Mathematical Adventures in Performance Analysis: From Storage Systems, Through Airplane Boarding, to Express Line Queues

296 Pages·2014·2.33 MB·English

by  Eitan Bachmat (auth.)

Modeling and Simulation in Science,
Engineering and Technology
Eitan Bachmat
Mathematical
Adventures in
Performance
Analysis
From Storage Systems, Through
Airplane Boarding, to Express Line
Queues

ModelingandSimulationinScience,EngineeringandTechnology
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Moreinformationaboutthisseriesathttp://www.springer.com/series/4960
Eitan Bachmat
Mathematical Adventures
in Performance Analysis
From Storage Systems, Through Airplane
Boarding, to Express Line Queues
EitanBachmat
DepartmentofComputerScience
Ben-GurionUniversity
BeerSheva,Israel
ISSN2164-3679 ISSN2164-3725(electronic)
ISBN978-3-319-09512-7 ISBN978-3-319-09513-4(eBook)
DOI10.1007/978-3-319-09513-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014949349
MathematicsSubjectClassification(2010):00,68,90
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Preface
Thisbookisaboutvarioustopicsinperformanceanalysis.Theselectionoftopicsis
extremelysubjectiveandinvolvesexclusivelytopicswhichIhappenedtoworkon
inthelast15years.Theotherselectioncriterionwasthatthetopicactuallyhadto
involvesomereasonablyinterestingmathematics;therefore,someworkwhichwas
technologicallyveryrelevant,butmathematicallytrivial,didnotmakethecut.
I did try to be honest in the problem selection. Rather than trying to artificially
invent a performance analysis problem that will fit some math that I knew, I first
learnedabouttheproblemsfromaperformanceanalysisperspective,eitherthrough
realproductdevelopmentorbyreadingsomepapersonthesubject.Infact,Ilearned
alotofnewmathintheprocessoftryingtosolvetheproblemsorbetterunderstand
somepapers.
Thegoalofproducinganelementarytextofreasonablelengththatissuitablefor
advancedundergraduate/beginninggraduatestudentsmeansthatfullproofsarenot
always possible. However, reading the literature in the bibliography will fill those
gaps.Infact,amajormotivationforthebookistoencouragethereaderstoexplore
that literature. With that goal in mind, we end each chapter with notes that point
in related directions, putting the specific problem that we considered in a larger
perspective.
Chapter 1 considers a problem which came up during the design of a commer-
cially available software system that dynamically changes the configuration of a
storage system in order to improve performance [8]. The software could only use
verycoarsedataandtheproblemwastoguaranteethatitwillnotmakebaddecisions
asaresult.Itturnedoutthatthisproblemisrelatedtothetheoryofmetricspaces.
We introduce some basic notions in the theory, such as metric spaces, Euclidean
spaces,` spaces,and`2 spaces.Wethenexplainhowthesenotionstogetherwith
1 2
somebasicresultsinthefieldhelpedusbuildconfidenceinthedecisionsmadeby
theproduct.
v
vi Preface
Chapter2istheresultoftryingtobetterunderstandapaperbyWangetal.[210].
TheauthorsintroducedaverynicemodelforuserI/Oaccesspatternsandshowed
that it captures many of the characteristics of actual user access patterns. They
exploredthemodelempiricallyandmadesomeinsightfulstatements.Weshowhow
themodelcanbestudiedanalyticallyandhowtoactuallyprovethestatements.
Chapter 3 involves an attempt to analyze a practically important performance
featureofdiskdrives.TheycanreorderI/Orequeststoimproveservicetime.While
ourattempttoanalyzetheperformanceadvantagesofthisfeaturefellshort,wewere
abletosolveatoymodel.FollowingasuggestionofStevenSkiena,thetoymodel
canbeusedtomodelairplaneboardingaswell.Themodelseemstodoasomewhat
betterjobatmodelingairplaneboardingthantheoriginaldiskschedulingproblem.
Inastrangetwist,thesamemathhasalreadybeenusedtomodeltheuniverse,i.e.,
itformsthemathematicalbasisforrelativitytheory.Inthechapter,wewillconsider
the concept of discrete space time and study it. We will use this notion to provide
some insights on disk scheduling, airplane boarding, and perhaps the universe as
well.
Chapter 4 deals with the problem of optimally configuring a storage system
whereeachpieceofdatahastwocopiesforfaulttolerancepurposes.Thisscenario
canbeusedtoimprovetheperformanceofdatareadcommandssincewecanchoose
whichcopywillservicetherequest.Theproblemofchoosingagooddatalayoutwas
raisedbyYoavRazatabriefinginwhichIexplainedthedesignofasoftwaretool
which was developed to manage read requests in such a system. It turned out that
one can construct optimal layouts by considering graphs which were constructed
byLubotzky,Phillips,andSarnak.Thesegraphsarerelatedtothearithmeticofthe
quaternions. We also show that some randomized constructions can provide good
solutionsaswell.
Chapter5istheproductofanattempttobetterunderstandacoupleofverynice
papersinqueueingtheory[76,91].Queueingtheoryisastapleofperformanceanal-
ysiswhichdealswithqueuessuchastheoneswemayexperienceinsupermarkets
orbanks.Specifically,wetrytoanalyzeamini-marketwithtwocheckoutcounters,
one serving as an express checkout line. A basic equation in queueing theory, the
Pollaczek–Khinchineequation,allowsustoexpresstheaveragewaitingtimeinsuch
queuesandourmissionistooptimizethistargetfunction.Aswewillshow,queues
ofthistypesatisfyasymmetrywhichgoesbacktotheworksofJacobiandRiemann
onthethetaandzetafunctions.Weexplainthebasicqueueingtheoreticanalysisof
expresslinesandsimilarqueuesandalsopresentJacobiandRiemann’sworkinits
number theoretic context. The end result is that some functions with nice number
theoreticpropertiesalsohavesomenicequeueingtheoreticproperties.
As the alert reader may have noticed, most of the chapters involve the perfor-
mance of storage systems. This is not the first book to mathematically analyze
storage systems. Two inspiring books on the subject have already appeared. The
first [218], by Wong, uses mostly combinatorial methods. Apart from disk-based
systems it analyzes automated tape libraries and drum-based storage. The second
[11], by Aven et al., uses stochastic methods and differential equations to study
cache, drum storage, and disks. The problems we consider are entirely different,
Preface vii
fromthosestudiedinthesemonographs,partlybecausethetechnologyissueshave
changed.
On the subject of queueing theory there are many excellent books. My favorite
basic introduction is [117]. Apart from storage and queues, performance analysis
deals with many other systems including communications systems and networks.
ThereadermaywanttoconsulttheproceedingsoftheannualconferenceSIGMET-
RICSwhichisdevotedtothesubject.
Chapters1,3,4,andtosomeextent,Chap.5aswellarebasedonvariouspapers
that I co-authored [3,12,15–18]. I would like to thank all my colleagues who
participated in these research projects, and in particular Noga Alon, Dani Berend,
Tao-Kai Lam, Luba Sapir, Hagit Sarfati, and Steven Skiena. I learned a lot about
storagesystemsfromDanArnon,NatanVishlitzky,andMosheYanai.BarakWeiss
helpedmewithsomeofthematerialinChap.2.Ihavehadmanyusefuldiscussions
about the material of Chap.3 with Ofer Zeitouni and Percy Deift. Equally helpful
withthematerialinChap.5wereMorHarchol-Balteronthequeueingtheoryside
andPeterSarnakwiththenumbertheory.IamalsoverygratefultoYuriTschinkel
whohasmadenumeroushelpfulcommentsontheentirebook.
I would like to pay special tribute to Avner Magen who made crucial con-
tributions to the material in Chap.1. Avner tragically died in May 2010 in a
mountaineeringaccident.Ihopethisbookcontributesinamodestwaytopreserving
thelegacyofthiswonderfulperson.
I thank the Israeli Science Foundation for supporting my work on this book
through grant 580/11. In particular the research that led to Chap.5 of this book
wassupportedbythisgrant.
Finally I want to thank my parents Yehuda and Menuha and last but certainly
notleastmywonderfulfamily,HagitmywifeandmykidsKatya,Zoe,andDaniel,
withoutwhomlifewouldsimplybealongmarch.
BeerSheva,Israel EitanBachmat

Contents
1 AClassicalModelforStorageSystemActivity........................... 1
1.1 ModelingDiskDrives................................................. 2
1.1.1 CoordinatesforDataLocations............................... 2
1.1.2 SeekTimeFunctions.......................................... 3
1.2 ModelingAccessPatterns............................................. 4
1.2.1 TheIRM ....................................................... 5
1.3 EstimatingSeekTimesintheIRM ................................... 7
1.4 MetricSpaces.......................................................... 10
1.5 TheIRMSeekEstimateIsPessimistic ............................... 12
1.6 NotesforChap.1 ...................................................... 19
2 AFractalModelforStorageSystemActivity............................. 27
2.1 DefinitionoftheModel................................................ 28
2.1.1 MarginalSpatialandTemporalMeasures.................... 30
2.2 AverageSeekDistanceCalculations.................................. 30
2.3 CacheHitRatiosandEntropy......................................... 33
2.3.1 BasicPropertiesofEntropy................................... 34
2.3.2 StaticCacheAlgorithms ...................................... 36
2.4 AsymptoticHitRatioComputations.................................. 36
2.5 Incompatibilityofq-aryandp-aryBiasModels..................... 40
2.6 NotesforChap.2 ...................................................... 47
3 AirplaneBoarding,DiskScheduling,andLorentzianGeometry ...... 51
3.1 TheDiskSchedulingProblem ........................................ 52
3.2 TheABZAlgorithm................................................... 53
3.2.1 PartiallyOrderedSets......................................... 53
3.2.2 PartiallyOrderedSetsandtheDiskSchedulingProblem... 54
3.2.3 EfficientComputationoftheABZAlgorithm ............... 57
ix