1. 1 Farsighted users harness network time-diversity P. Key, L. Massoulié, M. Vojnović Microsoft Research Cambridge, United Kingdom IEEE Infocom 2005, Miami, FL, March 2005.

2. 2 Premise: time-diversity Network congestion state fluctuates in time Origins: –Flows arrive and depart –Link failures –Wireless interference

3. 3 Time-diversity in the Internet: empirical evidence 1.Zhang et al (ACM IMC, 2001): 2375 hours of measurements in ’99- ’00 on 31 NIMI hosts (80% US), Poisson probes 10 pkts/sec Timescale = minutes

4. 4 Time-diversity in the Internet: empirical evidence (cont’d) 2. Markopoulou et al (ToN, Oct 2003): loss and delay e2e measurements collected in ’01 @ 5 US cities, 7 distinct providers, small probes of 50 bytes each 10 ms (see also Athina’s thesis) Time-diversity due to link failures or route updates or router bugs

5. 5 Who may benefit being farsighted ? long-lived bulk data transfers –insensitive to short-timescale delays may receive small rate at some times, but “win” in long run examples: –distribution of movie files –database synchronizations –file system backups –software distribution/updates Farsighted flow (loose def): a flow that adapts to congestion state fluctuations by –“avoiding” bad states –compensating by being somewhat “unfair” when state is good –however, being “fair” in long run

6. 6 Problem How can an adaptive flow exploit time-diversity of congestion state of network path ? Understand fairness of rate allocation for networks with time-diversity –equilibrium points and their properties –relation to TCP-friendliness norm Construct algorithms achieving equilibrium points (decentralized, end-points only, no special feedback)

7. 7 Traditional flows = myopic It means that: at each time, balance U’(x) = p utility function of a flow flow-perceived price per unit flow (implicit: loss event rate) send rate The balance condition necessarily holds for solution of M-USER: maximise U(x) – p x

8. 8 Farsighted flows Objective: Remark: p x = flow-perceived average charge per unit time Ob. farsighted flow “cares” only about average send rate achieved in long run F-USER: maximise U(x) – p x time-average send rate flow-perceived long-run average price (loss event rate)

9. 9 Microeconomics formulation Network state takes a finite set of phases U A phase u occurs  (u) fraction of time Assumption: phases occur as a stationary and ergodic process with time-stationary distribution  Notation: –x r (u) = rate of flow r in phase u –time-average rate of flow r: –average price of flow r:

10. 10 SYSTEM problem utility function (vector-valued argument) link cost function routing matrix in phase u A flow r rate allocation specified by: flows in phase u Remark 1: Myopic: ; farsighted: Remark 2: If all flows are myopic, SYSTEM separates into |U| independent problems

11. 11 Multi-path analogy SYSTEM formally equivalent to multi-path problem posed by Gibbens and Kelly (’02) –Each flow optimally splits its flow across paths indexed by phases u What’s different ? –index u corresponds to temporal phase, not “spatial” network path –paths interchange over time (time- vs. space-diversity) phase 1 phase 2phase 3phase N...

12. 12 Equilibria Property: for a farsighted user r, there exists a critical price such that: Moreover, for each phase u: Property: Price-equalisation in good phases “good phase”

13. 13 Illustration: single link Farsighted Myopic 1 2 u Result:

14. 14 Relation to TCP-friendliness TCP-friendliness usual (loose) definition: throughput of a source not larger than of a TCP flow under same circumstances Conservative source (V.-Le Boudec, ToN ’05): –given function f(x), flow r obeys: x r <= f(p r ) Conservative  with Conservative 

15. 15 Relation to TCP-friendliness (cont’d) Property: A farsighted flow is conservative –Verifies both: Remark: farsighted strategy can be seen as throughput- maximising subject to the condition of conservativeness

16. 16 Properties of equilibria Throughput of farsighted strategy is always at least that of myopic Consider a farsighted flow f and myopic flow m f and m compete for the same set of network resources f and m have a common utility function Result:

17. 17 Properties of equilibria (cont’d) Diminishing returns for switching to farsighted n flows (k farsighted, n-k myopic) flows use same routes = throughput of a farsighted flow for given k Result: decreases with k Rephrase: the larger the fraction of farsighted flows, the smaller their per-flow throughput Result: Switching flows to farsighted sometimes beneficial to a myopic flow, but not always

18. 18 Properties of equilibria (cont’d) Mean download time for myopic flows Farsighted flows reduce their send rates to 0, whenever the number of competing myopic flows becomes sufficiently large Intuition: The mean download time for a myopic flow would be smaller, than if the farsighted flow were myopic Result: One infinitely-lived flow Myopic flows arrive at times of a homogeneous Poisson Each myopic flow is a transfer of a file of size ~ Exp Single link (usual stability condition: load in (0,1)) mean number of competing myopic flows with infinitely-lived flow = farsighted … infinitely-lived flow = myopic … & (Little) => T F < T M (T’s are mean download times for myopic flows)

19. 19 Farsighted flows made low-priority Low-priority: configure farsighted flow to have positive send rate only when no myopic flows compete Result: Consider f farsighted flows competing with myopic flows for a single link (H1) link characterized by increasing, convex cost function C() (H2) utility function of a myopic flow assumed strictly concave Farsighted flows are low-priority for any C() that satisfies (H1), if and only if:

20. 20 Algorithms for farsighted strategy Three timescales: 1.Fast ~ round-trip time rounds (~ 1-100’s ms) 2.Medium ~ congestion state fluctuations (~ minutes) 3.Slow ~ congestion state averaging (>> medium time units) Algorithm F (farsighted flow r): (Fast): (Slow): with

21. 21 Farsighted algorithm: what is new ? Two-timescale adaptation, as opposed to only fast timescale adaptation of myopic: Requires no knowledge of current phase and distribution of phases –direct application of Gibbens and Kelly ’02 would require both Aggregate-price feedback at the source sufficient –as with standard controllers

22. 22 Farsighted algorithm: convergence One phase –farsighted algorithm converges from any initial value to equilibrium point that solves SYSTEM Set of phases –convergence under certain hypothesis

23. 23 Approximate farsighted algorithms Farsighted algorithm (introduced earlier) is of bang-bang type Problem: find approximate controllers that –have always a positive rate –are conservative –exact asymptotically with respect to some parameter Approximate algorithms: (Fast) Approx-1:Approx-2: (Slow)

24. 24 Approximate farsighted algorithms (cont’d) Approx-1 achieves: Approx-2 achieves: (C2) implies conservativeness only if  <= 1 (Jensen) Approx-1 and Approx-2 are same and equality in (C2) holds for  = 1 Approx-1 and Approx-2 are asymptotically farsighted algorithm as  tends to 0 (C2) (C1)

25. 25 Numerical examples: Two myopic flows For all numerical examples: background flows are non-persistent myopic arriving at times of homogeneous Poisson, file size ~ Exp

26. 26 Numerical examples: 1 farsighted & 1 myopic flow

27. 27 Numerical examples: 1 farsighted & 1 myopic flow Farsighted flow = Approx-1

28. 28 Summary Microeconomics form. for networks with time-diversity Equilibrium points and their properties for networks with farsighted and myopic flows –Price equalisation –Farsighted flow is conservative –Farsighted flow’s throughput at least as that of myopic –Diminishing returns of switching to farsighted; sometimes beneficial to myopic flows, but not always –Farsighted flow may induce smaller mean download time to competing myopic flows, than if the flow were myopic –Farsighted flows can be made low-priority Farsighted strategy implemented by algorithms at end- points that require no network special feedback