2011 International Conference on Electronic & Mechanical Engineering and Information Technology

Grid Resource Scheduling Strategy Based On Sequential Game

Zhongwen Zhao, Nan Song

Equipment and Command Technology Academy

National Key Laboratory

Beijing China

songnansoft@ 163 .com

Abstract—Resource price prediction is one of the most

important problems in resource scheduling optimization in

grid. But price status is difficult to estimate accurately due to

the dynamic nature and heterogeneity of grid resource. In

response to this issue, a resource scheduling strategy which

uses sequential game method to predict resource price for time

optimization in a proportional resource sharing environment

is proposed. The problem of multiple users bidding to

compete for a common computational resource is formulated

as a multi-player dynamic game. Through finding the Nash

equilibrium solution of the multi-player dynamic game,

resource price is predicted. Using this price information, a

set of users' optimal bids are produced to partition resource

capacity according to proportional sharing mechanism. The

experiments are performed based on GridSim toolkits and the

results show that the proposed strategy could generate

reasonable user bids, reduce resource processing time,

hence overcome the deficiency of Bredin's strategy that it

doesn't consider resource price variation. Conclusion indicates

that employing sequential game method for price prediction is

feasible in grid resource scheduling and adapts better to the

dynamic nature of heterogeneous resource in grid

environment

Key words-resource allocation ; sequential game ;

proportional resource sharing; grid

I Introduction

As existing technology can not deal with some problems

of the virtual organization, grid computing came into our

sight [1]. However, as an emerging field, grid faces many

challenges, among which resource scheduling is a complex

one [2]. The main difference between grid resource

scheduling and local system resource scheduling is:

resources scheduled in grid are distributed in different

administrative domains. This distribution causes

heterogeneity when configuring and managing similar

resources, what is more, the supply-demand relationship of

grid resource is changing dynamically. Hence, economic

principles are referenced to solve grid resource scheduling

problems in many researches which can be divided into two

categories approximately: one is general-equilibrium based

grid resource scheduling; another approach is

Nash-equilibrium based grid resource scheduling. However,

these scheduling strategies are all based on history resource

price information without considering the future change of

the resource price. Hence they can't get rational resource

price, not to mention resource scheduling optimization.

This paper proposes a simple and effective price

prediction method based on sequential game strategy which

formulates the problem of multiple users compete for one

computational resource in proportional resource sharing

environment as a multi-player dynamic game problem, then

get resource's future price information via finite sequential

game, forms the combination of final resource price and

user's optimal bid, in the end implements the dynamic

optimal scheduling of grid resource.

II Proportional resource sharing model in grid

Proportional resource sharing model can settle the

question of one user occupies one resource for a long period

of time. Assume there are n grid users compete for finite

computational resources, the higher bid a user offers, the

larger proportion it can use. Grid user needs to purchase grid

resources to execute its jobs (a sequence of different kinds of

tasks). The execution time of the job is the sum of execution

978-1-61284- -8/ll/$26.00 ©2011 IEEE 2884 12-14 August, 2011

time of all tasks. There are k kinds of resources, each user

can only execute one task on one specific kind of resource,

hence define parameters as below:

UA : user's task sequence, tasks must be executed in

accordance with their sequence, among which s[

represents the size of ith user's kth task;

a I : ith user's ability to get resource to execute

kth task;

b'k: ith user's bid per second for kth resource;

Ck : the collection of all users;

Bk: sum of all users' bids for kth resource;

B7*: sum of all users' bids for k resource except , k * th * ith

user, that is B7* = Y bl •

The amount of resources ith user gets is:

Qk = ak (1)

Assume each user has information of resource price,

that is Bkl is known. Execution time of ith user's kth

task is:

While the cost is :

* i x bk *[{K + B~k

l)

(2)

(3)

III Dynamic optimal strategy of grid resource scheduling

A Nash equilibrium in grid resource scheduling

In proportional resource sharing grid model, each grid

user must ascertain its best offer with reference to other

users' bids, which means each user's bid is affected by other

users'. This competition and decision-making behavior is the

very kind of question that game theory focuses on. The

result of the competition is a Nash equilibrium solution,

which might not be the maximization of the whole interest

but a corollary with the consideration of the given

information. The problem of grid user's time optimization is

to finish task under finite budget Z. as soon as possible,

which can be represented as:

m in V t! , s .t.y* z ' < Z . Z-! k =1 k ' Z-! k = 1 k '

(4)

Equation (4) can be solved by Lagrange method, its

Lagrange equation is:

^ = IL^(IL^-^) (5)

Get partial derivatives of b\ in (5), order it equal to 0,

then we get the relationship between two arbitrary task bids

b\ and blj offered by one user.

dL _ s[B-kl

db[ ■ + X ——

al(bl)2 a, 0

X ( B ; ) '

K = b'j^JB-' IB-'

(6)

(7)

(8)

Again, get partial derivatives of X in (5), order it equal

to 0, use expressions which contain b[ to replace

{^},=2>menweget

-T(K + B;') + ak

Then introduce 3 variables

(9)

k*\ ak a \ k*\ &k

Get solution of (9), then we have b\

2ri i + J i +

4r; (a; - PIBXY

(11)

Among which, Bx e (o, a[ I P[\ > if Bx exceed this

region, b\ = 0 . In (9), b\ represents optimal bid of grid

user i against job 1. Figure 1 shows the formation process of

resource price with reference to different quantity of users.

The increment of competitive users makes resource price

rise, what is more, with the rise of the resource price, the

total bids of users first rise then fall, this is because when

resource price exceeds user's budget (o,a[ I fi[) , some

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Fig. 1 Users' total bids based on resource prices Fig. 2 Bid convergence process under three competing users

users choose to quit bidding, i.e. 0

B Sequential game based price prediction In the above, the resource price is predicted using the

average of the history price, in order to adapt to the dynamic

nature of grid, we proposes a resource scheduling

optimization strategy which utilizes finite sequential game to

predict resource price. Sequential game uses sequential

rationality, i.e. no matter what happens in the past,

participants should optimize its policy at the very moment of

current gaming process. Participants are grid users, their

strategies are based on other users' bids, the Nash

equilibrium of the combination of resource price and user's

bid can be inferred from B = \N_b\B) • As the

coefficients of the user's bid function a\ and y[ are

functions of resource price, therefore the diversity of

resource price prediction may generate different Nash

equilibrium solutions (resource allocation schemes), hence

affects the job execution time. While in finite sequential

game, user adjusts its bid unceasingly according to the latest

gaming result, this gradually forms an equilibratory situation

which drives resource price to steady state.

Sequential game belongs to dynamic game from the

perspective of time, each game uses the result of the latest

game result as the input of current game, hence generates

different Nash equilibrium result which differs from the last

one. Let G denote the stage of the sequential game, let G(n)

denote the game of stage n, which generates the resource

price of BG{n). First use the average of history resource

price BG(0) and time optimization strategy to perform stage

1 gaming, then bring resource price BG(0) into (11), in the

end solve B = J] f_j V (B), i.e:

* = I.L (a'{BG{0))-(3'B)2

2(f(BG^)f - 1 + 1 -

i(rl{BGm))2 ~X\ (12)

{a'(BG{0))-p'B)

Hence, it gets the average of resource price BG(l), which

is the resource price predicted by stage 1 gaming. Then

perform stage 2 gaming, bring resource price BG(l) into (10)

and solve equation B = ^._j# ' (B) ,i.e.:

[a'(BGm)-p'B)r

* = E« 2(r'(B^>)f -1+. 1+- H*G<I))) (a'(Baw)-0'B)

(13)

JJ

G(2) Here it gets the average of resource price B ( } , which

is the resource price predicted by stage 2 gaming. Repeat

this process until the variation of the predicted price is less

than the given threshold. Assume stage n-1 is the last stage

of the gaming whose predicted resource price is BG(n~l),

thus we can calculate the optimal combination (Nash

equilibrium solutions) of all the users as well as resource's

ultimate resource price BG(n).

#(£G(w)) = max 0, {ct-pEF™)1

m 4 / ) B°M

-1+J1+ v ; . r( n , \ id -{?BG{n-l)f

(14)

Fig. 2 shows the curves of users' bids which vary

according to the stage of the game when there are 3 users.

IV Simulation, results and performance

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Fig. 3 Resources' turnover rate of 3 price prediction methods

A Price predicion methods

In the above, we propose a resource scheduling strategy

based on sequential game (SG). In order to illustrate its

efficiency, we compare its simulation result with 2 most

commonly used price predict methods at present.

One step ahead (OSA) prediction. It predicts next

moment's price based on the trend of past time sequence, if

the current price rises (or falls), then next moment's price

rises (or falls). While using this model, a question of to

what extent the price should rise (or fall) needs to be settled.

Let a constant denote the extent of variation, its

initialization value is 0.5, and it is adjusted based on the

difference between current price and latest price.

History mean (HM) prediction. Let the average of the

past 5 minutes' (also called time window) history resource

price represent current price. If the price variation extent is

large, we can shrink the time window, otherwise we can

loosen it.

B Simulation settings

Gridsim is a Java based event-driven toolkit package for

grid simulation [8], its main purpose is to study

computational economy based resource scheduling through

simulating. We simulate 20 grid users to compete for

resource. Each user contains 2-4 jobs whose length are

2000MI (million instructions) while vary from 0% to 10%

[9]. User's budget is set with reference to the law of 80/20,

that is 20% users own 80% wealth.

C Result and discussion

Figure 3 shows the turnover rate of bids under 3 price

prediction methods [10].The execution time of SG basically

lies on the top of the 3 curves, this is because after

continuously gaming, each user has an approximately

understanding about other users' bids. Hence it predicts the

resource price which can makes user's bid more reasonable,

thus the turnover rate increases. While OSA is seriously

unstable for the reason that it can only predict short-term

variation other than long-term variation. The result of HM

is the worst.

V Conclusion

This paper proposes a time optimization resource

scheduling strategy based on budget restriction which is

more adaptable to the dynamic nature of heterogeneous grid

resources. Simulation results show its efficiency.

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