A Solving Model of etwork esource

Wang Jian-hua, Zhang Jun, Ji Wei-dong, Zhang Long Computer Science and Information Engineering College

Harbin Normal University Harbin, China

wjh@vip.163.com, dzxxzj@yahoo.com.cn, kingjwd@126.com, zlwalkman@sina.com

Abstract—A kind of network model is presented in this paper it analyzes the optimized problem of network resources based on this model, and converts the computer network resources problem into shape transformed process of computing multi-dimension flexible cyberspace. This model and method can depict complicated social mutual behavior among the base nodes in the multi-dimension flexible cyberspace it also can depict respectively adopted dynamic strategy and autonomy behavior which changes with the situation.

Keywords—Network Model ; Dynamic Strategy ; Autonomy Action

I. INTRODUCTION With the quick development of computer and network

technology, Internet becomes an opening complex distributed system, so the contradiction between supply and demand of network resources is becoming more and more serious. At present, some method which allocates the limited resources through pricing has improved the resources utility in some extend such as basing on priority, boundary, but the above method cannot dynamically reflect the demanding and the usage of system resources. The presented model of multi-dimensional elastic network which has better parallel , interaction and dynamic can fully reflect complicated social mutual behavior among the basic class node in the multi-dimensional elastic cyberspace, also can depict respective adopted dynamic strategy and autonomic behavior with the situation changes.

II. MULTI-DIMENSIONAL ELASTIC NETWORK MODEL Internet is a complex huge system, and we can take

internet as cyberspace possessing elastic deformation .The feature of the cyberspace is elastic deformation and own growth.

We research on the system and take every network entity as basis growth, called basis entity which can divided into demand group and supply group .Under the condition of elastic network ,every basis entity having better parallel, interaction and dynamic can fully embody social mutual behavior among the basic and describes the dynamic strategy and autonomy behavior adopted by all kinds of basic with the change of situation, also reflects the dynamic changes of the usage of network resources. Basis entity randomly locates at every place of network, but owing to the activity of

themselves and dynamic functioning among basis entity, they are in orderly status. The nonlinear behaviour among the basis entity can express by recombination properties, and the basis entity can be moved by all kinds of resultant forces.

The supply group 1 2, ... nE E E E and the demand group 1 2, ... nD D D D . The resource vector held by the supply

group iE is expressed by i i i ir =(r (1),r (2) ,...,r (k)) . The

resource vector required by the demand group jD is expressed by j j j jd =(d (1),d (2) ,...,d (k)) . The allocation among n supply groups and m demand groups can be showed by table

1. In the table, ija denotes the resource vector allocated to the demand group Dj by the supply group Ei. The pay vector which the demand group Dj would like to provide to the specific resource vector of the supply group Ei is

ij ij ij ijp =(p (1),p (2) ,...,p (k)) . The supply group denotes some kind of entity in the network, such as host server, router, wideband link, etc. the resources may be the processing capacity of host server or router, the bandwidth of link, the information in web server, or other usable resources in network. The question of optimizing network resource allocation is to maximize self interests through all kinds of complex social interaction behaviors, including competition, cooperation and induction, among all supply groups, and at the same time, every demand group will adjust continuously its pricing strategy according to actual condition and the optimizing resource allocation under a certain condition is obtained finally.

TABLE I. ALLOCATION OF RESOURCE AND PAY AMONG THE SUPPLY GROUP AND THE DEMAND GROUP IN COMPUTER NETWORK

Entity D1 … Dj … Dm

E1 … Ei … En

a11, p11 a11, p1j a1m , p1m

ai1, pi1 aij , pij aim , pim

an1, pn1 anj , pnj anm , pnm

Physical model of multi-dimensional elastic network cyberspace:

The narrow-defined network cyberspace can be imagined easily as a sphere, which has elastic deformation and can expand gradually. The basic unit of it is the basic entity, i.e.

2009 Fifth International Joint Conference on INC, IMS and IDC

978-0-7695-3769-6/09 $26.00 © 2009 IEEE

DOI 10.1109/NCM.2009.27

39

2009 Fifth International Joint Conference on INC, IMS and IDC

978-0-7695-3769-6/09 $26.00 © 2009 IEEE

DOI 10.1109/NCM.2009.27

39

2009 Fifth International Joint Conference on INC, IMS and IDC

978-0-7695-3769-6/09 $26.00 © 2009 IEEE

DOI 10.1109/NCM.2009.27

39

the demand group and the supply group. The basic nodes in same kind of groups distribute around the circle center in equal distance and angle. In order to analyze the issue conveniently, we divide the sphere into a horizontal section and a vertical section. The supply group distributes on the horizontal section and the demand group distributes on the vertical section.

According to the rules of object moving, developing and changing, we make the following provisions: The distance between Dj, the vertical demand group, and the circle center is inversely proportional to the current pay of Dj. The distance between the horizontal supply node and the circle center is directly proportional to the current gain of Ei. All nodes exist in a huge circle-formed attracting field and move along the radial direction towards to the edge of attracting loop as far as possible. This means that the supply group attempts to obtain maximum gain of their own and the demand group attempts to pay as little as possible. Social mutual behaviors existing among same kind of groups, such as competition, collaboration or induction, can be expressed with corresponding expansion coefficients. Different expansion coefficients can express the strength of competition or collaboration. In this model, the groups move under the effect of circle attracting field and the combining forces from other groups.

Each supply group wants to provide resources to the demand which has higher pay, at the same time it is also subject to the cost limits of their own specific resource vector and the cost limits of coordination among several demands. The less this two spending of the supply group, the stronger it’s competition is, the bigger the attracting force of the supply applied by the circle edge is, as well as the stronger the moving trend towards to the circle edge is. In the mean while, each supply group can decide whether or not to compete or cooperate with other supply groups or make use of them according to its own strategy. In this way, each supply group moves along the radial direction under the combination of a variety of forces, and finally achieves a balance state in the condition of current pay. In this state, the node with higher pay and lower other expending is more close to the circle edge. Then, each demand group will adjust its pricing strategy timely according to the allocating condition of supply group resources and will pay as little as possible under the premise of ensuring that it’s request can be met. Therefore, the demand which obtain fewer resources will increase the pay providing to the supply group, while the demand which was assigned excessive resources will decrease appropriately the pay it providing. This is reflected by moving of each demand group along radial direction towards to the circle edge under the attracting force from the circle. Meanwhile, same as the supply group, there are corresponding interaction among the demand groups. They also achieve a balance state in the condition of current supply resources allocation under the combination of a variety of forces. Then, each supply group will make a further adjustment of its resource allocation according to the adjusted pay condition. Such a cycle operates, until the positions of the supply group nodes and the demand group nodes don’t change obviously, i.e. they all achieve a more

stable balance state. Thus the stable solution of network resource optimization will be obtained.

Basing on the above mentioned physical model, a mathematic model for multi-dimensional elastic network cyberspace is defined here: For a given supply group E and a given demand group D, the energy function of the supply group in a multi-dimensional elastic network cyberspace is defined as follows:

JE=EEI EEI DD

ijJ

a

DDJ EEij

i

a

jiEEE

ijji

ji ee 1

ji ee 2=(cos(i 2 /n) R (1 fi) cos(j 2 /n)

R (1 fj))2+ (sin(i 2 /n) R (1 fi)sin(j 2 /n) R (1 fj))2 2 Where, E, , and E are weight value coefficients.

Parameter is applied to avoid minimizing local energy. is the coordinate of the supply group . is an evaluation function value from the supply group for current condition.

, the more the gain of the supply group is.

, is the function of energy interaction between supply group and . expresses a kind of distance measurement of the vector. R is the radius of attracting force circle in the physical model.

The evaluation function form the supply group for current condition is defined as below:

1 1 1

1

( ) ( )

1

m m m

ij ij ij ij iij j j

a p comm a a cf

Its denominator expresses the difference of total income

and total expenditure of the supply group node Ei, i.e. the net gain. ci is the expenditure vector for specific resource of node Ei. is the inner product operation among vectors.

m

jijacomm

1)( expresses the expenditure for coordinating

communication of resource vectors which are allocated to the demand group Ei by the supply group. The less the fi, the more the gain of the supply group Ei is.

j, the evaluation function of the demand group Dj, can be defined as the pay offered by the demand, that is,

j

n

iijij pa

1)(

The first item in formula (1) is expressed by 1J . In 1J ,

when Ek tends to 0,

2

2( )

22 2

1max ( ) 2 ln

i

E

f xnk

i Ei if x k e

, then

21min min ln ( , ) min(max ( ))

ij iji

E i E ia a x iE EJ k f k f x . From

In case of denominator>1 In case of denominator 1

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this formula, it can be showed that 1min 0

ijaJ is equal to

2min(max ) 0ij i

ia Ef . That is to say, the less the 1J , the less

the corresponding to the supply group is and the more it’s gain is as possible.

The second item and the third item are expressed together by 2J . They deal with the constraint condition of node’s

own resource. The four item is expresses by 3J . Similarly, the energy function for the the demand group

in multi-dimensional elastic network cyberspace can be defined as: JD=

DDJ

j

jiDDD

ijji

ji ee

3 Where, j is the evaluation function value of the demand

group Dj under current condition. The less the j, the cost paid by the demand group is. In order to reduce JE and JD which are the energy function, aij , the change amount of aij and pij, the change amount of pij, should be expressed as below:

aij EkE

ij

E

aJ

EkE

EE EE ij

q

qijij

q

qq qaf

fJ

aJ

af

fJ 321

E

EE ij

qqEE

Q

q af

f

2 EkE

EEjij

DDijij

Jj

dara

EkEEE ij

q

q

jiEEE

jiij

X

JI

af

f

ee )(

pij DkD

ij

D

pJ

D

DD ij

qqDD

q

q ag

g DkD

DD ij

q

q

jiDDD

jiij

q

JI

pg

g

ee )(

qE

EE

kf

k

f

q

E

v

E

q

e

e

2

2

2

2

2

2

; qD

DD

kg

k

g

q

D

v

D

q

e

e

2`

2

2

2

2

2

Where, E is a positive constant and D is a learning coefficient.

III. ALGORITHM AND APPLICATION OF MULTI-DIMENSIONAL ELASTIC NETWORK CYBERSPACE

A. Algorithm of Multi-Dimensional Elastic Network Cyberspace The algorithm is stated as below: Step 1: Set in parallel

1( )ija t , the resource allocation

vector of the supply group Ei on moment of t1 and 2( )ijp t , the providing pay vector of the demand group on moment of t2. Initialize the weight value coefficients in formula (1), (2), (3), (4) and (5), as well as z1 and z2, the numbers of predicted reiteration steps.

Step 2: Calculate in parallel 1( )if t , the evaluation

function value of the supply group Ei and 1( )ie t , the

coordinate of entity node according to 1( )ija tthe resource

allocation and 2( )ijp t, the pay vector; Calculate 1( )EJ t

through formula (1).

Step 3: Calculate ija in parallel through formula (4),

renew ija in parallel, then obtain 1( 1)ija t

, the resource

allocation vector; Calculate in parallel 1( 1)if t , the evaluation function value of the supply group Ei, and

1( 1)ie t , the coordinate of the supply group; Calculate JEt1+1 and JE through formula (1); Step 4: If EJ is less than specified domain value, go to

step 5; Otherwise, if t1<z1, then specify t1= t1+1 and go to step 2; If t1=z1, then reduce parameter kE, specify t1=0 and go to step 2.

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Step 5: If the result meets the precision requirements specified by constraint 2J , go to step 6; Otherwise, modify the parameters and , specify t1 0, and go to step 2;

Step 6: According to 1( )ija t , the resource allocation and

2( )ijp t , the pay vector , calculate in parallel 2( )jg t the evaluation function value of the demand group Dj, and

2( )je t , the coordinate of the demand node; Calculate JD(t2) through formula (3).

Step 7: Calculate in parallel ijp through formula (5),

renew in parallel ijp , then obtain 2( 1)ijp t , the pay vector.

By use of 1( )ija t and 2( 1)ijp t , calculate in parallel

2( 1)jg t , the evaluation function value of demand group

Dj, and 2( 1)je t , the coordinate of demand group;

Calculate JD(t2 1) and DJ through formula (3).

Step8: If DJ is less than specified domain value, go to step 9; Otherwise, if t2<z2, then specify t2 =t2+1 and go to step 9; If 2 2t z , then reduce parameter Dk , specify t2=0 and go to step 6.

Step9: If the changes of EJ and DJ are all less than specified domain values, the algorithm ends successfully; Otherwise, specify t1=0, t2=0, and go to step 2.

In this algorithm, because every group node is defined with a corresponding evaluation function, the evaluation function value of every group node can be calculated in parallel respectively. The calculations corresponding to each

ija in every supply group node and each ijp in every demand group node can also be carried out in parallel. The calculations corresponding to every ija and every ijp can be performed by use of a processing cell unit.

B. Question Application Typical questions about computer network resource

optimization include the load balancing of cluster server, the dynamic bandwidth allocation of wideband network, as well as the load distribution and route selection of router. Here we discuss an example for the network bandwidth allocation optimization.

In a wideband network, many wide-distributing LANs need to be connected one another by means of a kind of half-permanent virtual channel which has certain bandwidth. The virtual channel between two nodes in the network may consist of several virtual paths. Every virtual path is formed by connecting several links serially. One link in a network can be regarded as a supply group. One virtual path in a network can be regarded as a demand group. Several interactions, such as competition, collaboration and induction, exist among some links and among some virtual paths. When optimizing questions by use of above-mentioned algorithm,

the constraint conditions of elastic potential energy function are showed as below:

j i ijb =min {a }for all pj, the virtual path, 1 j m;

m

jjb

1 Ci for all Li, the links, 1 i n; )(

)1(

hq

qjjb

qBfor all Pq, the virtual channels;

Where, qB indicates the total bandwidth demand of qP

, the virtual channel between some two nodes. Pq={pq(1)

pq(2) … pq(n)}, including virtual paths with quantity of h.

jb indicates the bandwidth allocated on pj, the virtual bath.

Ci indicates the maximum bandwidth of link Li. Here take the network topology in Fig. 1 as an example.

The maximum bandwidths for links from 1L to 9L are 10, 15, 30, 16, 20, 18, 20, 27 and 25 respectively. Connecting virtual channels between following nodes should be established for following nodes: 1-4, 1-5, 1-6, 3-4, and 3-6. The bandwidths required to be allocated are 25, 10, 12, 15 and 8 respectively. Every virtual channel includes virtual paths as below:

1 1 2 3 4 5, , , ,P p p p p p, 2 6 7,P p p

,

3 8 9 10, ,P p p p, 4 11 12,P p p

, 5 13P p.

Where, every virtual path is formed by connecting following links serially:

1p [L1,L5], 2p [L3,L8], 3p =[L2,L6,L9,L8],

4p =[L2,L7], 5p =[L1,L4,L9,L8], 6p =[L1,L4], 7p =[L2,L6],

8p =[L3], 9p =[L2,L6,L9], 10p =[L1,L4,L9], 11p =[L7],

12p =[L6,L9,L8], 13 6 9,p L L .

Figure 1. Network topology structure required to be allocated with

bandwidth (bandwidth allocation optimization is required among 5 virtual channels, 13 virtual paths and 9 links)

It is assumed that different virtual path at initial moment provides same pay for specific bandwidth. Then we can

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solve according to the algorithm. In the solution process, the weight value of node evaluation function in the network follows mainly the principle of shortest path first, i.e. the less the number of links contained in virtual path coming to the destination, the bigger it’s weight value is. For interaction, the mutual competitions among links and among virtual paths should be considered chiefly. Since total bandwidth supply of links in this example is more than the demand, along with the proceeding of algorithm iterative process, virtual paths reduce their providing pay in different degrees. This is showed by continuous expanding of internal demand nodes. For most links, the overall gains can maintain approximately at a constant. However, for some special links, such as 5L , because it can only provide bandwidth to virtual

path 1p , the result of competition with other links causes obvious gain change. After the iterative process comes into stability, the result shows as below:

1P ={0.9, 13.3, 1.1, 7.2, 2.5}, 2P {5.7, 4.3}, 3P ={11.2,

0.0, 0.8}, 4P ={10.4, 4.6}, 5P ={8.0}.

IV. CONCLUSION The presented model of the distributed optimization for

distributed network supply and demand resources converts the problem about optimizing allocation computer network supply and demand resources into shape-transformed process of computing multi-dimension elastic cyberspace. The analysis shows that the model and the method can depict complicated social mutual behavior among the basic class node in the multi-dimensional elastic cyberspace, also it can depict respective adopted dynamic strategy and autonomic behavior with the situation changes.

ACKNOWLEDGMENTS

This study is supported by the Youth Science and Technology Special Foundation of Heilongjiang Province of China (QC2008C27), the Higher Education Reform Project of Heilongjiang Provincial Education Department (S08-28) and the Heilongjiang Provincial Key Laboratory of Intelligence Education and Information Engineering.

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