Game Theory Based Bandwidth Allocation Scheme for Network Virtualization

Abstract

Running multiple virtual networks over a shared physical network is a promising way to support diverse applications, consequently network virtualization is viewed as the keystone of the next-generation architecture. However, decoupling the role of traditional ISPs into Infrastructure Providers (InPs) and Service Providers (SPs), also brings some new challenges to us. For example, how to fairly and efficiently share the sacred physical resources of InPs among multiple SPs is a key problem. The interaction between InPs and SPs, such as cooperation and competition, makes this topic even more complicated. In this paper, we develop a novel approach to encourage efficient behavior in solving the interaction between InPs and SPs by introducing economic incentives, in the form of Game Theory. Based on the non-cooperative game model, a bandwidth allocation scheme in the network virtualization environment is established, using the concept of the Nash Equilibrium. Then we propose an iterative algorithm to find the Nash Equilibrium and solve the bandwidth allocation problem. Finally, we demonstrate the convergence and the effectiveness of our scheme in the experiments.

Full text

Page 1

Game Theory Based Bandwidth Allocation Scheme
for Network Virtualization
Ye Zhou, Yong Li, Guang Sun, Depeng Jin, Li Su, Lieguang Zeng
State Key Laboratory on Microwave and Digital Communications
Tsinghua National Laboratory for Information Science and Technology
Department of Electronic Engineering, Tsinghua University, Beijing 100084, China
Email: jindp@mail.tsinghua.edu.cn
Abstract—Running multiple virtual networks over a shared
physical network is a promising way to support diverse applica-
tions, consequently network virtualization is viewed as the key-
stone of the next-generation architecture. However, decoupling
the role of traditional ISPs into Infrastructure Providers (InPs)
and Service Providers (SPs), also brings some new challenges to
us. For example, how to fairly and efficiently share the sacred
physical resources of InPs among multiple SPs is a key problem.
The interaction between InPs and SPs, such as cooperation and
competition, makes this topic even more complicated. In this pa-
per, we develop a novel approach to encourage efficient behavior
in solving the interaction between InPs and SPs by introducing
economic incentives, in the form of Game Theory. Based on the
non-cooperative game model, a bandwidth allocation scheme in
the network virtualization environment is established, using the
concept of the Nash Equilibrium. Then we propose an iterative
algorithm to find the Nash Equilibrium and solve the bandwidth
allocation problem. Finally, we demonstrate the convergence and
the effectiveness of our scheme in the experiments.
I. INTRODUCTION
In the last few years, network virtualization has attracted
a great deal of attention in the debate on how to model the
next-generation Internet. Deploying new Internet services is
difficult without the cooperation among all stakeholders, as
radical changes of the Internet architecture is not allowed
[1][2][3]. Researchers believe that network virtualization can
overcome the ossification of the current Internet and stimulate
innovation [4][5]. Furthermore, it is considered as the keystone
of the next-generation architecture [4].
In network virtualization, the role of the traditional ISPs are
decoupled into two parts: Infrastructure Providers (InPs) are
in charge of physical networks, and Service Providers (SPs)
deploy customized Virtual Networks (VNs) [5][6]. Then the
deployment of VNs should be observed from two different
perspectives: the former is the perspective of InPs, which
mean to maximize their own revenue by allocating physical
resources; while the later one regards to SPs, which mean to
obtain the contracted resources.
As a result, network virtualization faces a fundamental
challenge of fairly and efficiently sharing physical resources
among multiple VNs. On one hand, InPs should focus on how
to keep balance between ensuring the fairness and maximizing
their own revenue. On the other hand, SPs just focus on how
to gain enough resources, by competing with each other or
selecting from available InPs. In summary, SPs may cooperate
or compete with each other to suffice their own requirements,
especially when they are controlled by multiple parties. Due to
the complex interaction between InPs and SPs, we introduce
economic incentives into the resource allocation scheme in
the network virtualization environment, with the hope of
encouraging efficient behavior. In this paper, we consider a
model that consists of an InP and multiple SPs, and focus on
the interaction among multiple SPs in the framework of Game
Theory. Game Theory is a field of applied mathematics that
analyzes interactive decision situations, and provides analytical
tools to predict the outcome of complex interactions among
rational entities. We believe Game Theory will be applicable
to analyze the network virtualization environment.
In this paper, we consider the bandwidth allocation scheme
based on Game Theory. Moreover, we only focus on how InPs
allocate the limited bandwidth among multiple VNs. It’s obvi-
ously that the sharing of bandwidth has the most direct impact
on the performance of VNs, as well as the total performance
of the network virtualization environment. Therefore, our goal
is to efficiently allocate the bandwidth among multiple VNs
in the framework of the non-cooperative game model. The
contributions of our work can be summarized as follows:
• We propose a bandwidth allocation scheme based on the
non-cooperative game model to describe the interactions
among multiple VNs. We are the first to develope a
case for the application of Game Theory in the network
virtualization environment.
• The non-cooperative game model we set satisfies the
situation that the total bandwidth requirements of multiple
VNs exceed the capacity of the physical network. The
pricing scheme in our model associate the congestion
control with their payments, encouraging the efficient
behavior of VNs. Furthermore, the InP introduces dif-
ferential pricing scheme according to the characteristic
of different VNs.
• We prove that our scheme can achieve the Nash Equilib-
rium [7], and develop an iterative algorithm to implement
our scheme. Besides, the effectiveness and the conver-
gence of the Nash Equilibrium are also demonstrated in
our experiments.
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The remainder of this paper is organized as follows. After
presenting the related work in section II, we describe the
system model, propose our bandwidth allocation scheme, and
develop an iterative algorithm in section III. Then we study
the performance of our experiments in section IV. Finally, we
conclude the paper in section V.
II. RELATED WORK
To the best of our knowledge, there is no existing work
in the literature that tries to solve the bandwidth allocation
problem in non-cooperative game model. But there are several
proposals that have motivated as well as influenced our work.
Resource allocation in the network virtualization environ-
ment aims to fairly and efficiently share physical resource
among VNs. For the static resource allocation scheme could
cause worse performance than existing solutions, the authors in
[8] propose an adaptive network virtualization, with dynamic
resource allocation scheme. In this architecture, resource are
periodically reassigned among VNs, aided by optimization
theory. As mentioned in [8], if any of the VNs exhibit greedy
or malicious behaviors, unfair allocation results.
Then this problem is discussed in [9], the authors analyze
the problem of allocating bandwidth among multiple VNs,
especially when there are some greedy applications. More-
over, the mechanism to offer a fair bandwidth distribution is
introduced. Though the authors propose a strategy that tries
to fairly distribute the bandwidth among competing VNs, the
scheme itself is static and only applied to VNs with only one
service class. In order to efficiently use physical resources,
the authors in [10] propose a runtime, distributed, local view
approach to manage physical resources. An associated self-
organizing algorithm is used to reallocate resources of VNs
along different physical nodes.
The authors in [11] introduce three branches of Game
Theory, leader-follower, cooperative, and two-person nonzero
sum games, to the study of the Internet pricing issue. In
addition, both non-cooperative and cooperative game are ap-
plied to the Internet pricing framework, especially the resource
allocation problem. Recently, many researchers have used
game theoretical methods to analyze the resource allocation
problem in computer networks, especially wireless network.
The authors in [12] proposed a non-cooperating power control
game based on a specific energy efficient utility function that
is common to all users, and proved that the game has a unique
Nash Equilibrium. Then in [7], the concept of Pareto efficiency
was introduced in the game to gain better overall performance.
Based on the model mentioned above, the authors in [13]
generalized the game model to consider quality-of-service con-
straints. In [14], the authors summarize the game theoretical
approaches used for energy efficient resource allocation in
wireless network. Hence we believe Game Theory’s ability
to simplify analysis of interaction among different players
can help handling problems from network virtualization, for
example the resource allocation problem.
III. RESOURCE ALLOCATION SCHEME BASED ON
NON-COOPERATIVE GAME MODEL
A. System Model
In this section, we introduce the system model of the
network virtualization environment. The basic entity is the
VN, a collection of virtual nodes and virtual links. Hence,
here comes the problem of allocating the physical resource
among multiple VNs. In this paper we don’t consider how
to embed VNs onto the physical network, and we assume
that the mapping results are already known. Furthermore, we
ignore the admission control of VNs, assuming that a number
of VNs are already allowed to use the physical network. With
all these assumptions we can set up the model of our network
virtualization environment.
We model the physical network managed by the only InP
as a weighed undirected graph and denote it as G = (N, L),
where N is the set of physical nodes and L is the set of
physical links, denoted by L = {1,2,...,n}(n ≥ 2). The InP
uses C = {c1,c2,...,cn} to represent the bandwidth capacity
on every physical link. In order to obtain revenue from VNs,
the InP also denotes the price vector as p = {p1,p2,...,pn}.
Another vector β = {β1,β2,...,βn} is introduced to state
the importance of every physical link to the InP. We let
∀1 ≤ i ≤ n, 0 ≤ βi ≤ 1 and set the most important link’s
congestion price equal to 1.
The VNs are denoted by V = {1,2,...,m}(m ≥ 2), which
means that m VNs co-exist in the model. For the k-th VN,
the bandwidth allocated on physical links is described with
the vector yk= {yk
constraint of ∀1 ≤ i ≤ n,0 ≤ yk
InP sets the price-weighed vector wk= {wk
each VN, in order to realize differential pricing scheme. For
example, InP may set small price-weighed factor to encourage
bandwidth on low-propagation-deley links to be allocated to
VNs with delay-sensitive traffic class. In order to further
describe the workload of physical links, the path rate used
by the k-th VN is denoted as zk= {zk
constraint of ∀1 ≤ i ≤ n,0 ≤ zk
both allocated bandwidth and path rate of each VN, we also
denote the vector xkas xk= {yk
In Table I, the key notations used through the paper are
summarized.
In our model, the payoff function of the k-th VN includes
the following three parts:
• The Utility Function: we use Uk(yk,zk) to represent the
utility function of the k-th VN. As we can see, Ukde-
pends on both the path rate and the bandwidth allocated.
We follow the common practice in study of Internet that
utility function is defined as a convex function.
• The Pricing Function: we use Pk(yk) to represent the
pricing function of the k-th VN. The k-th VN should pay
pjwk
link. So we present the pricing function as the following
1,yk
2,...,yk
n}, which naturally follows the
i
≤ ci. Moreover, the
1,wk
2,...,wk
n} for
1,zk
i. In order to represent
2,...,zk
n}, with the
i≤ yk
1,yk
2,...,yk
n,zk
1,zk
2,...,zk
n}.
jas total price for the assigned bandwidth on the j-th
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TABLE I
LIST OF KEY NOTATION IN THE NETWORK VIRTUALIZATION MODEL.
VariableDescription
n
Index of physical links
m
Index of VNs
cl
Bandwidth capacity of the physical link l
pl
Price of the bandwidth on the physical link l
βl
Congestion price of the physical link l
xk
Bandwidth allocated and path rate of the k-th
VN
yk
l
Bandwidth assigned to the k-th VN on physical
link l
zk
l
Path rate of the k-th VN on physical link l
wk
l
Price-weighed factor of the k-th VN on physical
link l
equation:
Pk(yk) =
n
?
j=1
pjwk
jyk
j.
(1)
• The Congestion Function: we use Ck(yk,zk) to represent
the congestion function of the k-th VN, measuring the
congestion cost according to the assigned bandwidth and
actual path rate. If the total path rate of all VNs are less
than the capacity of the physical link, no congestion cost
will be charged. Otherwise, we denote the congestion
price as the following equation:
Ck(yk,zk) =
n
?
m
?
cj
j=1
βjyk
j[
m
?
i=1,i?=k
zi
j
cj
− 1]+(−zk
j
cj).
(2)
In Equation (2), [
m
?
cj
i=1,i?=k
zi
j
−1]+(−zk
j
cj) is defined as follows:
?
cj
if
i=1,i?=k
zi
j
− 1 < −zk
j
cj, then [
m
i=1,i?=k
zi
j
− 1]+(−zk
j
cj) = 0,
m
?
cj
which implies that no congestion cost is charged; if
?
cj
implies that the congestion cost is charged. We charge every
VN assigned with the bandwidth on the congested link. In
our model, when we consider the congestion function of the
k-th VN, we assume that zk
congestion, because we already pay for yk
the pricing function.
With these three parts, we can obtain the payoff function
Φkof the k-th VN, denoted by:
i=1,i?=k
zi
j
−
1 ≥ −zk
j
cj, then [
m
i=1,i?=k
zi
j
−1]+(−zk
j
cj) =
m
?
i=1,i?=k
zi
j
cj
−1, which
jis not the reason that causes the
jwhen we calculate
Φk(yk,zk) = Uk(yk,zk) − Pk(yk) − Ck(yk,zk).
B. The Resource Allocation Scheme
A non-cooperative game is one in which players are unable
to make enforceable contracts outside of those specifically
modeled in the game. In our model, individual VN does
not communicate with others to modify its own strategy.
Then the only InP play the role of enforcing VNs to modify
their strategies in the form of pricing scheme. To analyze
(3)
the outcome of the game, we adopt the well-known Nash
Equilibrium concept, in which every player will select a utility-
maximizing strategy given the strategies of other players.
From the perspective of each VN, there are two ways to
increase its own revenue: firstly, each VN is assigned with
certain bandwidth to run its own path rate, the more the better;
secondly, since congestion on certain links will absolutely
decrease the revenue of all VNs on those links, VNs have to
avoid congestion. Formally, we model our proposed scheme
as the following non-cooperative game:
• Players: K = {0,1,2,...,m}, where the 0-th player is the
only InP and k = 1,2,...,m stands for the k-th VN.
• Action space: P = Q × P1 × P2... × Pm, where
Q = [0,¯ Q] and Pk represents the action space of
the k-th VN. We denote¯Q as¯Q = {c1,c2,...,cn} to
represent the bandwidth capacity on every physical link,
and Pk= {yk
resource allocated to the k-th VN.
• Payoff function: we use Φk, ∀k = 1,2,...,m to represent
the final revenue of the k-th VN. They compete with each
other for the scared physical resources, aiming to obtain
more resources to increase their revenue as well as avoid
congestion cost when unnecessary.
In our scheme, we try to solve the problem by finding an
Nash Equilibrium, and we prove the existence of the Nash
Equilibrium in Theorem 1.
Theorem 1: There exists a Nash Equilibrium for the prob-
lem stated in Equation (3).
Proof: First we rewrite the Equation (3) as follows:
1,yk
2,...,yk
n,zk
1,zk
2,...,zk
n} to represent the
Φk(yk,zk) = Uk(yk,zk) − Pk(yk) − Ck(yk,zk)
= Uk(yk,zk) −
n
?
j=1
pjwk
jyk
j−
n
?
j=1
βjyk
j[
m
?
i=1,i?=k
zi
j
cj
− 1]+(−zk
j
cj).
(4)
In Equation (4), if there is no congestion, the congestion
function will be equal to 0, which simplifies the problem.
However, we will consider the most complicated situation,
which means congestion occurs in every link and VN. Hence
we can simplify the Equation (4) as follows:
Φk(yk,zk) = Uk(yk,zk) −
n
?
j=1
pjwk
jyk
j−
n
?
j=1
βjyk
j(
m
?
i=1,i?=k
zi
j
cj
− 1).
(5)
The action space of the only InP is a closed subset of Rn,
and with the constraints of ∀i, ∀k, 0 ≤ zk
action space of every VN is also a closed subset of Rn. It is
obvious that all three parts of Φkare continuous, thus Φkis
continuous in the action space. Furthermore, we mean to prove
the convexity of the Φk in Equation (5). Because the utility
function is a convex function, and the pricing function is a
linear function, the convexity of Φkdepends on the congestion
function. When we consider the congestion function, we only
focus on the path rate caused by the other VNs, and we denote
i≤ yk
i≤ ci, the
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αk
j=
m
?
i=1,i?=k
zi
j
cj
− 1. We notice that αk
jis a constant as far as
the k-th VN is concerned, then it is obviously that
n ?
j=1βjyk
jαk
j
is also a linear function. In a word, the payoff function Φk
is a convex function, which means it is also quasi-convex.
According to the Ref. [7], the Nash Equilibrium exists.
Then we develop the Algorithm 1 to implement the scheme
mentioned above and reach the Nash Equilibrium, aided by
the concept of iteration.
Algorithm 1 Bandwidth allocation scheme based on the non-
cooperative game model
State0represents the initial bandwidth allocation
while Statei ?= Statei−1 (The Nash Equilibrium isn’t
arrived) do
Compute the i+1-th iteration
for Every VN in the model do
Find the best response by solving the problem stated
in Equation (6)
end for
The result of the i+1-th iteration is denoted as Statei+1
end while
In our iterative algorithm, we use the vector Statei =
{x1
VNs after the i-th iteration. Initially, we assign resources to
all VNs, denotes by State0. In the i-th iteration, we update
the response of each VN one by one, based on the existing
bandwidth allocation situation. For example, we suppose the
bandwidth allocation of all the other VNs are fixed, and try to
find the best response of the k-th VN, denoted by xk
following equation describes our work in detail:
i,x2
i,...,xm
i} to represent the bandwidth allocation of all
i+1. The
xk
i+1= max
xk∈Pk(xk,x−k
i,xk−1
i
,xk+1
i
i),
(6)
where x−k
out the best response of all VNs, we finish the i-th iteration
and the bandwidth allocation of all VNs can be updated to
Statei+1. Finally, we can get an convergent state, which stands
for the final results of our scheme. Now, we have proposed
a bandwidth allocation scheme based on the non-cooperative
game model, and presented an iterative algorithm to implement
our scheme.
i
= {x1
i,x2
,...,xm
i}. After we figure
IV. THE PERFORMANCE OF THE RESOURCE ALLOCATION
SCHEME
Based on the algorithm mentioned above, we study our
experiments in the MATLAB environment and focus on the
convergence of the Nash Equilibrium, which matters the
convergence of our scheme. In our experiments, we use the
topology in Figure 1, in which Link1 has low bandwidth as
well as small propagation-delay, and Link2 has high band-
width as well as large propagation-delay. Though the topology
we used here is simple, it can distinguish the preference of
the following two traffic classes: delay-sensitive traffic and
12
Fig. 1. Topology with two links and two nodes.
throughput-sensitive traffic. Furthermore, the authors in [8]
also used it to evaluate the performance of the architecture
proposed. Then we employ two VNs with different objectives
on our topology. Virtual Network 1 (V N1) is used to run
delay-sensitive traffic, and Virtual Network 2 (V N2) is used to
run throughout-sensitive traffic. It is clearly that V N1 prefers
Link1 and V N2 prefers Link2.
Following Ref. [15], V N1 tries to minimize average end-
to-end delay:
n
?
where lj is the propagation delay on link l, l0 = 1ms is
the fixed delay on every link, and l0exp(zk
queueing delay as a function of the link utilization. Following
Ref. [15], V N2 tries to minimize:
j=1
zk
j(lj+ l0exp(zk
j
yk
j
)),
(7)
j
j) stands for the
yk
n
?
j=1
log(zk
j) − q
n
?
j=1
exp(zk
j
yk
j
),
(8)
where V N2 is maximizing its utility as a logarithmic function
of its path rate, and q = 0.5 keeps a balance between maximal
utility and minimal congestion. Moreover, we define the utility
functions of these two VNs as follows:
U1(y1,z1) = −
2
?
j=1
z1
j(lj+ l0exp(z1
j
y1
j
)),
(9)
U2(y2,z2) =
2
?
j=1
log(z2
j) − q
2
?
j=1
exp(z2
j
y2
j
).
(10)
U1(y1,z1) and U2(y2,z2) are both convex functions, so ac-
cording to the Theorem 1 the Nash Equilibrium exists.
We set aggregate bandwidth requirement as 600Mbps
for both VNs, but the capacity of the physical network is
1000Mbps. Taking into account of the fairness, we assume
that each VN will be allocated 500Mbps in the end. Initially,
we set β1= 1 and β2= 0.8, which implies that Link1 is more
important. Furthermore, we set the price-weighed factor for
V N1 and V N2 as follows: w1= {1,10} and w2= {10,1}.
Then we used our algorithm to demonstrate the convergence
of the Nash Equilibrium.
Figure 2 illustrates the situation when congestion happens
on both links. The V N1 is initially assigned 130Mbps on
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12345678910
−50
0
50
100
150
Iteration Number
Resourc Allocation On Link 1
12345678910
400
420
440
460
480
500
Iteration Number
Resourc Allocation On Link 2


VN 1
VN 2
Fig. 2.
both links.
Convergence of the Nash Equilibrium when congestion happens on
12345678910
0
20
40
60
80
100
Iteration Number
Resourc Allocation On Link 1
12345678910
400
450
500
550
600
Iteration Number
Resourc Allocation On Link 2


VN 1
VN 2
Fig. 3. Convergence of the Nash Equilibrium when congestion happens only
on the second link.
Link1 and 470Mbps on Link2, while the V N2 is initially
assigned 110Mbps on Link1 and 490Mbps on Link2. This
setup leads to the congestion on both links. As seen in
Figure 2, after one iteration V N1 is allocated with all the
bandwidth on Link1, and also 400Mbps on Link2. This is
due to the large difference between the price-weight factor
and the delay properties of the two links. Meanwhile, V N2 is
assigned 500Mbps on Link2. The convergence of the Nash
Equilibrium is demonstrated and our algorithm can quickly
find the Nash Equilibrium Point. Similar behavior is observed
in Figure 3. We set the bandwidth assigned on V N1 to be
30Mbps, and V N2 is allocated with 90Mbps as the initial
state. Again our algorithm easily find the Nash Equilibrium
Point, and confirms the fact that the existence of the Nash
Equilibrium is independent of initial values.
V. CONCLUSION
In this paper, we introduce the economic incentives to
analyze the complicated interaction between InPs and SPs,
aiming to encourage the efficient behavior among multiple
VNs. As far as we know, we are the first to develop a case
for the application of Game Theory in network virtualization
environment. In our model, an InP and multiple SPs achieve
efficient bandwidth allocation scheme in the framework of
the non-cooperative game. We propose an algorithm to solve
the scheme above, and demonstrate the convergence and the
effectiveness of our scheme with the experiments based on
Figure 1. In future, we will focus on the model with multiple
InPs and SPs, in which SPs can obtain physical resources
from multiple InPs, leading to the competition among multiple
InPs. Of course, the model mentioned above requires more
complicated bandwidth allocation scheme.
ACKNOWLEDGMENT
This work is supported by National Major Scientific and
Technological Specialized Project
002-02),National Basic
2007CB310701)and National High Technology Research
and Development Program (No. 2008AA01Z107 and No.
2008AA01A331), PCSIRT and TNLIST.
(No. 2010ZX03004-
ProgramResearch (No.
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