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, IEEE
IEEETRANSACTIONS ON COMPUTERS, VOL. 57, NO. 10,
OCTOBER 2008
Present by Ting-Wei, Chen
1
Index
 Introduction
 Cooperative Game Framework
 Pareto Optimal and Fair Job Allocation
Algorithm
 Experiments
 Conclusion
2
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
3
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
4
Cooperative Game Framework (cont.)
 Broker → Provider
Constraint
n
m
   
i 1
i
j 1
j   j
j
 Signal
k  User
i  Bro ker
j  Pr ovider
  Average _ arrival _ Rate _ of _ job
  Average _ proces sin g _ rate _ of _ job
  Rate _ of _ job _ sent _ to _ processor
j  0
5
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
6
Cooperative Game Framework (cont.)
 Model the grid load-balancing problem
– The m players are the service provider
– Each player has a performance function f j ( x )
0
u
– Each player has a minimum initial performance j
0
f
(
x
)

u
– x
j
– Solve the optimization problem
max  ( f j ( x)  u 0j ), x  X 0
x
jJ
– Equivalent optimization problem
max  ln( f j ( x)  u 0j ), x  X 0
x
jJ
7
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 The average processing time of a job
– Waiting time at the queue at a provider
Fj  h j 
h2j   j
2(1  h j   j )
– The expected transfer time of a job from any
player to provider
b
Lj 
cj
8
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
– The average completion time of jobs for
provider
h  j
2
j
b
D j  Fj  L j  ( h j 
)
2(1  h j   j ) c j
9
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Maximum expected service time
h2j   max
j
b
D  hj 

max
2(1  h j   j ) c j
0
j
 Maximum rate of jobs a provider
b
 hj )
cj

b
0
2h j ( D j   h j )  h 2j
cj
2( D 0j 
 max
j
 Note: 0   max
 j
j
10
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Nash bargaining solution is determined by
solving the following optimization problem
m
h2j   j
b
max D   ln( D  h j 
 )

2(1  h j   j ) c j
j 1
0
j
11
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 D in terms of 
0
j
max
j
 j   jmax
j  0
n
m
   
i 1
i
j 1
j
m
max D   ln(

j 1
h2j   max
j
2(1  h j   max
)
j

h2j   j
2(1  h j   j )
)
…(17)
 Concave function
12
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
h2j   max
j
h2j   j
n
m

L   ln(

)    j   i 
max
2(1  h j   j ) 2(1  h j   j )
j 1
i 1
 j 1

m
13
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 A necessary condition
L
0
 j
…(19)
 Solving (19), get
hj
1

   0, 1  j  m
max
j  j
1  hj  j
…(20)
14
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Solve for  j
 max
1
j 
 j 
2
2h j
max
h j max

1
h
(

 4)  
j
j
j
2  hj
n
m
i 1
j 1
…(22)
 Using (22) and constraint  i   j
m
max
h j max

1
h
(

 4)  
j
j
j
j 1
 hj
→ 
m
n
1 m max
     j  2 i
j 1 h j
j 1
i 1
15
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Setting
j  0

1

max
j
into (20)
 hj
16
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Cooperative Job Allocation
– According to the time equation
tj 
1

max
 hj
j
– Calculate two variables α and d
• α’s equation
d
max
h j max

1
h
(

 4)  
j
j
j
j 1
 hj

d
d
n
1
max
     j  2 i
j 1 h j
j 1
i 1
17
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
• d (1  d  m)
n
d
is the maximum positive integer
d
d
1
2 i      max

j
i 1
j 1 h j
j 1
j 1
max
h j max

1
h
(
t

 4)  td
j d j
j
td h j
 Finally
 max
1
j
j 


2
2h j
max
h j max

1
h
(

 4)  
j
j
j
2  hj
1 j  d
18
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
19
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Fairness index
( i 1 Ti )2
n
FI 
n i 1 Ti 2
n
 Provides a fair allocation for each brokers
i , j  i
j

n
i 1
i
 Amount of jobs to be sent from broker i to
provider j
i , j  0, and
m

j 1
i, j
i
20
Pareto Optimal and Fair Job
Allocation Algorithm (cont.)
 Periodically calculates an optimum job
allocation strategy
 Remain in equilibrium until the system’s
states change
21
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)
22
Experiments (cont.)
 Proportional-scheme
– Allocates jobs to providers in proportion to its
computing power
– The faster providers are sent more jobs by the
brokers
i , j  i
j

m
j 1
i
– Can’t take into account the communication
delays incurred in transferring job
23
Experiments (cont.)
 Noncooperative game algorithm
– Players are brokers
– Minimize their own average job completion
time
24
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
m
i  i      j
j 1
  The _ required _ overall _ average _ system _ loading
i  The _ relative _ job _ arrival _ rate _ of _ bro ker_ i
25
Experiments (cont.)
 Response Times
Average job completion time for each broker.
26
Experiments (cont.)
 Effect of System Loads
Average job’s completion time versus system load
27
Experiments (cont.)
 Effect of Service Time
– The Bounded Pareto distribution
f ( x) 
 k   x 1

1  (k / p)
kx p
k  The _ min imum _ job _ execution _ time
p  The _ max imum _ job _ execution _ time
28
Experiments (cont.)
 The mean (first moment) of the distribution

k
1
1
h

(  1   1 )

  1 1  ( k / p) k
p
 The second moment

k
1
1
2
h 

(  2   2 )

  2 1  ( k / p) k
p

29
Conclusion
 Fair to all users
 Represents a Pareto optimal solution to the
QoS objective
30
Thank you for your attention
31