Staffer Day Template

1
Myopic and Optimal Distributed Routing
Wei Chen and Sean P. Meyn ECE CSL University
of Illinois
2
MaxWeight Issues Raised before and after 2007

• Issues addressed by 12/07
• Why does MW work? Answer led to h-MW policies
  introduced in CTCN Performance evaluation
  and approximate optimality
• Improved robustness
• Specialization to routing

• Routing requires information. In the MaxWeight
  policy, this information is obtained through
  local queue length values. This can lead to
  irrational behavior when information is scarce

• Issues addressed in 2008
• Understanding dynamic multi-commodity flow
  problem, providing workload relaxation
• Distributed implementation in wireless setting
  based on message passing
• Simulation and analysis

Example (and Leith, 2007 Subramanian,
submitted). MaxWeight or Backpressure routing
will send packets upstream!
Better performance through geometric insight and
better information sharing
3
Context of the reported work
Work shows how important global information can
be used if available. Generally, amount of
global information required for approximate
optimization is low
4
Optimizing MaxWeight for Routing

• What is the state of the art and what are its
  limitations?
• MW routing inflexible with respect to
  performance improvement requires global
  information (wireless) very little flexibility
• New h-MW technique requires information, but is
  highly flexible, and approximately optimal in
  certain cases

• MAIN RESULT
• Decentralized implementation, use of consensus
  algorithms
• Simulation study for wireless model
• Multiple cuts/bottlenecks addressed

• Performance impact of consensus algorithms, and
  other learning schemes used in this project
• Simulation study in realistic wireless setting
• Full performance analysis of multiple
  bottlenecks
• Integration with Network Coding projects

• KEY NEW INSIGHTS
• Goal develop insights of 07
• Distributed implementation via message-passing
• New learning technique based on Ojas algorithm
• Learning cuts/workload through stochastic
  approximation techniques
• Construction of h for models with multiple
  bottlenecks based on DP insights for w.
  relaxation

HOW IT WORKS Step 1 Estimation of workload
generalization of network cuts Step 2
Estimation of congestion Step 3 Choice of h0 -
piecewise quadratic Special case Single
dominant destination gives h0 quadratic function
of workload, cost, and effective cost w.r.t.
workload relaxation

• Un-consummated union challenge Integrate
  coding and resource allocation
• Generally, solutions to complex decision
  problems should offer insight

Algorithms for dynamic routing Visualization and
Optimization
5
Simulations for Multiple Traffic Streams

• Network approximately 100 nodes. Single
  destination, multiple sources
• MaxWeight compared to policy based on logarithmic
  perturbation of
• Simulation for high load 50 improvement over
  greedy, 25 over MW

Greedy
Approximation of DP solution
Source of performance loss in MW Cycling back
and forth across bottleneck network cut leads to
higher workload values
Performance improves for functions h that more
closely approximate DP solution
6
Summaries and challenges
CHALLENGES Implementation requires learning on
several levels from global architecture, to
local policy parameters. CONCLUSIONS In
wireless model with multiple bottlenecks we again
see 25 delay improvement in simulation.
Logarithmic perturbation gives universally
stabilizing policies, with no evident performance
loss in experiments. Learning insights 1.
Message passing can lead to performance loss -
analysis underway. 2. Stochastic approximation
Ojas algorithm completed. Application to
network decomposition. 3. Local capacity learning
investigated. SCIENTIFIC FOUNDATIONS Stochastic
Lyapunov theory workload relaxation to
approximate DP solution
PERFORMANCE Without attention to bottlenecks, a
decentralized routing algorithm will create
inefficiency through cycling. Stability has been
established for logarithmic perturbation -
results from simulation studies give optimism.
Analysis is required. LEARNING Distributed
learning of workload in a dynamic
setting Distributed learning of control
parameters Learning of channel characteristics
Research bottlenecks Learning dynamic
bottleneck location and workload
7
References

• S. P. Meyn. Stability and asymptotic optimality
  of generalized MaxWeight policies. To appear,
  SIAM J. Control Opt. (Preliminary version to
  appear at the 46th IEEE Conference on Decision
  and Control, December 2007).
• W. Chen and S. P. Meyn Optimizing MaxWeight For
  Routing. In preparation.
• S. P. Meyn. Control Techniques for Complex
  Networks. Cambridge University Press, 2007.
  Chinese edition to appear.
• S. P. Meyn and R. L. Tweedie, Markov Chains and
  Stochastic Stability. CUP Mathematical Library
  edition, to appear 2008.
• C. Lin, V. V. Veeravalli, and S. P. Meyn.
  Distributed beamforming with feedback
  Convergence analysis. Submitted to the IEEE
  Transactions on Information Theory.
  http//arxiv.org/abs/0806.3023, 2008 (adaptation
  at nodal level)
• V. Borkar and S. Meyn. Ojas algorithm for graph
  clustering and Markov spectral decomposition. In
  Proceedings of VALUETOOLS, Athens, Greece, 2008
  (adaptation at network level)
• References in on-going research CTCN and
• Iterative Scheduling Algorithms, M. Bayati, B.
  Prabhakar, D. Shah and M. Sharma, Proceedings
  of IEEE Infocom 2007
• Distributed Subgradient Methods for Multi-agent
  Optimization Angelia Nedic and Asuman Ozdaglar.
  To appear in IEEE Transactions on Automatic
  Control.
• Polynomial Complexity Algorithms for Full
  Utilization of Multi-hop Wireless Networks Atilla
  Eryilmaz, Asuman Ozdaglar and Eytan Modiano.
  INFOCOM 2007.

8
Optimizing MaxWeight From 12/07 Meeting
What is the state of the art and what are its
limitations? MW routing inflexible with respect
to performance improvement MW corresponds to
h-myopic, with h quadratic. Key geometric
property of quadratic identified by Meyn prior to
July meeting.
MAIN RESULT h-myopic policy is universally
stabilizing Application to policy synthesis for
approximately optimal performance (delay or
backlog) in heavy traffic, with log regret

• Decentralized implementation, use of consensus
  algorithms
• Wireless models Apply D. Shahs insights on
  maxproduct convergence
• Full analysis of multiple bottlenecks
• Integration with Network Coding projects Can
  we code around network hot-spots?

Numerical study underway
Investigate performance and feasibility 100
nodes, multiple arrivals. Only wireline models
investigated to-date. Excellent performace as
predicted by theory

• KEY NEW INSIGHTS
• New perturbation technique
• Application to routing refinements for
  decentralization
• Heavy traffic optimality
• Taylor series approximation gives interpretation
  as adaptive MaxWeight - Diagonal matrix adapts to
  varying congestion

Decentralized implementation appears feasible.
HOW IT WORKS Step 1 Estimation of network
cuts Step 2 Estimation of congestion on
either side Step 3 Choice of h0 - piecewise
quadratic Special case Single dominant
destination gives h0 quadratic function of
workload, cost, and effective cost w.r.t.
workload relaxation

• Un-consummated union challenge Integrate
  coding and resource allocation
• Generally, solutions to complex decision
  problems should offer insight

Algorithms for dynamic routing Visualization and
Optimization