Multidimensional Game Theory Based Resource Allocation Scheme for Network Virtualization

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1 Journal of Computational Information Systems 9: 9 (2013) Available at Multidimensional Game Theory Based Resource Allocation Scheme for Network Virtualization Xiaochuan SUN 1,, Hongyan CUI 1, Bin XU 2, Jianya CHEN 1, Yunjie LIU 1 1 Beijing Key Laboratory of Network System Architecture and Convergence, Beijing University of Posts and Telecommunications, Beijing , China 2 Hebei United University, Tangshan , China Abstract A major challenge in network virtualization environment is how to fairly and efficiently allocate the various sacred physical resources of infrastructure providers among multiple virtual networks. In this paper, we propose a resource allocation scheme based on multidimensional game theory for network virtualization. We introduce multidimensional game theory to virtual resource allocation, aimed at investigating the interactions among multiple VNs. Firstly, we formulate the payoff function of noncooperative VNs, consisting of the utility function, price function and congestion function. Secondly, in the non-cooperative multidimensional resource game scheme, we prove the existence and uniqueness of its multidimensional Nash equilibrium (NE), and then derive the NE solution. Finally, the experimental results show the effectiveness and convergence of our model, and validate the derived NE solution. Keywords: Resource Allocation; Network Virtualization; Multidimensional Game Theory 1 Introduction Currently, deploying new Internet services is being more and more difficult. The existence of multiple stakeholders with conflicting goals and policies, alterations, even necessary ones, does not allow radical changes of the Internet architecture. To overcome the current ossification of the Internet [1, 2], network virtualization has emerged as a new technology to provide versatile customized services over a shared substrate network. Moreover, it has been recognized as an integral part of the next-generation networking paradigm. In a network virtualization environment, the roles of the Internet Service Providers are divided into different entities [3, 4]: the infrastructure providers (InPs) and the service providers (SPs). Project supported by the National Basic Research Program of China (973 program)(no. 2012CB315805); the Fundamental Research Funds for the Central Universities, (No. 2009RC0124); the National Key Science and Technology Projects, (No. 2010ZX ), the National Natural Science Foundation of China, (No ). Corresponding author. address: [email protected] (Xiaochuan SUN) / Copyright 2013 Binary Information Press DOI: /jcis5819 May 1, 2013

2 3432 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) The InPs, allocating various physical resources (e.g. CPU, bandwidth, delay, storage), charge SPs in order to maximize benefits, while the SPs, creating virtual networks (VNs) to provide reliably end-to-end services, focus on how to gain enough resources for maximum benefits. Hence, network virtualization confronts with great challenges how to fairly and efficiently allocate physical resources among multiple virtual networks (VNs) [5, 6, 7, 8]. To tackle the problem, game theory is a natural and powerful tool, which studies how the SPs cooperate or compete with each other to suffice their own requirements. Essentially, it is a multidimensional game problem that multiple SPs (players) obtain multiple resources (strategies). Recently, many researchers have utilized game theory to handle the resource allocation problem in computer networks, especially wireless networks. Zhou etal. [9] proposed a bandwidth allocation scheme based on the non-cooperative game model to describe the interactions among multiple VNs. The authors in [10] proposed a cooperative game approach to Internet pricing, and demonstrated that the leadercfollower game may lead to a solution that is not Pareto optima. The authors in [11] present a non-cooperative game for concurrent instances of a virtual router in order to better allocate the router s physical resources, and use a market-based approach and game theory to maximize the utilization of router s resources. In [12], the authors proposed a non-cooperative game with strict incomplete information based on the energy-efficient criterion to handle the problem of resource allocation for the uplink of a CDMA wireless communication network. In [13], the authors gave an overview of game-theoretic approaches to energy-efficient resource allocation in wireless data networks. However, the above schemes are not able to consider multidimensional game situation [14], that is, multiple players play games in multiple areas simultaneously. In this paper, we propose a resource allocation scheme for network virtualization based on multidimensional game theory. We consider a model that consists of an InP and multiple SPs. It is generally known that the fairness and efficiency of resource allocation directly affect the performance of VNs, even influencing the whole performance of the network virtualization environment. Based on the argument, our research motivation is focus on how to allocate various limited resources among multiple VNs in the framework of multidimensional game theory. The main contributions of the paper are: We propose a resource allocation scheme based on non-cooperative multidimensional game theory in order to investigate the interactions among VNs. More importantly, we firstly apply the multidimensional game theory to the allocation of multiple resources in network virtualization environment. In the framework of multidimensional game, we formulate the payoff function of VNs, consisting of the utility function, price function and congestion function. Additionally, the InPs charge VNs according to the unified price for certain physical resource. We prove that our resource allocation scheme can achieve multidimensional Nash equilibrium. Furthermore, by the Lagrange multiplier method, we demonstrate the existence and uniqueness of the multidimensional Nash equilibrium, and derive the unique NE solution. Finally, the experimental simulation validates the theoretical result. The rest of the paper is organized as follows. In section 2, we model a resource allocation scheme based on non-cooperative multidimensional game theory in network virtualization environment. In section 3, we prove the existence and uniqueness of the multidimensional Nash equilibrium in our model. Section 4 shows the performance results from running the proposed multidimensional game model. Finally, section 5 gives the short conclusion about our work.

3 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) System Model In network virtualization environment, e.g. Global Environment for Network Innovations (GENI) [15], components are the primary building block of the architecture. For example, a component, corresponding to an edge computer, a customizable router, or a programmable access point, encapsulates a collection of resources (e.g. CPU, memory, disk, bandwidth, delay, etc). Hence, here comes the problem of how to allocate various physical resources in numerous VNs. This is essentially a multidimensional game problem that multiple VNs play games for different resource to obtain maximum benefits. Consider a physical network that consists of M different components, denoted by M = {1, 2,, m}(m 2). There exists N = {1, 2,, n}(n 2) different resources in every component. We use C = {c ij, 1 i m, 1 j n} to represent the capacity of the resource j in the component i. Especially, the scalar c ij = 0 means that the component j does not contain the resource i. Subsequently, we model our scheme as the following multidimensional non-cooperative game. The VNs, i.e. players, are denoted by V = {1, 2,, v}(v 2), which implies that v VNs coexist in our scheme. For the k-th VN, the amount of n resources allocated in m components is expressed as the M -dimensional vector (s 1,, s m ) T k = {(xk 11,, x k 1n); ; (x k m1,, x k mn)}, which naturally follows the restriction of 1 i m, 1 j n, and 1 k v, 0 x k ij c ij. Likewise, x k ij = 0 means that the k-th VN does not share the resource j in the component i. In our scheme, s k =(s 1,, s m ) T k is essentially the strategy space of the k-th VN. Based on the strategies, the v VNs compete with each other for the n limited physical resources, aiming at obtaining more resource to maximize their revenues. Moreover, it is an effective way to increase revenues that the VNs should avoid congestion on certain components as much as possible. Formally, we formulate the payoff function of k-th VN, including the three parts as follows. 1) We consider VN utility to be logarithmic similar to Ref. [16, 17]. In our scheme, the utility function U(s k ) of the k-th VN is defined by extending one-dimensional resource allocation to the M-dimensional case, yielding: U(s k )= m i=1 n αj k log (1+ j=1 Θ jx k ij x l ij ) (1) where Θ j is the decoupling factor for the resource j which is used to distinguish sufficiently the demands of different VNs (Θ j > 1), and αj k > 0 is the utility factor of the resource j for the k-th VN which converts utility units to currency. For all the VNs k = 1, 2,, v, we set x ij = v and ᾱ j = v αj, l respectively. l=1 x l ij l=1 2) The price function P (s k ) defines the total price that the k-th VN should pay for using the assigned resource, given by: m n P (s k )= βj k x k ij (2) i=1 where βj k denotes the price per unit of the resource j for the k-th VN. 3) The congestion function C(s k ) [9] defines the congestion cost due to resource overload of certain components for the k-th VN. If the amount of certain resource on certain component, j=1

4 3434 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) obtained by all VNs, is more than the resource capacity, the congestion cost will be charged. Accordingly, the congestion price is expressed as the following equation: m n C(s k ) = γj k [x k ij (c ij x l ij)] (3) i=1 j=1 where γj k denotes the congestion price per unit of the resource j for the k-th VN. x k ij (c ij x l ij) is used to determine whether to generate congestion, defined as follows: if x k ij< c ij c ij occurs. x l ij, then x k ij (c ij v v x l ij, then x k ij (c ij x l ij) = 0, which indicates that no congestion occurs; if x k ij v x l ij) = c ij v x l ij, which indicates that the congestion With the above three parts, the payoff function Ψ(s k ) of the k-th VN eventually takes the following form: Ψ(s k ) = U(s k ) P (s k ) C(s k ) = m n αj k log (1+ Θ jx k ij ) m n β x l j k x k ij ij (4) m n γj k [x k ij (c ij v x l ij)] 3 Analysis of Multidimensional Nash Equilibrium In this section, we propose the definition of multidimensional Nash equilibrium (MNE)for virtual resource allocation, and prove the existence and uniqueness of MNE. Moreover, we give the MNE solutions for unified price scheme of the InP. 3.1 MNE In our multidimensional non-cooperation game model,the k-th VN s optimization problem is to maximize its payoff. Hence, its objective is to find M-dimensional vector s k = (s 1,, s m) T k that solves the following problem: max Ψ(s k) (5) s k >0 In other words, in pursuing the solution for the above multidimensional game, we have to compute the Multidimensional Nash Equilibrium Point (MNEP). Each VN have incentive to reach its MNEP as far as possible, aiming to maximize its payoff, as stated in the following definition. Definition 1 (Multidimensional Nash Equilibrium Point) Let the vector s k be a solution of the problem (5) for the k-th VN. s k = (s 1,, s m) T k is the MNEP for the multidimensional game if for any s k, we have Ψ(s k) Ψ(s k), k = 1, 2,, v (6) where s k is the set of resource strategies selected by the k-th VN except the MNEP s k.

5 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) Existence and uniqueness of MNE So far we have defined the MNEP, and analyzed how to solve it. Subsequently, we prove the existence and uniqueness of multidimensional Nash equilibrium in Theorem 1. Theorem 1 For multidimensional resource allocation in virtualized environments, there exists a multidimensional Nash equilibrium in Equation (4), and its solution is unique. Proof In this case, we consider that the InP charges VNs at unified prices, i.e. for k = 1, 2,, v, β k j = β j and γ k j = γ j. We use Lagrange multiplier method [18] to optimize the function (5), yielding: L = m m n αj k log(1+ Θ jx k ij x k ij v n γj k [x k ij (c ij λ( v x l ij c ij ) l=1 ) m x l ij)] n βj k x k ij where λ is the Lagrange multiplier. For the k-th VN, we take the first order derivative of L with respect to the resource j on the component i, i.e. x k ij, yielding: for j = 1, 2,, n, (7) L x k ij = Θ j αj k x l ij + Θ jx k ij β j γ j λ (8) By solving L x k ij = 0, we have: x k ij = Θ j Θ j 1 α k j β j + γ j + λ 1 Θ j 1 x ij (9) Moreover, we calculate the sum of x k ij(k = 1, 2,, v) over all the VNs, obtaining: x k ij = k=1 which gives the following equation: x ij = Θ j Θ j 1 k=1 After some manipulation, we deduce that α k j β j + γ j + λ 1 Θ j 1 x ij (10) Θ j ᾱ j Θ j 1 β j + γ j + λ v Θ j 1 x ij (11) k=1 x ij = Θ j ᾱ j Θ j + v 1 β j + γ j + λ (12) which is substituted into Eq.(9), obtaining: x k ij = Θ j 1 ᾱ j Θ j 1 β j + γ j + λ (αk j Θ j + v 1 ) (13)

6 3436 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) Additionally, we also calculate the first order derivative of L with respect to Lagrange multiplier λ as follows: L λ = 0 x ij = c ij (14) and then from Eq.(12) and for λ 0, we obtain: λ = Θ j ᾱ j γ j β j (15) Θ j + v 1 c ij We observe that λ is irrelevant for the variable x k ij. Hence, the second partial derivatives with respect to the x k ij are given by: (x k ij )2 (x = Θ 2 jαj k k ij )2 2 (16) ( x l ij + Θ jx k ij ) and based on the Young s Theorem, the mixed second order partial derivatives x k ij xk ij are given by: x k ij xk ij = 2 L x k ij x k ij x k ij xk ij and = 0 (j j) (17) To test the second order optimality condition, consider the N N Hessian matrix H of the payoff function Ψ(s k ) with the constraint (see (5)) for the component i, written as: H = = (x k i1 )2 x k i2 xk i1 x k i1 xk i2 x k in xk i1 x k i1 xk in 2 L (x k i2 )2 x k i2 xk in... x k in xk i2 (x k in )2 Θ 2 j αk j ( x l i1 +Θ jx k 2 0 i1 )... 0 Θ 2 j αk j ( x l in +Θ jx k 2 in ) Existence : From the Eq.(18), observe that the Hessian matrix H is a diagonal matrix in which all the elements of the principal diagonal are negative. Hence, all n-order leading principal minors has the following characteristics: (18) ( 1) k H k > 0, k = 1, 2,, n (19) where those k-order leading principal minors are negative if k is odd and positive if k is even.

7 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) The authors in [19] proposed the second-order sufficient condition for extremum. That is, for a given objective function f(x), if all leading principal minors of its corresponding Hessian matrix are positive, then f(x) obtains a minimum; if they duly alternate in signs, the first one being negative, then f(x) obtains a maximum. Hence, according to Eq.(19), Ψ(s k ) has the maximum for the component i (i = 1, 2,, m). It means that there exists m NEs on m components, which constitutes a m-dimensional NE. Uniqueness : In [20], Cachon and Netessine proposed the following two theorems to prove the uniqueness of Nash equilibrium (NE). Theorem 2 If the best response mapping is a contraction on the entire strategy space, there is unique NE in the game. Theorem 3 The mapping f(x), R n R n is a contraction if and only if the Hessian matrix of f(x) is the diagonally dominant matrix, that is: n 2 f(x) x i x k < 2 f(x), k = 1, 2,, n (20) i=1, i k x 2 k From Eq.(16)-(17), we can clearly observe that for i = 1, 2,, m, n x k ij xk ij < j=1,j j (x k ij )2 (21) which indicates that the Hessian matrix H satisfies diagonally dominant condition. Hence, from Theorem 3, we can prove that the payoff function Ψ(s k ), subjecting to the constraint given in (5), is a contraction on the entire strategy space of the component i. According to Theorem 2, there exists a unique NE on the component i. Moreover, substituting Eq.(15) into Eq.(13), we obtain the unique NE solution on the component i, given by: for j = 1, 2,, n, x k ij = c ij Θ j 1 Θ j + v 1 (αj k ᾱ j ᾱ j Θ j + v 1 ) (22) For all the components i (i = 1, 2,, m), due to independent among m component, there exists a unique m-dimensional NE, consisting of m unique NEs on m different components. 4 Simulation Results In this section, we evaluate the proposed multidimensional game model (MGM) for virtual resource allocation to provide better understanding of VNs behaviors to achieve maximum payoffs. Moreover, we compare it with the game theory model (GTM) [9] on bandwidth allocation. The models are simulated numerically in the MATLAB environment.we focus on the convergence of the multidimensional NE as regards our model, and investigate the payoffs of VNs with various price parameters. In our experiment, we consider the network topology with four components in Figure 1. Component 1 and 2 (e.g. network links) have bandwidth and delay, while component 3 and 4 (e.g. server)have CPU and storage. Furthermore, we employ two VNs to evaluate the performance of the proposed model on our topology, which implies that both VNs play 4-dimensional game, and every dimension includes two resources.

8 3438 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) Table 1: The list of parameters for the multidimensional game model in both price cases Resource Bandwidth Delay CPU Storage VN VN1 Parameter VN2 VN1 VN2 VN1 VN2 VN1 VN2 Θ α 20 $ 10 $ 5 $ 6 $ 10 $ 15 $ 9 $ 11 $ β 0.05 $/Mbps 0.08 $/ms $/MIP S/GHz 0.03 $/GB γ 0.02 $/Mbps $/MIP S/GHz 0.01 $/GB Component 1 200Mbps, 10ms Component 3 Component MIPS/GHz 500Gb MIPS/GHz 100Gb 800Mbps, 15ms Component 2 Fig. 1: Network topology with four components To evaluate the feasibility of the proposed model, we consider the most complicated case, that is, congestion occurs in every components. Hence, the Eq.(4) can be rewritten as: Ψ(s k ) = U(s k ) P (s k ) C(s k ) = m n αj k log (1+ Θ jx k ij ) m m n γ j (c ij x l ij l=1, v l k x l ij) n β j x k ij (23) We set bandwidth requirement to be 600Mbps for both VNs, but the total capacity for bandwidth is 1000Mbps in the component 1 and 2, which leads to congestion. We assume that if the bandwidth of certain component is required, the corresponding delay is also obtained. Likewise, we set CPU and storage requirements to be MIP S/GHz (MIPS: Million Instructions Per Second) and 350GB for both VNs, respectively, which also leads to congestions. Moreover, as shown in Table 1, we set decoupling factor Θ and price parameters α, β, γ for V N1 and V N2. Initially, the VN1 is allocated 210Mbps on component1, 390Mbps on component2, MIP S/GHz and 250GB on component 3, MIP S/GHz and 60GB on component 4, while the VN2 is allocated 130M bps on component1, 470M bps on component2, MIP S/GHz and 260GB on component 3, MIP S/GHz and 80GB on component 4.

9 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) Bandwidth allocation (Mbps) Mbps 140Mbps 60Mbps 0Mbps VN1 for MGM VN2 for MGM VN1 for GTM VN2 for GTM Iteration number Bandwidth allocation on component 2 (Mbps) Mbps 550Mbps 250Mbps VN1 for MGM VN2 for MGM VN1 for GTM VN2 for GTM 200Mbps Iteration number (a) Component 1 (b) Component 2 Fig. 2: Bandwidth allocation for MGM and GTM in unique prices case. CPU allocation on component 3 (MIPS/GHz) 1.6 x (12350MIPS/GHz, 308GB) (7650MIPS/GHz, 192GB) CPU allocation for VN1 Storage allocation for VN1 CPU allocation for VN2 Storage allocation for VN Iteration number Storage allocation on component 3(GB) CPU allocation on component 4 (MIPS/GHz) (6150MIPS/GHz, 60GB) CPU allocation for VN1 Storage allocation for VN1 CPU allocation for VN2 Storage allocation for VN2 (3850MIPS/GHz, 40GB) Iteration number Storage allocation on component 4 (GB) (a) Component 3 (b) Component 4 Fig. 3: CPU and storage allocation on for MGM in unique prices case. We firstly consider the MNE situations of MGM on four components for the unique prices case. Fig.2(a)-2(b) show bandwidth allocation of MGM and GTM on component 1 and 2, respectively. It should be noted that after one iteration, VN1 and VN2 in both models achieve NEs. In Fig.2(a), the NEPs of VN1 and VN2 for MGM are (140Mbps, 10ms) and (60Mbps, 10ms) on component 1, approximate to the theoretical value (138M bps, 10ms) and (62M bps, 10ms) (from Eq.(22)), while for GTM, the NEPs of VN1 and VN2 are (200Mbps, 10ms) and (0Mbps, 10ms). It implies that compared with the GTM, the proposed MGM can more fairly allocate bandwidth, which helps keep network reliability. The similar situation appears on component 2, as shown in Fig.2(b). Fig.3(a)-3(b) show CPU and storage allocation of VN1 and VN2 for MGM on component 3 and 4. It is also observed that after one iteration, VN1 and VN2 achieve NEs, and the corresponding NEPs are marked in Fig.3(a)-3(b). By computation (from Eq.(22)), we obtain the NE solutions of VN1 and VN2, i.e., ( MIP S/GHz, 188.9GB) and ( MIP S/GHz, 311.1GB) on component 3, ( MIP S/GHz, 37.8GB) and ( MIP S/GHz, 62.2GB) on component 4. We see that there is a close agreement between the simulation results and theoretical values.

10 3440 X. Sun et al. /Journal of Computational Information Systems 9: 9 (2013) VN1 VN2 140 Total payoff ($) Iteration number Fig. 4: Total payoffs for VN1 and VN2 Fig.4 shows the total payoffs of VN1 and VN2 versus iteration number. It is seen from the figure that when achieving NEs on component 3 and 4 after one iteration, both VNs obtain maximum payoffs. 5 Conclusions This paper presented a resource allocation scheme based on multidimensional game theory in network virtualization environment. Using the tools of game theory, this problem was recognized as a non-cooperative multidimensional game, in which the VNs compete with each other for multiple resources so as to maximize their payoff functions. We formulate the payoff function of VNs and prove the existence and uniqueness of MNE. Furthermore, when the InPs charge different VNs for certain resource at the unified price scheme, we derive the unique NE solution. Finally, through the experiment and analysis, we show the effectiveness and convergence of NE for our model. In future, we will focus on how to model multidimensional game based resource allocation scheme when the InPs charge different VNs at differentiated price. References [1] Anderson T, Peterson L, Shenker S, Turner J, Overcoming the Internet impasse through virtualization, Computer, vol. 38, no. 4, 2005, pp [2] Chowdhury N M M K, Boutaba R, Network virtualization: state of the art and research challenges, Communications Magazine, IEEE, vol. 47, no. 7, 2009, pp [3] Turner J S, Taylor D E, Diversifying the Internet, Global Telecommunications Conference, GLOBECOM 05. IEEE, 2005, pp [4] Jeamster N, Gao L, Rexford J, How to lease the Internet in your spare time, ACM SIGCOMM Computer Communication Review, vol. 37, no. 1, 2007, pp [5] Yu M, Yi Y, Rexford J, Chiang M, Rethinking Virtual Network Embedding: Substrate Support for Path Splitting and Migration, ACM SIGCOMM Computer Communication Review, vol. 38,

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