Cloud Networking Mean Field Games - IEEE Xplore

2012 IEEE 1st International Conference on Cloud Networking (CLOUDNET)

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Cloud Networking Mean Field Games Ahmed Farhan Hanif † , Hamidou Tembine ‡ , Mohamad Assaad ‡ , Djamal Zeghlache † Institut Mines-Télécom, Télécom SudParis, UMR CNRS 5157, RS2M Department, France ‡ Telecommunications Department, Ecole Superieure d’Electricite (Supelec), France {ahmedfarhan.hanif,djamal.zeghlache}@it-sudparis.eu,{hamidou.tembine,mohamad.assaad}@supelec.fr †

Abstract—In this paper we analyze a distributed resource sharing problem for cloud networking. Each user would like to maximize a given payoff based on its demand and the total demand on the cloud. The problem is formulated as a game where the action of each player is represented by its requested demand. We develop a distributed algorithm for each node which only requires mean demand from the cloud to update its respective demand, thus reducing overhead. We also prove the convergence of our algorithm to Nash equilibrium. For large scale systems, we analyze the performance for ‘selfish’ and ‘social’ user strategies with symmetric price, and present a non feedback based distributed algorithm. We compare the performance of our algorithm with existing algorithms. Finally we present numerical results which compares the convergence of feedback vs non feedback algorithms. Index Terms—Cloud Networking, Resource Sharing, Mean field game

I. I NTRODUCTION Foster (2008) defines the ambiguous cloud as “A largescale of distributed computing paradigm that is driven by economies of scale, in which a pool of abstracted virtualized, dynamically-scalable, managed computing power, storage, platforms, and services are delivered on demand to external customers over the Internet” [1]. The concept of cloud computing has emerged to satisfy the uneven demand of users wanting cheap computing resources ranging from storage to processing. Every aspect of a computer system can essentially be offered as a service to various users. Thanks to advances in virtualization and scheduling techniques it is possible to have multiple instances of software running on the same physical system. Amazon EC2 [2] in one of the leading providers of computing resources. Cloud networking has some new challenges compared to the classical OSI system. The cloud network layer includes web applications (in the cloud) which makes an additional component compared to the traditional OSI system where the network is layer 3 and application is layer 7. Cloud networking is not a separate layer. It goes belong the boundaries and couples with IaaS (infrastructure as a service) and SaaS (software as a service). Cloud networking aims at reducing energy consumption cost leading to a green cloud. However, traditional approaches to energy-efficiency such as sleep mode cannot be used for the cloud since severs have to be available 24 hours and 7 days a week. In cloud networking there are several important problems that need to be addressed. The research of Ahmed Farhan Hanif and Djamal Zeghlache has received partial funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement SACRA n 249060.

978-1-4673-2798-5/12/$31.00 ©2012 IEEE

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In this paper we consider a resource sharing problem, where several users try to access the computing resources of the cloud. In the context of cloud networking a resource could mean CPU computing power, storage space, access to content or access to a specialized service e.t.c. The main concept of cloud computing is offering software as a service (SaaS). There are several processor intensive softwares which require higher processing power and it is beneficial for users to have on demand access to computing resources. In cloud networking there are hundreds of nodes and it is difficult to do central resource optimization as the each user node behaves independently. An efficient way would be to have a distributed algorithm running on each of these nodes such that each node is able to make decisions for itself in order to best manage its demand and maximize its rewards. For large scale distributed scenarios, it can also be useful to incorporate the concepts from Mean Field games. The aggregate nature of the resource sharing problems suggests an adapted tool that is able to exploit the structure of the problems. Here the aggregative terms are the congestion levels and the total demand of the clients. Mean field games are well-adapted for both finite (but large) and infinite systems. In contrast to the other classical tools for large-scale systems, the Mean Field approach incorporates the dynamics which allows online demand management (payas-you-use scheme). It allows optimization and studies the interactive nature of cloud networking, leading to strategic decision-making problems (so-called games). Recent studies in cloud networking have already adopted a Mean Field regime. We construct a distributed resource sharing framework for large scale systems using a generic reward function. This reward can be modeled to work for various type of above mentioned services by varying certain design parameters. A. Related Work There have been several works dealing with cloud computing and cloud networking and most of them are quite recent. We only refer to some of them here. Game theoretic tools for cloud networking have been recently proposed in [3]. The authors in [3] study cloud resource allocation games. The concept of cloud computing is very old but it has recently become a reality. There are several aspects of cloud computing that still need to be discussed. In [4] Wei et.al. present a game-theoretic method of fair resource allocation for cloud computing services. In [5] Ma et.al. present a resource allocation for cloud users using Kelly Approach. In [6] Kantere et.al. study optimal service


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pricing for cloud cache services. In [7] Yehuda et.al. study the framework of Resource-as-a-Service (RaaS). In [8] Khan et.al. study non-cooperative, semi-cooperative, and cooperative games-based grid resource allocation. Most of these works assume a system model where each user has complete information about the total demand of each server, which is not true in practice due to various limitations centering around privacy and security. B. Contribution The main contributions of this paper are as follows. We present a framework where users compete (i.e. selfish behavious) for cloud computing resources. A distributed iterative algorithm is presented which converges to Nash equilibrium. The numerical results corroborate the above claims. We cover the cases where each user has same and different prices. We then develop an iterative learning algorithm for large scale systems using Mean Field approach and prove its convergence to the Nash equilibrium. The Mean Field framework is then extended to the case of a social user and the difference between the performance of a social v.s. selfish user is considered. The convergence proofs for all the presented algorithms are contained in the paper. The rest of the paper is organized as follows. Section II states the problem. Section III present the Nash equilibrium characterization and the proposed algorithm that converges to the Nash equilibrium. In Section IV we present the analysis for Mean Field scenario, in Section V we extend our reward for Mean Field scenario to the social users case. Section VI contains the numerical results. In Section VII we discuss the generality of the results presented and Section VIII concludes the paper.

user can access all the resources at one time. It is important to mention here that the set of active users (i.e. a set of users {j|xj > 0}) need to be at least two for the reward to make sense. Fig 1 shows a cloud network where each user has access to some information from the cloud and is able to update its respective action xj according to some update equation. Later we will consider cases regarding the type of information supplied from the cloud, and if such information is really necessary for the update. We are constructing this problem in abstract terms with fixed number of users, all competing for the same Cn resources from the same cloud. There can be scenarios where different users can turn on and off, also there can be multiple clouds providing resources at competitive rates. Such scenarios are not considered here.

C , pj Rn

R1 R2 n R3

r0

U1

Let there be n ≥ 2 users all seeking resources from a cloud network with total capacity Cn . The payoff for user j is given by rj (x) where xj ≥ 0 is the action/demand of user j. pj is the price per unit of action/demand of user j.

U2

r2 x2 xj x0 x1rj rn

U3

...

xn

Un

Fig. 1: Cloud Networking Scenario The interactive strategic decision-making problem between the users can be formulated as follows: sup xj ≥0

II. P ROBLEM F ORMULATION

r1

rj (x), ∀ j ∈ N , {1, . . . , n}

(2)

It should be mentioned here that the above problem formulation represents selfish user behaviors where each user wants to maximize its respective reward. Hence, the problem has multiple objectives (each user has its objective). It is also important to mention here that the demand from each user xj ≥ 0 as there is no concept of negative demand.

% of total capacity

rj (x) = | {z } payoff

Cn |{z}

available capacity

z

}| { Cost for user j z }| { xj − xj pj D |{z} |{z} |{z}

Total demand

III. E QUILIBRIUM (1)

demand price

Where x = (x1P , . . . , xn ) ∈ [0, +∞)n is the demand profile. n Let D , ǫn + k=1 xk and ǫn > 0 is a reservation price parameter necessary to keep the denominator greater than zero. Cn > 0 is a constant which represents available capacity. It is clear from the expression of rj (x) that the reward of user j depends on the total demand of all the users. So this is an interactive strategic decision-making problem where all users compete for the same resources and try to maximize their respective rewards in the process. The reward rj (.) of each user depends on the percentage of total capacity Cn it is demanding minus the cost that it has to pay for those resources. An alternate interpretation of xj /D could be the probability of getting access to Cn , if only one

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AND

A LGORITHMS

In this section, we characterize the Nash equilibrium and find an iterative distributed algorithms that converges to the Nash equilibrium. A. Equilibrium Characterization The solutions of the problem (2) are called pure Nash equilibrium of the game defined by the triplet G , (N , X , rj )j∈N , where X , R+ represents the action space i.e. xj ∈ X . We seek the Nash equilibrium of the game G. We first start with the existence problem. For each player j the reward function is concave in its decision variable, i.e., xj 7−→ rj (x) is concave. From the expression of rj (.) we can see that very high demand will be suboptimal as it will result in a negative reward. This implies that the user’s optimal response will be in a compact set. Thus, the existence of an equilibrium follows


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by continuity (see [9] for more details). Such equilibrium are called best-response strategies. The next Lemma provides a generic expression of the best-response of user j given the decisions of the other users. Lemma 1. The best-response of user j given the other users’ demand profile x−j = (x1 , . . . , xj−1 , xj+1 , . . . , xn ), is given by the formula s !! Cn ∗ D−j − D−j , (3) xj = max 0, pj where D−j , ǫn +

Pn

k6=j xk .

Proof of Lemma 1 follows the standard optimality of rj (.). In the next subsection we develop distributed iterative algorithms for seeking Nash equilibrium. B. Algorithms To solve the system of equations in Lemma 1 we develop a distributed iterative algorithm [10]. It is useful to have an iterative algorithm as the best-response in Lemma 1 is nonlinear, and we are dealing with a distributed scenario. 1) Banach-Picard Algorithm: One of the first iterative scheme for finding fixed-point is the so-called BanachPicard iterate [11]. It consists of iterating a function many times starting from an arbitrarily point x0 in the domain. xk+1 = f (xk ) = f (f (xk−1 )) = . . . = (f of o . . . of )(x0 ). {z } | k times

Here f (.) represents the best-response where f (x) = [f1 (x), . . . , fn (x)]T . For different initial condition xj,0 and price pj > 0 from (3) we can write the update equation for user j as

xj,k+1 = fj (xk ) (4) q Cn where fj (xk ) = max 0, . The pj D−j,k − D−j,k equation in (4) is called a Banach-Picard update equation. Unfortunately the update Equation in (4) does not converge to the optimal solutions, as can be seen from the plots in Fig 3, because fj (x) is not monotone and is not a contraction. For these reasons, we propose an Ishikawa based update algorithm [12] which is presented in the next section. 2) Ishikawa Algorithm: We slightly modify the update equations and introduce a small parameter λ were λ ∈ [0, 1], where we weigh the update between the new and the old value. This type of equations is called Ishikawa Update Equation which can be written as xk+1 = f (xk )λ + xk (1 − λ)

(5)

Note that if x∗ is a fixed-point of f i.e, f (x∗ ) = x∗ then f (xk )λ + xk (1 − λ) = x∗ λ + x∗ (1 − λ) = x∗ . Then x∗ is a fixed point of (5). ¯ < 1 sufficiently Theorem 1. There exists learning rate 0 < λ ¯ small such that for all 0 < λ < λ, the Ishikawa algorithm converges to a pure Nash equilibrium. The proof follows from [13].

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¯ in order to It should be noted that we need the value of λ ¯ which depends on the structure implement (5) where λ < λ, of the reward function and the number of users. In general ¯ therefore to ensure convergence to it is hard to compute λ, 1 Nash equilibrium, in this paper we use a vanishing λk = k+1 ¯ However, the convergence proof which is independent of λ. of Theorem 1 from [13] does not hold in this case. Therefore, we prove the convergence of (5) to the Nash equilibrium for vanishing λk in the Appendix. IV. M EAN F IELD A SYMPTOTICS In this section we analyze (3) in the context of large systems and construct an algorithm that doesn’t require feedback but is still able to converge to Nash equilibrium. Let Gmf , (N , X , rmf (.)) define a Mean Field Game. From here on we will be considering same price for all users i.e. p = pj ∀j. The reward for large scale system from (1) can be written in terms of the mean as Cx ˆj rmf (ˆ xj , m) = −x ˆj p (6) m+ǫ Pn where m , limn−→+∞ n1 k=1 x ˆk and ǫ , limn−→+∞ ǫnn . rjmf (.) is a linear function because of the influence of xˆj in m is negligible as n −→ +∞, hence by differentiating rjmf (ˆ xj , m) w.r.t xˆj and putting it equal to zero, m∗s , Cp − ǫ, where m∗s represent the mean for same price. Let f mf(.) represents the best Mean Field response to Gmf , with only the knowledge of Cp − ǫ. f mf(.) can be interpreted as a limit of f (.) function.  ˆj < Cp − ǫ  M, x mf f (ˆ xj ) = 0, (7) x ˆj > Cp − ǫ  ´ C M, x ˆj = − ǫ p

´ represents any value between 0 < M ´ < M. (7) Where M means that when x ˆj,k < Cp −ǫ, increase demand to some large number M > Cp and when xˆj,k > Cp − ǫ, reduce demand to zero. We can write the update algorithm from (5) using (7) as x ˆj,k+1 = f mf(ˆ xj,k )λ + x ˆj,k (1 − λ)

(8)

We will show in Lemma 2 that the algorithm in (8) converges to the trajectory of algorithm given in (5). Here we use a 1 vanishing lambda where λk = k+1 which gives us much better convergence to the fixed point as compared to fixed λ. The ODE (ordinary differential equation) for the above iterative algorithm can be written as x ˆ˙ t = f mf(ˆ xt ) − x ˆt

(9)

Lemma 2. MF Trajectory: The gap between the trajectory of the ODE xˆ˙ t and the optimal solution xˆ∗ is |ˆ xt − x ˆ∗ | vanishes as time goes to ∞ i.e. lim |ˆ xt − x ˆ∗ | = 0

t−→∞

(10)

Proof: Lemma 2 For x ˆ0 ∈ (0, M ), V (ˆ x) = |ˆ xt − xˆ∗ |2 ≥ ∗ ˙ ∗ mf ˙ 0, V (ˆ x) = 2(ˆ xt − x ˆ )(x ˆt ) = 2(ˆ xt − x ˆ )(f (ˆ xt ) − x ˆt ). From here we can argue that V˙ (ˆ x) ≤ 0 when x ˆt > x ˆ∗ and when


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x ˆt < x ˆ∗ , V (ˆ x∗ ) = 0, hence xˆ∗ is a stable equilibrium point of ODE in (9). In the next section we provide analysis for the social scenario were each user tries to maximize the mean reward for p = qpj . The gap between the finite and the infinite regime Cn p ∗ is x ( (n−1)x − p) which goes to zero as n −→ +∞. ∗ +ǫ n

for the same price scenario where p = 11. It can be clearly seen that all the users converge to the same demand and each has the same reward. The dotted lines in Fig 2 and Fig 3 shows the trajectory of using Banach-Picard algorithm and the solid lines are for the Ishikawa Iteration. It is clear from the plots that Banach-Picard algorithm dose not converge to a fixed point and oscillate around the Nash equilibrium in case of same price. The legend contains the price paid by V. S OCIAL U SERS :M EAN -F IELD G LOBAL O PTIMIZATION each user. Fig 4 presents the plot for the evolution of the All the work developed up to this point has been for selfish demand x for n = 100 users for the same price scenario j users where each user is trying to maximize their own reward. where p = 11. The red line shows the MeanField limit. The Here we will develop an optimization over the mean reward plots are generated using Our MeanField Algorithm in (8) function for large number of users to achieve a higher social (represented by the dotted lines) and Ishikawa algorithm in welfare for p = pj and m = xˆj . Lets write the average reward (5) represented by the solid line. It can be clearly seen that function rmfs (m) as, both trajectories converge to the same point which is given by Cp − ǫ. Our proposed MeanField algorithm converges to Z +∞ n X 1 the same point, and requires little (mean) or no information rmfs (m) = lim rj (.) = rjmf (ˆ x, m)µ(dˆ xj ) (11) exchange (in case of same starting point) to converge to the n−→+∞ n 0 j=1 same equilibrium point. Where, µ(.) is the distribution of xˆj . using (6) in (11) we get Simulation parameters are contained in the title of each plot. Cm mfs r (m) = − mp (12) Ishikawa and Banach Picard>Demand for C =400, N=4, λ=0.25,ε=0.1 m+ǫ 10 n

p1=18.1472

rmfs (m),

sup

8 demand xj

The social resource sharing problem for large scale systems can be written as

p2=19.0579 p3=11.2699

6

p4=19.1338

4 2

(13)

0

m≥0

0

50

100

150

Number of Iternations k

2 mfs optimal point of (13) it needs to satisfy ∂m (m∗mfs ) < 0 2r 2 mfs ∗ ∗ −3 which is ∂m2 r (mmfs ) = −2(ǫ + mmfs) ǫ < 0, thus satisfying the condition. Hence m∗mfs is the global maximizer for large scale systems. q Putting the value of m∗mfs back in (12)

we get rmfs (m∗mfs ) = (

Cǫ p

− ǫ)( qCCǫ − p) which goes to C

Ishikawa and Banach Picard>Reward for Cn=400, N=4, λ=0.25,ε=0.1 200 p1=18.1472 150 reward rj

In order to solve the problem in (13) the following two conditions should be satisfied i.e. the ∂m rmfs (m) = 0, 2 mfs ∂m r (m) < 0. Taking the firstq partial derivative and putting ∗ ∗ it equal to zero we get mmfs = Cǫ p − ǫ. For mmfs to be an

p2=19.0579 p3=11.2699

100

p4=19.1338

50 0 −50

0

50

100

150


Fig. 2: Ishikawa and Banach-Picard Iteration for Different Price

p

lim |rmfs (m∗mfs ) − rjmf (m∗s )| = C

ǫ−→0

Ishikawa and Banach Picard>Demand for Cn=400, N=4, λ=0.25,ε=0.1 10 p1=11 8 demand xj

when ǫ −→ 0. Even for a small ǫ, rmfs (m∗mfs ) ≃ C. Now let us compare the two performance of social user (from equation 12) v.s. selfish user(from equation 2). We note that m∗mfs < m∗s if ǫ < Cp , also rjmf (m∗s ) = 0 but q rmfs (m∗mfs ) > 0, |m∗s − m∗mfs | = Cp − Cǫ p

p2=11 p3=11

6

p4=11

4 2

(14)

0

0

50

100

150

Number of Iternations k Ishikawa and Banach Picard>Reward for C =400, N=4, λ=0.25,ε=0.1 n

100 p1=11 80

Fig 2 presents the plot for the evolution of the demand xj for n = 4 users. Each of them is charged a different price by the cloud. It can be clearly seen that the user who is charged the highest price has the lowest demand and its reward eventually goes to zero. The one who is charged the lowest price has the highest reward and has the highest demand. Fig 3 presents the plot for the evolution of the demand xj for n = 4 users

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reward rj

VI. N UMERICAL R ESULTS

p =11 2

p3=11

60

p4=11

40 20 0

0

50

100

150


Fig. 3: Ishikawa and Banach-Picard Iteration for the Same Price


5

Reward for Cn=10000, N=100, λ=0.25,ε=0.1,p=11

Taking limit such that λ −→ 0 we get

60 Ishikawa Algorithm Our Meanfield Algorithm Equilibrium Point

demand xj

50

x˙ j,t = fj (xt ) − xj,t = gj (xt )

Where x˙ j,t = dxj,t /dt. Here we can see that the trajectory of the ODE in (15) represents the solution of (3). This shows that our algorithm converges to x∗ as a stationary point.

30 20 10 0

0

50

100

150

Number of Iternations k Reward for Cn=10000, N=100, λ=0.25,ε=0.1,p=11 100 Ishikawa Algorithm Our Meanfield Algorithm Equilibrium Point

0

j

−100 reward r

(15)

40

−200 −300 −400 −500 −600

0

50

100

150


Fig. 4: M=N,Our MeanField Iteration for the Same Price N = 100 VII. D ISCUSSION The method presented for characterization of equilibrium can be extended to other type of reward function for other applications. The best-response fixed point algorithm developed in Section IV is more general and has applications to other type of reward functions, for example if r(x) is unknown or is a complicated expression which cannot be explicitly solved by each node due to computational limitations. In cloud networks the service provider may not want to share the information about the internal structure of the reward function due to several privacy/policy limitations. In such scenarios it is convenient to just share the mean at each time with all the users and all users try to converge to this mean, by updating their actions based on the best-response Equation in (8). The reward functions and the methodology considered in this paper is very generic and can be applied to other type of resource sharing problems. VIII. C ONCLUSION In this paper we present an iterative algorithm which is able to perform distributed resource sharing (for same and different prices) for a certain type of reward function. We show the stationary points of our algorithms converge to Nash equilibrium. We also develop a distributed iterative algorithm for large scale systems using Mean Field theory and apply them to finite scale systems. In large scale systems for the same price we were able to show that the reward for a social users in better than a selfish user. A PPENDIX Here we present the convergence of the algorithm in (5) for vanishing lambda to an ODE (ordinary differential equation) and then we show that this ODE converges to the stable equilibrium point. A. Convergence to ODE Here we will show that as λ becomes smaller the algorithm in (5) converges to the optimal solution presented by (3).

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B. Stability of Equilibrium We also need to show that the ODE in (15) has a stable equilibrium point at x∗ . Let J be the Jacobian matrix of g(x) evaluated at x = x∗ , where g(x) = [g1 (x), . . . , gn (x)]T . We can write the elements of the Jacobian matrix J as Jjj = ∂xj gj (x∗ ), andJjj ′ = ∂x′j gj (x∗ ), j ′ 6= j. Takingq the partial derivative of gj (x) w.r.t xj and evaluating ∗ = D ∗ at equilibrium) we get ∂ g (x∗ ) = it at x∗ ( Cpjn D−j xj j −1, Similarly taking the partial derivative of gj (x) w.r.t xj ′ and evaluating it at x∗ we get ( Cn , Cpjn > D−j ∗ ∗ ∂xj′ gj (x ) = 2D pj Cn 0, pj ≤ D−j For x∗ to be a stable equilibrium point, the eigenvalues of J should have negative real parts which is equivalent to proving yJy T < 0 ∀ y ∈ R1×n /0. For the same P price pj = p∀j P where Cn T 2 2 β , 2D ∗ p , we have y Jy = −(1 − β)( j yj ) − β j yj ≤ Cn ∗ 0. If 0 < β < 1 i.e. 2D∗ p < 1, hence x is a stable equilibrium point. This completes the proof. R EFERENCES [1] I. Foster, Y. Zhao, I. Raicu, and S. Lu, “Cloud computing and grid computing 360-degree compared,” in Grid Computing Environments Workshop, 2008. GCE ’08, nov. 2008, pp. 1 –10. [2] A. Inc, Amazon Elastic Compute Cloud (Amazon EC2). http://aws.amazon.com/ec2/#pricing: Amazon Inc., 2008. [Online]. Available: http://aws.amazon.com/ec2/#pricing [3] I. G. M. C. Virajith Jalaparti, Giang Nguyen, “Cloud resource allocation games,” Dept of Computer Science, Tech. Rep., Dec 2010. [4] G. Wei, A. V. Vasilakos, Y. Zheng, and N. Xiong, “A game-theoretic method of fair resource allocation for cloud computing services,” J. Supercomput., vol. 54, no. 2, pp. 252–269, Nov. 2010. [Online]. Available: http://dx.doi.org/10.1007/s11227-009-0318-1 [5] R. T. Ma, D. M. Chiu, J. C. Lui, V. Misra, and D. Rubenstein, “On resource management for cloud users: A generalized kelly mechanism approach,” Electrical Engineering, Tech. Rep., May 2010. [6] V. Kantere, D. Dash, G. Francois, S. Kyriakopoulou, and A. Ailamaki, “Optimal service pricing for a cloud cache,” Knowledge and Data Engineering, IEEE Transactions on, vol. 23, no. 9, pp. 1345 –1358, sept. 2011. [7] O. Agmon Ben-Yehuda, M. Ben-Yehuda, A. Schuster, and D. Tsafrir, “The resource-as-a-service (RaaS) cloud,” in USENIX Workshop on Hot Topics in Cloud Computing (HotCloud ’12), 2012. [8] S. Khan and I. Ahmad, “Non-cooperative, semi-cooperative, and cooperative games-based grid resource allocation,” in Parallel and Distributed Processing Symposium, 2006. IPDPS 2006. 20th International, april 2006, p. 10 pp. [9] G. Ellison, “Learning, local interaction, and coordination,” The Econometric SocietyStable, Econometrica, vol. 61, no. 5, pp. 1047–1071, Sep 1993. [Online]. Available: http://www.jstor.org/stable/2951493 [10] H. Tembine, “Distributed strategic learning for wireless engineers,” CRC Press, Taylor & Francis Inc., ISBN 9781439876374, 496 pages, 2012. [11] S. Willard, General topology. Reading, MA: Addison-Wesley, 1970. [12] S. Ishikawa, “Fixed points by a new iteration method.” Proc. Am. Math. Soc., vol. 44, pp. 147–150, 1974. [13] K. P. R. Sastry and G. V. R. Babu, “Approximation of fixed points of strictly pseudocontractive mappings on arbitrary closed, convex sets in a Banach space,” Proceedings of The American Mathematical Society, vol. 128, pp. 2907–2910, 2000.