Comparison of the Discriminatory Processor Sharing Policies

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Comparison of the Discriminatory Processor Sharing

Policies

Natalia Osipova

To cite this version:

Natalia Osipova. Comparison of the Discriminatory Processor Sharing Policies. [Research Report]

RR-6475, INRIA. 2008, pp.19. �inria-00264111v4�

inria-00264111, version 4 - 25 Apr 2008

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Thème COM

Comparison of the Discriminatory Processor

Sharing Policies

Natalia Osipova

N° 6475

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

NataliaOsipova

∗ †

ThèmeCOMSystèmes ommuni ants ProjetMAESTRO

Rapportdere her he n°6475June200717pages

Abstra t: Dis riminatory Pro essor Sharing poli y introdu ed by Kleinro k is of a great in-terestin many appli ation areas,in ludingtele ommuni ations, web appli ationsand TCPow modelling. Under theDPS poli y the job priority is ontrolled by ave torof weights. Varying the ve torof weights, it is possible to modify the servi erates of thejobs and optimize system hara teristi s. In the present paper we present results on erning the omparison of two DPS poli ieswithdierentweightve tors. Weshowthemonotoni ityof theexpe tedsojourntimeof thesystem depending onthe weightve torunder ertain onditionon thesystem. Namely, the system hasto onsist of lasses with means whi h are quite dierent from ea h other. For the lasseswithsimilarmeanswesuggesttosele tthesameweights.

Key-words: Dis riminatoryPro essorSharing,exponentialservi etimes,optimization.

∗

INRIASophiaAntipolis,Fran e,e-mail:Natalia.Osipovasophia.inria.fr

†

TheworkwassupportedbyFran eTele omR&DGrantModélisationetGestionduTra RéseauxInternet no.42937433.

Résumé : L'ordrede servi e DPS(Dis riminatory Pro essor Sharing)qui était introduit par Kleinro kestunproblèmetrèsintéressantetpeutêtreappliquédansbeau oupdedomaines omme lestélé ommuni ations,lesappli ationswebetlamodélisationdeuxTCP.Ave leDPS,lesjobs qui viennentdanslesystème sont ontrlés parunve teurde poids. En modiant leve teur de poids,ilestpossiblede ontrlerlestauxdeservi edesjobs,donnerlaprioritéà ertaines lasses de jobs et optimiser ertaines ara téristiques dusystème. Le problème du hoix despoids est don trèsimportantettrèsdi ileenraisondela omplexitédusystème. Dansleprésentpapier, nous omparonsdeuxpolitiquesDPSave lesve teursdepoidsdiérentset nousprésentonsdes résultats sur la monotoni ité dutemps moyen de servi edu système en fon tiondu ve teur de poids,sous ertaines onditionssurle système. Lesystème devrait onsisterenplusieurs lasses ave desmoyennestrèsdiérentes. Pourles lassesquiontunemoyennetrèspro heilfaut hoisir lesmemepoids.

1 Introdu tion

TheDis riminatoryPro essorSharing(DPS)poli ywasintrodu edbyKleinro k[11℄. Underthe DPS poli y jobs are organized in lasses, whi h share a single server. The apa ity that ea h lass obtainsdepends onthe numberof jobs urrentlypresentedin all lasses. All jobs present in the system are served simultaneously at rates ontrolled by the ve tor of weights

g

k

> 0

,

k = 1, . . . , M }

, where

M

isthe numberof lasses. If there are

N

j

jobs in lass

j

,then ea h job of this lassis servedwith the rate

g

j

/

P

M

k=1

g

k

N

k

. Whenall weights areequal, DPSsystemis equivalenttothestandardPS poli y.

TheDPSpoli ymodelhasre entlyre eivedalotofattentionduetoitswiderangeof appli a-tion. Forexample,DPS ouldbeappliedto modelowlevelsharingofTCPowswithdierent ow hara teristi ssu hasdierentRTTsandpa ketlossprobabilities. DPSalsoprovidesa nat-uralapproa h to model theweightedround-robindis ipline, whi h isused in operatingsystems for tasks heduling. In theInternet one an imaginethesituation that serversprovidedierent servi ea ordingtothepaymentrates. Formoreappli ationsofDPSin ommuni ationnetworks see[2℄,[4℄,[5℄, [7℄,[12℄.

VaryingDPSweightsitispossibletogiveprioritytodierent lassesattheexpenseofothers, ontrol their instantaneous servi e rates and optimize dierent system hara teristi s as mean sojourntimeandsoon. So,theproperweightsele tionisanimportanttask,whi hisnoteasyto solvebe auseofthemodel's omplexity.

The previouslyobtained resultsonDPS modelare the following. Kleinro k in [11℄ was rst studying DPS. Then the paper of Fayolle et al. [6℄ provided results for the DPS model. For the exponentially distributed required servi e times the authors obtained the expression of the expe ted sojourn time as a solution of a system of linear equations. The authors show that independentlyoftheweightstheslowdownfortheexpe ted onditionalresponse timeunder the DPSpoli ytendstothe onstantslowdownofthePSpoli yastheservi erequirementsin reases toinnity.

RegeandSenguptain [13℄provedade omposition theoremfor the onditionalsojourntime. For exponential servi e time distributions in [14℄ they obtained higher moments of the queue lengthdistributionasthesolutionsoflinearequationssystemandalsoprovidedatheoremforthe heavy-tra regime.VanKesseletal. in[8℄,[10℄studytheperforman eofDPSinanasymptoti regimeusingtimes aling. Forgeneraldistributionsoftherequiredservi etimestheapproximation analysis was arried outby Guoand Matta in [7℄. Altman et al. [2℄ study the behavior of the DPSpoli y in overload. Mostof theresultsobtainedfortheDPS queuewere olle tedtogether inthesurveypaperofAltman etal. [1℄.

Avra henkovet al. in [3℄ provedthat themean un onditional response time of ea h lass is niteundertheusualstability ondition. Theydeterminetheasymptoteofthe onditionalsojourn timeforea h lassassumingnite servi etimedistribution withnitevarian e.

The problem of weights sele tion in the DPS poli y when the job size distributions are ex-ponentialwasstudied by Avra henkovet al. in [3℄and by Kim and Kim in [10℄. In [10℄ it was shownthattheDPSpoli yredu estheexpe tedsojourntimein omparisonwithPSpoli ywhen theweightsin reaseintheoppositeorderwiththemeansofjob lasses. Alsoin[10℄ theauthors formulate a onje ture about themonotoni ity ofthe expe ted sojourn timeof the DPS poli y. Theideaof onje tureis that omparing twoDPSpoli ies, onewhi hhasaweightve tor loser totheoptimalpoli yprovidedby

cµ

-rule,see [15℄,hassmallerexpe tedsojourntime. Usingthe method des ribedin [10℄in thepresentpaperweprovethis onje turewithsomerestri tionson thesystemparameters. Therestri tionsonthesystemaresu hthattheresultistrueforsystems for whi h the valuesof the job size distribution means are very dierent from ea h other. The restri tion anbeover omebysettingthesameweightsforthe lasses,whi hhavesimilarmeans. The onditiononmeansisasu ient,butnotane essary ondition. Itbe omeslessstri twhen thesystemislessloaded.

Thepaperisorganizedas follows. InSe tion 2wegivegeneraldenitions oftheDPSpoli y andformulatetheproblemofexpe tedsojourntimeminimization. InSe tion 3weformulatethe

main Theorem and prove it. In Se tion 4wegive the numeri al results. Some te hni al proofs anbefoundintheAppendix.

2 Previous results and problem formulation

We onsider the Dis riminatory Pro essor Sharing (DPS) model. All jobs are organizedin

M

lassesand share asingle server. Jobsof lass

k = 1, . . . , M

arrivewith aPoisson pro ess with rate

λ

k

andhaverequiredservi e-timedistribution

F

k

(x) = 1 − e

−µ

k

x

withmean

1/µ

k

. Theload of the system is

ρ =

P

M

k=1

ρ

k

and

ρ

k

= λ

k

/µ

k

,

k = 1, . . . , M

. We onsider that the systemis stable,

ρ < 1

. Letusdenote

λ =

P

M

k=1

λ

k

.

The stateof thesystem is ontrolled by ave torof weights

g = (g

1

, . . . , g

M

)

, whi hdenotes thepriorityforthejob lasses. Ifinthe lass

k

thereare urrently

N

k

jobs,thenea hjobof lass

k

is servedwiththerateequalto

g

j

/

P

M

k=1

g

k

N

k

, whi h depends onthe urrentsystemstate,or onthenumberofjobsin ea h lass.

Let

T

DP S

betheexpe tedsojourntimeoftheDPSsystem. Wehave

T

DP S

=

M

X

k=1

λ

k

λ

T

k

,

where

T

k

areexpe tedsojourntimesfor lass

k

. Theexpressions fortheexpe tedsojourntimes

T

k

,

k = 1, . . . , M

anbefoundasasolutionofthesystemof linearequations,see[6℄,

T

k



1 −

M

X

j=1

λ

j

g

j

µ

j

g

j

+ µ

k

g

k





−

M

X

j=1

λ

j

g

j

T

j

µ

j

g

j

+ µ

k

g

k

=

1

µ

k

,

k = 1, . . . , M.

(1)

Letusnoti ethat forthestandardPro essorSharingsystem

T

P S

=

m

1 − ρ

.

OneoftheproblemswhenstudyingDPSistominimizetheexpe tedsojourntime

T

DP S

with someweightsele tion. Namely,nd

g

∗

su has

T

DP S

(g

∗

_{) = min}

g

T

DP S

(g).

This isageneralproblemand tosimplify it thefollowingsub aseis onsidered. Tondaset

G

su hthat

T

DP S

(g

∗

_{) ≤ T}

P S

_,

_∀g

∗

_{∈ G.}

(2)

Forthe asewhenjob size distributions are exponentialthe solutionof(2) is givenby Kimand Kimin [10℄andisasfollows. Ifthemeansofthe lassesaresu h as

µ

1

≥ µ

2

≥ . . . ≥ µ

M

,then

G

onsistsofallsu h ve torswhi hsatisfy

G = {g| g

1

≥ g

2

≥ . . . ≥ g

M

}.

Usingtheapproa hof[10℄wesolvemoregeneralproblemaboutthemonotoni ityoftheexpe ted sojourntimeintheDPSsystem,whi hweformulatein thefollowingse tionasTheorem1.

3 Expe ted sojourn time monotoni ity

Theorem1. Letthejobsizedistributionforevery lassbeexponentialwithmean

µ

i

,

i = 1, . . . , M

andweenumerate theminthe followingway

µ

1

≥ µ

2

≥ . . . ≥ µ

M

.

Letus onsider twodierentweightpoli ies fortheDPSsystem,whi hwedenoteas

α

and

β

. Let

α, β ∈ G

, or

α

1

≥ α

2

≥ . . . ≥ α

M

,

β

1

≥ β

2

≥ . . . ≥ β

M

.

The expe tedsojourntimeof the DPSpoli ies withweightve tors

α

and

β

satises

T

DP S

(α) ≤ T

DP S

(β),

(3) if theweights

α

and

β

are su hthat:

α

i+1

α

i

≤

β

i+1

β

i

,

i = 1, . . . , M − 1,

(4)

andthe following restri tionissatised:

µ

j+1

µ

j

≤ 1 − ρ,

(5)

for every

j = 1, . . . , M

.

Remark 2. If for some lasses

j

and

j + 1

ondition (5 ) is not satised, then in pra ti e, by hoosing the weights ofthese lasses tobeequal,we an stilluse Theorem 1. Namely, for lasses su has

µ

j+1

µ

j

> 1 − ρ

,wesuggesttoset

α

j+1

= α

j

and

β

j+1

= β

j

. Remark 3. Theorem 1 shows that the expe tedsojourn time

T

DP S

(g)

is monotonousa ording tothe sele tion of weightve tor

g

. The loser isthe weight ve tortothe optimal poli y, provided by

cµ

-rule, the smaller is the expe ted sojourn time. This isshown by the ondition (4 ), whi h shows thatve tor

α

is loser tothe optimal

cµ

-rule poli ythen ve tor

β

.

Theorem 1is provedwith restri tion (5). This restri tion isa su ient andnot a ne essary onditiononsystemparameters. Itshowsthatthemeansofthejob lasseshavetobequitedierent fromea hother. Thisrestri tion anbeover ome,givingthesameweightstothejob lasses,whi h mean valuesaresimilar. Condition (5) be omeslessstri tasthe systembe omeslessloaded.

ToproveTheorem1letusrstgivesomenotationsandproveadditionalLemmas.

Letusrewritelinearsystem(1)inthematrixform. Let

T

(g)

= [T

(g)

1

, . . . , T

(g)

M

]

T

betheve tor of

T

(g)

k

,

k = 1, . . . , M

. Hereby

[ ]

T

wemean transposesign,so

[ ]

T

is ave tor. By

[ ]

(g)

wenote that thiselementdependsontheweightve torsele tion

g ∈ G

. Letus onsider thatlaterin the paperve tors

g, α, β ∈ G

,iftheoppositeisnotnoti ed. Letdene matri es

A

(g)

and

D

(g)

in the followingway.

A

(g)

=











λ

1

g

1

µ

1

g

1

+µ

1

g

1

λ

2

g

2

µ

1

g

1

+µ

2

g

2

. . .

λ

M

g

M

µ

1

g

1

+µ

M

g

M

λ

1

g

1

µ

2

g

2

+µ

1

g

1

λ

2

g

2

µ

2

g

2

+µ

2

g

2

. . .

λ

M

g

M

µ

2

g

2

+µ

M

g

M

. . .

λ

1

g

1

µ

M

g

M

+µ

1

g

1

λ

2

g

2

µ

M

g

M

+µ

2

g

2

. . .

λ

M

g

M

µ

M

g

M

+µ

M

g

M











(6)

D

(g)

=











P

i

λ

i

g

i

µ

1

g

1

+µ

i

g

i

0

. . . 0

0

P

i

λ

i

g

i

µ

2

g

2

+µ

i

g

i

. . . 0

. . .

0

0

. . .

P

i

λ

i

g

i

µ

M

g

M

+µ

i

g

i











(7)

Then(1)be omes

(E − D

(g)

− A

(g)

)T

(g)

=

 1

µ

1

. . . .

1

µ

M



T

.

(8)

Weneedtondtheexpe tedsojourntimeoftheDPSsystem

T

DP S

(g)

. A ordingtothedenition of

T

DP S

(g)

andequation(8)wehave

T

DP S

(g) =

1

λ

[λ

1

, . . . , λ

M

]T

(g)

=

1

λ

[λ

1

, . . . , λ

M

](E − D

(g)

_{− A}

(g)

₎

−1

 1

µ

1

, . . . ,

1

µ

M



T

.

(9)

Letus onsider the asewhen

λ

i

= 1

for

i = 1, . . . , M

. Thisresults anbeextended for the asewhen

λ

i

aredierent,weproveitfollowingtheapproa hof[10℄inProposition10attheend ofthe urrentSe tion. Equation(9)be omes

T

DP S

(g)

= 1

′

_{(E − D}

(g)

_{− A}

(g)

₎

−1

_[ρ

1

, . . . , ρ

M

]

T

λ

−1

.

(10) Letusgivethefollowingnotations.

σ

(g)

_ij

=

g

j

µ

i

g

i

+ µ

j

g

j

.

Then

σ

(g)

ij

havethefollowingproperties.

Lemma4.

σ

(g)

ij

and

σ

(g)

ji

satisfy

σ

_ij

(g)

g

i

= σ

(g)

ji

g

j

,

σ

(g)

_ij

µ

i

+

σ

(g)

ji

µ

j

=

1

µ

i

µ

j

.

(11)

Proof. Followsfromthedenitionof

σ

(g)

ij

. Thenmatri es

A

(g)

and

D

(g)

givenby(6)and(7) anberewrittenin thetermsof

σ

(g)

ij

.

A

(g)

_i,j

= σ

_ij

(g)

,

i, j = 1, . . . , M,

D

(g)

_i,i

=

X

j

σ

(g)

_ij

,

i = 1, . . . , M,

D

(g)

_i,j

= 0,

i, j = 1, . . . , M,

i 6= j.

Forweightve tors

α, β

thefollowingLemma istrue. Lemma5. If

α

and

β

satisfy (4),then

α

j

α

i

≤

β

j

β

i

,

i = 1, . . . , M − 1,

∀ j ≥ i.

(12) Proof. Letusnoti ethatif

a < b

and

c < d

,then

ac < bd

when

a, b, c, d

arepositive. Alsoif

j > i

thenthereexist su h

l > 0

that

j = i + l

. Then

α

i+1

α

i

≤

β

i+1

β

i

,

α

i+2

α

i+1

≤

β

i+2

β

i+1

,

. . .

α

i+l

α

i

+l−1

≤

β

i+l

β

i

+l−1

,

i = 1, . . . , M − 2.

Multiplyingleftand rightpartsofthepreviousinequalitieswegetthefollowing:

α

i+l

α

i

≤

β

i+l

β

i

,

i = 1, . . . , M − 2,

Lemma6. If

α

and

β

satisfy (12),then

σ

_ij

(α)

≤ σ

(β)

_ij

,

i ≤ j,

σ

_ij

(α)

≥ σ

(β)

_ij

,

i ≥ j.

Proof. As(12)then

α

j

α

i

≤

β

j

β

i

,

i ≤ j,

α

j

µ

i

β

i

≤ β

j

µ

i

α

i

,

i ≤ j,

α

j

(µ

i

β

i

+ µ

j

β

j

) ≤ β

j

(µ

i

α

i

+ µ

j

α

j

),

i ≤ j,

α

j

µ

i

α

i

+ µ

j

α

j

≤

β

j

µ

i

β

i

+ µ

j

β

j

,

i ≤ j,

σ

_ij

(α)

≤ σ

_ij

(β)

,

i ≤ j.

Weprovethese ondinequalityofLemma 6in asimilarway.

Lemma7. If

α

,

β

satisfy(4), then

T

DP S

(α) ≤ T

DP S

(β),

whenthe elementsof ve tor

y = 1

′

_{(E − B}

(α)

₎

−1

_M

aresu hthat

y

1

≥ y

2

≥ . . . ≥ y

M

. Proof. Letusdenote

B

(g)

_{= A}

(g)

_{+ D}

(g)

,

g = α, β

. Thenas(10)

T

DP S

(g) = λ

−1

₁

′

_{(E − B}

(g)

₎

−1

_[ρ

1

, . . . , ρ

M

]

T

,

g = α, β.

Followingthemethoddes ribedin [10℄wegetthefollowing.

T

DP S

(α) − T

DP S

(β)

=

λ

−1

₁

′

_{(E − B}

(α)

₎

−1

_[ρ

1

, . . . , ρ

M

]

T

− λ

−1

1

′

(E − B

(β)

)

−1

[ρ

1

, . . . , ρ

M

]

T

=

=

λ

−1

₁

′

_{((E − B}

(α)

₎

−1

_{− (E − B}

(β)

₎

−1

_{) [ρ}

1

, . . . , ρ

M

]

T

=

=

λ

−1

₁

′

_{((E − B}

(α)

₎

−1

_(B

(α)

_{− B}

(β)

_{)(E − B}

(β)

₎

−1

_{) [ρ}

1

, . . . , ρ

M

]

T

.

Letusdenote

M

asadiagonalmatrix

M = diag(µ

1

, . . . , µ

M

)

and

y = 1

′

_{(E − B}

(α)

₎

−1

_M.

(13) Then

T

DP S

(α) − T

DP S

(β)

=

1

′

_{(E − B}

(α)

₎

−1

_{M M}

−1

_(B

(α)

_{− B}

(β)

_)T

(β)

₌

=

yM

−1

_(B

(α)

_{− B}

(β)

_)T

(β)

₌

=

X

i,j

 y

j

µ

j

σ

(α)

_ji

+

y

i

µ

i

σ

(α)

_ij

−

 y

j

µ

j

σ

(β)

_ji

+

y

i

µ

i

σ

_ij

(β)



T

(β)

_j

=

=

X

i,j

y

j

σ

(α)

ji

µ

j

−

σ

(β)

ji

µ

j

!

+

y

i

µ

i

(σ

(α)

_ij

− σ

_ij

(β)

)

!

T

(β)

_j

.

As(11):

σ

_ji

(g)

µ

j

=

1

µ

i

µ

j

−

σ

(g)

ij

µ

i

,

g = α, β,

then

T

DP S

(α) − T

DP S

(β)

=

X

i,j

−y

j

σ

(α)

ij

µ

i

−

σ

(β)

ij

µ

i

!

+

y

i

µ

i

(σ

_ij

(α)

− σ

_ij

(β)

)

!

T

(β)

j

=

=

X

i,j



−

y

j

µ

i



σ

_ij

(α)

− σ

(β)

_ij



+

y

i

µ

i

(σ

_ij

(α)

− σ

(β)

_ij

)



T

(β)

j

=

=

X

i,j



_

σ

_ij

(α)

− σ

(β)

_ij



(y

i

− y

j

)

1

µ

i



T

(β)

_j

.

UsingLemma6weget



σ

(1)

_ij

− σ

(2)

_ij



(y

i

− y

j

)

isnegativefor

i, j = 1, . . . , M

when

y

1

≥ y

2

≥ . . . ≥

y

M

. This provesthestatementofLemma7. Lemma8. Ve tor

y

given by (13 )satises

y

1

≥ y

2

≥ . . . ≥ y

M

,

if thefollowing istrue:

µ

i+1

µ

i

≤ 1 − ρ,

for every

i = 1, . . . , M

.

Proof. Theproof ouldbefoundintheappendix.

Remark9. Forthejob lassessu has

µ

i+1

µ

i

> 1−ρ

weprovethattomake

y

i

≥ y

i+1

itissu ient tosetthe weightsof these lasses equal,

α

i+1

= α

i

.

Combining the results of Lemmas 5, 6, 7and 8 we prove the statement of the Theorem 1. Remark9givestheRemark2afterTheorem1.

Proposition 10. The resultof Theorem1isextendedtothe asewhen

λ

i

6= 1

.

Proof. Letusrst onsider the asewhenall

λ

i

= q

,

i = 1, . . . , M

. It anbeshownthatforthis asetheproofofTheorem1isequivalenttotheproofofthesameTheorembutforthenewsystem with

λ

∗

i

= 1

,

µ

∗

i

= qµ

i

,

i = 1, . . . , M

. Forthis newsystemthe resultsof Theorem 1isevidently trueandrestri tion(5)isnot hanged. Then, Theorem1istruefortheinitialsystemaswell.

If

λ

i

arerational,thenthey ouldbewritten in

λ

i

=

p

i

q

,where

p

i

and

q

arepositiveintegers. Thenea h lass anbepresentedas

p

i

lasseswithequalmeans

1/µ

i

andintensity

1/q

. So, the DPSsystem anbe onsideredastheDPSsystemwith

p

1

+ . . . + p

K

lasseswiththesamearrival rates

1/q

. TheresultofTheorem1isextended onthis ase.

If

λ

i

,

i = 1, . . . , M

are positive and real we apply the previous ase of rational

λ

i

and use ontinuity.

4 Numeri al results

Let us onsider a DPS system with

3

lasses. Let us onsider the set of normalized weights ve tors

g(x) = (g

1

(x), g

2

(x), g

3

(x))

,

P

3

i=1

g

i

(x) = 1

,

g

i

(x) = x

−i

_/(

P

3

i=1

x

−i

)

,

x > 1

. Everypoint

x > 1

denotes aweightve tor. Ve tors

g(x), g(y)

satisfyproperty(4) when

1 < y ≤ x

, namely

g

i+1

(x)/g

i

(x) ≤ g

i+1

(y)/g

i

(y)

,

i = 1, 2

,

1 < y ≤ x

. OnFigures 1,2weplot

T

DP S

withweights

ve tors

g(x)

asafun tionof

x

,theexpe tedsojourntimes

T

P S

forthePSpoli yand

T

opt

forthe optimal

cµ

-rulepoli y.

OnFigure 1weplottheexpe tedsojourntimeforthe asewhen ondition(5)issatisedfor three lasses. Theparametersare:

λ

i

= 1

,

i = 1, 2, 3

,

µ

1

= 160

,

µ

2

= 14

,

µ

3

= 1.2

,then

ρ = 0.911

.

1

2

3

4

5

6

7

8

9

10

3.1

3.15

3.2

3.25

3.3

3.35

3.4

3.45

3.5

T

DPS

(g(x))

T

PS

T

opt

Figure 1:

T

DP S

(g(x))

,

T

P S

,

T

opt

fun tions, onditionsatised.

1

2

3

4

5

6

7

8

9

10

3.7

3.72

3.74

3.76

3.78

3.8

3.82

3.84

3.86

3.88

3.9

T

opt

T

PS

T

DPS

(g(x))

Figure 2:

T

DP S

(g(x))

,

T

P S

,

T

opt

fun tions, onditionnotsatised

On Figure 2 we plotthe expe ted sojourntime for the ase when ondition(5)is not satised for three lasses. The parametersare:

λ

i

= 1

,

i = 1, 2, 3

,

µ

1

= 3.5

,

µ

2

= 3.2

,

µ

3

= 3.1

, then

ρ = 0.92

. One ansee that

T

DP S

(g(x)) ≤ T

DP S

(g(y))

,

1 < y ≤ x

evenwhentherestri tion(5) isnotsatised.

5 Con lusion

We study the DPS poli y with exponential job size distributions. One of the main problems studyingDPSistheexpe tedsojourntimeminimizationa ordingtotheweighssele tion. Inthe present paper we ompare twoDPS poli ies with dierent weights. We show that theexpe ted sojourn timeis smaller forthe poli y with the weighve tor loser to theoptimal poli y ve tor, provided by

cµ

-rule. So, weprove the monotoni ity of the expe ted sojourntime for the DPS poli ya ordingto theweightve torsele tion.

Theresultisproved withsomerestri tionsonsystemparameters. Thefound restri tionson the system parameters are su h that the result is true for systems su h as the mean valuesof the job lass size distributions are very dierent from ea h other. Wefound, that to prove the main resultit issu ienttogivethesameweightstothe lasseswithsimilar means. Thefound restri tionis asu ientand notane essary onditiononasystem parameters. Whentheload ofthesystemde reases,the onditionbe omeslessstri t.

A knowledgment

I wouldlike tothank K.Avra henkov, P. Brown and U. Ayesta forfruitfuldis ussions and sug-gestions.

6 Appendix

Inthe followingproof in thenotationswedonotuse thedependen yofthe parameterson

g

to simplify the notations. We onsider that ve tor

g ∈ G

, or

g

1

≥ g

2

. . . ≥ g

M

. To simplify the notationsletususe

P

k

insteadof

P

M

k=1

. Lemma8. Ve tor

y = 1

′

_{(E − B)}

−1

_M

satises

y

1

≥ y

2

≥ . . . ≥ y

M

,

ifthefollowingis true:

µ

i+1

µ

i

≤ 1 − ρ,

forevery

i = 1, . . . , M

.

Proof. Using theresults ofthe followingLemmas we provethestatement ofLemma 8and give theproofforRemark9.

Letusgivethefollowingnotations

˜

µ = µ

T

_{(E − D)}

−1

_,

(14)

˜

A = M

−1

_{AM (E − D)}

−1

_.

(15)

Letusnoti ethefollowing

(E − D)

−1

j

=

1

1 −

P

k

g

k

µ

j

g

j

+µ

k

g

k

=

1

1 − ρ +

P

k

µ

j

g

j

µ

k

(µ

j

g

j

+µ

k

g

k

)

> 0,

j = 1, . . . , M,

˜

A

ij

=

µ

j

g

j

µ

i

(µ

i

g

i

+µ

j

g

j

)

1 −

P

k

g

k

µ

j

g

j

+µ

k

g

k

=

µ

j

g

j

µ

i

(µ

i

g

i

+µ

j

g

j

)

1 − ρ +

P

k

µ

j

g

j

µ

k

(µ

j

g

j

+µ

k

g

k

)

> 0,

i, j = 1, . . . , M

Letusgivethefollowingnotation

f (x) =

X

k

x

µ

k

(x + µ

k

g

k

)

.

Then

(E − D)

−1

j

=

1

1 − ρ + f (µ

j

g

j

)

,

j = 1, . . . , M,

˜

A

ij

=

µ

j

g

j

µ

i

(µ

i

g

i

+ µ

j

g

j

)(1 − ρ + f (µ

j

g

j

))

,

i, j = 1, . . . , M.

LetusrstproveadditionalLemma.

Lemma11. Matrix

˜

A = M

−1

_{AM (E − D)}

−1

isapositive ontra tion.

Proof. Matrix

A

˜

isapositiveoperatoraselementsofmatri es

M

and

A

arepositiveandelements ofmatrix

(E − D)

−1

arepositive. Let

Ω = {X|x

i

≥ 0, i = 1, . . . , M }

. If

X ∈ Ω

, then

AX ∈ Ω

˜

. Thentoprovethatmatrix

A

˜

isa ontra tionitisenoughtoshowthat

∃ q,

0 < q < 1,

|| ˜

AX|| ≤ q||X||,

∀ X ∈ Ω.

(16) As

X ∈ Ω

,thenwe antake

||X|| = 1

′

_{X =}

P

i

x

i

. Then

1

′

_AX

˜

₌

X

j

x

j

X

i

˜

A

ij

=

X

j

x

j

P

i

µ

j

g

j

µ

i

(µ

j

g

j

+µ

i

g

i

)

(1 − ρ + g(µ

j

g

j

))

=

=

X

j

x

j

f (µ

j

g

j

)

1 − ρ + f (µ

j

g

j

)

=

X

j

x

j



1 −

1 − ρ

1 − ρ + f (µ

j

g

j

)



=

=

X

j

x

j

− (1 − ρ)

X

j

x

j

1 − ρ + f (µ

j

g

j

)

.

Letusndthevalueof

q

, whi hsatises ondition(16).

1

′

_{AX ≤ 1}

˜

′

_X,

X

j

x

j

− (1 − ρ)

X

j

x

j

1 − ρ + f (µ

j

g

j

)

≤ q

X

j

x

j

1 − (1 − ρ)

P

j

x

j

1−ρ+f (µ

j

g

j

)

P

j

x

j

≤ q.

As

f (µ

j

g

j

) > 0

then

0 < 1 − (1 − ρ)

P

j

x

j

1−ρ+f (µ

j

g

j

)

P

j

x

j

< 1.

Letusdene

δ

in thefollowingway:

δ =

1

1 − ρ + max

j

f (µ

j

g

j

)

<

P

j

x

j

1−ρ+f (µ

j

g

j

)

P

j

x

j

.

Then

1 − (1 − ρ)

P

j

x

j

1−ρ+f (µ

j

g

j

)

P

j

x

j

< 1 − (1 − ρ)δ.

Letusnoti ethat

max

j

f (µ

j

g

j

)

alwaysexistsasthevaluesof

µ

j

g

j

,

j = 1, . . . , M

arenite. Then we ansele t

q = 1 − (1 − ρ)δ,

0 < q < 1.

Whi h ompletestheproof.

Lemma12. If

y

(0)

₁

= [0, . . . , 0],

(17)

y

(n)

= ˜

µ + y

(n−1)

A,

˜

n = 1, 2, . . . ,

(18) then

y

(n)

_{→ y}

,when

n → ∞

.

Proof. Letuspresent

y

inthefollowingway. As

B = E − A − D

,then

y = 1(E − B)

−1

_M,

yM

−1

_{(E − D − A) = 1,}

yM

−1

_{(E − D) = −yM}

−1

_{A + 1,}

y(E − D)

−1

_{M = −yM}

−1

_{A(E − D)}

−1

_{M + 1(E − D)}

−1

_M.

Asmatrixes

D

and

M

arediagonal,the

M D = DM

andthen

y = µ

T

(E − D)

−1

_{+ yM}

−1

_{AM (E − D)}

−1

_,

where

µ = [µ

1

, . . . , µ

M

]

. A ordingtonotations(14)and(15)wehavethefollowing

y = ˜

µ + y ˜

A.

Letusdenote

y

(n)

_{= [y}

(n)

1

, . . . , y

(n)

1

]

,

n = 0, 1, 2, . . .

andletdene

y

(0)

1

and

y

(n)

by(17)and(18). A ording to Lemma 11 ree tion

A

˜

is a positive ree tion and is a ontra tion. Also

µ

˜

i

are positive. Then

y

(n)

_{→ y}

Lemma13. Let

y

(n)

isdenedby (18 )and

y

(0)

isgiven by(17), then

y

(n)

1

≥ y

(n)

2

≥ . . . ≥ y

(n)

M

,

n = 1, 2, . . .

(19) if

µ

i+1

µ

i

≤ 1 − ρ

forevery

i = 1, . . . , M

.

Proof. We prove the statement (19) by indu tion. For

y

(0)

the statement (19) is true. Let us assumethat(19)istrueforthe

(n−1)

step,

y

(n−1)

1

≥ y

(n−1)

2

≥ . . . ≥ y

(n−1)

M

. Toprovetheindu tion statementwehavetoshowthat

y

(n)

1

≥ y

(n)

2

≥ . . . ≥ y

(n)

M

,whi hisequaltothat

y

(n)

j

≥ y

(n)

p

,if

j ≤ p

. As

y

(n)

_j

= ˜

µ

j

+

M

X

i=1

y

(n−1)

_i

A

˜

ij

,

then

y

(n)

_j

− y

_p

(n)

=

µ

˜

j

+

M

X

i=1

y

_i

(n−1)

A

˜

ij

−

µ

˜

p

+

M

X

i=1

y

(n−1)

_i

A

˜

ip

!

=

=

µ

˜

j

− ˜

µ

p

+

M

X

i=1

y

i

(n−1)

( ˜

A

ij

− ˜

A

ip

).

Toshowthat

y

(n)

j

− y

(n)

p

weneedtoshowthat

µ

˜

j

− ˜

µ

p

≥ 0

and

P

M

i=1

y

(n−1)

i

( ˜

A

ij

− ˜

A

ip

) ≥ 0

,when

j ≤ p

. Letus show that to provethat

P

M

i=1

y

(n−1)

i

( ˜

A

ij

− ˜

A

ip

) ≥ 0

,

j ≤ p

itis enoughto prove that

P

r

i=1

( ˜

A

ij

− ˜

A

ip

) ≥ 0

,

j ≤ p

,

r = 1, . . . , M

. Ifweregroupthissumwe angetthefollowing

M

X

i=1

y

(n−1)

_i

( ˜

A

ij

− ˜

A

ip

) =

M

X

i=1

(y

_i

(n−1)

− y

_i+1

(n−1)

+ y

_i+1

(n−1)

− . . . − y

_M

(n−1)

+ y

(n−1)

_M

)( ˜

A

ij

− ˜

A

ip

) =

=

M −1

X

i=1

(y

(n−1)

_i

− y

(n−1)

i+1

)

h

( ˜

A

1j

− ˜

A

1p

) + ( ˜

A

2j

− ˜

A

2p

) + . . . + ( ˜

A

ij

− ˜

A

ip

)

i

+

+y

(n−1)

_M

(( ˜

A

1j

− ˜

A

1p

) + . . . + ( ˜

A

_(M−1)j

− ˜

A

_(M−1)p

) + ( ˜

A

M j

− ˜

A

M p

)) =

=

M −1

X

i=1

(y

_i

(n−1)

− y

i+1

(n−1)

)

r

X

k=1

( ˜

A

kj

− ˜

A

kp

) + y

(n−1)

M

M

X

k=1

( ˜

A

kj

− ˜

A

kp

).

As

y

(n−1)

i

≥ y

i+1

(n−1)

,

i = 1, . . . , M

,a ordingtotheindu tionstep,thentoshowthat

P

M

i=1

y

i

(n−1)

( ˜

A

ij

−

˜

A

ip

) ≥ 0

,

j ≤ p

itisenoughtoshowthat

P

r

i=1

( ˜

A

ij

− ˜

A

ip

) ≥ 0

,

j ≤ p

,

r = 1, . . . , M

. Weshowthis in Lemma15. InLemma14 weshow

µ

˜

j

≥ ˜

µ

p

,

j ≤ p

, when

µ

i+1

µ

i

≤ 1 − ρ

forevery

i = 1, . . . , M

. Thenweprovetheindu tion statementandsoprovethestatementofLemma13.

Lemma14.

˜

µ

1

≥ ˜

µ

2

. . . ≥ ˜

µ

M

,

(20) if

µ

i+1

µ

i

≤ 1 − ρ

forevery

i = 1, . . . , M

.

Proof. Let us ompare

µ

˜

j

and

µ

˜

p

,

j ≤ p

. If

µ

j

= µ

p

and

g

j

= g

p

, then

µ

˜

j

= ˜

µ

p

and (20) is satised. Letusdenote

f

2

(x) =

X

k

g

k

x + µ

k

g

k

,

whi hhasthefollowingproperties

0 < f

2

(x) < ρ.

(21) Then

˜

µ

i

=

µ

i

1 −

P

j

g

j

µ

i

g

i

+µ

j

g

j

=

µ

i

1 − f

2

(µ

i

g

i

)

.

Letusnd

˜

µ

j

− ˜

µ

p

=

µ

j

1 − f

2

(µ

j

g

j

)

−

µ

p

1 − f

2

(µ

p

g

p

)

=

µ

j

− µ

p

− (µ

j

f

2

(µ

p

g

p

) − µ

p

f

2

(µ

j

g

j

))

(1 − f

2

(µ

j

g

j

))(1 − f

2

(µ

p

g

p

))

.

As(21)then

µ

j

f

2

(µ

p

g

p

) − µ

p

f

2

(µ

j

g

j

) < µ

j

ρ.

Then

˜

µ

j

− ˜

µ

p

>

(µ

j

− µ

p

)

(1 − f

2

(µ

j

g

j

)(1 − f

2

(µ

p

g

p

)))



1 − ρ



µ

j

µ

j

− µ

p



=

=

(µ

j

− µ

p

)

(1 − f

2

(µ

j

g

j

)(1 − f

2

(µ

p

g

p

)))

1 − ρ

1

1 −

µ

p

µ

j

!!

≥ 0,

when

µ

p

µ

j

≤ 1 − ρ.

So, if

µ

p

µ

j

≤ 1 − ρ

and

g

j

≥ g

p

, then

µ

˜

j

≥ ˜

µ

p

. Let us show that if

µ

j

> µ

p

and

g

j

= g

p

, then

˜

µ

j

≥ ˜

µ

p

. Inthis ase

˜

µ

j

− ˜

µ

p

=

µ

j

1 − f

2

(µ

j

g

j

)

−

µ

p

1 − f

2

(µ

p

g

p

)

=

=

µ

j

− µ

p

− (µ

j

f

2

(µ

p

g

j

) − µ

p

f

2

(µ

j

g

j

))

(1 − f

2

(µ

j

g

j

))(1 − f

2

(µ

p

g

j

))

=

∆

1

(1 − f

2

(µ

j

g

j

))(1 − f

2

(µ

p

g

j

))

.

Letusndwhen

∆

1

> 0

.

∆

1

= µ

j

− µ

p

−

µ

j

M

X

k=1

g

k

µ

p

g

j

+ µ

k

g

k

− µ

p

M

X

k=1

g

k

µ

j

g

j

+ µ

k

g

k

!

=

= µ

j

− µ

p

−

M

X

k=1

g

k

(g

j

(µ

2

j

− µ

p

2

) + µ

k

g

k

(µ

j

− µ

p

))

(µ

p

g

j

+ µ

k

g

k

)(µ

p

g

j

+ µ

k

g

k

)

!

= (µ

j

− µ

p

)

1 −

M

X

k=1

g

k

(g

j

(µ

j

+ µ

p

) + µ

k

g

k

)

(µ

p

g

j

+ µ

k

g

k

)(µ

p

g

j

+ µ

k

g

k

)

!

.

As

0 < µ

j

µ

p

g

2

j

,

k = 1, . . . , M,

g

k

µ

k

(g

j

(µ

j

+ µ

p

) + µ

k

g

k

) < (µ

j

µ

p

g

j

2

+ g

j

(µ

j

+ µ

p

)µ

k

g

k

+ µ

2

k

g

2

k

),

k = 1, . . . , M

g

k

µ

k

(g

j

(µ

j

+ µ

p

) + µ

k

g

k

) < (µ

j

g

j

+ µ

k

g

k

)(µ

p

g

j

+ µ

k

g

k

),

k = 1, . . . , M

g

k

(g

j

(µ

j

+ µ

p

) + µ

k

g

k

)

(µ

j

g

j

+ µ

k

g

k

)(µ

p

g

j

+ µ

k

g

k

)

<

1

µ

k

,

k = 1, . . . , M.

Then

∆

1

> (µ

j

− µ

p

)

1 −

M

X

k=1

1

µ

k

!

= 1 − ρ > 0.

Thenweprovedthefollowing:

If

µ

j

= µ

p

, g

j

= g

p

,

then

µ

˜

j

= ˜

µ

p

,

If

µ

j

> µ

p

, g

j

= g

p

,

then

µ

˜

j

> ˜

µ

p

,

If

µ

p

µ

j

≤ 1 − ρ, g

j

≥ g

p

,

then

µ

˜

j

≥ ˜

µ

p

.

Setting

p = j + 1

and rememberingthat

µ

1

≥ . . . ≥ µ

M

, we get that

µ

˜

1

≥ ˜

µ

2

. . . ≥ ˜

µ

M

is true when

µ

i+1

µ

i

≤ 1 − ρ

forevery

i = 1, . . . , M

. ThatprovesthestatementofLemma 14.

Returning ba k to the main Theorem 1, Lemma 14 gives ondition(5) asa restri tionon a systemparameters.

Letusnoti ethatforthejob lassessu hforwhi hthemeansaresu has

µ

i+1

µ

i

< 1 − ρ

,ifthe weightsgivenforthese lassesareequal,thenstill

µ

˜

i

≥ ˜

µ

i+1

. This onditiongivesus asaresult Remark9andRemark2.

Lemma15.

r

X

i=1

˜

A

i1

≥

r

X

i=1

˜

A

i2

≥ . . . ≥

r

X

i=1

˜

A

iM

,

r = 1, . . . , M.

Proof. Letusremember

A = M

˜

−1

_{AM (E − D)}

−1

. Thenas

ρ =

P

M

k=1

1

µ

k

,then

r

X

i=1

˜

A

ij

=

P

r

i=1

µ

j

g

j

µ

i

(µ

j

g

j

+µ

i

g

i

)

1 −

P

M

k=1

g

k

µ

j

g

j

+µ

k

g

k

=

P

r

i=1

µ

j

g

j

µ

i

(µ

j

g

j

+µ

i

g

i

)

1 − ρ +

P

M

k=1

µ

j

g

j

µ

k

(µ

j

g

j

+µ

k

g

k

)

Letusdene

f

3

(x) =

P

r

i=1

µ

i

(x+µ

x

i

g

i

)

1 − ρ +

P

M

k=1

µ

k

(x+µ

x

k

g

k

)

=

h

1

(x)

1 − ρ + h

1

(x) + h

2

(x)

,

where

h

1

(x) =

r

X

i=1

x

µ

i

(x + µ

i

g

i

)

> 0,

h

2

(x) =

M

X

j=r+1

x

µ

j

(x + µ

j

g

j

)

> 0.

Let us show that

f

3

(x)

is in reasing on

x

. Forthat it enough to showthat

df

3

(x)

dx

≥ 0

. Letus onsider

df

3

(x)

dx

=

h

′

1

(x)(1 − ρ) + h

′

1

(x)h

2

(x) − h

1

(x)h

′

2

(x)

(1 − ρ + h

1

(x) + h

2

(x))

2

Sin e

h

′

1

(x) > 0

and

1 − ρ > 0

:

df

3

(x)

dx

≥ 0

if

h

′

1

(x)h

2

(x) − h

1

(x)h

′

2

(x) ≥ 0.

Letus onsider

h

′

1

(x)h

2

(x) − h

1

(x)h

′

2

(x)

=

r

X

i=1

g

i

(x + µ

i

g

i

)

2

M

X

k=r+1

x

µ

k

(x + µ

k

g

k

)

−

r

X

i=1

x

µ

i

(x + µ

i

g

i

)

M

X

k=r+1

g

k

(x + µ

k

g

k

)

2

=

=

r

X

i=1

M

X

k=r+1



g

i

x

(x + µ

i

g

i

)

2

(x + µ

k

g

k

)µ

k

−

g

k

x

µ

i

(x + µ

i

g

i

)(x + µ

k

g

k

)

2



=

=

r

X

i=1

M

X

k=r+1

x

(x + µ

i

g

i

)(x + µ

k

g

k

)



g

i

µ

k

(x + µ

i

g

i

)

−

g

k

µ

i

(x + µ

k

g

k

)



=

=

r

X

i=1

M

X

k=r+1

x

(x + µ

i

g

i

)(x + µ

k

g

k

)

 µ

i

g

i

(x + g

k

µ

k

) − µ

k

g

k

(x + µ

i

g

i

)

µ

i

µ

k

(x + µ

k

g

k

)(x + µ

i

g

i

)



=

=

r

X

i=1

M

X

k=r+1

x

2

_(µ

i

g

i

− µ

k

g

k

)

(x + µ

i

g

i

)

2

(x + µ

k

g

k

)

2

µ

k

µ

i

≥ 0,

Then

df

3

(x)

dx

≥ 0

and

f

3

(x)

is anin reasingfun tion of

x

. As

µ

j

g

j

≥ µ

p

g

p

,

j ≤ p

, then weprove thestatementofLemma15.

Referen es

[1℄ E.Altman,K.Avra henkov,U.Ayesta,Asurveyondis riminatorypro essorsharing, Queue-ingSystems,Volume53,Numbers1-2,June2006,pp.53-63(11).

[2℄ E. Altman, T. Jimenez, and D. Kofman, DPS queueswith stationary ergodi servi e times andtheperforman eofTCPinoverload,inPro eedingsofIEEEInfo om,Hong-Kong,Mar h 2004.

[3℄ K. Avra henkov, U. Ayesta, P. Brown, R. Nunez-Queija, Dis riminatoryPro essor Sharing Revisited,INFOCOM2005.24thAnnualJointConferen eoftheIEEEComputerand Com-muni ationsSo ieties.Pro eedingsIEEE,Vol.2(2005),pp.784-795vol.2.

[4℄ T.Bu andD. Towsley, FixedpointapproximationforTCPbehaviourin anAQMnetwork, in Pro eedingsofACM SIGMETRICS/Performan e,pp.216-225,2001.

[5℄ S. K.Cheung, J.L.vanden Berg,R.J. Bou herie,R.Litjens, andF. Roijers,An analyti al pa ket/ow-levelmodelling approa h for wireless LANs with quality-of-servi e support, in Pro eedingsofITC-19,2005.

[6℄ G. Fayolle, I. Mitrani, R. Iasnogorodski, Sharing a Pro essor Among Many Job Classes, JournaloftheACM(JACM),Volume27,Issue 3(July1980),pp.519-532,1980, ISSN:0004-5411.

[7℄ L.GuoandI.Matta,S hedulingowswithunknownsizes: approximateanalysis,in Pro eed-ingsofthe2002ACM SIGMETRICSinternational onferen eonMeasurementandmodeling of omputersystems,2002,pp.276-277.

[8℄ G. van Kessel, R. Nunez-Queija, S. Borst, Dierentiated bandwidth sharing with disparate owsizes,INFOCOM2005.24thAnnualJointConferen eoftheIEEEComputerand Com-muni ationsSo ieties.Pro eedingsIEEE,Vol.4(2005),pp.2425-2435vol.4.

[9℄ G. van Kessel, R. Nunez-Queija, S. Borst,Asymptoti regimes and approximationsfor dis- riminatorypro essorsharing,SIGMETRICS Perform.Eval.Rev.,vol.32, pp.44-46,2004.

[10℄ B.KimandJ.Kim,Comparisonof DPSand PSsystemsa ordingtoDPSweights, Com-muni ationsLetters,IEEE,vol.10,issue7,pp.558-560,ISSN:1089-7798,July2006.

[11℄ L. Kleinro k, Time-shared systems: A theoreti al treatment, J.ACM, vol.14, no. 2, pp. 242-261,1967.

[12℄ Y. Hayel and B. Tun, Pri ing for heterogeneous servi es at a dis riminatory pro essor sharingqueue,inPro eedingsofNetworking,2005.

[13℄ K.M.Rege,B.Sengupta,Ade ompositiontheoremandrelatedresultsforthedis riminatory pro essorsharingqueue,QueueingSystems,SpringerNetherlandsISSN0257-0130,1572-9443, IssueVolume18,Numbers3-4/September,1994,pp.333-351,2005.

[14℄ K. M. Rege, B. Sengupta, Queue-Length Distribution for the Dis riminatory Pro essor-SharingQueue,OperationsResear h,Vol.44,No.4(Jul.-Aug.,1996),pp.653-657.

[15℄ RhondaRighter,"S heduling",inM.ShakedandJ.Shanthikumar(eds),"Sto hasti Orders", 1994.

Contents

1 Introdu tion 3

2 Previous resultsand problemformulation 4

3 Expe tedsojourn time monotoni ity 4

4 Numeri alresults 8

5 Con lusion 9

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