Compact Markov-modulated models for multiclass trace fitting

ContentslistsavailableatScienceDirect

European

Journal

of

Operational

Research

journalhomepage:www.elsevier.com/locate/ejor

Stochastics

and

Statistics

Compact

Markov-modulated

models

for

multiclass

trace

fitting

Giuliano

Casale

a,∗

_,

_Andrea

_Sansottera

b

_,

_Paolo

_Cremonesi

b

a Department of Computing, Imperial College London, 180 Queen’s Gate, SW7 2AZ, London, UK. b Politecnico di Milano, DEIB, Via Ponzio 34/5, 20133 Milan, Italy

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 8 May 2014 Accepted 6 June 2016 Available online xxx Keywords: Counting process

Marked Markov-modulated Poisson process Trace

Fitting

a

b

s

t

r

a

c

t

Markov-modulatedPoissonprocesses(MMPPs)arestochasticmodelsforfittingempiricaltracesfor sim-ulation,workloadcharacterizationand queueinganalysis purposes.Inthispaper, wedevelop thefirst countingprocessfittingalgorithmforthemarkedMMPP(M3PP),ageneralizationoftheMMPPfor mod-elingtraces withevents ofmultipletypes.We initiallyexplain howtofittwo-stateM3PPs to empiri-caltracesofcounts.Wethenproposeanovelformofcomposition,called interposition,whichenables theapproximatesuperpositionofseveraltwo-stateM3PPswithoutincurringintostatespaceexplosion. Comparedtoexactsuperposition,wherethestatespacegrowsexponentiallyinthenumberofcomposed processes,ininterpositionthestatespacegrowslinearlyinthenumberofcomposedM3PPs. Experimen-talresultsindicatethattheproposedinterpositionmethodologyprovidesaccurateresultsagainstartificial andreal-worldtraces,withasignificantlysmallerstatespacethansuperposedprocesses.

© 2016TheAuthors.PublishedbyElsevierB.V. ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).

1. Introduction

The Markov-modulated Poisson process (MMPP) is a general fitting tool for traces with correlated arrivals (Fischer & Meier-Hellstern,1993)whichfindsapplicationinmodelingnetwork traf-fic (Okamura, Dohi, & Trivedi, August 2009), disk I/O patterns (Verma & Anand, December 2007), gridand cloud workloads(Li, Muskulus, & Wolters, 2006), and self-adaptive software systems (Perez-Palacin,Merseguer, &Mirandola, 2012). Acentral property ofMMPPsisthecomposabilitywithotherMarkovmodels,suchas queueingsystems(Horváth,Horváth,&Telek,2009;Horváth,2012; Houdt, 2012) or stochastic Petri nets (Perez-Palacin et al., 2012), whichenables usingnon-renewalprocessesinperformance mod-els.Thisisimportantbecauseautocorrelationandtemporal depen-dence can greatly affect system performance andtherefore need to be taken into account in system modeling (Mi, Zhang, Riska, Smirni,&Riedel,2007).

The MMPP is a special caseof the Markovian Arrival Process (MAP) (Neuts, 1979) in which the departure process is a modu-lation of Poisson processes and the active process is chosen ac-cordingtothestateofacontinuous-timeMarkovchain.Inthis pa-per, we proposeageneralization oftheMMPP, whichwe callthe

markedMMPP(M3PP),anddevelopascalablemethodologyfor

fit-∗ _{Corresponding author. Tel.: +44 20 759 42920.}

E-mail addresses: [email protected] (G. Casale), [email protected] (A. Sansottera), [email protected] (P. Cremonesi).

tingM3PPstoempiricaldatasets.TheM3PPcanbe regardedasa specializationoftheMMAP,themarkedextensionoftheMAP(He &Neuts,1998).Amarkedpointprocessassociatestoeacharrivala classlabel(He&Neuts,1998;He,2001),thusitisusefultomodel traceswitheventsofmultipleclasses(e.g.,readandwriterequests indiskdrives).Markedprocessesarealsoimportantinthe analy-sisofpriorityqueuesandmulticlassmodels(Buchholz,Kemper,& Kriege, 2010; Horváth et al., 2009; Horváth, 2012; Houdt, 2012). However, few techniquesexist fortheir fittingand they all focus onmatchingmomentsoftheinter-arrivaltimeprocessformarked MAPs(MMAPs)(Buchholzetal.,2010;Horváthetal.,2009). There-fore,notechniquesexistyetforfittingmarkedMarkov-modulated processesto countdata.Still, tolimit monitoringoverheads, only countdatacanbe extractedfromcertainclassesofcomputerand communications systems(e.g., network links).This motivates the investigationintomethodologiestofitmarkedcountdata.

In this paper, we fill this gap by developing approximate fit-ting algorithms for the counting process of the M3PP. In partic-ular,we first explain how to approximatelyfit a two-stateMMPP counting process and then extend the idea to two-state M3PPs. Theproposed approachisapplicable totraceswithaggregatedor coarse-grainedmeasurements,whichcannotbeanalysedusing ap-proachesbasedoninter-arrivaltimes,sincetheserequiremoments froma tracerecordingallthe arrivals.The maindrawbackisthat counting processes are mathematically less tractable than inter-arrivalprocesses,thereforeonenormallyfocusesonlow-order mo-mentsofcounts.

http://dx.doi.org/10.1016/j.ejor.2016.06.005

0377-2217/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ ).

Two-state models are often insufficient to fit complex traces, therefore we also study the approximate fitting of large M3PPs. In the single class setting, a known limitation of MMPPs is the inabilityto simultaneously fit many statistical descriptors dueto the non-linearity of their underlying equations (Bodrog, Heindl, Horváth,& Telek,2008; Heindl,Horváth,& Gross, 2006; Horváth &Telek,2009).Thishasledtothedefinitionofseveralapproaches to fit complex traces by composing multiple small-sized MMPPs or MAPs using Kronecker operators (Andersen & Nielsen, 1998; Casale,Zhang,&Smirni,2010;Horváth&Telek,2002).These meth-ods employ composition operators for moment fitting, offering a differenttrade-off betweencomputationalcostandfittingaccuracy comparedto fittingmethods basedon theEM algorithm(Breuer, 2002;Horváth&Okamura,2013;Klemm,Lindemann,&Lohmann, 2003).In particular,thesuperposition operatorallows one to de-scribeatrace bythestatisticalmultiplexingofseveralMMPPs,at theexpenseofan exponential growthofthenumberof statesin theresultingprocess(Sriram& Whitt,1986).Thisstate space ex-plosionis anobstacle fortheapplicationofMMPPs andMAPsto modeling real systems; forexample it considerably slows down, oreven renders infeasible, the numerical evaluation ofqueueing modelsbymatrixgeometricmethods(Bini,Meini,Steffé,Pérez,& Houdt,2012;Pérez,Velthoven,&Houdt,2008).

In this paper, we tackle the state space explosion problem ofsuperposition by showingthat M3PPs admit a particular form of composition, which we call interposition, that enables several MMPPsto sharethesame state spacewithoutmutuallyaffecting themarginaldistributionsoftheircountingprocesses.However, in-terpositionintroduces spurious covariancesbetweenclass arrivals that may be seen as the cost of the state-space reduction. The interposition method defines an original approach to build large Markov-modulatedprocesses,inwhichasetofJtwo-stateM3PPs is merged into a singleM3PP process withjust J₊1 states and without affecting the marginal counting processes of the initial M3PPs.We identify general conditions forinterposition to result in a valid MMAP andobserve that these conditions can be sat-isfiedby M3PPs,butnot by generalMMAPs. The abilityto inter-poseprocessesmakes a case forusingM3PPs, insteadof general MMAPs,forfittingcountdata.Wethen define amethodto auto-maticallyidentifytheM3PPstobeinterposedandamixed-integer linear program (MILP) that can help in automatically identifying theorderofinterpositionoftheM3PPs.

We conclude the paperby reporting fittingexperiments fora setofartificialandreal-worldtraces.Weshowthat theproposed M3PPfittingalgorithms are widely applicable andrun efficiently even in the caseof approximate fitting.We also find that inter-positionismuchmorescalablethansuperposition,whileretaining comparableaccuracy.

Summarizing,ourmaincontributionsareasfollows:

• Section 3 defines fitting algorithms for the counting process oftwo-stateM3PPs witharbitrarynumber ofclasses.Our for-mulas are in closed-form for exact fitting and use quadratic programming for approximate fitting. As a by-product, our approach also introduces the first infeasibility adjustment methodologyforapproximatefittingofunmarkedMMPPs. • Section4introducesthenewnotionofinterposition.Thisisan

aidtocomposemultipleM3PPs,whilepreservingtheir statisti-calproperties,aswerigorouslyestablishinTheorem1. • Section 5 develops a methodology for fitting empirical traces

usingtheinterpositionoperator.

The paper reports in Section 6 an experimental study on random models and a real trace validating the effectiveness of the proposed models and algorithms. In addition to the above, a description of necessary background is given in Section 2. Section7concludesthepaper.

2. Background 2.1. Modelandnotation

An m-state MMPP is specified by a continuous-time Markov chain(CTMC)withirreducibleinfinitesimalgeneratorQ,havingm

states,andratevector(

λ

(1),_...,

λ

(m)).WhentheCTMCisinstate

j,theMMPPgeneratesarrivalsaccordingtoaPoissonprocesswith rate

λ

(j). The effect of the modulating action of the underlying CTMC is to enable the modeling of non-Poisson, possibly nonre-newal, arrival processes. We assume the underlying CTMC to be initializedaccordingtoitssteady-statedistributionsothatthe pro-cessistime-stationary.Foreaseofcomparisonwiththeliterature, weusethroughoutthepapertheMAP(D0,D1)notation,wherefor

aMMPPD1=diag

(

λ

(

1

)

,. . .,

λ

(

m

)

)

andD0=Q− D1.

Wedefinea M3PP[K]asaMMPPinwhicharrivalsaremarked withoneoutofKavailableclasses.Thismaybeseenasamarking ofthePoissonprocessesoftheMMPPwhereonedecomposeseach rate

λ

(j),1≤ j≤ m, intoarrivalratesqj,k

λ

(j),1 ≤ k≤ K,subject to_kq_j,k=1_,q_{j,k≥ 0.}WhenanarrivalisgeneratedbyaPoisson process withrate qj,k

λ

(j) itis said to be ofclass k. Equivalently, inmatrixnotation, augmentinga MMPP(D0,D1)withmarks

de-fines a set of matrices D_1,k=diag

(

q_1,k

λ

(

1

)

_{,. . .,}q_m,k

λ

(

m

)

)

_, such thatD1=Kk=1D1,k.Thetuple

(

D0,D1,1,...,D1,K

)

iscalledthe

rep-resentationoftheM3PP[K].Also,(D0,D1) istheembeddedMMPP,

whichwerefertoastheunmarkedprocess.

In the restof the paper we often deal withprocesses having

m=2states.Inthiscase,forreadability,weuse

λ

and

λ

inplace of

λ

(1)and

λ

(2)andqkandqk inplaceofq1,kandq2,k.

2.2. Problemstatement

Let us consider a M3PP[K] and let nk(t) ≥ 0 be the num-ber of arrivals of class k at time t after initialization, subject to

n_k

(

0

)

₌0forallclasses.ThecountingprocessoftheM3PP[K]isthe CTMC with state n

(

t

)

=

(

n1

(

t

)

,n2

(

t

)

,...,nK

(

t

))

. We shalldenote by n

(

t

)

=K

k=1nk

(

t

)

the aggregate arrival count. Given an initial state probabilityvector, theevolution overtime ofthisprocess is characterized bya matrixP(n,t), withelementpi,j(n, t) inrowi andcolumnjrepresenting theprobability thata M3PPinitialized instate iisinstate j attimet withanarrival countn(t).We re-ferto P(n,t) asthecounting processmatrix.The countingprocess evolvesaccordingtotheKolmogorovforwardequations

∂

P

(

n,t

)

∂

t =P

(

n,t

)

D0+ K  k=1 P

(

n− ek,t

)

D1,k, (1) where ek is a column vector of zeros except for a one in posi-tionk.Fromthisequation,itissimpletoderivefactorialmoment functions, from which moments of the counting process can be obtained in closed form He and Neuts (1998). For example,this methodyieldsthemeanarrivalcountofclasskattimetas

μ

k

(

t

)

=E[nk

(

t

)

]=

μ

kt,

μ

k=

π

D1,k1, (2) where

π

is the stationary distribution of the embedded CTMC,

π

Q=0,

π

1=1,

μk

isthemeanarrivalrateofclasskinthe time-stationaryprocess,0and1arerespectivelycolumnvectorsofzeros andones. The variance ofclass k incounts (also called variance-timecurve)is(He&Neuts,1998)

Var[nk

(

t

)

]=



μ

k− 2

μ

2k+2ckD1,k1



t− 2ck



I− eQt



_d k, (3)

whereck=

π

D1,k

(

1

π

− Q

)

−1anddk=

(

1

π

− Q

)

−1D1,k1.Theabove formulas and notations are also valid for the unmarked process (K=1),butinthatcasewe omitthedependenceonclass k.The

covarianceofcountsforclasseskandhisgivenby

Cov[nk

(

t

)

,nh

(

t

)

]= 1

2

(

Var[nk

(

t

)

+nh

(

t

)

]−Var[nk

(

t

)

]− Var[nh

(

t

)

]

)

, (4) whereVar[nk

(

t

)

+nh

(

t

)

]canbeobtainedfrom(3)byreplacing

μk

with

μk

+

μh

andD1,kwithD1,k+D1,h.

The M3PP[K] fitting problem is to find a representation

(

D0,D1,1.....,D1,K

)

such that (2)–(4) match, to the best possible extent,thecorrespondingempiricalmomentsatgiventimescalest. Such a representation needs to be valid, meaning that all rates needbenon-negativerealnumbersandthegeneratorQofthe em-beddedCTMCmustbevalidandirreducible.

2.3. Two-stateMMPPfitting

Inordertofitatwo-stateM3PP[K],weproposetoseparatelyfit theMMPPfortheunmarkedprocessandtheclassmarkingsofthe M3PP. SinceMMPP fittingiswell-understood, we justsummarize theMMPPfittingmethodproposedinHeffesandLucantoni(1986). This algorithm receives in input the following descriptors of the countingprocess:

μ

= E[n

(

t

)

] t , I

(

t

)

= Var[n

(

t

)

] E[n

(

t

)

] , I=tlim→∞I

(

t

)

,

μ

(3)

₍

_t

₎

_=E



₍

_n

₍

_t

₎

₋

_μ

_t

₎

3



_,

where

μ

istherate,I(t)istheindexofdispersionforcounts(Sriram &Whitt,1986) atagiventimescalet,Iistheasymptoticindexof dispersionvaluefort_→∞,and

μ

(3)_{(t)isthethirdcentralmoment}

ofcountsattimescalet.Theobjectiveistofittheratesusedinthe MMPPrepresentation D0=



−

(

λ

+ r

)

r r −

(

λ

_+r

₎

, D1=diag

(

λ

,

λ



)

,

subjecttoalltheparametersbeingnon-negativeandto

λ

₊

λ

_>0. Noteinparticularthatweexcludethetrivialcases

λ

=

λ

_andr=

r=0,wheretheMMPPdegeneratesintoaPoissonprocess,which doesnotrequireatwo-statemodel.

Fitting theabove descriptors toa two-stateMMPP requiresto solve forthe unknownsr, r,

λ

,and

λ

 usinga nonlinear system composed by the following four equations (Heffes & Lucantoni, 1986)

μ

=

λ

r+

λ

r x , I=1+ 2

(

λ

−

λ



₎

2_rr x2

(

λ

_r₊

λ

_r

)

, I− I

(

t

)

I− 1 = 1− e−xt xt , h=

(

λ

−

λ



)

x, (5)

where x=r+r, x=r− r, t is an arbitrary finite timescale at which we want to fit the index of dispersion, and h=

(

g(3)

₍

_1,t

₎

_{− k}

1− k2+k3

μ

− k4

μ

x

)(

k4+

(

k3/x

)

− k5

)

−1.The

param-eters in the last equation are k1=

μ

3t3, k2=3

μ

2

(

I− 1

)

t2, k3=

3

μ

(

I− 1

)

/xt, k4=3

μ

(

I− 1

)

te−xt/x2, k5=6

μ

(

I− 1

)(

1− e−xt

)

/x3,

andg(3)_{(1,t)isthethirdfactorialmomentofcountsattimescalet.}

WepointtoHeffesandLucantoni(1986,Eq.(14d))forexplicit ex-pressionsofg(3)_(1,t).

Heffes and Lucantoni (1986) propose the following fitting

method.First,computex₌r₊r_,solvingatanarbitrarytimescale

t=t1thefixedpointequation

x=1 t

_{I− 1} I− I

(

t

)



1− e−xt



_{, (6)}

obtained by rewriting the third equation appearing in (5). Then, compute hatasecondarbitrarytimescalet=t2.Ifh=0,the

fit-tingformulasarethengivenby r=₂x



1+

1 4y+1



, r=x− r,

λ

=

μ

−_xhr_x,

λ

=

μ

+_xhr_x.

wherey=

(

I− 1

)

μ

x3

₍

_2h2

₎

−1_{.Ifh=0,theexplicitfittingformulas}

insteadbecome r=r=x 2,

λ

=

μ

− 1 2

2

(

I− 1

)

μ

x,

λ

=

μ

+1 2

2

(

I− 1

)

μ

x. Themaindrawbackofthismethodisthattheformulascanreturn negativeratesforsome combinations oftheinput parameters. In thiscase, approximatefitting techniquesare required.Yet, to the bestofourknowledgethesearenot availableintheliterature on MMPPcountingprocessfitting.

3. Fittingthecountingprocessofatwo-stateM3PP[K]

Inthissection,wedevelopamethodtofittwo-stateM3PP[K]s withrepresentation D0=



−

(

λ

+ r

)

r r −

(

λ

_+r

₎

, D1,k=diag

(

qk

λ

,qk

λ



)

, where

λ

+

λ

_>0,K k=1qk=Kk=1qk=1,qk ≥ 0,qk≥ 0,forall1 ≤ k≤ K.Asmentionedbefore,theavailabilityofexactmethodsto fitunmarkedMMPPsjustifiesanapproachwhereweseparatelyfit theembedded MMPPfirst, followedby the markingprocess that definestheM3PP. Inpractice,thismeansthat we firstfit D0 and

D1 usingaMMPPfittingalgorithmappliedtotheunmarkedtrace.

ThenwedeterminetheindividualD1,kmatricesusingthemarked traceandsubjectto theconditionD1=Kk=1D1,k,whichensures thattheembeddedMMPPdoesnotchange.InSection3.1,we dis-cusshowtoperformapproximatefittingoftheunmarkedprocess using an extension ofthe algorithm by Heffes and Lucantoni. In Section3.2,wediscussthefittingofthemarkedprocess.

3.1. Step1:approximateMMPPfitting

The fitting algorithm of Heffes and Lucantoni described in Section2.3cannotbe appliedtocaseswheretheinput setof de-scriptors

μ

,I(t),Iand

μ

(3)_{(t)isinfeasiblefortheconsideredMMPP.}

Wehavefoundthisto happenfrequentlyinrealtraces,andeven though one may repeatedly attemptto fit different timescalest1

andt2 until finding a feasible set ofdescriptors, we believe that

bettersolutionsareneededandpossible.Wehavenotfound pre-vious worksaddressing thisproblem, atleast forfittingcounting processes.Ourapproximationconsistsinsacrificing thedegree of freedomofthethirdmomentofcounts

μ

(3)_{(t)torestorefeasibility}

ofallsecond-orderdescriptors.Asbefore,wefocusoncaseswhere theM3PPprocess is notPoisson, thus we assume

λ

_=

λ

,r_> 0,

r>0.

Webeginbycharacterizingthefeasibilityregionoftheindexof dispersionforatwo-stateMMPP.

Proposition1. Ina two-stateMMPPwith

λ

=

λ

_,theindexof

dis-persionsatisfiesI>I(t)>1foranytimescalet.

Proof. Fromthesecond equationin(5),itreadilyfollowsthatI_>

1since

λ

=

λ

_{.Moreover,sincex>0,itfollowsfrom(6)thatI>}

I(t).Notingthatforx>0itis 0<1− e−xt

xt <1,

∀

t>0,x>0,

by(5)andtheconstraintI_>I(t)weconcludethatI(t)_>1. _ The previous proposition states the well-known fact that MMPPscanonlyfittracesthathavegreatervariabilitythana Pois-son process. We now show that asking for r> 0 and r > 0, to excludethecasewheretheMMPPisaPoissonprocess,isimplied bythepreviousstatementandthusalwaysverified.

Proposition2. ForI>1,theconditionsr>0andr>0arealways satisfied.

Proof.Forthe caseh= 0,observethat y > 0ifI> 1.Therefore 1/

4y+1<1thatimpliesr>0andr>0.Sincexispositive,it alsofollowsthat0<r<xand0<r<x.

For the case h₌0_, the result follows immediatelygiven that

x> 0since r=r=0would implya Poissonprocess that would haveI=1againsttheassumption. 

Therefore, providedthat I>I(t)> 1,feasibilityfollows by en-suringthatarrivalratesarenon-negativeand

λ

+

λ

_>0.Using(5), wecanreformulatethisrequirementas:

−

μ

x r ≤ h x ≤

μ

x r . (7)

Notethat thisexpression implicitly characterizes the feasible re-gionfortherate

μ

and,viathehterm,forthethirdmomentg(3)_(1,

t).Ifoneoftheseconditionsisnotmet,weproposetorelaxthe fit-tingmethodbyignoringthematchingofthethirdmomentg(3)_(1,

t)asfollows.Considerfirstthefollowinglowerboundonr.

Proposition3. Withoutloss ofgenerality,assume

λ

>

λ

_.Thenina

feasibleMMPP(2)itisr≥ u,where

u=

(

I− 1

)

x2

2

μ

+

(

I− 1

)

x. (8)

inwhichx=r+r isthesolutionof(6).

Proof.Wesubstitutethefirstequation in(5)intothesecondone andfindaftersimplealgebra

λ

=

λ

+



(

I− 1

)

μ

x3

2rr . (9)

Obtaining

λ

fromthefirstequationin(5)andequatingto(9),we getthefeasibilityconstraint

λ

₌

_μ

₋r x



(

I− 1

)

μ

x3

2rr ≥ 0. (10)

Theresultfollows usingr=x− r andsolving forr, whichyields thelowerboundu. 

Anyvalueu≤ r<xleadstoafeasiblesolution,thuswecanfor examplechoosethemiddlepointofthisintervalr=u+

(

u− x

)

/2. Oursuggestionofthemiddlepointisconvenienttokeep the for-mulas in closed-form, however other choices are possible. After choosing r, the other parameters are easily obtained by setting

r=x− r andusing(9)and(10)toobtain

λ

and

λ

.

Summarizing, provided that I _> I(t1) > 1, the approximate

methodwe havedefinedfits

μ

,I(t1)andIexactlyattheexpense

ofsacrificingan infeasiblethirdmoment

μ

(3)_(t

2),wheret1 andt2

arearbitrarytimescales.

3.2.Step2:fittingthemarkingprocess

In the second step of the fitting algorithm we determine the D1,kmatrices.After Step1,allthestatisticalpropertiesofthe un-markedprocessareconstrainedbytheD0andD1 matricesofthe

MMPP,thusthefocusofthisstepistofittheclass-specific proper-tiesandtheclasscovariances.Weconsiderbothexactand approx-imatefittingmethods.

There are several ways of choosing which empirical descrip-tors to fit with a M3PPand each choice leadsto equations that maydifferintractability comparedtoother choices. Wehave ex-perimentedwithseveralpossibilities, andwe havefoundthat fit-tingasetof centralmoments, suchasmeanclass rates

μk

,class variances,or covariances,typicallyleadsto adifficult non-convex formulation. Conversely, we have found more efficient to fit a two-stateM3PP[K]usingthemeanclassratesandper-class contri-butionstotheasymptoticindexofdispersionI.Otherefficient ap-proachesarepossible,suchasfittingmeanclassratesandrelative

covariancesorfittingmeanclassratesanddifferencesbetweenthe classvariances.However,weherefocusonthefirstmethod,which we believe providesthe bestcombinationofefficiency (quadratic programming)andeaseofinterpretation.

3.2.1. Fittingmeanclassratesandper-classcontributionstothe indexofdispersion

We begin by considering the more general problem of fitting the mean class rates

μk

and gk

(

t

)

=Var[nk

(

t

)

]+ Cov[n_k

(

t

)

_,_i=kn_i

(

t

)

].The g_k(t) termshave thesimple interpreta-tion ofmodeling thecontribution ofclass k to thevariance-time curve of the unmarked process, since

σ

(

t

)

=Var



K_k=1nk

(

t

)



= K

k=1gk

(

t

)

.Wethenspecializethemethodtotheindexof disper-sion,usingthefactthatI

(

t

)

=

σ

(

t

)

/E[n

(

t

)

]=K

k=1gk

(

t

)

/E[n

(

t

)

]. OurgoalistocomputetheD_1,kmatricesfromtheg_k(t)values, given D0 and D1=Kk=1D1,k. Recall that D1,k=diag

(

qk

λ

,qk

λ



)

. The problem under consideration is to determine the values of theprobabilitiesqk andqk thatuniquelydefinetheD1,k matrices, giventhat

λ

and

λ

areknownfromtheD1 matrixfittedinStep1.

Exact matching. We now give formulas to fitthe probabilities qk andq_kfrom

μk

andg_k(t).Notethatthearbitrarytimescaletused inthestatementdoesnotneedtobeequaltothetimescalesused forfittingtheembeddedMMPP.

Proposition4. Given(D0,D1)and,foreachclassk,

μk

andgk(t)at

anarbitrarytimescalet,theparametersoftheM3PPcanbecomputed asfollows: qk=f1,1

μ

k+f1,2gk

(

t

)

, (11) q_k=f2,1

μ

k+f2,2gk

(

t

)

, (12) where f1,1=−F2x/F, f1,2=

λ

r/F, f2,1=F1x/F, f2,2=−

λ

r/F, in whichF=

(

F1

λ

r− F2

λ

r

)

, F1=

λ

rx−4

(

2

(

λ

−

λ



)

re−xt+tx

(

x2− 2

(

λ

−

λ

)

r

)

+2

(

λ

−

λ

)

r

)

, F2=

λ

rx−4

(

2

(

λ

−

λ

)

re−xt+tx

(

x2− 2

(

λ

−

λ



)

r

)

+2

(

λ

−

λ



)

r

)

,

andx=r+risthesolutionof(6).

Proof. The expressions of

μk

and gk(t) for a two-state M3PP[K] processarefoundfromthedefinitionstobe

μ

k=

λ

qkr+

λ

qkr

x , (13)

gk

(

t

)

=F1qk+F2qk, (14)

whereF1andF2areconstantcoefficientsgivenD0,D1andthe

ref-erencetimescalet.Solving(13)forqkweobtain qk=

μ

kx−

λ

qkr

λ

r . (15)

Substituting(15)into(14),weobtainthefittingformulas(11)and (12). 

We are now ready to determine the contributions to the in-dex of dispersion. Note that Proposition 4 may also be used to fitthe contributionofclassk toI(t) intheembeddedMMPP, i.e.,

Gk

(

t

)

=gk

(

t

)

/E[n

(

t

)

]. This is because we can rewrite the fitting formulasas

qk=c1,1

μ

k+c1,2Gk

(

t

)

, (16)

q_k=c2,1

μ

k+c2,2Gk

(

t

)

, (17)

where c1,1= f1,1, c1,2=f1,2

μ

t, c2,1= f2,1, c1,1=f2,2

μ

t. The

asymptoticexpressionsofthecoefficientsast→∞arethen read-ilyobtainedas

c1,1=− 2

(

λ

−

λ

)

r+x2 2

λ

r

(

λ

−

λ



₎

, c1,2=

μ

x2 2

λ

r

(

λ

−

λ



₎

, c2,1= 2

(

λ

−

λ



₎

_r+x2 2

λ

r

(

λ

−

λ



₎

, c2,2=−

μ

x2 2

λ

r

(

λ

−

λ



₎

.

Combining these expressions with the requirements q_{k ≥ 0} and

q_k≥ 0,wefindaftersimplepassagesthislowerboundrequiredto holdforthefeasibilityoftheper-classcontributionstothe asymp-toticindexofdispersion

Gk≥

μ

_μ

k



1+2max

((

λ

−

λ

)

r,

(

λ

−

λ



)

r

)

x2



. (18)

For two-state M3PPs,the minimum value ofthe right-hand side is achievedforPoisson processes,where Gk=

μk

/

μ

andkGk=

I=1.

Approximatematching. Somevaluesofthedescriptorsmayleadto unfeasiblevaluesoftheparameters(e.g.,negativeprobabilities).In thiscase,wechoosetofittheper-classarrivalrates

μk

exactlyand findthefeasiblevaluesG˜kthatminimizethefollowingL2-norm

K  k=1



_˜ Gk− Gk Gk



2 =1 2x T_Hx+fT_{x+K, (19)} with x=

(

G˜1,...,G˜K

)

, H=diag

(

2/G21,...,2/G2K

)

, fT=

(

−2/G1,. . .,−2/GK

)

,K being the numberofclasses ofthe M3PP, andsubjecttotheconstraints

−ci,2G˜k≤ ci,1

μ

k

∀

i=1,2; k=1,...,K (20) c_i,2 K  k=1 ˜ Gk=1− ci,1

μ

∀

i=1,2. (21)

Theseconstraintsarederived usingequations(16)and(17)forq_k

andq_k.Inparticular,(20)stemsdirectlyfromtheconstraintsqk≥ 0andq_k≥ 0,whereas(21)followsfromtheconditionsK_k=1qk= 1andK_k=1q_k=1.

The above optimization program mayalso be used forfitting mean class rates

μk

and the contributions gk(t) to the index of dispersion I(t).Inordertodoso,itissufficienttoreplacethe co-efficients c_i,j withthe coefficientsf_i,j,G_k withg_k(t), andG˜_k with

˜

gk

(

t

)

.

In either case, the program has a convex quadratic objective function,thusitsminimizercanbeefficientlyobtainedusing stan-dard quadratic programmingsolvers. Once the problemis solved andthefeasiblevaluesGkorg˜k

(

t

)

areobtained,theparametersqk andq_karereadilyfittedusing(16)and(17).

4. CompositionalfittingofM3PP[K]s

Intheprevioussection,wehavedefinedageneralpurpose fit-ting methodfortwo-state M3PPs.We nowconsider the problem ofexploitingtheadditionaldegreesoffreedomofaM3PP[K]to in-creasetheflexibilityofthefitting.Inthiscase,exactfitting meth-ods capable ofexploitingall available degreesoffreedom do not existevenforunmarkedprocesses.Thereforewefocuson compo-sitionalfitting,whereonebuildsalargeprocessbycompositionof smallerprocessesthat aresimplerto fittocountdata.The draw-backofcompositionalapproachesofthiskindisthattheyusejust afewdegreesoffreedomofageneralMMAPinreturnforeaseof fitting.

We first review andgeneralize superposition methods for un-marked MMPPs. Afterwards, we introduce a novel form of com-position,calledinterposition,whichoffersadifferenttrade-off be-tween accuracyandcompactnessof therepresentation.Notethat we donotconsidermethods thatare specificto inter-arrival pro-cesses, such as theKronecker product composition (Casaleet al., 2010).

4.1. Superposition

Unmarked case. Consider a set of JMMPPs

(

D₀j,D₁j

)

, 1 ≤ j ≤

J, their superpositionis the MMPPprocess

(

D₀+_,D+₁

)

where D+_{0 =} J j=1D j 0, D+1 = J j=1D j

1, in which  denotes the Kronecker sum

operator(Brewer,1978).Superpositionnaturallyarisesin network-ing to describe the traffic process obtained by merging separate traffic flows, each describedby a MMPP.The superposedprocess definesthemultiplexingofJchannels,eachwithinter-arrivaltimes describedbythej-thMMPP.Thisprocesshasmeanarrivalrateand variance-time curve equal to the sum of mean arrival rates and variance-timecurves of the individual MMPPs. The index of dis-persion is a weightedsumof the IDCsof the individual MMPPs, i.e., I

(

t

)

₌_j

(

μj

_/

μ

)

I_j

(

t

)

_, where

μ

₌_j

μj

is the mean arrival rateofthesuperposition.Also,ifMMPPjhasmjstates,the super-posedprocesshasJ_j=1mjstates,thusitssizegrowsexponentially withJ.

Marked case. Let _K=

{

1_,...,K

}

be a set of classes. We con-siderJM3PPsandassumethat M3PPjgeneratesarrivalofclasses

Kj⊂ K,withK1∪K2∪· · · ∪KJ=K.Inthiscase,someM3PPs con-tributetoarrivalsofclassk,thusD+₀ =J

j=1D j 0,D+1,k= J j=1D j 1,k,

∀

k∈K,wherewedefineD₁j_,k=0ifk∈/Kj,where0isherea ma-trixofall zeros ofordermj.The statisticalproperties ofthe em-beddedunmarkedprocess areobtainedasinanunmarked super-position. Lastly, note that if each M3PP has two states, the re-sultingprocess is a M3PPwith2J _{statesand K classes.Thus, the} main drawback of the superposition method is the state-space explosion, which limits the composition to a small number of processes.

4.2.Interposition

We now propose a new form of composition for Markov-modulatedprocesses that tackles thestate-space explosion prob-lemofsuperposition.Informally,ourideaistodefinean operator bywhichseveralM3PPscanbydefineduponthesamestatespace, without interfering with their respective marginal counting pro-cesses.Thisallowsustopreserveintheinterpositionthecounting processpropertiesfittedinisolationforeachcomposedM3PP.We characterizethemainfeatureofthisnewcompositionoperatorin Theorem1,givenlater.Notethattheresultsinthissectionarealso applicabletounmarkedMMPPs,andthusrepresentadvances also forunmarkedprocesses.

ConsiderasetofJtwo-stateM3PPswheretheithprocesshasKi classes.Assumewithoutlossofgeneralitythatclassesarelabelled suchthateachclassappearsinoneandonlyoneM3PP.TheM3PPs haverepresentation Di₀=



−ri−

λ

i ri r_i −r_i−

λ

 i

, Di_1,k=diag

(

qi,k

λ

i,qi,k

λ

i

)

, k=1. . .,Ki,

where we define the probabilities Ki

k=1qi,k=1, qi,k ≥ 0, and K_i

k=1qi,k=1,qi,k≥ 0,andtheratesri=Jj=i

α

jandri= i

j=1

βj

,

forgivenconstants

αj

≥ 0and

βj

≥ 0.Equivalently,giventhe val-uesofthe ri andri constants, we can define theratedifferences

αi

=r_{i− ri+1} and

βi

₌r_i_{− r}_i−1,withboundaryvalues

β

1=r1 and

αJ

=rJ.

WenowdefinetheinterpositionoperatorforM3PPs.Giventhe setof Jtwo-stateM3PPs

(

D0,Di1,1,...,Di1,Ki

)

, theinterposed

pro-cess

(

D∗₀,D∗_1,1,...,D∗_1,K

)

istheM3PPwithJ+1states,K=J i=1Ki

classes,andrepresentation D∗0=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

−



α

1

α

2 ...

α

J

β

1 −



α

2 ...

α

J . . . ... ₋

_

... ...

β

1 ...

β

J−1 −



α

J

β

1 ...

β

J−1

β

J −



⎤

⎥

⎥

⎥

⎥

⎥

⎦

, D∗1,k=



qi,c

λ

iIi 0 0 q_i,c

λ

_iIJ+1−i

,

whereIn istheidentitymatrixofordern,classk=ij−1=1Kj+c is theclass inthe interposedprocess associated to classc of thei -thcomposedM3PP,andthediagonalelementsofD∗₀ aresuchthat

(

D∗₀+kD∗1,k

)

1=0.Throughoutthissection,wedenotebyKithe setofclassindexesintheinterposedprocessassociatedtothei-th composedM3PP,andby_K¯_i=

{

1,...,K

}

\

Kiitscomplement.

TheinterposedprocessmaybeseenasaM3PPmodulatedbya CTMCwithaninfinitesimalgeneratorQ∗=D∗0+Kk=1D∗1,kthat al-lowsexactCTMCaggregation(Bolch,Greiner,deMeer,andTrivedi, 1998,Chap 4). Specifically, foreach of the initial M3PPs,we can defineapartitionofthestatesoftheinterposedprocesssuchthat, byexactaggregationoftheCTMCQ∗onerecoversthe correspond-ingCTMC ofthe initial M3PP. Forexample,we mayconsider the aggregation Q∗=

⎡

⎢

⎢

⎢

⎢

⎢

⎢

⎣

−r1

α

1

α

2 ...

α

J

β

1 −r2− r1

α

2 ...

α

J . . . ... ... ... ...

β

1 ...

β

J−1 −rJ− rJ−1

α

J

β

1 ...

β

J−1

β

J −rJ

⎤

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⇒

⎡

⎢

⎣

−r1 J  i=1

α

i

β

1 −

β

1

⎤

⎥

⎦

=



−r1 r1 r₁ −r1

=Q1, whereQ1=D10+ 

kD11,k isthe infinitesimalgenerator of theith composedM3PP.Similarly,foreachpartition,thedefinitionofD∗_1,k ensures that the departure rates are identical tothe ones in the composedM3PP.

In order to prove that the marginal counting process of each composed M3PP, for all classes k_∈Ki, is preserved by the in-terposition operator, we require additional notation. Let P(ni, t),

ni=

(

n1,. . .,nK_i

)

,bethe2× 2countingprocessmatrixforthei-th M3PPattimet.Similarly,letP∗(n,t)bethe

(

J+1

)

×

(

J+1

)

count-ingprocessmatrixforthecomposedM3PP,wheren=

(

n1,...,nJ

)

. Let0n and 1n be rowvectors ofsize n ofall zeros andall ones, respectively,anddefinetheaggregationmatrix(Buchholz,1994)

Vi=



1i 0J+1−i 0i 1J+1−i

.

Foran arbitrary

(

J+1

)

×

(

J+1

)

matrixX,the2× 2matrix Xi=

WiXVTi,whereWi=diag

(

1JT+1VTi

)

−1Vi,givestheaggregationofX intothetwostatesassociated tothei-thcomposed M3PP. There-fore

P∗i

(

n,t

)

=WiP∗

(

n,t

)

VTi,

isthecountingprocessmatrixaggregatedontothestatesofthei -thcomposedM3PP.Fromthismatrix,wecanreadilycomputethe marginalcountingprocessmatrixofthei-thM3PPas

P∗i

(

ni,t

)

=  nh:h∈K¯i

P∗i

(

n,t

)

,

wherethesummationisoverallclassesnot associatedtothei-th M3PP.Usingthisnotation,weprovethemaincharacterization re-sultfortheinterposedprocess,whichstatesthattheinterposition operatordoesnotaffectthemarginalcountingprocessforthei-th M3PP.

Theorem1. Assumeallprocessestobetime-stationary,then P

(

ni,t

)

=P∗i

(

ni,t

)

,

∀

t≥ 0,ni≥ 0. (22)

Proof. Forthe composed M3PPit is simple to verifythat V_i has thefollowingproperties

Q∗VTi =ViTQi, D∗1,kVTi =VTiDi1,k,

∀

k∈Ki. (23) Note that conditions of this kind naturallyarise in the study of minimal representations of Markovian and Rational Arrival Pro-cesses(Buchholz&Telek,2013).

LetusthenconsidertheKolmogorovforwardequationforP∗(n,

t). Pre-multiplying(1)by Wi andpost-multiplying byVTi,we ob-tain

∂

P∗_i

(

n,t

)

∂

t =WiP ∗

₍

_n,t

₎

_D∗ 0V T i +  k∈Ki P∗_i

(

n− ek,t

)

Di1,k + k∈_K¯i WiP∗

(

n− ek,t

)

D∗1,kVTi,

wherewehaveused(23)tosimplifytheexpressionandthe nota-tionn− ek indicates theremovalofa job ofclassk fromn. Now pluggingtheidentityD∗₀=Q∗−K

k=1D∗1,kandusingagain(23)we get

∂

P∗_i

(

n,t

)

∂

t =P ∗ i

(

n,t

)(

Qi−  k∈Ki Di_1,k

)

+ k∈Ki P∗_i

(

n− ek,t

)

Di1,k + k∈_K¯i Wi

(

P∗

(

n− ek,t

)

−P∗

(

n,t

))

D∗1,kVTi =P∗i

(

n,t

)

Di0+  k∈Ki P∗i

(

n− ek,t

)

Di1,k + k∈_K¯i Wi

(

P∗

(

n− ek,t

)

− P∗

(

n,t

))

D∗1,kVTi.

WenowconsiderthemarginalprobabilityP∗_i

(

ni,t

)

,forwhichthe lastexpressionimplies

∂

P∗i

(

ni,t

)

∂

t =  nh:h∈K¯i P∗i

(

n,t

)

Di0+  nh:h∈K¯i  k∈Ki P∗i

(

n− ek,t

)

Di1,k +  nh:h∈K¯i  k∈K¯i Wi

(

P∗

(

n− ek,t

)

− P∗

(

n,t

))

D∗1,kVTi. (24) Since theinfinitesummations inthelast termof(24)are onthe sameclassindexes(_K¯_isets),andP∗

(

n− ek,t

)

=0whennk=0,by symmetrythedoublesummationvanishes.Thus(24)reducesto

∂

P∗_i

(

ni,t

)

∂

t =P ∗ i

(

ni,t

)

Di0+  k∈Ki P∗_i

(

ni− ek,t

)

Di1,k,

which is identical to the Kolmogorov forward equation for the counting process of the i-th composed M3PP. In order to prove this statement, we just need to show that the initial conditions oftheKolmogorov forwardequationsarethesame,i.e.,P

(

ni,0

)

=

P∗_i

(

n_i,0

)

_,

∀

n_i.Sinceweareconsideringtime-stationaryprocesses, the initial state of the interposed process is determined by the equilibrium distribution of the CTMC with generator Q∗. Con-versely, forthe i-thM3PP thisisgiven bythe equilibrium distri-butionoftheCTMCwithgeneratorQi.By(23)weseethatforan arbitraryvector

π

Wi

π

Q∗VTi =Wi

π

VTiQi=

πi

Qi.

Therefore, if we choose the initial distribution to be the time-stationary distribution

π

Q∗=0,

π

1=1, we readily find that its aggregationcorresponds tothetime-stationarydistributionofthe

i-th M3PP, i.e.,

π

_iQ_i=0 and

π

_i1₌1_, where the last condition holdssinceweuseexactaggregation.Thisconcludestheproof.  We remarkthat(23)provides aconditionforageneralMMAP to define a valid interposition. It is simple to see that the diag-onal structure of theD1,k matricesin a M3PP satisfiesthese as-sumptions.Whileouranalysisdoesnotexcludethatotherkindsof MMAPs maybe usedforinterposition,the conditionsrequiredto satisfy(23)donotseemtoreadilysuggestalternatives otherthan theM3PP.ThismotivatestheuseofM3PPsasabuildingblockfor interposition.

4.2.1. Feasibilityofaninterposition

We now give examples of valid and invalid interposed pro-cesses.Considerthefollowingtwo-stateM3PP[2]s:

A0=



−6 3 1 −5

, B0=



−3 1 4 −7

, C0=



−9 7 2 −5

, E0=



−7 4 1 −3

,

where A1,1=diag

(

1,3

)

, A1,2=E1,1=diag

(

2,1

)

, B1,1=C1,1=

diag

(

1_,2

)

_,andB1,2=C1,2=E1,2=diag

(

1,1

)

.Itiseasytoseethat

the interposition of A=

(

A0,A1,1,A1,2

)

and B=

(

B0,B1,1,B1,2

)

satisfiestheassumptionsonthenon-negativityoftherates

αj

and

βj

andyieldsthecomposed3-stateM3PP[4]

D∗0=



_{−8 2 1} 1 −8 1 1 3 −11



,

with D∗_1,1=diag

(

1,3,3

)

, D∗_1,2=diag

(

2,1,1

)

, D∗_1,3=diag

(

1,1,2

)

,

D∗_1,4=diag

(

1,1,1

)

. The per-class arrival rates and variances of counts inthe interposedprocess matchthe corresponding statis-ticsofthetwoclassesofAandthetwoclassesofB.

Conversely, the interposition of processes A and C=

(

C0,C1,1,C1,2

)

is invalid because the non-diagonal rates of C0

areboth greaterthanthecorrespondingratesinA0.Similarly,the

interposition of A andE=

(

E0,E1,1,E1,2

)

is invalid, even though

BandE arethesameprocess,becausetheirstatesareorderedin a differentwayandthisaffectsthecomputationofthe

αj

and

βj

coefficients. Thislast example provides intuitionon thefact that, toobtain afeasibleinterposed process,onemayneedtore-order the states of the M3PPs. In the next section, we describe an algorithm tofindafeasible interpositionofagivenset ofM3PPs, ifoneexists.

4.2.2. Classcovariances

While theinterposed process preservesthe marginalcounting propertiesofthecomposedM3PPs,whencomparedto superposi-tionthiscomesattheexpenseofintroducingspuriouscovariances betweenarrivalsofdifferentM3PPs.Thisisbecause,bydefinition oftheinterposedprocess,atransitioninthestatespaceofa com-posedM3PPalsochangesthecurrentstate oftheothercomposed M3PPs. In order to quantify the magnitude of thesecovariances, welookattheasymptoticcovariances,whichcontributetothe in-dex ofdispersion.Observe that,by plugging (3)into(4), wefind aftersomesimplifications

σ

k,h=_tlim_→∞Cov[nk

(

t

)

,nh

(

t

)

]=−2

μ

k

μ

h+

π

D1,h

(

1

π

− Q

)

−1D1,k1 +

π

D1,k

(

1

π

− Q

)

−1D1,h1.

Let P=

(

1

π

− Q

)

−1 _{and observe that this by construction is a}

stochastic matrix with equilibrium vector

π

. Using the fact that

π

D1,k=

μk

,foranyclassk,andafterdeterminingthestructureof

thePmatricesfortheinterposedprocess,itispossibletocompute their Jordan canonical form, which after algebraic simplifications yieldstheformula

σ

k,h=

r_irj

(

qj,h

λ

j− qj,h

λ

j

)(

qi,k

λ

i− qi,k

λ

i

)(

xj+xi

)

x2

jx2i

, (25)

whereitisassumedthatk∈Kiandh∈Kj,andi<j. Someremarksontheformulasareasfollows:

• WhenanyofthetwoM3PPstendstoaPoissonprocess,apair ofdepartureratesatthenumeratorof(25)annihilatesand

σ

k,h goestozero.

• Theorderofthedenominatorsuggeststhatawaytoreducethe covariance introduced by interposition may consist in spend-ing degrees of freedom to maximize xi and xj in the em-bedded MMPPs, for fixed arrival rates. When applyingsuch a scheme, one shouldhowever take intoaccount the boundsin Proposition 3,since anincrease ofthevalue ofxialsoreduces fittingflexibilityintheembeddedMMPP.

• Since x_i>0foranyi,wenote thatthe signofthecovariance isdeterminedonlybythedifferencesbetweentheper-class ar-rivalrateswithineachofthecomposedM3PPs.

5. Fittingtheinterposedprocess

In thissection we consider two issues that arise in composi-tional fitting based on interposition. First, we consider the deci-sionprobleminvolvingthe mappingofa markedtrace intoa set oftwo-statesM3PPs.Then weshowthattheproblemoffindinga feasibleinterpositionofasetofJsecond-orderM3PPs,wherethe

i-thM3PPmodelsanysubsetKioftheKclasses,canbeformulated asaMILPandhencesolvedusinganintegerprogrammingsolver.

5.1. FittingamarkedtraceintoasetofM3PPs

Givenatrace witharrivalsofKclasses,iftheclassarrivalsare independentitispossibletofitthetraceusingasuperpositionof

KindependentMMPPs,oneforeachclass,andthenreducethesize oftheresultingprocessusinginterposition.However,iftheclasses haveasignificantlylargecovariance,thismethodmaynotproduce goodresults.Therefore,we proposetwo heuristicfittingmethods to address thesetwo cases.We assessthe effectiveness of these methodslaterinSection6.

5.1.1. IndependentMethod

Inthismethod, we ignoreclass covariances andfit each class intoa separate two-stateMMPP.The method issimilar to super-position,but returns a much smaller M3PP[K] withK₊1 states, instead of 2K _{states. Interposition uses the order of composition} obtainedfromtheMILPmethodthatwepresentinSection5.2.

5.1.2. Covariance-basedmethod

We initially build the asymptotic co-variance matrix



= [

σk

_,h]=limt→∞Cov[nk

(

t

)

,nh

(

t

)

] betweeneach pairofclasses.We approximatetheasymptotictimescalewiththelargest finitetfor whichwecan averagecountsoveratleast100 samples.Theuser isthenrequestedtospecifyacovariancethreshold

δ

.Wethen it-erateontheclasses,indecreasingorderofvariance

σ

_k,k.Foreach class k, we first determine all the classes h = k with

σ

h,k ≥

δ

andrecordthedecisiontofittheseclassesintothesameM3PP[K]. The algorithm then continues by analyzing in a similar way the other classes not already planned for fitting in any M3PP. After thisstage, each class k is mapped to a unique M3PP i. In order toobtaina representationforeachM3PPi,we runthe algorithm giveninSection 5.2,whichreturns theparametersofthe embed-ded MMPPs and their order of composition in the interposition.