1 A Cooperative Game Framework for QoS Guided Job Allocation Schemes in Grids Riky Subrata, Member,...

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A Cooperative Game Framework for QoS Guided Job Allocation Schemes in Grids

Riky Subrata, Member, IEEE, Albert Y. Zomaya, Fellow, IEEE, and Bjorn Landfeldt, Senior Member, IEEEIEEETRANSACTIONS ON COMPUTERS, VOL. 57, NO. 10, OCTOBER 2008

Present by Ting-Wei, Chen

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Index

Introduction Cooperative Game Framework Pareto Optimal and Fair Job Allocation

Algorithm Experiments Conclusion

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Introduction

Game theoretic solution to the QoS sensitive grid job allocation problem

Model the QoS-based grid job allocation problem as a cooperative game

Present the structure of the Nash Bargaining Solution

4

Cooperative Game Framework (cont.)

Send jobs to more than one broker

The broker decides which provider will process the job

Sends the job to that provider

First-come-first-serve

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Broker → Provider Constraint

Signal

0

j j

j

1 1

n m

i ji j

ker

Pr

_ _ _ _

_ sin _ _ _

_ _ _ _ _

k User

i Bro

j ovider

Average arrival Rate of job

Average proces g rate of job

Rate of job sent to processor

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Nash Bargaining Game– Two players– If the two proposals sum to no more than the

total good– Then both players get their demand– Otherwise, get nothing

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Model the grid load-balancing problem– The m players are the service provider– Each player has a performance function– Each player has a minimum initial performance– – Solve the optimization problem

– Equivalent optimization problem

0ju

( )jf x

0( )x jf x u

00max ( ( ) ),j j

xj J

f x u x X

00max ln( ( ) ),j j

xj J

f x u x X

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Pareto Optimal and Fair Job Allocation Algorithm (cont.)

The average processing time of a job– Waiting time at the queue at a provider

– The expected transfer time of a job from any player to provider

2

2(1 )j j

j j

j j

hF h

h

jj

bL

c

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– The average completion time of jobs for provider

2

( )2(1 )

j jj j j j

jj j

h bD F L h

ch

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Pareto Optimal and Fair Job Allocation Algorithm (cont.) Maximum expected service time

Maximum rate of jobs a provider

Note:

2 max

0

max2(1 )j

j

j

j jjj

h bD h

ch

0

max

0 2

2( )

2 ( )j

j

j jj

j j jj

bD h

c

bh D h h

c

max0j j

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Nash bargaining solution is determined by solving the following optimization problem

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1

max ln( )2(1 )

mj j

j jj jj j

h bD D h

ch

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Pareto Optimal and Fair Job Allocation Algorithm (cont.) in terms of

…(17)

Concave function

0jD

maxj

2 max 2

max1

max ln( )2(1 ) 2(1 )

j

j

mj j j

j j j j

h hD

h h

max

1 1

0

j j

j

n m

i ji j

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Pareto Optimal and Fair Job Allocation Algorithm (cont.) First, maximize the objective function (17) Lagrangian is a function that summarizes

the dynamic of the system

2 max 2

max1 1 1

ln( )2(1 ) 2(1 )

j

j

m m nj j j

j ij j ij j j

h hL

h h

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A necessary condition

…(19)

Solving (19), get

…(20)max

10,

1j

j

j j j

h

h

0j

L

1 j m

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Solve for

…(22)

Using (22) and constraint

→

j

max maxmax 1 ( 4)1

22 2

j jj j j

j

j j

h h

h h

1 1

n m

i ji j

max max

max

1 1 1 1

1 ( 4) 12j j

j

m m m nj j

ij j j ijj

h h

hh

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Setting into (20)0j

max

1j

j

h

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Cooperative Job Allocation– According to the time equation

– Calculate two variables α and d• α’s equation

max

1

j

j jt h

max max

max

1 1 1 1

1 ( 4) 12j j

j

d d d nj j

ij j j ijj

h h

hh

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• is the maximum positive integer

Finally

max max

max

1 1 1 1

1 ( 4)12 j j

j

n d d dj j d d

ii j j jj d j

h h t t

h t h

(1 )d d m

max maxmax 1 ( 4)1

22 2

j jj j j

j

j j

h h

h h

1 j d

19


Simultaneously maximize the QoS level of all the providers

User’s fairness criterion– Concerned in the users and brokers– The average job completion time for brokers

are the same

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Pareto Optimal and Fair Job Allocation Algorithm (cont.) Fairness index

Provides a fair allocation for each brokers

Amount of jobs to be sent from broker to provider

2

1

2

1

( )

i

n

iin

i

TFI

n T

,

1

ji j i n

ii

, 0,i j and ,1

m

i j ij

ij

21


Periodically calculates an optimum job allocation strategy

Remain in equilibrium until the system’s states change

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Experiments (cont.)

QoS goals on the average job completion time– CG generally gives better performance than NG

and PS• CG (Cooperative Game Algorithm)

• PS (Proportional-Scheme Algorithm)

• NG (Noncooperative Game Algorithm)

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Experiments (cont.)

Proportional-scheme– Allocates jobs to providers in proportion to its

computing power– The faster providers are sent more jobs by the

brokers

– Can’t take into account the communication delays incurred in transferring job

,

1

ji j i m

ij

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Experiments (cont.)

Noncooperative game algorithm– Players are brokers – Minimize their own average job completion

time

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Experiments (cont.)

Concerned about the aggregate number of jobs arriving at each broker

Not the individual jobs from each user The actual arrival rate of each broker

1

m

i i jj

_ _ _ _ _

_ _ _ _ _ _ ker_i

The required overall average system loading

The relative job arrival rate of bro i

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Experiments (cont.)

Response Times

Average job completion time for each broker.

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Experiments (cont.)

Effect of System Loads

Average job’s completion time versus system load

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Experiments (cont.)

Effect of Service Time– The Bounded Pareto distribution

1

( )1 ( / )

k xf x

k p

k x p

_ min _ _ _

_ max _ _ _

k The imum job execution time

p The imum job execution time

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Experiments (cont.)

The mean (first moment) of the distribution

The second moment

1 1

1 1( )

1 1 ( / )

kh

k p k p

22 2

1 1( )

2 1 ( / )

kh

k p k p

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Conclusion

Fair to all users Represents a Pareto optimal solution to the

QoS objective

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Thank you for your attention

1 A Cooperative Game Framework for QoS Guided Job Allocation Schemes in Grids Riky Subrata, Member,...

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