Intuitions on Proportional Fairness

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.


2

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


3

Maximizer of aggregate log utility

r

rxf )log(

r r

r

x

dxdf )(

0*

r

rr

rx

xx0df *

rx is the maximizer


4

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


5

Packet Decoding

Distance

Ch

ann

el E

rror

Pro

bab

ility

100%

Transmission range


6

Carrier Sensing

Distance

Pro

bab

ility Ca

rrier is

sensed

100%

Interference range


7

AIS in real networks

B b

A a


8

Simulations with 802.11 protocol

Measurements

every 400 ms

X = two-way handshake

= four-way handshake

Long term unfair !

Fair !

Short term Unfair !


9

Modeling AIS (general equations)


10

Modeling AIS (Non-backlogged case)

e is the probability that the transmission queue is empty.


11

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


12

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


13

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.


14

00.10.20.30.40.50.60.70.80.9

1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pro

bab

ility

(c

on

dit

ion

ed

)

Normalized distance (sender-receiver distance/ transmission range)

SCAISSIS

Probabilities of 3 groups of scenarios

• Problematic scenarios are highly likely to occur !


15

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 !


16

Transition probability for SIS class


17

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


18

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


19

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


20

TCP throughput

• J. Padhye, V. Firoiu, D. Towsley, and J. Kurose. Modeling TCP throughput: a simple model and its empirical validation. ACMSIGCOMM, September 1998.


21

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


22

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


23

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


24

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


25

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


26

Unfair Contention in Mesh

Two TCP flows contend.

GW

A

B

TCP traffic using Iperf v.1.7.0


27

Prior Work Related to Unfairness Analysis

• Two classes of prior work related to our analysis on unfairness:

• Studies on fairness with perfect, TDMA or Slotted Aloha MAC: – [Radunovic TMC 04 ] [Huang MobiHoc 01 ] [Chen Infocom 06] [Chen Infocom 05] [Tan IEEE Comm. Letters 06]

[Tassiulas INFOCOM 02] [Kar IEEE Transactions on Automatic Control 04].

• Studies on fairness with CSMA or IEEE 802.11 MAC.– Papers reporting poor performance of IEEE 802.11.

• [Sundaresan, Ad Hoc Networks Journal 04] [Nandagopal MOBICOM 00] [Chen MOBICOM 06] [Luo MOBICOM 00] [Karn ARRL/CRRL ARCNC 90] [Bharghavan SIGCOMM 94] [Kanodia MobiHoc 02] [Wang INFOCOM 05] [Carvalho,MOBICOM 04]

• We systematically study all possible two-flow scenarios, and analytically capture unfairness contention between the two flows.

– Papers reporting poor performance of TCP. • [Gerla, WMCSA 99] [Tang MMTWCW 99] [Raniwala INFOCOM 07] [Xu IEEE Communications Magazine, 01] [Xu

WOWMOM 02] [Xu MOBICOM 03] [Holland MOBICOM 99] [Fu INFOCOM 03] [Yu MOBICOM 04] [Gambiroza MOBICOM 04]

• We identify unfair contention in the basic scenario, and develop analytical models to study two flow contention.


28

Prior Work Related to Our Solution

• Prior work on the use of multiple channels.– [Adya Broadnets 04] [ Bahl MobiCom04] [Jain IC3N01] [Nasipuri 99] [So MobiHoc 04] [Wu I-

SPAN 00]– All these protocols are designed to improve fairness, and do not provided any sort of lower

throughput bound for individual flows.

• Prior work on contention window policy.– [Cali TON 00] [Kuo INFOCOM’ 03] [Chen INFOCOM 2001] [Nafaa WCNC 05] [Romdhani

WCNC 03]– None of these identified the role of 1st-hop contention window in shifting queuing of mesh

network and improving fairness.


29

Multi-hop flow topology

IEEE 802.11 networks, Ns 2, 50 nodes, 10 flows, 1m/s, 1000x1000m UDP load: 30 pkts/s


30

Multi-channels to solve starvation, multi-hop flows

• Multi-channel protocols do not necessarily address starvation.

• Our objective: improves per-flow throughput

0 2 4 6 8 100

10

20

30

40

50

Flow ID

Thr

ough

put (

pkt/

s)

80211MMAC


31

Challenges in solving starvation

• Single channel starvation problem– Several transmissions can occur on one channel, thus inherit single-

channel starvation problems.

• Multi-channel coordination problem– Separate transmissions to reduce interference.

– Coordinate their transmission.

– How to achieve these two goals.


32

Multi-channel coordination:missed channel reservation

• 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


33

Multi-channel coordination:receiver on different channel

• Receiver is missing (on a different channel)

A B C

Example

Hard to synchronize channel hopping schedule.


34

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


35

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


36

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


37

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


38

AMCP Principle 3

– Self-learning channel hopping • Stick to the channel given successful transmission• Contend for a different channel given collision

success collision


39

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

…


40

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.


41

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.


42

Protocol PerformanceSingle-hop flows, multi-hop topology

12 data channels, 100 nodes, 50 one-hop flows 1000mx1000m area

Flows starve with 80211Log U = -90.9

MMAC, Log U = -3.7

AMCP = 13.2

Maximum Log U = 34.65


43

Protocol performance (multi-hop flows with mobility)

50 nodes, 10 flows, 1m/s, UDP traffic: 30 pkts/s

AMCP outperforms802.11 and MMAC

Log: 802.11 = -24.2MMAC = -21.05AMCP = -15.3Max = -10.20 2 4 6 8 10

0

10

20

30

40

50

Flow ID

Thr

ough

put (

pkt/

s)

AMCPMMAC80211


44

Protocol performance (multi-hop flows with mobility)

AMCP outperforms802.11 and MMAC

Log: 802.11 = -56.2MMAC = -74.3AMCP = -54.5Max = -32.4

Scenario: 20 nodes, gateway download to each node. Gateway is saturated.


45

Channel switching overhead


46

Inefficiency due to channel switching constraints

Some packets may be stuck in the queue due to incapabilities of swift channel switching

A

BCC

B

C

C

Example

Documents

Intuitions on Proportional Fairness