Worst-case analysis of Non-Cooperative Load Balancing

Transcription

1 Worst-case analysis of Non-Cooperative Load Balancing O. Brun B.J. Prabhu LAAS-CNRS 7 Av. Colonel Roche, Toulouse, France. ALGOGT 2010, Bordeaux, July 5, Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

2 Outline 1 Introduction 2 Problem statement 3 Worst Case Scenario 4 Lower bound on Price of Anarchy 5 Conclusion Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

3 Server Farms Diverse applications : e-service industry, database systems, grid computing,... What is the optimal routing policy? The complexity of the system may call for decentralized routing algorithms. DISPATCHER DISPATCHER DISPATCHER DISPATCHER Centralized architecture. Decentralized architecture. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

4 Non-cooperative Load Balancing Routing decisions are made by each traffic source independently, according to its own individual objectives, Game theory provides the systematic framework to study and understand such problems, Convergence to equilibrium points where unilateral deviation does not help any traffic source to improve its performance Can we provide performance guarantees for these decentralized routing algorithms? Study the Price of Anarchy [Papadimitriou 1998] Worst case ratio between the performance (mean delay) obtained by the non-cooperative setting equilibrium and the global optimal solution. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

5 Contributions Identification of the worst-case scenario for decentralized routing Worst-case scenario is a potential game Procedure for computing lower bounds on the Price of Anarchy Lower bounds for two specific functions Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

6 Notations Set C of traffic classes Set S of servers λ λ i i x i,j j r j, c j, u j = cj r j y j = i x i,j λ K K S Total traffic intensity : λ = i λ i. Assumptions : u 1 u 2... u S and λ 1 λ 2... λ K. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

7 Non-Cooperative Routing Each class seeks to minimize its own (weighted) total cost λ1 1 1 λi i xi,j j Min xi D i K = j S u j x i,j φ(ρ j ) λk K S s.t. j S x ij = λ i x ij 0 j S Existence of a unique NEP (Nash Equilibrium Point) : x i = argmin z D i K (x 1,...,x i 1, z, x i+1,...,x K ) i C Global cost at the NEP : D K (λ, r, c) = i C Di K = j S u j y j φ(ρ j ) Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

8 Price of Anarchy Centralized setting (globaly optimal solution) 1 λ 1 y j j Min yj D 1 = j S u j y j φ(ρ j ) S s.t. j S y j = λ y j 0 Price of Anarchy : D K (λ, r, c) PoA = sup λ,r,c ; PoA [1, ) D 1 (λ, r, c) Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

9 Type of cost functions Type-B functions [Orda et al. 1993] Type I φ : IR + IR + with lim φ φ(ρ) Examples : polynomial, exponential Type II φ : [0, 1) IR + with lim φ 1 φ(ρ) Examples : M/G/1/PS delay function, M/Pareto/1/SRPT in heavy traffic Existence and Uniqueness of NEP follows from [Orda et al 1993] Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

10 Properties of the Nash equilibrium C j = { } i C : xij > 0 : classes which route traffic to server j. S i = {j S : x ij > 0} : servers to which class i routes traffic. Lemma The Nash equilibrium is such that, λ i < λ k x ij < x kj, j S i S k and, u n < u m C m C n Water-filling structure : there exist thresholds S 1 S 2... S K such that S i = {1,...,S i }. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

11 Worst Case Theorem Assume λ = K i=1 λ i is fixed. The global cost D K (., r, c) achieves its maximum value when λ = λ =, i.e. λ i = λ/k for all i (symmetric case). PoA Numerical Example : K = 2 classes S = 2 servers α (%) λ 2 /λ 1 (%) Consequence : PoA = sup λ,r,c D K (λ, r, c) D 1 (λ, r, c) = sup r,c D K (λ =, r, c) D 1 (λ, r, c) Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

12 Outline of the proof Transformation λ ˆλ : increase λ 1 and decrease λ K Start by comparing D K (λ, r, c) and D K (ˆλ, r, c) with S i = Ŝ i. ŷ 1 > y 1, ŷ 2 < y 2 D K (ˆλ) > D K (λ) DK(λ) DK(λ) ŷ 1 = y 1, ŷ 2 = y 2 D K (ˆλ) = D K (λ) 0 λ ˆλ 1/2 λ 2 /λ 1 S 1 S 2. S 1 = S 2. Show that we can find a step for the transformation such that the cost increases even if the set of servers changes. 0 λ 2 /λ 1 λ ˆλ 1/2 Show that the symmetric vector can be reached from any vector using a finite number of such transformations. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

13 Analysis of the Symmetric Game Proposition In the symmetric case, the offered loads y j at the NEP are the optimal solutions of the following convex optimization problem, Min y s.t. ρj c j ρ j φ(ρ j ) + (K 1)c j φ(z)dz j S j S r jρ j = λ 0 ρ j 1 j S 0 Note that when K = 1, the above problem reduces to the global optimization problem and when K with λ fixed, we obtain the potential function of the Wardrop equilibrium. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

14 Procedure for Lower bound on Price of Anarchy Main ideas : We can restrict the analysis to the symmetric game. Using the KKT conditions, define the parameters such that for K > 1 only the first server (that is, the least costly server) is used, whereas for K = 1 more than one server is used. By splitting the traffic over several servers, centralized does much better than the decentralized setting. Global cost at NE is easy to compute. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

15 Procedure for Lower bound on Price of Anarchy Specific steps for Type-I functions Take the number of servers to be equal to λ. Compute the costs in the limit λ. Specific steps for Type-II functions Finite number of servers. Compute the costs in the limit λ r 1. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

16 Lower bound for PoA φ(ρ) = (1 ρ) m Theorem For m > 0, PoA(K) K (1 + m)k 1/(m+1) m. For m = 1, we retrieve the bound for the M/G/1/PS delay function in [Ayesta et al 2010]. For fixed m, LB is O(K m/(m+1) ), i.e, PoA is infinite for the Wardrop game. For fixed K, LB is K/ log(k) for large m. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

17 Lower bound for PoA φ(ρ) = 1 + ρ m Theorem For m > 1, PoA(K) (1 + m/k) 1 ( 1+m/K 1+m (1 + m/k) 1 ) m+1 m + m 1 log ( 1+m 1+m/K ). For K =, LB is O(m/ log(m)). Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

18 Conclusion We have established several properties of the NEP. We have shown that the worst performances are obtained for the symmetric game. We have shown that in the symmetric case the non-cooperative routing game is a potential game. We have given lower bounds on the PoA for φ(ρ) = (1 ρ) m and φ(ρ) = 1 + ρ m. Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

19 Perspectives Does our first theorem hold for general network topologies (and not only parallel links)? Integrate communication costs (see Kameda). Conjecture : the lower bound gives the right order for the PoA. (For the M/G/1/PS delay function it does!) Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21

20 Questions? Brun, Prabhu (LAAS-CNRS) Non-Cooperative Load Balancing ALGOGT / 21