IEEE Conference on Decision and Control

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Link Level 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 Network

• Optical fibers transmit light
• WDMs Multiplex Channels
• EDFAs amplify signals, introduce ASE noise
• Optical Cross Connects (OXC) reroute channels

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Optical Network Links
EDFA
EDFA
OXC
EDFA
WDM MUX
WDM DEMUX
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Optical Network System

• Inputs are signal powers at sources
• Outputs are optical signal to noise (OSNR) values
  at receivers
• Feedback returns OSNR value to sources
• 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
channel gain at 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
?i tunable parameter at sources
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Time Delay Example 1

• ?i,j time delay from source j to OSNR i
• ?i,B time delay from OSNR i to source i

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SSSL Time Delay System
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Example 1 Block Diagram
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Closed Loop System
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Problem Statement

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

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Research Context

• Single source single link results in general
  communications (Z.Wang et. al., R. Johari et.
  al., S. Deb et. al.)
• no coupling in utility functions (Z.Wang et. al.,
  J.T. Wen et. al.,T.Alpcan et. al.,R.La et. al.)
• no study of time delay in optical networks

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Time-Dependent Stability
For the following range of ai,
Closed loop stability is ensured given ?i
satisfies
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Time-Independent Stability
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Derivation of Results

• Modify control algorithm as function of
  time-delay
• Linear System - apply frequency domain stability
  analysis (Wang and Paganini,2001)
• Apply Gershgorins Theorem to exponential matrix
• Exploit shape of Nyquist plot

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Nyquist Plot Details
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Time-Dependent Simulation
Design 10 OAs cascaded, delay10ms, OSNR
targets?23dB
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Future Research

• Tighten the sufficient conditions for stability
  of closed loop system.
• Include dynamic link pricing
• Razumikhin Theorem
• Krasovskii Theorem
• Extend OSNR model to include time-varying
  parameters

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Thank You!

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Extra Slides
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Channel Pricing Algorithm

• For pricing parameters ?i
• solve vi for OSNR target
• select all ?i 1

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Time-Independent Simulation
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Time-Dependent Simulation
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Time-Dependent Instability
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Time-Dependent ai Range

• Valid ai region for closed loop stability