Flow Control KAIST CS644 Advanced Topics in Networking

1. Flow Control KAIST CS644Advanced Topics in Networking Jeonghoon Mo <jhmo@icu.ac.kr> School of Engineering Information and Communications University

2. Acknowledgements • Part of slides is from • tutorial of R. Gibbens and P. Key at SIGCOMM 2000 • S. Low’s OFC presentation

3. Overview • Problem • Objectives • Kelly’s Framework - Wired Data Networks • Extensions • Quality of Service • Wireless Network • High Speed Network: Aggregated Flow Control

4. Problem Flows share links: How to share the links bandwidth?

5. Problem • How to control the network to share the bandwidth efficiently and fairly?

6. Link Model • Set of resources, J; set of routes, R • A route r is a subset r J. • Let • Capacity of resource j is Cj. A’x  c x  0

7. A Few System Objectives • Max Throughput • Max-min Fairness (Most Common) • Proportional Fairness (Kelly) • -Fairness (Mo, Walrand)

8. Maximize: x(1) + x(2) + x(3) x*= (0,6,6) maximizes the total system throughput. However, user 1 does not get anything. => unfair 6 6 x1 x3 x2 Max System Throughput • Two links with capacity 6 • Three users: 1,2,3 • x(i) : bandwidth to user i • x(1)+x(2) <= 6 • x(1)+x(3) <= 6

9. Most commonly used definition of fairness. Maximize Minimum of the x(i), recursively. x*= (3,3,3) is the max-min allocation. However, user 1 uses more resources. 6 6 x1 x3 x2 Max-Min Fairness

10. 6 6 x1 x3 x2 Proportional Fairness • Proposed by Frank Kelly • Social Welfare: Sum of Utilities of Users • Maximize the Social Welfare • x*= (2,4,4) is the Proportional Fair Allocation. • Can be generalized into “Utility Fairness”.

11. x(1-) 1- Max i pi -Fairness • Generalized Fairness Definition • System Objective: • includes proportional-fair, max-min-fair, max throughput •  = 0 : Maximum allocation (p=1) •   1 : Proportional fair allocation •  = 2 : TCP-fair allocation •    : Max-min fair allocation (p=1)

12. -Fairness • Trade-off between Fairness and Efficiency • Bigger  favors Fairness • Smaller  favors Efficiency (source: Is Fair Allocation Inefficient, INFOCOM 04)

13. Fairness and Efficiency (Infocom 04) • Counter-Example

14. Algorithms How to achieve those system objectives?

15. Players • source • controls its rate or window based on (implicit or explicit) network feedback • router (link) • Generate (implicit) feedback or controls packets

16. Source Algorithm • TCP Vegas, RENO, ECN • XCP

17. Active Queue Management (AQM) • Priority Queue • WFQ • RED • REM • XCP Router

18. Kelly’s Model and Algorithm

19. User: rate and utility • Each route has a user: if xr is the rate on route r, then the utility to user r is Ur(xr). • Ur() --- increasing, strictly concave, continuously differentiable on xr [0 , ) --- elastic traffic • Let C=(Cj, j J), x=(xr, r  R) then Ax  C.

20. System problem • Maximize aggregate utility, subject to capacity constraints

21. User problem • User r chooses an amount to pay per unit time wr, and receives in return a flow xr = wr/r

22. Network problem • As if the network maximizes a logarithmic utility function, but with constants (wr, rR) chosen by the users

23. Three optimization problems • SYSTEM(U,A,C) • USERr(Ur;r) • NETWORK(A,C;w)

24. Decomposition theorem • There exist vectors  , w and x such that • wr = rxr for r  R • wr solves USERr(Ur; r) • x solves NETWORK(A, C; w) The vector x then also solves SYSTEM(U, A, C).

25. Thus the system problem may be solved by solving simultaneously the network and user problems

26. Result • A vector x solves NETWORK(A, C; w) if and only if it is proportionally fair per unit charge

27. Solution of network problem • Strategy: design algorithms to implement proportional fairness • Several algorithms possible: try to mimic design choices made in existing standards

28. Primal algorithm

29. Interpretation of primal algorithm • Resource j generates feedback signals at rate j(t) • signals sent to each user r whose route passes through resource j • multiplicative decrease in flow xr at rate proportional to stream of feedback signals received • linear increase in flow xr at rate proportional to wr

30. Related Work • Optimization Flow Control (S. Low) • Window based Model (Mo, Walrand)

31. Optimization Flow Control • Distributed algorithm to share network resources • Link algorithm: what to feed back • RED • Source algorithm: how to react • TCP Tahoe, TCP Reno, TCP Vegas Source alg Link alg

32. Welfare maximization Primal problem: • Capacity can be less than real link capacity • Primal problem hard to solve & does not adapt

33. x1 c1 c2 x2 x3 Model • Network: Links l each of capacity cl • Sources s:(L(s), Us(xs), ms, Ms) L(s) - links used by source s Us(xs) - utility if source rate = xs

34. Distributed Solution Dual problem: BW price along path of s • Given sources can max own benefit individually • indeed primal optimal if is dual optimal • Solve dual problem!

35. Distributed Solution (cont…) • Dual problem: • Grad projection alg: • Update rule: • A distributed computation system to solve the dual problem by gradient projection algorithm

36. Source Algorithm Decentralized: Source s needs only and

37. Router (Link) Algorithm • Decentralized • Rule of supply and demand • Any work-conserving service discipline • Simple aggregate source rate

38. Random Exponential Marking (REM) • Source algorithm • Identical but does not communicate source rate • Link algorithm • At update time t, sets price to a fraction of buffer occupancy: • Theorem: Synchronous convergence • Under same conditions (with possibly smaller ) : • Price update maintains descent direction • Gradient estimate converge to true gradient • Limit point is primal-dual optimal

39. RED • Idea: early warning of congestion • Algorithm Link: Source (Reno): marking window 1 queue time B

40. rate fraction of marks 1 RED • Idea: marks for estimation of shadow price • Algorithm Link Source Global behavior of network of REM: stochastic gradient algorithm to solve dual problem marking 1 queue Q

41. d1 q11 q21 w1 x1 c2 c1 x3 x2 q23 q12 d2 d3 A’x  c Q(c - A’x) = 0 w = X(d + qA) Window-based Model [Mo,Walrand] Q = diag{qi }; X = diag{xi }. xi  0, i = 1, 2, 3 qi  0, i = 1, 2, x1+x2 c1 q1(c1 - x1 - x2) = 0 w1=x1d1 + x1 q1 + x1 q2

42. Window-based Algorithm Theorem:[Mowlr98] Let dwi si := wi - xi di - pi di si = - k ti := end-to-end delay dt ti wi Then x(t) -> unique weighted -fair point x* Proof: 2 The function (si /wi ) i is a Lyapunov function

43. Extensions • Aggregated Flow Control • Quality of Service • Wireless Network • Maxnet and Sumnet

44. Aggregate Flow Control • Motivations: • High Capacity of Optical Fiber • Idea: • player are core routers and access routers. • access router: regulates the rate of aggregated flow • core router: provide feedbacks to access routers

45. Quality of Service • Only bandwidth is modeled. • QoS is affected by • loss and delay also • How to incorporate other parameters?

46. Considered sigmoidal utility function Non-convex optimization problem =>duality gap Non-Convex Utility Function (Lee04) (source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)

47. Non-Convex Utility Functions Dual Algorithm with Self-Regulating Property Without Self-Regulation With Self-Regulation (source: J. Lee et. al. Non-convexity Issues, INFOCOM 04)

48. Wireless Ad-Hoc Network [RAD04] • Physical Model:Rate r is an increasing function of SINR. • MAC : Each time slot determines power pn,which determines rate xn • Routing matrix R and flow to path matrix F are given.

49. Random Topology Results 100m x100m grid 12 random node, with 6 pairs of transmissions

50. In the wireless Ad-hoc Networks • The max-min fair rate allocation of any network has all rates equal to the worst node. • The capacity maximization objective leads to starving users. • Proportional Fair Allocation give reasonable trade-off between fairness and efficiency. • The worst node does not starve.