Load Balance vs Energy Efficiency in Traffic Engineering: A Game Theoretical Perspective


 Gervase Garrison
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1 Load Baance vs Energy Efficiency in Traffic Engineering: A Game Theoretica Perspective Yangming Zhao, Sheng Wang, Shizhong Xu and Xiong Wang Schoo of Communication and Information Engineering University of Eectronic Science and Technoogy of China Chengdu, P. R. China, 673 Emai: {zhaoyangming, wsh_eyab, xsz, wangxiong Xiujiao Gao and Chunming Qiao Department of Computer Science and Engineering State University of New Yor at Buffao Buffao, NY, USA, Emai: {xiujiaog, Abstract In this paper, we study the tradeoff between two important traffic engineering objectives: oad baance and energy efficiency. Athough traditiona commony used mutiobjective optimization methods can yied a Pareto efficient soution, they need to construct an aggregate objective function (AOF) or mode one of the two objectives as a constraint in the optimization probem formuation. As a resut, it is difficut to achieve a fair tradeoff between the both objectives. Accordingy, we induce a Nash bargaining framewor which treats the two objectives as two virtua payers in a game theoretic mode, who negotiate how traffic shoud be routed in order to optimize both objectives. During the negotiation, each of them announces its performance threat vaue to reduce its cost, so the mode is regarded as a threat vaue game. Our anaysis shows that no agreement can be achieved if each payer sets its threat vaue sefishy. To avoid such a negotiation breadown, we modify the threat vaue game to have a repeated process and design a mechanism to not ony guarantee an agreement, but aso generate a fair soution. In addition, the insights from this wor are aso usefu for achieving a fair tradeoff in other mutiobjective optimization probems. Index Terms Traffic Engineering; Load Baance; Energy Efficiency; Nash Bargaining; MutiObjective Optimization. I. INTRODUCTION Many objectives exist in networ traffic engineering [7], such as minimizing EE (end to end) deay or hop count, maximizing throughput, baancing in oad, and reducing tota energy consumption. Most of these objectives are conficting with each other, in that improving one objective hurts the other. How to achieve a tradeoff between such conficting objectives is an interesting probem in networ traffic engineering. In this paper, we focus on achieving a fair tradeoff between two important traffic engineering objectives: oad baance and energy efficiency. Load baance is a cassic objective in traffic engineering [4]. The main goa of oad baance is to enhance the performance of networ traffic whie utiizing networ resource economicay. To achieve oad baance, traffic shoud be distributed among a the ins uniformy, so as to reduce the carried traffic on each in. It coud improve the performance of networ traffic, in terms of reduced queueing deay, and enhanced networ scaabiity. Energy efficiency is a reativey new but increasingy important traffic engineering objective [57], whose goa is to save energy consumption even though it might resut in unbaanced traffic. More specificay, there are two commony used modes to determine the networ energy consumption. One is the powering down mode, and the other is the speed scaing energy mode. In the powering down mode [6], we shoud use as a few ins to route the traffic as possibe, so that the corresponding networ eements can be turned off to save energy consumption. Obviousy, this objective contradicts to that of oad baance which is to distribute traffic to as many ins as possibe. Even in the speed scaing energy mode [5, 7], where the energy consumption of each in is characterized by its energy curve, the traffic distribution achieving the highest energy efficiency woud hardy be the same as that achieving oad baance. The energy curve is a nondecreasing function of traffic oad on each in. It coud be convex or concave depending on its physica architecture. But it is unie to be the same as congestion cost of each in. Since carriers/operators are interested in both objectives above instead of ony one of them, how to optimay route traffic becomes a mutiobjective optimization probem. There are two commony used traditiona methods to sove the optimization probem with mutipe objectives [9]. One is to treat a objectives except the most favorite one as constraints, and then optimize the favorite one. Such a method might wor when there were specific performance goas considered desirabe for a the other objectives, in the form of the threshod vaues used to set the corresponding constraints in the optimization probem formuation. However, this is often not the case as there is usuay no hard imit on the performance of these objectives. For exampe, a carrier does not now (nor wants to set) the desired specific performance of oad baance (or energy efficiency). As a resut, most iey some ad hoc performance threshods wi be specified in the corresponding constraints, and accordingy, ony the favorite objective wi achieve the best performance at the expense of a the other objectives. If a the objectives need to be pursued without restricting any to its ad hoc performance threshod, such a method is not suitabe. The other traditiona method is to construct an aggregate objective function (AOF), such as the wenown weighted inear sum of the objectives. It wi yied a Pareto optima soution in theory, but it is difficut to determine the appropriate weight for each objective. This is because these objective vaues have not ony different performance metrics representing different dimensions of interest (e.g., oad baance and energy efficiency), but aso have different scaes or orders of magnitude. Note that Langarian reaxation has simiar imitations in that it may be difficut to determine the appropriate ad hoc performance threshods for different objectives. Ideay, when we pursue mutipe objectives in traffic engineering, we do not want to discriminate against any objective by improving the performance of some objectives more significanty than that of the others due to ad hoc constraints or
2 weights assigned for various objectives. In other words, we aim to achieve a fair tradeoff among the objectives we are pursuing under a rationa guideine. For instance, from the viewpoint of achieving fairness between mutipe objectives, the ones with a reativey arger optimization space shoud obtain more performance improvement than the ones with a reativey smaer optimization space. Unfortunatey, the traditiona mutiobjective optimization methods cannot guarantee the fair tradeoffs among different objectives. In the first method mentioned above, a the objectives treated as constraints in the optimization probem are in a weaer position whie the one treated as the optimization objective is in a stronger position, resuting in unfair tradeoffs. In the second method mentioned above, the objectives whose vaues have higher orders of magnitude are iey to be in a stronger position than the objectives that are of ower order of magnitude. To overcome the difficuty in achieving a fair tradeoff among mutipe objectives in traffic engineering, e.g., oad baance and energy efficiency, we propose a framewor based on Nash bargaining to jointy optimize both objectives and guarantee the fairness between them. We treat the objectives of oad baance and energy efficiency as two virtua game payers who are negotiating the soution of traffic engineering. In this framewor, we assume that each payer changes its threat vaue (performance threshod) to improve its performance. Accordingy, the interaction of each payer can be modeed as a threat vaue game. Our anaysis shows that ) there are an infinite number of Pareto efficient Nash equiibriums in this threat vaue game, and ) a payer can improve its performance by uniateray reducing its threat vaue. This means that if we were to mode this probem as a static game, both payers woud announce the threat vaue as ow as possibe to improve its performance, which woud prevent an agreement. To ensure an agreement, we modify the threat vaue game to be a repeated procedure where each payer changes its threat vaue stepwise. Based on this repeated procedure mode, we design a mechanism which can not ony guarantee an agreement, but aso achieve a fair tradeoff. The main contributions of our wor can be summarized as foows: We anayze the probem of achieving a fair tradeoff between mutipe conficting objectives in traffic engineering and provide usefu insights into the genera mutiobjective optimization probems. We aso propose a new method to achieve a fair tradeoff between mutipe conficting objectives in traffic engineering. Our method overcomes the difficuties of traditiona methods in assigning appropriate performance threshods to the objectives or determining appropriate weights in the AOF. The rest of the paper is organized as foows. Section II briefy describes the reated wor. Section III and Section IV present the commony used modes for oad baance and energy efficiency, respectivey. In Section V, we introduce the Nash bargaining theory which forms the foundation of our mode. After that, we anayze the fair tradeoff probem in detai and describe the motivation for using Nash bargaining to sove the probem in Section VI. In Section VII, we propose a method to achieve a fair tradeoff between oad baance and energy efficiency, and determine the initia (and subsequent) threat points which can induce a fair soution. We aso present case studies of our method in Section VIII and concude our paper in Section IX. II. RELATED WORK In the past decades, a ot of wors have been done on the traffic engineering in networs. Some have focused on traffic oad baance [4], whie others have tried to minimize the energy consumption [57]. Load baance can be reaized by optimizing the in weight in the OSPF networ [] or by configuring LSPs (abeswitched path) in MPLS networs [0]. None of such wors, however, considered reducing the energy consumption in the networ. On the other hand, recent wors in [5] and [] pursued energy efficiency, but neither of them considered oad baance as an objective. To the best of our nowedge, there is no existing wor on traffic engineering that pursues both oad baance and energy efficiency simutaneousy. To dea with mutipe objectives, the commony used traditiona mutiobjective optimization approach [9] creates an AOF or treats one of the objectives as the primary objective and expresses a other (secondary) objectives as constraints of the optimization probem. Though both of these methods can yied a Pareto efficient soution, it is difficut to seect the appropriate weights when constructing the AOF or to determine the appropriate performance threshod vaues to be used in the constraints for the secondary objectives. As a resut, a fair tradeoff between among the objectives cannot be achieved. Game theory is a usefu too to sove many networ optimization probems. There are an increasing number of researchers who appy it to address routing issues in mutiayer networ [4], cooperation (or competition) among mutipe autonomous systems [4], and content provider seection [5]. To the best of our nowedge, there has been no existing wor which has anayzed the tradeoff between mutipe objectives of a singe operator in a game theoretic perspective, et aone any wor on achieving fair tradeoff between oad baance and energy efficiency. The wor which is most simiar to our wor is the Nash arbitration scheme [7]. In this scheme, an AOF having the same form as that in Nash bargaining was introduced to derive the optimization soution. But it approached the probem mosty from a mutiobjective optimization perspective whie we wi approach it from a game theoretic perspective in this paper. In addition, it has been proved that the soution of Nash arbitration depends on the threat vaue of each objective and an objective that is the farthest away from its threat vaue tends to improve most significanty [6]. However, it neither showed how the threat vaue affects the outcome of the optimization, nor how to set the threat vaues. III. LOAD BALANCE MODEL In this section, we describe the networ mode and formuate the standard oad baance optimization probem. Consider a networ represented by a directed graph G=(V, E), where V denotes the set of nodes and E denotes the set of directed physica ins. Let P ={p } denote the set of a the paths from i to j, where p denote the th path from i to j and i, j V. A in on the th path from i to j wi be referred to as p.we aso use x to denote the rate of fow on the th path from i to j, x the rate of
3 fow on in, and d the demand from i to j. The capacity of in is c >0. The goa of oad baance is to enhance the networ performance by reducing congestion and improving scaabiity. In practice, networ operators contro routing either by changing OSPF in weights [] or by estabishing MPLS abeswitched paths [0]. The atter one is assumed in our mode. It is not ony because it is optima, i.e. it gives the routing with minimum congestion cost, but aso due to the fact that it can be reaized easiy by routing protocos that use MPLS tunneing. Based on the above discussion, the traffic engineering for oad baance can be formuated as foows: minimize f( x ) () subject to x = x, i, j: i j : p x c E x = d i, j i where f ( ) represents the congestion cost of in. In this paper, we assume that f ( ) is a convex, continuous and nondecreasing function of x. Using such a cost function in the optimization objective wi penaize high in utiization and baance the oad in the networs. More specificay, a queueing theory stye congestion cost function such as f (x )=x /(c x ) is usuay adopted for this purpose, as it has the desired properties of being convex, continuous and nondecreasing with x. In addition, by using this congestion cost function, the in that had higher oad (or utiization) wi have a higher cost than the ins with a ower oad, so that traffic wi be distributed uniformy in the networ in the optima soution. IV. ENERGY EFFICIENCY MODEL Whie the oad baance probem usuay assumes that the congestion cost function of each in is a convex, continuous and nondecreasing function of the amount of traffic carried by the in, there are two popuar modes that reate power consumption to traffic oad: speed scaing and powering down. In the former mode, the processing (or transmission) speed of a networ eement is adjusted (and accordingy, the corresponding energy consumption aso varies) according to the carried traffic oad. In the atter mode, one tries to turn down any eements carrying no traffic oad at a to save energy. We focus on the speed scaing mode in this paper because it is more reaistic. In addition, the powering down mode focuses on optimizing an individua eement in isoation [5], but we want to examine optimization probems that arise in a networ consisting of mutipe networ eements. In particuar, we assume that the energy consumption in the networ can a be represented in terms of the energy consumption of the ins, which can be characterized by energy curve g (x ). The goa of traffic engineering to achieve energy efficiency is thus to minimize the tota energy cost of a the ins in the networ. Accordingy, the optimization probem can be formuated as foows: minimize g( x ) () subject to x = x, i, j: i j : p x c E x = d i, j i Note that, the energy curve is often modeed by a poynomia function g (x )= μ x α, where μ and α are device specific parameters. For exampe, the vaue of the α is.,.66, and.6 for Inte PXA 70, a TCP offoad engine, and Pentium M 770, respectivey [8]. Accordingy, we wi set α to be.5 in the case studies of the paper. V. NASH BARGAINING Since our method is based on the Nash bargaining soution [7] to derive a tradeoff between oad baance and energy efficiency, we briefy introduce the Nash bargaining soution and anayze its properties in this section. Consider two payers, abeed i=,, that are trying to achieve an agreement over a strategy space Χ. And the utiity function u i of each payer i is defined over the space X {T}, where T is the strategy of the two payers that eads to a faied agreement. Define the space S to be the set of a possibe utiities that the two payers can achieve, i.e., S = {( u( x), u( x)) x = ( x, x) X} Let d=(u (t,t ), u (t,t ))=(d, d ) be the pair of utiity expected to be obtained by the two payers when they fai to achieve an agreement, i.e. the disagreement point or threat point. We aso say d and d are the threat vaues of pay and payer, respectivey. The threat vaue can aso be comprehended as the minima utiity a payer expected to obtain in the agreement. A bargaining probem is defined as the pair (S, d) where S R and d S such that S is a convex and compact set There is some s S such that s>d, by which we mean s i d i for i=, and s i >d i for i= or. The Nash bargaining soution we are interested in is a mapping f: (S, d) S for every bargaining probem (S, d) (note that f i (S,d) is used to represent the utiity vaue of payer i) which satisfies the foowing four properties:. Pareto efficiency: A bargaining soution f(s,d) is Paretoefficient means that there is no point (s, s ) S such that s i f i (S,d) for a i and s i > f i (S,d) for some i.. Symmetry: If (S, d) is symmetric around s =s, i.e. (s, s ) S iff (s, s ) S and d = d, then f (S,d)= f (S,d). 3. Invariance to equivaent utiity representation: Assume the soution of Nash bargaining (S, d) is (s, s ), if it is transformed to (S, d ) by taing s i =α i s i β i and d i =α i d i β i, where α i >0, the soution of (S, d ) is (α s β, α s β ). 4. Independence of irreevant aternatives: Given two bargaining probem (S, d) and (S, d), where S S, if f(s,d) S, there must be f(s,d)= f(s,d). Nash s resut [6] shows that there is a unique bargaining soution that satisfies the four properties, which is the soution of the foowing optimization probem: maximize ( s d)( s d) (3) subject to ( s, s) S 3
4 ( s, s) ( d, d) If the utiity function of each payer is defined to be the opposite number of its cost function, the tradeoff probem can be modeed as the foowing Nash bargaining form: maximize ( d f )( d g ) subject to (4) EE x = x, i, j: i j : p x c E x = d i, j i f d g d EE where d and d EE are the performance threshods (maxima cost toerated by each payer) set by the oad baance and energy efficiency objectives, respectivey. It shoud be noted that these performance threshods correspond to the worstcase performance which can be easiy determined. However, they cannot be used by the conventiona mutioptimization approach whereby different performance threshods corresponding to desirabe performance are needed for the constraints, not the worstcase possibe performance, which is too oose as a performance bound. VI. PROBLEM ANALYSIS In this section, we wi anayze the desirabe properties of the soution obtained by our method and expain why the Nash bargaining framewor is suitabe to our probem. For each of the optimization probems in () and (), x in the objective function can be substituted based on the first equation (constraint), so f ( ) and g ( ) can be treated as a function of x={x }. From now on, we aso use f (x) and g (x) to denote the cost function of oad baance and energy efficiency for in. A. Desirabe Properties a. Pareto efficiency: In our probem, the two objectives (oad baance and energy efficiency) are pursued by one operator, so that the soution x shoud not be worse than any soution x for both objectives, i.e. there exist no feasibe soution x such that f ( x) < f g ( x) < g and. It aso means that the soution of our method shoud ie on the Pareto frontier (See Definition 6) and hence no other soution can improve at east one payer s performance without hurting the performance of the other one. b. Fairness: Load baance and energy efficiency are both pursued and any one of them are not preferred more than the other one, so that we shoud treat them equitaby. The fairness is defined as foows: Definition : Let x and x EE be the soutions to the optimization probems () and () respectivey, then we define the best case and worst case oad baance cost to be best = f ( x ) and worst = f ( xee ) respectivey. Simiary, we define the best case and worst case costs of energy efficiency to be 4 EEbest = g ( xee ) and EEworst = g ( x ) respectivey. Definition (Proportiona Fairness): Assume that s and s EE are the objective vaues of oad baance and energy efficiency corresponding to a soution that achieves some tradeoffs between the two, then the soution is proportionay fair if and ony if it satisfies the foowing equation: worst s EEworst see =. worst best EEworst EEbest Definition means that in the soution with a fair tradeoff, both the oad baance and energy efficiency objectives obtain the same percentage (or reative) improvement. It is worth noting that athough the vaues of the two utiity functions may have significanty different orders of magnitude, and/or their optimization spaces have different sizes, the above definition of a fair tradeoff uses a reative term and as a resut, each objective function wi resut in a proportiona improvement over its worst case performance. Definition 3 (MaxMin Fairness): Assume that s and s EE are the objective vaues of oad baance and energy efficiency corresponding to a soution that achieves some tradeoffs between the two, then, the soution is MaxMin fair iff worst s EEworst s ( s, see ) = arg max min{, }. ( s, s) S worst best EEworst EEbest In other words, a MaxMin fair soution maximizes the reative performance improvement of the objective who gets ess reative performance improvement, and accordingy, tries to minimize the performance gap between the two objective functions (in terms of their reative performance improvement). Theorem : For a Pareto efficient soution (s, s EE ), if this soution is proportiona fair, it must aso be maxmin fair. We prove it by contradiction by assuming that a Pareto efficient soution (s, s EE ) is proportiona fair but not maxmin fair. Let another Pareto soution (s, s EE ) (s, s EE ) be maxmin fair soution such that worst s EEworst see min{, } > worst best EEworst EEbest worst s EEworst see min{, } =, worst best EEworst EEbest worst s EEworst see = worst best EEworst EEbest the foowing inequations must be satisfied worst s worst s > worst best worst best EEworst see EEworst see and >. EEworst EEbest EEworst EEbest These two inquaities impy that (s, s EE ) is not a Pareto efficiency soution, which is a contradiction. Theorem means that we can focus on finding a tradeoff soution which is Pareto efficient and satisfies the proportiona fairness property hereafter.
5 B. Why Nash Bargaining The ey idea of our wor is to find a utiity aocation method such that the fairness between mutipe objectives can be guaranteed. We adopt the Nash bargaining framewor because it not ony is a cassica cooperative game framewor which pursues the fairness between the payers in the game, but aso wi obtain a Pareto efficient soution. Since a soution using Nash bargaining is determined by the threat points which can be treated as the performance threshods of each payer, a ey issue is to determine the threat point of a Nash bargaining probem such that the fairness between different objectives can be guaranteed. VII. TRADING OFF LOAD BALANCE AGAINST ENERGY EFFICIENCY USING NASH BARGAINING In this section, we reaize the tradeoff between oad baance and energy efficiency based on Nash bargaining framewor. In such a framewor, each payer announces its threat vaue to improve its own performance, so that we ca it threat vaue game. In Subsection VII.A, we introduce this game and anayze it in depth. Our anaysis shows that such game has an infinite number of Nash equiibriums and when each payer sefishy determines threat vaue, it wi prevent the agreement. To achieve the agreement, we modify the threat vaue game to be a repeated process and design a mechanism to guarantee the agreement in Subsection VII.B. Through this mechanism we can easiy determine the initia threat point (and a subsequent threat points) which can resut in fair soution. Since the optimization probem (5) is not in a convex form, we wi show how to transate it into a convex form which can be soved more efficienty in Subsection VIII.C. A. Nash Bargaining Mode and Threat Vaue Game Let a and a EE be some chosen threat vaues of oad baance and energy efficiency, respectivey. The Nash bargaining soution can be derived by the foowing optimization probem: maximize ( a f )( a g ) subject to (5) EE x = x, i, j: i j : p x c E x = d i, j i f a g a EE Obviousy, both payers can change its threat vaue to improve its own performance. But neither of them shoud be aowed to change the threat vaue arbitrariy, otherwise it may prevent the agreement (eading to no feasibe soution to (5)). To anayze such a game in more depth, we first have the foowing definition. Definition 4: A threat vaue game is a tupe G=(N, (A i ) i {,EE}, (c i ) i {,EE}), where A i is the set of avaiabe strategies for payer i {,EE}. In our mode, A =[ best, worst ] and A EE =[EE best, EE worst ]. We use a i to denote a specia strategy for payer i. c i is the cost for payer i {,EE}. The vaue of c i depends on the soution of optimization probem (5). 5 The threat vaue can aso be treated as the performance threshod of each payer to sign the agreement. If there exist no feasibe soution to (5) (which means no agreement can be achieved), c i = for each payer. Otherwise, c = f and cee = g where x ={x } is the soution of (5). Lemma : For each payer in the threat vaue game, reducing its threat vaue uniateray wi improve its performance or prevent the agreement. Without oss of generaity, we assume that payer energy efficiency reduces its threat vaue uniateray, and as a resut, the threat point is moved from (a, a EE ) to (a, a EE), where a EE > a EE. If there exists no feasibe soution for the threat point (a, a EE), the agreement wi be broen, and the cost for both payers is. Otherwise, we denote the optima soution before and after payer energy efficiency reduces its threat vaue by x and x respectivey, then we now: ( a f ( x))( aee g ( x)) > (6.) ( a f )( a g ) and EE ( a f ( x))( a g ( )) EE x < ( a f )( a g ( )) EE x From (6.)/(6.), we can see aee g ( x) aee g > a g ( x) a g EE EE (6.3) can be converted to be ( a a )( g g ( x)) > 0 EE EE Since a EE > a EE, we obtain g ( x) > g (6.) (6.3) (6.4) Definition 5: Let S be the set of a the possibe cost pairs for two payers. We say that the cost pair (a, a EE ) S is dominated by (a, a EE) S iff: a L B a and a EE a EE a L B>a or a EE >a EE Definition 6: A Pareto frontier is a subset of S, such that a the points in the Pareto frontier are not dominated by any points in S. Theorem : Every point (a, a EE ) in Pareto frontier is a Nash equiibrium point in the threat vaue game. From Lemma, we now that if a payer increases its threat vaue uniateray, it wi increase its cost, so that it is cost inefficient for any payer to increase its threat vaue uniateray. On the other hand, if a payer reduces its threat vaue, there wi be no feasibe soution, which can be proven by contradiction as foows: Without oss of generaity, we assume that payer energy efficiency reduces its threat vaue to a EE < a EE and there exists a feasibe soution x to (5). From Lemma, we now that
6 Compute traffic routing by soving probem (5) Soution to probem (5) g < g( x). Combining with (6.), we have f > f ( x) (7.) where x is the optima soution to (5) before the payer energy efficiency reduces its threat vaue. Since (a, a EE ) is not dominated by any point in S, a = f ( x) (7.) and aee = g ( x) must be satisfied. Otherwise (a, a EE ) wi be dominated by the cost pair associated with the soution of (5). Substituting (7.) into (7.) eads to f > a Threat point which contradicts to the constraint of (5). B. How to derive a fair soution From Lemma, when each payer pursues its performance sefishy, each of them wants to set its threat vaue to be the minima vaue in its own strategy space. Unfortunatey, the threat vaue combined with the best performance of each payer wi mae the optimization probem in (5) infeasibe and prevent the agreement achievement. In order to avoid such an undesirabe situation, we modify the mode formuated in the previous subsection to be a repeated Nash bargaining probem as shown in Fig.. In this repeated Nash bargaining mode, each payer changes its threat vaue stepwise to optimize it performance. Let (a (), a () EE) denote the threat point during the th iteration, and x () be the optima soution to (5) corresponding to threat point (a (), a () EE),. We assume that each payer updates its threat vaue during the th iteration as a function of its current threat vaue (a () for payer and a () EE for payer EE) and the optima soution to (5) (x () ). Therefore, we have, ( ) ( ) ( ) a = h ( a, x ) (8.) ( ) ( ) ( ) a = h ( a, x ) (8.) EE EE EE Update threat vaue to optimize performance Initia threat point Fig. Procedure of repeated threat vaue game To guarantee the fairness of the soution, we first find a threat point to induce a fair soution as foows. Theorem 3: If the cost function of both oad baance and energy efficiency, i.e. f ( ) and g ( ) for a, are continuous, then the threat point of probem (5) ( worst, EE worst ) wi yied a fair soution. Let x={x } be the soution to (5) with (a, a EE )= ( worst, EE worst ) and we denote the cost of oad baance and energy updating of each payer to satisfy 6 efficiency associating with the soution to probem (5) by s and s EE respectivey. Construct a new bargaining probem (S, d ), by setting see = αsee β and dee = αdee β, where worst best best EEworst worst EEbest α = and β = EEworst EEbest EEworst EEbest then the cost of oad baance and energy efficiency in (S, d ) are denoted by s and s EE, respectivey. In such a bargaining probem, we have EE = and EE =, worst worst best best where best and worst (or EE best and EE worst ) are in the best and worst case oad baance (or energy efficiency) cost probem respectivey. Due to the fact that f ( ) and g ( ) are continuous and nondecreasing for a, the cost function of oad baance and energy efficiency have the same range in (S, d ). If (y, y EE ) is a feasibe soution of (S, d ), then we consider the foowing equation group f( x) = yee g( x) = y (9) x = d i, j i x = x, i, j: i j : p Because there are V V p variabes but ony i= j=, j i E { } E V ( V ) constraints in equation group (9), a soution must exist. It means that (y EE, y ) is aso a feasibe outcome of (S, d ). Hence, s =s EE must be satisfied in (S, d ). Based on the Nash bargaining s property of invariance to equivaent utiity representation, we obtain best = α EEbest β (0.) s = α see β (0.) worst = α EEworst β (0.3) From (0.3)(0.), worst s = α( EEworst see ) (.) Simiary, (0.3)(0.) can yied worst best = α( EEworst EEbest ) (.) From (.)/(.), we obtain worst s EEworst see = EE EE worst best worst best Theorem 3 gives an initia threat point which can yied a fair tradeoff between oad baance and energy efficiency. Such a threat point is easy to determine as it correspond to the worstcase performance of oad baance and energy efficiency, respectivey, independent of the other objective. The foowing mechanism is designed to prevent payers from deviating from such a fair soution when they are optimizing its performance sefishy by changing its threat vaue. Mechanism : Let (a (), a () EE) denote the threat point during the th iteration and x () be the optima soution to (5), we initiaize the threat point at ( worst, EE worst ) and constraint the threat vaue
7 a f a a ( a f ) (.) ( ) ( ) ( ) ( ) E ( ) ( ) = ( ) ( ) aee g ( ) ( ) E ( ( ) ( ( ) a EE aee = aee g x )) (.) With the above two constraints, each payer can ony caim to occupy haf of the performance gap between its threat vaue and its cost corresponding to the soution of (5). Lemma : Let x be the optima soution of optimization probem (5) associated with threat point (a, a EE ), then x is aso the optima soution of probem (5) associated with the threat point ( ( a f ), ( aee g )). x is a feasibe soution for the probem (5) when the threat point is (a, a EE ), so f a then Simiary, f x a f x ( ) ( ( )) g x a g x ( ) ( EE ( )) The above means that x is aso a feasibe soution to the optimization probem (5) associated with the threat point ( ( a f ), ( aee g )). On the other hand, the objective of (5) is equivaent to maximize og( a f ) og( a g ) EE Since x is the optima soution of (5) associated with the threat point (a, a EE ), so we have f g = 0 (3) a f a g EE When the threat point is ( ( a f ), ( aee g )), we shoud maximize F( x) = og( ( a f ) f ( x)), og( ( aee g g ( x)) The gradient of F(x) is f ( x) F( x) = ( ( a f x )) f g ( x) ( ( aee g x ) g 7 We shoud note that x=x wi yied F( x) = 0. Due to the fact that there is ony one optima soution for each Nash bargaining probem, accordingy, x is aso the optima soution of (5) associated with the threat point ( ( a f ), ( aee g )). Theorem 4: If each payer optimizes its performance sefishy by changing its threat vaue under Mechanism, traffic routing wi be constant as if the threat point was never changed. During th iteration, each payer can at most reduce its threat vaue to ( ( ) ( ( ) a f x )) and ( ( ) ( ( ) aee g x )) which are arger than ( ) ( ) f and g respectivey. So the reduction of threat vaue wi not ead to agreement breadown and an infinite cost to the payers. According to Lemma, payer wi reduce its threat vaue to ( ) ( ) be ( ( a f x )) and payer EE wi reduce its threat vaue to be ( ( ) ( ( ) aee g x )) during th iteration. From Lemma, we now that the optima soution of (5) wi be the same as if the threat point was never changed. Coroary : If we set the initia threat point at any points on the ine connecting threat point (a, a EE ) and the outcome at the optima soution of (5) corresponding to this threat point, we wi derive the same soution as if the initia threat point is (a, a EE ). This coroary can be proven in the same method as Lemma. Assume that x is the optima soution to (5) corresponding to the threat vaue (a, a EE ) and its associated outcome is ( f ( x), g ( x)). A point on the ine connecting (a, a EE ) ( f( x), g) can be represented by ( λa ( λ) f ( x), λaee ( λ) g ) and weher λ [0,] Setting above point as the threat point, the objective of (5) can be varied to be { λ[ a f ] ( λ)[ f ( x) f ]} { λ[ a g ] ( λ)[ g ( x) g ]} We shoud maximize Gx ( ) = og{ λ[ a f( x)] ( λ)[ f( x) f( x)]} og{ λ[ a g ] ( λ)[ g ( x) g ]} The gradient of G(x) is
8 g( x) f( x) Gx ( ) = λ[ a f ] ( λ)[ f ( x) f ] λ[ a g ] ( λ)[ g ( x) g ] Since x is the optima soution of (5) associated with the threat point (a, a EE ), so we have f( x) g( x) = 0 a f ( x) a g ( x) EE Let x=x, we have f( x) g( x) Gx ( ) = = 0 λ[ a f ( x)] λ[ a g ( x)] Due to the fact that there is ony one optima soution for each Nash bargaining probem, accordingy, x is aso the optima soution of (5) associated with the threat point ( λa ( λ) f ( x), λa ( λ) g ). EE Coroary tes us that a the points on the ine connecting ( worst, EE worst ) wi ead to a fair soution by setting it as the threat point of (5). Coroary : If the initia threat point is not set as ( worst, EE worst ) or any point aong the ine connecting ( worst, EE worst ) to the outcome at the optima soution of (5), then the soution wi not converge to the fair tradeoff. This coroary can be proofed by contradiction. Assume that the threat vaue (a, a EE ) is not on the ine connecting ( worst, EE worst ) to the outcome at the optima soution of (5), but it can induce a fair tradeoff. That is to say and f g EE = 0 a f a g f g worst worst = 0 f EE g where x is the fair trade off soution. The above two equations can induce a f aee g = f EE g f x g x and ( worst, EE worst ) worst worst It means (a, a EE ), ( ( ), ( )) are on the same ine, where contradiction occurs. Intuitivey, this is because starting with a different threat point than ( worst, EE worst ) hides some of the payers optimization space information. Note that athough this coroary states that the fair soution cannot be derived from any arbitrary initia threat points, it has no negative impication as one can easiy find (and use) the initia threat point ( worst, EE worst ). 8 C. Conversion to Convex Optimization Form The optimization probem (5) is not in a convex optimization form. To sove it more efficienty, we shoud convert it into the form of standard convex optimization form without changing its soution. Theorem 5: If the optimization probem (5) is feasibe, its soution wi be the same as probem (4): maximize og( a t) og( aee tee) (4) subject to: The constraints in (5) t f( x) tee g( x) To maximize objective of (4), the variabe t and t EE shoud be as itte as possibe. Therefore, t = f and t = g must be satisfied in the optima soution. So that EE probem (4) is equivaent to maximize og( a f ) og( a g ) (5) EE When the probem (5) is feasibe, we have f( x) a and g( x) aee. In this case, the variabe maximizing the objective of (5) aso maximize (5), because og( ) is an increasing function of its argument on R. Theorem 6: If f ( ) and g ( ) are both convex functions, probem (4) is a convex optimization probem. Theorem 6 can be verified easiy by checing that it does satisfy the definition of convex optimization. Theorem 5 and Theorem 6 guarantee the Nash bargaining soution can be soved efficienty. VIII. CASE STUDIES In this section, we wi present two case studies of our methods. We wi first appy our method in a simpe parae ins networ to reaize a fair tradeoff between oad baance and energy efficiency. It not ony presents how our method wors but aso verifies its correctness. We wi then appy our method in a reaistic networ NSFNET to show its practicabiity. A the computations are carried out on a computer with DuoCore.0 GHz inte CPU using CVX. [8]. A. Case Study in a Simpe Networ In this subsection, we appy our mode in a simpe parae ins networ as shown in Fig. to iustrate how our method wors. In this networ, there are 3 parae ins from node s to node d with capacity of 000, 000 and 500 (units) respectivey. We aso assume that there is a demand from s to d for 900 units of capacity in the networ. As discussed in Sections III and IV, we set the cost of oad x baance and energy efficiency to be f ( x) = and c x
9 x.5 g = ( ) for a. From the optimization probem (), 000 we can find that the best case cost for oad baance is and this soution corresponds to the worst case cost for energy efficiency of Simiary, by soving probem () we can aso determine the best case cost for energy efficiency and worst case cost for oad baance to be and respectivey. Fig. 3 shows the Pareto frontier and these two imitation scenarios. Obviousy, such two cases are the bounds of Pareto frontier. Note that we choose to use this simpe exampe where the costs for oad baance and energy efficiency have comparabe vaues in order to show (beow) that even in such a case, our method is more usefu than the conventiona method that uses an AOF. To get a fair soution, we set the threat point to be (0.8555, 0.63) and use the optimization probem (5) to derive a fair soution. In the soution, the traffic carried by each in is (x, x, x 3 ) = (5.603, , ) and the cost for oad baance and energy efficiency are 0.79 and 0.57 respectivey. We shoud note the foowing reationship: = = 77% It means that each payer improve its performance with the same percentage of its optimization space, which shows that the soution derived by our method is fair to both objectives. To verify the correctness of our mechanism designed for the repeated threat vaue game, we aso appy it in the simpe networ shown in Fig. with the initiaized threat point ( worst, EE worst ). Fig. 3 shows how the threat point moves when one iteration executed after another and the performance of each payer in each iteration. It aso verifies that as ong as the initia threat point is chosen as ( worst, EE worst ), the resut wi converge to a fair tradeoff soution. If one uses the cassic AOF method to get a tradeoff for this probem, she may set the same weight to both objectives because they have the same order of magnitude. In this case, the resuting cost for oad baance is and the cost for energy efficiency s , c =000, c =000 3, c 3 =500 Fig. Parae ins networ used in case study Pareto Frontier Convergence Trace of Threat Vaue Outcome of Our Method Limiation Scenarios d is Though this soution is Pareto efficient, energy efficiency obtains ess performance improvement (70%) than oad baance (83%). In our method, we set the performance threshod of each objective to be its worst case performance. If one uses the conventiona approach whereby one objective is treated as a constraint, she obviousy cannot use the worsecase performance as the threshod vaue for any of the objective, but what maes such an approach difficut is that she aso wi not now which other threshod vaue is the most appropriate. She may choose a medium vaue between the worst and best cases for exampe, but such a choice is ad hoc at the best and cannot be considered fair. B. Case Study in a NSFNET In this section, we wi use our proposed method to reaize a fair tradeoff between oad baance and energy efficiency in the NSFNET bacbone networ (shown in Fig. 4). In this networ, each in is bidirectiona and without oss of generaity, we assume that the capacity of each in is 45Mbps (such capacity is offered by NSFNET during 99 and 995 [9] and was chosen to simpify our computation ony). To route the demand in the networ, we find two in disjointed routes between each node pairs and we add a 0Mpbs demand between every pair of the six supercomputer sites (SDSCNET, NCSA, CNSF, PSCNET, JVNC and NCAR). In order to show that our method can derive a fair soution even when each objective has a different order of x.5 magnitude, we set the energy curve of in to be g = ( ). 45 In this case study scenario, the best case cost for oad baance is whie the worst case cost is , which is much worse than the best case. For energy efficiency, the best and worst case costs are 5. 4 and respectivey. This means that the optimization space of oad baance is much arger than that of energy efficiency. Note that if the conventiona approach based AOF were to be used in this case, one woud not now how to set an appropriate weight for each objective. More specificay, if she uses more or ess the same weights for the two objectives, the oad baance wi get much more performance improvement than energy efficiency, because it has a arger optimization space than energy efficiency, which woud be unfair to energy efficiency. In short, a fair soution is difficut to obtain by using the AOF method. In our method, we set the threat point to be ( , 5.833) to derive a fair soution. In the soution, the cost of oad baance is whie the cost of energy efficiency is very cose to (the accuracy is to 04 ). This represents that both objectives get % optimization space. Energy Efficiency Cost BARRNet NorthwestNet Westnet NCAR/USAN CNSF/NYSERNet Merit PSCNET JVNC SDSCNET MIDnet NCSA/UIUC Sesquinet SURAnet Load Baance Cost Fig.3 Trace of threat point changing 9 Fig. 4 Topoogy of NSFNET
10 IX. CONCLUSION In this paper, we have studied how to achieve a fair tradeoff between oad baance and energy efficiency in traffic engineering. Different from the traditiona mutiobjective optimization methods which either construct an aggregate objective function (AOF) or treat one of the objectives as a constraint of the probem, we have anayzed such a probem from a game theoretic perspective. More specificay, we have treated the two objectives as two virtua payers in a socaed threat vaue game, who negotiate with each other in order to achieve an agreement under the Nash bargaining framewor. In such a game, each payer can announce its threat vaue to optimize its performance and our anaysis have shown that the number of Nash equiibriums can be infinite and each payer determines its threat vaue wi prevent an agreement, so as to induce an infinite cost to both of them. To avoid such an undesirabe outcome, we have designed a mechanism that can not ony reach an agreement but aso ead to a fair tradeoff between oad baance and energy efficiency. In addition, it is very easy to find a the initia threat points which can be used to get the fair tradeoff soution in our method. Athough this wor focuses on achieving a fair tradeoff between oad baance and energy efficiency in traffic engineering, it aso provides some usefu insights into the other mutiobjective optimization probems. Pubications [7] J.F. Nash, The bargaining probem, Econometrica, vo. 8, pp. 55 6, 950. [8] M. Grant and S. Boyd, cvx Users Guide for cvx version., [9] L.M. David and H. Braun, The NSFNET Bacbone Networ, [0] Y. Zhao et a., "Load Baance vs Energy Efficiency in Traffic Engineering: A Game Theoretica Perspective", Technica Report 00, Department of CSE, SUNY Buffao (avaiabe at pdf) REFERENCES [] D. Awduche, MPLS and Traffic Engineering in IP Networs, IEEE Communications Magazine, vo. 37, no., pp. 4 47, Dec [] B. Fortz and M. Thorup, Internet Traffic Engineering by Optimizing OSPF Weights, in Proc. 9 th IEEE Conf. on Computer Communications (INFOCOM), 000, pp [3] S. Secci, K. Liu, K. Rao, and B. Jabbari, Resiient Traffic Engineering in a TransitEdge Separated Internet Routing, in Proc. IEEE ICC, 0. [4] G. Shrimai, A. Aea and A. Mutapcic, Cooperative Interdomain Traffic Engineering Using Nash Bargaining and Decomposition, IEEE Trans. on Netw. vo. 8, no., pp , Apri 00 [5] M. Andrews, A.F. Anta, L. Zhang and W. Zhao, Routing for Energy Minimization the Speed Scaing Mode, in Proc, 9 th IEEE Conf. on Computer Communication (INFOCOM), 00, pp. 9 [6] E. Yetginer and G.N. Rousas, Power Efficient Traffic Grooming in Optica WDM Networs, in Proc. 5 th IEEE Goba Teecommunication Conf. (GLOBECOM), 009, pp. 6 [7] A. Wierman, Lachan L. H. Andrew, and Ao Tang. Poweraware speed scaing in processor sharing systems, in Proc. 8 th IEEE Conf. on Computer Communication (INFOCOM), 009, pp. 9 [8] Enhanced Inte Speed Step Technoogy for the Inte Pentium M processor. Inte White Paper , 004 [9] R. E. Steuer. Mutipe Criteria Optimization: Theory, Computation, and Appication. New Yor: John Wiey & Sons, Inc. ISBN X, 986. [0] D. Awduche, J. Macom, J. Agogbua, M. O De, and J.McManus, RFC 70: Requirements for Traffic Engineering Over MPLS, September 999 [] M. Xia, M. Tornatore, Y. Zhang, P. Chowdhury, C.U. Marte and B. Muherjee. Green Provision for Optica WDM Networs, IEEE Jour. of See. Top. in Quan. Eec. vo. 7, no., pp , Match 0 [] S. Seetharaman, V. Hit, M. Hofmann, and M. Ammar, Resoving CrossLayer Confict between Overay Routing and Traffic Engineering, IEEE/ACM Tran. Netw. vo.7, pp , 009 [3] L. Qiu, R. Y. Yang, Y. Zhang, and S. Shener, On sefish routing in internetie environments, in Proc. ACM SIGCOMM, 003, pp [4] Y. Liu, H. Zhang, W. Gong, and D. Towsey, On the interaction between overay routing and traffic engineering, in Proc. IEEE INFOCOM, 005, pp [5] W. Jiang, Z.S. Rui, J.Rexford, M. Chiang, Cooperative Content Distributed and Traffic Engineering in an ISP Networ, in Proc. SIGMETRICS/Performance, 009, pp. , June 009. [6] M.D. Davis. Game Theory, A Nontechnica Introduction. New Yor: Dover 0
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