Analysis of TTL-based Cache Networks

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Nicaise Choungmo Fofack, Philippe Nain, Giovanni Neglia, Don Towsley

To cite this version:

Nicaise Choungmo Fofack, Philippe Nain, Giovanni Neglia, Don Towsley. Analysis of TTL-based

Cache Networks. [Research Report] RR-7883, INRIA. 2012. �hal-00676735�

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RESEARCH

REPORT

N° 7883

March 2012

Analysis of TTL-based

Cache Networks

Nicaise Choungmo Fofack, Philippe Nain, Giovanni Neglia, Don

Towsley

RESEARCH CENTRE

SOPHIA ANTIPOLIS – MÉDITERRANÉE

Ni aise ChoungmoFofa k

∗

, Philippe Nain

∗

, GiovanniNeglia

∗

, Don Towsley

†

Proje t-TeamMAESTRO

Resear hReport n° 7883Mar h201224pages

Abstra t: Thispaperprovidesbuildingblo ksfortheperforman eevaluationofContent Centri -likeNetworks(CCNs). InCCNsifa a here eivesarequestfora ontentitdoesnotstore(miss), itforwardstherequesttoahigher-level a he,ifany,ortotheserver. Whenlo ated,thedo ument is routed on thereverse-pathand a opy is pla edin ea h a he along the path. In this paper we onsider a a herepla ement poli y based on Time-to-Lives(TTLs) like in aDNS network. A lo al TTL is set when the ontent is rst stored at the a heand is renewed every time the a he an satisfy a request for this ontent (at ea h hit). The ontent is removed when the TTL expires. Under the assumption that requests follow a renewal pro ess and the TTLs are exponentialrandomvariables,wedetermineexa tformulasfortheperforman emetri sofinterest (average a heo upan y,hitandmissprobabilities/rates)forsomespe i ar hite tures(alinear networkand atree network withoneroot node and

N

leaf nodes). Formoregeneraltopologies andgeneralTTL distributions,wepropose anapproximate solution. Numeri al resultsshowthe approximationsto be a urate, with relativeerrors smallerthan

10

−3

and

10

−2

respe tively for exponentiallydistributedand onstantTTLs.

Key-words: Ca hear hite ture, ontent- entri network,timer,Markovmodel, renewaltheory.

∗

Email: {ni aise. houngmo_fofa k,philippe.nain,giovanni.neglia}inria.fr. Postal address: INRIA,B.P.93, 06902Sophia Antipolis,Cedex,Fran e. ThisauthorwassupportedinpartbyOrangeLabsunderGrantCRE CCN number5376.

†

Email: towsley s.umass.edu. Postal address: Department of Computer S ien e, University of Mas-sa husetts, Amherst, MA 01002, USA.This author was supported inpart byInria and by NSFunder grant CNS-1040781.

Résumé : Cet arti le développe des briques de base pour l'évaluation des performan es de réseauxorientés ontenus. Dans es réseauxlorsqu'un n÷udou a he reçoitune requêtepour un ontenu qu'il ne possède pas il la transmet à un ou plusieurs a hes de niveau supérieur. Unefoisle ontenulo aliséil estenvoyéet sto kéàtousles a hesqui ontreçularequêteainsi qu'àl'utilisateur. Dans et arti le nousnous intéressonsà une politique degestion des a hes qui utilise des temporisateurs(TTL pour Time-to-Live). A haquearrivée d'un ontenu dans un a heuntemporisateurestdé len hé. Chaque nouvellerequêtepour e ontenu régénérele temporisateur. Dès qu'un temporisateur expirele ontenu orrespondantest ea é du a he. Nous al ulonsdemanièreexa tediérentesmesuresdeperforman e(o upationmoyennedes a hes,probabilitéettauxdesu ès)pourdesar hite turesparti ulières(réseaulinéaire,réseau arbores ent omposé d'unera ineet de

N

feuilles)dans le as oùlesrequêtessu essivesaux feuillesformentdespro essusderenouvellementetoùlestemporisateurssontexponentiellement distribués. Desapproximationstrèspré ises(erreursrelativesdel'ordrede

10

−2

)sontproposées pourdesar hite turesplusgénéraleset/oudesdistributionsarbitrairesdesTTL.

Mots- lés: Ar hite turede a hes,réseauorienté ontenus,temporisateur,modèledeMarkov, théoriedurenouvellement.

1 Introdu tion

Ca hesare widelyused in networks anddistributed systemsfor improvingperforman e. They areintegral omponentsofthe Web[4℄, DNS[15℄, andContentDistributionNetworks(CDNs) [23℄. More re ently there has been a growingemphasis on ontent networks where ontent is addressableandhost-to- ontentintera tionisthe ommon ase[22℄. Manyofthesesystemsgive risetohierar hi al(tree) a hetopologiesandtoevenmoregeneralirregulartopologies.

The design, onguration, and analysis of these a he systems pose signi ant hallenges. An abundant literature exists on the performan e (e.g. hit probability, sear h ost) of a sin-gle a he running the Least Re ently Used (LRU) repla ementpoli y

1

or, its ompanion, the Move-to-Front (MTF) poli y (see [20, 16, 9, 2, 1, 8, 6, 10, 11℄ for iid requests and [5, 13, 12℄ for orrelated requests). With fewex eptions, exa tmodelsof evensingle a hesare omputa-tionallyintra table, resultinginthe relian eonapproximations[6,11℄. Networksof a hesare signi antlymoredi ulttoanalyzeandnoexa tsolutionhasbeenobtainedsofarforeventhe simple ongurationof two LRU a hes in series. A few approximations havebeenproposed, instead,forasimpletwo-levelLRU a henetwork[4℄andageneralLRU network[21℄;however theira ura ies anbepoor(relativeerrorupto

16%

reportedin [21℄).

Inthispaperwefo usona lassof a hes,referredtoasTime-To-Live(TTL) a hes. When anun a heddataisbroughtba kintothe a hedueto a a hemiss,aTTLisset. Allrequests tothatdatabeforetheexpirationoftheTTLaresu essful( a hehit);therstrequestforthat datatoarriveaftertheTTLexpirationwill yielda a hemiss. TTL-based a hesareknownto exhibithigh hitratesand goods alingpropertiesandareusedin DNSforthatreason.

Wedevelopasetofbuildingblo ksfortheperforman eevaluationofhierar hi alTTL a he networks where TTLs are set with every request. These building blo ks allow one to model exogenous requests to dierent nodes as independent renewalpro esses and to allowfor TTL durationsto bedes ribed byan arbitrarydistribution solongastheyare independent of ea h other.

Thebuildingblo ks onsistof

ˆ arenewaltheoreti model of a single ontent TTL a he when fed by a renewalrequest stream,

ˆ arenewalpro ess approximationofthesuperpositionofindependentrenewalpro esses.

The rstblo kforms the basisfor al ulating a hemetri ssu h asmiss and hit probabilities and rates while the se ond blo k is used to represent thesuperposition of exogenous requests andthoseresultingfromamissatanupstream a heasarenewalpro ess.

Weapply these blo ksprimarily to the ase that TTL durationsare either onstant or ex-ponentiallydistributed. Furthermore,wefo usprimarilyonlinearTTLneworks,twolevelTTL tree networks and ombinationsofthetwo. Insome asesourresultsareexa tbut when they arenot,therelativeerrorsareextremelysmall(

< 10

−3

in the aseofexponentiallydistributed TTLsand

< 10

−2

inthe aseof onstantTTLs). Thusourapproa hisextremelypromisingand webelieve apable of a uratelymodelinga ri her lassof network topologies. Last, although theapproa happliestosingle ontent a hes,wedemonstratehowit anbeusedtooptimizea multi- ontent a henetwork.

Intheliteraturethepaper losertoourapproa his[14℄,wheretheauthors onsiderasingle TTL-based a hefedbyiidrequeststoasingledata. Theyobtainthehitratefora onstantTTL

1

Therepla ementpoli yistheruleusedtosele tthedatatoeje tfromthe a he. Otherpopularpoli iesare theMost-Re ently-Used(MRU)andRandomRempla ement(RR);MRUismoreee tivethan LRUfor y li a esspatternsandRRisusedinRISCar hite turesduetoitssimpli ity.

viathesolutionofarenewal-likeequation. Despitethein reasinginterestinCCNs,previouswork hasmainlyfo usedonglobalar hite turedesign. [3℄isprobablytherstattempttomodeldata transferin CCNs. Theauthors developapproximationsto al ulate thestationary throughput inanetwork ofLRU a hestakingintoa ounttheinterplaybetweenre eiverdriventransport andper- hunk a hing.

The paper is organized as follows. We introdu e notation, the model assumptions, and a key result from [19℄ regarding how to ompute the marginal interarrival distribution for a superposition ofarrivalstreams modeled asrenewalpro essesin Se tion 2. Se tion 3 ontains ourrenewaltheoreti modelalongwithitsappli ationtoanumberofnetworkswhereitleadsto exa tresults. Se tion4des ribesourapproa htomodelingthe ombinedexogenous/missrequest streamasarenewalpro essandtheresultingapproximationsforalarger lassoflinearandtree networks.Se tions5and6reportonthea ura yofthemodelsandthe omputational ostsof theirsolutions. Se tion7reportsanappli ationofourapproa hto ongureandoptimizeTTL a henetworks. Con lusionsarefoundinSe tion 8.

2 Denitions and assumptions

Throughoutthispaperwefo usonparti ularinstan esofTTL a hetreenetworkswithasingle data hunk(simply alledthedata). The aseofmultipledatawillbebrieydis ussedinSe tion 7. Fromnowonthe wordsnode and a he will beused inter hangeably. Also, a a hewill alwaysbeaTTL a heunlessotherwisespe ied.

Newrequestsforthedata anbegeneratedatanynodeofthenetworka ordingtomutually independentrenewalpro esses;these requestsare referredto asexogenousrequests orarrivals. Thetimeinstantsatwhi h exogenousrequestsarriveis alled theexogenous request pro essor the exogenous arrival pro ess. If upon thearrivalof a new request the data is notpresentin the a hetherequest isinstantaneouslyforwardedtothenextlevelof thetreeand thepro ess repeatsitself until thedata is found. In asethe data annot befoundalong thepath toward theroot, the root retrievesit from aserver. On e thedata is found, either at a a heorat a server,a opyofitisinstantaneouslytransmittedtoea h a healongthepathbetweenthe a he wherethedatawasfoundandthe a hethatissuedtherequest. AnewTTLissetforea hnew opyofthe dataandanewTTL isalsoset at the a he, ifany, wherethedata wasfound(by onvention,theTTLat theserverisinnite). Thisis in ontrastwiththemodelin [14℄ where thereisnoTTLresetupona a hehit(newTTLissetonlyupona a hemiss). Resettingthe TTLalsoatea h a hehitin reasestheo upan yandthehitprobabilityspe iallyforpopular ontents(high

λ

). This hoi eismotivatedbytheCCNparadigmofmovingpopulardo uments as loseaspossibleto theusers.

Wepointoutthatinthisdes riptionarequestisinstantaneouslysatisedwhetherthedatais foundlo ally,ataremote a heorataserver. This orrespondstoasituationwheretransmission timesarenegligiblewithrespe ttothefrequen yatwhi hthedataisrequested. Wedososin e our primary obje tive is to investigate the tra generated in the network in response to a requestforthedata.

Wedene themiss pro ess ata a heasthe su essivetime instantsat whi h misseso ur at this a he, namely, the timesat whi h thedata is requested and isnot found in the a he. Letus denote by

C(n)

theset of hildren of a he

n

. The (overall) requestpro ess, also alled thearrival pro ess, at a he

n

isthe superposition of themiss pro essesof a hes in

C(n)

and of theexogenous request pro ess at a he

n

, ifany. Weassume that su essiveTTLs at ea h a heareiidrvs,thatTTLsatdierent a hesaremutuallyindependent,andthatallTTLsare independentoftheexogenousarrivals.

λ

Arrivalrate(single a he)

1/µ

Expe tedTTL(single a he)

F (t)

CDFexogenousarrivals(single a he)

G(t)

CDFinter-misstimes(single a he)

T (t)

CDFTTLduration(single a he)

λ

n

Exogenousarrivalrateat a he

n

Λ

n

Overallarrivalrateat a he

n

Λ

n,k

Overallarrivalrateat a he

n

for ontent

k

1/µ

n

Expe tedTTLat a he

n

F

n

(t)

CDFexogenousarrivalsat a he

n

H

n

(t)

CDFoverallarrivalsat a he

n

H

n,k

(t)

CDFofoverallarrivalsat a he

n

for ontent

k

G

n

(t)

CDFinter-misstimesat a he

n

T

n

(t)

CDFTTLdurationat a he

n

h

P,n

, m

P,n

Hit,miss probabilityresp. at a he

n

h

R,n

, m

R,n

Hit,miss rateresp. at a he

n

π

n

O upan yof a he

n

(stationaryprobability ontentisin a he

n

)

π

n,k

O upan yof a he

n

for ontent

k

q

n

Averagesize of a he

n

(

= π

n

ifsingle ontentin network)

h

P

, m

P

Hit,miss probabilityresp. (single a he)

h

R

, m

R

Hit,miss rateresp. (single a he)

C(n)

Setof hildrenof a he

n

χ

∗

_(s)

LST ofCDF

χ(t)

Table1: Glossaryofmain notation

Throughout the paper

h

P,n

(resp.

m

P,n

= 1

− h

P,n

) and

h

R,n

(resp.

m

R,n

) denote the stationary hit (resp. miss) probability and the stationary hit (resp. miss) rate at a he

n

, respe tively. We denoteby

π

n

the steady-stateprobabilitythat thedata is in a he

n

and we allittheo upan yof a he

n

.

If

Λ

n

is thearrivalrate ofrequests at a he

n

, thehit rateis given by

h

R,n

= Λ

n

h

P,n

and themissrateby

m

R,n

= Λ

n

(1

− h

P,n

)

. Asaresulton ewehave al ulated

h

P,n

and

Λ

n

alsothe hit/missprobabilityandhit/missrateat a he

n

aredetermined.

Foranynon-negativerv

X

withCumulativeDistributionFun tion(CDF)

χ(t) = P (X < t)

(

t

≥ 0

),

χ

⋆

_{(s) = E[e}

−sX

_{] =}

R

∞

0

e

−st

_dχ(t)

(

s

≥ 0

)denotesitsLapla e-StieltjesTransform(LST), and

χ(s) =

ˆ

R

∞

0

e

−st

_χ(t)dt

(

s > 0

)denotestheLapla eTransform(LT)of

χ(t)

. Theidentity

χ

∗

(s) = s ˆ

χ(s),

s > 0,

(1) willbeextensivelyusedthroughout.

Foranynumber

a

∈ [0, 1]

,

¯

a := 1

− a

. Inparti ular, if

χ(t)

isaCDF,

χ(t) = 1

¯

− χ(t)

isthe orrespondingComplementaryCumulativeDistributionFun tion(CCDF).

Additionalnotationanddenitionswill begivenwhenstudyingspe i a henetworks. Thefollowingresult,takenfrom[19,Formula(4.1)℄,willberepeatedlyusedthroughoutthis paper.

Theorem2.1 TheCCDF,denotedby

R(t)

¯

,oftheinter-eventtimesofthepointpro essresulting from the superposition of

K

mutuallyindependent renewal pro esses, labeled,

1, . . . , K

,is given

by

¯

R(t) =

K

X

k=1

α

k

P

K

k=1

α

k

¯

R

k

(t)

K

Y

j=1,j6=k

α

j

Z

∞

t

¯

R

j

(u)du,

(2) with

R

k

(t)

and

α

k

> 0

the CDF of the inter-event times and the arrival rate of pro ess

k

, respe tively.

Weobservethatsu hasuperposition isnotin generalarenewalpro essitself.

3 Exa t results

3.1 Single a he with renewal arrivals and general TTLs

We onsider asingle TTL a he. Requests arriveat the a hea ordingto a renewal pro ess. Without lossofgenerality, weassumethat the rstrequest arrivesat time

t = 0

and ndsan empty a he. We denote by

X

ageneri inter-arrivaltime with CDF

F (t) = P (X < t)

and density

f (t) = dF/dt

. Wealsodenoteby

T

ageneri TTLduration,withCDF

T (t) = P (T < t)

. Sin esu essiveinter-arrivaltimesandsu essiveTTLsformtwoindependentrenewalsequen es andsin eamisstriggersanewTTL,misstimesareregenerationpointsofthestateofthe a he. Thisimpliesthatmisstimesformarenewalpro ess,withgeneri inter-misstimedenotedby

Y

andCDF

G(t) = P (Y < t)

.

Thestationaryhitprobability,hitrateandmissratedenotedby

h

P

,

h

R

and

m

R

,respe tively, aregivenby

h

P

= P (X

≤ T ) =

Z

∞

0

F (t)dT (t),

(3)

h

R

= λh

P

,

m

R

= λ(1

− h

P

)

(4) respe tively, with

λ := 1/E[X]

thearrivalrate. Themiss rateis alternativelygivenby

m

R

=

1/E[Y ]

andthehitrateby

h

R

= λ

− m

R

.

Proposition3.1givesthe a heo upan y

π

(i.e.thesteady-stateprobabilitythatthe ontent isinthe a he),forarbitraryinter-arrivaltimeandTTLdistributions.

Proposition 3.1(Stationary a he o upan y)

π := λE

"

Z

X

0

(1

− T (t))dt

#

.

(5)

Proof. Let

V

bethetimeduring whi h thedo umentisin the a hebetweentwo onse utive requestarrivals. Wehave

π = E[V ]/E[X] = λE[V ]

byrenewaltheory. Letusnd

E[V ]

. Dene thebinaryrv

U (t)

tobeoneifthedo umentisinthe a heattime

t

andzerootherwise. Without lossofgenerality onsidertheinterval

[0, X]

orrespondingtotheinter-arrivaltimebetweenthe rstandthese ondrequest. Wehave

E[V ] = E

"

Z

X

0

U (t)dt

#

= E

X

"

Z

X

0

E[U (t)

_|X]dt

#

= E

X

"

Z

X

0

(1

_{− T (t))dt}

#

IfTTLsareexponentiallydistributed withrate

µ

π =

λ(1

− F

∗

_(µ))

µ

(6)

from (5). IfarrivalsarePoisson(withrate

λ

)

π = 1

− T

∗

_(λ)

. IfarrivalsarePoissonandTTLs areexponentiallydistributed withrate

µ

then

π = λ/(λ + µ)

. Notethat

h

P

= π

ifarrivalsare Poisson,thankstothePASTAproperty.

Theproposition belowprovidesawayfor al ulating

G(t)

, theCDFof theinter-misstime, aquantitythat wewillneedlateron.

Proposition3.2 . The CDFof inter-misstimesis the uniqueboundedsolution ofthe integral equation

G(t) =

Z

t

0

G(t

_{− x)(1 − T (x))dF (x) +}

Z

t

0

T (x)dF (x).

(7)

Proof. Let

X

1

(resp.

Y

1

,

T

1

)denote therstinter-arrivaltime(resp. rstinter-misstime, rstTTL)after

t = 0

. Sin e

Y

1

≥ X

1

,theevent

{Y

1

< t

}

mayonlyo urif

X

1

< t

. Therefore,

G(t)

=

P (Y

1

< t, X

1

< t, X

1

≤ T

1

) + P (Y

1

< t, T

1

< X

1

< t)

=

P (Y

1

< t, X

1

< t, X

1

≤ T

1

) + P (T

1

< X

1

< t)

(8)

=

P (Y

1

< t, X

1

< t, X

1

≤ T

1

) +

Z

t

0

T (x)dF (x)

(9)

where(8)followsfromthefa tthat theevent

{Y

1

< t

}

istruewhen

T

1

< X

1

< t

. Itremainsto evaluatethe probability

P (Y

1

< t, X

1

< t, X

1

≤ T

1

)

in (9). By onditioningon

X

1

and

T

1

we obtain

P (Y

1

< t, X

1

< t, X

1

≤ T

1

) =

Z

t

x=0

Z

∞

τ=x

G(t

_{− x)dF (x)dT (τ)}

=

Z

t

0

G(t

− x)(1 − T (x))dF (x),

(10) where the rst equality is due to the fa t that the TTL is renewed at ea h request and then

Y

1

− X

1

onditionedto

X

1

≤ T

1

hasthesamedistributionof

Y

1

.

Suppose thatthere aretwosolutions

G

1

(t)

and

G

2

(t)

satisfying(7). Then

G

1

(t)

− G

2

(t) =

R

t

0

(G

1

(t)

−G

2

(t))(1

−T (x))dF (x)

. ByLapla etransformingbothsidesofthisequality,itappears evidentthat

G

∗

1

(s)

− G

∗

2

(s) = 0

andthenthesolutionisunique.

⋄

Dene

h(t) = f (t)T (t)

, wherewere allthat

f (t)

is thedensityof

F (t)

. TakingtheLapla e transformofbothsidesof(7)yields

ˆ

G(s) = (F

∗

(s)

_{− ˆh(s)) ˆ}

G(s) +

ˆ

h(s)

s

fromwhi hweget

ˆ

G(s) =

ˆ

h(s)

s(1

− F

∗

_{(s) + ˆ}

_h(s))

(11) and,from (1),

G

∗

(s) =

ˆ

h(s)

1

_{− F}

∗

_{(s) + ˆ}

_h(s)

.

(12)

IftheTTLisa onstant,equalto

T

,(7)be omes

G(t) =

Z

t∧T

0

G(t

_{− x)dF (x) + (F (t) − F (T ))1(t > T )}

(13) with

a

∧ b = min(a, b)

. Inthis asethehitprobabilityis

F (T )

.

We on ludethisse tionbyexaminingtwoparti ular ases:

Example1: Exponentiallydistributed TTLs For

T (t) = 1

− e

−µt

,

h

P

= F

∗

_(µ)

from(3), whi h inturn impliesfrom (4)that

h

R

= λF

∗

_(µ)

and

m

R

= λ(1

− F

∗

_(µ))

. Ontheotherhand,

ˆ

h(s) = F

∗

_(s)

− F

∗

_{(s + µ)}

sothat(12)be omes

G

∗

(s) =

F

∗

_(s)

_{− F}

∗

_{(s + µ)}

1

_{− F}

∗

_{(s + µ)}

.

(14) Themiss rate an also be obtainedfrom (14) as

m

R

= 1/E[Y ] =

−1/dG

∗

_(s)/ds

_|

s=0

= λ(1

−

F

∗

_(µ))

, onsideringthat

−dF

∗

_/ds

_|

s=0

= 1/λ

and

F

∗

_{(0) = 1}

.

Example 2: Poisson arrivals, exponentially distributed TTLs Let

T (t) = 1

− e

−µt

(exponentialTTLs)and

F (t) = 1

− e

−λt

(Poisson requestswitharrivalrate

λ

). FromEqs. (3)-(4)wend

h

P

= F

∗

_{(µ) = λ/(λ + µ)}

,

h

R

= λ

2

_{/(λ + µ)}

and

m

R

= λµ/(λ + µ)

. TheLST of

Y

is givenby(use(14)with

F

∗

_{(s) = λ/(s + λ)}

):

G

∗

(s) =

λ

λ + s

×

µ

µ + s

(15)

whi hshowsthattheinter-misstimeisthesumoftwoindependentexponentialrvswith param-eters

λ

and

µ

,withCDF

G(t) =

(

1

₋

µe

−λt

_µ−λ

−λe

−µt

if

λ

6= µ

1

− (1 + λt)e

−λt

if

λ = µ.

(16)

Thisresultisnotsurprising sin e,inthe aseofexponentialTTLs,itisequvalenttoregenerate or not the TTL at ea h hit, and then the inter-miss time is equal to the sum of a TTL (an exponentialrvwithrate

µ

)andof thetimeuntiltherstrequest arrivalaftertheexpirationof theTTL(anexponentialrvwithrate

λ

). Inparti ular,

m

R

=

1

E[Y ]

=

1

−

dG

ds

∗

(s)

|

s=0

=

λµ

λ + µ

=

1

E[T ] + E[X]

.

(17)

3.2 Line of a hes with exogenous arrivals at a single a he and

expo-nential TTLs

Consider the line network in Fig. 1 omposed of

N

TTL a hes labeled

1, . . . , N

, with no exogenous requests submitted at a hes

2, . . . , N

. Requests arrive to a he

1

a ording to a renewalpro esswithgeneri inter-arrivaltime

X

andarrivalrate

λ

. A ordingtothedes ription madeatthebeginningofSe tion2,uponamissat a he

1

therst a hetoholdthedo ument, say a he

n

≤ N

, returns a opy of the do ument to a hes

n

− 1, . . . , 1

and all TTLs are reinitialized. TTLs at all a hes are mutually independent and exponentially distributed rvs withrate

µ

n

at a he

n

.

Figure1: Line of a heswithexogenousarrivalsonlyat a he

1

(S=server)

Thearrivalpro essat a he

n

isthemisspro essat a he

n

− 1

sin etherearenoexogenous arrivals. Morever,itiseasily seenthatthemiss timesat a he

n

− 1

formregenerationpoints, sothatthemisspro essat this a he,and thereforethearrivalpro essat a he

n

,isarenewal pro ess. We an then apply re ursively theresults obtained for asingle a he. In parti ular, ifwedenote by

G

∗

n

(s)

theLST of the inter-misstimes at a he

n

,wemayapply formula (14) wheretheLST oftheinterarrivaltimesis

G

∗

n−1

. Weobtain:

G

∗

_n

(s) =

G

∗

n−1

(s)

− G

∗

n−1

(s + µ

n

)

1

− G

∗

n−1

(s + µ

n

)

(18) for

n = 1, . . . , N

, where

G

∗

0

(s) = F

∗

(s)

. The hit probability at a he

n

is given by

h

P,n

=

G

∗

n−1

(µ

n

)

,sin eahito ursat a he

n

ifthetimedurationbetweentwosu essiverequestsat this a he(withCDF

G

n−1

(t)

)doesnotex eedtheTTLduration.

Ifwedenote by

Y (n)

the generi inter-misstime at a he

n

, the miss rate at this a he is givenby

m

R,n

=

1

E[Y (n)]

=

1

−dG

∗

n

(s)/ds

|

s=0

,

(19) sothat byusing(18)

m

R,n

= m

R,n−1

(1

− G

∗

n−1

(µ

n

)) = λ

n−1

Y

i=0

(1

_{− G}

∗

i

(µ

i+1

))

(20) with

m

R,0

:= λ

by onvention. Thehit rateat a he

n

is

h

R,n

= m

R,n−1

h

P,n

= λG

∗

n−1

(µ

n

)

n−2

Y

i=0

(1

_{− G}

∗

i

(µ

i+1

))

(21) with

G

0

(t) = F (t)

by onvention. The o upan y of a he

n

(i.e. the stationary probability that the ontent isat a he

n

) is givenby (Hint: apply (6) with

F (t) = G

n−1

(t)

and

E[X] =

1/m

R,n−1

)

π

n

= m

R,n−1



₁

− G

∗

n−1

(µ

n

)

µ

n



.

(22)

Theunknown onstants

{G

∗

n−1

(µ

n

)

}

N

n=2

in(20)-(22) anbere ursively omputedfrom(18). 3.3 SimpletreenetworkwithPoisson exogenous arrivals and

exponen-tial TTLs

ConsiderthetreenetworkinFig. 2withoneroot(labeled

N + 1

)and

N

hildren(leaves)labeled

1, . . . , N

. Exogenousrequests arriveat a he

n = 1, . . . , N + 1

a ordingto aPoisson pro ess withrate

λ

n

. Ca hes

1, . . . , N

haveexponentialdistributedTTLswith rate

µ

n

at a he

n

. We assumethatTTLsat a he

N + 1

haveanarbitraryCDF

T

N

+1

(t)

,withLST

T

∗

N+1

(s)

. Forea h

Figure2: Simple treenetwork

hit/missprobability,and the hit/missrateat this a hearegivenin Se tion 3.1 (with

λ = λ

n

and

µ = µ

n

). Thetotalarrivalrateat a he

n

is

Λ

n

= λ

n

for

n = 1, . . . , N

,andthetotalarrival rateat a he

N + 1

is

Λ

N

+1

= λ

N

+1

+

N

X

i=1

ν

i

(23) with

ν

i

= λ

i

µ

i

/(λ

i

+ µ

i

)

from(17).

Let us fo us on a he

N + 1

. The arrivalpro essat a he

N + 1

is thesuperposition ofa Poisson pro esswithrate

λ

N

+1

andthemutuallyindependentmissrenewalpro essesof a hes

1, . . . , N

,withtheCDFoftheinter-misstimesat a he

n = 1, . . . , N

givenby(use(16))

G

n

(t) =

(

1

₋

µ

n

e

−λnt

−λ

n

e

−µnt

µ

n

−λ

n

if

λ

n

6= µ

n

1

_{− (1 + λ}

n

t)e

−λ

n

t

if

λ

n

= µ

n

.

TheCDF,denoted by

H

N

+1

(t)

, oftheoverallinter-arrivaltimes at a he

N + 1

isobtainedby usingTheorem2.1. Tedious(buteasy)algebrayields

H

N+1

(t) = 1

−

e

−λ

N+1

t

Λ

N

+1

N

Y

i=1

µ

2

i

e

−λ

i

t

− λ

2

i

e

−µ

i

t

µ

2

i

− λ

2

i

!

λ

N

+1

+

N

X

i=1

λ

i

µ

i

µ

i

e

−λ

i

t

− λ

i

e

−µ

i

t

µ

2

i

e

−λ

i

t

− λ

2

i

e

−µ

i

t

!

.

(24)

Withthehelpof theidentity

N

Y

i=1

(µ

2

i

e

−λ

i

t

− λ

2

i

e

−µ

i

t

) =

N

Y

i=1

(

_−λ

2

i

)

X

il∈{0,1}

l=1,...,N

N

Y

k=1

(

₋₁₎

i

k

 µ

k

λ

k



2i

k

!

e

−

P

N

k=1

(λ

ik

k

+µ

1−ik

k

)

t

(25)

we an express

H

N

+1

(t)

as a weighted sum of negative exponential terms, as shown in the appendix (see(43)). Thehit probabilityat a he

N + 1

isgivenby(use(43))

h

P,N+1

=

Z

∞

0

H

N+1

(t)dT (t)

(26)

=

1

₋

(

−1)

N

Λ

N+1

N

Y

i=1

λ

2

i

µ

2

i

− λ

2

i

"

λ

N

+1

X

il∈{0,1}

l=1,...,N

N

Y

k=1

(

₋₁₎

i

k

 µ

k

λ

k



2i

k

×T

N

∗

+1

(λ

N+1

+

N

X

k=1

(λ

i

k

k

+ µ

1−i

k

k

))

−

N

X

i=1

µ

i

λ

i

X

il∈{0,1}

l=1,...,N,l6=i

N

Y

k=1

k6=i

(

−1)

i

k

 µ

k

λ

k



2i

k

×

µ

i

T

N

∗

+1

(λ

N+1

+ λ

i

+

N

X

k=1

k6=i

(λ

i

k

k

+ µ

1−i

k

k

))

−λ

i

T

N

∗

+1

(λ

N+1

+ µ

i

+

N

X

k=1

k6=i

(λ

i

k

k

+ µ

1−i

k

k

))

!#

.

(27)

Thehitrate(resp. missrate)at a he

N +1

isgivenby

h

R,N+1

= Λ

N

+1

h

P,N

+1

(resp.

m

R,N

+1

=

Λ

N+1

(1

− h

P,N+1

)

).

Example3: Ca hes

1, . . . , N

identi al Assume that

λ := λ

n

and

µ := µ

n

for

n = 1, . . . , N

. Eq. (24)redu esto

H

N

+1

(t) = 1

−

N

X

n=0

a

n

e

−c

n

t

−

N−1

X

n=0

b

n

µe

−c

n+1

t

− λe

−c

n

t

(28) with

a

n

:=

N

n

 λ

N

+1

µ

2n

(

−λ

2

)

N−n

Λ

N

+1

×

1

(µ

2

_{− λ}

2

₎

N

b

n

:=

N

_{− 1}

n

 N λµ

2n+1

₍

−λ

2

₎

N

−1−n

Λ

N

+1

×

1

(µ

2

_{− λ}

2

₎

N

c

n

:= λn + µ(N

− n) + λ

N+1

.

Thehitprobabilityat a he

N + 1

is(use(26))

h

P,N+1

= 1

−

N

X

n=0

a

n

T

N

∗

+1

(c

n

)

−

N

−1

X

n=0

b

n

µT

N

∗

+1

(c

n+1

)

− λT

N+1

∗

(c

n

)
.

(29) 4 Approximate results

Theexa tresultsinSe tion3 annotbeeasilyextendedtogeneralnetworks.Themainproblem is that when the aggregate arrival pro ess at a a he is not a renewal pro ess, we an still determinethemain performan e metri sat the a heifwe an al ulatethe CDFofthe inter-arrivaltimes,but we annotapply Proposition 3.2 that allowsusto hara terizethemiss rate andthentostudy theupstream a hes.

In this se tion we develop an approximation method whi h produ es highly a urate ap-proximations under more general topologies for all metri s onsidered in Se tion 3 (hit/miss probabilities, hit/missrate, a he o upan y). Thequalityofthe approximationis assessedin Se tion5.

Ourapproximationisbasedonthefollowingassumption:

AssumptionA1: theoverallarrivalpro essat ea hnodeisarenewalpro ess.

A dire t onsequen eofAssumption A1is that we approximate alsothe miss pro ess at a nodeasarenewalpro ess.

With a slightabuse of notation we will use the notation of Se tion 3to denote the orre-sponding approximate values al ulated under Assumption A1. Forexample

H

n

(t)

is used to

denotetheapproximateCDFoftheoverallinter-arrivaltimes. Similarly

G

n

(t)

,

Λ

n

,

m

R,n

,

h

P,n

and

h

R,n

areusedto denoteapproximatequantities. Regardingthetotalrate

Λ

n

,notethat

Λ

n

= λ

n

+

X

i∈C(n)

m

R,i

(30)

where

C(n)

is theset of hildren of node

n

. As in Se tion 3, exogenous arrivals at ea h node formarenewalpro ess,withCDF

F

n

(t)

atnode

n

.

AssumptionA1allowsustoinvokeTheorem2.1 toget

H

n

(t) = 1

−

λ

n

Λ

n

¯

F

n

(t)

Y

i∈C(n)

ν

i

Z

∞

t

¯

G

i

(u)du

−

X

i∈C(n)

ν

i

Λ

n

¯

G

i

(t)λ

n

Z

∞

t

¯

F

n

(u)du

Y

j∈C(n)

j6=i

ν

j

Z

∞

t

¯

G

j

(u)du.

(31) An approximationof theCDF of theinter-misstimes at a he

n

isobtainedfrom Proposition 3.2

G

n

(t) =

Z

t

0

G

n

(t

− x)(1 − T

n

(x))dH

n

(x) +

Z

t

0

T

n

(x)dF

n

(x)

(32) where

T

n

(t)

theCDFof theTTLdurationat a he

n

.

Eqs(31)-(32)provideare ursivepro edure for al ulating,atleastnumeri ally, approxima-tionsfor the CDFs

G

n

(t)

and

H

n

(t)

for ea h a he

n

, from whi h we anderiveapproximate formulasfor the hit/miss probability, the hit/miss rate, and the o upan y at ea h a he. In parti ular, for a general tree network the pro edure requires al ulating the CDFs for all the a hesatthesamedepth,startingfromthosefarthest fromtheroot. Theinter-misstimeCDF at agivenleaf a he an be derivedfrom the exogenous request pro ess at the a hethrough renewalequation (32). Fora a heatthe levelabovetheinter-arrivalrequestCDF anbe al- ulated usingEq. (31) to ombine theCDFsof theinter-arrivaltimes ofits exogenous request pro ess and theinter-miss times forits hildren. We anapply again therenewal equation to hara terizetheoutputpro essatthis a heandsoon.

However, even for small networks the numeri al omplexity of this pro edure an be very highasitrequiressolvingintegralequations(seeEq.(32))and al ulatingintegralsoverinnite ranges(seeEq.(31)).

Inorder togetexpli itresultswenowfo uson aparti ular lassoftree networks, lass

N

. TTLsatea hnodeareexponentiallydistributed withrate

µ

n

. Anetworkbelongsto lass

N

if, inadditiontoassumptionA1,thefollowingassumptionholds:

AssumptionA2: Forea h

n

,node

n

isfedbythesuperpositionoftwoindependentrequest arrivalpro esses: one(stream 1) isthemiss rateof a hildof a he

n

and isageneri renewal pro essandtheotherone(stream2)isarenewalpro esswithCDFoftheform

K

n

(t) = 1

−

M

n

X

m=0

α

n,m

e

−β

n,m

t

(33) where

0

≤ M

n

<

∞

and

{β

n,m

}

m

isaset ofnonnegativenumbers.

Inwhat followswe assumewithoutlossof generalitythat stream 1originatesfrom a a he hildlabeled

n

− 1

andthenwedenotetheCDFoftheinter-misstimesinstream

1

as

G

n−1

(t)

andthemiss rateas

ν

n−1

. Sin e

η

n

:=

M

n

X

m=0

isthearrivalrateofstream2from(33),thetotalarrivalrateat node

n

is

Λ

n

= ν

n−1

+ η

n

.

(35) AssumptionsA1andA2togetheryieldthefollowingpro edureforapproximating

G

∗

n

(s)

and

H

∗

n

(s)

.

Proposition4.1 (Approximation for lass

N

) UnderAssumptions A1andA2, for ea hnode

n

,

H

n

∗

(s) = 1

− s

η

n

Λ

n

M

n

X

m=0

α

n,m

s + β

n,m

− s

2

η

n

ν

n−1

Λ

n

M

n

X

m=0

α

n,m

(1

− G

∗

n−1

(s + β

n,m

))

(s + β

n,m

)

2

β

n,m

(36) and

G

∗

_n

(s) =

H

∗

n

(s)

− H

n

∗

(s + µ

n

)

1

_{− H}

∗

n

(s + µ

n

)

.

(37) Proof.

H

n

(t) = 1

−

η

n

ν

n−1

Λ

n

M

n

X

m=0

α

n,m



¯

G

n−1

(t)

Z

∞

t

e

−β

n,m

u

_{du + e}

−β

n,m

t

Z

∞

t

¯

G

n−1

(u)du



fromwhi hwededu e(36). (37)isobtainedfrom (14).

⋄

Dierentiatingbothsidesof(37)wrtto

s

andletting

s = 0

yields

ν

n

= Λ

n

(1

− H

n

∗

(µ

n

)) = (ν

n−1

+ η

n

)(1

− H

n

∗

(µ

n

))

(38) wherethese ondequality omesfrom (35)(Hint:

Λ

n

=

−(dH

∗

n

(s)/ds

|

s=0

)

−1

),so that

ν

n

=

n

X

i=1

η

i

n

Y

j=i

(1

_{− H}

j

∗

(µ

j

))

(39) and,nally (using(35))

Λ

n

=

n−1

X

i=1

η

i

n−1

Y

j=i

(1

_{− H}

j

∗

(µ

j

)) + η

n

.

(40) Observethattherstequalityin (38) anbeobtainedwithoutany al ulationsin e

H

∗

n

(µ

n

)

is thehitprobabilityatnode

n

sothat

Λ

n

(1

− H

∗

n

(µ

n

))

isthemissrateatnode

n

.

Relations(36)-(37)and(39)-(40)provideare ursivepro edurefor al ulating

Λ

n

and

H

∗

n

(µ

n

)

forea h

n

,fromwhi hweobtainapproximationsforthehitprobability,hitrate,missrateand stationaryo upan yatnode

n

:

h

P,n

= H

n

∗

(µ

n

), h

R,n

= Λ

n

H

n

∗

(µ

n

),

(41)

m

R,n

= Λ

n

(1

− H

n

∗

(µ

n

)), π

n

= Λ

n

 1

_{− H}

∗

n

(µ

n

)

µ

n



.

Thelatterresultfollowsfrom (6).

Below,wepresenttwonetworksbelongingtothe lass

N

,i.e. withexponentiallydistributed TTLsandsatisfyingassumptionA2.

Figure3: Line of a heswithexogenousarrivals

4.1 Line of

N

TTL a hes with Poisson exogenous arrivals and expo-nential TTLs

Thisis thesamelinenetwork asin Se tion 3.2 withtheaddition ofPoissonexogenous arrivals atall a hes

1, . . . , N

withrate

λ

n

at a he

n

,seeFig. 3.

This networkbelongsto the lass

N

with

M

n

= 0

,

α

n,0

= 1

,

β

n,0

= λ

n

for ea h

n

,sothat

η

n

= λ

n

and

Λ

n

=

n−1

X

i=1

λ

i

n−1

Y

j=i

(1

_{− H}

∗

j

(µ

j

)) + λ

n

from(34)and(40).

Remark4.1 (Exa tresultsat nodes

1

and

2

)

Quantitiesinthe r.h.s. of (41 )givethe exa thit probability, hit rate, missrate, ando upan y atnode

n = 1, 2

for the linenetworkin Fig. 3sin e a hes

1

and

2

formasimple treenetwork forwhi h ouranalysis isexa t.

4.2 Line of simple tree networks

ConsiderthenetworkinFig. 4: node

n

isfedbynode

n

− 1

andbythesuperpositionofthemiss pro essesofsingle a hes(node

1

isonlyfedbysingle a hes;theanalysisbelowextends tothe asewherenode

1

isfedbyanadditionalPoissonstreamofrequests). Forthesakeofsimpli ity, herewe assume that there are

R

n

identi al TTL a hes feeding node

n

, but the analysis an be extended to the heterogeneous ase (see below). Ea h of these single a hes is fed by an exogenousstreamofPoissonrequestswithrate

δ

n

andhasexponentiallydistributedTTLswith rate

γ

n

. Nodes

1, . . . , N

haveexponentialTTLswithrate

µ

n

atnode

n

. Denoteby

S(n)

theset of

R

n

singleTTL a hesasso iatedwith a he

n

. Letusshowthat thisnetworkbelongsto

N

.

Figure4: Lineofsimpletreenetworks

Weneedtoshowthat theCDFoftheinter-arrivaltimesofrequestsjoining node

n

from a hes in

S(n)

(denotedby

K

n

(t)

)hastheform(33). ThisCDFhasalreadybeen al ulatedin Se tion 3.3useformula(28)with

N = R

n

,

λ

N

+1

= 0

,

λ = δ

n

,

µ = γ

n

,

Λ

N

+1

= R

n

δ

n

γ

n

/(δ

n

+ γ

n

)

. It istheneasilyseenthat (33)holdswith

β

n,m

=

δ

n

m + γ

n

(R

n

− m)

α

n,0

=

−δ

n

b

n,0

α

n,m

=

γ

n

b

n,m−1

− δ

n

b

n,m

,

m = 1, . . . , R

n

− 1

α

n,R

n

=

b

n,R

n

−1

γ

n

where

b

n,m

:=

R

n

− 1

m

 γ

2m

n

(

−δ

m

2

)

R

n

−1−m

(δ

m

+ γ

m

)

(γ

2

n

− δ

n

2

)

R

n

for

m = 0, . . . , R

n

. Asimilaranalysis anbe arriedoutwhen,forevery

n

,the

R

n

a hesfeeding node

n

arenotidenti al. Inthis ase,formula(43)inAppendixshouldbeusedinsteadof(28). Remark4.2 (Exa t resultsat node

1 &

leaves) Quantitiesinthe r.h.s. of(41 )give the exa t hitprobability, hitrate,miss rateando upan y atnode

1

andatallleaf nodesof the networkin Fig. 4. In fa tour analysisisexa t for asingle a heandfor asimpletreenetwork (asthat formedbynode

1

andits hildren).

5 Validation

In this se tion we investigate the a ura y of theapproximationmethod developed in Se tion 4. Re allthat themethod onsistsinassumingthat allinternalarrivalpro essesat anode(i.e. pro essesformed ofthemiss pro essesofthe node's hildren) arerenewalpro essesand touse Eq. (31) to al ulate the CDF of the inter-arrivaltimes of the superposed pro ess. The miss pro essat thisnode anthenbe hara terizedbyusingEq. (32)andthepro edure isrepeated atthenode'sparent.

Wefo us our validation on the asewhen theTTLs are exponentially distributed, but we alsoprovidesomeresultsfor onstantTTLs.

5.1 Exponential timers

We start by observing that it is possible to model a lass

N

network with

N

a hes as an irredu ibleMarkovpro ess,withstate

x(t) = (x

1

(t), . . . , x

N

(t))

∈ E = {0, 1}

N

, where

x

n

(t) = 1

(resp.

x

n

(t) = 0

)ifthedo umentispresent(resp. missing)attime

t

atnode

n

. On ethe steady-state probabilities(

p(x)

) havebeen al ulated, theexa t valuesof theperforman e metri sof interest anbeobtained by onveniently ombining thestationary probabilities and therates. Forexamplethe stationaryo upan yof a he

i

is

π

M

i

=

P

x∈E ,x

i

=1

p(x)

(the supers riptM standsforMarkov). Foraline of a hesthehit probabilityand themiss rateat a he

1

are respe tively

h

M

P,1

(1) = p(1,

∗)

and

m

M

R,1

= λ

1

p(0,

∗)

,whilefor a he2itholds

h

M

P,2

=

λ

1

p(0, 1,

∗) + λ

2

(p(0, 1,

∗) + p(1, 1, ∗))

λ

1

(p(0, 0,

∗) + p(0, 1, ∗)) + λ

2

and

m

M

R,2

= λ

1

p(0, 0,

∗)+λ

2

(p(0, 0,

∗)+p(1, 0, ∗))

,where

p(i,

∗) =

P

x

2

,...,x

N

∈{0,1}

p(i, x

2

, . . . , x

N

)

and

p(i, j,

∗) :=

P

x

3

,...,x

N

∈{0,1}

p(i, j, x

3

, . . . , x

N

)

arethestationaryprobabilitiesthat a he

1

is in state

i

∈ {0, 1}

and a hes

(1, 2)

are in state

(i, j)

∈ {0, 1}

2

, respe tively. Due to spa e onstraintsweomit the generalexpressions for these quantities for ageneri a he in the line andthoseforalineofsimpletreenetworksthat anbesimilarly al ulated.

In the rest of this se tion we ompare our approximate results versusthe exa t ones that an be obtained studying the Markov pro ess. A omparison of the omputational osts of

the twoapproa hes is in Se tion 6. We onsider rst the line network in Fig. 5: it has four nodes (

N = 4

), exogenous arrivals and exponentially distributed TTLs at node

n

with rate

λ

n

and

µ

n

, respe tively. Wehave al ulatedthe absoluterelativeerrorsat a he

n

forthe hit

Figure5: Lineoffour a hes

probability(

E

HP,n

), themissrate(

E

M R,n

)andtheo upan yprobability(

E

OP,n

). Theexa t valueis al ulatedthroughtheanalysisoftheMarkovpro ess,e.g.

E

HP,n

:=

|h

M

P,n

− h

P,n

|/h

M

P,n

. Fig. 6 shows the CDFs of the relative errors at a he

4

for

1001

dierent parameter ve tors

((λ

n

, µ

n

), n = 1, . . . , 4)

. Thevaluesoftheexogeneousarrivalrates(resp. TTLrates)havebeen sele ted in the interval

[0.001, 10]

(resp.

[0.1, 2]

) a ording to the FAST (Fourier Amplitude Sensitivity Test) method (see[18, Se . VI-C℄andreferen estherein). We anobservethat the approximationisverya urate: in 99%ofthedierentparametersettingsthe relative erroris smallerthan

2

∗ 10

−5

.

0

1

2

x 10

−4

0

0.5

1

E

HP,4

CDF

0

1

2

x 10

−4

0

0.5

1

E

_MR,4

CDF

0

1

2

x 10

−4

0

0.5

1

E

OP,4

CDF

Figure6: CDFof

E

HP,4

, E

M R,4

, E

OP,4

fornetworkin Fig.5

Figure7: Lineartreenetwork

Fig.8showshowtheerror hangesfordierentrequestloads.Inthis asewehave onsidered thehomogeneouss enariowhereallthe a heshavethesameTTLandthesameexogenousarrival rate, i.e.

µ

n

= µ

and

λ

n

= λ

for ea h

n

. The erroris shown asafun tion of thenormalized load

ρ = λ/µ

. We anobservethat thelargesterror(about

2

× 10

−4

)is obtainedwhenarrival ratesandtimerrateshave omparablevalues(

ρ

≈ 1

). Inthis asethedierentrequestpro esses superposed at a node have similar time s ales and then the inter-arrival times of the overall requestpro essaremore orrelated(seealso ommentsbelow).

0

5

10

0

2

4

x 10

−4

ρ

E

HP,4

0

5

10

0

1

2

x 10

−4

ρ

E

MR,4

0

5

10

0

1

2

x 10

−4

ρ

E

OP,4

Figure 8:

E

HP,4

, E

M R,4

, E

OP,4

for network inFig. 7with homogeneousnodes(

λ

n

= λ = ρµ =

ρµ

n

)

0

0.5

1

x 10

−3

0

0.5

1

E

HP,2

CDF

0

0.5

1

x 10

−3

0

0.5

1

E

MR,2

CDF

0

0.5

1

x 10

−3

0

0.5

1

E

OP,2

CDF

Figure9: CDFof

E

HP,2

,

E

M R,2

,

E

OP,2

fornetworkinFig. 7

We have also investigated the a ura y of the approximation for the line of simple tree networks(denedinSe tion4.2)showninFig. 7: wherenodes

11, 12, 13

(resp. nodes

21, 22, 23

) haveidenti alarrivalratesandidenti alTTLsrates.Sin etheapproximationresultsareexa tfor allnodesbut node

2

weonlyreportresultsforthatnode. Theempiri alCDFsof

E

HP,2

,

E

M R,2

and

E

OP,2

areshownin Fig. 9. LikeforFig. 6exa tresultshavebeenobtainedby onsidering theMarkovpro essasso iatedwiththislineofsimpletreesnetwork.DierentrequestandTTL rateshavebeensele teda ordingtotheFAST method respe tivelyin theintervals

[0.001, 10]

and

[0.1, 2]

. We used

4921

samples for ea h rate. Results are analogous to those for a line of a hes. The relativeerrors an be larger in this s enario, but they are probably negligible for most of the appli ations (

< 3

× 10

−4

in

99

% of the ases). We have also onsidered the homogenouss enarioalsoforthistopology,therelativeerrorshavethesameorderofmagnitude (

< 10

−3

).

Wehaveshown that AssumptionA1leadsto verya urateresultswhenexogenous arrival pro essesarePoissonand TTLareexponentiallydistributed. Thisletusthink that the super-position of therequest arrivalpro essesat every a he is very` lose'to arenewalpro ess. In ordertojustifysu hstatement,wehave al ulatedtherstauto orrelationlag(

r

1

)forthea tual arrivalpro essat node

2

inFig.5using Eq. (6.4)in [19℄. This auto orrelation lagdependson the arrival rates

λ

1

and

λ

2

and the timer

µ

1

. We havefound that for any possible hoi e of theseparameters

0 > r

1

>

−0.015

. Simulationresultsshowthattheauto orrelationisevenless signi antatlargerlags. We an then on ludethat theinter-arrivaltimesareweakly oupled.

5.2 Deterministi Timers

Whentimersare deterministi weneedto relyonthegeneralpro edure des ribedin Se tion 4 and based on Eqs (31) and (32). There are two sour es of errors in this pro edure. Firstly, theaggregaterequestpro ess ata a heisnotarenewalpro essand itisnot orre t toapply the renewal equation (32). Se ondly, both the steps (31) and (32) introdu e some numeri al

Figure10: Tree network

errors. Two parameters determine the entity of the numeri al error: 1) the time interval (

τ

) from whi h theCDFsamples aretaken, 2)the timedistan e betweentwo onse utivesamples (

∆

). Clearlythelarger

τ

andthesmaller

∆

thesmallerthenumeri alerror,but alsothelarger the omputational ost.

WehaveimplementedaMatlabnumeri alsolverthatiterativelydeterminestheCDFsinthe networkasdes ribedabove,andthenthemetri sofinterestfora a henetwork. Theintegrals appearing in Eqs. (31) and(32) are approximatedassimple sumsand for simpli ity thesame values

τ

and

∆

havebeen onsideredforalltheCDFs. Theseparametersaresele tedasfollows: oursolver rst obtainsan approximated solutionfor the whole network assuming that all the requestpro essesarePoissonandsettheparameter

τ

to5timesthelargestexpe tedinter-arrival time in the network. Theparameter

∆

is set to one thousandthof the minimum of the TTL valuesandtheexpe tedinterarrivaltimesoftheexogenousrequestpro esses.

Wepresentsomepreliminaryresultsforthetreenetwork inFig.10. Theexogenousrequest pro essesarePoissonpro esseswithrates

λ

i

(

i = 5, 6, 7, 8, 9

). TTLvalues

T

i

(

i = 1, 2, . . . 9

)are shown in the gure. In order to evaluate the relativeerror of the estimated metri s, wehave onsideredas orre tvaluesthoseobtainedthroughalongsimulation. Forexampleifourmethod predi tsthevalue

h

P,n

forthe hitprobability rateat node

n

and the 99% onden e interval, al ulatedbysimulation,is

[h

S

P,n

−ǫ, h

S

P,n

+ǫ]

,therelativeerroris al ulatedas

|h

P,n

−h

S

P,n

|/h

S

P,n

. Therelativein ertitudeof thesimulation(

ǫ/h

S

P,n

)is atmost

0.310

−4

. Foralltheperforman e metri sandallthe a hestherelativeerrorofourapproa hislessthan

10

−2

.

6 Computational Cost

Inthis se tionweperformapreliminaryanalysis ofthe omputational ostofourapproa h. Werstaddressthe aseofa lass

N

network,andinparti ularwe onsideralineofsimple treenetworkswith

N

treesand

M

nodesintotalasinFig.4. Sin ethe omputational ostfor allthemetri sisroughlythesame,wefo ushereonthehitprobability. Inorderto al ulatethe hitprobabilityatoneoftherootsofthesimpletrees,sayit a he

n

,weneedtoevaluatetheLST

H

∗

n

(µ

n

)

(Eq.(41)). Thisrequiresanumberofoperationsproportionaltothenumberof hildren of a he

n

(

R

n

)andtheevaluationoftheLSTofthemissrate omingfrom a he

n

− 1

in

µ

n

,i.e

G

∗

n−1

(µ

n

)

(Eq.(36)). Inturn

G

∗

n−1

(µ

n

)

anbe al ulatedevaluating

H

∗

n−1

(s)

intwopoints(

µ

n

and

µ

n

+ µ

n−1

)(Eq.(37))andsoonre ursively. Thisimpliesthatthe ostto al ulate thehit

probabilityat a he

n

is

O(αR

n

+ 2

n

₎

forsome onstant

α

. Whenevaluatingthehitprobability atother a hesthesameLSTsneedstobeevaluated,butingeneralatdierentpoints,thenwe havethatthetotal ostis

O(

P

N

n=1

αR

n

+ 2

n

) = O(αM + 2

N

)

. Then,dependingonthetopology ofthenetwork,the ost an bemainly linearinthenumberofnodes(foranetworkwithsmall depth, e.g. whenthere are afew treesea hwith alot nodes)orexponentialin thenumberof nodes(foranetworkwithlargedepth,e.g. forthelinearnetworkin Fig.3).

Itisinterestingto omparethis ostwithalternativeapproa hes. Forthelineofsimpletree networks,all themetri s anbeexa tly al ulated solvingaMarkov pro essas we mentioned in Se tion 5. The size of the statespa e is

2

M

, then the ost of determining the steady-state distribution by solvingthe linearequation system is

O(2

3M

₎

and this is mu h larger thanthe ostofourmethod

O(αM + 2

N

₎

. Adierentapproa histoobtainanapproximatedsteady-state distributionoftheMarkovpro essusinganiterativemethod. Thisapproa htakesadvantageof thefa t thatmostofthetransitionrateshavevaluezero. Infa tastate hangeistriggeredby anexogenousrequestarrivalata a hethatdoesnothavethedataorbyatimerexpirationata a hewiththedata,i.e. fromagivenstatewe anonlyrea hother

M

states. Thenthenumber ofnon-zeroratesisequalto

M 2

M

andea hiterationofthemethodrequires

O(M 2

M

₎

operations. Thetotal ostoftheiterativemethod isthen

O(KM 2

M

₎

,where

K

is thenumberof iterations until terminationanddepends onthespe tralgap ofthematrixused at ea h iterationandon the required pre ision, but in generalwe an expe t

O(KM 2

M

_{) << O(2}

3M

₎

. Assuming that thisisthe ase,we anobservethatourmethod,evenintheworst aseofthelinearnetwork,is stillmore onvenientthansolvingtheMarkovpro ess,be ause

O(2

M

_{) < O(KM 2}

M

₎

.

1

2

3

4

5

6

7

8

9

0

0.2

0.4

0.6

0.8

Number of Caches (N)

Time ratio = T

A

/ T

M

Figure11: Runningtime omparison

Fig.11showstheratio of the omputation times to al ulate ourapproximation(

T

A

)and tosolvetheMarkov hain(

T

M

)foralineof

N

a hes(with

N = 1, 2, . . . 9

). Boththemethods havebeenimplementedinMatlab,inparti ularthefun tionlinsolvehasbeenusedtodetermine thesteady-statedistributionoftheMarkov hain.

Letus now onsiderthe aseofageneraltreenetworkwith onstantTTLs(equalto

T

). In this ase there is no exa tsolution to ompare ourapproa h with, so we onsider simulations asan alternative approa h. Weperform an asymptoti analysis. A meaningful omparison of the omputational osts needs to take also into a ount the in ertitude of the solution: both thesimulationsandourmethod anprodu e abetter resultifoneis willingto aorda higher number of operations. In order to ombine these two aspe ts in our analysis we onsider as metri the produ t pre ision times number of operations. Intuitively the larger this produ t the more expensive is to get a given pre ision. Forthe simulationsthe omputational ost is at least proportional to thenumber ofeventsthat aregenerated, letus denote it by

n

E

. The in ertitudeonthenalresult anbeestimatedbytheamplitudeofthe onden einterval,that

de reasesas

1/√n

E

, then theprodu t pre isiontimes numberof operationsis proportional to

√

_n

E

forthesimulations. Inthe aseof ourapproa h, theheaviest operationis thesolutionof therenewalequation. Ifweadopt thesame

τ

and

∆

forall theintegrals,weneed to al ulate thevalueoftheCDFofthemissrate(

G(t)

)in

n

P

= τ /∆

pointsandthenweneedto al ulate

n

P

integrals. The integration interval is at most equal to the TTL duration

T

(see Eq. 13), thenea h integralrequires anumberof operations proportionalto

n

′

P

= T /∆

. If thevalue of

τ

is sele tedproportionallyto

T

, then the ost of ourmethod is proportionalto

n

2

P

. A naive implementation of the integral asa sum of the fun tion values leadsto an error proportional to theamplitude of thetime step and theninversely proportional to

n

′

P

or

n

P

. In on lusion theprodu t pre isiontimesthenumberof operationsis proportionalto

n

P

. Then, for agiven pre ision,ourmethodwouldrequireanumberofpointsmu hlargerthanthenumberofeventsto be onsideredin the orrespondingsimulation(atleastasymptoti ally). The omparisonwould thenleadtopreferthesimulationsatleastwhensmallin ertitudeisrequired(thenlarge

n

E

and

n

P

). Inrealityintegrals anbe al ulatedin moresophisti atedways,forexampleifweadopt Romberg'smethod,withaslightlylarger omputation ost,we angetapre isionproportional to

n

−2

P

. In this ase the produ t pre ision times number of operations is a onstant for our method,that shouldbepreferred.

7 Appli ations

Inthis se tion we rst show how ourmodel an predi tthe lo ationof a ontent in ageneral network.ThenweuseittotunetheTTLvaluesinordertominimizethesizeofevery a he.

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

Request rate at each leaf

Occupancy probability

root

intermediate node

leaf

Figure12: O upan yversusrequestrate

Werst onsiderabinarytree with7 a hes: oneroot, 2intermediate a hesand 4leaves. Requestsarriveonlyattheleavesa ordingtoiidPoissonpro esses. Thetimerisdeterministi and equal to 1 at all the a hes. This topology does not belong to lass

N

, but still Eqns (31)-(32)provideare ursivepro edurefor al ulatingnumeri allyallthequantitiesofinterest. Inparti ular, Fig.12showsthe a he o upan yat thedierent levelsof thetreefor dierent requestarrivalratesat theleaves.

Whentherequestrateat ea h leafissmall, requestsareveryunlikelytobesatisedatthe leavesandthe ontentismorelikelytobestoredathigher-level a hesinsidethenetwork( loser totheserver),thatre eiveahigheraggregaterequestrate. Astherequestratein reases,thehit rateat the leavesin reaseand themiss rate(

m

R

= λ

− h

R

) rstin reasesand thende reases asthe hit ratein rease be omes thedominantee t. As the rateof forwardedrequests starts de reasing,the o upan yat higher level a hesde reases. This onrms theintuition that in a a he network popular ontents are lo ated loser to the user. At the same time, even in su hasimplenetwork,there isnomonotoni relationamongo upan iesatdierentlevels: for examplefor smallrequest ratesthe ontentis morelo atedinside the network (the o upan y is in reasingmovingfromthe leavesto theroot),but forlarge requestrate,the ontentis less presentatintermediate a hesthanattherootorattheleaves.

Ifmultiple ontentsare presentin thenetwork,wedenote thesteady-stateprobabilitythat ontent

k

is present at a he

n

as

π

n,k

. The average number of ontents at a he

n

an be al ulatedsimplyas

P

k

π

n,k

. We allitthetotalo upan yof a he

n

andwedenoteitby

q

n

. Sowhen requestandTTLratesare known,ourmodel anbeused to al ulate thetotal a he o upan y andthen to sizethebuer at ea h node. Conversely,we ansetthe TTLvaluesat ea h a hein orderto optimizesomeperforman e metri . Forexampletheprovidermay want tosettheTTLsinordertominimizethemaximumaverage a heo upan y. Weshowhowour analysis anbeusedto solvethisproblemin a lass

N

network.

Letus onsider anetwork with

N

a hes and

K

do uments. We denote

Λ

n,k

and

H

∗

n,k

(s)

, respe tively, the total arrival rate and the LST of the CDF of the overall arrival pro ess for ontent

k

at a he

n

. Thenitfollowsthatthetotalo upan y at a he

n

is

q

n

=

K

X

k=1

π

A

n,k

=

K

X

k=1

Λ

n,k

1

_{− H}

∗

n,k

(µ

n

)

µ

n

.

Ourgoalis tosolvethefollowingoptimizationproblem:

min

µ

1

,µ

2

,...,µ

N

max

_{q

n

|n = 1, 2, . . . N}

s.t.

N

X

n=1

q

n

= Q,

(42) where

Q

isa onstraintonthetotal(expe ted)o upan yinthenetwork(thetotalbuerusage in thenetwork).

Itiseasyto he kthat under theoptimalsetting,ea h a hehasthe sametotalo upan y equalsto

Q/N

. We anthenapplyourre ursivepro edurestartingfromthenodesfarthestfrom theserveranddetermineforea h ofthem

µ

n

su hthat

q

n

=

K

X

k=1

Λ

n,k

1

_{− H}

∗

n,k

(µ

n

)

µ

n

=

Q

N

.

Thisequation anbesolvednumeri allybyusingtheNewtonmethod.

As anumeri alexample we have onsidered the line-of-simple-treesnetwork of Fig. 7with

K = 100

and iidrequest pro esses at theleaves. Therate of ea h request pro ess for agiven ontenthasbeendrawnuniformlyatrandomintheinterval

[0.01, 10]

. Fig.13showsthesele ted TTLratesforthenodes

1

and

2

andoneleaf(allofthemhavethesameTTLratesbe ausetheir requestarrivalpro essesareidenti al)fordierentvaluesofthetotalo upan yin thenetwork