Optimal Choice of Threshold in Two Level Processor Sharing

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Optimal Choice of Threshold in Two Level Processor

Sharing

Konstantin Avrachenkov, Patrick Brown, Natalia Osipova

To cite this version:

Konstantin Avrachenkov, Patrick Brown, Natalia Osipova. Optimal Choice of Threshold in Two Level

Processor Sharing. [Research Report] RR-6215, INRIA. 2007, pp.19. �inria-00153966v2�

inria-00153966, version 2 - 13 Jun 2007

a p p o r t

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Thème COM

Optimal Choice of Threshold in

Two Level Processor Sharing

Konstantin Avrachenkov — Patrick Brown — Natalia Osipova

N° 6215

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

Konstantin Avra henkov

∗

, Patri k Brown

†

, NataliaOsipova

‡ §

ThèmeCOMSystèmes ommuni ants ProjetMAESTRO

Rapportdere her he n°6215June200719pages

Abstra t: WeanalyzetheTwoLevelPro essorSharing(TLPS) s hedulingdis iplinewiththe hyper-exponentialjobsizedistributionandwiththePoissonarrivalpro ess. TLPSisa onvenient model tostudy thebenetofthelesize baseddierentiationin TCP/IPnetworks. Inthe ase of the hyper-exponential job size distribution with two phases, we nd a losed form analyti expression for the expe ted sojourn time and an approximation for the optimal value of the thresholdthatminimizestheexpe tedsojourntime. Inthe aseofthehyper-exponentialjobsize distributionwithmorethantwophases,wederiveatightupperboundfortheexpe ted sojourn time onditioned on the job size. We show that when the varian e of the job size distribution in reases,thegaininsystemperforman ein reasesandthesensitivitytothe hoi eofthethreshold nearitsoptimalvaluede reases.

Key-words: TwoLevelPro essorsharing,Hyper-Exponentialdistribution,Lapla etransform.

∗

INRIASophiaAntipolis,Fran e,e-mail:K.Avra henkovsophia.inria.fr

†

Fran eTele omR&D,Fran e,e-mail:Patri k.Brownorange-ftgroup. om

‡

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

§

TheworkwassupportedbyFran eTele omR&DGrantModélisationetGestionduTra RéseauxInternet no.42937433.

d'une Politique à Temps Partagé Ave Deux-Niveaux

Résumé : Nous étudions lale d'attente munie d'une politique aTemps Partagé ave Deux-Niveaux"TwoLevelPro essorSharing"ave distributionhyper-exponentielledestempsdeservi e et ave lepro essus d'arrivée Poisson. TLPSest unmodèle ommode pourordonnerl'a èsaux ressour esenfon tiondelatailledansunréseauTCP/IP.Dansle asoùladistributiondutemps deservi eestunedistributionhyper-exponentielleave deuxphases,noustrouvonsuneexpression analytiquepourletempsderéponsemoyen. Aussinoustrouvonsuneapproximationdevaleurde seuil optimalqui réduitauminimum letemps deréponse moyen. Dans le as oùladistribution dutempsdeservi eaplusquedeuxphases,noustrouvonsunebornesupérieurepourlafon tion detempsderéponse moyenquiest onditionnéeautempsdeservi e. Nousmontronsquequand lavarian edeladistributiondestempsdeservi eaugmente,legaindansl'exé utiondusystème est onsidérableetil n'yapasdesensibilitéau hoixduseuil sousoptimal.

Mots- lés : La le d'attente munie d'une politique à Temps Partagées ave Deux-Niveaux, réseauTCP/IP,distributionhyper-exponentielle,Lapla etransforment.

1 Introdu tion

Ithasbeenknownforalongtimethat a levers hedulingoftasks ansigni antlyimprove systemperforman e. Forinstan e, ShortestRemainingPro essing Time(SRPT)s heduling dis- ipline minimizes the expe ted sojourntime [15℄. However, SRPT requires to keeptra k of all jobs in the system and also requires the knowledge of the remaining pro essing times. These requirementsareoftennotfeasibleinappli ations. Theexamplesofsu happli ationsarelesize baseddierentiationinTCP/IPnetworks[3, 9℄andWebserverrequestdierentiation[10,11℄.

The Two Level Pro essor Sharing (TLPS) s heduling dis ipline [12℄ helps to over ome the abovementionedrequirements. Itusesthedierentiation ofjobsa ordingto athresholdonthe attainedservi eandgivespriorityto thejobswithsmallsizes. Adetaildes riptionoftheTLPS dis iplineispresentedintheensuingse tion. Of ourse,TLPSprovidesasub-optimalme hanism in omparisonwith SRPT. Nevertheless, aswasshown in [1℄, when thejob size distribution has ade reasinghazard rate, theperforman e of TLPSwith appropriate hoi e of thresholdis very lose to optimal. It turns out that the distribution of le sizes in the Internet indeed has a de reasinghazardrateandoften ouldbemodeledwithaheavy-taileddistributions. Itisknown, that the heavy-tailed distribution ould be approximatedwith a hyper-exponentialdistribution with asigni antnumberof phases[5, 8℄. Alsoin [7℄, it wasshown that the hyper-exponential distributionmodels well thelesize distribution in theInternet. Therefore,in thepresentwork weanalyzetheTLPSsystemwithhyper-exponentialjob sizedistribution.

Thepaperorganization and main results are asfollows. In Se tion 2we providethe model formulation, main denitions and equations. In Se tion 3westudy the TLPSdis ipline in the aseofthehyper-exponentialjobsizedistribution withtwophases. Itisknownthat theInternet onne tionsbelong to twodistin t lasseswith very dierentsizes of transfer. Therst lassis omposed of short HTTP onne tions and P2P signaling onne tions. The se ond lass orre-spondsto downloads(PDFles, MP3 les, MPEG les, et .). This fa t providesmotivation to onsiderrstthehyper-exponentialjobsize distributionwithtwophases.

Wendananalyti alexpressionfortheexpe tedsojourntimeintheTLPSsystem. Then,we present theapproximationof theoptimalthresholdwhi h minimizestheexpe tedsojourntime. Weshowthat theapproximatedvalueof thethresholdtends tothe optimalthresholdwhen the se ondmomentofthejobsizedistributionfun tiongoestoinnity.

WeshowthattheuseoftheTLPSs hedulingdis ipline anleadto45%gainin theexpe ted sojourntimein omparisonwith thestandardPro essorSharing. Wealsoshowthat thesystem performan eis nottoosensitivetothe hoi eofthethresholdarounditsoptimalvalue.

InSe tion4weanalyzetheTLPSdis iplinewhenthejobsizedistributionishyper-exponential with many phases. We provide an expression of the expe ted onditional sojourn time as the solutionofasystemoflinearequations. Alsoweapplyaniterationmethodtondtheexpression oftheexpe ted onditionalsojourntimeandusingtheresultingexpressionobtainanexpli itand tight upperbound for theexpe ted sojourntime fun tion. Inthe experimental resultsweshow thattherelativeerrorofthelatterupperboundwithrespe ttotheexpe tedsojourntimefun tion is6-7%.

Westudy the properties of the expe ted sojourntime fun tion when the parametersof the job size distribution fun tion are sele ted in a su h a way that with the in reasing number of phasesthe varian e in reases. We show numeri allythat with the in reasing number of phases therelativeerror of thefound upper bound de reases. We also show that when thevarian eof thejobsizedistributionin reasesthegaininsystemperforman ein reasesand thesensitivityof thesystemto thesele tionoftheoptimalthresholdvaluede reases.

2 Model des ription

2.1 Main denitions

We study the Two Level Pro essor Sharing (TLPS) s heduling dis ipline with the hyper-exponentialjobsize distribution. Letusdes ribethemodelindetail.

Jobsarrivetothesystema ordingtoaPoissonpro esswithrate

λ

. Wemeasurethejobsize in timeunits. Spe i ally,asthejobsize wedenethetime whi h would bespentbytheserver totreatthejobiftherewerenootherjobsin thesystem.

Let

θ

be a given threshold. The jobs in the system that attained aservi e less than

θ

are assignedtothehighpriorityqueue. Ifinadditiontherearejobswithattainedservi egreaterthan

θ

, su hajob isseparatedinto twoparts. Therstpartofsize

θ

isassigned to thehighpriority queueandthese ondpartofsize

x − θ

waitsinthelowerpriorityqueue. Thelowpriorityqueueis servedwhenthehighpriorityqueueisempty. Bothqueuesareserveda ordingtothePro essor Sharing(PS)dis ipline.

Let usdenote the jobsize distribution by

F (x)

. By

F (x) = 1 − F (x)

wedenotethe omple-mentary distribution fun tion. The mean job size is given by

m =

R

∞

0

xdF (x)

and the system loadis

ρ = λm

. Weassumethatthesystemisstable(

ρ < 1

)andisinsteadystate.

It isknownthat manyimportantprobabilitydistributions asso iatedwithnetworktra are heavy-tailed. Inparti ular,thelesizedistributionin theInternetisheavy-tailed.

Adistributionfun tionhasaheavytailif

e

ǫx

_{(1 − F (x)) → ∞}

as

x → ∞, ∀ǫ > 0

. The heavy-taileddistributionsarenotonlyimportantandprevalent,butalsodi ulttoanalyze. Oftenitis helpful tohavetheLapla etransformofthejobsize distribution. However,thereisevidentlyno onvenientanalyti expressionfortheLapla etransformsoftheParetoandWeibulldistributions, themost ommonexamplesofheavy-taileddistributions. In[5,8℄itwasshownthatitispossible to approximate heavy-tailed distributions by hyper-exponential distribution with a signi ant numberofphases.

Ahyper-exponentialdistribution

F

N

(x)

isa onvex ombinationof

N

exponents,

1 ≤ N ≤ ∞

, namely,

F

N

(x) = 1 −

N

X

i=1

p

i

e

−

_µ

_i

_x

,

µ

i

> 0, p

i

≥

0,

i = 1, ..., N,

and

N

X

i=1

p

i

= 1.

(1)

In parti ular, we an onstru t a sequen e of hyper-exponential distributions su h that it onvergestoaheavy-taileddistribution[5℄. Forinstan e,ifwesele t

p

i

=

ν

i

γ

1

,

µ

i

=

η

i

γ

2

,

i = 1, ..., N,

γ

1

> 1,

γ

1

−

1

2

< γ

2

< γ

1

−

1,

where

ν = 1/

P

i=1,..,N

i

−

γ

1

_,

_{η = ν/m}

P

i=1,...,N

i

γ

2

−

γ

1

_,

then the rst moment of the job size distributionisnite, but these ondmomentisinnitewhen

N → ∞

. Namely,therstand the se ondmoments

m

and

d

forthehyper-exponentialdistribution aregivenby:

m =

Z

∞

0

x dF (x) =

N

X

i=1

p

i

µ

i

,

d =

Z

∞

0

x

2

dF (x) = 2

N

X

i=1

p

i

µ

2

i

.

(2) Letusdenote

F

i

θ

= p

i

e

−

µ

i

θ

_,

_{i = 1, ..., N.}

(3) Wenotethat

P

i

F

θ

i

= F (θ)

. Thehyper-exponentialdistributionhasasimpleLapla etransform:

L

_F

_(x)

(s) =

N

X

i=1

p

i

µ

i

s + µ

i

.

Wewould like to note that the hyper-exponentialdistribution has ade reasing hazard rate. In [1℄ it was shown, that when a job size distribution has a de reasing hazard rate, then with thesele tionofthethresholdtheexpe tedsojourntimeoftheTLPSsystem ouldberedu edin omparisontostandardPSsystem.

Thus, in our work we use hyper-exponential distributions to represent job size distribution fun tions. Inparti ular, theappli ation of the hyper-exponential job size distribution with two phasesismotivatedbythefa tthatintheInternet onne tionsbelongtotwodistin t lasseswith verydierentsizes oftransfer. Therst lassis omposed of shortHTTP onne tionsand P2P signaling onne tions. The se ond lass orrespondsto downloads(PDF les,MP3 les, MPEG les,et .). So,intherstpartofourpaperwelookat the aseofthehyper-exponentialjobsize distributionwithtwophasesand inthese ondpartofthepaperwestudy the aseofmorethan twophases.

2.2 The expe ted sojourn time in TLPS system

Let usdenoteby

T

T LP S

(x)

theexpe ted onditionalsojourntime intheTLPSsystemfora job ofsize

x

. Of ourse,

T

T LP S

(x)

also depends on

θ

, butforexpe ted onditional sojourntime we only emphasizethe dependen e on thejob size. On theother hand, we denote by

T (θ)

the overall expe ted sojourntime in the TLPSsystem. Here we emphasizethedependen e on

θ

as laterweshalloptimizetheoverallexpe tedsojourntimewithrespe ttothethresholdvalue.

To al ulate the expe ted sojourn time in the TLPS system we need to al ulate the time spentbyajob ofsize

x

inthersthighpriorityqueueandin these ondlowpriorityqueue. For thejobswithsize

x ≤ θ

thesystemwillbehaveasthestandardPS systemwheretheservi etime distributionistrun ated at

θ

. Letusdenoteby

X

n

θ

=

Z

θ

0

ny

n−1

F (y)dy

(4)

the

n

-thmomentof thedistributiontrun atedat

θ

. Inthefollowingse tionswewill need

X

1

θ

= m −

N

X

i=1

F

i

θ

µ

i

,

X

2

θ

= 2

N

X

i=1

p

i

µ

2

i

−

2θ

m −

N

X

i=1

F

i

θ

µ

i

!

−

2

N

X

i=1

F

i

θ

µ

2

i

.

(5)

Theutilization fa torforthetrun ateddistributionis

ρ

θ

= λ X

θ

1

= ρ − λ

N

X

i=1

F

i

θ

µ

i

.

(6)

Then,theexpe ted onditionalresponsetimeisgivenby

T

T LP S

(x) =











x

1 − ρ

θ

,

x ∈ [0, θ],

W (θ) + θ + α(x − θ)

1 − ρ

θ

,

x ∈ (θ, ∞).

A ordingto[12℄,here

(W (θ) + θ)/(1 − ρ

θ

)

expressesthetimeneededtorea hthelowpriority queue. Thistime onsistsof thetime

θ/(1 − ρ

θ

)

spentin thehighpriorityqueue,wheretheow isserveduptothe threshold

θ

, plusthetime

W (θ)/(1 − ρ

θ

)

whi h isspent waiting forthehigh priorityqueuetoempty. Here

W (θ) = λX

2

θ

/(2(1 − ρ

θ

))

.

The remaining term

α(x − θ)/(1 − ρ

θ

)

is the time spent in the low priority queue. To nd

α(x)

we anusetheinterpretationofthelowerpriorityqueueasaPSsystemwithbat h arrivals [4,14℄. Aswasshownin[12℄,

α

′

(x) = dα/dx

isthesolutionofthefollowingintegralequation

α

′

(x) = λn

Z

∞

0

α

′

(y)B(x + y)dy + λn

Z

x

0

α

′

(y)B(x − y)dy + bB(x) + 1.

(7)

Here

n = F (θ)/(1 − ρ

θ

)

is the average bat h size,

B(x) = F (θ + x)/F (θ)

is the omplementary trun ated distributionand

b = b(θ) = 2λF (θ)(W (θ) + θ)/(1 − ρ

θ

)

is theaverage numberof jobs thatarrivetothelowpriorityqueuein additiontothetaggedjob.

Theexpe tedsojourntimein thesystemisgivenbythefollowingequations:

T (θ) =

Z

∞

0

T

T LP S

(x)dF (x),

T (θ) =

X

1

θ

+ W (θ)F (θ)

1 − ρ

θ

+

1

1 − ρ

θ

T

BP S

(θ),

(8)

T

BP S

(θ) =

Z

∞

θ

α(x − θ)dF (x) =

Z

∞

0

α

′

(x)F (x + θ)dx.

(9)

3 Hyper-exponential job size distribution with two phases

3.1 Notations

Intherstpartofourworkwe onsiderthehyper-exponentialjobsize distributionwithtwo phases. Namely,a ordingto(1)the umulativedistributionfun tion

F (x)

for

N = 2

isgivenby

F (x) = 1 − p

1

e

−

µ

1

x

−

p

2

e

−

µ

2

x

,

where

p

1

+ p

2

= 1

and

p

1

, p

2

> 0

.

Themeanjob size

m

,these ondmoment

d

,theparameters

F

i

θ

,

X

1

θ

,

X

2

θ

and

ρ

θ

aredenedas inSe tion 2.1andSe tion2.2byformulas(2),(3),(5),(6)with

N = 2

.

Wenotethat thesystemhasfour freeparameters. Inparti ular, ifwex

µ

1

,

ǫ = µ

2

/µ

1

,

m

, and

ρ

,theotherparameters

µ

2

,

p

1

,

p

2

and

λ

willbefun tionsoftheformerparameters.

3.2 Expli it form for the expe ted sojourn time

Tond

T

T LP S

(x)

weneedtosolvetheintegralequation(7). Tosolve(7)weusetheLapla e transformbasedmethod des ribedin[6℄.

Theorem1. The expe tedsojourn timein the TLPSsystemwith the hyper-exponential job size distributionwith twophases isgiven by

T (θ) =

X

1

θ

+ W (θ)F (θ)

1 − ρ

θ

+

m − X

1

θ

1 − ρ

+

b(θ)



µ

1

µ

2

(m − X

_θ

1

)

2

+ δ

ρ

(θ)F

2

(θ)



2(1 − ρ)F (θ) µ

1

+ µ

2

−

γ(θ)F (θ)

,

(10) where

δ

ρ

(θ) = 1 − γ(θ)(m − X

1

θ

) = (1 − ρ)/(1 − ρ

θ

)

and

γ(θ) = λ/(1 − ρ

θ

)

. Proof. We anrewriteintegralequation(7)inthefollowingway

α

′

(x)

= γ(θ)

Z

∞

0

α

′

(y)F (x + y + θ)dy + γ(θ)

Z

x

0

α

′

(y)F (x − y + θ)dy + b(θ)B(x) + 1,

α

′

(x)

= γ(θ)

X

i=1,2

F

i

θ

e

−

_µ

_i

_x

Z

∞

0

α

′

(y)e

−

_µ

_i

_y

dy + γ(θ)

Z

x

0

α

′

(y)F (x − y + θ)dy + b(θ)B(x) + 1.

Wenotethatinthelatterequation

R

∞

0

α

′

(y)e

−

µ

i

y

_dy

,

i = 1, 2

aretheLapla etransformsof

α

′

(y)

evaluatedat

µ

i

, i = 1, 2

. Denote

L

i

=

Z

∞

0

α

′

(y)e

−

_µ

_i

_y

dy,

i = 1, 2.

Then,wehave

α

′

(x) = γ(θ)

X

i=1,2

F

i

θ

L

i

e

−

_µ

_i

_x

+ γ(θ)

Z

x

0

α

′

(y)F (x − y + θ)dy + b(θ)B(x) + 1.

Nowtaking theLapla etransformoftheaboveequationandusing the onvolutionproperty,we get

L

α

′

(s) = γ(θ)

X

i=1,2

F

i

θ

L

i

s + µ

i

+ γ(θ)

X

i=1,2

F

i

θ

L

α

′

(s)

s + µ

i

+

b(θ)

F (θ)

X

i=1,2

F

i

θ

s + µ

i

+

1

s

,

=⇒

L

α

′

(s)



1 − γ(θ)

X

i=1,2

F

i

θ

s + µ

i



 = γ(θ)

X

i=1,2

F

i

θ

L

i

s + µ

i

+

b(θ)

F (θ)

X

i=1,2

F

i

θ

s + µ

i

+

1

s

.

Here

L

α

′

(s) =

R

∞

0

α

′

(x)e

−

sx

_dx

istheLapla etransformof

α

′

(x)

. Letus notethat

L

α

′

(µ

_i

) = L

_i

, i = 1, 2

. Then,ifwesubstituteintotheaboveequation

s = µ

1

and

s = µ

2

,we an get

L

1

and

L

2

asasolutionofthelinearsystem

L

1

=

1

µ

1

+ µ

2

−

γ(θ)F (θ)

δ

ρ

(θ)



b(θ)

2F (θ)



µ

2

(m − X

_θ

1

) + δ

ρ

(θ)F (θ)



+

1

µ

1

δ

ρ

(θ)

,

L

2

=

1

µ

1

+ µ

2

−

γ(θ)F (θ)

δ

ρ

(θ)



_b(θ)

2F (θ)



µ

1

(m − X

_θ

1

) + δ

ρ

(θ)F (θ)



+

1

µ

2

δ

ρ

(θ)

.

Next weneedto al ulate

T

BP S

(θ)

.

T

BP S

(θ) =

Z

∞

0

α

′

(x)F (x + θ)dx =

Z

∞

0

α

′

(x)

X

i=1,2

F

i

θ

e

−

µ

i

x

_{dx =}

X

i=1,2

F

i

θ

L

i

,

T

BP S

(θ) =

1 − ρ

θ

1 − ρ



m − X

1

θ

+

b(θ)



µ

1

µ

2

(m − X

_θ

1

)

2

+ δ

ρ

(θ)F

2

(θ)



2F (θ) µ

1

+ µ

2

−

γ(θ)F (θ)



 .

Finally,by(8)wehave(10).

3.3 Optimal threshold approximation

Weareinterestedintheminimizationoftheexpe tedsojourntime

T (θ)

withrespe tto

θ

. Of ourse,one andierentiatetheexa tanalyti expressionprovidedinTheorem1andsettheresult of the dierentiation to zero. However, this will give atrans endental equation for the optimal valueofthethreshold.

Inorder to nd anapproximate solutionof

T

′

(θ) = dT (θ)/dθ = 0

, we shall approximate the derivative

T

′

(θ)

bysomefun tion

T

e

′

(θ)

andobtainasolutionto

T

e

′

(˜

θ

opt

) = 0

.

Sin e in the Internet onne tions belong to two distin t lasses with very dierent sizes of transfer(see Se tion 2.1), then tond theapproximationof

T

′

(θ)

we onsider aparti ular ase when

µ

2

<< µ

1

. Letusintrodu easmallparameter

ǫ

su hthat

µ

2

= ǫµ

1

,

ǫ → 0,

p

1

= 1 −

ǫ (mµ

1

−

1)

1 − ǫ

,

p

2

=

ǫ (mµ

1

−

1)

1 − ǫ

.

Wenotethatwhen

ǫ → 0

these ondmomentofthejobsizedistribution goesto innity. Wethenverifythat

θ

˜

opt

indeed onvergestotheminimumof

T (θ)

when

ǫ → 0

. Lemma2. Thefollowing inequalityholds:

µ

1

ρ > λ

.

Proof. Sin e

p

1

> 0

and

p

2

> 0

, wehave the following inequality

mµ

1

> 1

. Then,

m >

1

µ

1

. Takingintoa ountthat

λm = ρ

weget

ρ

λ

>

1

µ

1

. Consequently,wehavethat

µ

1

ρ > λ

. Proposition 3. Thederivative of

T (θ)

anbeapproximatedbythe followingfun tion:

e

T

′

(θ) = −e

−

_µ

1

θ

_µ

1

c

1

+ e

−

_µ

2

θ

_µ

2

c

2

,

where

c

1

=

λ(mµ

1

−

1)

µ

1

(µ

1

−

λ)(1 − ρ)

,

c

2

=

λ(mµ

1

−

1)

(µ

1

−

λ)

2

.

(11) Namely,

T

′

(θ) − e

T

′

(θ) = O(µ

2

/µ

1

).

Proof. Using theanalyti al expression for both

T

′

(θ)

and

e

T

′

(θ)

, we get the Taylorseries for

T

′

(θ) − e

T

′

(θ)

withrespe tto

ǫ

,whi hshowsthatindeed

T

′

(θ) − e

T

′

(θ) = O(ǫ).

Thuswehavefoundanapproximationofthederivativeof

T (θ)

. Nowwe anndan approxi-mationoftheoptimalthresholdbysolvingtheequation

e

T

′

(θ) = 0

.

Theorem 4. Let

θ

opt

denote the optimal value of the threshold. Namely,

θ

opt

=

arg

min T (θ)

. The value

θ

˜

opt

given by

˜

θ

opt

=

1

µ

1

−

µ

2

ln



_(µ

1

−

λ)

µ

2

(1 − ρ)



approximates

θ

opt

sothat

T

′

(˜

θ

opt

) =

o

(µ

2

/µ

1

)

. Proof. Solvingtheequation

e

T

′

(θ) = 0,

wegetananalyti expressionfortheapproximationoftheoptimalthreshold:

e

θ

opt

= −

1

µ

1

(1 − ǫ)

ln



ǫ

µ

1

(1 − ρ)

(µ

1

−

λ)



=

1

µ

1

−

µ

2

ln



_(µ

1

−

λ)

µ

2

(1 − ρ)



.

Letus show that the abovethreshold approximationis greater thanzero. Wehaveto show that

(µ

1

−

λ)

µ

2

(1−ρ)

> 1

. Sin e

µ

1

> µ

2

and

µ

1

ρ > λ

(seeLemma 2),wehave

µ

1

> µ

2

=⇒

µ

1

(1 − ρ) > µ

2

(1 − ρ)

=⇒

λ < µ

1

ρ < µ

1

−

µ

2

(1 − ρ)

=⇒

(µ

1

−

λ) > µ

2

(1 − ρ).

Expanding

T

′

(e

θ

opt

)

asapowerserieswithrespe tto

ǫ

gives:

T

′

(e

θ

opt

) = ǫ

2

(const

0

+ const

1

ln ǫ + const

2

ln

2

ǫ),

where

const

i

, i = 1, 2

aresome onstantvalues 1

withrespe tto

ǫ

. Thus,

T

′

(e

θ

opt

) = o(ǫ) = o(µ

2

/µ

1

),

whi h ompletestheproof.

Inthenextpropositionwe hara terizethelimitingbehaviorof

T (θ

opt

)

and

T (e

θ

opt

)

as

ǫ → 0

. Inparti ular,weshowthat

T (e

θ

opt

)

tendstotheexa tminimumof

T (θ)

when

ǫ → 0

.

Proposition 5.

lim

ǫ→0

T (θ

opt

) = lim

ǫ→0

T (e

θ

opt

) =

m

1 − ρ

−

c

1

,

where

c

1

isgiven by(11 ). 1

Proof. Wendthefollowinglimit,when

ǫ → 0

:

lim

ǫ→0

T (θ) =

m

1 − ρ

−

λ(mµ

1

−

1)

µ

1

(µ

1

−

λ)(1 − ρ)

+

λ(mµ

1

−

1)e

−

µ

1

θ

µ

1

(µ

1

−

λ)(1 − ρ)

,

lim

ǫ→0

T (θ) =

m

1 − ρ

−

c

1

+ c

1

e

−

µ

1

θ

,

where

c

1

is given by (11). Sin e the fun tion

lim

ǫ→0

T (θ)

is ade reasing fun tion, the optimal thresholdforitis

θ

opt

= ∞

. Thus,

lim

ǫ→0

T (θ

opt

) = lim

θ→∞

lim

ǫ→0

T (θ) =

m

1 − ρ

−

c

1

.

Ontheotherhand,weobtain

lim

ǫ→0

T (e

θ

opt

) =

m

1 − ρ

−

c

1

,

whi hprovestheproposition.

3.4 Experimental results

InFigure1-2weshowtheplotsforthefollowingparameters:

ρ = 10/11

(defaultvalue),

m = 20/11

,

µ

1

= 1

,

µ

2

= 1/10

,so

λ = 1/2

and

ǫ = µ

2

/µ

1

= 1/10

. Then,

p

1

= 10/11

and

p

2

= 1/11

. In Figure 1 we plot

T (θ)

,

T

P S

and

T (e

θ

opt

)

. We note, that the expe ted sojourn time in thestandardPS system

T

P S

is equalto

T (0)

. Weobservethat

T (e

θ

opt

)

orresponds well to the optimumeventhough

ǫ = 1/10

isnottoosmall.

LetusnowstudythegainthatweobtainusingTLPS,bysetting

θ = e

θ

opt

,in omparisonwith thestandardPS.Tothisend, weplot theratio

g(ρ) =

T

P S

−

T

(e

θ

opt

)

T

P S

in Figure2. Thegainin the system performan e with TLPS in omparison with PS strongly depends on

ρ

, the loadof the system. One ansee, thatthegainoftheTLPSsystemwithrespe ttothestandardPS system goesupto 45%whentheloadofthesystemin reases.

Tostudythesensitivityof theTLPSsystemwith respe tto

θ

, wendthegainoftheTLPS system with respe t to the standardPS system, we plot in Figure 2

g

1

(ρ) =

T

P S

−

T

(

3

2

θ

e

opt

)

T

P S

and

g

2

(ρ) =

T

P S

−

T

(

1

2

θ

e

opt

)

T

P S

. Thus,evenwiththe50%errorofthe

θ

e

opt

value,thesystemperforman eis losetooptimal.

One anseethatitisbene ialtouseTLPSinsteadofPSinthe aseofheavyandmoderately heavyloads. Wealsoobservethat theoptimalTLPSsystemisnottoosensitiveto the hoi eof thethresholdnearitsoptimalvalue,whenthejobsizedistribution ishyper-exponentialwithtwo phases. Nevertheless,itisbetterto hooselargerratherthansmallervaluesofthethreshold.

0

5

10

15

20

25

30

35

40

0

2

4

6

8

10

12

14

16

18

20

θ

T(

θ

)

T(

θ

_opt

)

T

PS

Figure 1:

T (θ)

- solid line,

T

P S

(θ)

-dash dot line,

T (e

θ

opt

)

-dashline

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

ρ

g(

ρ

)

g

1

(

ρ

)

g

2

(

ρ

)

Figure 2:

g(ρ)

- solid line,

g

1

(ρ)

- dash line,

g

2

(ρ)

-dashdotline

4 Hyper-exponential job size distribution with more than two phases

4.1 Notations

In the se ond part of the presented work we analyze the TLPS dis ipline with the hyper-exponentialjobsizedistributionwithmorethantwophases. Aswasshownin[5,7,8℄,the hyper-exponentialdistributionwithasigni antnumberofphasesmodelswellthele sizedistribution in the Internet. Thus, in this se tion asthe job size distribution we takethehyper-exponential fun tion withmanyphases. Namely,a ordingto (1),

F (x) = 1 −

N

X

i=1

p

i

e

−

µ

i

x

,

N

X

i=1

p

i

= 1, µ

i

> 0, p

i

≥

0, i = 1, ..., N, 1 < N ≤ ∞.

Inthefollowingweshallwrite simply

P

i

insteadof

P

N

i=1

. The meanjob size

m

, the se ond moment

d

, theparameters

F

i

θ

,

X

1

θ

,

X

2

θ

and

ρ

θ

are dened as in Se tion 2.1 and Se tion 2.2 by formulas (2),(3),(5 ), (6) for any

1 ≤ N ≤ ∞

. The formulas presented in Se tion 2.2 anstill beused to al ulate

b(θ)

,

B(x)

,

W (θ)

,

γ(θ)

,

δ

ρ

(θ)

,

T

T LP S

(x)

,

T (θ)

.

Weshallalsoneedthefollowingoperatornotations:

Φ

1

(β(x))

=

γ(θ)

Z

∞

0

β(y)F (x + y + θ)dy + γ(θ)

Z

x

0

β(y)F (x − y + θ)dy,

(12)

Φ

2

(β(x))

=

Z

∞

0

β(y)F (y + θ)dy

(13)

foranyfun tion

β(x)

. Inparti ular, forsomegiven onstant ,wehave

Φ

1

(c)

=

c γ(θ)(m − X

_θ

1

) = c q,

(14)

Φ

2

(c)

=

c (m − X

_θ

1

),

(15) where

q = γ(θ)(m − X

1

θ

) =

λ(m − X

1

θ

)

1 − ρ

θ

=

ρ − ρ

θ

1 − ρ

θ

< 1.

(16)

Theintegralequation(7) annowberewrittenintheform

α

′

(x) = Φ

1

(α

′

(y))+

b(θ)

F (θ)

F (x + θ) + 1.

(17)

andequation(9)for

T

BP S

(θ)

takestheform

T

BP S

(θ) = Φ

2

(α

′

(x)).

(18)

4.2 Linear system based solution

Similarlyto therstpartoftheproofofTheorem 1,weobtainthefollowingproposition.

Proposition 6.

T

BP S

(θ) =

X

i

F

i

θ

L

i

,

with

L

i

= L

∗

i

+

1

δ

ρ

(θ)µ

i

,

wherethe

L

∗

i

arethe solutionof the linear system

L

∗

p

1 − γ(θ)

X

i

F

i

θ

λ

p

+ µ

i

!

= γ(θ)

X

i

F

i

θ

L

∗

i

λ

p

+ µ

i

+

b(θ)

F (θ)

X

i

F

i

θ

λ

p

+ µ

i

,

p = 1, ..., N.

(19)

Unfortunately,thesystem(19)doesnotseemtohaveatra tableniteformanalyti solution. Therefore, in the ensuing subse tions we proposed an alternativesolution basedon anoperator seriesand onstru tatightupperbound.

4.3 Operator series form for the expe ted sojourn time

Sin etheoperator

Φ

1

isa ontra tion[3,4℄,we aniterateequation(17)startingfromsome initial point

α

′

0

. The initial point ould besimply a onstant. As shownin [3, 4℄the iterations will onvergetotheuniquesolutionof(17). Spe i ally,wemakeiterationsin thefollowingway:

α

′

n+1

(x) = Φ

1

(α

′

n

(x))+

b(θ)

F (θ)

F (x + θ) + 1,

n = 0, 1, 2, ...

(20)

Ateveryiterationstepwe onstru tthefollowingapproximationof

T

BP S

(θ)

a ordingto(18):

T

BP S

n+1

(θ) = Φ

2

(α

′

n+1

(x)).

(21) Using(20)and(21),we an onstru ttheoperator seriesexpressionfortheexpe tedsojourn timeintheTLPSsystem.

Theorem7. Theexpe tedsojourntime

T (θ)

in theTLPSsystemwiththe hyper-exponentialjob sizedistributionisgiven by

T (θ) =

X

1

θ

+ W (θ)F (θ)

1 − ρ

θ

+

m − X

1

θ

1 − ρ

+

b(θ)

F (θ)(1 − ρ

θ

)

∞

X

i=0

Φ

2

Φ

i

1

(F (x + θ))

!

.

(22)

Proof. From(20)wehave

α

′

n

= q

n

α

′

0

+

n−1

X

i=1

q

i

₊

b(θ)

F (θ)

n−1

_X

i=1

Φ

i

1

(F (x + θ)) +

b(θ)

F (θ)

F (x + θ) + 1,

andthenfrom(21)and(14)itfollows,that

T

BP S

n

(θ) = (m − X

θ

1

)

q

n

α

′

0

+

n−1

_X

i=0

q

i

!

+

b(θ)

F (θ)

Φ

2

n−1

_X

i=0

Φ

i

1

(F (x + θ))

!!

.

Usingthefa ts(see(16)):

1. q < ρ < 1 =⇒ q

n

→

₀

as

n → ∞,

2.

∞

X

i=0

q

i

₌

1

1 − q

=

1 − ρ

θ

1 − ρ

,

we on ludethat

T

BP S

(θ) = lim

n→∞

T

BP S

n

(θ) = (m − X

θ

1

)



1 − ρ

θ

1 − ρ



+

b(θ)

F (θ)

∞

X

i=0

Φ

2

Φ

i

1

(F (x + θ))

!

.

Finally,using(8)weobtain(22).

Theresultingformula(22)isdi ulttoanalyzeanddoesnothavea learanalyti expression. Usingthis resultinthe nextsubse tionwend anapproximation,whi h isalsoanupperbound, oftheexpe tedsojourntimefun tionin amoreexpli itform.

4.4 Upper bound for the expe ted sojourn time

Letusstartwithauxiliaryresults.

Lemma8. For anyfun tion

β(x) ≥ 0

with

β

j

=

R

∞

0

β(x)e

−

_µ

_i

_x

dx

, if

d(β

j

µ

j

)

dµ

j

≥

0,

j = 1, ..., N

itfollows, that

Φ

2

(Φ

1

(β(x))) ≤ qΦ

2

(β(x)) .

Proof. SeeAppendix.

Lemma 9. For the TLPS system with the hyper-exponential job size distribution the following statementholds:

Φ

2

(Φ

1

(α

′

(x))) ≤ qΦ

2

(α

′

(x)) .

(23) Proof. We dene

α

′

j

=

R

∞

0

α

′

(x)e

−

µ

j

x

_{dx, j = 1, ...N}

. As was shown in [14℄,

α

′

(x)

has the followingstru ture:

α

′

(x) = a

0

+

X

k

a

k

e

−

_b

_k

_x

,

a

0

≥

0, a

k

≥

0, b

k

> 0,

k = 1, ..., N.

Then,wehavethat

α

′

(x) ≥ 0

and

α

′

j

=

a

0

µ

j

+

X

k

a

k

b

k

+ µ

j

,

j = 1, ..., N,

=⇒

d(α

′

j

µ

j

)

dµ

j

=

X

k

a

k

b

k

+ µ

j

−

X

k

a

k

µ

j

(b

k

+ µ

j

)

2

=

X

k

a

k

b

k

(b

k

+ µ

j

)

2

≥

0,

j = 1, ..., N,

LetusstatethefollowingTheorem:

Theorem10. An upper boundfor the expe tedsojourn time

T (θ)

in the TLPSsystemwith the hyper-exponential jobsizedistributionfun tion withmany phasesisgiven by

Υ(θ)

:

T (θ) ≤ Υ(θ) =

X

1

θ

+ W (θ)F (θ)

1 − ρ

θ

+

m − X

1

θ

1 − ρ

+

b(θ)

F (θ)(1 − ρ)

X

i,j

F

i

θ

F

j

θ

µ

i

+ µ

j

.

(24)

Proof. A ordingtothere ursion(20)wehavefor

α

′

n

(x)

weapproximate

α

′

(x)

withthefun tion

e

α

′

(x)

,whi hsatisesthefollowingequation:

e

α

′

(x) = e

α

′

(x)Φ

1

(1) +

b(θ)

F (θ)

F (x + θ) + 1.

Then,a ordingto(14)we anndtheanalyti alexpressionfor

α

e

′

(x)

:

e

α

′

(x) = q e

α

′

(x) +

b(θ)

F (θ)

F (x + θ) + 1,

=⇒

α

e

′

(x) =

1

1 − q



b(θ)

F (θ)

F (x + θ) + 1



.

Wetake

Υ

BP S

(θ) = Φ

2

(e

α

′

(x))

asanapproximationfor

T

BP S

(θ) = Φ

2

(α

′

(x))

. Then

Υ

BP S

(θ) = Φ

2

(e

α

′

(x)) =

(m − X

1

θ

)

1 − q

+

b(θ)

F (θ)

Φ

2

(F (x + θ)) =

(m − X

1

θ

)

1 − q

+

b(θ)

F (θ)

X

i,j

F

i

θ

F

j

θ

µ

i

+ µ

j

.

Letusprove,that

T

BP S

(θ) ≤ Υ

BP S

(θ),

orequivalently

T

BP S

(θ) − Υ

BP S

(θ) = Φ

2

(α

′

(x)) − Φ

2

(e

α

′

(x)) ≤ 0.

Letuslook at

Φ

2

(α

′

(x)) − Φ

2

(e

α

′

(x)) =

= Φ

2

(Φ

1

(α

′

(x))) + Φ

2



b(θ)

F (θ)

F (x + θ) + 1



−



qΦ

2

(e

α

′

(x)) + Φ

2



b(θ)

F (θ)

F (x + θ) + 1



= Φ

2

(Φ

1

(α

′

(x))) − qΦ

2

(α

′

(x)) + q (Φ

2

(α

′

(x)) − Φ

2

(e

α

′

(x)))

=⇒

Φ

2

(α

′

(x)) − Φ

2

(e

α

′

(x)) =

1

1 − q

(Φ

2

(Φ

1

(α

′

(x))) − qΦ

2

(α

′

(x))) .

Andfrom Lemma9andformula(8)we on ludethat(24)is true.

Inthissubse tionwefoundtheanalyti alexpressionoftheupperboundoftheexpe tedsojourn timeinthe asewhenthejobsizedistribution isahyper-exponentialfun tion withmanyphases. In theexperimental results of thefollowingsubse tion we show that theobtained upperbound isalso a loseapproximation. Theanalyti expressionof theupperbound whi h we obtainedis more lear and easier to analyzethen the expression of the expe ted sojourntime. It ould be usedinthefuture resear honTLPSmodel.

4.5 Experimental results

We al ulate

T (θ)

and

Υ(θ)

for dierent numbers of phases

N

of the job size distribution fun tion. Wetake

N = 10, 100, 500, 1000

. To al ulate

T (θ)

wendthenumeri alsolutionofthe systemof linearequations(19)using theGauss method. Then usingthe resultofProposition 6 wend

T (θ)

. For

Υ(θ)

weuseequation(24).

As wasmentionedin Subse tion 2.1, by usingthe hyper-exponentialdistribution with many phases,one anapproximateaheavy-taileddistribution. Inournumeri alexperiments,wex

ρ

,

m

,andsele t

p

i

and

µ

i

inasu haway,thatbyin reasingthenumberofphasesweletthese ond moment

d

(see(2))in reaseaswell. Here wetake

ρ = 10/11, λ = 0.5, p

i

=

ν

i

2.5

, µ

i

=

η

i

1.2

,

i = 1, ..., N.

Inparti ular,wehave

X

i

p

i

= 1,

=⇒ ν =

1

P

i

i

−

2.5

,

X

i

p

i

µ

i

= m, =⇒ η =

ν

m

X

i

i

−

1.3

_.

InFigure3one anseetheplotsoftheexpe tedsojourntimeanditsupperboundasfun tions of

θ

when

N

variesfrom10upto 1000. InFigure4weplottherelativeerroroftheupperbound

∆(θ) =

Υ(θ) − T (θ)

T (θ)

,

when

N

variesfrom10upto 1000. As one ansee, theupperbound(24)isverytight.

Wendthemaximumgainoftheexpe tedsojourntime oftheTLPSsystemwithrespe tto thestandardPS system. Thegainisgivenby

g(θ) =

T

P S

−

T

(θ)

T

P S

. Here

T

P S

isanexpe tedsojourn timeinthestandardPSsystem. Letusnoti e,that

T

P S

= T (0)

. ThedataandresultsaresummarizedinTable1.

N

η

d

θ

opt

max

θ

g(θ)

max

θ

∆(θ)

10 0.95 7.20 5 32.98

%

0.0640 100 1.26 32.28 12 45.75

%

0.0807 500 1.40 113.31 21 49.26

%

0.0766 1000 1.44 200.04 26 50.12

%

0.0743

Table1: In reasingthenumberofphases

Withthein reasingnumberofphasesweobservethat 1. these ond moment

d

in reases;

2. themaximumgain

max

θ

g(θ)

in expe tedsojourntimein omparisonwithPS in reases; 3. therelativeerroroftheupperbound

∆(θ)

withtheexpe ted sojourntimede reasesafter thenumberofphasesbe omessu ientlylarge;

4. the sensitivityof thesystemperforman e withrespe t to thesele tionof thesub-optimal thresholdvaluede reases.

Thus theTLPS systemprodu esbetterand morerobustperforman e asthe varian eof the jobsize distributionin reases.

0

10

20

30

40

50

60

70

80

90

100

8

10

12

14

16

18

20

22

24

θ

N=10

N=100

N=500

N=1000

T

PS

ϒ

(

θ

)

T(

θ

)

Figure3: Theexpe tedsojourntime

T (θ)

and its upper bound

Υ(θ))

for

N = 10

,

100

,

500

,

1000

0

10

20

30

40

50

60

70

80

90

100

0%

1%

2%

3%

4%

5%

6%

7%

8%

9%

θ

N=10

N=100

N=500

N=1000

∆

(

θ

)

Figure 4: The relative error

∆(θ) = (T (θ) − Υ(θ))/T (θ)

for

N = 10

,

100

,

500

,

1000

5 Con lusion

WeanalyzetheTLPSs hedulingme hanismwiththehyper-exponentialjobsizedistribution fun tion.

InSe tion3weanalyzethesystemwhenthejobsizedistributionfun tionhastwophasesand ndtheanalyti alexpressionsoftheexpe ted onditionalsojourntimeandtheexpe tedsojourn timeoftheTLPSsystem.

Conne tions in the Internet belong to two distin t lasses: short HTTP and P2P signaling onne tionsand longdownloadssu has: PDF, MP3, and so on. Thus, a ordingto this obser-vation,we onsideraspe ialsele tionoftheparametersofthejobsizedistributionfun tion with twophasesandndtheapproximationoftheoptimalthreshold,whenthevarian eofthejobsize distributiongoesto innity.

Weshow,thattheapproximatedvalueofthethresholdtendstotheoptimalthreshold,when the se ond momentof the distribution fun tion goes to innity. We foundthat the gainof the TLPSsystem omparedtothestandardPSsystem ouldrea h45%whentheloadofthesystem in reases. Alsothesystemisnottoosensitivetothesele tionoftheoptimalvalueofthethreshold. In Se tion 4 we have studied the TLPS model when the job size distribution is a hyper-exponential fun tion with many phases. We provide an expression of the expe ted onditional sojourntimeasasolutionofthesystemoflinearequations. Alsoweapplytheiterationmethod tondtheexpressionoftheexpe ted onditionalsojourntimeintheformofoperatorseriesand usingtheobtainedexpressionweprovideanupperboundfortheexpe tedsojourntimefun tion. With theexperimental resultsweshowthat theupperbound isverytightand ouldbeusedas anapproximationoftheexpe tedsojourntimefun tion. Weshownumeri ally, thattherelative errorbetweentheupperboundandexpe tedsojourntimefun tion de reaseswhenthevariation ofthejobsizedistributionfun tionin reases. Theobtainedupperbound ouldbeusedtoidentify an approximationof the optimalvalueof the optimalthreshold for TLPSsystem when thejob sizedistributionisheavy-tailed.

Westudy the properties of the expe ted sojourntime fun tion, when the parametersof the job size distribution fun tion are sele ted in su h a way, that it approximates a heavy-tailed distribution as the number of phases of the job size distribution in reases. As the number of phasesin reasesthe gainof theTLPSsystem omparedwith the standardPS systemin reases andthesensitivityofthesystemwithrespe ttothesele tionoftheoptimalthresholdde reases.

6 Appendix: Proof of Lemma 8

Letustakeanyfun tion

β(x) > 0

anddene

β

j

=

R

∞

0

β(x)e

−

µ

j

x

_dx,

_{j = 1, ..., N.}

Letusshow for

β(x) ≥ 0

thatif

d(β

j

µ

j

)

dµ

j

≥

0,

j = 1, ..., N,

thenitfollowsthat

Φ

2

(Φ

1

(β(x))) ≤ qΦ

2

(β(x)) .

As

Z

∞

0

Z

x

0

β(y)F (x − y + θ)F (x + θ)dydx =

Z

∞

0

Z

∞

0

β(y)F (x

1

+ θ)F (x

1

+ y + θ)dx

1

dy

and

Φ

2

(Φ

1

(β(x)))

=

γ(θ)

Z

∞

0

Z

∞

0

β(y)F (x + y + θ)F (x + θ)dydx

+γ(θ)

Z

∞

0

Z

x

0

β(y)F (x − y + θ)F (x + θ)dydx,

then

Φ

2

(Φ

1

(β(x)))

= 2γ(θ)

Z

∞

0

Z

∞

0

β(x)F (x + θ)F (x + y + θ)dydx =

= 2γ(θ)

Z

∞

0

β(x)

X

i,j

F

i

θ

F

j

θ

µ

i

+ µ

j

e

−

µ

j

x

_{dx = 2γ(θ)}

X

i,j

F

i

θ

F

j

θ

µ

i

+ µ

j

β

j

.

Alsofor

Φ

2

(β(x))

,takingintoa ountthat

q = γ(θ)

P

i

F

i

θ

µ

i

,weobtain

qΦ

2

(β(x))

= γ(θ)

X

i

F

i

θ

µ

i

X

j

F

_θ

j

Z

∞

0

β(x)e

−

µ

j

x

_{dx = γ(θ)}

X

i,j

F

i

θ

F

j

θ

µ

i

β

j

.

Thus,asu ient onditionfortheinequality

Φ

2

(Φ

1

(β(x))) ≤ qΦ

2

(β(x))

tobesatisedisthat foreverypair

i, j

:

2

µ

i

+ µ

j

β

j

+

2

µ

j

+ µ

i

β

i

≤

1

µ

i

β

j

+

1

µ

j

β

i

⇐⇒

−

(β

j

µ

j

−

β

i

µ

i

)(µ

j

−

µ

i

) ≤ 0.

Theinequalityisindeedsatisedwhen

β

j

µ

j

isanin reasingfun tion of

µ

j

. We on ludethat

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Contents

1 Introdu tion 3

2 Model des ription 4

2.1 Maindenitions. . . 4

2.2 Theexpe tedsojourntimein TLPSsystem . . . 5

3 Hyper-exponentialjob size distributionwith twophases 7 3.1 Notations . . . 7

3.2 Expli itformfortheexpe tedsojourntime . . . 7

3.3 Optimalthresholdapproximation . . . 8

3.4 Experimental results . . . 10

4 Hyper-exponentialjob size distributionwith more thantwo phases 11 4.1 Notations . . . 11

4.2 Linearsystembasedsolution . . . 12

4.3 Operatorseriesform fortheexpe tedsojourntime . . . 12

4.4 Upperbound fortheexpe tedsojourntime . . . 13

4.5 Experimental results . . . 15

5 Con lusion 16

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