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Distributed Resource Allocation in Communication Networks: From Competition to Cooperation Jianwei Huang Princeton University Seminar at Electrical Engineering Department Columbia University J. Huang (Princeton University) Distributed


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SLIDE 1

Distributed Resource Allocation in Communication Networks: From Competition to Cooperation

Jianwei Huang Princeton University Seminar at Electrical Engineering Department Columbia University

  • J. Huang (Princeton University)

Distributed Resource Allocation

  • Jan. 2007

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SLIDE 2

Acknowledgement

Academia:

◮ Princeton University: Mung Chiang, Vincent Poor, Robert Calderbank,

Ruby Lee, Dahai Xu, Yung Yi, Chee-Wei Tan, Jeffrey Dwoskin

◮ Northwestern University: Michael Honig, Randall Berry, Aggelos

Katsaggelos, Rakesh Vohra

◮ Columbia University: Xiaodong Wang, Kai Yang ◮ California Institute of Technology: Kevin Tang ◮ Boise State University: Zhu Han ◮ University of Toronto (Canada): Wei Yu ◮ K.U.Leuven (Belgium): Marc Moonen ◮ Yonsei University (Korea): Jang-Won Lee ◮ HKUST (Hong Kong): Daniel Palomar

Industry:

◮ AT&T: Russ Bellford ◮ Fraser Research Lab: Alexander Fraser ◮ Motorola: Rajeev Agrawal, Zhu Li ◮ Hamilton Institute (Ireland): Vijay Subramanian ◮ Marvell (Hong Kong): Raphael Cendrillon

  • J. Huang (Princeton University)

Distributed Resource Allocation

  • Jan. 2007

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SLIDE 3

Communication Networks Are Everywhere ...

SGSN Internal Network Inter net ISP

Fast growth and successful deployment of

◮ Internet: carrying data, voice, video, ... ◮ Broadband access networks: DSL, cable, WiMAX, ... ◮ Wireless networks: cellular, Wi-Fi, bluetooth, ...

  • J. Huang (Princeton University)

Distributed Resource Allocation

  • Jan. 2007

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SLIDE 4

... but Difficult to Design and Control

Design Objectives:

◮ Efficient utilization of the network resource ◮ Fair opportunities for network access

Research Challenges:

◮ Physically distributed: self-interest, difficult to control centrally ◮ Performance coupling: shared resource, mutual interactions

  • J. Huang (Princeton University)

Distributed Resource Allocation

  • Jan. 2007

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SLIDE 5

Example 1: Cognitive Radio Network (More Later)

U . S . D E P A R T M E N T O F C O M M E R C E N A T I O N A L T E L E C O M M U N I C A T I O N S & I N F O R M A T I O N A D M I N I S T R A T I O N MOBILE (AERONAUTICAL TELEMETERING) S) 5.68 5.73 5.90 5.95 6.2 6.525 6.685 6.765 7.0 7.1 7.3 7.35 8.1 8.195 8.815 8.965 9.040 9.4 9.5 9.9 9.995 10.003 10.005 10.1 10.15 11.175 11.275 11.4 11.6 11.65 12.05 12.10 12.23 13.2 13.26 13.36 13.41 13.57 13.6 13.8 13.87 14.0 14.25 14.35 14.990 15.005 15.010 15.10 15.6 15.8 16.36 17.41 17.48 17.55 17.9 17.97 18.03 18.068 18.168 18.78 18.9 19.02 19.68 19.80 19.990 19.995 20.005 20.010 21.0 21.45 21.85 21.924 22.0 22.855 23.0 23.2 23.35 24.89 24.99 25.005 25.01 25.07 25.21 25.33 25.55 25.67 26.1 26.175 26.48 26.95 26.96 27.23 27.41 27.54 28.0 29.7 29.8 29.89 29.91 30.0

UNITED STATES

THE RADIO SPECTRUM

NON-GOVERNMENT EXCLUSIVE GOVERNMENT/NON-GOVERNMENT SHARED GOVERNMENT EXCLUSIVE RADIO SERVICES COLOR LEGEND ACTIVITY CODE NOT ALLOCATED RADIONAVIGATION FIXED MARITIME MOBILE FIXED MARITIME MOBILE FIXED MARITIME MOBILE Radiolocation RADIONAVIGATION FIXED MARITIME MOBILE Radiolocation FIXED MARITIME MOBILE FIXED MARITIME MOBILE AERONAUTICAL RADIONAVIGATION AERONAUTICAL RADIONAVIGATION Aeronautical Mobile Maritime Radionavigation (Radio Beacons) MARITIME RADIONAVIGATION (RADIO BEACONS) Aeronautical Radionavigation (Radio Beacons) 3 9 14 19.95 20.05 30 30 59 61 70 90 110 130 160 190 200 275 285 300 3 kHz 300 kHz 300 kHz 3 MHz 3 MHz 30 MHz 30 MHz 300 MHz 3 GHz 300 GHz 300 MHz 3 GHz 30 GHz Aeronautical Radionavigation (Radio Beacons) MARITIME RADIONAVIGATION (RADIO BEACONS) Aeronautical Mobile Maritime Radionavigation (Radio Beacons) AERONAUTICAL RADIONAVIGATION (RADIO BEACONS) AERONAUTICAL RADIONAVIGATION (RADIO BEACONS) Aeronautical Mobile Aeronautical Mobile RADIONAVIGATION AERONAUTICAL RADIONAVIGATION MARITIME MOBILE Aeronautical Radionavigation MOBILE (DISTRESS AND CALLING) MARITIME MOBILE MARITIME MOBILE (SHIPS ONLY) MOBILE AERONAUTICAL RADIONAVIGATION (RADIO BEACONS) AERONAUTICAL RADIONAVIGATION (RADIO BEACONS) BROADCASTING (AM RADIO) MARITIME MOBILE (TELEPHONY) MARITIME MOBILE (TELEPHONY) MOBILE (DISTRESS AND CALLING) MARITIME MOBILE LAND MOBILE MOBILE FIXED STANDARD FREQ. AND TIME SIGNAL (2500kHz) STANDARD FREQ. AND TIME SIGNAL Space Research MARITIME MOBILE LAND MOBILE MOBILE FIXED AERONAUTICAL MOBILE (R) STANDARD FREQ. AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) AERONAUTICAL MOBILE (R) FIXED MOBILE** Radio- location FIXED MOBILE* AMATEUR FIXED FIXED FIXED FIXED FIXED MARITIME MOBILE MOBILE* MOBILE* MOBILE STANDARD FREQ. AND TIME SIGNAL (5000 KHZ) AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) STANDARD FREQ. Space Research MOBILE** AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) FIXED MOBILE* BROADCASTING MARITIME MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) FIXED Mobile AMATEUR SATELLITE AMATEUR AMATEUR FIXED Mobile MARITIME MOBILE MARITIME MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) FIXED BROADCASTING FIXED STANDARD FREQ. AND TIME SIGNAL (10,000 kHz) STANDARD FREQ. Space Research AERONAUTICAL MOBILE (R) AMATEUR FIXED Mobile* AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) FIXED FIXED BROADCASTING MARITIME MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (OR) RADIO ASTRONOMY Mobile* AMATEUR BROADCASTING AMATEUR AMATEUR SATELLITE Mobile* FIXED BROADCASTING STANDARD FREQ. AND TIME SIGNAL (15,000 kHz) STANDARD FREQ. Space Research FIXED AERONAUTICAL MOBILE (OR) MARITIME MOBILE AERONAUTICAL MOBILE (OR) AERONAUTICAL MOBILE (R) FIXED FIXED BROADCASTING STANDARD FREQ. Space Research FIXED MARITIME MOBILE Mobile FIXED AMATEUR AMATEUR SATELLITE BROADCASTING FIXED AERONAUTICAL MOBILE (R) MARITIME MOBILE FIXED FIXED FIXED Mobile* MOBILE** FIXED STANDARD FREQ. AND TIME SIGNAL (25,000 kHz) STANDARD FREQ. Space Research LAND MOBILE MARITIME MOBILE LAND MOBILE MOBILE** RADIO ASTRONOMY BROADCASTING MARITIME MOBILE LAND MOBILE FIXED MOBILE** FIXED MOBILE** MOBILE FIXED FIXED FIXED FIXED FIXED LAND MOBILE MOBILE** AMATEUR AMATEUR SATELLITE MOBILE LAND MOBILE MOBILE MOBILE FIXED FIXED MOBILE MOBILE FIXED FIXED LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE Radio Astronomy RADIO ASTRONOMY LAND MOBILE FIXED FIXED MOBILE MOBILE MOBILE LAND MOBILE FIXED LAND MOBILE FIXED FIXED MOBILE MOBILE LAND MOBILE AMATEUR BROADCASTING (TV CHANNELS 2-4) FIXED MOBILE FIXED MOBILE FIXED MOBILE FIXED MOBILE AERONAUTICAL RADIONAVIGATION BROADCASTING (TV CHANNELS 5-6) BROADCASTING (FM RADIO) AERONAUTICAL RADIONAVIGATION AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE AERONAUTICAL MOBILE AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (R) AERONAUTICAL MOBILE (R) MOBILE FIXED AMATEUR BROADCASTING (TV CHANNELS 7-13) MOBILE FIXED MOBILE FIXED MOBILE SATELLITE FIXED MOBILE SATELLITE MOBILE FIXED MOBILE SATELLITE MOBILE FIXED MOBILE AERONAUTICAL RADIONAVIGATION
  • STD. FREQ. & TIME SIGNAL SAT. (400.1 MHz)
  • MET. SAT.
(S-E) SPACE RES. (S-E) Earth Expl. Satellite (E-S) MOBILE SATELLITE (E-S) FIXED MOBILE RADIO ASTRONOMY RADIOLOCATION Amateur LAND MOBILE Meteorological Satellite (S-E) LAND MOBILE BROADCASTING (TV CHANNELS 14 - 20) BROADCASTING (TV CHANNELS 21-36) TV BROADCASTING RADIO ASTRONOMY RADIOLOCATION FIXED Amateur AERONAUTICAL RADIONAVIGATION MOBILE** FIXED AERONAUTICAL RADIONAVIGATION Radiolocation Radiolocation MARITIME RADIONAVIGATION MARITIME RADIONAVIGATION Radiolocation Radiolocation Radiolocation RADIO- LOCATION RADIO- LOCATION Amateur AERONAUTICAL RADIONAVIGATION (Ground) RADIO- LOCATION Radio- location
  • AERO. RADIO-
NAV.(Ground) FIXED SAT. (S-E) RADIO- LOCATION Radio- location FIXED FIXED SATELLITE (S-E) FIXED AERONAUTICAL RADIONAVIGATION MOBILE FIXED MOBILE RADIO ASTRONOMY Space Research (Passive) AERONAUTICAL RADIONAVIGATION RADIO- LOCATION Radio- location RADIONAVIGATION Radiolocation RADIOLOCATION Radiolocation Radiolocation Radiolocation RADIOLOCATION RADIO- LOCATION MARITIME RADIONAVIGATION MARITIME RADIONAVIGATION METEOROLOGICAL AIDS Amateur Amateur FIXED FIXED SATELLITE (E-S) MOBILE FIXED SATELLITE (E-S) FIXED SATELLITE (E-S) MOBILE FIXED FIXED FIXED FIXED MOBILE FIXED SPACE RESEARCH (E-S) FIXED Fixed MOBILE SATELLITE (S-E) FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) FIXED SATELLITE (E-S) FIXED SATELLITE (E-S) FIXED SATELLITE (E-S) FIXED SATELLITE (E-S) FIXED FIXED FIXED FIXED FIXED FIXED FIXED MET. SATELLITE (S-E) Mobile Satellite (S-E) Mobile Satellite (S-E) Mobile Satellite (E-S) (no airborne) Mobile Satellite (E-S)(no airborne) Mobile Satellite (S-E) Mobile Satellite (E-S) MOBILE SATELLITE (E-S) EARTH EXPL. SATELLITE(S-E) EARTH EXPL.
  • SAT. (S-E)
EARTH EXPL. SATELLITE (S-E) MET. SATELLITE (E-S) FIXED FIXED SPACE RESEARCH (S-E) (deep space only) SPACE RESEARCH (S-E) AERONAUTICAL RADIONAVIGATION RADIOLOCATION Radiolocation Radiolocation Radiolocation Radiolocation MARITIME RADIONAVIGATION Meteorological Aids RADIONAVIGATION RADIOLOCATION Radiolocation RADIO- LOCATION Radiolocation Radiolocation Amateur Amateur Amateur Satellite RADIOLOCATION FIXED FIXED FIXED FIXED FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) Mobile ** SPACE RESEARCH (Passive) EARTH EXPL.
  • SAT. (Passive)
RADIO ASTRONOMY SPACE RESEARCH (Passive) EARTH EXPL. SATELLITE (Passive) RADIO ASTRONOMY BROADCASTING SATELLITE AERONAUTICAL RADIONAV. Space Research (E-S) Space Research Land Mobile Satellite (E-S) Radio- location RADIO- LOCATION RADIO NAVIGATION FIXED SATELLITE (E-S) Land Mobile Satellite (E-S) Land Mobile Satellite (E-S) Fixed Mobile FIXED
  • SAT. (E-S)
Fixed Mobile FIXED Mobile FIXED MOBILE Space Research Space Research Space Research SPACE RESEARCH (Passive) RADIO ASTRONOMY EARTH EXPL. SAT. (Passive) Radiolocation RADIOLOCATION Radiolocation FX SAT (E-S) FIXED SATELLITE (E-S) FIXED FIXED FIXED MOBILE EARTH EXPL.
  • SAT. (Passive)
MOBILE Earth Expl. Satellite (Active) Standard Frequency and Time Signal Satellite (E-S) Earth Exploration Satellite (S-S) MOBILE FIXED MOBILE FIXED Earth Exploration Satellite (S-S) FIXED MOBILE FIXED SAT (E-S) FIXED SATELLITE (E-S) MOBILE SATELLITE (E-S) FIXED SATELLITE (E-S) MOBILE SATELLITE (E-S) Standard Frequency and Time Signal Satellite (S-E)
  • Stand. Frequency
and Time Signal Satellite (S-E) FIXED MOBILE RADIO ASTRONOMY SPACE RESEARCH (Passive) EARTH EXPLORATION
  • SAT. (Passive)
RADIONAVIGATION RADIONAVIGATION INTER-SATELLITE RADIONAVIGATION RADIOLOCATION Radiolocation SPACE RE. .(Passive) EARTH EXPL.
  • SAT. (Passive)
FIXED MOBILE FIXED MOBILE FIXED MOBILE Mobile Fixed FIXED SATELLITE (S-E) BROAD- CASTING BCST SAT. FIXED MOBILE F X SAT(E-S) MOBILE FIXED EARTH EXPLORATION SATELLITE FI XED SATELLITE (E-S) MOBILE SATELLITE (E-S) MOBILE FIXED SPACE RESEARCH (Passive) EARTH EXPLORATION SATELLITE (Passive) EARTH EXPLORATION
  • SAT. (Passive)
SPACE RESEARCH (Passive) INTER- SATELLITE RADIO- LOCATION SPACE RESEARCH FIXED MOBILE FIXED MOBILE SATELLITE (E-S) MOBILE SATELLITE RADIO NAVIGATION RADIO- NAVIGATION SATELLITE EARTH EXPLORATION SATELLITE FIXED SATELLITE (E-S) MOBILE FIXED FIXED SATELLITE (E-S) AMATEUR AMATEUR SATELLITE AMATEUR AMATEUR SATELLITE Amateur Satellite Amateur RADIO- LOCATION MOBILE FIXED MOBILE SATELLITE (S-E) FIXED SATELLITE (S-E) MOBILE FIXED BROAD- CASTING SATELLITE BROAD- CASTING SPACE RESEARCH (Passive) RADIO ASTRONOMY EARTH EXPLORATION SATELLITE (Passive) MOBILE FIXED MOBILE FIXED RADIO- LOCATION FIXED SATELLITE (E-S) MOBILE SATELLITE RADIO- NAVIGATION SATELLITE RADIO- NAVIGATION Radio- location EARTH EXPL. SATELLITE (Passive) SPACE RESEARCH (Passive) FIXED FIXED SATELLITE (S-E) SPACE RESEARCH (Passive) RADIO ASTRONOMY EARTH EXPLORATION SATELLITE (Passive) FIXED MOBILE MOBILE INTER- SATELLITE RADIO- LOCATION INTER- SATELLITE Radio- location MOBILE MOBILE SATELLITE RADIO- NAVIGATION RADIO- NAVIGATION SATELLITE AMATEUR AMATEUR SATELLITE Amateur Amateur Satellite RADIO- LOCATION MOBILE FIXED FIXED SATELLITE (S-E) MOBILE FIXED FIXED SATELLITE (S-E) EARTH EXPLORATION SATELLITE (Passive) SPACE RES. (Passive) SPACE RES. (Passive) RADIO ASTRONOMY FIXED SATELLITE (S-E) FIXED MOBILE FIXED MOBILE FIXED MOBILE FIXED MOBILE FIXED MOBILE FIXED SPACE RESEARCH (Passive) RADIO ASTRONOMY EARTH EXPLORATION SATELLITE (Passive) EARTH EXPLORATION
  • SAT. (Passive)
SPACE RESEARCH (Passive) INTER- SATELLITE INTER- SATELLITE INTER- SATELLITE INTER- SATELLITE MOBILE MOBILE MOBILE MOBILE SATELLITE RADIO- NAVIGATION RADIO- NAVIGATION SATELLITE FIXED SATELLITE (E-S) FIXED FIXED EARTH EXPLORATION SAT. (Passive) SPACE RES. (Passive) SPACE RESEARCH (Passive) RADIO ASTRONOMY EARTH EXPLORATION SATELLITE (Passive) MOBILE FIXED MOBILE FIXED MOBILE FIXED FIXED SATELLITE (S-E) FIXED SATELLITE(S-E) FIXED SATELLITE (S-E) EARTH EXPL.
  • SAT. (Passive)
SPACE RES. (Passive) Radio- location Radio- location RADIO- LOCATION AMATEUR AMATEUR SATELLITE Amateur Amateur Satellite EARTH EXPLORATION SATELLITE (Passive) SPACE RES. (Passive) MOBILE MOBILE SATELLITE RADIO- NAVIGATION RADIO- NAVIGATION SATELLITE MOBILE MOBILE FIXED RADIO- ASTRONOMY FIXED SATELLITE (E-S) FIXED 3.0 3.025 3.155 3.230 3.4 3.5 4.0 4.063 4.438 4.65 4.7 4.75 4.85 4.995 5.003 5.005 5.060 5.45 MARITIME MOBILE AMATEUR AMATEUR SATELLITE FIXED Mobile MARITIME MOBILE STANDARD FREQUENCY & TIME SIGNAL (20,000 KHZ) Space Research AERONAUTICAL MOBILE (OR) AMATEUR SATELLITE AMATEUR
  • MET. SAT. (S-E)
  • MOB. SAT. (S-E)
SPACE RES. (S-E) SPACE OPN. (S-E)
  • MET. SAT. (S-E)
  • Mob. Sat. (S-E)
SPACE RES. (S-E) SPACE OPN. (S-E)
  • MET. SAT. (S-E)
  • MOB. SAT. (S-E)
SPACE RES. (S-E) SPACE OPN. (S-E)
  • MET. SAT. (S-E)
  • Mob. Sat. (S-E)
SPACE RES. (S-E) SPACE OPN. (S-E) MOBILE FIXED FIXED Land Mobile FIXED MOBILE LAND MOBILE LAND MOBILE MARITIME MOBILE MARITIME MOBILE MARITIME MOBILE MARITIME MOBILE LAND MOBILE FIXED MOBILE MOBILE SATELLITE (E-S) Radiolocation Radiolocation LAND MOBILE AMATEUR MOBILE SATELLITE (E-S) RADIONAVIGATION SATELLITE
  • MET. AIDS
(Radiosonde) METEOROLOGICAL AIDS (RADIOSONDE) SPACE RESEARCH (S-S) FIXED MOBILE LAND MOBILE FIXED LAND MOBILE FIXED FIXED RADIO ASTRONOMY RADIO ASTRONOMY METEOROLOGICAL AIDS (RADIOSONDE) METEOROLOGICAL AIDS (Radiosonde) METEOROLOGICAL SATELLITE (s-E) Fixed FIXED
  • MET. SAT.
(s-E) FIXED FIXED AERONAUTICAL MOBILE SATELLITE (R) (space to Earth) AERONAUTICAL RADIONAVIGATION
  • RADIONAV. SATELLITE (Space to Earth)
AERONAUTICAL MOBILE SATELLITE (R) (space to Earth) Mobile Satellite (S- E) RADIO DET. SAT. (E-S) MOBILESAT(E-S)
  • AERO. RADIONAVIGATION
  • AERO. RADIONAV.
  • AERO. RADIONAV.
RADIO DET. SAT. (E-S) RADIO DET. SAT. (E-S) MOBILE SAT. (E-S) MOBILE SAT. (E-S) Mobile Sat. (S-E) RADIO ASTRONOMY RADIO ASTRONOMY MOBILE SAT. (E-S) FIXED MOBILE FIXED FIXED (LOS) MOBILE (LOS) SPACE RESEARCH (s-E)(s-s) SPACE OPERATION (s-E)(s-s) EARTH EXPLORATION
  • SAT. (s-E)(s-s)
Amateur MOBILE Fixed RADIOLOCATION AMATEUR RADIO ASTRON. SPACE RESEARCH EARTH EXPL SAT FIXED SAT. (S-E) FIXED MOBILE FIXED SATELLITE (S-E) FIXED MOBILE FIXED SATELLITE (E-S) FIXED SATELLITE (E-S) MOBILE FIXED SPACE RESEARCH (S-E) (Deep Space) AERONAUTICAL RADIONAVIGATION EARTH
  • EXPL. SAT.
(Passive) 300 325 335 405 415 435 495 505 510 525 535 1605 1615 1705 1800 1900 2000 2065 2107 2170 2173.5 2190.5 2194 2495 2501 2502 2505 2850 3000 RADIO- LOCATION BROADCASTING FIXED MOBILE AMATEUR RADIOLOCATION MOBILE FIXED MARITIME MOBILE MARITIME MOBILE (TELEPHONY) MARITIME MOBILE LAND MOBILE MOBILE FIXED 30.0 30.56 32.0 33.0 34.0 35.0 36.0 37.0 37.5 38.0 38.25 39.0 40.0 42.0 43.69 46.6 47.0 49.6 50.0 54.0 72.0 73.0 74.6 74.8 75.2 75.4 76.0 88.0 108.0 117.975 121.9375 123.0875 123.5875 128.8125 132.0125 136.0 137.0 137.025 137.175 137.825 138.0 144.0 146.0 148.0 149.9 150.05 150.8 152.855 154.0 156.2475 157.0375 157.1875 157.45 161.575 161.625 161.775 162.0125 173.2 173.4 174.0 216.0 220.0 222.0 225.0 235.0 300 ISM – 6.78 ± .015 MHz ISM – 13.560 ± .007 MHz ISM – 27.12 ± .163 MHz ISM – 40.68 ± .02 MHz ISM – 24.125 ± 0.125 GHz 30 GHz ISM – 245.0 ± 1GHz ISM – 122.5 ± .500 GHz ISM – 61.25 ± .250 GHz 300.0 322.0 328.6 335.4 399.9 400.05 400.15 401.0 402.0 403.0 406.0 406.1 410.0 420.0 450.0 454.0 455.0 456.0 460.0 462.5375 462.7375 467.5375 467.7375 470.0 512.0 608.0 614.0 698 746 764 776 794 806 821 824 849 851 866 869 894 896 901901 902 928 929 930 931 932 935 940 941 944 960 1215 1240 1300 1350 1390 1392 1395 2000 2020 2025 2110 2155 2160 2180 2200 2290 2300 2305 2310 2320 2345 2360 2385 2390 2400 2417 2450 2483.5 2500 2655 2690 2700 2900 3000 1400 1427 1429.5 1430 1432 1435 1525 1530 1535 1544 1545 1549.5 1558.5 1559 1610 1610.6 1613.8 1626.5 1660 1660.5 1668.4 1670 1675 1700 1710 1755 1850 MARITIME MOBILE SATELLITE (space to Earth) MOBILE SATELLITE (S-E) RADIOLOCATION RADIONAVIGATION SATELLITE (S-E) RADIOLOCATION Amateur Radiolocation AERONAUTICAL RADIONAVIGATION SPA CE RESEARCH ( Passive) EARTH EXPL SAT (Passive) RADIO ASTRONOMY MOBILE MOBILE ** FIXED-SAT (E-S) FIXED FIXED FIXED** LAND MOBILE (TLM) MOBILE SAT. (Space to Earth) MARITIME MOBILE SAT. (Space to Earth) Mobile (Aero. TLM) MOBILE SATELLITE (S-E) MOBILE SATELLITE (Space to Earth) AERONAUTICAL MOBILE SATELLITE (R) (space to Earth) 3.0 3.1 3.3 3.5 3.6 3.65 3.7 4.2 4.4 4.5 4.8 4.94 4.99 5.0 5.15 5.25 5.35 5.46 5.47 5.6 5.65 5.83 5.85 5.925 6.425 6.525 6.70 6.875 7.025 7.075 7.125 7.19 7.235 7.25 7.30 7.45 7.55 7.75 7.90 8.025 8.175 8.215 8.4 8.45 8.5 9.0 9.2 9.3 9.5 10.0 10.45 10.5 10.55 10.6 10.68 10.7 11.7 12.2 12.7 12.75 13.25 13.4 13.75 14.0 14.2 14.4 14.47 14.5 14.7145 15.1365 15.35 15.4 15.43 15.63 15.7 16.6 17.1 17.2 17.3 17.7 17.8 18.3 18.6 18.8 19.3 19.7 20.1 20.2 21.2 21.4 22.0 22.21 22.5 22.55 23.55 23.6 24.0 24.05 24.25 24.45 24.65 24.75 25.05 25.25 25.5 27.0 27.5 29.5 29.9 30.0 ISM – 2450.0 ± 50 MHz 30.0 31.0 31.3 31.8 32.0 32.3 33.0 33.4 36.0 37.0 37.6 38.0 38.6 39.5 40.0 40.5 41.0 42.5 43.5 45.5 46.9 47.0 47.2 48.2 50.2 50.4 51.4 52.6 54.25 55.78 56.9 57.0 58.2 59.0 59.3 64.0 65.0 66.0 71.0 74.0 75.5 76.0 77.0 77.5 78.0 81.0 84.0 86.0 92.0 95.0 100.0 102.0 105.0 116.0 119.98 120.02 126.0 134.0 142.0 144.0 149.0 150.0 151.0 164.0 168.0 170.0 174.5 176.5 182.0 185.0 190.0 200.0 202.0 217.0 231.0 235.0 238.0 241.0 248.0 250.0 252.0 265.0 275.0 300.0 ISM – 5.8 ± .075 GHz ISM – 915.0 ± 13 MHz INTER-SATELLITE RADIOLOCATION SATELLITE (E-S) AERONAUTICAL RADIONAV. PLEASE NOTE: THE SPACING ALLOTTED THE SERVICES IN THE SPEC- TRUM SEGMENTS SHOWN IS NOT PROPORTIONAL TO THE ACTUAL AMOUNT OF SPECTRUM OCCUPIED. AERONAUTICAL MOBILE AERONAUTICAL MOBILE SATELLITE AERONAUTICAL RADIONAVIGATION AMATEUR AMATEUR SATELLITE BROADCASTING BROADCASTING SATELLITE EARTH EXPLORATION SATELLITE FIXED FIXED SATELLITE INTER-SATELLITE LAND MOBILE LAND MOBILE SATELLITE MARITIME MOBILE MARITIME MOBILE SATELLITE MARITIME RADIONAVIGATION METEOROLOGICAL AIDS METEOROLOGICAL SATELLITE MOBILE MOBILE SATELLITE RADIO ASTRONOMY RADIODETERMINATION SATELLITE RADIOLOCATION RADIOLOCATION SATELLITE RADIONAVIGATION RADIONAVIGATION SATELLITE SPACE OPERATION SPACE RESEARCH STANDARD FREQUENCY AND TIME SIGNAL STANDARD FREQUENCY AND TIME SIGNAL SATELLITE RADIO ASTRONOMY FIXED MARITIME MOBILE FIXED MARITIME MOBILE Aeronautical Mobile STANDARD FREQ. AND TIME SIGNAL (60 kHz) FIXED Mobile*
  • STAND. FREQ. & TIME SIG.
  • MET. AIDS
(Radiosonde) Space Opn. (S-E) MOBILE.
  • SAT. (S-E)
Fixed Standard
  • Freq. and
Time Signal Satellite (E-S) FIXED STANDARD FREQ. AND TIME SIGNAL (20 kHz) Amateur MOBILE FIXED
  • SAT. (E-S)
Space Research ALLOCATION USAGE DESIGNATION SERVICE EXAMPLE DESCRIPTION Primary FIXED Capital Letters Secondary Mobile 1st Capital with lower case letters U.S. DEPARTMENT OF COMMERCE National Telecommunications and Information Administration Office of Spectrum Management October 2003 MOBILE BROADCASTING TRAVELERS INFORMATION STATIONS (G) AT 1610 kHz 59-64 GHz IS DESIGNATED FOR UNLICENSED DEVICES Fixed AERONAUTICAL RADIONAVIGATION SPACE RESEARCH (Passive) * EXCEPT AERO MOBILE (R) ** EXCEPT AERO MOBILE WAVELENGTH BAND DESIGNATIONS ACTIVITIES FREQUENCY 3 x 107m 3 x 106m 3 x 105m 30,000 m 3,000 m 300 m 30 m 3 m 30 cm 3 cm 0.3 cm 0.03 cm 3 x 105Å 3 x 104Å 3 x 103Å 3 x 102Å 3 x 10Å 3Å 3 x 10-1Å 3 x 10-2Å 3 x 10-3Å 3 x 10-4Å 3 x 10-5Å 3 x 10-6Å 3 x 10-7Å 10 Hz 100 Hz 1 kHz 10 kHz 100 kHz 1 MHz 10 MHz 100 MHz 1 GHz 10 GHz 100 GHz 1 THz 1013Hz 1014Hz 1015Hz 1016Hz 1017Hz 1018Hz 1019Hz 1020Hz 1021Hz 1022Hz 1023Hz 1024Hz 1025Hz THE RADIO SPECTRUM MAGNIFIED ABOVE 3 kHz 300 GHz VERY LOW FREQUENCY (VLF) Audible Range AM Broadcast FM Broadcast Radar Sub-Millimeter Visible Ultraviolet Gamma-ray Cosmic-ray Infra-sonics Sonics Ultra-sonics Microwaves Infrared P L S X C Radar Bands LF MF HF VHF UHF SHF EHF INFRARED VISIBLE ULTRAVIOLET X-RAY GAMMA-RAY COSMIC-RAY X-ray

ALLOCATIONS FREQUENCY

BROADCASTING FIXED MOBILE* BROADCASTING FIXED BROADCASTING FIXED Mobile FIXED BROADCASTING BROADCASTING FIXED FIXED BROADCASTING FIXED BROADCASTING FIXED BROADCASTING FIXED BROADCASTING FIXED BROADCASTING FIXED BROADCASTING FIXED FIXED FIXED FIXED FIXED FIXED LAND MOBILE FIXED AERONAUTICAL MOBILE (R) AMATEUR SATELLITE AMATEUR MOBILE SATELLITE (E-S) F I X E D F i x e d M o b i l e R a d i o - l o c a t i o n F I X E D M O B I L E LAND MOBILE MARITIME MOBILE FIXED LAND MOBILE FIXED LAND MOBILE RADIONAV-SATELLITE FIXED MOBILE FIXED LAND MOBILE
  • MET. AIDS
(Radio- sonde) SPACE OPN. (S-E) Earth Expl Sat (E-S) Met-Satellite (E-S) MET-SAT. (E-S) EARTH EXPL
  • SAT. (E-S)
Earth Expl Sat (E-S) Met-Satellite (E-S) EARTH EXPL
  • SAT. (E-S)
MET-SAT. (E-S) LAND MOBILE LAND MOBILE FIXED LAND MOBILE FIXED FIXED FIXED LAND MOBILE LAND MOBILE FIXED LAND MOBILE LAND MOBILE LAND MOBILE LAND MOBILE MOBILE FIXED MOBILE FIXED BROADCAST MOBILE FIXED MOBILE FIXED FIXED LAND MOBILE LAND MOBILE FIXED LAND MOBILE AERONAUTICAL MOBILE AERONAUTICAL MOBILE FIXED LAND MOBILE LAND MOBILE LAND MOBILE FIXED LAND MOBILE FIXED MOBILE FIXED FIXED FIXED MOBILE FIXED FIXED FIXED BROADCAST LAND MOBILE LAND MOBILE FIXED LAND MOBILE METEOROLOGICAL AIDS FX Space res. Radio Ast E-Expl Sat FIXED MOBILE** MOBILE SATELLITE (S-E) RADIODETERMINATION SAT. (S-E) Radiolocation MOBILE FIXED Amateur Radiolocation AMATEUR FIXED MOBILE B-SAT FX MOB Fixed Mobile Radiolocation RADIOLOCATION MOBILE ** Fixed (TLM) LAND MOBILE FIXED (TLM) LAND MOBILE (TLM) FIXED-SAT (S-E) FIXED (TLM) MOBILE MOBILE SAT. (Space to Earth) Mobile ** MOBILE** FIXED MOBILE MOBILE SATELLITE (E-S) SPACE OP. (E-S)(s-s) EARTH EXPL.
  • SAT. (E-S)(s-s)
SPACE RES. (E-S)(s-s) FX. MOB. MOBILE FIXED Mobile R- LOC. BCST-SATELLITE Fixed Radio- location B-SAT R- LOC. FX MOB Fixed Mobile Radiolocation FIXED MOBILE** Amateur RADIOLOCATION SPACE RES..(S-E) MOBILE FIXED MOBILE SATELLITE (S-E) MARITIME MOBILE Mobile FIXED FIXED BROADCAST MOBILE FIXED MOBILE SATELLITE (E-S) FIXED F I X E D MARITIME MOBILE FIXED FIXED MOBILE** FIXED MOBILE** FIXED SAT (S-E)
  • AERO. RADIONAV.
FIXED SATELLITE (E-S) Amateur- sat (s-e) Amateur MOBILE FIXED SAT(E-S) FIXED FIXED SATELLITE (S-E)(E-S) FIXED FIXED SAT (E-S) MOBILE Radio- location RADIO- LOCATION FIXED SAT.(E-S) Mobile** Fixed Mobile FX SAT.(E-S) L M Sat(E-S) AERO RADIONAV FIXED SAT (E-S) AERONAUTICAL RADIONAVIGATION RADIOLOCATION Space Res.(act.) RADIOLOCATION Radiolocation Radioloc. RADIOLOC. Earth Expl Sat Space Res. Radiolocation BCST SAT. FIXED FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) EARTH EXPL. SAT. FX SAT (S-E) SPACE RES. FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) FIXED SATELLITE (S-E) MOBILE SAT. (S-E) FX SAT (S-E) MOBILE SATELLITE (S-E) FX SAT (S-E) STD FREQ. & TIME MOBILE SAT (S-E) EARTH EXPL. SAT. MOBILE FIXED SPACE RES. FIXED MOBILE MOBILE** FIXED EARTH EXPL. SAT. FIXED MOBILE** RAD.AST SPACE RES. FIXED MOBILE INTER-SATELLITE FIXED RADIO ASTRONOMY SPACE RES. (Passive) AMATEUR AMATEUR SATELLITE Radio- location Amateur RADIO- LOCATION Earth Expl. Satellite (Active) FIXED INTER-SATELLITE RADIONAVIGATION RADIOLOCATION SATELLITE (E-S) INTER-SATELLITE FIXED SATELLITE (E-S) RADIONAVIGATION FIXED SATELLITE (E-S) FIXED MOBILE SATELLITE (E-S) FIXED SATELLITE (E-S) MOBILE FIXED Earth Exploration Satellite (S-S) std freq & time e-e-sat (s-s) MOBILE FIXED e-e-sat MOBILE SPACE RESEARCH (deep space) RADIONAVIGATION INTER- SAT SPACE RES. FIXED MOBILE SPACE RESEARCH (space-to-Earth) SPACE RES. FIXED
  • SAT. (S-E)
MOBILE FIXED FIXED-SATELLITE MOBILE FIXED FIXED SATELLITE MOBILE SAT. FIXED SAT MOBILE SAT. EARTH EXPL SAT (E-S) Earth Expl. Sat (s - e) SPACE
  • RES. (E-S)
FX-SAT (S-E) FIXED MOBILE BROAD- CASTING BCST SAT. RADIO ASTRONOMY FIXED MOBILE** FIXED SATELLITE (E-S) MOBILE SATELLITE (E-S) FIXED SATELLITE (E-S) MOBILE RADIONAV. SATELLITE FIXED MOBILE
  • MOB. SAT(E-S)
RADIONAV.SAT. MOBILE SAT (E-S). FIXED MOBILE F X SAT(E-S) MOBILE FIXED INTER- SAT EARTH EXPL-SAT (Passive) SPACE RES. INTER- SAT SPACE RES. EARTH-ES INTER- SAT EARTH-ES SPACE RES. MOBILE FIXED EARTH EXPLORATION
  • SAT. (Passive)
S P A C E RES. MOBILE FIXED INTER
  • SAT
FIXED MOBILE INTER- SAT RADIO- LOC. MOBILE FIXED EARTH EXPLORATION
  • SAT. (Passive)
MOBILE FIXED INTER- SATELLITE FIXED MOBILE** MOBILE** INTER- SATELLITE MOBILE INTER- SATELLITE RADIOLOC. Amateur Amateur Sat. Amateur RADIOLOC. AMATEUR SAT AMATEUR RADIOLOC. SPACE RESEARCH (Passive) EARTH EXPL SAT. (Passive) FIXED MOBILE INTER- SATELLITE SPACE RESEARCH (Passive) EARTH EXPL SAT. (Passive) Amatuer FIXED MO- BILE INTER- SAT. SPACE RES. E A R T H EXPL . SAT INTER- SATELLITE INTER-SAT. INTER-SAT. MOBILE FIXED FX-SAT (S - E) BCST - SAT. B- SAT. MOB** FX-SAT SPACE RESEARCH SPACE RES.. This chart is a graphic single-point-in-time portrayal of the Table of Frequency Allocations used by the FCC and NTIA. As such, it does not completely reflect all aspects, i.e., footnotes and recent changes made to the Table of Frequency Allocations. Therefore, for complete information, users should consult the Table to determine the current status of U.S. allocations.

Radio-domain-aware wireless network Project: Smart Markets for Smart Radio (Northwestern, Motorola) Physically distributed: devices learn and adapt to the environment Performance coupling: mutual interferences

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slide-6
SLIDE 6

Example 2: DSL Network (More Later)

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

Copper-based last-mile broadband access network Project: FAST Copper (Princeton, Stanford, Fraser Research, AT&T) Physically distributed: phone lines terminate at different equipments Performance coupling: mutual interferences

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slide-7
SLIDE 7

Example 3: Mobile Ad Hoc Network (MANET)

Infrastructureless mobile wireless network Project: Control-based MANET (large DARPA Team, incl. Princeton) Physically distributed: nodes join and leave, move around Performance coupling: multihop relay, mutual interferences

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slide-8
SLIDE 8

We Need Good Resource Allocation Algorithms

Need to design good resource allocation algorithms

◮ Distributed ◮ Low complexity ◮ Local computation ◮ Limited or no message passing

Turn competition into cooperation

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slide-9
SLIDE 9

How to Achieve The Desired Solutions?

Mathematical Approaches

◮ Optimization theory ◮ Game theory ◮ Distributed computation and control ◮ Microeconomics

Engineering Implications

◮ Realistic network deployments and tests ◮ Engineering problem structure

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slide-10
SLIDE 10

Outline

1

Introduction

2

Case I: Cognitive Radio Network

3

Case II: DSL Network

4

Summary

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slide-11
SLIDE 11

Outline

1

Introduction

2

Case I: Cognitive Radio Network

3

Case II: DSL Network

4

Summary

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slide-12
SLIDE 12

What is Cognitive Radio?

Fixed Radio: transmit parameters (i.e., frequency, modulation) determined by hardware Adaptive Radio: parameters determined by software, easy to adapt to anticipated events Cognitive Radio: sense their environment and learn how to adapt

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slide-13
SLIDE 13

Why Cognitive Radio?

Improve spectrum utilization

◮ Licensed bands: IEEE 802.22 (Television band) ◮ Unlicensed bands: IEEE 802.19

Improve reliability

◮ Emergency networks ◮ Military networks

Support interoperability Reduce costs of wireless communications

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slide-14
SLIDE 14

Spectrum Sharing for Licensed Band

Primary owner: the exclusive licensee of the spectrum Secondary users: the cognitive radio devices Primary owner gets extra revenue by allowing secondary users to transmit in an unharmful way

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slide-15
SLIDE 15

Spectrum Sharing for Licensed Band

Primary owner: the exclusive licensee of the spectrum Secondary users: the cognitive radio devices Primary owner gets extra revenue by allowing secondary users to transmit in an unharmful way Interference temperature constraint:

◮ Maximum allowed interference measured at a measurement point ◮ Equivalent to a total received power constraint

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slide-16
SLIDE 16

Network Model

3 2

T

1

User Receivers Measurement Point T T

More general model (with no new math challenges) will be discussed later

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slide-17
SLIDE 17

Mathematical Model

3 1

h3

2

h P1 P2 P h

3 2

T

1

User Receivers Measurement Point T T

Multi-user interference channel User n’s

◮ Transmission power: pn ◮ Channel gain: hn ◮ Signal-to-interference plus noise ratio (SINR):

SINRn (p) = pnhn n0 + 1

B

  • m=n pmhm
  • Interference temperature constraint:

n pnhn ≤ P

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slide-18
SLIDE 18

Utility Function

n

U (SINR )

n

SINRn

Characterize QoS as function of SINR Increasing and strictly concave: elastic data application Private user information ⇒ challenges to distributed solution

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slide-19
SLIDE 19

Network Objective I: Efficiency

Efficiency: maximize the total network utility: Efficiency Problem maximize

  • n

Un(SINRn (p)) subject to

  • n

pnhn ≤ P variables pn ≥ 0, ∀n Example: Un(SINRn) = θn log(1 + SINRn)

◮ Maximizing total weighted rate

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slide-20
SLIDE 20

Network Objective II: Fairness

Fairness: fair share of resource, independent of location Fairness Problem maximize SINR1 (p) subject to U′

n(SINRn (p)) = U′ m(SINRm (p)), ∀m = n

  • n

pnhn ≤ P variables pn ≥ 0, ∀n Example: Un(SINRn) = θn log(SINRn)

◮ Weighted max-min fair

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slide-21
SLIDE 21

Technical Challenges

Non-convexity:

◮ SINR and utility may not be concave in power

Physically distributed:

◮ Local information: utility functions, channel gains ◮ Selfish objectives

Performance coupling:

◮ Mutual interference ◮ Shared received power at measurement point

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slide-22
SLIDE 22

Technical Challenges

Non-convexity:

◮ SINR and utility may not be concave in power

Physically distributed:

◮ Local information: utility functions, channel gains ◮ Selfish objectives

Performance coupling:

◮ Mutual interference ◮ Shared received power at measurement point

Our solution: auction-based resource allocation algorithm

◮ Distributed in nature ◮ Capture interactions between users

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slide-23
SLIDE 23

Background on Auction

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slide-24
SLIDE 24

Background on Auction

Example: painting auction

◮ Highest bidder gets the good and pays the bid

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slide-25
SLIDE 25

Background on Auction

Example: painting auction

◮ Highest bidder gets the good and pays the bid

Elements of auction:

◮ Good: resource ◮ Auctioneer (manager): representing seller of the good ◮ Bidders (users): buyers of the good

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slide-26
SLIDE 26

Background on Auction

Example: painting auction

◮ Highest bidder gets the good and pays the bid

Elements of auction:

◮ Good: resource ◮ Auctioneer (manager): representing seller of the good ◮ Bidders (users): buyers of the good

Rules of auction:

◮ Bids: what the bidders submit to the auctioneer ◮ Allocation: how auctioneer allocates the good to the bidders ◮ Payments: how the bidders pay the auctioneer

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slide-27
SLIDE 27

Background on Auction

Example: painting auction

◮ Highest bidder gets the good and pays the bid

Elements of auction:

◮ Good: resource ◮ Auctioneer (manager): representing seller of the good ◮ Bidders (users): buyers of the good

Rules of auction:

◮ Bids: what the bidders submit to the auctioneer ◮ Allocation: how auctioneer allocates the good to the bidders ◮ Payments: how the bidders pay the auctioneer

Types of auction

◮ Indivisible auction ◮ Divisible auction: suitable for communication resource allocation

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slide-28
SLIDE 28

Auction-based Comm. Resource Allocation

Network Coupling Bid Efficiency

  • Rep. Paper

Wireline simple complex Yes Lazar-Semret’98 Wireline simple simple Yes Maheswaran-Basar’98 Wireline simple simple No Johari-Tsitsiklis’04

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slide-29
SLIDE 29

Auction-based Comm. Resource Allocation

Network Coupling Bid Efficiency

  • Rep. Paper

Wireline simple complex Yes Lazar-Semret’98 Wireline simple simple Yes Maheswaran-Basar’98 Wireline simple simple No Johari-Tsitsiklis’04 Wireless simple simple No Sun-Modinao-Zheng’03 Wireless simple complex No Dramitinos et al.’04

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slide-30
SLIDE 30

Auction-based Comm. Resource Allocation

Network Coupling Bid Efficiency

  • Rep. Paper

Wireline simple complex Yes Lazar-Semret’98 Wireline simple simple Yes Maheswaran-Basar’98 Wireline simple simple No Johari-Tsitsiklis’04 Wireless simple simple No Sun-Modinao-Zheng’03 Wireless simple complex No Dramitinos et al.’04 Wireless complex simple Yes Huang-Berry-Honig’04

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slide-31
SLIDE 31

Auction-based Comm. Resource Allocation

Network Coupling Bid Efficiency

  • Rep. Paper

Wireline simple complex Yes Lazar-Semret’98 Wireline simple simple Yes Maheswaran-Basar’98 Wireline simple simple No Johari-Tsitsiklis’04 Wireless simple simple No Sun-Modinao-Zheng’03 Wireless simple complex No Dramitinos et al.’04 Wireless complex simple Yes Huang-Berry-Honig’04

We design two auctions: efficient and fair allocation Proposed framework is general

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slide-32
SLIDE 32

Divisible Auction for Spectrum Sharing

Initialization: manager announces A fixed reserve bid β > 0: to make the auction result unique A price π: to determine the payment

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slide-33
SLIDE 33

Divisible Auction for Spectrum Sharing

Initialization: manager announces A fixed reserve bid β > 0: to make the auction result unique A price π: to determine the payment Rules: Bids: user n submits bn ≥ 0 to the manager

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slide-34
SLIDE 34

Divisible Auction for Spectrum Sharing

Initialization: manager announces A fixed reserve bid β > 0: to make the auction result unique A price π: to determine the payment Rules: Bids: user n submits bn ≥ 0 to the manager Allocation: the manager allows user n to generate interference at the measurement point with pnhn = bn

  • n bn + β P

◮ Weighted proportional allocation rule ◮ Positive reserve bid makes sure that the values of bids count, not just

the ratio

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slide-35
SLIDE 35

Two Different Payments

SINR auction: user n pays Cn(π) = π × SINRn

◮ User-centric payment ◮ Proportional to user’s achieved QoS (SINR) ◮ Leads to fair allocation

Power auction: user n pays Cn(π) = π × pnhn

◮ Network-centric payment ◮ Proportional to the allocated resource (power) ◮ Leads to efficient allocation

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slide-36
SLIDE 36

Best Response and Nash Equilibrium

Users participate in a non-cooperative game

◮ User’s payoff (benefit) = utility - payment ◮ Both utility and payment depend on bn and b−n (bm, ∀m = n)

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slide-37
SLIDE 37

Best Response and Nash Equilibrium

Users participate in a non-cooperative game

◮ User’s payoff (benefit) = utility - payment ◮ Both utility and payment depend on bn and b−n (bm, ∀m = n)

A user n wants to choose bid to maximize its own payoff

◮ Best response: Bn(b−n)

Bn(b−n) = arg max

bn [Un(SINRn(bn; b−n)) − Cn (π, bn; b−n)]

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slide-38
SLIDE 38

Best Response and Nash Equilibrium

Users participate in a non-cooperative game

◮ User’s payoff (benefit) = utility - payment ◮ Both utility and payment depend on bn and b−n (bm, ∀m = n)

A user n wants to choose bid to maximize its own payoff

◮ Best response: Bn(b−n)

Bn(b−n) = arg max

bn [Un(SINRn(bn; b−n)) − Cn (π, bn; b−n)]

Solution of the game: everyone is happy with the result

◮ Nash Equilibrium (N.E.): b∗ = {b∗

n, ∀n}

◮ Fixed point solution of all users’ best responses

b∗

n = B

  • b∗

−n

  • , ∀n

◮ Stable outcome of the game

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slide-39
SLIDE 39

N.E. N.E. Solutions Divisible Auction (Non−cooperative Game) SINR Auction Power Auction Solutions Efficient Allocation Fair Allocation (Optimization Problems) Network Objectives Efficiency Problem Fairness Problem

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slide-40
SLIDE 40

What Do We Want to Know?

When does an N.E. exist? Is it unique? What are the properties of the N.E.? (Fairness? Efficiency?) How to achieve the N.E. in a distributed fashion?

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slide-41
SLIDE 41

What Do We Want to Know?

When does an N.E. exist? Is it unique? What are the properties of the N.E.? (Fairness? Efficiency?) How to achieve the N.E. in a distributed fashion?

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slide-42
SLIDE 42

Sufficiently Large Price Leads to Unique N.E.

Theorem (Sufficiently Large Price Leads to Unique N.E.) In SINR Auction, there is a threshold price πth, s.t. π > πth ⇒ unique N.E. π < πth ⇒ no N.E.

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slide-43
SLIDE 43

Sufficiently Large Price Leads to Unique N.E.

Theorem (Sufficiently Large Price Leads to Unique N.E.) In SINR Auction, there is a threshold price πth, s.t. π > πth ⇒ unique N.E. π < πth ⇒ no N.E. Proof: matrix analysis.

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slide-44
SLIDE 44

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

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slide-45
SLIDE 45

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

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slide-46
SLIDE 46

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)
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slide-47
SLIDE 47

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)

4

Determine spectral radius through max-min operation ρ(K(π)) = max

x≥0,x=0 min m,xm=0

1 xm

N

  • m=1

kmn(π)xn

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slide-48
SLIDE 48

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)

4

Determine spectral radius through max-min operation ρ(K(π)) = max

x≥0,x=0 min m,xm=0

1 xm

N

  • m=1

kmn(π)xn

5

Show ρ(K(π)) is continuous and nonincreasing in price π.

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slide-49
SLIDE 49

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)

4

Determine spectral radius through max-min operation ρ(K(π)) = max

x≥0,x=0 min m,xm=0

1 xm

N

  • m=1

kmn(π)xn

5

Show ρ(K(π)) is continuous and nonincreasing in price π.

6

Find a price π > maxm U′

m

  • Phmm

n0h0m

  • , s.t. ρ(K(π)) < 1.
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slide-50
SLIDE 50

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)

4

Determine spectral radius through max-min operation ρ(K(π)) = max

x≥0,x=0 min m,xm=0

1 xm

N

  • m=1

kmn(π)xn

5

Show ρ(K(π)) is continuous and nonincreasing in price π.

6

Find a price π > maxm U′

m

  • Phmm

n0h0m

  • , s.t. ρ(K(π)) < 1.

7

Find a price π > 0, s.t. ρ(K(π)) > 1.

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SLIDE 51

Proof Outline

1

User n’s payoff is strictly quasi-concave in bn ⇒ unique best response

2

Linear best responses in matrix form B(b) = K(π)b + k0(π)β

3

Unique fixed point if spectral radius ρ(K(π)) < 1 b∗ = B(b∗) ⇒ b∗ = ∞

  • i=0

Ki

  • k0(π)β (N.E.)

4

Determine spectral radius through max-min operation ρ(K(π)) = max

x≥0,x=0 min m,xm=0

1 xm

N

  • m=1

kmn(π)xn

5

Show ρ(K(π)) is continuous and nonincreasing in price π.

6

Find a price π > maxm U′

m

  • Phmm

n0h0m

  • , s.t. ρ(K(π)) < 1.

7

Find a price π > 0, s.t. ρ(K(π)) > 1.

8

There exists πth ∈ [π, π], s.t. ρ(K(π)) < 1 iff π > πth.

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SLIDE 52

Unique (Large Price) (Positive Reserve Bid) N.E. N.E. Solutions Divisible Auction (Non−cooperative Game) SINR Auction Power Auction Solutions Efficient Allocation Fair Allocation (Optimization Problems) Network Objectives Efficiency Problem Fairness Problem

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slide-53
SLIDE 53

What Do We Want to Know?

When does a unique N.E. exist? What are the properties of the N.E.? (Fairness? Efficiency?) How to achieve the N.E. in a distributed fashion?

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slide-54
SLIDE 54

SINR Auction: N.E. Achieves Fair Allocation

Theorem (SINR auction: Fair Allocation) Under properly chosen price, the unique N.E. leads to a power allocation that is arbitrary close to the fair allocation.

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slide-55
SLIDE 55

SINR Auction: N.E. Achieves Fair Allocation

Theorem (SINR auction: Fair Allocation) Under properly chosen price, the unique N.E. leads to a power allocation that is arbitrary close to the fair allocation. Implication: Positive reserve bid β leads to resource waste This waste can be made very small by properly chosen price Argument can be made precise by defining an ǫ-system Proof:

1

Satisfy fairness conditions (equalizing marginal utility)

2

Minimum SINRn can not be further improved

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SLIDE 56

Power Auction: N.E. Achieves Efficient Allocation

Large bandwidth ⇒ the efficiency problem becomes convex

◮ Example: logarithmic utility Un(SINRn) = log(SINRn)

B > P/n0

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slide-57
SLIDE 57

Power Auction: N.E. Achieves Efficient Allocation

Large bandwidth ⇒ the efficiency problem becomes convex

◮ Example: logarithmic utility Un(SINRn) = log(SINRn)

B > P/n0

Theorem (Power Auction: Efficient Allocation) Under properly chosen price, the unique N.E. leads to a power allocation that is arbitrary close to the efficient allocation. Proof:

1

Power allocation at the N.E. satisfies KKT conditions.

2

KKT conditions are necessary and sufficient for efficient allocation.

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slide-58
SLIDE 58

(Under Proper Price) Arbitrarily Close Unique (Large Price) (Positive Reserve Bid) N.E. N.E. Solutions Divisible Auction (Non−cooperative Game) SINR Auction Power Auction Solutions Efficient Allocation Fair Allocation (Optimization Problems) Network Objectives Efficiency Problem Fairness Problem

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slide-59
SLIDE 59

What Do We Want to Know?

When does a unique N.E. exist? What are the properties of the N.E.? (Fairness? Efficiency?) How to achieve the N.E. in a distributed fashion?

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slide-60
SLIDE 60

Best Response Updates

For user n at time iteration t, update bid as b(t)

n

= B

  • b(t−1)

−n

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slide-61
SLIDE 61

Best Response Updates

For user n at time iteration t, update bid as b(t)

n

= B

  • b(t−1)

−n

  • SINR auction

b(t)

n

=

  • m=n

kmnb(t−1)

m

+ k0nβ

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slide-62
SLIDE 62

Best Response Updates

For user n at time iteration t, update bid as b(t)

n

= B

  • b(t−1)

−n

  • SINR auction

b(t)

n

=

  • m=n

kmnb(t−1)

m

+ k0nβ Theorem (BR Updates Locally Computable) Best response update of user n can be written as b(t)

n

= g(t−1)

n

b(t−1)

n

where coefficient g (t−1)

n

  • nly depends on

◮ Common information: P, n0 and π. ◮ Local information: Un, hn and SINR(t−1)

n

.

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slide-63
SLIDE 63

Convergence of Best Response Updates

SINR Auction Theorem (Global Convergence of BR Updates) Under a fixed price, best response updates globally and geometrically converge to the unique N.E.

◮ Only limited explicit message passing: ⋆ bid (user to manager) and resource allocation (manager to user) ◮ No need to know anything about other users

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slide-64
SLIDE 64

Convergence of Best Response Updates

SINR Auction Theorem (Global Convergence of BR Updates) Under a fixed price, best response updates globally and geometrically converge to the unique N.E.

◮ Only limited explicit message passing: ⋆ bid (user to manager) and resource allocation (manager to user) ◮ No need to know anything about other users

Power auction

◮ Similar arguments also apply for the power auction ◮ Only works for specific utility functions ⋆ Examples: log(1 + SINRn) and log(SINRn)

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slide-65
SLIDE 65

Distributed BR Updates (Under Proper Price) Arbitrarily Close Unique (Large Price) (Positive Reserve Bid) N.E. N.E. Solutions Divisible Auction (Non−cooperative Game) SINR Auction Power Auction Solutions Efficient Allocation Fair Allocation (Optimization Problems) Network Objectives Efficiency Problem Fairness Problem

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slide-66
SLIDE 66

Extension to General Network Model

All results related to SINR auction go through

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slide-67
SLIDE 67

Extension to General Network Model

All results related to SINR auction go through

  • R

1

R

2

R

3

T3 T M

1

T2

Network Topology

5 10 15 20 16 17 18 19 20 21 22 23 24 Iterations Achieved SINR (dB) user 1 user 2 user 3

Convergence of SINR

SINR Auction Same logarithmic utility function ⇒ same SINR allocation

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slide-68
SLIDE 68

Summary

Topic: spectrum sharing in licensed bands Key idea: simple divisible auction Performance

◮ SINR auction: fair allocation ◮ Power auction: efficient allocation ◮ Large system: two auctions are both efficient

Algorithm: best response updates: distributed, provable convergence Practice: a first step towards building a flexible cognitive-radio based spectrum sharing network Main contribution: a new modeling framework and solution methodology for distributed resource allocation in a coupled system

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slide-69
SLIDE 69

Outline

1

Introduction

2

Case I: Cognitive Radio Network

3

Case II: DSL Network

4

Summary

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slide-70
SLIDE 70

Digitial Subscriber Line (DSL) Networks

Wireline communications networks based telephone copper lines Cost-effective broadband access network More than 160 million users world-wide

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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slide-71
SLIDE 71

Digitial Subscriber Line (DSL) Networks

Wireline communications networks based telephone copper lines Cost-effective broadband access network More than 160 million users world-wide Speed is the bottleneck

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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slide-72
SLIDE 72

How DSL Works?

Copper line can support signal transmissions over a large bandwidth Voice transmission: up to 3.4 KHz DSL transmissions: up to 30 MHz

◮ Multi-carrier transmissions: Discrete Multitone Modulation

Frequency (KHz) 3.4 Voice DSL

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slide-73
SLIDE 73

Network and Channel Model

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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Mathematical model: multi-user multi-carrier interference channel Each telephone line is a user (transmitter-receiver pair) Generate mutual crosstalks over multiple frequency tones

slide-74
SLIDE 74

Network and Channel Model

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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Physical model: mixed CO/RT case Channel attenuates with distance Central Office (CO) connect customers who are reasonably close Remote Terminal (RT) connect customers who are farther away

slide-75
SLIDE 75

Network and Channel Model

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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Frequency-Dependent Channel Direct channel gain decreases with frequency Crosstalk channel gain increases with frequency

slide-76
SLIDE 76

Network and Channel Model

crosstalk TX TX RX RX CO RT (Remote Terminal) (Central Office) Customer Customer

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Frequency-Dependent Channel Direct channel gain decreases with frequency Crosstalk channel gain increases with frequency Lead to near-far problem

◮ RT generates strong crosstalk to CO line, especially in high tones ◮ CO generates little crosstalk to RT in all tones

slide-77
SLIDE 77

Network Objective: Maximize Rate Region

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Rate Region: set of all achievable rate vectors

1

R Rate Region

2

R

slide-78
SLIDE 78

Network Objective: Maximize Rate Region

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Problem A: (Find One Point On the Rate Region Boundary) maximize

{pn∈Pn}n

  • n

wnRn User n’s achievable rate Rn =

k log

  • 1 + SINRk

n

  • .

◮ No multi-user joint decoding

User n chooses a power vector pn ∈ Pn =

  • k pk

n ≤ Pmax n

, pk

n ≥ 0

  • .

Rate Region: set of all achievable rate vectors

1

R Rate Region

2

R

slide-79
SLIDE 79

Network Objective: Maximize Rate Region

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Problem A: (Find One Point On the Rate Region Boundary) maximize

{pn∈Pn}n

  • n

wnRn User n’s achievable rate Rn =

k log

  • 1 + SINRk

n

  • .

◮ No multi-user joint decoding

User n chooses a power vector pn ∈ Pn =

  • k pk

n ≤ Pmax n

, pk

n ≥ 0

  • .

Changing different weights trace the entire rate region boundary Rate Region: set of all achievable rate vectors

1

R Rate Region

2

R

slide-80
SLIDE 80

Network Objective: Maximize Rate Region

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Problem A: (Find One Point On the Rate Region Boundary) maximize

{pn∈Pn}n

  • n

wnRn User n’s achievable rate Rn =

k log

  • 1 + SINRk

n

  • .

◮ No multi-user joint decoding

User n chooses a power vector pn ∈ Pn =

  • k pk

n ≤ Pmax n

, pk

n ≥ 0

  • .

Changing different weights trace the entire rate region boundary A suboptimal algorithm leads to a reduced rate region Rate Region: set of all achievable rate vectors

R Rate Region

2

R1

slide-81
SLIDE 81

Properties of Problem A

Technical difficulties

◮ Non-convexity: total weighted rate not concave in power. ◮ Physically distributed: local channel information ◮ Performance coupling: across users (interferences) and tones (power

constraint)

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slide-82
SLIDE 82

Properties of Problem A

Technical difficulties

◮ Non-convexity: total weighted rate not concave in power. ◮ Physically distributed: local channel information ◮ Performance coupling: across users (interferences) and tones (power

constraint)

Difference from the wireless case

◮ Static channels ◮ Multi-carrier transmissions ◮ Typical network topology ◮ Unique channel frequency responses with good empirical models ◮ Cannot decode explicit message passing

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slide-83
SLIDE 83

Properties of Problem A

Technical difficulties

◮ Non-convexity: total weighted rate not concave in power. ◮ Physically distributed: local channel information ◮ Performance coupling: across users (interferences) and tones (power

constraint)

Difference from the wireless case

◮ Static channels ◮ Multi-carrier transmissions ◮ Typical network topology ◮ Unique channel frequency responses with good empirical models ◮ Cannot decode explicit message passing

Our Solution: ASB algorithm

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slide-84
SLIDE 84

Dynamic Spectrum Management (DSM)

State-of-art DSM algorithms:

◮ IW: Iterative Water-filling [Yu, Ginis, Cioffi’02]

IW

2

R1 R

Algorithm Operation Complexity Performance IW Autonomous O (KN) Suboptimal

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slide-85
SLIDE 85

Dynamic Spectrum Management (DSM)

State-of-art DSM algorithms:

◮ IW: Iterative Water-filling [Yu, Ginis, Cioffi’02] ◮ OSB: Optimal Spectrum Balancing [Cendrillon et al.’04] ◮ ISB: Iterative Spectrum Balancing [Liu, Yu’05] [Cendrillon, Moonen’05]

OSB/ISB IW

2

R1 R

Algorithm Operation Complexity Performance IW Autonomous O (KN) Suboptimal OSB Centralized O

  • KeN

Optimal ISB Centralized O

  • KN2

Near Optimal

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slide-86
SLIDE 86

Dynamic Spectrum Management (DSM)

State-of-art DSM algorithms:

◮ IW: Iterative Water-filling [Yu, Ginis, Cioffi’02] ◮ OSB: Optimal Spectrum Balancing [Cendrillon et al.’04] ◮ ISB: Iterative Spectrum Balancing [Liu, Yu’05] [Cendrillon, Moonen’05] ◮ ASB: Autonomous Spectrum Balancing [Huang et al.’06]

/ASB OSB/ISB IW R

1

R

2

Algorithm Operation Complexity Performance IW Autonomous O (KN) Suboptimal OSB Centralized O

  • KeN

Optimal ISB Centralized O

  • KN2

Near Optimal ASB Autonomous O (KN) Near Optimal

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slide-87
SLIDE 87

Key Idea: Reference Line

Reference line: static pricing for static channel

◮ A virtual line representative of the typical victim in the network ◮ Good choice: the longest CO line ◮ Parameters (power, noise, crosstalk) are publicly known

Each user will choose its transmit power to protect the reference line

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slide-88
SLIDE 88

Reference Line

CP RT RT RT CP CO CP CP

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slide-89
SLIDE 89

Reference Line

Actual Line Reference Line CO CP CO RT CP RT RT CP CP CP

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slide-90
SLIDE 90

Reference Line’s Rate

User n’s obtains the reference line parameters locally

Length & Location Reference Crosstalk: Reference Noise: Reference Power: Operator Reference Line Database pk,ref σk,ref αk,ref

n

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slide-91
SLIDE 91

Reference Line’s Rate

User n’s obtains the reference line parameters locally

Length & Location Reference Crosstalk: Reference Noise: Reference Power: Operator Reference Line Database pk,ref σk,ref αk,ref

n

The reference line rate Rref

n

=

  • k

log

  • 1 +

pk,ref αk,ref

n

pk

n + σk,ref

  • ◮ Interference only depends on user n’s transmit power pk

n

◮ Locally computable without explicit message passing

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slide-92
SLIDE 92

Payoff and Best Response

User n’s payoff Sn

  • pn; p−n
  • Rref

n (pn) + wnRn

  • pn; p−n
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slide-93
SLIDE 93

Payoff and Best Response

User n’s payoff Sn

  • pn; p−n
  • Rref

n (pn) + wnRn

  • pn; p−n
  • Best response

B

  • p−n
  • arg max

pn∈Pn Sn

  • pn; p−n
  • ◮ Requires solving a nonconvex optimization problem

◮ Duality gap is zero (under large number of tones) ◮ Satisfied in real DSL networks (ADSL: 256 tones, VDSL: 4096 tones) ◮ Can be solved using dual decomposition

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slide-94
SLIDE 94

Frequency Selective Water-filling

Under high SNR approximation of the reference line

Bk

n

  • p−n
  • =

  wn λn + αk,ref

n

/σk,ref · 1{pk,ref>0} −

  • m=n

hk

n,m/hk n,npk m − σk n

 

+

◮ Reference line is not active in high frequency tones

Special case: traditional water-filling (ignore αk,ref

n

/σk,ref)

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slide-95
SLIDE 95

Frequency Selective Water-filling

Under high SNR approximation of the reference line

Bk

n

  • p−n
  • =

  wn λn + αk,ref

n

/σk,ref · 1{pk,ref>0} −

  • m=n

hk

n,m/hk n,npk m − σk n

 

+

◮ Reference line is not active in high frequency tones

Special case: traditional water-filling (ignore αk,ref

n

/σk,ref)

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Power Traditional Water−Filling Frequency Interference & Noise

slide-96
SLIDE 96

Frequency Selective Water-filling

Under high SNR approximation of the reference line

Bk

n

  • p−n
  • =

  wn λn + αk,ref

n

/σk,ref · 1{pk,ref>0} −

  • m=n

hk

n,m/hk n,npk m − σk n

 

+

◮ Reference line is not active in high frequency tones

Special case: traditional water-filling (ignore αk,ref

n

/σk,ref)

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Power Active Reference Line Frequency−Selective Water−Filling Frequency Interference & Noise

slide-97
SLIDE 97

Convergence of ASB Algorithm

ASB Algorithm: users update their individual power allocation according to best responses either sequentially or in parallel Theorem ASB algorithm globally and geometrically converges to the unique N.E. if the crosstalk channel is small, i.e., max

n,m,k

hk

n,m

hk

n,n

< 1 N − 1. Independent of the reference line parameters. Recover the convergence of iterative water-filling as a special case. Proof: contraction mapping

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slide-98
SLIDE 98

Proof Outline

1

Key Lemma: min-max of an increasing function and an decreasing function is achieved at the intersection.

2

Construct two such functions based on the ASB algorithm.

3

Show the maximum difference between the PSD during adjacent iterations is decreasing. max

n

max

  • k
  • pk,t+1

n

− pt,t

n

+ ,

  • k
  • pk,t+1

n

− pk,t

n

  • ≤ max

n

max

  • k
  • pk,t

n

− pk,t−1

n

+ ,

  • k
  • pk,t

n

− pk,t−1

n

  • ◮ Sequential updates: bound the maximum eigenvalue of the mapping

matrix.

◮ Parallel updates: more realistic with cleaner proof.

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slide-99
SLIDE 99

ASB Performance

4 ADSL lines. Mixed CO/RT deployment. Practical channel and background noise models. User 4

CP RT CP CP CO CP

5Km 4Km 3.5Km 3Km 2Km 3Km 4Km

RT RT

User 1 User 2 User 3

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slide-100
SLIDE 100

ASB Performance

4 ADSL lines. Mixed CO/RT deployment. Practical channel and background noise models. Both users 2 and 3 acheive fixed rates 2Mbps. Examine the rate region in terms of users 1 and 4’s rates. User 4

CP RT CP CP CO CP

5Km 4Km 3.5Km 3Km 2Km 3Km 4Km

RT RT

User 1 User 2 User 3

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slide-101
SLIDE 101

Achievable Rate Regions of Different Algorithms

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 User 4’s Rate (Mbps) User 1’s Rate (Mbps)

Optimal (OSB) Best Available Today (IW) ASB

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slide-102
SLIDE 102

Summary

Topic: spectrum management in DSL multiuser interference channels Key idea: static pricing using reference line Algorithm: ASB: autonomous, low complexity, and robust Performance: close to optimal, provable convergence Practice: achieve significantly larger rate region compared with the state-of-the-art distributed algorithm

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slide-103
SLIDE 103

Summary: Key Technical Challenges

Performance Bottleneck: Non-convexity

◮ Mutual interference ◮ Integer constraints of channel utilization in OFDMA systems ◮ Discrete choices of multimedia coding and content selection

Complexity Bottleneck: Physically Distributed

◮ Local utility information ◮ Local channel/buffer information ◮ Local multimedia content information ◮ Self interests

Coupled Performance

◮ Mutual interference ◮ Shared total received power ◮ Shared total downlink transmission power ◮ Shared relay resources or source budgets ◮ Correlation between traffic arrival and departure

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slide-104
SLIDE 104

More Information http://www.princeton.edu/∼jianweih

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