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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
10/26/2005 CCW053
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
10/26/2005 CCW054
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
10/26/2005 CCW055
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
10/26/2005 CCW056
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
10/26/2005 CCW057
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
10/26/2005 CCW058
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
10/26/2005 CCW059
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
10/26/2005 CCW0510
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= −
10/26/2005 CCW0511
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=
+ = ∑
10/26/2005 CCW0512
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
10/26/2005 CCW0513
Packet-level Simulation
10/26/2005 CCW0514
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