Utility, Fairness, TCP/IP
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Transcript of Utility, Fairness, TCP/IP
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Utility, Fairness, TCP/IP
Steven Low
CS/EEnetlab.CALTECH.edu
Feb 2004
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Acknowledgments Caltech
Bunn, Choe, Doyle, Jin, Newman, Ravot, Singh, J. Wang, Wei
UCLA Paganini, Z. Wang
CERN Martin
SLAC Cottrell
Internet2 Almes, Shalunov
Cisco Aiken, Doraiswami, Yip
Level(3) Fernes
LANL Wu
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Protocol Decomposition
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, … Topology, power control Maximize capacity
Shortest-path routing Minimize path costs
Duality model (Kelly, Low et al) Maximize aggregate
utility
HOT (Doyle et al) Minimize user response
time Heavy-tailed file sizes
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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Congestion control
xi(t)
pl(t)
Example congestion measure pl(t) Loss (Reno) Queueing delay (Vegas)
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TCP/AQM
Congestion control is a distributed asynchronous algorithm to share bandwidth
It has two components TCP: adapts sending rate (window) to congestion AQM: adjusts & feeds back congestion information
They form a distributed feedback control system Equilibrium & stability depends on both TCP and AQM And on delay, capacity, routing, #connections
pl(t)
xi(t)TCP: Reno Vegas
AQM: DropTail RED REM/PI AVQ
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Network model
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
liRlif link uses source if 1
liR lib link uses source if 1
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Network model
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
lieR lislif link uses source if
lieR lislib link uses source if E
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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MethodologyProtocol (Reno, Vegas, RED,
REM/PI…)
Equilibrium Performance
Throughput, loss, delay
Fairness
Dynamics Local stability Global stability
))( ),(( )1())( ),(( )1(txtpGtptxtpFtx
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Network model
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
Reno, Vegas
DT, RED, …
liRli link uses source if 1 IP routing
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Duality model
))( ),(( )1())( ),(( )1(txtpGtptxtpFtx
Primal-dual algorithm:
)( )( max )( min
subject to )( max
00
0
:Dual
:Primal
ll
ll
sss
xp
sss
x
xcpxUpD
cRxxU
s
s
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Duality Model of TCP
))( ),(( )1())( ),(( )1(txtpGtptxtpFtx
Primal-dual algorithm:
Reno, Vegas
DropTail, RED, REM
Source algorithm iterates on rates Link algorithm iterates on prices With different utility functions
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Summary: duality model
cRx
xUs
ssxs
subject to
)( max0
Flow control problem (Kelly, Malloo, Tan 98)
TCP/AQM Maximize utility with different utility functions
Primal-dual algorithm
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
Reno,
VegasDropTail, RED, REM
Result (L 00): (x*,p*) primal-dual optimal iff 0 ifequality with ** lll pcy
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Example utility functions
1 log1 )1(
:General
log : Vegas32
log1 :2-Reno
3/2tan23 :1-Reno
11
1
i
i
ii
ii
ii
i
iii
xx
xTxTx
T
TxT
/
FAST, STCP
(Mo, Walrand 00)
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MethodologyProtocol (Reno, Vegas, RED,
REM/PI…)
Equilibrium Performance
Throughput, loss, delay
Fairness
Dynamics Local stability Global stability
))( ),(( )1())( ),(( )1(txtpGtptxtpFtx
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222
2
3
33
)1(4
)1 )(2
-(NcN
c
Theorem (Low et al, Infocom’02) Reno/RED is locally stable if
Stability: Reno/RED
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
TCP: Small Small c Large NRED: Small Large delay
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Stability: scalable control
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
lll
l ctyc
tp )(1 )()(
)(tq
mii
iii
i
extx
Theorem (Paganini, Doyle, L, CDC’01) Provided R is full rank, feedback loop is locally stable for arbitrary delay, capacity, load and topology
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Linear Stability: scalable control
Theorem (Paganini, Doyle, Low, CDC’01) Provided R is full rank, feedback loop is locally stable for arbitrary delay, capacity, load and topology
2
1
( )fy R s x ( )Tbq R s p
0i ii i
i
xx qM
1l l
l
p yc
Globally stable in presence of delay?
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Stability: Stabilized Vegas
)()(1)( tan)(
1 )()(1-2 tqtt
tTx iid
tqtxi ii
ii
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
lll
l ctyc
tp )(1 )(
Theorem (Choe & L, Infocom’03) Provided R is full rank, feedback loop is locally stable if
),( max aTx ii
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Stability: Stabilized Vegas
)()(1)( tan)(
1 )()(1-2 tqtt
tTx iid
tqtxi ii
ii
F1
FN
G1
GL
Rf(s)
Rb’(s)
TCP Network AQM
x y
q p
lll
l ctyc
tp )(1 )(
Application Stabilized TCP with current routers Queueing delay as congestion measure has right scaling Incremental deployment with ECN
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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Reno TCP Packet level
Designed and implemented first Flow level
Understood afterwards Flow level dynamics determines
Equilibrium: performance, fairness Stability
Design flow level equilibrium & stability Implement flow level goals at packet level
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Packet level ACK: W W + 1/W
Loss: W W – 0.5W
Reno AIMD(1, 0.5)
ACK: W W + a(w)/W
Loss: W W – b(w)W
HSTCP AIMD(a(w), b(w))
ACK: W W + 0.01
Loss: W W – 0.125W
STCP MIMD(a, b)
RTT
baseRTT W W :RTT FAST
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Flow level: Reno, HSTCP, STCP, FAST
Similar flow level equilibrium
= 1.225 (Reno), 0.120 (HSTCP), 0.075 (STCP)
pkts/sec
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Flow level: Reno, HSTCP, STCP, FAST
Different gain and utility Ui They determine equilibrium and stability
Different congestion measure pi Loss probability (Reno, HSTCP, STCP) Queueing delay (Vegas, FAST)
Common flow level dynamics
windowadjustment
controlgain
flow levelgoal=
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Implementation strategy Common flow level dynamics
windowadjustment
controlgain
flow levelgoal=
Small adjustment when close, large far away Need to estimate how far current state is wrt target Scalable
Window adjustment independent of pi Depends only on current window Difficult to scale
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FAST TCP
Theorem (Jin, Wei, L ‘03) In absence of delay at a single link Mapping from w(t) to w(t+1) is contraction Global exponential convergence Full utilization after finite time Utility function: i log xi (proportional fairness)
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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Network
(Sylvain Ravot, caltech/CERN)
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FAST TCPutil: 95%
Linux TCPutil: 19%
1Gbps path; 180 ms RTT; 1 flowJin, Wei, Ravot, etc (Caltech, Nov 02)
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Dynamic sharing: 3 flowsFAST Linux
Dynamic sharing on Dummynet capacity = 800Mbps delay=120ms 3 flows iperf throughput Linux 2.4.x (HSTCP: UCL)
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Dynamic sharing: 3 flowsFAST Linux
HSTCP STCP
Steady throughput
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FAST Linux
throughput
loss
queue
STCPHSTCP
Dynamic sharing on Dummynet capacity = 800Mbps delay=120ms 14 flows iperf throughput Linux 2.4.x (HSTCP: UCL)
30min
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FAST Linux
throughput
loss
queue
STCPHSTCP
30min
Room for mice !
HSTCP
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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Network model
F1
FN
G1
GL
R
RT
TCP Network AQM
x y
q p
))( ),(( )1())( ),(( )1(
tRxtpGtptxtpRFtx T
Reno, Vegas
DT, RED, …
liRli link uses source if 1 IP routing
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Motivation
ll
li l
lliRiiixp
iii
xR
cppRxxU
cRxxU
ii
max)( max min
subject to )( maxmax
00
0
:Dual
:Primal
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Motivation
Can TCP/IP maximize utility?
ll
li l
lliRiiixp
iii
xR
cppRxxU
cRxxU
ii
max)( max min
subject to )( maxmax
00
0
:Dual
:Primal
Shortest path routing!
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TCP-AQM/IP Theorem (Wang, et al 03) Primal problem is NP-
hard
Ai
iAi
i cc
Proof Reduce integer partition to primal problem
Given: integers {c1, …, cn}Find: set A s.t.
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TCP-AQM/IP Theorem (Wang, et al 03) Primal problem is NP-
hard
Achievable utility of TCP/IP?
Stability? Duality gap?Conclusion: Inevitable tradeoff
between achievable utility routing stability
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Ring networkdestination
r
Single destination Instant convergence of
TCP/IP Shortest path routing
Link cost = pl(t) + dl
price static
TCP/AQM
IPr(0)
pl(0)
r(1)
pl(1)
… r(t), r(t+1) , …
routing
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Ring networkdestination
r
TCP/AQM
IPr(0)
pl(0)
r(1)
pl(1)
… r(t), r(t+1) , …
Stability: r ?
Utility: V ?r* : optimal routing
V* : max utility
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Ring networkdestination
r
link cost = pl(t) + dl
0
0||*
*
VV
rr
Theorem (Infocom 2003) Solve primal problem
asymptoticallyas
Stability: r ?
Utility: V ?
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Ring networkdestination
r
link cost = pl(t) + dl
Theorem (Infocom 2003) large: globally unstable small: globally stable medium: depends on r(0)
Stability: r ?
Utility: V ?
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General networkConclusion: Inevitable tradeoff
between achievable utility routing stability
random graph20 nodes, 200 links Achievable
utility
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Outline Network model FAST TCP
Equilibrium Stability Implementation Experiments
TCP/IP interaction Fairness-efficiency
Applications
TCP/AQM
IP
Transmission
WWW, Email, Napster, FTP, …
Ethernet, ATM, POS, WDM, …
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TCP/AQM: duality model Flow control problem (Kelly, Malloo, Tan 98)
TCP/AQM Maximize utility with different utility functions (L 00): (x*,p*) primal-dual optimal iff
Primal-dual algorithmReno, Vegas, FAST
DT, RED, REM/PI, AVQ
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Fairness
Identify allocation with An allocation is fairer if its is
larger
(Mo, Walrand 00)
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Fairness
maximum throughput
proportional fairness
min delay fairness infinity maxmin
fairness
(Mo, Walrand 00)
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Efficiency
Unique optimal rate x() An allocation is efficient if T() is
large
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Conjecture
Conjecture T() is nonincreasingi.e. a fair allocation is always
inefficient
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Example 1
Conjecture T() is nonincreasingi.e. a fair allocation is always
inefficient
1/(L+1)
L/(L+1)
1/2
1/2
maxmin proportional
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Example 1
Conjecture T() is nonincreasingi.e. a fair allocation is always
inefficient
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Example 2
Conjecture T() is nonincreasingi.e. a fair allocation is always
inefficient
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Example 3
)(T
Conjecture T() is nonincreasingi.e. a fair allocation is always
inefficient
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Intuition
“The fundamental conflict between achieving flow fairness and maximizing overall system throughput….. The basic issue is thus the trade-off between these two conflicting criteria.”
Luo,etc.(2003), ACM MONET
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Results Theorem: Necessary & sufficient
condition for general network
Corollary 1: true if N(R)=1
1/(L+1)
L/(L+1)
1/2
1/2
maxmin proportional
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Results Theorem: Necessary & sufficient
condition for general network
Corollary 1: true if N(R)=1
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Results Theorem: Necessary & sufficient
condition for general network
Corollary 2: true if N(R)=2 2 long flows pass through same# links
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Counter-example There exists a network such that
dT/d > 0 for almost all >0 Intuition
Large favors expensive flows Long flows may not be expensive
Maxmin may be more efficient than proportional fairness
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Counter-example Theorem: Given any 0>0, there exists
network where
Compact example
0
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netlab.caltech.edu/FAST