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1
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
2
Index
Introduction Cooperative Game Framework Pareto Optimal and Fair Job Allocation
Algorithm Experiments Conclusion
3
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
5
Cooperative Game Framework (cont.)
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
6
Cooperative Game Framework (cont.)
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
7
Cooperative Game Framework (cont.)
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
8
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
9
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
– The average completion time of jobs for provider
2
( )2(1 )
j jj j j j
jj j
h bD F L h
ch
10
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
11
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
Nash bargaining solution is determined by solving the following optimization problem
20
1
max ln( )2(1 )
mj j
j jj jj j
h bD D h
ch
12
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
13
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
14
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
A necessary condition
…(19)
Solving (19), get
…(20)max
10,
1j
j
j j j
h
h
0j
L
1 j m
15
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
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
17
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
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
18
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
• 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
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
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
20
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
Pareto Optimal and Fair Job Allocation Algorithm (cont.)
Periodically calculates an optimum job allocation strategy
Remain in equilibrium until the system’s states change
22
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)
23
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
24
Experiments (cont.)
Noncooperative game algorithm– Players are brokers – Minimize their own average job completion
time
25
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
28
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
29
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