Table Of ContentSPRINGER BRIEFS IN MATHEMATICS
Peter Buchholz
Jan Kriege
Iryna Felko
Input Modeling
with Phase-Type
Distributions and
Markov Models
Theory and
Applications
123
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Peter Buchholz • Jan Kriege • Iryna Felko
Input Modeling with
Phase-Type Distributions
and Markov Models
Theory and Applications
123
PeterBuchholz JanKriege
DepartmentofComputerScience DepartmentofComputerScience
TechnicalUniversityofDortmund TechnicalUniversityofDortmund
Dortmund,Germany Dortmund,Germany
IrynaFelko
DepartmentofComputerScience
TechnicalUniversityofDortmund
Dortmund,Germany
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-3-319-06673-8 ISBN978-3-319-06674-5(eBook)
DOI10.1007/978-3-319-06674-5
SpringerChamHeidelbergNewYorkDordrechtLondon
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Mathematics Subject Classification (2010): 60J22, 60J28, 62M05, 65C40, 68M20, 68U20, 60K25,
65C10,65C40,90B15,90B18,90B25
©PeterBuchholz,JanKriege,IrynaFelko2014
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Preface
Nowadays system analysis of man-made systems, like computer systems, com-
munication networks, and manufacturing plants, and also of natural systems, like
biological or social systems, is often model based. To capture the complexity of
real systems stochastic discrete event models are used in many application areas.
Oneofthe keyaspectsinbuildingsuchmodelsisthe adequatedescriptionofreal
processesandeventstreamsinastochasticmodel.Oftensimpledistributionsarenot
sufficientforthispurposebecauseobserveddistributionsaremultimodalandevents
arecorrelated.
One class of stochastic models, which allows one to describe multimodal
distributions and correlated event times, are Markov processes with marked tran-
sitions. Since Markov processes can be analyzed with numerical methods and
with stochastic simulation, they are an ideal candidate to describe event times
in stochastic models. However, the big disadvantage of using Markov processes
insteadofsimpledistributionsorstochasticprocesses,likeautoregressiveormoving
averagetimeseries,istheparameterizationeffort.Usually,thefindingofadequate
parametersofaMarkovmodel,tocapturesomeobservedbehavior,isanon-linear
optimization problem with many parameters and non-unique representations of a
givenstochasticdistributionorprocess.Thisoftenprohibitsthewideruseofthose
models, in particular in stochastic simulation, and is the reason that only fairly
simple phase-type distributions can be found in textbookson stochastic modeling
orsimulation.
Thisbookssummarizesourworkontheparameterizationofphase-typedistribu-
tionsandMarkovianarrivalprocesses,whicharethecommonlyusedmodeltypes
inmodelingeventstreams.Tothebestofourknowledgeitisthefirsttimethatthe
availablemethodsarecollectedinatextbook.Wehopethatthishelpstosupportthe
useofthementionedmodelsinstochasticmodelingandinparticularinstochastic
simulation.
We thankourcolleaguesAxelThümmler,AndriyPanchenko,andFalko Bause
whoworkedtogetherwithusonseveralaspectsofparameterfittingforMarkovian
arrivalprocesses.PeterKemperfromthecollegeofWilliamandMarry(Williams-
burg, VA, USA) spent his sabbatical in our group and worked together with us
v
vi Preface
on the parameterizationofMarkovianarrivalprocesseswith multiple eventtimes.
We learned a lot about phase-type distributions and Markovian arrival processes
fromMiklósTelek,GáborHorváth,andLeventeBodrogfromtheStochasticMod-
ellingLaboratoryoftheDepartmentofTelecommunications,TechnicalUniversity
Budapest. We thank them for a long lasting and very fruitful cooperation. The
research work has been supported by the Deutsche Forschungsgemeinschaft and
bytheDeutscheAkademischeAustauschdienst.
Dortmund,Germany PeterBuchholz
Dortmund,Germany IrynaFelko
Dortmund,Germany JanKriege
March2014
Contents
1 Introduction .................................................................. 1
2 Phase-TypeDistributions ................................................... 5
2.1 BasicDefinitions........................................................ 5
2.1.1 MarkovChains.................................................. 5
2.1.2 AbsorbingMarkovChainsandPhase-TypeDistributions.... 7
2.1.3 AnalysisofPhase-TypeDistributions ......................... 10
2.2 SimilarityandEquivalence............................................. 12
2.2.1 SimilarityTransformations..................................... 13
2.2.2 LumpingandGeneralEquivalence............................ 14
2.3 AcyclicPhase-TypeDistributions...................................... 15
2.3.1 ErlangandHyper-ErlangDistributions........................ 16
2.3.2 CoxianDistributions............................................ 20
2.3.3 CanonicalRepresentations..................................... 21
2.4 Properties................................................................ 24
2.5 ConcludingRemarks.................................................... 27
3 ParameterFittingforPhaseTypeDistributions ......................... 29
3.1 TraceBasedFitting ..................................................... 29
3.1.1 DefinitionofTracesandDerivedMeasures................... 30
3.1.2 ExpectationMaximizationApproachforGeneral
PHDistributions................................................ 33
3.1.3 Expectation Maximization Approach
forHyper-ErlangDistributions................................. 40
3.1.4 Expectation Maximization Approach
forCanonicalRepresentations ................................. 46
3.1.5 DensityBasedParameterFitting............................... 49
3.2 MomentsBasedFitting................................................. 51
3.2.1 ClosedFormEquations......................................... 52
3.2.2 LeastSquaresBasedTechniques............................... 60
3.3 ConcludingRemarks.................................................... 62
vii
viii Contents
4 MarkovianArrivalProcesses .............................................. 63
4.1 DefinitionandBasicResults ........................................... 63
4.1.1 DefinitionofMAPs............................................. 63
4.1.2 AnalysisofMAPs .............................................. 66
4.1.3 EquivalentRepresentationsofMAPs.......................... 67
4.1.4 MAPsasCountingProcesses .................................. 68
4.2 MAPsofOrder2........................................................ 70
4.3 BMAPsandMMAPs ................................................... 71
4.4 Properties................................................................ 73
4.5 ConcludingRemarks.................................................... 74
5 ParameterFittingofMAPs................................................. 75
5.1 MomentandJointMomentBasedFitting ............................. 75
5.1.1 ParameterFittingforMAPswithTwoStates ................. 76
5.1.2 ACompositionalApproach.................................... 79
5.2 TraceBasedFittingofMAPs........................................... 80
5.3 TwoPhaseApproaches................................................. 84
5.3.1 JointMomentFitting ........................................... 84
5.3.2 AutocorrelationFitting ......................................... 87
5.3.3 IterativeEMApproaches....................................... 89
5.4 FittingoftheCountingProcess ........................................ 90
5.5 ConcludingRemarks.................................................... 93
6 StochasticModelsIncludingPHDistributionsandMAPs.............. 95
6.1 QueueingSystems ...................................................... 95
6.1.1 SingleQueues................................................... 95
6.1.2 QueueingNetworks............................................. 98
6.2 ModelingReliabilityandAvailability.................................. 102
6.2.1 ModelingFailureandRepairTimes ........................... 102
6.2.2 ReliabilityandAvailabilityModels............................ 104
6.3 SimulationModels...................................................... 105
6.3.1 GeneratingRandomNumbersfromPHDsandMAPs........ 105
6.3.2 SimulationModelswithPHDsandMAPs .................... 107
6.4 ConcludingRemarks.................................................... 110
7 SoftwareTools................................................................ 111
7.1 SoftwareToolsforGeneratingPHDistributions...................... 111
7.2 SoftwareToolsforGeneratingMAPs.................................. 112
7.3 SoftwareToolsforAnalyzingModelswithPHDsandMAPs........ 113
8 Conclusion.................................................................... 115
References......................................................................... 117
Index............................................................................... 125
Acronyms
ALS Alternatingleastsquaresapproach
APHD Acyclicphase-typedistribution
BMAP BatchMarkovianarrivalprocess
BS BasicseriesofanAPH
cdf Cumulativedistributionfunction
CTMC ContinuoustimeMarkovchain
EM Expectationmaximization
HErD Hyper-Erlangdistribution
IPP InterruptedPoissonprocess
KPC Kroneckerproductcomposition
MAP Markovianarrivalprocess
ME Matrixexponential
ML Maximum-likelihood
MMAP MarkedMarkovianarrivalprocess
MMPP MarkovmodulatedPoissonprocess
MRAP Markedrationalarrivalprocess
NNLS Non-negativeleastsquaresapproach
PH Phase-type
PHD Phase-typedistribution
QBD Quasi-birth-deathprocess
QN Queueingnetwork
RAP Rationalarrivalprocess
ix
