A Flexible Bandwidth Reservation Framework for Bulk Data Transfers in Grid Networks

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Data Transfers in Grid Networks

Bin Bin Chen, Pascale Primet

To cite this version:

Bin Bin Chen, Pascale Primet. A Flexible Bandwidth Reservation Framework for Bulk Data Transfers

in Grid Networks. [Research Report] 2006. �inria-00078069�

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inria-00078069, version 1 - 12 Jun 2006

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Thème NUM

A Flexible Bandwidth Reservation Framework for

Bulk Data Transfers in Grid Networks

Bin Bin Chen — Pascale Primet

N° ????

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Bin BinChen , Pas ale Primet

ThèmeNUMSystèmesnumériques

ProjetsRESO

Rapportdere her he n°???? June200626pages

Abstra t: Ingridnetworks,distributedresour esareinter onne tedbywideareanetwork

to support ompute and data-intensive appli ations, whi h require reliable and e ient

transfer of gigabits (even terabits) of data. Dierent from best-eort tra in Internet,

bulkdatatransferingridrequiresbandwidthreservationasafundamentalservi e. Existing

reservations hemessu hasRSVParedesignedforreal-timetra spe iedbyreservation

rate,transferstart timebut with unknownlifetime. In omparison, bulkdata transfer

re-questsare dened in terms of volume and deadline, whi h provide moreinformation, and

allowmoreexibilityin reservations hemes,i.e.,transferstarttime anbeexibly hosen,

andreservationforasinglerequest anbedividedintomultipleintervalswithdierent

reser-vation rates. We dene aexible reservation framework using time-ratefun tion algebra,

andidentify aseriesof pra ti alreservation s hemefamilieswithin reasinggeneralityand

potential performan e, namely, FixTime-FixRate, FixTime-FlexRate, FlexTime-FlexRate,

and Multi-Interval. Simple heuristi s are used to sele t representative s heme from ea h

family for performan e omparison. Simulation results show that the in reasing

exibil-ity anpotentiallyimprovesystemperforman e,minimizingbothblo kingprobabilityand

meanowtime. Wealsodis ussthedistributed implementationofproposedframework.

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Résumé: Danslesréseauxdegrilles,lesressour esdistribuéessontinter onne téespardes

réseauxlonguesdistan epourexé uterdesappli ationsintensivesde al uloudetraitement

dedonnées,quiné essitentdestransfertsablesete a esdevolumesdedonnéesdel'ordre

deplusieursgigao tetsoutero tets. Letransfertsmassifsdanslesgrilles, ontrairementau

tra besteortdel'Internet,requierentunservi ederéservationdebande-passante. Les

s hémasderéservationexistants,telsRSVP,ontété onçuspourdutra temps-réeletpour

lequelonspé ieundébitréservé,une datededébutdetransfertmaisonnepré isepasla

durée. En omparaison, lestransferts massifs de grillessont dénisen termes de volumes

et de date limite, e qui ore plus d'informations et autorise des s hémas de réservation

plus exibles. Le début ee tif du transfert peut être hoisi, une réservation pour une

même requête peut être divisés en plusieurs intervals ave des débits réservés diérents.

Nous dénissonsun adre exiblede réservation de bande passante àl'aide d'unealgèbre

defon tionstemps-débitet identionsunesérie defamilles des hémasderéservation,que

nousnommonsFixTime-FixRate,FixTime-FlexRate,FlexTimeFlexRate,etMulti-Interval,

présentantunegénéralitéetunpotentieldeperforman e roissants. Desheuristiquessimples

sontutiliséespourséle tionneruns hémareprésentatifdans haquefamillepour omparer

lesperforman es. Lesrésultatsde simulationmontrentquel'augmentationdelaexibilité

peut potentiellement augmenter lesperforman es dusystème, minimiser laprobabilité de

blo ageet laduréemoyennedesux. Nousdis utonsaussidel'implantationdistribuée du

adreproposé.

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1 Introdu tion

Grid omputingisapromisingte hnologythatbringstogetherlarge olle tionof

geographi- allydistributedresour es(e.g., omputing,storage,visualization,et .) tobuildaveryhigh

performan e omputingenvironmentfor omputeand data-intensiveappli ations[7℄. Grid

networks onne t multiple sites,ea h omprisinga numberof pro essors,storagesystems,

databases, s ienti instruments, and et . In grid appli ations, like experimental

analy-sisandsimulationsin high-energyphysi s, limatemodeling,earthquakeengineering,drug

design,andastronomy,massivedatasetsmustbesharedbya ommunityofresear hers

dis-tributed in dierent sites. These resear herstransfer largesubsets ofdata a rossnetwork

for pro essing. Thevolume of dataset an usually bedetermined from task spe i ation,

and astri t deadline is often spe ied to guaranteein-time ompletion ofthe whole task,

also to enfor e e ientuse of expensive grid resour es,not only network bandwidth, but

alsothe o-allo atedCPUs,disks,and et .

WhileInternetbulkdatatransferworkswell withbest-eortservi e,high-performan e

grid appli ations require bandwidth reservation for bulk data transfer as a fundamental

servi e. Besides stri t deadline requirement and expensive o-allo ated resour es as we

dis ussedabove, thesmaller multiplexing level of gridnetworks omparedto Internet also

serves as a main driving for e for bandwidth reservation. In Internet, the sour e a ess

rates are generally mu h smaller (

2M bps

for DSL lines) than the ba kbonelink apa ity (hundredsto thousands of Mbps, say). Coexisten e of many a tive owsin a single link

smoothes thevariationof arrivaldemandsdue to thelawof largenumber,and thelink is

not abottlene k until demand attains above

90%

of its apa ity [13℄. Thus no proa tive admission ontrol isused in Internetfor bulkdatatransfer. Instead,distributed transport

proto ols, su h asTCP,are usedto statisti allyshare available bandwidthamongowsin

a fair way. Contrarily, in grid ontext, the apa ity of a single sour e (

c = 1Gbps

) is omparableto the apa ityof bottlene k link. Forasystemwith smallmultiplexing level,

ifno pro-a tiveadmission ontrol is applied, burst of loadgreatlydeterioratesthesystem

performan e.

A on rete example is given in Se tion 2 to demonstrate the importan e of resour e

reservationforgridnetworks. Throughtheexample,wealsoshowthatexistingRSVP-type

frameworkis notexible enoughforbulkdatatransferreservation. InSe tion3, wedene

a exible reservation framework using time-rate fun tion algebra. Se tion 4 identies a

seriesofpra ti alreservations hemefamilieswith in reasinggenerality,andweusesimple

heuristi sto sele trepresentatives heme from ea h family. InSe tion 5,simulationresult

of hosen s hemes are presented and the impa t of exibility is analyzed. A distributed

ar hite ture is proposed in Se tion 6. In Se tion 7, we briey review related works on

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2 Motivation

InFigure 1,wesimulate asingle link with apa ity

C

. Bulkdata transferrequests arrive a ordingtoaPoissonpro esswithparameter

λ

. Requestvolumeisindependentofarrival time, and follows an exponential distribution with parameter

µ

. Simulations with other arrival pro essesand tra volume distributiones reveal similar trend, whi h are not

pre-sented herefor brevity. Load

ρ = λ/(C ∗ µ)

. Requestshavemaximaltransferrate

R

max

. InInternet setting

R

Internet

max

= C/100

, and in gridsetting

R

grid

max

= C/10

. Ideal transport

proto ol is assumed, so that if there are no more than

C/R

max

a tive ows, all of them transferat full rate

R

max

. If there are

n > C/R

max

a tiveows, theyalltransferat rate

max

/2

for all requests). In Internet-NoAC setting (AC standsfor AdmissionControl), thefail probabilityis low until load

ρ

attains above

95%

. In grid-NoAC setting, however, the fail probability is nonnegle table even under a mediumload,anditdeterioratesrapidlyasloadin reases. Thuswe onsiderusingasimple

reservation s heme, whi h enfor es requeststo reserve

R

min

bandwidthwhen they arrive, sothat alla eptedrequestsareguaranteedto ompletebefore deadline(failprobabilityis

0

). Requests are blo ked if the numberof a tive reservations rea hes

C/R

exp

. This kind ofreservation anbesupportedbyexisting reservations hemes,forexample,RSVP[3℄. In

grid-AC setting, westill assumeideal transportproto ol, i.e., a eptedrequests areableto

fairlyshareunreserved apa ityin additiontotheirreservedbandwidth. Blo kprobability

ofgrid-AC settingismu hlowerthanfailprobabilityofgrid-NoAC setting.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.5

0.6

0.7

0.8

0.9

1 blocking / fail probability

load

grid-NoAC

grid-AC

Internet-NoAC

M/M/m/m

Figure1: Fail/blo kprobabilityunderdierentmultiplexinglevel

In Figure 1, we also plot a variation of grid-AC setting in whi h ows an only use

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mod-eledasa standard

M/M/m/m

queuing systemwith

m = C/R

min

= 10

. Comparingthis

M/M/m/m

settingagainstgrid-NoAC setting, simplereservation s hemewith dull

trans-portproto ol anstilloutperformnoadmission ontrolsettingwithidealtransportproto ol

whenloadisrelativelyhigh. Thisagaindemonstratesthebenetofreservation. Meanwhile,

thebigperforman egapbetween

M/M/m/m

settingandgrid-AC settingshowsthatwhen transportproto olis dull,aRSVP-typereservation doesnotfullyexploit thesystem's

a-pa ity. Thetransport proto ol design forhigh speednetwork is still an ongoingresear h.

Complementarily,we onsiderhowto improvesystem'sperforman ebyusingmoreexible

reservations hemesinthispaper.

RSVP is designed forreal-time tra whi h normally requests for aspe iedvalue of

bandwidthfrom axedstarttime. Their lifetimeis unknown, thus reservation remainsin

ee tforanindenitedurationuntilexpli itTeardownsignalisissuedorsoftstateexpires.

Instead,bulkdatatransferrequestsarespe iedbyvolume anddeadline. Thisallowsmore

exibility in the designof reservation s hemes. As volume is known, the ompletion time

anbe al ulated by s hedulerand keptin time-indexedreservationstates. Ifthere is not

enoughbandwidth at themoment arequest arrives, transfer anbes heduledto start at

somefuture timepointaslongasit an omplete before deadline. Bandwidth reservation

analso omprisesub-intervalswithdierentreservedrates.

0 Time

(h)

Rate

(Tbph)

1

2

3

4

5

6

7 Link

capacity C

f

₁

f

₂

f

₃

f

₄

No

solution

4

3

2

1 One

solution

Multiple

solutions

Figure2: Flexiblereservations hemesexample

Limitation of RSVP-type reservation for bulk data transferis illustrated in Figure 2.

Inthisexample, we onsider alinkwith apa ity

C = 4T bph

. Requests arriveonlinewith varying volume, theirmaximaltransferrateis

R

max

= 2T bph

andtheirminimum average transferrate is

R

min

= 1T bph

. A request arrivesat time

t

with volume

v

has adeadline

t + v/R

min

. Assume at urrenttime

0h

, there are four a tive reservations ea h reserving

1T bph

bandwidth. Their terminationtimes are known and marked in the gure. A new

requestarrivesat

0h

withvolume

v = 4T b

,anditsdeadlineis

0h + 4T b/R

min

= 4h

. Sin e thereisnobandwidthleftattime

0h

,thisrequestwillbereje tedbyRSVP-typereservation s heme. Thisunne essaryreje tion anbeavoided,ifweusemoreexiblereservations heme

(9)

andexploit thetime-indexed reservation stateinformation. A feasiblereservation solution

isto reserving

1T bph

forthe requestfrom time

1h

(otherthanfrom

0h

)until

3h

,followed byadierentreservationrateof

2T bph

until

4h

.

In the ase of

v = 4T b

, this is the only solution to a ept the request and guarantee its su essful ompletion without preempting any existing reservations. However, if the

requesthasvolume

v = 2T b

and thus deadline

2h

, nofeasiblesolutionexists to a eptthe newrequestunlesspreemption isallowed. The on eptofpreemption isborrowedfrom job

s hedulingliterature,whi hmeansthemodi ation(in ludingteardown)ofthereservation

state of an already-a epted request by system. Compared to non-preemptive s hemes,

preemptive s hedulers enjoyhigherde isionexibility whi h impliespotentialperforman e

gain. Buttheyhavesomedrawba ksin luding:

Droppinga eptedrequest ausesmoredissatisfa tionthanblo kingnewone;

Dynami hange(QoSdegration)ofreservationstatehurtsservi epredi tability,whi h

isimportantbe ausebandwidthis o-allo atedwithotherresour es.

Also, it is hallengingto design adistributed preemptivereservationar hite ture. In this

paper,wefo usonanon-preemptive reservation framework.

There maybemultiplefeasiblesolutionstoa eptarequest, forexampleiftherequest

here is with volume

v = 6T b

and deadline

6h

. The algorithm to sele t a solutionout of all feasible solutions depends on the obje tive fun tions of reservation s hemes. Besides

in reasing a ept probability, there are other important performan e riteria. Borrowing

on ept again from job s heduling, ow time is dened as the time between a request's

arrival and its ompletion. For bulk data transfers, espe ially in grid appli ations, it is

desirable to minimize ow time. Smaller ow time not only improvesusers' satisfa tion,

but also releases all o-allo ated resour es earlier ba k to sharing pool. Fairness among

owsisalsoanimportantperforman e riteria. Forexample,bulkdatatransfermaydene

fairnessovertheiraverage throughput. These riteriamay be oni ting with ea h other.

For example, the solution to minimize ow time here is to reserve

1T bph

from

1h

to

3h

,

and

2T bph

from

3h

to

5h

sothat therequest anbenished at

5h

5h

,followedby

2T bph

(

R

max

)from

5h

to

6h

,sothat theremainedbandwidthvariation alongtimeaxisisminimized.

It is very di ult (if not totally impossible) to identify the optimal solution in both

o-lineandon-linesetting. Sometimesitispreferabletoreje tarequestevenwhenfeasible

solution exists. In this paper, we don't emphasis the hoi e of obje tive fun tions and

optimalsolutions. Instead,wefo usonformalizingaexibleyetpra ti alsolutionspa e,so

thatapotential andidatesolutionwillnotbemissedbe auseofthelimitationinreservation

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3 Flexible reservation framework

3.1 System model

We model grid networks asa set of resour es inter onne ted by wide area network. The

underlying ommuni ationinfrastru tureof grid networksis a omplexinter onne tion of

enterprizedomainsandpubli networksthatexhibitpotentialbottlene ksandvarying

per-forman e hara teristi s. Forsimpli ity,weassumea entralizeds hedulermanages

reserva-tionstateve tor

L

foralllinksinthesystem. Wewilldis ussthedistributedimplementation inSe tion 6.

Wedenearequestasa6-tuple:

r = (s

r

, d

r

, v

r

, a

r

, d

r

, R

max

r

R

max

r

is the maximum ratethat request

r

an support, onstraintedbyeitherlink apa ityofend nodes,appli ationortransportproto ol.

Request r

Plus

func

Min

System

State L(t)

Decision

D

_r

(t)

Constraint

C

_r

(t)

Figure3: Reservations hemesalgorithmframework

Abandwidths hedulermakesde isionforrequestbasedonsystemstate

L(t)

andrequest spe i ation

r

. Asshownin Figure3, as hedulerrst al ulates onstraintfun tion

C

r

(t)

for the reservation, onsidering both request spe i ation and urrent system state

L(t)

. Cal ulationof onstraintis amin operationovertime-rate fun tion whi h will be dened

below. Constraintfun tion

C

r

(t)

thenisusedto makereservation de ision

D

r

(t)

.

D

r

(t)

is theoutputofs heduler,andisalsousedinternallyto updatelinkstate

L(t)

.

3.2 Time-rate fun tion algebra

Wedenotetheset ofalltime-ratefun tions as

F

,andwedene Min-Plus algebraover

F

:

(11)

(f

1 + f

2 )(t) = f

1 (t) + f

2 (t)

(3) WhileMin-algebraisasemigroup,Plus-algebraisagroupwithidentityelement

f

0 _{(t) =}

0, ∀t ∈ (−∞, ∞)

. Wedene

≤

relationover

F

as:

f

1 ≤ f

2

,i

f

1 (t) ≤ f

2 (t), ∀t ∈ (−∞, ∞)

(4)

Notethat

F

with

≤

isapartialorderset notsatisfying omparability ondition.

a

r

, d

r

, R

max

r

inrequestspe i ationdeterminesatime-ratefun tion,whi h anbeviewed astheoriginal onstraintfun tion imposedbyrequestspe i ation:

C

request

r

(t) = R

max

r

h(t − a

r

) − R

max

r

h(t − d

r

)

(5) where:

h(t) =

1 t ∈ [0, ∞)

0

otherwise (6)

is theHeaviside stepfun tion (unistep fun tion). Translation of

h(t)

is indi atorfun tion forhalf-openinterval.

The onstraint al ulationstage shownin Figure 3 isto onsider both

C

request

r

(t)

and systemreservation state

L(t)

, sothat theresulted

C

r

(t)

returns themaximumbandwidth that anbeallo atedto request

r

attime

t

:

C

r

(t) = (C

r

request

min L

1 min L

2 min . . . min L

k

)(t)

(7)

whereweassumelinks

L

1 , L

2 , . . . L

k

formpathfrom

source[r]

to

dest[r]

,and

L

i

(t)

isthe time-(remainedbandwidth) fun tionforlink

L

i

. Themin operationisillustratedinFigure 4withtwolinks

L

1

and

L

2

inrequest

r

'spath:

a

_r

time

rate

L

1 (t)

L

₂

(t)

C

_r

request

_(t)

C

_r

(t)

0 R

max

r

d

_r

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Reservation de ision fun tion

D

r

(t)

returns the reserved data rate for

r

at time

t

. If s heduler reje ts the request, no bandwidth will be reserved for the request in the whole

timeaxis. Thusreje tionde ision anberepresentedby

f

0

.

D

r

(t)

satises:

D

r

(t) ≤ C

r

(t)

(8)

Z

d

r

t=a

r

D

r

(t)dt = v

r

,if

D

r

6= f

0

(9)

InthesystemstateupdatestageshowninFigure3:

L

i

(t) = (L

i

− D

r

)(t), ∀L

i

∈

pathof

r

(10)

At time

τ

, anempty link

L

i

withoutanyreservationhas

L

i

(t) = B[L

i

]h(t − τ )

, where

B[L

i

]

isthetotal apa ityoflink

L

i

.

3.3 Step time-rate fun tions

General time-ratefun tions are not suitablefor implementation, thus werestri t our

dis- ussionto aspe ial lassoftime-ratefun tions,i.e.,thesteptime-rate fun tions,whi hare

easytobestoredandpro essed.

Formally,afun tion is alledstepfun tion ifit anbewritten asanite linear

ombi-nationofindi atorfun tions of half-open intervals. Informallyspeaking,astepfun tion is

apie ewise onstant fun tion havingonly nitely manypie es. Atime step fun tion

f (t)

anberepresentedas:

f (t) = a

1 h(t − b

1 ) + a

2 h(t − b

2 ) + · · · + a

n

h(t − b

n

)

(11) Wedenotetheset of allstepfun tionsas

F

s

⊂ F

. A stepfun tion with

n

non- ontinuous points an be uniquely represented by a

2 × n

matrix

a

1 . . . a

n

b

1 . . .

b

n

with elements in

rst row non-zero, and elements in se ond row stri tly in reasing. All step fun tions

0 s

= {f

0 }

.

F

1 s

=

{

alltranslationsof

h(t)}

. Allnon-regressivelinear ombinationoftwodierentelementsin

F

1 s

form

F

2 s

. For

f

2 _{∈ F}

2 s

, if

a

1 + a

2 = 0

,

f

2

and

f

0

en ompass are tangular intime-rate

oordinate. Allsu h

f

2

formthere tangularfun tionset

F

rec

. Wealsodenegeneral

n

-step fun tion set

G

n

_{= F}

0 _{∪ F}

1 _{. . . F}

n

.

Followingdis ussionsrestri treservations hemestomakede isioninstepfun tionform,

i.e.,

D

r

∈ F

s

. For

f

n

_{(t) ∈ F}

n

s

and

f

m

_{(t) ∈ F}

m

s

, itiseasyto showthat

(f

n

_{min f}

m

_{)(t) ∈}

F

n+m

s

, and

(f

n

_{+ f}

m

_{)(t) ∈ F}

n+m

s

, i.e., both min and plus operations are losed in

F

s

, thus onstraintfun tion

C

r

(t)

andtime-(remainedbandwidth) fun tion

L

i

(t)

arealsostep fun tions. The omputationand spa e omplexityformin, plus andorder operationsover

fun tion

f

n

_(t)

and

f

m

_(t)

are

O(n + m)

. Wedis uss al ulationof

D

r

(t)

basedon

C

r

(t)

and

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S hemes a eptde ision exibility

FixTime-FixRate

D

r

Multi-Interval

D

r

(t) ∈ G

n

2n-2

Table1: Reservations hemes

4 Reservation s hemes

4.1 S hemes taxonomy and heuristi s

Existing RSVP-type reservation s hemes only supports reservation of a xed bandwidth

fromaxedstarttime,whi hwenameasFixTime-FixRate s hemes. Slightlymoregeneral

areFixTime-FlexRate s hemes, whi h still enfor es axedstart time, but allows heduler

to exibly determine the reservation bandwidth. To further generalize the idea, we have

FlexTime-FlexRate s hemes, whi h allowsreservation startsfrom any time in

[a

r

, d

r

]

and reservesanyrate(but needto be onstant) ontinuously untiltransfer ompletes. Finally,

byallowingreservation ompriseofmultiple (

n ≤ 1

)sub-intervalswithdierentreservation bandwidths, we have Multi-Interval s hemes. Regarding their solution spa e,

FixTime-FixRate

⊂

FixTime-FlexRate

⊂

FlexTime-FlexRate

⊂

MultiRate. Theirdierentexibilities aresummarizedinTable1.

The exibilitymakesithard to hooseasuitable de ision

D

r

(t)

ifmultiple andidates areavailable. AsmentionedinSe tion2,therearemultipleperforman e riteria,in reasing

a eptprobability,minimizingowtime,andensuringfairnessamongows,justnameafew.

Infa t,evenforRSVP-typereservations hemewithonlytwo hoi es(reje t,ora eptthe

requestwithxedrateatxedstarttime),itishardtomakeanoptimalsele tionasproved

in[12℄. Instead,weusesimpleheuristi stosele trepresentatives hemefromea hfamilyfor

performan e omparison. Athreshold-basedrate-tuningheuristi isusedto hoose andidate

fromFixTime-FlexRate s hemeswhi hwillbedetailedinSe tion5. SimpleGreedy-A ept

and Minimize-FlowTime heuristi s are used to hoose andidate from FlexTime-FlexRate

familyand Multi-Interval family.

Greedy-A ept means: Ifthereisatleastonefeasiblesolutiontoa epta omingrequest,

the requestshould notbe reje ted. Greedily a eptnew request is notoptimal in an

o-linesense,be ausesometimesitmaybebetterto Early-Reje t arequestevenwhenfeasible

solutionexists,sothat apa ity anbekeptformorerewarded-requestswhi h arrivelater.

Despitethis,itisaninterestingheuristi tostudy,be ause:

Greedy-A ept heuristi anbeusedorthogonallywithtrunkreservationtomimi the

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Greedy-A ept introdu esastri t prioritybasedonarrivingorder,whi hbyitselfisa

reasonableassignmentphilosophy.

Minimize-FlowTime means: Iftherearemultiplefeasiblesolutionsinthesolutionspa e,

theonewithminimal ompletiontimewillbe hosen. Besidesthestraightforwardbeneton

minimizingowtime,thisphilosophyalsohelpsmaximizetheutilizationofresour einnear

future,whi hotherwiseismorelikelytobewastedifnonewrequest omessoon. However,

sin ethe near future is moredensely pa kedwith reservation, assuming all requests have

identi al

R

exp

, then a small volume request with short life span is easier to get reje ted thanalarge volume requestwith longlife span. This unfairness analsobe addressedby

volume-basedtrunkreservation.

4.2 FixTime-FixRate s hemes

In FixTime-FixRate s hemes, request spe ies its desired reservation rate. S heduler an

onlyde idetoa eptorreje t. Asshownin[1℄,redu ingreservationratein reasessystem's

Erlang apa ity. Thusa andidate FixTime-FixRate s heme tomaximizea eptrateis to

enfor e:

D

r

=

R

min

r

(h(t − a

r

) − h(t − d

r

))

if

R

min

r

≤ C

r

(a

r

)

f

0

otherwise (12) Here

R

min

r

=

v

r

d

r

_−a

r

satisfy Equation(9). Inthis s heme,everya eptedrequest

om-pletestransferexa tlyatitsdeadline,ifadulltransferproto olisused. Thisisthe

reserva-tions hemeusedinFigure1. Noti ethatforFixTimes hemeswithoutadvan ereservation,

Equation(8)issimplied to onsider onstraintfun tion

C

r

(t)

'svalueat

a

r

only,be ause: FixTime s hemes'reservationisenfor edtobeginfrom

a

r

;

UnderFixTime s hemeswithoutadvan eresearvation,time-(remained apa ity)

fun -tion

L

i

(t)

foranylink

L

i

isnon-de reasingalongtimeaxis.

4.3 FixTime-FlexRate s hemes

FixTime-FlexRate s hemes still enfor e transferstart at

a

r

, thus

D

r

(t) ∈ F

rec

must have

term

h(t − a

r

fora eptedrequest

r

fromitsarrivaltime

a

r

toits ompletiontime

a

r

+

v

r

R

r

:

D

r

(t) = R

r

(h(t − a

r

) − h(t − a

r

−

v

r

R

r

))

(13)

These ondtermaboveis al ulatedusingEquation(9). WhileEquation(8)issimplied

as:

a

r

+

v

r

R

r

≤ d

r

thus

R

r

≥

v

r

d

r

_−a

r

(15)

, t

start

r

+

v

r

R

r

] ⊆ [a

r

, d

r

]

. The

D

r

anbefully hara terizesbyapair

(t

start

r

, R

r

)

. Completiontimeis al ulatedusingEquation(9).

Tosimplify Equation(8), wedene onstraint re tangular fun tion set

F

constraint

rec

and

Pareto optimal re tangular fun tion set

F

P areto

rec

for onstraintfun tion

C

r

(t)

:

F

constraint

rec

= {f (t)|f (t) ∈ F

rec

and

f (t) ≤ C

r

(t)}

(14)

F

rec

P areto

=

{f (t)|f (t) ∈ F

constraint

rec

and

!∃ g(t) ∈ F

constraint

rec

, g(t) > f (t)}

(15)

Pareto optimal re tangular fun tion set of a n-step onstraint fun tion

C

r

(t)

an be al ulatedin

O(n

2 )

asillustratedinFigure5,

F

P areto

rec

ontains

O(n

2 )

elements.

Left

1 Left

2 Left

3 Right

11 Right

12 Right

21 Right

31 time

Rate

C

_r

(t)

Rectangle

Side

Diagonal

a

_r

d

_r

Figure5: ParetoOptimalRe tangularfun tion set

Apply Greedy-A ept and Minimize-FlowTime heuristi s here: a request

r

is reje ted, if and only if there is no

f (t) ∈ F

P areto

rec

with integration no less than

v

r

; otherwise, all Pareto optimal re tanglarfun tions with large enoughintegration are he kedto identify

theoneprovidingminimumowtime. GivenaParetooptimalre tangularfun tion

f (t) =

a

1 (h − t

1 ) − a

1 (h − t

2 )

,theminimumowtimeit anprovideis

t

1 +

v

r

a

₁

. Theimplementation ofthiss hemeisdetailed inTable2.

4.5 Multi-Interval s hemes

Comparedtoallaboves hemes,reservationde isionin Multi-Interval s hemes anbe

(16)

stru ttime-rate{

doubletime;

doublerate;

booleanunVisited=true;

};

Input: 6-tuplerepresentationofrequest

r

andits onstraintfun tion

C

r

(t)

,whi hisa

n

-step fun tionrepresentedbyatime-rateve tor

v

. For

i ∈

[0, . . . , n − 1]

:

v[i℄.timeis the

(i + 1)

th

non ontinuouspointsof

C

r

(t)

, v[i℄.rate=

C

r

d.time){

d.time=v[left℄.time+r.volume/resv-rate;

d.rate=resv-rate; break; } resv-rate=v[right℄.rate; } }

(17)

aredierentfrompreemptives hemes. Althoughmultiplerates anbeusedinMulti-Interval

s hemes, and owsare probably s heduledto transferin two dis ontinuous intervals, this

de isionisdetermined atthemomenttherequest arrives,andis not hanged(preempted)

afterthat.

time

rate

v

_r

C

_r

(t)

a

r

r

(t) =

C

r

(t)

t ≤ τ

0 t > τ

(16)

where time

τ

satises:

R

τ

t=a

r

C

r

(t)dt = v

r

.

D

r

(t) = f

0

if no su h

τ

exists. As shown in Figure6,when

C

r

(t) ∈ F

n

s

isa

n

-stepfun tion, omputational omplexityof MR-MaxPa k-MinDelays hemeis

O(n)

,and

5 Performan e evaluation

5.1 Simulation setup

Weusesimulationto demonstratethepotentialperforman egainfromthein reasing

ex-ibility. We onsider the performan e of both blo king probability and mean ow time for

followings hemes:

FixTime-FixRate-

R

max

s hemeisaFixTime-FixRate s hemewithreservationrateof

of link apa ityin this simulation), and reservates

R

min

otherwise;

Greedy-A ept andMinimize-FlowTime heuristi intheFlexTime-FlexRate family;

Greedy-A ept andMinimize-FlowTime heuristi intheMulti-Interval family.

Forallabovesettings,dulltransportproto ol isassumed,whi husesandonlyusesreserved

bandwidth.

Tosimplifythedis ussiononthepotentialgainofin reasingexibility,weideallyassume

thatbulkdatatransferrequestsarriveonlinea ordingtoaPoissonpro esswithparameter

λ

, allrequests havethesame volume

v

,

R

max

= C/10

and

R

min

= C/20

, where

C

isthe link apa ity. Observation in thissimple settingalso helpsexplain thesystembehaviorin

moregeneralsettings, whi hmayhavedierentarrivalpro ess, volumedistribution,

R

max

and

R

min

.

5.2 Single Link setting

Werst onsiderthe aseofsinglebottlene klink. Performan eofaboves hemesisplotted

underin reasingload.

Figure 7 shows that in terms of blo king probability, FixTime-FixRate-

R

min

s heme performsbetterthanFixTime-FixRate-

R

max

s heme. Whenreservationratede reases,two oni ting ee ts happen: On one hand, more requests an be a epted simultaneously;

on the other hand, ea h request takes alonger time to nish. [1℄ shows that de reasing

reservationratesin reasesystem'sErlang apa ity,whi hisveriedinthisFigure. However,

asFixTime-FixRate-

R

min

always onservativelyreserve

R

min

,itsrequestowtimeisalways

v

r

/R

Exploitingtheexibilityofsele tingreservationrates,Threshold-FixTime-FlexRates heme

(19)

0

0.05

0.1

0.15

0.2

0.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 blocking probability

load

Rmax

Rmin

threshold 10%

FlexTime-FlexRate

Multi-Interval

Rmin + Ideal Transport Protocol

Figure7: Blo kingprobabilityofreservations hemes

1

1.2

1.4

1.6

1.8

2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 mean flow time

load

Rmax

Rmin

threshold 10%

FlexTime-FlexRate

Multi-Interval

Rmin + Ideal Transport Protocol

Figure8: Meanowtimeofreservations hemes

time. When load is low, a new request reserves full rate

R

max

, so that its ow time is minimized. Although the new request agressively seizes bandwidth, the threshold

statis-ti ally ensures that there are still abundant bandwidth left. Thus the probability is low

that inanear future omingowsareblo keddueto this aggressiverequest. Instead,the

new request exploits the resour e whi h will otherwise be wasted, and also it is able to

release network resour emore qui kly, whi h benets the system at a middle-range time

(20)

FixTime-FixRate-

R

max

s heme. Howeverwhenloadin reases,links areoften run in satu-ratedstate,anewrequesthashigherprobabilitytondremained apa itybelowthreshold.

Thus in this region, Threshold-FixTime-FlexRate s heme automati ally adapts its

behav-iortoperformsimilarto FixTime-FixRate-

max

. Wealso ondu t extensive simulationsovermore generalvolume,

R

benetother o-allo atedresour es. This resultstrongly motivates thestudy of advan ed

(21)

5.3 Grid network setting

Wealsoevaluatedierents hemes'performan einanetworksetting. Weusethetopologyas

shownin Figure9.

n

ingresssitesand

n

egresssitesare inter onne tedbyover-provisioned ore networks. Ea h site omposed of a luster of grid nodes, and is onne ted to ore

networkwith a link of apa ity

C

. The maximal aggregatebandwidthdemands from the ulstermayex eed

C

, makingthese linkspotentialbottlene ks. Forsimpli ity, we assume that the ore network is over-provisioned, like the visioned Grid5000 networks in Fran e

[5℄. Corenetwork an beprovisioned,forexample,using hosemodel[6℄. Whengenerating

request, its sour e is randomly sele ted from ingress sites, then a random destination is

sele tedindependentlyamongegresssites. Allsiteshavethesameprobabilitytobe hosen.

Figure9: Topology

Figure 10 andFigure 11 plotthe performan e when there are

10

ingress nodesand

10

egressnodesin the network. Compared to Figure 7and Figure8, three phenomenonsare

observed:

Overall,performan eofs hemesdegradesslightly;

FlexTime-FlexRate s heme'sblo kingprobabilityshowsabigin rease,andits

perfor-man eisnolonger losetoMulti-Intervals heme;

Multi-Interval s heme'smeanowtimeperforman edeterioratesobviously.

Theoverallperforman edegration anbetra edtothefa tthatreservationinanetwork

needto onsidermultiplelinks(bothingressandegresslinkinthistopology). Areservation

request is blo ked orits ow time be omes longer when any oneof them is ongested. If

weassumethat ongestionstatesintwolinksareindependentlyandidenti allydistributed,

withmean ongestionprobability

p

,theprobabilitythatthereisatleastoneofthembeing ongestedis

2p − p

2

(22)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 blocking probability

load

Rmax

Rmin

threshold 10%

FlexTime-FlexRate

Multi-Interval

Figure 10: Blo kingprobabilityofreservations hemes

1

1.2

1.4

1.6

1.8

2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 mean flow time

load

Rmax

Rmin

threshold 10%

FlexTime-FlexRate

Multi-Interval

Figure11: Meanowtime ofreservations hemes

The performan e degration of FlexTime-FlexRate s heme's blo king probability and

Multi-Interval s heme's mean ow time anbe explained using asimple example in

Fig-ure12.

Inthisexample,therearetwoingresslinksandtwoegresslinks inter onne tedby

₁

Ingress I

₂

Egress E

₁

Egress E

₂

r

₁

r

₁

r

₂

r

₂

r

₃

r

₃

Figure12: Afragmentationexample

s hemes (FixTime-FixRate-

R

max

s heme, FixTime-FixRate-

R

min

s heme and Threshold-FixTime-FlexRate s heme), theyare notallowedto a ept

r

3

sin ebandwidth is fully re-served for the urrent time. This prevents fragmentation as shown in the Figure when

bothFlexTime-FlexRates hemeandMulti-Intervals hemeexploittheirexibilitytoa ept

r

3

. Thistime-axisframentationin reasesFlexTime-FlexRate s heme'sblo kingprobability, sin eFlexTime-FlexRate s heme anonlyallo atea ontinuoustimeinterval. Ontheother

hand,blo kingrateofMulti-Interval s hemeisnotae tedasmu h asFlexTime-FlexRate

s heme be auseMulti-Interval s heme anmakeuse of multiple (dis ontinuous)intervals.

HoweverMulti-Interval s heme'smeanowtimeisae ted.

In aboveexamples, Multi-Interval s hemesoften give thebest perfroman e. However,

usingmultiple intervals omesata ost. Figure13showsthein reasetrendofsub-interval

numberwhen network sizeisin reased. Itisshownthatthis numberbe omesquitestable

aroundasmalllevel,whenthenumberofnodesgrowslargerthanthemultiplexinglevelof

asinglelink,whi his

C/R

max

. Thisresultholdsfordierentloadlevels. Thisobservation showsthefeasibilityofexploiting Multi-Interval s heme.

(24)

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0

5

10

15

20

25

30 mean number of sub intervals per flow

number of in(e)gress nodes

load=0.9

load=0.8

load=0.7

Figure13: Meannumberofintervalsperowin Multi-Interval s heme

6 System ar hite ture

Thelogi frameworkshownin Figure 3 orrespondsto a entralizeds heduler, whi h may

notbedesirablebe ause:

links maybeunder ontrolofdierentauthorities;

whennetworksizegrows,the entralizeds heduleritselfmaybe omeabottlene k;

Centralizeds hedulerpresentsanone-failure-point.

source

Scheduler

L

₁

....

Decision

D

_r

(t)

System

State L

1 (t)

Plus

Request

Min

Request

Reply

Decision

D

_r

(t)

System

State L

1 (t)

Plus

Constraint

C

_r

(t)

Min

func

dest

Scheduler

L

₂

Scheduler

L

_n

(25)

Thuswepresentasimpledistributedar hite utreasshowninFigure14. Inthis

ar hite -ture,everybottlene klinkisasso iatedwithalo albandwidths heduler,whi h maintains

thelo alreservationstate. Requestgeneratedfromthesour erstarrivesatlink

L

1

,whose s heduler uses

min

operation to ombine its lo al link state onstraint into the request spe i ation. The updated request spe i ationis forwardsto the nexthop. In this way,

the onstraintfun tion isupdated hop byhop:

C

i

r

(t) = (L

i

min C

r

i−1

)(t)

. Whenrequest rea hes the last hop

L

n

, the onstraint fun tion

C

r

(t)

L

n

.

Singleoutalo als heduler,itslogi anstillbeinterpretedusingthelogi frameworkof

Figure3. Theonlydieren eisthatfors hedulersnotinthelasthop,theirfun operation

(26)

7 Related works

Admission ontroland bandwidthreservationhavebeenstudied extensivelyin multimedia

networking. A real-time ownormally requests aspe ied value of bandwidth. Existing

reservations hemessu h asRSVP[3℄ attemptto reservethespe iedbandwidth

immedi-ately when request arrives. Reservation remains in ee t for an indenite duration until

expli itTeardown signalisissuedorsoftstateexpires. Notime-indexedreservationstate

iskept.

Time-indexed reservationisneededwhen onsideringadvan ereservationofbandwidth

[15℄,whi hallowsrequestingbandwidthbeforea tualtransferisreadytohappen. For

exam-ple,as heduledtele- onferen emayreservebandwidthforaspe iedfuture timeinterval.

[4℄showsthatadvan ereservation ausesbandwidthfragmentationintimeaxis,whi hmay

signi antly redu e a ept probability of requests arrivinglater. Toaddressthe problem,

theyproposethe on eptofmalleablereservation,whi hdenesadvan ereservationrequest

withexiblestarttimeandrate.

Optimal ontrolandtheir omplexityisstudiedfordierentlevelsofexibility. [2℄

stud-ies alladmission ontrolin aresour e-sharingsystem,i.e. howto usethereje texibility

regardingdierent lassesoftra . Optimal poli y stru ture isidentiedfor somespe ial

ase. [12℄provedthat inanetworkwithmultipleingressandegresssites,o-line

optimiza-tion of a ept rate for uniform-volume uniform-rate requests with randomly spe ied life

spanisNP- omplete. Theyalso onsider exibletuningof reservationrate. [1℄studiesthe

in reaseofErlang apa ityofasystembyde reasingtheservi erate. Initsessential,su h

servi erate s aling is identi al to the apa itys aling, whi h is studied by[10℄ and [9℄ to

approximatelargelossnetworks.

There is alsoalargeliterature ofonline job s hedulingwithdeadline, for example,[8℄,

[11℄, [14℄. A job monopolizes pro essor for thetime it's being s heduled, whi h maps

ex-a tlyto pa ket level s heduling,while in ow level, wemust onsider multiple owsshare

(27)

8 Con lusion

Inthis paper, westudy thebandwidth reservationproblem for bulkdatatransfersin grid

networks. Wemodelgridnetworksasmultiple sites inter onne ted bywideareanetworks

withpotentialbottlene ks. Datatransferrequestsarriveonlinewithspe iedvolumes and

deadlines, whi h allowmoreexibility in reservation s hemes design. Weformalize a

gen-eral non-preemptive reservation framework,and use simulation to examine the impa t of

feasibility over performan e. We also propose a simple distributed ar hite ture for the

givenframework.Thein reasedexibility anpotentiallyimprovesystemperforman e,but

theenlargeddesignexibilityalsoraisesnew hallengesto identifyappropriatereservation

(28)

Contents

1 Introdu tion 3

2 Motivation 4

3 Flexible reservation framework 7

3.1 Systemmodel . . . 7

3.2 Time-ratefun tion algebra. . . 7

3.3 Steptime-ratefun tions . . . 9

4 Reservation s hemes 10 4.1 S hemestaxonomyandheuristi s . . . 10

4.2 FixTime-FixRate s hemes . . . 11 4.3 FixTime-FlexRate s hemes . . . 11 4.4 FlexTime-FlexRate s hemes . . . 12 4.5 Multi-Interval s hemes . . . 12 5 Performan e evaluation 15 5.1 Simulationsetup . . . 15

5.2 SingleLinksetting . . . 15

5.3 Gridnetwork setting . . . 18

6 Systemar hite ture 21

7 Related works 23

8 Con lusion 24

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