Cross-layer Control of Wireless Networks: From Theory to Practice

Korea Advanced Institute of Science and Technology

Network Systems Lab.

Cross-layer Control of Wireless Networks:

From Theory to Practice

Professor Song ChongNetwork Systems Laboratory

EECS, [email protected]



Multi-user Opportunistic Communication

Multi-user diversity

In a large system with users fading independently, there is likely to be a user with a very good channel at any time. Long-term total throughput can be maximized by always serving the user with the strongest channel.



Capacity Region: A Realization of Channel Consider a single channel realizationCDMA downlink with two users

θ: orthogonality factor in [0,1]

Capacity region [Kum03]

1 1 2 21 2 2 2

1 2 1 2 1 2

log 1 , log 1g p g p

r rg p n g p n

1 2p p P

0 0.5 0.5 1

User 1 User 2

1 2g p1 1g p

2 1g p2 2g p

1r

2rConvex

1r

2rNonconvex



Long-term Capacity RegionTime-varying achievable rate region

Long-term rate region

: long-term rate region

(can be shown to be convex)

F

1R

2R

F

1r

2r

1r

2r

1r

2r

t

: long-term rate of user iR i



Convexity Proof [Stol05]

Case of finite channel states and scheduling policiesNotation

S: finite set of channel statesSequence of channel states s(t)∈S, t=0,1,... forms an irreducible Markov chain with stationary distribution

K(s): set of all possible scheduling decisions for given channel state s∈Sri

s(k)≥0: rate allocated to user i for channel state s∈S and scheduling decision k∈K(s)rs(k): rate vector, i.e., rs(k)=[ri

s(k),∀i]

For each channel state s, a probability distribution φs=[φsk,∀k∈K(s)] is fixed, i.e.,

, , , , 1s s ss S

s S s S

( )

0, ( ), , 1sk skk K s

k K s s S



Convexity Proof [Stol05]

Rate vector for a set of distributions φ=[φs,∀s∈S]

If we interpret φsk as the long-term average fraction of time slots when the channel state is s and the rate allocation is k, then R(φ) is the corresponding vector of long-term average service rates

The long-term rate region FF is defined as the set of all average service rate vectors R(φ) corresponding to all possible φThe convexity of FF immediately follows as it is a convex hull of all possible instantaneous rates

Consider which is a convex combination of all possible rate vectors rs(k), ∀k∈K(s), ∀s∈S

( )

( )ss sk

s S k K s

R r k

( )

( )ss sk

s S k K s

R r k



Long-term NUM

Utility function [Mo00]

Network Utility Maximization (NUM)

max ( )

where = long-term rate of user i

= long-term capacity region

iR F

i

i

U R

R

F

11

, if 0 & 11log , if 1

ii i

i

RU R

R

α→0: throughput maximizationα=1: proportional fairness (PF)α→∞: max-min fairness



Maximization of sum of weighted rates

Both problems yield an unique and identical solution if we set , where is the optimal solution of the long-term NUM problem.

Sum of Weighted Rates (SWR)

max i iR F

i

R

*'( )i iU R *iR

2R

1R

F

*2R

*1R

1 2( ) ( )U R U R J

0

1 1 2 2R R K



Gradient-based SchedulingAssuming stationarity and ergodicity, we have

The long-term NUM problem can be solved if we solve with at each state sThe resource allocation problem during slot t

where Ri(t) is the average rate of user i up to time t and is the replacement of Ri

* which is unknown a prioriConvergence of Ri(t) to Ri

* can be proved by stochastic approximation theory [Kush04] or fluid limit technique [Stol05].

( )max maxi i s i iR F r F s

i i

R E r

where rate of user at state

( ) capacity region at state ir i s

F s s

( )max i ir F s

i

r

*'( )i iU R

( )

max '( ( )) ( )

subject to ( ) ( )

i ir t

i

U R t r t

r t F s t



Gradient-based Scheduling

This coincides with the optimality condition given by directional derivative

Consider

The optimality condition is given by

The optimal solution to the following problem is R*

Thus we set

* arg max iR F

i

R U R

* *

* * *

0,

,

i i ii

i i i ii i

U R R R R F

U R R U R R R F

*max i iR F

i

U R R

*'( )i iU R



HDR PF Scheduler

PF scheduler is a special case of gradient-based scheduler

Logarithmic utility functionFeasible region (TDMA)

PF scheduler serves user i* such that

( ), 1*

max ' ( ) ( )arg max

( ) subject to ( ) ( )

i ir ti t i

ii

U R t r tri

R t r t F s t

logi iU R R

1, 1

2, 1

, 1

00 0

0 0( ) , , , ...,

0

0 0 0

t

t

M t

r

rF s t

r

, 1where available data rate of user during slot i tr i t



Opportunistic Communication in OFDMA Downlink

Exploit multi-user diversity in time and frequencyIn a large system with users fading independently, there is likely to be a user with a very good channel at some carrier frequency for each time. Long-term total throughput can be maximized by always serving the user with the strongest channel.Challenge is to share the benefit among the users in a fair way.

User M

MobileUser 1

frequency

Channel

gain

frequency

Channel

gain

Fading channel



Frequency Selectivity in Channel

Frequency response in multipath environment

Delay spread

Coherence bandwidth Bc

Frequency separation at which the attenuation of two frequency-domain samples becomes decorrelated

For given delay spread,Frequency-selective channel if B>>Bc

Frequency-flat channel if B<<Bc

1

2cd

BT

2 ( )( ; ) ( ) ij f ti

i

H f t a t e ( ) : attenuation on path ( ) : propagation delay on path i

i

a t it i

,max ( ) ( )d i ji j

T t t

( ), ( )i ia t t

Bc

B

freq.

gain

B



Long-term NUM Problem in OFDMA Downlink

max ii

U R

User 1

User 2

User 3

User 3

User 4

User 3

Frequency (subcarrier)

Time slot

P

Power allocation

Subcarrier allocation

(user selection)Dynamic subcarrier and power allocation achieving



Joint OptimizationConsider M mobile users and N subcarriersJoint optimization of subcarrier and power allocation at each time slot t

Mixed integer nonlinear programming

, 1 , 1

, 1 , 1, 1 2

, 1

, 1

, 1

, 1 , 1

maximize ' ( )

subject to log 1

1,

0, 0,1 , ,

j ji i t i t

i M j N

ji t j tj

i t ji t

j tj

ji t

i

jj t i t

U R t r x

g pBr

N N

p P

x j

p x i j

( )F s t

, 1 where ( )i tr r F s t



Suboptimal Allocation [Lee08]

Iteratively solve two subproblemsFor fixed p, subcarrier allocation problem

Opportunistic scheduling over each subcarrier

For fixed x, power allocation problemConvex optimization (water filling)

Each subproblem is easy to solve

Frequency-selective power allocation (FPA)

Equal power allocationEqual power allocation

Subcarrier allocation for givenpower allocation

Subcarrier allocation for givenpower allocation

Power allocation for givensubcarrier allocation

Power allocation for givensubcarrier allocation

While subcarrier allocationis changing

InitializationEqual power allocation



Subproblem I: Opportunistic Scheduling

Find x for a fixed power vector p0

Separable w.r.t. subcarriersFor each subcarrier j, select user ij* such that

0max log 1

subject to 1,

0,1 , ,

ij i ij jx

i M j N

iji M

ij

x w g p

x j

x i j

* 0arg max log 1j i ij ji M

i w g p



Subproblem II: Water FillingFind p for a fixed subcarrier allocation x0

Convex optimizationWater filling is optimal

λ is a nonnegative value satisfying

( ) ( )max log 1

subject to

0,

m j m j j jp

j N

jj N

j

w g p

p P

p j

( )*

( )

1m jj

m j j

wp

g

*j

j N

p P

subcarrier

( )m jw

*jp

( )

1

m j jg



FPA vs. EPA

FPA gives significant throughput gain (up to 40%) in OFDMA downlink when

Sharing policy becomes more fairness-oriented Delay spread (frequency selectivity) increasesSystem bandwidth becomes wider

00.5

12

510

50100

12

34

56

78

1

1.1

1.2

1.3

1.4

s

No

rma

lize

d T

ota

l Th

rou

gh

pu

t (T F/T

E)

0 0.51

25

10 50100

12

34

56

78

1

1.1

1.2

1.3

1.4

s

No

rma

lize

d T

ota

l Th

rou

gh

pu

t (T F/T

E)B=5MHz B=20MHz

MT

MM

MT

MM



Impact of α: InterpretationEfficiency-oriented policy (α=0)

Only best user for each subcarrier

Fairness-oriented policy (α→∞)Bad-channel users are also selected

( ) 1, m j jg j N *j

Pp

N

FPA ≈ EPA

subcarrier

*jp

subcarrier

*jp

High, medium, low gm(j)j’s

*jp

P

N



Impact of System Bandwidth: Interpretation

Narrowband (less frequency-selective)

Wideband (more frequency-selective)

frequency

Channelgain

B

frequency

Channelgain

Bsubcarrier

*jp

Extreme case(frequency flat)

*j

Pp

N

subcarrier

*jp

*jp

P

N



Impact of SNR Distribution

EPA is comparable to FPA only when all the mobiles are located in high SNR regime

0 0.5

1 2

5 10

50 100

020

4060

80100

11.11.21.31.41.51.6

Low SNR Percentage (%)

No

rma

lize

d T

ota

l Th

rou

gh

pu

t (T F/T

E)

B=20MHzs=6

MT

MM

Low SNR: gij<5dB



Impact of SNR Distribution: Interpretation

High SNR

Mix of high and low SNR

Low SNR

*jp

subcarrier

*jp

p

( )log(1 )m j j jg p

p

( )log(1 )m j j jg p

Sensitive to power variation

Insensitive to power variation

Subcarriers with low SNR usersare more sensitive to power thanhigh SNR users

*jp

subcarrier

subcarrier



Throughput-optimal Scheduling and Flow Control

Joint scheduling and flow control— Stabilize the system whenever the long-term

input (demand) rate vector lies within the capacity region

— Stabilize the system while achieving throughput optimality even if the long-term input (demand) rate vector lies outside of the capacity region

Long-term NUM for arbitrary input rates [Nee05]

max

subject to (stability constraint)

0 (demand constraint)

iR

i

U R

R F

R D

where = long-term capacity region

= long-term demand vector

F

D



Single-carrier Downlink Problem

AssumptionInfinite demandsInfinite backlog at every transport layer queue

Cross-layer controlJoint optimization of flow control and scheduling

1( )d t

2 ( )d t

( )Kd t

1( )x t

2 ( )x t

( )Kx t

1( )q t

2 ( )q t

( )Kq t

demands

Flow Control at Source

Base Station

Scheduling

fading channel

1( )r t

2 ( )r t

( )Kr tfeedback: achievable rates

1( )L t

2 ( )L t

( )KL t



Cross-layer Control Scheduling at BS

Flow control at source i

Algorithm Performance

*

( ) ( ( ))max ( ) ( ) arg max ( ) ( )i i i i

r t F s t ii

q t r t i q t r t

0 : control parameterV 1 ( )( ) max ,0i

i

q tx t U

V

1

0

1limsup ( )

t

it i

q O Vt

1

*

0

1 1liminf ( )

t

i it

i i

U E x U R Ot V

Stability

Optimality



Derivation of Cross-layer Control Primal problem

Dual problem

Dual decomposition

,max ( )

subject to x ,

1

, 0

iX

i

i i i

ii

U x

r i

X

,min max ( ) ( )

subject to 1

, 0

i i i i iQ X

i i

ii

U x q x r

X

1 ,i ix U q i

1max

ii

i i ii

q r

,i i iiq x r i



Multi-hop Wireless Networks:Cross-layer Control



Multi-hop Wireline Network

Network Utility Maximization

―Link capacity is given and constant―Rate allocation problem

( )

max

s.t. ,

0

sr

s

s ls S l

U r

r c l

r

1r

2r 3r

1c 2c



Functional DecompositionLagrangian function

Dual problem

Dual decomposition― Flow control at source

― Congestion price at link

TCP is an approximation of this dual decomposition

( )

, = s l s ls l s S l

L r U r r c

min max ,r

L r

1

( ) ( )

max s s l s lr

s l L s l L s

U r r r U

( ) ( )

min ll s l s ll s S l s S l

r c r c



Multi-hop Wireless Network

Long-term Network Utility Maximization

― Link capacity is time-varying and a function of resource control

― Joint rate, power allocation and link scheduling

, , max

s.t. ,

sR P I

s

U R

R F P I

1R

2R 3R

1 , ,C P I h 2 , ,C P I h : power allocation: link scheduling: channel state

PIh



Functional DecompositionFor a realization of channelsLagrangian function

Dual problem

Dual decomposition― Flow control at source

― Scheduling/power control at link

― Congestion price at link

Joint MAC and transport problemDistributed scheduling/power control is a challenge

( )

, , , = , ,s l s ls l s S l

L r P I U r r C P I h

, ,

min max , , ,r P I

L r P I

1

( ) ( )

max s s l s lr

s l L s l L s

U r r r U

( ) ( )

min , , , ,ll s l s ll s S l s S l

r C P I h r C P I h

,

max , ,l lP I

l

C P I h



Per-link Queueing Case

User 0

User 2

)( 11 xU

)( 00 xU

)( 22 xU

cA=1 cB=1

a is the fraction oftime link A is used

0 1 2, ,

0 1

1 2

max ( )

subject to

1

, 0

i ix x x

i

a

b

a b

U x

x x

x x

x



Lagrange Multipliers

0 1,

0 2

max ( ) ( )

( )

1

, 0

i i A ax

i

B b

a b

U x p x x

p x x

x



Functional DecompositionCongestion control (sources and nodes)

MAC or scheduling (network)

0 10

0 1

max ( ) ( )

( )

i i Ax

i

B

U x p x x

p x x

1maxA B

A A B Bp p



Per-flow Queueing Case

User 0

User 1

User 2

)( 11 xU

)( 00 xU

)( 22 xU

cA=1 cB=1

a0 is the fraction oftime link A is used foruser 0

, 0

0 0

1 1

0 0

2 2

max ( )

subject to

1

i ix

i

a

a

a b

b

ij

U x

x

x

x



Functional DecompositionCongestion control (sources)

MAC or scheduling (network)

x0 μa0 μb0

x1 μa1 x2 μb2

pa0 pb0

pa1 pb2

0 0 1 1 2 20

max ( )i i a a ax

i

U x p x p x p x

0 0 0 0 0 1 1 2 21

max ( )i

i

a a b b b a a b bp p p p p



Interference Model

2r

1 2 3 4

5

node

link

5 2

1 3

4

1r

Network connectivity graph G

Conflict graph CG

- Links in G = nodes in CG- CG-Edge if links in G interfere with each other



Interference Model Maximal independent set model

- Only one maximal independent set can be active at a time

- - NUM problem

{1, 4}, {2}, {3}, {4, 5}

5 2

1 3

4

CGMaximal independent sets

( )I t I where {1, 4}, {2}, {3}, {4, 5}I

2

1

max ( )ii

U r

1 1

1 2 2

1 2 3 41 2 3

41 2 4

52

4

1

subject to independent set constraints

00 0

00 0

00 0

0 0

0 0 0

1ii

r c

r r c

a a a ar r c

cr r c

cr

a



Jointly Optimal Power and Congestion Control

NUM at particular state s [Chiang05]

is a nonconcave function of p Assuming high SINR regime, i.e,

can be converted into a concave function of p through a log transformation (geometric programming) Joint optimization of congestion control and power control

,

( )

max ( )

subject to

( , ) ,

where

ip r

i

i li I l

U r

r C p s l

( , )lC p slog(1 ( , )) log( ( , ))l lSINR p s SINR p s

( , )lC p s

0

( )( , ) log(1 ( , )) log 1

( )l ll

l lk kl

k Lk l

p sC p s SINR p s

N p s



Jointly Optimal Power and Congestion Control

Flow control at source

Power control at link

Congestion price at link

Interpretation

1

( )i l

l L i

r U

( ) ( ),( )

( ) ( )k k

kk kk

t SINR p t sm t

p t s

( )

( 1) ( ) ( ) ( )( )

ll l kl k

k Llk l

tp t p t s m t

p t

( )

( 1) ( ) ( ) ( ),l l i li I l

t t r t C p t s

Physical layer

r C

Transport layer

Source NodeFlow Control

LinkPower Control

Congestion

Price

r



Routing and Network Layer Queueing1 2

3

45

transport

layer

network

layer

(3)1A

(4)1A

(3)5A

(1)4A (1)

4R (1)4U

= set of commodities in the network = the amount of new commodity c data that exogenously arrives to node i during slot t = the amount of commodity c data allowed to enter the network layer from the transport layer at node i during slot t = the backlog of commodity c data stored in the network layer queue at node i during slot t

K( ) ( )ciA t

( ) ( )ciR t

( ) ( )ciU t

( and )i N c K



Dynamic Control for Network Stability

The stabilizing dynamic backpressure algorithm [Tassiulas92]

- An algorithm for resource allocation and routing which stabilizes the network whenever the vector of arrival rates lies within the capacity region of the network

Resource allocation- For each link , determine optimal commodity and optimal weight by

- Find optimal resource allocation action by solving

)(* tCab

)(* tab

]0,[max)(

)]()([maxarg)(

))(())((*

)()(

}),|({

*

** tCb

tCaab

cb

ca

Lbacab

abab

c

UUt

tUtUtC

)(

*

)(

)(.

))(),(()(max

ts

ababab

tI

ItIts

tstICt

),( ba

)(* tI



Dynamic Control for Network Stability Routing

- For each link such that , offer a transmission rate of to data of commodity .

The algorithm requires in general knowledge of the whole network state. However, there are important special cases where the algorithm can run in a distributed fashion with each node requiring knowledge only of the local state information on each of its outgoing links.

Interpretation- The resulting algorithm assigns larger transmission rates to links with larger differential backlog, and zero transmission rates to links with negative differential backlog.

),( ba 0)(* tab))(,)(()( ** tstICt abab )(* tCab



Dynamic Control for Infinite DemandsAssumption

Infinite backlog at every transport layer queueCross-layer control

Flow control at node iEach time t, set Ri

(c)(t) to

Routing and resource allocationSame as previous

Performance

Tradeoff between utility and delay

0 : control parameterV ( )

( ) 1( ) max ,0c

c ii

UR t U

V

1

( )

0

1limsup ( )

tc

it i

U O Vt

1

( ) *

, 0 ,

1 1liminf ( )

tc

i it

i c i c

U E R U R Ot V



References[Kum03] K. Kumaran and L. Qian, “Uplink Scheduling in CDMA Packet-Data Systems,” IEEE INFOCOM 2003.[Mo00] J. Mo and J. Walrand, “Fair End-to-End Window-Based Congestion Control,” IEEE/ACM Trans. Networking, Vol. 8, No. 5, pp. 556-567, Oct. 2000.[Kush04] H. J. Kushner and P. A. Whiting, “Convergence of Proportional-Fair Sharing Algorithms Under General Conditions,” IEEE Trans. Wireless Comm., vol. , no., 2004.[Stol05] A. L. Stolyar, “On the Asymptotic Optimality of the Gradient Scheduling Algorithm for Multiuser Throughput Allocation,” Operations Research, vol. 53, no. 1, pp. 12-25, Jan. 2005.[Lee08] H. W. Lee and S. Chong, "Downlink Resource Allocation in Multi-carrier Systems: Frequency-selective vs. Equal Power Allocation", IEEE Trans. on Wireless Communications, Vol. 7, No. 10, Oct. 2008, pp. 3738-3747.[Nee05] M. J. Neely et al., “Fairness and Optimal Stochastic Control for Heterogeneous Networks,” IEEE INFOCOM 2005. [Chiang05] M. Chiang, “Balancing Transport and Physical Layers in Wireless Multihop Networks: Jointly Optimal Congestion Control and Power Control,” IEEE J. Sel. Areas Comm., vol. 23, no. 1, pp. 104-116, Jan. 2005.[Tassiulas92] L. Tassiulas and A. Ephremides, “Stability Properties of Constrained Queueing Systems and Scheduling Policies for Maximum Throughput in Multihop Radio Networks,” IEEE Trans. Automatic Control, vol. 37, no. 12, Dec. 1992.

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Cross-layer Control of Wireless Networks: From Theory to Practice

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