1. Mathematical models of the Internet Frank Kelly www.statslab.cam.ac.uk/~frank Hood Fellowship Public Lecture University of Auckland 3 April 2012

2. Source: CAIDA, Young Hyun

3. Source: CAIDA - Young Hyun, Bradley Huffaker (displayed at MOMA)

4. Physical technology Applications TCP/IP Functional view

5. Outline End-to-end congestion control –square-root formula Utilities and fairness –optimization formulation Multipath routing –reliability, robustness

6. End-to-end congestion control 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 senders receivers T = round-trip time, p = packet drop probability. (Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd and Fall 1999)

7. Feedback mechanism Now: dropped packets (acknowledgements carry information from receivers to senders) Standards track: Explicit Congestion Notification (Ramakrishnan, Floyd and Black RFC3168) - routers mark packets at queues as overload approached

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

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

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

11. 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

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

13. The user problem User r chooses an amount to pay per unit time, w r, and receives in return a flow x r =w r / r USER r (U r ; r ):

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

15. Problem decomposition Theorem: the system problem may be solved by solving simultaneously the network problem and the user problems

16. Max-min fairness Rates {x r } are max-min fair if they are feasible: and if, for any other feasible rates {y r },

17. Proportional fairness Rates {x r } are proportionally fair if they are feasible: and if, for any other feasible rates {y r }, the aggregate of proportional changes is negative:

18. Weighted proportional fairness A feasible set of rates {x r } are such that are weighted proportionally fair if, for any other feasible rates {y r },

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

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

21. 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)

22. 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?

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

24. - shadow price for the resource j capacity constraint Observe alignment with square-root formula when Solution

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

26. Example 0 11 maximum flow: 1/3 2/3 1/2 max-min fairness: proportional fairness:

27. Rate control algorithms Can rate control algorithms be interpreted within the optimization framework? Several types of algorithm possible, based on –congestion indication using linear increase and multiplicative decrease rules –explicit rates inversely proportional to network shadow prices

28. Notation - set of resources - set of routes - resource j is on route r - weight of route r - flow rate on route r at time t - shadow price, or rate of congestion indication, at resource j at time t

29. A primal algorithm x r (t) - rate changes by linear increase, multiplicative decrease p j (.) - proportion of packets marked a function of flow through resource

30. 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 {w r }, and solves the network problem, when

31. TCP We’ve seen that TCP’s square root formula is interpretable in terms of an optimization problem Next we show that its dynamic behaviour is interpretable as a form of primal algorithm

32. TCP (Jacobson’s congestion avoidance algorithm) Source maintains a window of sent, but not yet acknowledged, packets - size cwnd cwnd incremented by 1/cwnd on positive ack cwnd decremented by cwnd/2 on congestion change in rate x per unit time is about

33. Differential equations for TCP

34. Equilibrium point check: the equilibrium of the dynamical system recovers the inverse square root formula for TCP note again the round-trip time bias

35. Consequences of RTT bias Question: where to place cache? long uncongested link, T’ short congested links, T user cache? If then short RTT preferred under TCP, even though twice the resource usage

36. Multipath routing Suppose a source-destination pair has access to several routes across the network: resource route source destination - set of source-destination pairs - route r serves s-d pair s Combined multipath routing and congestion control: a robust Internet architecture. Key, Massoulié & Towsley 2005

37. Routing and optimization formulation subject to maximize Suppose is chosen to ( H is an incidence matrix, showing which routes serve a source-destination pair )

38. Example of multipath routing Routes, as well as flow rates, are chosen to optimize over source-destination pairs s

39. First cut constraint Cut defines a single pooled resource

40. Second cut constraint Cut defines a second pooled resource

41. Conclusions Mathematical models of end-to-end congestion control show it to be a distributed algorithm that: solves an optimization problem; computes a fair allocation; and finds a market clearing equilibrium. The models provide a framework for engineering efforts to improve reliability and robustness.

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