10/26/2005 CCW051

Resolving Greedy Users via Stateless AQM

Murat AlanyaliBoston University

Joint work with Ashraf Al Daoud

10/26/2005 CCW052

Queue Management

• Goal: fair and efficient bandwidth sharing among sessions.

• WFQ: Perfect isolation of traffic. Stateful.

• RED: Proactive packet discarding. Stateless. – Extensions well-protect less aggressive sessions

(e.g. CHOKe -- Pan et al. 2000, Tang et al. 2003)– Well-studied under realistic settings (e.g. Hollot et

al, Srikant et al, Low et al, with control-theoretic viewpoint).

session i

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Related work on Analysis of Conflict

• Single link

• Akella et al 2002: “TCP game”, under RED, CHOKe.

• Dutta et al. 2003:– Strategic sessions, action = pkt generation rate,

utility = goodput.– Efficiency: tradeoff between goodput and packet loss.

• Altman et al. 2004:– Differentiated service to sessions– Utility: goodput – loss – price

Packet discard prob

Queue length

1

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Overview

• Extend to richer topologies

• Consider packet discarding and pricing to stabilize good regimes

• Illustrate how known principles in circuit-switching may help

• Emphasis: on stateless AQM schemes

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Model: Network

• Node operation:– RED-like: discard packet w/p D(packet arrival rate).

D( ) arbitrary, subject to certain technical conditions.

– Charge sessions for accepted packets – same price for all sessions.

• Topology: Feed-forward

n1

session

n2

n3 n4 n5

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Model: Packet Losses

• Bn(v) : Packet discard probability at node n

• In general:

• Feed-forward topologies: B(v) is uniquely defined and it is smooth in v.

1 2( ) ( )B D v=v

: :

( ) (1 ( ))n s js n s j j n on r

B D v B∈ <

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠∑ ∏v v

n1

v2

n2

n3 n4 n5

v1

v4

v3

2 1 2 1( ) ( (1 ( )))B D v v B= + −v v

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Model: Sessions

• Greedy, non-strategic sessions.

• vs: Rate of packet generation by session s.

• Continuous time, deterministic model for vs.

• Assumption:– Delivered packets have monetary value

– Change in vs is proportional to rate of profit generation:

( ):

(1 ( )) mean cost per unit time

goodput, call it ( )

s s nn n s

s

dv v B

dt

G∈

= − −∏1 4 42 4 43

v

v

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Efficiency

• Social welfare: (not concave)

• Theorem: There exist parameters cn(v) such that

Related work: Kelly (1988) on circuit-switched networks.

• Idea: Charge cn(v) per admitted packet at node n :

in turn local maxima of SW(v) are locally stable.

( ) ( )ss

SW G∑@v v

:

( ) (1 ( )) ( ) (1 ( ))n n jn sn s j j n on ss

SW B c Bv ∈∈ ≤

∂= − − −

∂ ∑∏ ∏v v v v

1 2( ) ( ) ( ) 0s s ss ss

d d dSW v SW v v

dt dt v dt−∂

= = ≥∂∑ ∑v v

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Price Computation

Theorem:: :

,( ) (( ) ( ) ( ) )n s js n s j n j on

n s js

c G cγ β∈ <

⎛ ⎞= −⎜ ⎟

⎝ ⎠∑ ∑v vv v v

Locally measurable:

2

: :

,:

( ) ' (1 ( )) (1 ( ))

( ) (1 ( ))

k s j ks k s j j k on s

s k s jj j k on s

D v B B

v B

γ

β

−

∈ <

≤

⎛ ⎞= − −⎜ ⎟⎜ ⎟

⎝ ⎠

= −

∑ ∏

∏

v v v

v v

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Price Computation

Theorem:: :

,( ) (( ) ( ) ( ) )n s js n s j n j on

n s js

c G cγ β∈ <

⎛ ⎞= −⎜ ⎟

⎝ ⎠∑ ∑v vv v v

Locally measurable:

1 6 6 1,G c Gγ=

2 5 2,5G cμ β= −

5 5 sc Gγ= ∑2 3 4, ,G G G

( )2 2c γ η μ= +

n1

v2

n2

n3 n4 n5

v1

v4

v3

n61 6 1,6G cη β= −

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Price Computation (2)

• Prices can be computed in a stateless manner:

,:

( ) (1 ( )) s k s jj j k on s

v B sβ≤

= − =∏v v rate of admitted packets f rom session

2 5 2,5G cμ β= −

( )2 2c γ η μ= +

1 6 1,6G cη β= −

n1

v2

n2

n3 n4 n5

v1

v4

v3

n661 c−

51 c−ACK, sent per accepted pkt

( )# [0, ]

1

1 ACKs in T

ii

value on ACKT

η μ=

+ = ∑

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Numerical Evaluation

• Packet-level simulation: – slotted time– time-slot = 1ms– Poisson arrivals per slot– rate-update step size =

0.1

v1

v2

v3

v4 v5 v6

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Packet-level Simulation

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Extensions

• Alternative measures of social welfare:– Fairness oriented:– Service differentiation:

• Alternate routes per session.

• Loopy topologies: – No conclusive result whether the fluid model is well-

defined.– Special cases can be worked out:

log( ( ))ss

G∑ v

( ), 0s s ss

w G w >∑ v

D(x)

x

1