Max{tij} Ti or

m

1j

eij Mi, the play (game) hasn't

a simple solve.

Algorithm 3. Evolutionary optimization

Input : Matrix a Evoloptimize (a) {

i = 1 flag = True while flag {

if I = = 1flage = False

obtain multiplexing resource vector of task si , denote as ms


605

order ms by tij dessecnt ; For all j ms{

q = Min Global (a,j) ; if q=-1 {

P = Min single (a,q,j) ; execute ralloc (q,j,p);

Flag = True ;

}//end of it }//end of for if (i = n) {

if flag = = false exit ; else

i = i +1; }//end of while }//end of Evoloptimize

Theory 3. if the final solve in the compulsory term use to all of the tasks Nash equilibrium that it is a kind of

source allocation play (game).

Providing – let us suppose that (a1) is a strategy of (si) task and (a-i) is a strategy to other tasks. Suppose that

(si) has a possible strategy and one another is based on (a'i).ui (a'i, a-i ) > ui (ai,a-i) suppose to change of a'i is a

(a'i) that one of the change optimization. There is only one secondary task of (ai) in (a'i) that, they are different.

Whit out generality we suppose that ai =(ai1…aip,…aiq,…aim) and a'i = (ail…a'ip…a'iq,…aim). When aip shifted to aiq.

There are 2 situation before shifting

1)

1i

n

1i

n 0aiq1aip

In this situation , Rp depend on only to si task , because

1i

n 1aip . In this source Rq hasn't any

task , because

1i

n 0aiq . the allocations of Rp and Pq are a kind of two another sides. The si

operation continues ant it's task will graw. It means u-I (a-i) to shift a secondary task of Rp to Rq isn't changed ,

because aip = 1 is a final connected of allocation , first processing and optimization isn't any Rq., that can {q:

q*1,…,m+

n

1iiq )0apq } u-i (a'-1)>u-1(a-1) and u1 (a'1)>u1(a1). It means that ui (a'i, a-1)> ui

(ai,a-i) is against to above hypothesis.

2) 1aor1a iq

n

1iip

n

1i

In this situation , you suppose that a secondary task a shift of Rp to Rq, couse that graw the value of u i(ai).

(ui(a')i>ui(ai). there for the value of ui(a')i. increase , because )a(uif,is1a ii

n

1iiq

to be a growing


606

distant and )a(u ii to be a reducing distant , we use to optimization Algorithm processing that is a

.)a(u)a(u iiii this is against above theories.

6- experience and operation evaluation :

To show, optimization algorithm , the used state of the design , in the part 3 specify and the first result of a

10111

11100

11000

is a bilateral allocation and the sources of R3, R4,R5 are bilateral too.

So, allocation to every task is the time of secondary task that it is bilatcral by secondary task.

In the execution time Matrix we have

2/704/65/34

4/862/700

97000

tij and ti5 =

Maxj *1,…,5+ ,tij- that there is on the tij line. Optimization algorithm that show to find other optimization

allocation (with a set of the strategies). These stages are :

Stage 1. you find the acceptable value of SRAT and SRRT , if there isn't , you exit of program.

The value of (a) and (tij) select to again allocation. a1 is the best s1 strategy.

To x2 , minimum Vlues SRRT and SRAT when i=2 , is there . it means the si task strategy is a2=(0,0,1,1,1) and

this equation changes a'2=(0,1,1,1,0). The value of SRAT si task equal -0/001 and SRRT is – 0/0062. to a3 source

strategy negative number isn't to SRAT.

Stage2. when you have a allocation again, and to all of new operation task, you should calculate them.

Consider to stage 1 the new allocation Matrix is

10111

01110

11000

'a and

operations consist of 0/403 , 0/513 and a 0/412. we see the new allocation is more than the former allocation.

Stage 3. you go to the stage1. and you repeat to when the number gets to SRAT and SRRT.

In this term allocation Matrix is a find result.


607

10111

01110

11000

a

The tasks strategies consist of a'1=(0,0,0,1,1) , a'2 = (0,1,1,1,0) and a3=(1,1,1,0,1) . table 1 show to compare

former allocations and their changes. The results show to all contestant after optimization , con increase their

change operations. When a' is a state of select a'i new strategy that operations increased and other task

decreased. In this game all operations are a kind of studying to proposal methods , that we explain with first

optimization and the subject allocations.

First optimization of programming issue is binary. The contortion completion of former optimization in all

of the task consist of Oo(mm2) and like part 5 , the time contortion of change optimization consist of

o(mnlogm). These optimization in 2 stages serially and consecutively. So the time of contortion is a general

solve and it consist of max [o(mnlogm) , Oo(mm2)]- the essay presented the issues about source allocation by

Nisan and Ronen [10,9]. According to the time, when the change approximately between 2, n). In theory 3

cristdoilo , show the minimum optimization to the number 3 or more than source and it means 21 .

Task Intial strategy Final strategy Initial utility Final utility

S1

S2

S3

(0,0,0,1,1)

(0,0,1,1,1)

(1,1,1,0,1)

(0,0,0,1,1)

(0,1,1,1,0)

(1,1,1,0,1)

0/0469

0/0403

0/0400

0/0518

0/413

0/0403

Total utility 0/1272 0/1334

Table1. compare between former allocation and changed

Many people study about these cases and they use to machine viewpoint (vision) The proposed mechanism

at this essay, to study scientific issues. The real important of this result is time and least contortion of

mechanisms that operational . [12,11,10,9,8,7,5,4,3,2,1].

However approximate number there is that it can solve NP issue , but in this essay doesn't point it.

7-Result

A theoric Method of games select to task timing compating that is equall with Qos needs.

In this computer service , the cost of service based on computing number. Every task of computer have

homogenous sub tasks , they delicated to time performance. At the first , one programming way selected to


608

take former independent game , that it doesn't consider to blintary of sources. So , consider to first result this

mechanism designed to optimization solve. When we present, conception of SRAT and SRRT , 3 algorithm For

evaluation Strategy, all of the attendants use to minimize their affect the equilibrium Nash if the game of

source allocation , had a possible solve, optimization , issu , improve some more of computer tasks.

This method is a good theory to analyse that to specify timing optimization in the dynamic and contorion

issues , and it's a multi secondary task for gria computer networks and reserve of data is best.

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