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Mathematical models of the Internet

Mathematical models of the Internet. Frank Kelly www.statslab.cam.ac.uk/~frank Hood Fellowship Public Lecture University of Auckland 3 April 2012. Source: CAIDA, Young Hyun. Source: CAIDA - Young Hyun, Bradley Huffaker (displayed at MOMA). Functional view. Applications. TCP/IP.

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Mathematical models of the Internet

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  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. Functional view Applications TCP/IP Physical technology

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

  6. senders receivers 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 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 • 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

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

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

  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 SYSTEM(U,A,C): Maximize aggregate utility, subject to capacity constraints

  13. 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

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

  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 {xr}are max-min fair if they are feasible: and if, for any other feasible rates {yr},

  17. 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:

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

  19. 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

  20. 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)

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

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

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

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

  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 xr(t)- rate changes by linear increase, multiplicative decrease pj(.)- 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{wr}, 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? short congested links, T user long uncongested link, T’ cache? 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: route source resource 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 Suppose is chosen to maximize subject 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

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