throughput optimization of urban wireless mesh networks peng wang department of electrical and...

Throughput Optimization of Urban Wireless Mesh

Networks

Peng WangDepartment of Electrical and Computing Engineering

University of Delaware

Outline

• Scenario of interest• Optimal Scheduling

– Reduce the optimization space– Computation complexity of MWIS– Interference model

• Conclusions

Scenario: Urban Mesh Networks

Many fixed wireless relays (routers) and few wired base stations (gateways).– Lamppost mounted routers– Indoor routers– E.g., Mountain View CA, Philadelphia, SF, Corpus

Christi, …

destination

base station

destination

fixed relay

Real Urban Maps and Realistic Propagation

The map is from UDEL mobility model.

• Only outdoor routers

Outline



• Conclusions

maximize utility

Example objective functions

max w log kf ,k

maximize minimum e2e data rate

max min w kf ,k

is an end-to-end connection

Each connection is spread among one or more flows. The kth flow is denoted (,k)

f,k is the data rate along flow (,k)

kf,k is the total data rate for connection

w administrative weight for connection

is the set of all connections

P(,k) is the path for flow (,k), i.e., it is a set of links traversed by flow f,k

is an end-to-end connection

Each connection is spread among one or more flows. The kth flow is denoted (,k)

f,k is the data rate along flow (,k)

kf,k is the total data rate for connection

w administrative weight for connection

is the set of all connections

P(,k) is the path for flow (,k), i.e., it is a set of links traversed by flow f,k

Notation

The total data rate across link x is

,k |xP ,k f ,k

Notation: Assignments

v denotes an assignment. It specifies:1. which links are transmitting2. their transmission power3. the bit-rates

Assignments

V denotes the set of all considered assignments

•v is a vector of zeros and ones

•vx=1 link x is transmitting during assignment v

•There are 2L assignments, where L is the number of

links

In the simple case (no power control, only a single bit-rate, SISO)

Notation: Data RatesR(v,x) denotes the data rate over link x during assignment v

In general, R(v,x) depends on the whole assignment.

A schedule is a convex combination of assignments.

{v | vV} such that vVv1 and 0 v

Notation: Schedules

In some simple cases (no power control, only a single bit-rate, SISO),

0 otherwiseR(v,x) =Rx if vy=0 for all y(x)

Nominal data rate over link x

Conflict set of link x

The data rate provided by schedule {v} over link x is

vV vRv,x

Problem Definition of Optimal Scheduling

Maximize Throughput

maxU f

such that: ,k |xP ,k

f ,k vV

vRv,x for each link x

vV

v 1

0 v for all v V

Maximize utility

max w log kf ,k

Maximize min e2e data rate

max min w kf ,k

Total flow rate cross link x

Link capacity provided by schedule v at link x

Simple Example of Wireless Network

AssigAssignmenmentsnts

L1L1 L2L2 L3L3 L4L4 ScheScheduledule

ss

V1V1 1 0 0 0 α1

V2V2 0 1 0 0 α2

V3V3 0 0 1 0 α3

V4V4 0 0 0 1 α4

V5V5 1 0 1 0 α5

V6V6 1 0 0 1 α6

V7V7 0 1 0 1 α7

RateRate R1R1 R2R2 R3R3 R4R4

},,,min{max 4321 ffff

D1 D2 D3SL1

f2

D4L2 L4L3

f1f4

f3

Objective Function:

vV

vRv,x

Constraint for link L2:

21 ff ,k |xP ,k

f ,k

7

12,2

iiivR

7

12,221

iiivRff

Feasible Assignments

Problem Definition of Optimal Scheduling

Maximize Throughput

maxU f


f ,k vV


vV

v 1

0 v for all v V

Maximize utility

max w log kf ,k

Maximize min e2e data rate

max min w kf ,k

Total flow rate cross link x

Link capacity provided by schedule v at link x

Should schedule v include all assignments? 2L

Key observation: Not all assignments are needed.

The Space of Assignments

Caratheodory’s Theorem The optimal schedule is the combination of L assignments.

The optimization problems can easily be solved if V only has L elements.

The challenge is finding these special L assignments. Reduce the optimization space without loss of throughput

Idea: Given a set of assignments, find a better assignment

maxU f


f ,k vV


vV

v 1

0 v for all v V

Finding Better Assignments

Link constraints Lagrange multiplier x

Time (bandwidth) constraints Lagrange multiplier

x is the price/bit to transmit data across link x.

xR(v,x) x is the revenue generated by assignment v.

is the revenue generated by the best assignments in V.

Assignments that are better than all considered assignments must satisfy

xR(v,x) x >

Algorithm for Optimal Scheduling

“Toward tractable computation of the capacity of multihop wireless networks.” infocom 2007.

Given the initial set of assignments V

Solve Optimization over V

Search for New Assignment v+

v+= argmax xR(v, x) x

μx* and

λ* V= v+ V

Optimal Schedule

xR(v+, x) x

≤

v

xR(v+, x) x

>

We can find the optimal schedule for network with more than 2000

links.

maxU f

suchthat: ,k |xP ,k

f ,k vV

vRv,xlink x

vV

v 1

0 v forall v V


Optimization Solver

max xR(v,x) x

λ* and μx

*

New assignment v+

v)(max fU

s.t. Considered assignments V

)(log)( ,

k kfwfU

k kfwfU ,min)(

Convergence of the algorithm

Algebraic convergence

Geometrical convergence

Idea of Proof of Convergence

Vfor

xlinkeachforxRf

fU

V

VPx

v

v

v

vv

vv

0

1

),(

)(min

)}(|{

Primal Problem

s.t

Dual Problem

0)(

),(

),(max

μ

vv

μ

v

h

VforxR

g

xV

s.t

Cutting plane method


Optimization Solver

max xR(v,x) x

λ* and μx

*

New assignment v+

In the worst case, MWIS is NP-Hard.

v

This work provides empirical evidence that the MWIS problem that arises in

scheduling is not computationally difficult.

This is equivalent to the maximum weighted independent set problem (MWIS)

)(max fU

s.t. Considered assignments V

)(log)( ,

k kfwfU

k kfwfU ,min)(

Convergence of the algorithm

Algebraic convergence

Geometrical convergence

Outline



• Conclusions

The Conflict Graph

• Each link in the network is a vertex in the conflict graph• If y(x), then there is an edge between x and y

– Neighboring vertices in the conflict graph cannot simultaneously transmit.

• Each vertex x has weight wx

(a) = {b,c,d}

(b) = {a,c,d}

(c) = {a,b,d,e}

(d) = {a,b,c,f}

(e) = {c}(f) = {d}

ab

c

d

e

f

Topology

e f

a

c

b

we

wawb

wc

wf

wd

d

Weighted Conflict Graph Cliqu

e

The Maximum Weighted Independent Set

• An independent set is a selection of vertices such that no two vertices in the selection are neighbors.– Example: {a,e,f}, {b,e,f}, {d,e}, {c,f},…

• Maximum Weighted Independent set– The independent set with maximum weight– Example: W{b,e,f}=7, W{d,e}=7

e f

a

c

b

3

1 2

3

2

4

d

Weighted Conflict Graph

wa = 1

wb = 2

wc = 3

wd = 4

we = 3

wf = 2

Computing a MWIS

many constraints

maxvx

Rx xvx

such that: vx vy 1 if y x

vx 0,1

{Qi i =1,2,…,I} be any set of cliques such that if y(x), then there exists an i such that xQi and yQi

Clique Decomposition of the Conflict Graph

maxvx

Rx xvx

such that: xQ i

vx 1 for i 1,2, . . . I

vx 0,1 10 improvement

Construction of Random Wireless Networks

•Topology parameters– Propagation Models

• Two-ray (nodes are uniformly distributed)• Two-ray with lognormal shadowing (nodes are uniformly distributed)• Ray-tracing with UDEL model (realistic propagation)

– The number of nodes : n {32, 64, 128, 256, 512, 1024, 2048}– The target number of neighbors (at the target bit-rate): Δ {3, 6, 9, 12, 15,

18, 24}– The number of the gateways: NGW n / {8, 16, 32} – The target bit-rate of links: r* {6, 9, 12, 18, 24, 36, 48, 54} Mbps

•At least 40 topology samples for each set of the above parameters. Over 10000 topologies in all.

•Max-flow interference aware routing– Details are in the dissertation– It turns out that this LP problem is the most computational complex problem

of this investigation

Large number of computational

experiments on random wireless networks

Empirical Evidence

NO theoretic analysis of the complexity of MWIS

problem

Propagation• Ray-tracing

(outdoors)• Attenuation Factor

Model (indoors)

City Map• 2 km2 downtown Chicago• ~ 500 outdoor lamppost

mounted nodes• ~ 6500 indoor

infrastructure nodes

The Time to Compute the MWIS (low degree case)

50 100 500 1000200010

-4

10-3

10-2

10-1

100

Urban Propagation

50 100 500 1000200010

-4

10-3

10-2

10-1

100

50 100 500 1000200010

-4

10-3

10-2

10-1

100

Mean

Tim

e t

o C

om

pu

te M

WIS

(se

c)

Num of Nodes

Two-Ray Two-Ray with Shadowing

10-6.7n1.97

Computation time Computation time Computation time

10-6.6n1.85 10-6.1n1.75

• The time to solve MWIS is quite small (one second for 2048 nodes topology)

• The time is polynomial in the number of nodes

AxnB + T0 secs

• Do not depend on the propagation models

Δ = 6 Target Bit rate = 24 Mbps NGW = number of nodes / 16

only 6 neighbors/node

A Model of the Computation Time

K: depends on the number of nodes and propagation model

α, β: Only depend on the propagation model

Mean degree: encapsulates other parameters

T0: overhead to load CPLEX solver

Time to find a MWIS – T0 ≈ K x Mean degree of the conflict graph

≈ α x nβ x Mean degree of the conflict graph

The number of links that a link is in conflict with, which is the degree of vertex in the conflict graph

• Δ • NGW • r*

Computation time versus the mean degree of the conflict graph

• As Δ increases, the mean degree increases.

• As the number of gateways increases, the mean degree slightly decreases.

• Linear fit: Computation time ≈ K * mean degree.

10 20 301

2

3

4

5

6x 10

-3

128 nodes, GWs=[16]

20 40 600

0.05

0.1

0.15

0.2512 nodes, GWs=[16 32 64]

20 40 60 800

0.2

0.4

0.6

0.8

1024 nodes, GWs=[64]

Mean

Tim

e t

o C

om

pu

te M

WIS

(se

c)

=3 =6 =9 =12 =15 =18 =24

Red: 16 GWs Blue: 32 GWs Green: 64 GWs

0.00018 Degree 0.0012 Degree 0.008 Degree

Mean Degree in Conflict Graph

Δ: varies from 3 to 24 NGW{16, 32, 64} Target Bit rate: 24 Mbps Urban Propagation Model

fixed

Behavior is the same for other propagation

models

Computation time versus the mean degree of the conflict graph

• The mean degree increases with the target bit-rate.

High bit-rate is more susceptible to interference.

• Linear fit: Computation time ≈ K * mean degree.

r*=6Mbps

Red: 16 GWs Blue: 32 GWs Green: 64 GWsr*=12Mbps r*=18Mbps r*=24Mbps r*=36Mbps r*=48Mbps r*=54Mbps

10 15 20 25 30 350

0.02

0.04

0.06

0.08

0.1

Mean Degree in Conflict Graph

512 nodes, GWs=[16 32 64]M

ean

Tim

e t

o C

om

pu

te M

WIS

(se

c)

0.0012 Degree

Δ = 6 NGW{16, 32, 64} Target Bit rate {6, 12, 18, 24, 36, 48, 54} Mbps Urban Propagation Model

fixedBehavior is the same for

other propagation models

Slope of Computation Time versus Mean Degree

Previous slides Computation time ≈ K mean degree

K versus the number of nodes in the network

urban prop. Two-ray Two-ray w/ shadowing128 256 512 1024

10-4

10-3

10-2

Num of Nodes

K (

slop

e)

Computed

1.77×10-8×n1.88

K (

slop

e)

K (

slop

e)

128 256 512 102410

-4

10-3

10-2

10-1

Num of Nodes

Computed

1.09×10-7×n1.64

128 256 512 102410

-4

10-3

10-2

10-1

Num of Nodes

Computed

7.87×10-8×n1.75

Computation time ≈ K mean degree.

≈ α nβ mean degree.

The time to compute MWIS can be modeled as

Propagation Model α β

Urban 1.77*10-8 1.88

Two-ray 1.09*10-7 1.64

Two-ray with Shadowing 7.87*10-8 1.75

?? Computation is polynomial in the number of nodes

• The mean degree also varies with the number of nodes.

• There is no simple relationship between the mean degree and the number of nodes

(128,8)(256,16) (512,32) (1024,64)10

20

30

40

50

60

70

80

90

(Num Nodes, Num GWs)

Mean

deg

ree i

n t

he c

on

flic

t g

rap

h

Δ = 3

Δ = 6Δ = 9

Δ = 12Δ = 15

Δ = 18

Δ = 24

Urban propagationGateway density fixedTarget bit-rate = 24 Mbps

• Computing maximum weighted independent set when computing optimal schedules in wireless mesh networks is not computationally difficult

• E.g., it takes one second when there are 2048 nodes

• With the mean degree fixed, the time to compute the MWIS grows polynomially with the number of nodes

• With the number of nodes fixed, the MWIS grows linearly with the mean degree of the conflict graph

Outline



• Conclusions

SINR Protocol Model

NOISEHHH

HxSINR

xcxbxa

xxcba

,,,

,,,, :)(

If links a,b,…,c are transmitting,

T(x): SINR threshold to achieve the desired data rate Rx at link x

)()(

)()()(

yTySINR

xTxSINRyx

x

y

Conflict set of link x:

.0

)(,...,)(,...,, otherwise

xcbaifRxR x

cba

The data rate of link x:

Protocol Model with Multi-Conflict Problem

X

y

z

Suppose that y (x) and z (x)

Hence, • x and y can transmit simultaneously• x and z can transmit simultaneously

•But x, y and z, cannot all transmit simultaneously.

The conflict graph can only represent binary conflicts, but there may be multi-conflicts

Co-channel interference

• Most previous works neglect the co-channel interference– Steve Low (Caltech), “Cross-layer congestion control, routing

and scheduling design in ad hoc wireless networks”– R. Srikant (UIUC), “Joint congestion control and distributed

scheduling in multihop wireless networks with a node-exclusive interference model”

– N. Shroff (Purdue), “Joint congestion control and distributed scheduling for throughput guarantees in wireless networks”

– Xiaojun Lin (Purdue), “The impact of imperfect scheduling on cross-layer congestion control in wireless networks”

– B. Hajek and G. Sasaki, “Link scheduling in polynomial time”– A. Kashyap, S. Sengupta, R. Bhatia, and M. Kodialam, “Two-

phase routing, scheduling and power control for wireless mesh networks with variable traffic”

– …

• Our work can deal with co-channel interference

Correcting Multi-Conflict

• Adding Multi-Conflicts Constraints

}.1,0{

...

,...,2,11:

max

1

1

i

yyx

Qxx

L

xxxx

v

v

kvvv

Miforvst

vR

k

i

Where y1,y2,…,yk are the minimum number of links that cause the inactive of link x.

All of the following work uses this technique.

Numerical Experiments of Optimal Scheduling

Numerical Experiments

Network Topologies

1 GW 36 Destinations 6 GW 18 Destinations

The topology is a set of trees. No communication among the trees except interference.

Variation in Throughput as Assignments are Added

Topology (1024 Nodes, 992 links, 32 gateways)

log(kf,k)

Each iteration a better assignment is added, increasing the performance.

min k f,k

0 50 100 1500.05

0.1

0.15

IterationM

in F

low

Rate

(Mb

ps)

0 50 100 150

-2200

-1600

-800

Iteration

Uti

lity

Num of Iterations Until Algorithm 1 Converges

log(kf,k)

The number of iterations increases polynomially with the number of nodes.

min k f,k

64 128 256 512 1024 204810

1

102

103

Number of NodesM

ean

Nu

mb

er

of

Itera

tion

s

64 128 256 512 1024 204810

1

102

103

Number of Nodes

Mean

Nu

mb

er

of

Itera

tion

s

The number of Multi-Conflicts

The number of multi-conflicts is much smaller than the number of iterations.

64 128 256 512 1024 204810

-1

100

101

102

Number of Nodes

Mean

Nu

mb

er

of

Mu

lti-

Con

flic

ts

Time to perform Clique Decomposition

64 128 256 512 1024 2048

100

Number of Nodes

Mean

Cli

qu

e D

eco

mp

. T

ime (

sec)

Time to perform clique decomposition is quite small. Clique decomposition executes once.

Comparison to 802.11 with CSMA/CA

• “Small” topologies– 6x6 block regions of downtown Chicago– Lamppost mounted routers– 1, 2, 3, 4, 5, 6 wired gateways– 18, 36, 56, 72, 90 fixed wireless routers– 10 samples each (300 topologies total)

• Realistic propagation– Ray-tracing with UDelModels

• 802.11a data rate to SNR relationship

Comparison to 802.11 with CSMA/CA

Improvement by a factor of 10 when there are many gateways.

Improvement by a factor of 4 when there are few gateways.

Small topologies (66 block region)

min k f,k

20 40 60 802

4

6

8

10

12

Op

tim

al

Th

rou

gh

pu

t

80

2.1

1 T

hro

ug

hp

ut

1 GW 2 GWs 3 GWs 6 GWs1 GW1 GW 2 GWs2 GWs GWsGWs 6 GWs6 GWs

Number of Destinations

Optimal Routing

MWIS-based technique developed to infer the link costs of unused

links.

Given the initial set of paths P(Φ)

Solve Optimal Scheduling

Search for New Path P(Φ,k+) for each connection

P(Φ,k+) = argmin xP(Φ,k)x

μx*

P(Φ) = P(Φ,k+) P(Φ)

Optimal Routes

),(),( okPx

xkPx

x

),(),( okPx

xkPx

x

• Link cost x must be known for each link x.• Each link x must be used by at least one path.

Computation Complex

• Use the Lagrange multipliers, x

• Given an initial set of paths, find a new flow for each connection

Power Control and Bit Rate Selection

100 101 102 1031

1.02

1.04

1.06

1.08

1.1

1.121 Gateways

Com

pu

ted

Th

rou

gh

pu

t (r

ati

o)

Computation Time (ratio)100 101 102 103 104 105 106

1

1.1

1.2

1.3

1.46 Gateways

Computation Time (ratio)

18 Routers36 Routers54 Routers72 Routers90 Routers

100 101 102 103 104 1051

1.05

1.1

1.15

1.2

1.25

1.33 Gateways

Computation Time (ratio)

Allowing nodes to transmit with different power and use different bit-rates can increase the resulting throughput, but also increases the complexity of the optimization problem.• We developed a set of schemes that trade-off complexity and

throughput.• In general, using 2-4 bit-rates and 2 transmissions powers

achieves a good trade-off between complexity and throughput.

•Peng Wang and Stephan Bohacek. Computational Aspects of Optimal Scheduling with Power Control and Multiple Bit Rates. (under review).

Conclusions• Optimal schedules can be efficiently

computed for realistic mesh networks– Iterative approach to significantly reduce the

optimization space– The algorithm has good convergence property.– MWIS that arises from wireless mesh network can

be solved quickly. • It takes 1 second to solve MWIS problem for

network with 2000 nodes.– A general SINR protocol model is proposed to

accurately model the interference. • Multi-conflicts can be removed easily.

Papers• Peng Wang and Stephan Bohacek. Practical Computation of Optimal

Schedules and Routing in Multihop Wireless Networks. IEEE/ACM Transactions on Networking. (under review).

• Peng Wang and Stephan Bohacek. Computational Aspects of Optimal Scheduling with Power Control and Multiple Bit Rates. (under review).

• Peng Wang and Stephan Bohacek. On the practical complexity of solving the maximum weighted independent set problem for optimal scheduling in wireless networks. WICON 2008 (Hawaii, USA, 2008).

• Peng Wang and Stephan Bohacek. Communication Models for Capacity Optimization in Mesh Networks. ACM PE-WASUN 2008 (Vancouver, Canada, 2008).

• Peng Wang and Stephan Bohacek. An Overview of Tractable Computation of Optimal Scheduling and Routing in Mesh Networks. ACM SIGMETICS Performance Evaluation Review, 2007.

• Peng Wang and Stephan Bohacek. The Practical Performance of Subgradient Computational Techniques for Mesh Network Utility Optimization. NET-COOP 2007, LNCS. (Avignon, France, 2007)

• Stephan Bohacek and Peng Wang. Toward Tractable Computation of the Capacity of Multihop Wireless Networks. Proc. IEEE Infocom 07 (Anchorage, Alaska, 2007).

Papers• Wang, P. and D.L. Mills, Further Analysis of XCP Equilibrium

Performance. Proc. IEEE Globecom 06 (San Francisco, CA, 2006).

• Wang, P. and D.L. Mills, Simple Analysis of XCP Equilibrium Performance. Proc. CISS 2006 (Princeton NJ, 2006), 585-590.

• Wang, P., and D.L. Mills. Speeding Up the Convergence of Estimated Fair Share in CSFQ. Proc. Fourth IASTED International Conference on Communications, Internet and Information Technology (Cambridge MA, October 2005), 14-20.

• Wang, P. and D.L. Mills, A Probabilistic Approach for Achieving Fair Bandwidth Allocations in CSFQ. Proc. IEEE NCA 05 (Cambridge MA, 2005).

Questions

??Thank You Thank You !!!!!!

throughput optimization of urban wireless mesh networks peng wang department of electrical and...

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