Network Bandwidth Allocation

1. Network Bandwidth Allocation (and Stability) In Three Acts

2. Problem Statement How to allocate bandwidth to users? How to model the network? What criteria to use?

3. Act I Modeling

4. Host 1 Host 2 Host 3 Host 4 Host 5 A Physical View Router : interconnect, where links meet. Host : multi-user, endpoint of communication. Link / Resource : bottleneck, each has finite capacity Cj.

5. Host 1 Host 2 Host 3 Host 4 Host 5 System Usage Route : static path through network, supporting Ni(t) flows with Li(N(t)) allocated bandwidth. Flows / Users : transfer documents of different sizes, evenly split allocated bandwidth along route. Dynamic. Not directed.

6. Simplification Extraneous elements have been removed.

7. Abstraction Routes are just subsets of links / resources. Represented by [Aji] : whether resource j is used by route i. Capacity constraint:

8. 220 020 120 130 110 121 Stochastic Behavior Model N(t) as a Markov process with countable state space. Poisson user arrivals at rate ni. Exponential document sizes with parameter mi. Define traffic intensity ri = ni / mi.

9. Act II PerformanceCriteria

10. Allocation Efficiency • An allocation L is feasible if capacity constraint satisfied. • A feasible allocation L is efficient if we don’t have m³L for any other feasible m. • Defined at a point in time, regardless of usage.

11. Stability • Stable « Markov chain positive recurrent. • Returns to each state with probability 1 in finite mean time. • Necessary, but not sufficient condition: • How tight this is gives us an idea of utilization. • Does not uniquely specify allocation.

12. 10 10 10 Maximize Overall Throughput • That is, max • No unique allocation. • Could get unexpected results.

13. 12 12 Max-Min Fairness • Increase allocation for each user, unless doing so requires a corresponding decrease for a user of equal or lower bandwidth to satisfy the capacity constraints. • Uniquely determined. • Greedy algorithm. Not distributed.

14. Proportional Fairness • L is proportionally fair if for any other feasible allocation L* we have: • Same as maximizing: • Interpret as utility function. • Distributed algorithms known.

15. a-Fair Allocations Maximize Subject to With ki=1, a® 0 : maximize throughput a = 1 : proportional fairness a® ¥ : max-min fairness With ki = 1 / RTTi2, a = 2 : TCP

16. TCP Bias • Congestion window based on additive increase / multiplicative decrease mechanism. • Increase for each ACK received, once every Round Trip Time. • Timeouts based on RTT. • Bias against long RTT. RTT timeout

17. Properties of a-Fair Allocations Assume Ni(t) > 0. Let L(N(t)) be a solution to the a-fair optimization. • The optimal L exists and is unique. • It’s positive: L > 0. • Scale invariance: L(rN) = L(N), for r > 0. • Continuity: L is continuous in N. • System is stable when

18. Act III Fluids &Formalities

19. Fluid Models Decompose into non-decreasing processes: Ni(0) : initial condition Ei(t) : new arrival process Ti(t) : cumulative bandwidth allocated Si(t) : service process Consider a sequence indexed by r > 0:

20. Fluid Limit : Visual

21. Fluid Limit : Math Look at slope: By strong law of large numbers for renewal processes: Thus with probability 1.

22. Fluid Model Solution A fluid model solution is an absolutely continuous function so that at each regular point t and each route i and for each resource j

23. Fluid Analysis is Easier Definition A complex function f is absolutely continuous on I=[a,b] if for every e > 0 there is a d > 0 such that for any n and any disjoint collection of segments (a1,b1),…,(an,bn) in I whose lengths satisfy Theorem If f is AC on I, the f is differentiable a.e. on I, and

24. Visualizing Fluid Flow

25. For Stability • If fluid system empties in finite time, then system is stable. • Show that • In general, what happens as t ®¥ when some of the resources are saturated? • We approach the invariant manifold, aka the set of invariant states

26. Towards a Formal Framework • Interested in stochastic processes with samples paths in DÂ[0, ¥), the space of right continuous real functions having left limits. • Well behaved. At most countably many points of discontinuity.

27. Why we need a better metric. … … What goes wrong in Lp ? L¥?

28. Skorohod Topology Let L be the set of strictly increasing Lipschitz continuous functions l mapping [0,¥) onto [0,¥),such that Put (standard bounded metric) For functions mapping to any Polish (complete, separable, metric) space.

29. Prohorov Metric Let (S,d) be a metric space, B(S) the s-algebra of Borel subsets of S, P(S) family of Borel probability measures on S. Define The resulting metric space is Polish.

30. Fluid Limit Theorem from Gromoll & Williams

31. Outline of Proof • Apply functional law of large numbers to load processes. • Derive dynamic equations for state and bounds. • State contained in compact set with probability 1 in limit. • State oscillations die down with probability 1 in limit. • Sequence is C-tight. • Weak limit points are fluid solutions with probability 1.

32. Papers 1995 2000 2005 Dai Bonald, Massoulié Gromoll, Williams Kelly, Maulloo, Tan Kelly Kelly, Williams Mo, Walrand Massoulié