A Unified Framework for Max -Min and Min-Max Fairness with application Bozidar Radunoic, Member, IEEE, and Jean-Yves Le Boudec, Fellow, IEEE IEEE/ACM TRANSACTIONS ON NETWORKING, VOL. 15, NO. 5, OCTOBER 2007 R96725005 謝友仁 R96725025 王猷順 R96725037 陳冠瑋 指導老師：林永松 老師

A Unified Framework for Max -Min and Min-Max Fairness with application Author Bozidar Radunoic Received the B.S. degree in electrical engineering from the University of Belgrade, Serbia, in 1999, and the Doctorate in 2005 from EPFL, Switzerland. Participated in the Swiss NCCR project terminodes.org. 第一位作者在1999年時取得電機學士，到2005年時於EPFL取得博士學位，除此之外，該作者也參與Swiss NCCR project。他的研究領域在於architecture and performance of wireless ad-hoc networks His interests are in the architecture and performance of wireless ad-hoc networks. 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

A Unified Framework for Max -Min and Min-Max Fairness with application Author Jean-Yves Le Boudec Graduated from Ecole Normale Superieure de Saint-Cloud, Paris, and received the Doctorate in 1984 from the University of Rennes, France. In 1987, he joined Bell Northern Research, Ottawa, Canada. In 1988, he joined the IBM Zurich Research Laboratory. 第二位作者在1984年時取得博士學位，1987年加入Bell Northern Research，1988年十，也參與IBM Zurich Research Laboratory，至1994年，於EPFL擔任教授的職務 In 1994, he became a Professor at EPFL, where he is now a full Professor. 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

A Unified Framework for Max -Min and Min-Max Fairness with application Abstract Max-min fairness is widely used Based on the notion of bottlenecks A unifying treatment of max-min fairness First, we observe that the existence of max-min fairness A geometric property of the set of feasible allocations 點出Max-min主要是利用bottleneck的觀念，在尋找的過程中以bottleneck為跳板 該篇也對Max-min fairness 提出一個統一的架構 就Max-min fairness 的存在性而言，是以feasible allocations的幾何特性決定 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

A Unified Framework for Max -Min and Min-Max Fairness with application Abstract (cont’d) Second, we give a general purpose centralized algorithm Max-min Programming Complexity is the order of N linear programming steps in RN Free disposal property and Water Filling Algorithm Mutatis mutandis to min-max fairness 介紹名詞Max-min Programming 詳細的複雜度分析在後面介紹 稍微題Free disposal 、Water Filling 和Max-min Programming的關係 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

A Unified Framework for Max -Min and Min-Max Fairness with application Max-Min Fairness Allocating as much as possible to users with low rates x1 7 用圖代Water Filling的觀念 8 3 x2 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application 8

Microeconomic Approaches to Fairness Max-min fairness is closely related to leximin ordering. Free disposal property A bottleneck argument can be made 稍為題leximin ordering並以前頁圖的斜線為例 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Bottleneck and Water-Filling If each flow has a bottleneck link, then the rate allocation is max-min fair. The bottleneck argument is often used to prove the existence of max-min fairness. 說明Bottleneck 和 Water-Filling的關係 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Water-filling algorithm (WF) Rates of all flows are increased at the same pace, until one or more links are saturated. 正試舉例介紹Water-filling 1 0.5 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

When Bottleneck and Water-Filling Become Less Obvious Point-to-point multi-path routing scenario. All links have capacity 1. 介紹Water-filling不能直接解決(須轉換)的例子 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

When Bottleneck and Water-Filling Become Less Obvious (cont’d) Re-interpret the original multi-path problem as a virtual single path problem. x1, x2, y1, y2≥0, x1=y1+y2, y2+x2≤1, y1≤1 x2≤1, x1+x2≤2 y1 y2 描述轉換方法，並recall water-filling的斜線精神 x2 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

When Bottleneck and Water-Filling Do Not Work Equalize load on the servers while satisfying the capacity constraints. Min-max fair Cannot decrease a load on a server without increasing a load of another server that already has a higher load. 7 8 3 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

A Unified Framework for Max -Min and Min-Max Fairness with application Our Findings The existence of max-min fairness On algorithms to locate the max-min fair allocation Max-min Programming (MP) Relation between the general MP algorithm and the existing WF algorithm Free disposal property Form of feasible set 小結 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

Definitions and Uniqueness If is a min-max fair vector on , then is max-min fair on and vice versa If a max-min fair vector exists on a set , then it is unique 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Max-Min Fairness and Leximin Ordering Leximin maximum is not necessarily unique, if a vector is leximin maximal, it is also Pareto optimal If a max-min fair vector exists on a set , then it is the unique leximin maximal vector on it follows that the max-min fair vector, if it exists, is Pareto optimal 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Existence and Max-Min Achievable Sets Max-min fair vector does not exist on all feasible sets A set is max-min achievable if there exists a max-min fair vector on In the example on the left, both points (1, 3) and (3, 1) are leximin maximal in this example. In the example on the right, point (3, 1) is the single leximin maximal point. 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

The Max-Min Programming (MP) Algorithm finds the smallest component of the max-min fair vector by maximizing the minimal coordinate minimal coordinate is fixed, and the dimension corresponding to the minimal coordinate is removed Repeated until all coordinates are fixed vector obtained in such way is indeed the max-min fair one 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

The Max-Min Programming (MP) Algorithm (Cont’d) finds the smallest component minimal coordinate is fixed 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

The Max-Min Programming (MP) Algorithm (Cont’d) maximizes the minimal coordinate in each step until all coordinates are processed If is compact and max-min achievable, the above algorithm terminates and finds the max-min fair vector on in at most steps. 第一部分 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Max-Min Programming Numerical Example 1 Step 1: Step 2: 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Max-Min Programming Numerical Example 2 Step 1: The min-max fair rate allocation is thus (4,3) 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Max-Min Programming with Max-min does not exist max-min fair allocation does not exist Step 1: This point is neither leximin maximal, nor Pareto optimal 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

The Water-Filling (WF) Algorithm free disposal applies to sets where each coordinate is independently lower-bounded Let be a max-min achievable set that satisfies the free disposal property. Then, at every step, the solutions to problems and are the same WF terminates and returns the same result as MP, namely the max-min fair vector if it exists 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

The Water-Filling (WF) Algorithm (Cont’d) set each xi to be the smallest component or its original minimum 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

The Water-Filling (WF) Algorithm (Cont’d) 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Water-Filling Algorithm Numerical Example Free disposal property is a sufficient but not a necessary condition for MP to degenerate to WF C1 = 3, C2 = 3, C3 = 4, X1 + X2 ≥ 3 same shape and orientation as in Fig. 4, but it is translated to the left such that it touches both X2 and X1 axes Without the free disposal property, still solved by WF in the single step 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Complexity of the Algorithms in Case of Linear Constraints Χ is a n-dimensional feasible set defined by m linear inequalities Max-Min Programming Complexity the overall complexity is O(nLP(n,m)) LP(n,m) is the complexity of linear programming Linear programming the worst case: exponential complexity most practical cases: polynomial complexity Water-Filling Algorithm Complexity N steps, each of complexity O(m), overall O(nm) 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Complexity of the Algorithms in Case of Linear Constraints Χ is defined implicitly, with an ι–dimensional slack variable (ex: multi-path case) Max-Min Programming on implicit sets The complexity is O(nLP(n,m)) Water-Filling Algorithm on implicit sets it might be possible to construct an implicitly defined feasible set that cannot be converted to an explicit form in a polynomial time paper interested in explicitly finding the values of the slack variables at the max-min fair vector, and the complexity is O(nm) 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

A Unified Framework for Max -Min and Min-Max Fairness with application Example Scenarios Load Distribution in P2P Systems a minimal rate a user needs to achieve an upper bound on each flow given by a network topology and link capacities Maximum Lifetime Sensor Networks to minimize the average transmitting powers of sensors we look for min-max fair vector of average power consumptions of sensors. 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Load Distribution in P2P Systems represent the feasible rate set as be the total loads on the servers the flows from the servers to clients the total traffic received by clients the capacities of links the minimum required rates of the flows where A, B, C ≥ 0 are arbitrary matrices defined by network topology and routing. 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Load Distribution in P2P Systems Each server is interested in minimizing its own load It is natural to look for the min-max fair vector Since set is convex, it is min-max achievable Use Max-min Programming to solve this problem Y1 Y2 A代表為所有capacity的矩陣，B代表所有server的loading矩陣，C代表為client所收到的總traffic矩陣。 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Maximum Lifetime Sensor Networks A network has a certain minimal amount of data to convey to a sink consider different scheduling and routing strategies that achieve this goal Each of these strategies yields different average power consumptions Look for min-max fair vector of average power consumptions of sensors 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Maximum Lifetime Sensor Networks The maximum rate of information can achieve is Denote with the average rate of sd link throughout a schedule Χ is a set of feasible , that is such that there exists a schedule and power allocations that achieve those rates calculate the average power dissipated by a node during a schedule Time Power Rate Total interference 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Maximum Lifetime Sensor Networks 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Maximum Lifetime Sensor Networks P = 1 N = 1 hS1D1 = 1 hS1D2 = 1 hS2D1 = 10 hS2D2 = 0.7 M1 = 0.6 M2 = 0.4 P1=0, P2= -0.62; min(P1, P2)=-0.62 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

Agenda Introduction Max-min and Min-max Fairness Max-min Programming and Water-filling Example Scenarios Conclusion

A Unified Framework for Max -Min and Min-Max Fairness with application Conclusion We have elucidated the role of bottleneck arguments in the water-filling algorithm, and explained the relation to the free disposal property We have given a general purpose algorithm (MP) for computing the max-min fair vector whenever it exists, and showed that it degenerates to the classical water-filling algorithm, when free disposal property holds We have focused on centralized algorithms for calculating max-min and min-max fair allocations 2017/4/24 A Unified Framework for Max -Min and Min-Max Fairness with application

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