Throughput Optimization of Urban Wireless Mesh Networks

1
Throughput Optimization of Urban Wireless Mesh
Networks
Peng Wang Department of Electrical and Computing
Engineering University of Delaware
2
Outline

• Scenario of interest
• Optimal Scheduling
• Reduce the optimization space
• Computation complexity of MWIS
• Interference model
• Conclusions

3
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,

fixed relay
base station
destination
destination
4
Real Urban Maps and Realistic Propagation

• The map is from UDEL mobility model.
• Only outdoor routers

5
Outline

• Scenario of interest
• Optimal Scheduling
• Reduce the optimization space
• Computation complexity of MWIS
• Interference model
• Conclusions

6
Notation
? 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
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Notation Assignments
Assignments

• v denotes an assignment. It specifies
• which links are transmitting
• their transmission power
• the bit-rates

V denotes the set of all considered assignments
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Notation Data Rates
R(v,x) denotes the data rate over link x during
assignment v
In general, R(v,x) depends on the whole
assignment.
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Problem Definition of Optimal Scheduling
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Simple Example of Wireless Network
Feasible Assignments
Assignments L1 L2 L3 L4 Schedules
V1 1 0 0 0 a1
V2 0 1 0 0 a2
V3 0 0 1 0 a3
V4 0 0 0 1 a4
V5 1 0 1 0 a5
V6 1 0 0 1 a6
V7 0 1 0 1 a7
Rate R1 R2 R3 R4
Objective Function
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Problem Definition of Optimal Scheduling
Should schedule ?v include all assignments? 2L
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The Space of Assignments
Caratheodorys 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

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Finding Better Assignments
?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.
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Algorithm for Optimal Scheduling
We can find the optimal schedule for network with
more than 2000 links.
Toward tractable computation of the capacity of
multihop wireless networks. infocom 2007.
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Algorithm for Optimal Scheduling
? and µx
Optimization Solver
max ?xR(v,x) ?x
v
New assignment v
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Idea of Proof of Convergence
Primal Problem
Dual Problem
s.t
s.t
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Algorithm for Optimal Scheduling
? and µx
Optimization Solver
max ?xR(v,x) ?x
v
New assignment v
Algebraic convergence
In the worst case, MWIS is NP-Hard.
Geometrical convergence
This work provides empirical evidence that the
MWIS problem that arises in scheduling is not
computationally difficult.
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Outline

• Scenario of interest
• Optimal Scheduling
• Reduce the optimization space
• Computation complexity of MWIS
• Interference model
• Conclusions

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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

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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 Wb,e,f7, Wd,e7

Weighted Conflict Graph
2
1
a
b
3
2
f
c
d
e
3
4
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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 Wb,e,f7, Wd,e7

Weighted Conflict Graph
2
1
a
b
3
2
f
c
d
e
3
4
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Computing a MWIS
? many constraints
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Construction of Random Wireless Networks
Large number of computational experiments on
random wireless networks
NO theoretic analysis of the complexity of MWIS
problem
Empirical Evidence

• 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)? ? ? 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

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The Time to Compute the MWIS (low degree case)
only 6 neighbors/node
? 6 Target Bit rate 24 Mbps NGW
number of nodes / 16

• 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

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A Model of the Computation Time
Time to find a MWIS T0 K x Mean degree of the
conflict graph a x nß x Mean
degree of the conflict graph
K depends on the number of nodes and propagation
model a, ß Only depend on the propagation
model Mean degree encapsulates other
parameters T0 overhead to load CPLEX solver
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Computation time versus the mean degree of the
conflict graph
Behavior is the same for other propagation models
fixed
? varies from 3 to 24 NGW?16, 32,
64 Target Bit rate 24 Mbps Urban
Propagation Model

• As ? increases, the mean degree increases.
• As the number of gateways increases, the mean
  degree slightly decreases.
• Linear fit Computation time K mean degree.

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Computation time versus the mean degree of the
conflict graph
Behavior is the same for other propagation models
fixed
? 6 NGW?16, 32, 64
Target Bit rate ? 6, 12, 18, 24, 36, 48, 54
Mbps Urban Propagation Model

• The mean degree increases with the target
  bit-rate.
• High bit-rate is more susceptible to
  interference.
• Linear fit Computation time K mean degree.

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Slope of Computation Time versus Mean Degree
Previous slides ? Computation time K ? mean
degree
 Propagation Model a ß
Urban 1.7710-8 1.88
Two-ray 1.0910-7 1.64
Two-ray with Shadowing 7.8710-8 1.75
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?? 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

Urban propagation Gateway density fixed Target
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

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Outline

• Scenario of interest
• Optimal Scheduling
• Reduce the optimization space
• Computation complexity of MWIS
• Interference model
• Conclusions

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SINR Protocol Model
If links a,b,,c are transmitting,
T(x) SINR threshold to achieve the desired data
rate Rx at link x
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Protocol Model with Multi-Conflict Problem
Suppose that y ??(x) and z ??(x)
y

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

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

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Correcting Multi-Conflict
All of the following work uses this technique.
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• Numerical Experiments of Optimal Scheduling

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Numerical Experiments
Network Topologies
6 GW 18 Destinations
1 GW 36 Destinations

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

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Variation in Throughput as Assignments are Added
Topology (1024 Nodes, 992 links, 32 gateways)
????log(?kf?,k)
min??? ?k f?,k
Each iteration a better assignment is added,
increasing the performance.
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Num of Iterations Until Algorithm 1 Converges
????log(?kf?,k)
min??? ?k f?,k

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

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The number of Multi-Conflicts

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

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Time to perform Clique Decomposition

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

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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

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Comparison to 802.11 with CSMA/CA
Small topologies (6?6 block region)
min??? ?k f?,k

• Improvement by a factor of 10 when there are many
  gateways.
• Improvement by a factor of 4 when there are few
  gateways.

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Optimal Routing

• Use the Lagrange multipliers, ?x
• Given an initial set of paths, find a new flow
  for each connection

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

MWIS-based technique developed to infer the link
costs of unused links.
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Power Control and Bit Rate Selection
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).

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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.

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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).

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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).

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Questions

• ?

Thank You !!!