Analyzing markov chains using kronecker products : theory and applications

97 Pages·2012·0.649 MB·English

by  Tuğrul Dayar

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Tugˇrul Dayar
Analyzing Markov Chains
using Kronecker Products
Theory and Applications
123
TugˇrulDayar
DepartmentofComputerEngineering
BilkentUniversity
Ankara,Turkey
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-1-4614-4189-2 ISBN978-1-4614-4190-8(eBook)
DOI10.1007/978-1-4614-4190-8
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2012939483
MathematicsSubjectClassification(2010):60J27,60J10,60-04,15A69,15B51,15-04,65C20,65C40,
65F50, 65F10, 65F08, 65F05, 65-04, 37A30, 37A50, 37B25, 93D30, 37-04, 93-04, 60J28, 60J20,
68M15,90B25,68M20,68-04,90B22,90-04,80A30,80-04,92E20,92C40,92C42,92C45,92D25,
92-04
©TugˇrulDayar2012
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To thememoryof myfather

Preface
This book has grown out of work spanning the last 15 years. Billy Stewart and
BrigittePlateauintroducedmetostochasticautomatanetworks(SANs)duringmy
visit to Grenoblein June 1996.A study on robotictape librariesusing SANs was
carriedoutin Augustthatyear with OdysseasPentakalosand BrookeStephensin
Greenbelt,Maryland.ThehelpreceivedduringthisprocessfromPauloFernandes
regarding the software was instrumental. Later I had many interesting talks with
Jean-MicheleFourneauandFranckQuessetteoverSANsinVersaillesandAnkara
from1998to2000.NihalPekerginwasalsopresentduringthesevisitsandbrought
her expertise on stochastic comparison into the picture. In June 2000, we had
enjoyablediscussionswithIvoMarekandPetrMayerinPragueontheconvergence
properties of iterative aggregation–disaggregation. In the academic year 2002–
2003,I gota chanceto learn abouthierarchicalMarkovianmodels(HMMs) from
Peter Buchholz in Dresden. Our discussions on HMMs continued in Dortmund
in June 2005 and at Dagstuhl in February 2007. It was in Dortmund and then
Ankara where we investigated compositional Markovian models for symmetries
withPeterKemperin2005.KishorTrivedivisitedAnkarain1997;hewasalways
availablebye-mailin2008whilewewerewritingajointpaperandwasreadyfor
further discussions later that year in Seattle. In early 2010 in Saarbru¨cken, I was
convincedby VerenaWolf of the difficultiesassociatedwith analyzingsystemsof
stochastic chemical kinetics using SANs and HMMs. This also led to stimulating
exchangeswith Holger Hermanns, Werner Sandmann, and David Spieler. I thank
them all for providing a scholarly atmosphere in which to carry out research and
for their time. I am grateful to Bilkent University for being very understanding
and generous in granting these research leaves without which it would not have
beenpossibletowritethisbook.Iamalsofortunatetohaveworkedwithanumber
of students at Bilkent who foundcompositionalKroneckermodelsfor Markovian
systems such as SANs and HMMs interesting and exciting: Ertugˇrul Uysal, Oleg
Gusak, Akın Meric¸, ˙Ilker Nadi Bozkurt,and Muhsin Can Orhan.Grants available
in one form or another over the years allowed me to continue working on the
subjectfromtheTurkishScientificandTechnologicalResearchCouncil,theFrench
National Scientific Research Center, the Center of Excellence in Space Data and
vii
viii Preface
InformationSystems,theCzechTechnicalUniversity,theAlexandervonHumbodt
Foundation, the Turkish Academy of Sciences, and the Cluster of Excellence in
Multimodal Computing and Interaction. Finally, I thank Donna Chernyk and her
teamatSpringer,NewYorkforinitiatingthebookproject,walkingmethroughthe
process, and making it happen. I also thank Lesley Poliner from Springer for the
editorial process she conducted and Udaiyar Rekha from SPi Technologies, India
forhereffortduringthe productionof thebook.Partsofthe bookappearedin the
proceedingsoftheMarkovAnniversaryMeetingin2006[55].Ihopeyouenjoythe
outcome.
Ankara,Turkey TugˇrulDayar
Contents
1 Introduction ................................................................... 1
2 Preliminaries .................................................................. 9
2.1 KroneckerRepresentation............................................... 10
2.2 Vector–KroneckerProductMultiplicationAlgorithm.................. 14
2.3 Preprocessing............................................................. 17
3 IterativeMethods ............................................................. 21
3.1 SplittingtheSmallerMatrices........................................... 22
3.2 BlockIterativeMethods ................................................. 25
3.3 PreconditionedProjectionMethods..................................... 29
3.4 MultilevelMethods ...................................................... 30
4 DecompositionalMethods.................................................... 37
4.1 ExactVersusApproximative ............................................ 37
4.2 HandlingUnreachableStates............................................ 41
4.3 CaseStudyfromClosedQueueingNetworks .......................... 44
5 Matrix-AnalyticMethods.................................................... 57
5.1 Level-DependentQuasi-Birth-and-DeathProcesses ................... 57
5.2 HandlingInfiniteness .................................................... 59
5.3 CaseStudyfromStochasticChemicalKinetics ........................ 60
6 Conclusion..................................................................... 75
References.......................................................................... 77
Index................................................................................ 85
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