scheduling for multiple antenna wireless systems: capacity ...€¦ · scheduling for multiple...

Scheduling for Multiple AntennaScheduling for Multiple AntennaWireless Systems: Capacity andWireless Systems: Capacity and

User Performance LimitsUser Performance Limits

Prof. Robert W. Heath Jr.Joint work with Manish Airy and Prof. Sanjay Shakkottai

Wireless Networking and Communications Group (WNCG)Dept. of Electrical and Computer Engineering

The University of Texas at Austin

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OutlineOutline Introduction to the WNCG @ UT Austin

Scheduling in MIMO ChannelsScheduling in MIMO Channels

User-Level Performance LimitsUser-Level Performance Limits

Limited FeedbackLimited Feedback

Conclusions

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WNCG at The University of Texas AustinWNCG at The University of Texas Austin Wireless Networking and Communications Group

14 faculty 60 graduate students Networking, wireless, devices

7 Industrial affiliates Metrowerks, Motorola, National Instruments, SBC, Texas

Instruments, Time Warner Cable, U.S. Department of Defense Symposium in October

October 20 - 22, 2004 Wed-Fri before the Info Theory Workshop in San Antonio!

www.wncg.org

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StudentsStudentsCurrent Students Manish Airy: Scheduling for multi-user MIMO systems Robert Daniels: TBD Antonio Forenza: Channel modeling & antenna design for MIMO systems Caleb Lo: Interference management in MIMO ad hoc networks Bishwarup Mondal: Adaptive feedback / data mining application in comm Muhammad Farooq Sabir: Image and video in MIMO systems Roopsha Samanta: Frame theory & channel quantization Ahmad Sheikh: TBD Taiwen Tang: MAC design, MIMO ad hoc networks Chwan-Ming Wang: Handoff in MIMO-OFDM systems

Graduated Students David J. Love: Limited feedback MIMO communication systems

Starting as an Assistant Professor at Purdue Fall 2004

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Conclusions

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Scheduling in the MIMO BCScheduling in the MIMO BC

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Prior WorkPrior Work

Core idea ofCore idea of multiuser multiuser diversity diversity Exploit independence of the channels between multiple users toExploit independence of the channels between multiple users to

obtain diversity and thus better ratesobtain diversity and thus better rates Early work on Early work on multiusermultiuser diversity (not MIMO) diversity (not MIMO)

MultiuserMultiuser diversity on the uplink [KnoHum diversity on the uplink [KnoHum’’95]95] MultiuserMultiuser diversity on the downlink [Tse diversity on the downlink [Tse’’97]97]

Characterizing delay-capacity tradeoffsCharacterizing delay-capacity tradeoffs Infinite file size and fixed users [VisTseLarInfinite file size and fixed users [VisTseLar’’02], [VisVenHua02], [VisVenHua’’03]03] Throughput optimal scheduling [Throughput optimal scheduling [ShaSriStoShaSriSto]]

On limiting scheduling modelsOn limiting scheduling models ““FastFast”” channel limit [Bor channel limit [Bor’’03],[PraVee03],[PraVee’’03],[AirShaHea03],[AirShaHea’’03]03] ““SlowSlow”” channel limit [AirShaHea channel limit [AirShaHea’’03]03]

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MultiuserMultiuser Diversity & MIMO Diversity & MIMO Fixed number of users

Single user (time sharing) [Tel’95] Multiuser broadcast [VisJinGol’02]

Capacity increase with number of users With multiplexing [HocVis’02] With time-sharing [ChuHwaKimKim’03] With linear receivers (mux / sharing) [HeaManPau’01] In cellular systems [VisVenHua’03] With reduced feedback [GesAlo’03]

Others as well Boche, El Gamal, etc.

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Motivation: BC Multiplexing AlternativesMotivation: BC Multiplexing Alternatives

How do we allocate substreams to users?

Single-user (time-sharing) spatial multiplexing Multiple data streams to single user Spatially-greedy if single best user

Multi-user transmission Every scheduling instant transmit to multiple users “Collectively” greedy multiplexing if to the “best” users

Average delay as choice of QoS metric Scheduling policy enforces the QoS constraint

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System ModelSystem Model

i is the index of the i-th user s(t) is the transmitted signal Hi(t) is the Mr x Mt channel matrix for user i

Assumed known perfectly at the MS Assumed known perfectly at the BTS

vi(t) is AWGN

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Single User MultiplexingSingle User Multiplexing Capacity for user i is given by

Where are the eigenvalues of andis obtained from waterfilling

Spatial greedy (MaxRate) scheduler achieves

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MIMO BC CapacityMIMO BC Capacity

Sum Capacity of MIMO BC, [VisJinGolSum Capacity of MIMO BC, [VisJinGol’’02]02]

Sum capacity scheduler achieves this rateSum capacity scheduler achieves this rate

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Multiuser Multiuser Sum CapacitySum Capacity

Ergodic sum capacity increases with users

MaxRate (multi-user diversity)Single user transmissionto rate maximizing channel

MaxSumRateMulti-user transmission tosum rate maximizing channels

Example2x2, Uncorrelated H

0 20 40 60 80 1000

1

2

3

4

FCFSMaxRateMaxSumRate

SNR = 0 dB

Number of users

Normalized Ergodic Capacity

Asymptotically in number of users, ergodic capacity for MaxRate and MaxSumRate scheduling scales identically.

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Number of Number of ActiveActive Users Users

How many active users account for sum capacity?

2 3 4 5 6 7 8 90

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Number of Active Users

Probability Mass Function

Distribution of Number of Active Users

------- 2x2

------- 4x4

SNR = 0 dB, Total user population K = 20 Example Uncorrelated H, 100 trials Roughly linear increase in

total active users withnumber of antennas

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Rates of Rates of ActiveActive Users Users

What is the allocated rate to each active user? Overall throughput increases Each user gets lower rate when scheduled

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Distribution of Individual Rates

Rate (Normalized to ergodic channel capacity)

Prob(Rate > x)

2x2, User 12x2, User 22x2, User 34x4, User 14x4, User 24x4, User 34x4, User 44x4, User 54x4, User 6

0 dB SNR

Example Uncorrelated H, 100 trials At 0 dB SNR, with MT

antennas, ~85% of mediansum capacity achieved byMT users

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Conclusions

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Conflicting objectives

Maximize overall throughput (or sum capacity)

Maintain acceptable user experienced delay

Multi-user sum capacity grows with number of users True for both single- and multi-user greedy scheduling Each user competes with many users for access Each user gets lower allocated rate than single-user scheduling

With an average delay constraint, can we still expect overall capacityincrease with number of users?

User Level PerformanceUser Level Performance

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File Model and AssumptionsFile Model and Assumptions Arrivals at BTS are variable length files

Assume arrivals are geometric Arbitrary file size distribution Implication is that there are a varying number of users

Other assumptions Symbol period is smaller than all events of interest Need this with certain fluid limits

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System state S(t) = {N(t), H(t), A(t), F(t)} N(t) is the number of active users H(t) = {H1(t), H2(t), …, HN(t)(t)} is the channel process A(t) = {A1, A2, …, AN(t)} are the arrival times F(t) = {F1(t), F2(t), …, FN(t)(t)} are the remaining file sizes

System evolution in discrete time

Discrete-time System ModelDiscrete-time System Model

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νν Allocated rate, RAllocated rate, Rii, where i=1, 2, , where i=1, 2, ……, N(t), N(t)

νν State evolutionState evolutionνν FFii(t+(t+ΔΔt) = t) = FFii(t) (t) –– ΔΔt Rt Rii(t)(t)νν If If FFii(t+(t+ΔΔt) = 0 then N(t+t) = 0 then N(t+ΔΔt) = Nt) = N(t)(t) –– 1 1

Allocated RateAllocated Rate

Multi-userMulti-user

Single-userSingle-user

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νν Scheduling as a user selection processScheduling as a user selection process((drop the time index for notational simplicitydrop the time index for notational simplicity))νν FCFSFCFS

νν SRPTSRPT

νν MaxRateMaxRate

νν MaxSumRateMaxSumRate

Scheduling AlgorithmsScheduling Algorithms

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Over a large enough time interval system is stationary Follows from stationarity of the channel and arrival process

“Fast” Channel evolution (upper bound) Slot duration → 0 Channel evolution independent every time-slot

channel bandwidth → ∞ / coherence time → 0 “Slow” Channel evolution (lower bound) Slot duration → 0 Channel state fixed for the duration of each job channel coherence time → ∞

Limiting ModelsLimiting Models

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Limiting Analysis - SummaryLimiting Analysis - Summary

SimulationProcessor sharingMaxSumRate

Preemptive Resume(M/G/1)Processor sharingMaxRate

Exact (SRPT delaycalculations)ApproximateSRPT

PK formula (M/G/1)PK formula (M/G/1)FCFS

“Slow” Channel Limit“Fast” Channel Limit

Limiting anls. -> different limiting queueing models Can calculate user data rate & average job delay

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Some Details (fast limit)Some Details (fast limit)

SRPT and FCFS calculations are straightforward For MaxRate and MaxSumRate use continuous-

time Markov chain Arrivals are Departures are and are state dependent

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Some Details (continued)Some Details (continued) Average job delay per user

Steady state distribution of users

Expected file length

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Average ThroughputAverage Throughput Use uniform convergence of DT Markov chain to

a CT Markov chain [PraVee]

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Numerical ResultsNumerical Results

Assumptions Downlink 0 dB or 10dB SNR, 2 x 2 MIMO Bandwidth 1.25 MHz Perfect channel knowledge at BTS Web downloads with mean 12 Kbytes

Limiting Scenarios “Fast” channel evolution “Slow” channel evolution

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““FastFast”” Channel (0dB) Channel (0dB)

0 1 2 3 4 5 6 70

0.5

1

1.5

FCFSSRPT-SimulatedSRPTMaxRateMaxSumRateAverage Delay threshold = 1 sec

"Fast" Channel Variation, SNR = 0 dB

Throughput (Mbps)

Average Delay (seconds)

At 1 sec average file delay:φ

SRPT = 1.2 φFCFSφ

SRPT (simulated) = 1.5 φFCFSφ

MaxRate = 2.7 φFCFSφ

MaxSumRate = 4.1 φFCFS

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““FastFast”” Channel (10dB) Channel (10dB)

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““SlowSlow”” Channel (0dB) Channel (0dB)"Slow" Channel Variation, SNR = 0 dB

0

0.5

1

1.5

0 0.5 1 1.5 2

Throughput (Mbps)

Ave

rage

Del

ay (

seco

nds)

FCFSSRPTMaxRateMaxSumRate

At 1 sec average file delay:φMaxSumRate = 1.4 φFCFS

Average Delay = 1 sec

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““SlowSlow”” Channel (10dB) Channel (10dB)

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An Asymptotic ResultAn Asymptotic Result

For the For the multiuser multiuser collectively greedy schedulingcollectively greedy schedulingpolicy, apolicy, asymptotically,symptotically,

IntuitionIntuition Channel Channel ““hardeninghardening””, [HocMar, [HocMar’’03]03]

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DiscussionDiscussion “Fast channel” limit (maximal opportunistic gains)

Both greedy schedulers yield lower delay than the SRPT andFCFS scheduler at all points

“Slow channel” limit (minimal opportunistic gains) At low SNR

MaxRate delay slightly better than MaxSumRate scheduler SRPT delay slightly better than MaxRate Better to serve and remove users than to wait for higher rate

At high SNR MaxSumRate similar as fast channel regime

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Conclusions

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Problem StatementProblem Statement

Iterative AlgorithmIterative Algorithm


DiscussionDiscussion

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Problem StatementProblem Statement

How to achieve an arbitrary rate vector within theHow to achieve an arbitrary rate vector within thecapacity region of the MIMO BC?capacity region of the MIMO BC? Without full channel knowledge at the BTS?Without full channel knowledge at the BTS?

Capacity RegionCapacity Region

Sum Capacity PointSum Capacity PointCovariances Covariances for for this rate vector?this rate vector?

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Solutions, Prior WorkSolutions, Prior Work

Trivial SolutionTrivial Solution Appropriate time-sharing of scheduling policies thatAppropriate time-sharing of scheduling policies that

allocate rates achieved by using transmit allocate rates achieved by using transmit covariancescovariancescorresponding to the boundary of the capacity regioncorresponding to the boundary of the capacity region((requires full channel knowledgerequires full channel knowledge))

Indirect Solution, [VisVenHuaIndirect Solution, [VisVenHua’’03]03] Solve the uplink rate-constrained problemSolve the uplink rate-constrained problem

((requires full channel knowledgerequires full channel knowledge)) Use duality transformations to get the downlinkUse duality transformations to get the downlink

covariances covariances corresponding to the given rate constraintcorresponding to the given rate constraint

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Proposed SolutionProposed Solution

Main IdeaMain Idea Start with arbitrary downlink transmit Start with arbitrary downlink transmit covariancescovariances Using Using eigenspace eigenspace rotations, iteratively update transmitrotations, iteratively update transmit

covariances covariances until the rate constraint is satisfieduntil the rate constraint is satisfied Algorithm has a limited feedback interpretationAlgorithm has a limited feedback interpretation

(similar to fast power control for CDMA)(similar to fast power control for CDMA)

Other Other eigenspace eigenspace algorithms are available,algorithms are available,[YeBlum[YeBlum’’03], [PopPopRos03], [PopPopRos’’02]02]

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Basic RotationsBasic Rotations

Consider the complex space, Consider the complex space, CCMMTT

There are There are MMTTCC22 orthogonal 2-dimensional subspaces orthogonal 2-dimensional subspaces Example, in Example, in CC33 there are 3 orthogonal 2-D subspaces there are 3 orthogonal 2-D subspaces For any For any vv11, , vv22 ∈∈ CC33, there exists , there exists QQ such that such that vv11==QQ vv22

QQ may be expressed in terms of may be expressed in terms ofrotations confined to the rotations confined to the MMTTCC22

2-D subspaces of 2-D subspaces of CC33

A A basicbasic rotation rotation QQ, is such that, is such thatd(d(vv, , QQssvv) ) ≤≤ γγ, s = 1, 2, , s = 1, 2, ……, , MMTTCC22

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Iterative Iterative Multiuser Multiuser AlgorithmAlgorithm

Assume arbitrary user encoding order (FCFS)Assume arbitrary user encoding order (FCFS) Because of Because of ““dirty paperdirty paper”” encoding, user k experiences encoding, user k experiences

interference from users with index j<kinterference from users with index j<k Estimate interference covariance for user k, Estimate interference covariance for user k, σσ22

int,kint,k RRkk

Estimate effective channel for user k, Estimate effective channel for user k, __kk=(=(II + + RRkk))-1/2-1/2HHkk

Waterfilling Waterfilling over over __k k yields optimum transmit eigenvectors andyields optimum transmit eigenvectors andpower allocation per eigenvectorpower allocation per eigenvector

Rate constraint imposed by choosing appropriate Rate constraint imposed by choosing appropriate waterfilledwaterfilled““levellevel””

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Limited Feedback InterpretationLimited Feedback Interpretation

The transmitter broadcasts a The transmitter broadcasts a ““dirty paperdirty paper””encoded encoded ““interference estimationinterference estimation”” waveform waveform(similar to channel training periods)(similar to channel training periods)

User k computes effective whitened channelUser k computes effective whitened channel Feedback per user is Feedback per user is MMTTCC2 2 + M+ MT T bitsbits

1 bit required to indicate choice of 1 bit required to indicate choice of QQss,, s=1, 2, s=1, 2, ……, , MMTTCC22

1 bit required to indicate power per eigenvector1 bit required to indicate power per eigenvector

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Illustrating convergence, 2-user BC Illustrating convergence, 2-user BC

0 0.5 1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate Allocated, Power

Prob(Rate > x) or Prob(Power > x)

2x2, PMAX = 10, Rate constraint = [1, 1]

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Iteration Count

Rate or Power

2x2, PMAX = 10, Rate Constraint = [1, 1]

single channel realization single channel realization 100 channel realizations 100 channel realizations

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DiscussionDiscussion

Presented an iterative algorithm that achieves aPresented an iterative algorithm that achieves aspecified rate vector in the capacity region of aspecified rate vector in the capacity region of aMIMO Broadcast ChannelMIMO Broadcast Channel Convergence demonstrated (see CISS Convergence demonstrated (see CISS ‘‘04 paper)04 paper)

Proposed algorithm has a limited feedbackProposed algorithm has a limited feedbackinterpretationinterpretation Incremental transmit covariance updates obtainedIncremental transmit covariance updates obtained

from each receiver using from each receiver using MMTTCC2 2 + M+ MT T bits per userbits per user

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Conclusions

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ConclusionsConclusions Proposed a model for scheduling in MIMO BCs

Evaluated the performance of various schedulers Max sum rate scheduling yields good delay perf

Note this does not include file length or delay For low SNR values there is a crossing point

Max rate or shortest remaining file size may be better

Proposed iterative feedback algorithm forachieving points in BC rate region Algorithm has a limited feedback interpretation

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Ongoing WorkOngoing Work Evaluation of other scheduling disciplines Effect of low complexity receivers on performance

Extend our work in [HeaAirPau’01] to include delay Evaluate the performance of other receivers

Transmit beamforming versus DPC Iterative feedback with other quantized feedback

scheduling for multiple antenna wireless systems: capacity ...€¦ · scheduling for multiple...

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