Nash Equilibrium Load Balancing

Transcription

1 Nash Equilibrium Load Balancing Computer Science Department Collaborators: A. Kothari, C. Toth, Y. Zhou

2 Load Balancing A set of m servers or machines. A set of n clients or jobs. Each job can be run only on a subset of servers. Load balancing problem: find an optimal assignment of jobs to servers. Some objective functions: maximum load on a server L 2 -norm: sum of the load squares Long history in multi-processor scheduling, storage allocation, circuit routing etc.

3 Selfish Clients What if clients/jobs are selfish, strategic players? Humans strategize in choosing queues at grocery store checkout, airport, banks etc. Internet: TCP game and selfish routing. Rationality-based computing [Nisan, Papadimitriou] Our motivation comes from Peer to Peer (P2P) networks. Multiple sites to download music, video, source code etc. Each client concerned only about his/her latency. A non-cooperative game: a client s latency depends on other clients choices.

4 P2P Networks Decentralized, distributed data-sharing networks. Napster, Gnutella, Freenet, CAN, Chord, Pastry, Tapestry, Morpheus, KaZaa, Farsite, Jxta, OceanStore... P Q R Internet Cloud U "A Beautiful Mind" No central authority: all nodes autonomous and same functionality (democracy of peers). Resource sharing by direct exchange between peers.

5 P2P is Popular Big part of Internet traffic now P2P: KaZaa, Gnutella etc. As distributed systems, P2P paradigm has many pluses: No expensive centralized hardware or administration Self-organizing, incremental scaling Redundancy, fault tolerance, robustness Resource sharing: distributed.net, OceanStore, Farsite, DMBS, archives. It works. But how about performance? How much worse off are we without coordination? What is the price of anarchy?

6 Balancing Selfish Clients Each client has choice of many servers to download from, and is free to choose. Client are rational, strategic players, who want to minimize their own latency. A client s latency depends on the server load. No coordinating scheduler to enforce an assignment: anarchic load balancing. A non-cooperative game among clients strategies are server choices, payoff is latency. Nash assignment: no client motivated to switch unilaterally. Worst-case cost ratio between Nash and optimal assignments; competitive analysis.

7 The Model A bipartite graph between n clients and m servers. A matching assigns each client to an adjacent server. v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 u1 u 2 u 3 u 1 u 2 u 3 An assignment u 1 u 2 u 3 A Nash assignment A server s load l grows linearly with the number of clients assigned to it. A client s latency λ equals its server s load. cost(m) = n i=1 λ(u i).

8 Server Models 1. Integral Assignment Each client matched to at most 1 server. Server j has speed v j. If server j has deg(j) clients, then its load is l j = deg(j). Each client of server j experiences latency l j /v j. 2. Fractional Assignment A client can split its job among multiple servers: x ij. Server j s load is l j = i x ij. Server j completes all its assigned jobs at time l j. Client i s latency is λ i = max j {l j x ij > 0}. Fractional model similar to one used by KaZaa. 3. L p norm latency functions.

9 Client Latency vs. Server Load Lemma: Sum of client latencies equals sum of squares of server loads: n m l 2 j λ i =. v j i=1 j=1 In the integral model, server j services l j clients, each of whom suffers latency l j /v j. Similarly, summation over job fractions in the fractional model.

10 L 2 -norm Load Balancing Traditionally, load balancing has concentrated on optimizing server load. The greedy job assignment has competitive ratio (1 + 2) for L 2 -norm load balancing. Awerbuch et al. (FOCS 95). But greedy assignment is not at Nash equilibrium. Equilibrium helps! Nash assignment has much better competitive ratio.

11 Summary of Results 1. Integral server model (identical or different speeds): cost(nash) 5 2 cost(opt). For identical servers, cost(nash) cost(opt) + 2n. Lower bound: The competitive ratio is Fractional server model: Nash = Optimal. 3. Individual Payoffs: In the integral server model, a client s latency under Nash can be Θ(log n/ log log n) times its latency in the optimal. 4. Structural Theorem for Optimal Assignment.

12 Optimal vs. Nash With identical server speeds, optimal always Nash, but not vice versa. v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 u1 u 2 u 3 u 1 u 2 u 3 u 1 u 2 u 3 Optimal: cost = 3 Nash: cost = 5 Proof: Suppose a client can switch from server i to j. cost(m ) cost(m opt ) = ( (d j + 1) 2 + (d i 1) 2) (d 2 i + d2 j ) = 2(d j d i + 1) < 0.

13 Optimal vs. Nash But with different server speeds, optimal is not always Nash. 1 1/4 1 1/4 1 1/4 u1 u 2 u 3 u1 u 2 u 3 u 1 u 2 u 3 Optimal: cost = 8 Nash: cost = 9 cost(opt) = ; cost(nash) =

14 Lower Bound on Worst-case Nash A tree structure: each node has a client-server pair. Edges between neighboring levels only. In optimal, each client assigned to its server in the pair, for a total cost of n. In Nash, each client matched to a server at higher level. The total cost becomes 2n 1. server client Optimal Cost = 11 Worst Case Nash Cost = 21 ( )

15 The Upper Bound Proof sketch for identical servers, integral server model. n jobs, and m servers. Let S i (resp. T i ) be the set of jobs assigned to server i in the optimal (resp. Nash). Then, v 1 v 2 v 3 v 1 v 2 v 3 v 1 v 2 v 3 u1 u 2 u 3 u 1 u 2 u 3 u 1 u 2 u 3 S 3 = {u 3 }; T 3 = {u 2, u 3 }. Optimal: cost = 3 Nash: cost = 5

16 Upper Bound cost(nash) = m i=1 T i 2 ; cost(opt) = m i=1 S i 2. Claim: S i T j > 0, for i j, implies that T j T i + 1. The job in the common intersection can be assigned to both servers i and j. If T j > T i + 1, then this job could improve its latency by switching to i. v i v j T i T j u

17 Upper Bound cost(nash) = = = = m m m T j 2 = T j S i T j j=1 m i=1 m i=1 j=1 i=1 m S i T j T j j=1 m ( T i + 1) S i T j j=1 m ( T i + 1) i=1 m S i T j j=1 m S i T i + i=1 i=1 m S i

18 Upper Bound m i=1 T i 2 m i=1 S i T i + m i=1 S i. But m i=1 S i = m i=1 T i = n. Using Cauchy s inequality, xy 1 2 (x2 + y 2 ), we get m T j 2 j=1 m S j 2 + 2n j=1 Thus, cost(nash) cost(opt) + 2n. Ratio bound: cost(nash) 5 2 cost(opt). Holds even with varying server speeds.

19 Fractional Server Model A client can split its job among multiple servers. x ij is fraction of client i s job assigned to j. Server j s load is l j = i x ij. Server j completes all its assigned jobs at time l j. Client i s latency is λ i = max j {l j x ij > 0}. Fact: If x ij > 0, then l j = λ i. Theorem: A fractional Nash assignment is optimal.

20 Individual Client Latency A lower bound construction. server client Optimal Worst Case Nash root deg = 1 root degree = 3 Theorem: In the worst-case, for any server v, ( ) d nash (v) log n = Θ d opt (v) log log n

21 Uniqueness of Optimal Matching Optimal matching is not unique. v 1 v 2 v 1 v 2 v1 v 2 u1 u 2 u 3 u 1 u 2 u3 u 1 u 2 u 3 Two different optimal matchings To which optimal shall we compare a Nash matching? Theorem: Over all optimal matchings, a server s degree varies by at most 1.

22 nash vs. greedy Nash assignment always greedy, but not vice versa. v 1 v 2 v 3 v 1 v 2 v 3 u1 u 2 u 3 u 1 u 2 u 3 A greedy assignment For every Nash, there is an ordering of clients, for which the greedy assignment is same as the Nash. Figure shows a greedy matching, which is not Nash.

23 Sequencing Clients Clients Servers A Nash Matching Only the servers degree sequence matters. Sequence the clients row by row. If Nash deg of client is d, then it has no edge to a server of deg < d 1.

24 Closing Remarks Computing Nash assignments. Online model.

25 State of the Art in P2P Load Balancing Napster, Gnutella etc. leave it to users. KaZaa splits each request across multiple hosts. CUP, Controlled Update Propagation, (Stanford) randomizes across servers depending on their load or speed. Adaptive Peer Selection (UMass) scheme suggests trying a few servers, then using learning theory to choose. Problem very much a game theoretic problem, but has not been studied with that perspective.