Game Theory-Based Nonlinear Bandwidth Pricing for Congestion Control in Cloud Networks

Transcription

1 216 IEEE 8th Internationa Conference on Coud Coputing Technoogy and Science Gae Theory-ased Noninear andwidth Pricing for Congestion Contro in Coud Networs Abouzar Ghavai, Zhuozhao Li and Haiying Shen Departent of Eectrica and Coputer Engineering, Ceson University, Ceson, SC Departent of Coputer Science, University of Virginia, Charottesvie, VA 2294 Eai: Abstract In the coud, the networ ins are shared aong tenants, which aes the easy to get fuy congested (overoaded). Overoaded ins degrade the perforance of tenants appications, and ipose additiona costs to the coud provider. In this paper, we propose a noninear bandwidth pricing poicy for congestion contro in the coud networ. In order to axiize socia wefare (i.e., axiize the tota satisfaction of the tenants whie iniizing the congestion over the in), the coud provider uses the noninear pricing poicy that increases the unit price with increent of bandwidth usage. Each tenant copetes for bandwidth aocation to axiize its utiity (i.e., both axiize its own individua satisfaction and iniize its bandwidth payent cost). We design a gae between tenants and the coud provider, and show that there exists a unique optia bandwidth schedue (Nash equiibriu) that jointy axiizes the socia wefare and the utiity of each tenant at the sae tie. In order to find the optia schedue, we use an asynchronous-based best response strategy, in which each tenant updates its optia bandwidth aocation based on the updated bandwidth payent function fro the coud provider. We prove that the updated bandwidth aocations converge to the optia bandwidth schedue. In our siuation study and rea ipeentation, we verify the perforance of our proposed pricing echanis under different scenarios. I. INTRODUCTION Many of today s web services (e.g., Dropbox and Netfix) are depoyed on the coud. The coud providers own and aintain the arge-scae coputation, storage and networ resources, and the tenants rent the resources on deand instead of purchasing the costy servers, software, datacenter space or networ equipents [1]. Many web services such as transaction processing web appications [2] and video-on-deand (VoD) appications require predictabe networ perforance to ensure the perforance of their end-users. Thus, it is iportant for the coud providers to guarantee the quaity of service (QoS) of the tenants in order to attract and aintain tenants, and hence earn profit by eeping a steady ong-ter business. Recenty, any bandwidth aocation poicies have been proposed [3] [8]. Optia bandwidth aocation is addressed extensivey in the previous studies. Key [9] and Low [1] propose decentraized optiization ethods using inear pricing to axiize the tota satisfaction of users in a networ with iited capacity ins. Niu et a. [3] proposed a inear pricing ode to jointy axiize the tota satisfaction of the tenants and iniize the reservation payent cost of the coud provider. The authors found the optia bandwidth reservation through two different distributed and step-sizefree optiization ethods, Chaotic price update and cuttingpane ethod. However, by charging tenants at a fat rate for bandwidth usage, the above wors do not ai to contro congestion and avoid overoading on the ins, which is very essentia for guaranteeing the QoS of the tenants. Our wor aows the coud provider to contro congestion and avoid overoading in the coud networ. In this paper, we axiize the socia wefare of the tenants that aso iniizes the networ congestion in addition to axiize the tota satisfaction of the tenants in the networ. Our bandwidth payent ethod is aso ore transparent to the tenants than the proposed ethods in [9] and [1], where the bandwidth payent for each tenant is updated based on the payent rate at the previous update step and the tota bandwidth consuption of the tenants. In our proposed ethod, the pricing echanis at each iteration step is evauated based on a pre-deterined congestion cost function, that is nown for a the tenants before the first step of bandwidth update process, and it does not change during the update process. The bandwidth cost function is aso deterined based on the current tota oad in the networ that is observabe by the tenants. Furtherore, our proposed ethod provides ore generaized noninear pricing functions for the coud provider to contro congestion in the networ rather than inear pricing rates. We have aso extended our proposed non-inear pricing ethod to utipe switch cores in [11]. The noencature of the notations used in this paper is suarized in tabe I. The rest of this paper is organized as foows. We present the preiinaries overview in Section II. In Section III, we foruate the probe to axiize the socia wefare and in Section IV, we propose the pricing poicy that effectivey avoids congestion and axiizes tenants utiity. In Section V, we define a gae between the tenants, and we investigate the Nash equiibriu properties. We find the optia bandwidth schedue using the best response strategy in Section VI, and we discuss the tota payents of the tenants in Section VII. In Section VIII, we present our siuation and rea ipeentation resuts to verify the perforance of our proposed pricing echanis. We concude our paper with rears on our future wor in Section IX /16 $ IEEE DOI 1.119/CoudCo

2 TALE I: Noencature of the notations used in this paper b U ( ) V( ) π( ) b ax η C Z( ) S( ) ψ ( ) F (,ψ ( )) R( ) bandwidth of tenant Satisfaction function of tenant The congestion cost function The unit bandwidth price at a specific bandwidth usage Lin The bandwidth schedue of the tenants The feasibe set of a bandwidth schedues Tota bandwidth of the tenants on in Tota bandwidth capacity on in Aowed in capacity percentage Overoad cost function on in Tota overoad and congestion cost function on in Socia wefare of the tenants Tota payent of tenant Utiity function of tenant Revenue of the coud provider (a) Fig. 2: (a) Congestion cost increents at different bandwidth usages and (b) Congestion-based bandwidth pricing. (b) III. PROLEM FORMULATION Fig. 1: The bandwidth usage of tenants in in. II. PRELIMINARIES AND OVERVIEW ased on recent wors on sharing data center networs [12], [13], we abstract the connections of tenants into a hose ode [6], [14], where the networ throughput of each tenant is boced by a shared access in (denoted by in ). We consider a set of tenants, K = {1,..., K}, that use in to send data to their destinations in the networ. As shown in Fig. 1, ax denotes the bandwidth capacity of in, b denotes the aount of bandwidth that tenant aocates for sending data through in, and = K b denotes the tota bandwidth on the in. We denote as the congestion ax degree of in. Let b denote the aount of bandwidth aocated to tenant on in. Let us denote b =(b 1,..., b K ) as the bandwidth schedue for the tenants on in, and denote b as the bandwidth schedue for a the tenants excuding tenant on in. A tenant ay have a axiu required aount of bandwidth if it is not wiing to pay ore than a specific axiu bandwidth deand based on its appication usage. Thus, we assue that each tenant aocates a axiu aount of bandwidth denoted by b ax, i.e. b b ax. We have b ax = ax for a tenant without axiu bandwidth aocation. Let us denote = [,b ax ], and denote = 1... K as the set of a feasibe bandwidth aocations of the tenants. The set is a copact and convex set. The coud provider defines a safety argin η 1, and tries to constrain the bandwidth consuption on the in to no ore than η ax. A feasibe bandwidth schedue ust satisfy the in s capacity constraint: = K b η ax. (1) In this section, we foruate the probe to find the optia bandwidth schedue that satisfies tenants deands as uch as possibe and avoids congestion in in. To achieve this objective, the coud provider specifies a noninear pricing poicy that has a higher unit price when the current bandwidth usage is higher. This disincentivizes the tenants to use bandwidth when the bandwidth usage ( ) is higher (i.e., the in is ore iey to be congested). It is possibe for the coud provider to deterine the bandwidth payent according to a stricty convex function of the bandwidth usage. We refer this cost as congestion cost, and denote it as V( ). Let us denote π = V as the derivative of the pricing function V. Figure 2(a) shows the variations of the stricty convex congestion cost function as the bandwidth usage varies. It is seen that the congestion cost function is higher at the higher bandwidth usage. Figure 2(b) shows the increents of the bandwidth pricing function for user (denoted as ψ ) using the derivative of the congestion cost function, π. The coud provider needs to transfer this congestion cost to tenants by charging the on their individua bandwidth usages. When the current bandwidth usage is at a higher eve, tenants are ore disincentivized fro aocating ore bandwidths, thus avoiding the in to be fuy congested. Let us denote U (b ) as the satisfaction function of tenant fro using b aount of bandwidth. The satisfaction function, U ( ), is considered to be non-decreasing as each tenant desires high quaity of service and a higher bandwidth provision aes a tenant ore satisfied [3], [15]. Aso, the argina satisfaction of a user is non-increasing because a tenant s eve of satisfaction graduay gets saturated when the provisioned bandwidth increases [3], [15]. Therefore, we consider that U ( ) is a stricty increasing and stricty concave function, and its second derivative is continuous in. In order to attract and aintain ong-ter business, the coud provider needs to provide bandwidth to eet each tenant s satisfaction, and aso provide an un-congested networ support for a the tenants at the sae tie. Since these two factors affect the QoS to the tenants appications, which represents their wefare, we define socia wefare of tenants as a joint consideration of these two 215

3 factors in the foowing: ( ) K P(b) = U (b ) V b =1 K s.t. η ax, b. (2) The above equation considers the bandwidth capacity constraint as in (1). It aso considers the axiu deand constraint for each tenant by constraint b, i.e., b [,b ax ]. Then, the coud provider ais to find the bandwidth schedue that axiizes the socia wefare, which is the objective of this paper. To find the bandwidth schedue that axiizes the socia wefare, we transfor (2) to another objective that integrates the constraint in (2) [16]. When the tota bandwidth deands >η ax, in is considered as overoaded. In this case, soe bandwidth deands cannot be fuy satisfied, which ay ead to SLA penaty, reputation degradation and business oss to the coud provider. We ca these costs caused by the overoaded in overoad cost. We aso et C(x) denote the overoad cost function associated with in. We define C(x) as a stricty convex function with C(x) =for x<. Let us denote Z(x) =V(x)+C(x η ax ) (3) as the tota cost function and overoad penaty function of consuing x unit of bandwidth. The socia wefare of the tenants in (2) is rewritten as in the foowing: ( ) K S(b) = U (b ) Z b, =1 K b. (4) As U ( ) is stricty concave for each tenant, and V( ) and C( ) are stricty convex function, S( ) is a stricty concave function in. Definition 1. A feasibe bandwidth schedue is a sociay optia bandwidth schedue if it axiizes the socia wefare of the tenants as in (4). Note that in the case that coud provider nows satisfaction functions and axiu bandwidth requireents of the tenants, the optiization probe in (4) is soved using a singe step standard convex optiization ethod [16]. In this paper, we consider that the tenants do not revea their private inforation such as satisfaction function and axiu bandwidth requireents. In this case, the sociay optia bandwidth schedue is ony derived through an iterative decentraized bandwidth aocation process that is described in the next foowing sections. eow, we show how the coud provider transfers the congestion cost to the tenants by charging each tenant based on its contribution to the congestion cost and overoad cost, and in Section VI, we propose our decentraized optiization fraewor to find the sociay optia bandwidth aocation schedue. IV. PROPOSED PRICING POLICY We now introduce how the coud provider aes pricing poicy to transfer the congestion cost and overoad cost to the tenants in order to incentivize the tenants to vountariy constrain the costs. Let ψ (x ) denote the bandwidth payent function that tenant pays for x aount of bandwidth reservation for the next tie period T i. The tota payent of tenant for reserving b aount of bandwidth, ψ (b ),is cacuated as in the foowing: ψ (b )=Z j K {} b j + b Z b j j K {}. (5) Note that the ter Z( ) in (5) is independent fro b. These ters resut in an unbiased cost function for the tenants, i.e. ψ () =,. Next, we introduce the utiity of tenant that it tries to axiize when deterining its bandwidth deand. The gain and the cost of tenant by using b aount of bandwidth can be easured by its satisfaction U (b ) and bandwidth payent function, ψ ( ), respectivey. Thus, the utiity function of tenant, F (b,ψ ( )), for using b aount of bandwidth is cacuated as in the foowing: F (b,ψ ( )) = U (b ) ψ (b ), b. (6) ased on the proposed bandwidth payent function, ψ ( ) in (5), the utiity function of tenant, F (b,ψ ( )), is aso a function of b. We refer F (b,ψ ( )) as a function of b and b, and denote it as F (b,b ). Therefore, the tenant utiity function is cacuated as in the foowing: Z j K {} F (b,b )=U (b ) b j + b Z b j j K {},b. In order to find the sociay optia bandwidth schedue, we define a strategic gae between the tenants. In this gae, the coud provider proposes the bandwidth payent function ψ ( ) to the tenants, and each tenant responds its proposed strategy, i.e., its aocated bandwidth b, to the coud provider. In the next foowing sections, we derive the Nash equiibriu properties of this gae, where both the coud provider and tenants gain the axiu utiity. We show that Nash equiibriu is sociay optia and vice versa. We wi aso show that the best response strategy of the tenants converge to the sociay optia bandwidth schedue. V. PROPERTIES OF NASH EQUILIRIUM The bandwidth reservation aount chosen by rationa tenants, who aways try to axiize their individua profits as defined in (7), can be represented as a strategic gae K,, F. In this strategic gae, the strategy of each tenant (7) 216

4 ( K) is its bandwidth aocation, b. Tenant chooses its strategy, b, as a response to the bandwidth payent function, ψ (b ), that is deterined based on the bandwidth aocations of other tenants sharing in. The best response strategy of tenant is the bandwidth aocation that axiizes its individua utiity function, F ( ). In this section, we investigate the Nash equiibriu of this strategic gae. We show that the Nash equiibriu exists, and it is equa to the unique sociay optia bandwidth schedue. That is, each tenant ony has one choice of his bandwidth reservation in order to axiize its utiity. A. Existence of Nash Equiibriu. The foowing conditions hod for each tenant K: 1. The set is a non-epty, convex, and copact subset of a finite-diensiona Eucidean space. 2. The tenant utiity function, F (b,b ), is continuous in b as U ( ), V( ), and C( ) are continuous. 3. The tenant utiity function, F (b,b ), is a stricty concave function of b as U ( ) is stricty concave, and V( ) and C( ) are stricty convex functions. Then, based on the Debreu, Gicsberg, Fan theore [17], the gae K,, F has a pure strategy Nash equiibriu.. Socia Optiaity of Nash Equiibriu. Theore 1. The sociay optia bandwidth schedue is a Nash equiibriu, and vice versa. Proof: Reca b denotes a sociay optia bandwidth schedue that axiizes the socia wefare of the tenants as in (4). As S( ) is stricty concave over, fro the first order inequaity condition [16], we have: S(b ) T (b b )= K F (b,b )(b b ), b. (8) where F = F b denotes the partia derivative of F with respect to b. Let us set b =(b,b ) in (8), where b. Thus, for a K,wehave: F (b,b )(b b ), b. (9) As F is stricty concave over, fro the above Proposition, the condition in (9) is aso sufficient for b to axiize F (b ) over,. Therefore, no tenant deviates fro its bandwidth reservation choice b,. Thus, the sociay optia bandwidth schedue b is a Nash equiibriu. To prove the converse, et b is a Nash equiibriu bandwidth schedue that axiizes the utiity functions of ) the tenants, and tenants do not deviate fro it. As F ( b,b is stricty concave in, fro the first order inequaity condition, we have: F ( b,b )( b b ),, b. (1) Fig. 3: The iterative bandwidth update process. Writing the first order inequaity condition for S in (4) at b, and using ( (7) and the resut in (1), for a b,wehave: S( b) T b b ) = )( F ( b,b b b ). K (11) As S( ) is stricty concave in b, fro Proposition (b) in [16], b axiizes S(b) over. Therefore, the Nash equiibriu bandwidth schedue, b, is a sociay optia bandwidth schedue. The foowing Lea foows iediatey fro the stricty concavity of S( ) in. Lea 1. The sociay optia bandwidth schedue is unique. C. Uniqueness of the Nash Equiibriu Theore 2. The Nash equiibriu bandwidth schedue is unique. Proof: The proof foows iediatey fro Theore 1 and Lea 1. VI. ASYNCHRONOUS-ASED EST RESPONSE STRATEGY In this section, we find the sociay optia bandwidth schedue through a distributed ethod. As we entioned, the tenants usuay do not reease their private inforation, such as satisfaction function, U ( ), and axiu bandwidth requireent, b ax ( ), to the coud provider. Without nowing this inforation, the coud provider is not abe to find the sociay optia bandwidth schedue to axiize the socia wefare as in (4) in a centraized anner. Then, to find the optia bandwidth schedue, we propose a decentraized bandwidth aocation fraewor, in which the coud provider uses an asynchronous-based best response strategy process [18] to aocate the bandwidth for the tenants. A. Decentraized andwidth Aocation Figure 3 shows this asynchronous-based best response strategy process. In each iteration step of this process, a rando tenant updates its bandwidth aocation to axiize its utiity (F (b,ψ ( )) in (6)) based on the bandwidth payent function (ψ ( ) in (5)) fro the coud provider. Let b denote the updated bandwidth aocations of the tenants at iteration step. As entioned, we assue that the initia bandwidth deands of the tenants, b, is predictabe fro historica data [19]. 217

5 At the beginning of each step +1, the coud provider updates the bandwidth payent, ψ +1 ( ), for each tenant according to (5) based on the bandwidth aocations in the th step, b, cost function V( ), and the overoad cost function C( ) as in (12). ψ +1 (b )=Z b j j K {} + b Z b j j K {}. (12) Next, the coud provider announces the new bandwidth payent functions, ψ +1 ( ), to the tenants. Each tenant updates its bandwidth aocation, b +1, to axiize its individua utiity as in (13) and sends the updated bandwidth aocation to the coud provider. b +1 =argax b F (b,b ) =argax b U (b ) ψ +1 (b ) (13) We define cyce as the set of N successive updates that each tenant updates at east once in N + K nuber of successive updates and ca N cyce ength. After N updates, if a tenant fais to update its bandwidth aocation, the coud provider ass for its bandwidth update. If the tenant does not respond to the coud provider s aocation, the coud provider wi drop it out of the update process and ony aocates the in bandwidth aong other tenants. This ensures that each tenant updates its bandwidth at east once in N + K nuber of successive updates. This iterative process repeats unti a specific arge nuber of updates has reached and each tenant has updated its deand in order to achieve a sa convergence error to the Nash equiibriu bandwidth schedue. We show that the best response strategy of the tenants converges to the sociay optia bandwidth schedue, b = (b 1,..., b K ), under this decentraized asynchronous updates. The foowing Lea foows fro the stricty concavity of F (b,b ) in. Lea 2. At each update step, the optia bandwidth aocation of the updating tenant is unique.. est Reponse andwidth Update of Tenants In the proposed asynchronous-based best response update process, tenant soves (13) to find its own optia bandwidth aocation at step +1. This optia bandwidth aocation is derived as in the foowing Lea. Lea 3. The optia bandwidth aocation of tenant at ( +1) th update iteration is derived as in the foowing: (, F b, ) <, b +1 ( ) = b ax, F b,b ax >, (14) { ( ) } arg b F b,b =, O.W. Proof: The proof foows using the first order inequaity condition: ( F b,b +1 )( b b +1 ), b. (15) ( for three possibe cases: (i) F b, ) ( ) ( <, (ii) F b,b ax > and (iii) F b, ) ( ) and F b,b ax. For these three possibe cases, the unique soution to (13) that axiizes the utiity of tenant at ( +1) th update step is derived as in (14). Given the bandwidth payent function, ψ ( ), that is deterined based on the congestion cost function V( ), overoad cost function, C( ), and the bandwidth aocations of the tenants, b, the tenants find the unique Nash equiibriu bandwidth schedue through the best response strategy. As derived in Section V, the achieved Nash equiibriu is the sociay optia bandwidth schedue that uniquey axiizes the socia wefare of the tenants as in (4). The foowing theore foows fro the stricty concavity of S( ) and U( ) by odifying the Gauss-Siede agorith in [2]. Theore 3. The asynchronous update process converges to the sociay optia bandwidth schedue. VII. TOTAL ANDWIDTH PAYMENTS OF THE TENANTS In this section, we evauate the tota bandwidth payents of the tenants based on the proposed pricing echanis. We show that the tota bandwidth payents of the tenants is axiized at the fuy congested in, i.e. = ax. We consider the in capacity constraint as in (1) with the safety argin factor set at η =1. With satisfaction of the axiu in capacity constraint as in (1), each tenant pays R (b,b ) for using b aount of bandwidth: R (b,b )=V b j V j K b j j K/{}. (16) The tota bandwidth payents of the tenants, R(b), is cacuated as in the foowing: R(b) = V b j V K j K b j j K/{} = KV ( ) K V ( b ). (17) Fro stricty convexity of V, the tota bandwidth payents of the tenants, R(b, b ), is stricty convex and increasing in b,. Thus, it gets axiized at = ax. VIII. PERFORMANCE EVALUATION In this section, we conduct siuations and rea ipeentation experients on Aazon EC2 [21] to evauate the noninear pricing ode copared with the inear pricing ode. 218

6 Socia wefare Socia wefare Noninear, K = (a) Socia wefare (a) Socia wefare Tota bandwidth payent Tota bandwidth payent Noninear, K = Tota satisfaction (b) Tota bandwidth payents (c) Tota satisfaction Fig. 4: The siuation resuts of hoogeneous tenants Tota satisfaction (b) Tota bandwidth payents (c) Tota satisfaction Fig. 5: The siuation resuts of heterogeneous tenants Congestion degree Congestion degree Nuber of updates (d) Congestion degree Nuber of updates (d) Congestion degree A. Environent of Siuation and Ipeentation In both siuation and rea ipeentation, we consider the foowing scenario. Assue there are K tenants and each tenant has 2 VMs connected with each other on the sae in. Each tenant has an appication that sends data fro one VM to another VMs through in. Therefore, the K tenants copete the bandwidth resource in in. We consider x the stricty convex function V 1 (x) =α( ) as the inear ax x congestion cost function, and V 2 (x) = β( ) 2 as the ax noninear congestion cost function in our siuation studies. The coud provider deterines the cost factors, β and α, to achieve a specified congestion degree on in. The overoad cost function was set to C(x) =γx 2 for x>, and C(x) = for x. We set γ = 1β and γ = 1α to ae the overoad cost arge enough copared to the congestion cost function, V(x), for noninear and inear pricing poicies, respectivey. The capacity of in is ax =4Gbps. In order to use the axiu in capacity, we set the safety argin factor η =1. Uness otherwise specified, we set the nuber of updates in the best response strategy update process to 1, which is arge enough to converge to the optia bandwidth schedue. In our siuation, we tested the perforance when the nuber of tenants set to K = 8, 16 and 32. We set the cyce ength as N = 8K. WeuseC d = ax to denote the desired congestion degree that the coud provider ais to achieve. We consider two different scenarios: i) hoogeneous tenants with the sae satisfaction function and axiu bandwidth deand, and (ii) heterogeneous tenants with different satisfaction functions copeting to aocate their individua share of the bandwidth in in. For hoogeneous tenants, we consider stricty concave satisfaction functions of tenant, U = n(1 + b ), where b is easured in Gbps. For heterogeneous tenants, the satisfaction function of tenant is considered as U =n(1+(1+( K ))b ). In genera, the coud provider does not have the inforation of U of the tenants. In this case, it uses binary search agorith and the best response strategy to find the cost factors β and α that achieve the desired congestion degree C d. In our experients on Aazon EC2 [21], there are K =8 heterogeneous tenants and 16 t2.icro VM instances (1 CPU core and 1G eory). We used Python socet prograing to create a TCP counication between each of the 8 VM pairs of each tenant on a in with ax =4Mbps. We consider a scenario in which each tenant tries to send data fro one VM to the other VM. We set the nuber of updates in the asynchronous-based iterative bandwidth update process to 1, which is arge enough to converge to the optia bandwidth schedue. In order to euate the coud provider, we used another VM on Aazon EC2. Uness otherwise indicated, the settings in the rea experients are the sae as those in the siuation.. Experienta Resuts In the siuation and rea ipeentation, we easured the socia wefare, average bandwidth payent, and tota satisfaction of the tenants. We varied the desired congestion degree C d fro.1 to 1 with step increase of.1. Then, we deterined α and β as expained previousy to achieve 219

7 Socia wefare Noninear, K=8 2 Linear, K= Desired congestion degree (a) Socia wefare Tota bandiwdth payent Noninear, K=8 1 Linear, K= Desired congestion degree (b) Tota bandwidth payents Tota satisfaction Noninear, K=8 Linear, K= Desired congestion degree (c) Tota satisfaction Congestion degree Fig. 6: The experienta resuts of heterogeneous tenants in rea ipeentation. Noninear, K=8 Linear, K= Nuber of updates (d) Congestion degree this specified congestion degree. After the optia bandwidth schedue is deterined, we cacuated the socia wefare, the tota bandwidth payent and tota satisfaction of the tenants based on (4), (17). Figures 4(a) and 5(a) show the socia wefare of hoogeneous and heterogeneous tenants versus the desired congestion degrees in the siuation, respectivey. Figure 6(a) shows the socia wefare of heterogeneous tenants versus the desired congestion degree in the rea ipeentation. We see that the socia wefare is higher with noninear pricing ode than with inear pricing ode. It deonstrates that noninear pricing poicy provides better socia wefare, i.e., higher QoS, to the tenants. Through noninear pricing ode, the tenants are ore iey to be satisfied with the coud provider, and hence the coud provider can further attract ore tenants to use its services. Furtherore, we see that using the noninear pricing poicy, the socia wefare increases as the nuber of tenants increases for both hoogenous and heterogenous cases. However, using the inear pricing poicy, the socia wefare does not necessariy increase as the nuber of tenants increases for hoogeneous tenants. It indicates that with the noninear pricing poicy, ore tenants ead to higher socia wefare. Figures 4(b) and 5(b) show the tota bandwidth payent of hoogeneous and heterogeneous tenants versus the desired congestion degree. Figure 6(b) shows the tota bandwidth payent of heterogeneous tenants versus the congestion degree in the rea ipeentation. We see that the average bandwidth payent with the inear congestion cost function is sighty higher than that with the noninear congestion cost function. It shows that using the noninear pricing poicy, the coud provider assigns ess bandwidth cost for the tenants to achieve a desired congestion degree in in. Paying ess aount of the bandwidth payent by the tenants aes the ore satisfied with the service of the coud provider. Consequenty, it encourages the tenants to choose the coud provider for their services. Figures 4(c) and 5(c) show the tota satisfaction of the hoogeneous tenants and heterogeneous tenants in the siuation, respectivey. Figure 6(c) shows the tota satisfaction of the heterogeneous tenants in the rea ipeentation. We see that for both hoogeneous and heterogeneous tenants, the tota satisfaction is equa for both inear and noninear congestion cost functions, since at a fixed congestion degree, the ter V( b ) in (4) is fixed at the sae vaue for both inear and noninear pricing poicies. Thus, the optiization probe in (4) is equivaent with axiizing the first ter in (4) that is the tota satisfaction of the tenants. In addition, we see that at a fixed desired congestion degree, the tota satisfaction increases as the nuber of tenants increases due to the concavity of the satisfaction function. This shows that the coud provider prefers to have ore tenants using its service. We then easured the convergence speed for the noninear and inear pricing poicies when the desired congestion degree was set to 9%. We define congestion error as C d Cd C d, where Cd is the congestion degree for the bandwidth aocations at th update, and C d is the desired congestion degree. Figures 4(d) and 5(d) pot the congestion degree for the bandwidth aocations at th update versus the nuber of updates (i.e., ) to achieve the convergence error of 1%. Figures 4(d) and 5(d) show the congestion degree of in as the nuber of updates increases in the siuation. These figures are obtained by taing average of 1 ties of experient runs. Figure 6(d) shows the congestion degree as the nuber of updates increases for heterogeneous tenants in the rea ipeentation. As it is seen, the congestion degree converges faster using the noninear pricing poicy, and it needs fewer updates to converge than the inear pricing poicy for a different nuber of tenants. As we discussed in Section III, using noninear pricing poicy, the bandwidth payent cost increases as the congestion degree increases. Thus, for the ess congested in, the tenants pay ess for a specific aount of bandwidth. Therefore, the tenants aocate ore bandwidth when the in is ess congested. In contrast, the bandwidth payent does not change with the congestion degree when we use the inear pricing poicy. As a resut, the congestion degree converges to the desired congestion degree faster than using the inear pricing poicy as shown in figures 4(d), 5(d) and 6(d). In order to show the rea tie duration for the iterative update process, we easured the tota tie it too to copete the update 22

8 process. In this test, we set the desired congestion degree to.9 and varied the nuber of updates fro 2 to 1. Figure 7 shows the update process duration as the nuber of updates increases. We see that it taes around one second to copete 1 updates (a sufficient nuber of updates as shown in Figure 6(d)), which is a short tie to converge to the optia soution. The resut eans that for each shorttie period (e.g., 5 inutes), noninear pricing poicy spends around one second to deterine the bandwidth schedue for this period. As it is seen, the update process duration increases ogarithicay as the nuber of updates increases. Further nuerica experients aso show that the nuber of updates increases ineary as the nuber of tenants increases. Update process duration (s) Nuber of updates Fig. 7: Update process duration of a tenant in rea ipeentation. IX. CONCLUSION AND FUTURE WORK In this paper we proposed noninear bandwidth pricing for congestion contro in a counication in in the coud networs. We defined the socia wefare of the tenants as the tota satisfaction functions of the tenants inus the congestion cost over the in. We foruated the utiity optiization of the tenants as a strategic gae, and we showed different properties of the Nash equiibriu of the designated gae. We showed that the Nash equiibriu exists, and it is uniquey equa to the socia optiu bandwidth schedue that axiizes the socia wefare of the tenants. As the tenants are not obiged to truthfuy revea their private inforation such as axiu required bandwidth and satisfaction function, we ipeented a decentraized asynchronous-based best response strategy to find the Nash equiibriu bandwidth schedue. We showed that the best response strategies of the tenants converge to the Nash equiibriu bandwidth schedue under reasonabe id assuptions such as individuay sefish utiity axiizing tenants. We aso derived the tota bandwidth payents of the tenants, and discussed that the tota bandwidth payents is axiized at the fuy congested in. We verified our resuts with nuerica siuations for different nuber of the tenants. In this paper, we considered different tenants sharing one bandwidth iited in in the coud networ. We used the overoad cost function to satisfy the bandwidth axiu capacity constraint. In our future wor, we wi consider the scenario that the tenants share utipe ins in the networ. We wi aso use adaptive contro ethods to satisfy the bandwidth capacity constraint. X. ACKNOWLEDGEMENT This research was supported in part by U.S. NSF grants NSF , IIS , CNS-12546, IM Facuty Award and Microsoft Research Facuty Feowship REFERENCES [1] M. Arbrust, A. Fox, R. Griffith, A. D. Joseph, R. Katz, A. Konwinsi, G. Lee, D. Patterson, A. Rabin, I. Stoica, and M. Zaharia, A view of coud coputing, Coun. ACM, vo. 53, no. 4, pp. 5 58, Apr. 21. [2] D. Kossann, T. Krasa, and S. 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