01/16/2007 ECE department, Rice University Jingpu Shi 1 Intuitions on Proportional Fairness 0 * r r r r r x x x W Proportional fair rate per unit charge 0 * r r r r x x x Proportional fair rate tion between these two: replace users r by Wr identical users, construct the proportionally fair allocation over all users, and then provide to user r the aggregate rate cated to its sub-users. then the resulting rates are proporti per unit charge.
Intuitions on Proportional Fairness. Proportional fair rate. Proportional fair rate per unit charge. Relation between these two: replace users r by Wr identical sub-users, construct the proportionally fair allocation over all sub-users, and then provide to user r the aggregate rate - PowerPoint PPT Presentation
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01/16/2007 ECE department, Rice UniversityJingpu Shi
1
Intuitions on Proportional Fairness
0*
r
rrr
rx
xxW
Proportional fair rate per unit charge
0*
r
rr
rx
xxProportional fair rate
Relation between these two: replace users r by Wr identical sub-users, construct the proportionally fair allocation over all sub-users, and then provide to user r the aggregate rate allocated to its sub-users. then the resulting rates are proportional fair per unit charge.
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Intuitions on Proportional Fairness
Definition: A vector of rates x is proportionally fair if it is feasible and if for any other feasible vector x*, the aggregate of proportional changes is zero or negative
x2
x1
P1: x2 = x1 , Max-min fairness
P2: x2 = 3x1
P3: maximum aggregate throughput
12
3
xx
Aggregate change:
P1: maximum throughputP2: proportional fairnessP3: equal throughput
x2 + 3x1 = 0
0*
r
rr
rx
xx
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Maximizer of aggregate log utility
r
rxf )log(
r r
r
x
dxdf )(
0*
r
rr
rx
xx0df *
rx is the maximizer
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Proportional Fairness In CSMA networks: the case of two contending flows.
A a
B b
(1) Achievable log utility is bounded by P1.(2) If T(Aa)+T(Bb) = constant, P2 achieves maximum utility.(3) For achievable throughput, maximum is achieved around P3.
T(Aa)
T(Bb)
C
P1
P2P3
T(Aa) = T(Bb)
D
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Packet Decoding
Distance
Ch
ann
el E
rror
Pro
bab
ility
100%
Transmission range
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Carrier Sensing
Distance
Pro
bab
ility Ca
rrier is
sensed
100%
Interference range
01/16/2007 ECE department, Rice UniversityJingpu Shi
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AIS in real networks
B b
A a
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Simulations with 802.11 protocol
Measurements
every 400 ms
X = two-way handshake
= four-way handshake
Long term unfair !
Fair !
Short term Unfair !
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Modeling AIS (general equations)
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Modeling AIS (Non-backlogged case)
e is the probability that the transmission queue is empty.
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Validation – Model vs Simulation
0
200
400
600
800
1000
200 400 600 800 100012001400
Pa
cke
t Thr
oug
hpu
t (p
kt/s
)
Data Payload Size (bytes)
0
200
400
600
800
1000
200 400 600 800 100012001400
ns - Flow Bmodel - Flow B
ns - Flow Amodel - Flow A
With RTS/CTS Without RTS/CTS
TFA
TFA
ns - Flow Amodel - Flow A
TFA
ns - Flow Bmodel - Flow B
TFA
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Analysis of AIS
B bA a
B b B b
B bA a
B b B b
B b A a B b B b
• The collision probability of flow A a can be accurately computed assuming that the first packet arrives at a random point in time
•The collision probability of flow B b is zero
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Occurrence Probability
• We compute the occurrence probability of each scenario• Random throw two flows, given they are connected, what are the
probability that each of these scenarios occurs.
01/16/2007 ECE department, Rice UniversityJingpu Shi
• Problematic scenarios are highly likely to occur !
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Hop distance distribution in a multi-hop network
300 nodes - 2000 m x 2000 m – Random waypoint – DSDV
0
0.1
0.2
0.3
0.4
0.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Pro
bab
ility
Hop distance / TX range
Most of actively used hops are close to the maximum TX range !
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Transition probability for SIS class
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Model Vs. Simulation
System’s bi-stability, with large probability, the system is in one of the two stable states.
01
23
45
6
0 1 2 3 4 5 6stage A
00.020.040.060.080.1
0.12
Pro
bab
ility
stage B
0.140.16
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Two-hop Node’s severe TCP Penalty
First time segment is transmittedTCP retransmissions
TCP Congestion Window
TCP Timeouts
TCP ACK received (Accumulated ACK)
MAC Packet drop (Max Retry Limit reached)
295
300
305
310
315
320
325
330
335
80 85 90 95 100 105
Time [sec]
TC
P s
eque
nce
num
ber
[kB
]A B GW
TCP DATA
TCP ACK
TCP penalty
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Some Model Details
))(( Mp ii
ip Occurrence probability.
Markov stable state probability.
i Binary transmission matrix.
M Transition matrix.
66Tp
a
44Tpb
Throughput
Average channel state duration
T State duration
01/16/2007 ECE department, Rice UniversityJingpu Shi
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TCP throughput
• J. Padhye, V. Firoiu, D. Towsley, and J. Kurose. Modeling TCP throughput: a simple model and its empirical validation. ACMSIGCOMM, September 1998.
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Basic Topology RTS/CTS On
• Severe unfairness with Default CWmin, log utility = -0.6931• Improved fairness with increased CWmin at B, log utility = 0.6523• Log utility upper bound = 3.2917
Increase CWmin
A B GW
CWmin at 1st hop nodes
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Two Branches RTS/CTS ON
• Severe unfairness with default CWmin, log utility = -3.8• Improved fairness with larger CWmin at 1st hop nodes, log
utility = -1.23• Bounding log utility = 3.2
Increase CWmin Increase CWmin
B->GW
A->GW
C->GW
CWmin at 1st hop nodes
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Large Topologies: Long Hop Chain
1st hop CWmin = 128
• Severe unfairness with default CWmin, log utility = -11.9763• Improved fairness with larger CWmin, log utility = -6.1721• Log utility is bounded by 3.6931
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Large Topologies: Long Hop Chain (one queue)
• Severe unfairness with Default CWmin, log utility = -14.0015• Improved fairness with larger CWmin, log utility = -6.1415• Log utility is bounded by 3.6931
1st hop CWmin = 128
01/16/2007 ECE department, Rice UniversityJingpu Shi
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TFA Network
• 802.11 access and backhaul serving tier.
• Wireless card: • SMC 2532-b 802.11b 200 mW
power
• Antenna: • 15 dBi omni-directional
• Iperf
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Unfair Contention in Mesh
Two TCP flows contend.
GW
A
B
TCP traffic using Iperf v.1.7.0
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Prior Work Related to Unfairness Analysis
• Two classes of prior work related to our analysis on unfairness:
• Channel reservation of one flow may not be heard by its neighbors on a different channel.
AaBb
xxxxChannel N
A a B
(First identified by Junmin So etc, Mobihoc 04)
Example
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Multi-channel coordination:receiver on different channel
• Receiver is missing (on a different channel)
A B C
Example
Hard to synchronize channel hopping schedule.
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Challenges in solving all the problems
MMAC (Junmin So etc, Mobihoc 2004)Common time reference, infrastructure supported
t RTS/CTS/DATA/ACK (Channel 1)RTS/CTS/DATA/ACK (Channel 2)RTS/CTS/DATA/ACK (Channel 3)
Channel contention
phase
Data Transmission phase
Flow 1…Flow 2
Flow N…
Problems1) Duration of negotiation phase2) Receiver missing3) Single channel starvation problems
01/16/2007 ECE department, Rice UniversityJingpu Shi
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AMCP general description
– Asynchronous Multi-channel Coordination Protocol– Asynchronous– One common control channel, multiple data channels.
• Separate control exchange from data transmission.• Provide a common frequency reference for nodes.
Control channel
Data channel 1
Data channel 2
Data channel 3
RTS/CTS
DATA/ACK
RTS/CTS
DATA/ACK
RTS/CTS
DATA/ACK
01/16/2007 ECE department, Rice UniversityJingpu Shi
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AMCP Principle 1
– Reserve common channel and data channel differently.• Improve efficiency, avoid collision on data channels.
RTS/CTS
Data + ACK
Control channel
Data channel 1
Data channel 2
Defer transmissionon control channel
Reserve Data 2
01/16/2007 ECE department, Rice UniversityJingpu Shi
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AMCP Principle 2
– Only contend for channels clear of traffic
control
data + ACK
Control channel
Data channel 1
Data channel 2
t0 t1
Contendfor 2
Contend for 1, 2
Max Tx time
01/16/2007 ECE department, Rice UniversityJingpu Shi
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AMCP Principle 3
– Self-learning channel hopping • Stick to the channel given successful transmission• Contend for a different channel given collision
success collision
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Lower throughput bound analysis step 1
• Construct a worst-case low throughput scenario with N interferers: A cannot sense the activity of the interferers
A a
1
2
N
…
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Lower throughput bound analysisstep 2
– Assume aggregate transmission attempt distribution is Poisson.
NTTT
TT
DATACTSRTS
CTSRTS
ep
2
1
– Compute conditional collision probability perceived by this flow.
01/16/2007 ECE department, Rice UniversityJingpu Shi
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Lower throughput bound analysis step 3
• Use our single-channel CSMA analytical model to compute the (minimum) throughput of this flow.
M. Garetto, J. Shi, and E. Knightly. Modeling Media Accessin Embedded Two-Flow Topologies of Multi-hop WirelessNetworks. In Proc. ACM MobiCom, Cologne, Germany,August 2005.
01/16/2007 ECE department, Rice UniversityJingpu Shi