Outline

1. Network control Frank KellyUniversity of Cambridge www.statslab.cam.ac.uk/~frankConference on Information Science and SystemsPrinceton, 22 March 2006

2. Outline • Example: end-to-end congestion control in the Internet • square-root formula • Control of elastic network flows • utilities, fairness and optimization formulation • Stability, under • delays • noise • fluctuating demand

3. End-to-end congestion control receivers senders Senders learn (through feedback from receivers) of congestion at queue, and slow down or speed up accordingly. With current TCP, throughput of a flow is proportional to T = round-trip time, p = packet drop probability. (Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd and Fall 1999)

4. Control of elastic network flows • How should available resources be shared between competing streams of elastic traffic? • Conceptual problem: fairness • Pragmatic problem: stability of rate control algorithm flow resource user

5. utility, U(x) utility, U(x) rate, x rate, x Elastic flows elastic traffic - prefers to share (Shenker) inelastic traffic - prefers to randomize

6. route resource Network structure(J, R, A) - set of resources - set of routes - if resource j is on route r - otherwise

7. route resource Notation - set of resources - set of users, or routes - resource j is on route r - flow rate on route r - utility to user r - capacity of resource j - capacity constraints

8. The system problem SYSTEM(U,A,C): Maximize aggregate utility, subject to capacity constraints

9. The user problem USERr(Ur;lr): User r chooses an amount to pay per unit time, wr , and receives in return a flow xr =wr /r

10. The network problem NETWORK(A,C;w): As if the network maximizes a logarithmic utility function, but with constants{wr} chosen by the users

11. Problem decomposition Theorem: the system problem may be solved by solving simultaneously the network problem and the user problems Johari, Tsitsiklis 2005, Yang, Hajek 2006

12. Max-min fairness Rates {xr}are max-min fair if they are feasible: and if, for any other feasible rates {yr}, Rawls 1971, Bertsekas, Gallager 1987

13. Proportional fairness Rates {xr}are proportionally fair if they are feasible: and if, for any other feasible rates {yr}, the aggregate of proportional changes is negative:

14. Weighted proportional fairness A feasible set of rates {xr} are such that are weighted proportionally fair if, for any other feasible rates {yr},

15. Fairness and the network problem Theorem: a set of rates {xr} solves the network problem, NETWORK(A,C;w), if and only if the rates are weighted proportionally fair

16. Bargaining problem (Nash, 1950) Solution to NETWORK(A,C;w)withw = 1 is unique point satisfying • Pareto efficiency • Symmetry • Independence of Irrelevant Alternatives (General w corresponds to a model with unequal bargaining power)

17. Market clearing equilibrium (Gale, 1960) Find prices p and an allocation x such that (feasibility) (complementary slackness) (endowments spent) Solution solves NETWORK(A,C;w)

18. Fairness criteria We’ve seen three fairness criteria: • proportional fairness • max-min fairness • TCP-fairness (as described by the square-root formula) Can we unify these fairness criteria within a single framework?

19. (weighted -fair allocations, Mo and Walrand 2000) Optimization formulation Suppose the allocation x is chosen to maximize subject to

20. Solution - shadow price (Lagrange multiplier) for the resource j capacity constraint Observe alignment with square-root formula when

21. Examples of -fair allocations maximize subject to - maximum flow - proportionally fair - TCP fair - max-min fair

22. Example 1/2 1/2 max-min fairness: 1/2 2/3 2/3 proportional fairness: 1/3 1 1 maximum flow: 0

23. Rate control algorithms • Can rate control algorithms be interpreted within the optimization framework? • Several types of algorithm possible, based on, e.g., • congestion indication using increase and decrease rules at sources • explicit rates determined by shadow prices estimated by resources Low, Srikant 2004, Srikant 2004

24. route resource Network structure - set of resources - set of routes - resourcej is on route r - flow rate on route r at time t - rate of congestion indication, at resource jat timet

25. A primal algorithm xr(t)- rate changes by linear increase, multiplicative decrease pj(.)- proportion of packets marked as a function of flow through resource

26. Global stability Theorem: the above dynamical system has a stable point to which all trajectories converge. The stable point is proportionally fair with respect to the weights{wr}, and solves the network problem, when K, Maulloo, Tan 1998

27. What’s missing? The global stability results ignore: • time delays • stochastic effects

28. General TCP-like algorithm Source maintains window of sent, but not yet acknowledged, packets - size cwnd On route r, • cwnd incremented by ar cwnd non positive acknowledgement • cwnd decremented by br cwnd m for each congestion indication (m>n) • ar = 1, br = 1/2, m=1, n= -1 corresponds to Jacobson’s TCP

29. Differential equations with delays j r

30. Equilibrium point • ar = 1, br = 1/2, m=1, n= -1 corresponds to Jacobson’s TCP, and recovers square root formula • But what is the impact of delays on stability? Can we choose m, n,… arbitrarily?

31. Johari, Tan 1999, Massoulié 2000, Vinnicombe 2000, Paganini, Doyle, Low 2001 Delay stability Equilibrium is locally stable if there exists a global constant b such that condition on sensitivity for each resource j condition on aggressiveness for each route r

32. Consequences? Delay stability condition: • n = -1 (Jacobson’s TCP) • instability if congestion windows are too small, sluggishness if congestion windows are too big • n = 0 (Scalable TCP, Vinnicombe, T. Kelly) • condition independent of size of congestion window, • and choice of br can remove round-trip time bias

33. Stochastic stability The signalling of congestion is noisy: for example, the receiver might see ---1--0--0-----0------0----------1------0-- receiver

34. Ott 1999, K 2003, Baccelli, McDonald, Reynier 2002 Variance • If m=1, then: • variance does not depend on Tr • coefficient of variation (= standard deviation/mean) • does not depend on xr - scale invariance

35. Suggestion on single path congestion control? ar = a, n = 0, m = -1 is attractive, giving both delay stability and stochastic stability

36. route source resource destination Routing? Can we extend the algorithm to allow load balancing across routes? Danger: route flap. - set of source-destination pairs - route r serves s-d pair s

37. Combined rate control and routing algorithm On route r • xr(t) increased by a /Tron positive acknowledgement • xr(t) decreased by br ys(r)(t) /Tr for each congestion indication, where is rate of returning acknowledgements for s-d pair s at timet • s = {r} corresponds to Scalable TCP

38. Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005 Delay stability Equilibrium is locally stable if there exists a global constant b such that condition on sensitivity for each resource j condition on aggressiveness of sources

39. Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005 Delay stability Equilibrium is locally stable if there exists a global constant b such that impact of routing condition on sensitivity for each resource j condition on aggressiveness of sources

40. Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005 Key, Massoulié, Towsley 2006 Suggestion on routing? • Stable, scalable load balancing across paths, based on end-to-end measurements, can be achieved on the same time-scale as rate control • For load balancing, the key constraint on the responsiveness of each route is the round-trip time of that route. • While it is natural for structural information to be provided by the network layer, load balancing is more naturally part of the transport layer. Chiang, Low, Calderbank, Doyle 2003

41. average round-trip time of packets through resource j Low, Lapsley 1999 Katabi, Handley, Rohrs 2002 Liu, Basar, Srikant 2003, K 2003 Example: fair dual algorithm A sufficient condition for delay stability:

42. Stochastic stability For a single resource • neither coefficient of variation depends on • scale invariance with respect to • promising for ad-hoc networks, where there are no natural scalings for prices (of battery power, bandwidth, etc )

43. Flow level model Define a Markov process with transition rates at rate at rate where is an -fair allocation - Poisson arrivals, exponentially distributed file sizes - model originally due to Roberts and Massoulié 1998

44. Stability Suppose Then Markov process is positive recurrent De Veciana, Lee, Konstantopoulos 1999, Bonald, Massoulié 2001, Ye 2003, Kang, K, Lee, Williams 2004, Lin, Shroff, Srikant 2006, Massoulié 2006

45. C=1 C=1 What goes wrong without fairness? Suppose vertical streams have priority: then condition for stability is and not (Bonald and Massoulié 2001)

46. Conclusions • End-to-end congestion control can be viewed as a distributed algorithm that: solves an optimization problem computes a fair allocation finds a market clearing equilibrium • Choices of distributed algorithm are limited by the requirement of stability under delays noise fluctuating load

47. Further information and references are available at:www.statslab.cam.ac.uk/~frank