Sheduling and Multiuser Diversity

InSetion2.5,wepresentedshemesthatdealwiththeoptimizationoftheinputovariane

matriesorthepreodingdesign. Inthissetion,adierentapproahisfollowedandwetry

toidentifytheoptimalseletionofuserstobeserved. FollowingtheseminalworkofKnopp

andHumblet[1℄, multiuserdiversity reeivedan inreaseattention in theeld ofresoure

alloation for wireless networks, shattering the traditional viewof fading asdetrimental.

Inthis work,the authorsprovided novelinsights to thequestionof `whih user should be

servedinordertomaximizethe sumrate'andgaveriseto anovelsetoftehniques,oined

as opportunisti ommuniation. Simply speaking, opportunism reommends sheduling

thebest user(i.e. theuser withthe mostfavorablehannelonditions) in eahoherene

intervalinordertomaximizethesystemthroughput.

ConsideraMISO

K

^-user^broadast^hannel, ^for^whih ^the^sum ^rateâpaityîs ûpper

bounded by

Clearly,thesumrateismaximizedwhenonlythestrongestuserisassignednon-zeropower

P k = P

^,^i.e.

C ^BC = E

Traditionally, hannel fading was viewed as a soure of unreliability that has to be

mitigated. Animportantmeanstoopewithfadingisdiversity,whihanbeobtainedover

time (interleaving of oded bits), frequeny (ombining of multipaths in spread-spetrum

orfrequeny-hoppingsystems)andspae(multipleantennas). Thebasiideaistoimprove

performanebyreatingseveralindependentsignalpathsbetweenthetransmitterandthe

reeiver. Theseminal work of [1℄gavetheidea that in theontextof multiuser diversity,

fadingan be onsideredasasoure ofrandomizationthat anbeexploited. This isdone

bydynamially shedulingtransmissions (resoures)among the usersasa funtion of the

theirpeaks. Inonephrase,underopportunistitransmission,wetransmitwhenandwhere

thehannel isgood.

2.7.1 Asymptoti Sum-rate Analysis with Opportunisti

Shedul-ing

Multiuser diversity is a form of diversity inherent in wireless networks, provided by the

independent time-varyinghannels arossthedierentusers. Themultiuser diversitygain

omes from thefatthat theeetivehannelgain,denoted as

g k

^, ^is^improved^from

g k

^to

max 1 ≤ k ≤ K g k

^. ^Theâmountôf^multiuser^diversity^gain^depends ^ruiallyôn^the^tailôf ^the

distribution of

g k

^, împlying ^that ^the ^heavier ^the^tail, ^the ^more^likely^there îsâ ûser^with

a verystronghannel, andthe largerthemultiuser diversitygain. Therefore, thehannel

statistis has an impat on system throughput. In the following, we derive the sum-rate

growthfordierenthannelgaindistribution.

Fading

Weonsider thatthehannelsofallusersarei.i.d. Rayleighfading,thus

g k

^is^hi-squared

distributed with

2M

^degrees ^of ^freedom, ^i.e.

g k ∼ χ ² _(2M)

^if

g k = k h _k k ²

^or

g k ∼ χ ² ₍₂₎

^if

g k = | h _k | ²

^. ^The ^limiting distribution(l.d.) of ahi-squarerandomvariable is of Gumbel typeanditanbeshownthatthemaximumvalueof

K

^i.i.d.

g k ∼ χ ² _(2M ₎

^random^variables

satises[9℄

Pr { log K + (M − 2) log log K + O(log log log K)

≤ max

1 ≤ k ≤ K g k ≤ log K + M log log K + O(log log log K) }

≥ 1 − O 1

log K

(2.48)

Therefore,forlarge

K

max

1 ≤ k ≤ K g k

^behaves^as

log K

^with^highprobability,thus

R ≈ log log K+

log P + o(1)

^. ^The^larger^the^number^of^users,^the^stronger^tends^to^be^the^strongest^hannel

andthelargerthemultiuserdiversitygain.

Itan beeasilyshownthat thelimitingdistribution of Rieanand Nakagami fadingis

ofGumbeltype. However,themultiuserdiversitygainissigniantlysmallerintheRiean

ase omparedto theRayleighase. Exponentialandgammadistributions also belong to

themaximumdomainofattrationofaGumbeldistribution.

Log-normal Shadowing

Weonsidernowthattheeetivehannelgain

g k

^is^dominated^by^log-normal^shadowing.

It an be shown that the maximum value of

K

^i.i.d. ^log-normal distributed r.v. with logarithmimean

µ s

^and^variane

σ ² _s

^,^satises^[48℄

Pr { b K − a K log log K ≤ max

1 ≤ k ≤ K g k ≤ b K + a K log log K } ≥ 1 − O 1

log K

where

b K =

^exp

{ √

2 log Kσ s + µ s }

^and

a K = b K σ s / √ 2 log K

Hene,thethroughputsaleslike

R ≈ √

2 log Kσ s + µ s

Pathloss

Consider now a more realisti senario, in whih the users are loated randomly over a

ellgivenby a disk of radius

R

^around ^the ^serving ^BS. ^The ^hannel ^gain^onsists ⁱⁿ ^the

produtbetweenavariable representingthepath lossand avariable representingthefast

fadingoeient,i.e.

g k = L k γ k

^,^where

L k

^is^the^path^loss^between^user

k

^and^the^BS^(f.

Set.2.1),and

γ k

^is^theorrespondingnormalizedomplexfadingoeient.

We onsider auniform distributionof the population in eah ell. Thus the distane

betweenuser

k

^and^the^BS,

d k

^,^is^a^r.v. ^withnon-uniform distribution

f D (d)

^given^by

f D (d) = 2d/R ² , d ∈ [0, R]

^(2.49)

Further,therandomproess

d k

ân^beônsideredî.i.d. ârossûsersândêlls,îfûsersⁱⁿêah

ellaredroppedrandomlyineahdisk. Theonsideredoverageregionanbeassimilated

withtheinside areaofeah disk, inadisk-pakingregionofthe2Dplane. Usersdropped

outside the disks andropped from the analysis, as these will not aet the saling law.

Assuming

R = 1

^fornormalization,thedistributionof

L k = βd ⁻ _k ^ǫ

^is^given^by

f L (x) =

( ₂

ǫ ( ^x _β ) ⁻ ² ^ǫ ¹ _x

^with

x ∈ [β, ∞ ) 0

^with

x / ∈ [β, ∞ )

(2.50)

The distributionof

L k

^is ^remarkable ⁱⁿ ^that ^it ^diers ^strongly ^from ^fast ^fading

distribu-tions,due toitsheavy tailbehavior. Formally,

L k

^follows^aPareto-typedistribution andis aregularlyvaryingrandom variablewith exponent

− 2/ǫ

^, ^i.e.

lim

t →∞

1 − F α (x)

1 − F α (tx) → t ^2/ǫ

^. ^An

interestingaspet of regularly varying r.v. is that they are stable with respet to

multi-pliationwithotherindependent r.v. withnite momentsaspointedoutbythefollowing

theorem:

Theorem2.1[49℄: Let

X

^and

Y

^be^twoindependentr.v. suhthat

X

^is^regularly^varying

with exponent

− η

^. ^Assuming

Y

^has ^nite ^moment

E { Y ^η }

^, ^then ^the ^tail ^behavior ^of ^the

produt

Z = XY

^is^governed^by:

1 − F Z (z) → E { Y ^η } (1 − F X (z))

^when

z → ∞

^(2.51)

Theideabehindthistheoremisthatwhenmultiplyingaregularlyvaryingr.v. withanother

onewith nite moment, oneobtainsa heavy tailedr.v. whose tail behavioris similar to

therst one,upto a saling. Sine

γ k

^has^nite ^moments, ^the^tail^behavior^of

g k

^an^be

haraterizedby:

1 − F g (x) → E { γ _k ² ^ǫ } β

x ² ǫ

when

x → ∞

^(2.52)

Therefore,

g k

îsâlso^regularly^varying^withêxponent

− ² ǫ

^,^whih^implies^that^[50℄

K lim →∞

{ max

1 ≤ k ≤ K g k ≤ β E { γ _k ^2/ǫ } ^ǫ/2 K ² ^ǫ x } = e ⁻ ^x ^−2/ǫ ∀ x > 0,

^(2.53)

Usingtheaboveresult,wean showthatthethroughputsalesforasymptotiallylarge

K

R ≈ ǫ

2 log K

^(2.54)

Observe that a muh greater throughput growth than in the ase of fading is obtained.

loss aross the user loations in the ell. As the distribution of pathloss belongs to the

maximum domain of attration of Fréhet type, alogarithmi apaitygrowth with

K

^is

ahieved. However,theshedulingdeisionsaretakeninaquiteunfairfashion admittedly,

sinethesheduler tendstoseletusersloserto theaess pointasmoreusersareadded

to thenetwork.

Dans le document Multiuser multi-antenna systems with limited feedback (Page 51-54)

K

P k = P

C BC = E

g k

g k

max 1 ≤ k ≤ K g k

g k

g k

2M

g k ∼ χ 2 (2M)

g k = k h k k 2

g k ∼ χ 2 (2)

g k = | h k | 2

K

g k ∼ χ 2 (2M )

Pr { log K + (M − 2) log log K + O(log log log K)

≤ max

1 ≤ k ≤ K g k ≤ log K + M log log K + O(log log log K) }

≥ 1 − O 1

log K

K

max

1 ≤ k ≤ K g k

log K

R ≈ log log K+

log P + o(1)

g k

K

µ s

σ 2 s

Pr { b K − a K log log K ≤ max

1 ≤ k ≤ K g k ≤ b K + a K log log K } ≥ 1 − O 1

log K

b K =

{ √

2 log Kσ s + µ s }

a K = b K σ s / √ 2 log K

R ≈ √

2 log Kσ s + µ s

R

g k = L k γ k

L k

k

γ k

k

d k

f D (d)

f D (d) = 2d/R 2 , d ∈ [0, R]

d k

R = 1

L k = βd − k ǫ

f L (x) =

( 2

ǫ ( x β ) − 2 ǫ 1 x

x ∈ [β, ∞ ) 0

x / ∈ [β, ∞ )

L k

L k

− 2/ǫ

lim

t →∞

1 − F α (x)

1 − F α (tx) → t 2/ǫ

X

Y

X

− η

Y

E { Y η }

Z = XY

1 − F Z (z) → E { Y η } (1 − F X (z))

z → ∞

γ k

g k

1 − F g (x) → E { γ k 2 ǫ } β

x 2 ǫ

x → ∞

g k

− 2 ǫ

C ^BC = E

g k ∼ χ ² _(2M)

g k = k h _k k ²

g k ∼ χ ² ₍₂₎

g k = | h _k | ²

g k ∼ χ ² _(2M ₎

σ ² _s

f D (d) = 2d/R ² , d ∈ [0, R]

L k = βd ⁻ _k ^ǫ

( ₂

ǫ ( ^x _β ) ⁻ ² ^ǫ ¹ _x

1 − F α (tx) → t ^2/ǫ

E { Y ^η }

1 − F Z (z) → E { Y ^η } (1 − F X (z))

1 − F g (x) → E { γ _k ² ^ǫ } β

x ² ǫ

− ² ǫ

1 ≤ k ≤ K g k ≤ β E { γ _k ^2/ǫ } ^ǫ/2 K ² ^ǫ x } = e ⁻ ^x ^−2/ǫ ∀ x > 0,