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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
• Scenario of interest• Optimal Scheduling
– Reduce the optimization space– Computation complexity of MWIS– Interference model
• 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
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
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
such that: ,k |xP ,k
f ,k vV
vRv,x for each link x
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
Algorithm for Optimal Scheduling
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
Algorithm for Optimal Scheduling
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
• Scenario of interest• Optimal Scheduling
– Reduce the optimization space– Computation complexity of MWIS– Interference model
• 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
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
• Scenario of interest• Optimal Scheduling
– Reduce the optimization space– Computation complexity of MWIS– Interference model
• 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 !!!!!!