PrimalDual Power Control of Optical Networks with TimeDelay

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Primal-Dual Power Control of Optical Networks
with Time-Delay

• Nem Stefanovic and Lacra Pavel
• University of Toronto

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• Pictures of EDFAs

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Where/Why Use Optical Networks

• Backbone of Internet
• More bandwidth than any other communication
  medium
• No external electromagnetic interference

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Optical Network Operation

• Signal channels carried by light
• Channels are wavelength division multiplexed
  (WDM)
• Network delivers signal from one end of network
  to other
• Light has a propagation delay

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Components of Optical Links

• Optical fibers transmit light
• WDMs Multiplex Channels
• EDFAs amplify signals, introduce ASE noise

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Optical Link
EDFA
EDFA
EDFA
EDFA
WDM MUX
WDM DEMUX
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Optical Link as a System

• Inputs are signal powers at sources
• Outputs are optical signal to noise (OSNR) values
  at receivers
• Link algorithm computes the channel prices based
  on link utilization
• Control algorithm at sources adjusts channel
  powers for OSNR optimization

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OSNR Model
ui(n) input power ith channel ?i,j jth noise
gain of ith channel output n0,i noise at Tx
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OSNR Optimization as Nash Game

• Each channel w/ action ui is a player in a game
• Each player minimizes their own coupled, cost
  function

• Ui is a coupled utility function
• u-i is the u vector without the ith entry

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Utility Function

• ai channel dependent parameter

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Control Algorithm

• ? channel price

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Link Algorithm

• ? - channel price
• P0 - total link power, capacity constraint
• ? - step size

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Interconnected System
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Time-Delay in Optical Link

• ?fforward time-delay (transmitter-receiver)
• ?bbackward time-delay (receiver-transmitter)
• ? round-trip time, ?f ?b

?f
OPTICAL LINK
Input powers, ui (Tx)
Output OSNRi (Rx)
?b
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Simplifying Assumptions

• Modify control algorithm to increase the
  time-delay in ui(t) to ui(t-?)
• Move link algorithm to the sources
• Increase time-delay of channel price ?(t) to ?(t
  -?) in the control algorithm

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Modified Block Diagram
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Closed Loop System
where,

• zi ui-ui, x?-? ,, and ? is small
• .

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Problem Statement

• Determine the conditions for stability of the
  closed loop system for all time delays ? ? 0.

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Research Context

• Paganini et. al. (2003) studies the primal-dual
  control of a single flow using boundary layer
  techniques.
• Arcak et. al. (2004) use passivity and Lyapunov
  techniques to study primal-dual and positive
  projection gradient algorithms
• A useful reference for Lyapunov time-delay
  stability analysis in congestion control is
  Niculescu et. al. (2003)

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Stability Conditions
The closed loop system is exponentially stable if
and ? is small enough.
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Steps of Proof

• Break up the closed loop system into its reduced
  and boundary layer (Matrix) form
• where the reduced system has no delay, and the
  boundary layer system is linear

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Steps of Proof (Continued)

• Analyze the boundary layer system via
  Lyapunov-Razumikhin stability theory
• or equivalently, set PI and ?00,
• which manipulates to the given time bound

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Steps of Proof (Continued)

• Find Lyapunov functions for the reduced and
  boundary layer systems
• Sum the Lyapunov functions into a composite
  function
• Apply the composite Lyapunov function to the
  closed-loop system

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Simulation 1(?i0.1)
Design 10 OAs cascaded, delay10ms, OSNR
targets?23dB
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Simulation 2 (?i0.54)
Design 10 OAs cascaded, delay10ms, OSNR
targets?23dB
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Future Research

• Apply Lyapunov-Krasovskii theory instead of
  Lyapunov-Razumikhin theory
• Extend the theory from the single link case to
  general network configurations
• Improve the OSNR model to include time-varying
  parameters

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Thank You!