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Farsighted users harness network time-diversity

P. Key, L. Massoulié, M. Vojnović

Microsoft ResearchCambridge, United Kingdom

IEEE Infocom 2005, Miami, FL, March 2005.

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Premise: time-diversity

• Network congestion state fluctuates in time

• Origins:

– Flows arrive and depart– Link failures– Wireless interference

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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

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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

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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

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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)

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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

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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 rateflow-perceived long-run average price (loss event rate)

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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:– xr(u) = rate of flow r in phase u

– time-average rate of flow r:

– average price of flow r:

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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

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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 2 phase 3 phase N. . .

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EquilibriaProperty: 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”

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Illustration: single linkFarsighted

Myopic 1

2

u

Result:

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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: xr <= f(pr)

Conservative with

Conservative

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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

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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:

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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

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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 largeIntuition: 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) => TF < TM

(T’s are mean download times for myopic flows)

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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:

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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

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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

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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

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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)

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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)

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Numerical examples:Two myopic flows

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

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Numerical examples:1 farsighted & 1 myopic flow

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Numerical examples:1 farsighted & 1 myopic flowFarsighted flow = Approx-1

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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