Large Deviations for Performance Analysis Queues, Communication and Computing

564 Pages·6.402 MB·English

by  Adam Shwartz, Alan Weiss

Routledge Revivals
Large Deviations for Performance Analysis
Originally published in 1995, Large Deviations for Performance Analysis consists
of two synergistic parts. The first half develops the theory of large deviations
from the beginning, through recent results on the theory for processes with
boundaries, keeping to a very narrow path: continuous-time, discrete-state
processes. By developing only what is needed for the applications, the theory
is kept to a manageable level, both in terms of length and in terms of diffi-
culty. Within its scope, the treatment is detailed, comprehensive and self-
contained. As the book shows, there are sufficiently many interesting appli-
cations of jump Markov processes to warrant a special treatment. The second
half is a collection of applications developed at Bell Laboratories. The appli-
cations cover large areas of the theory of communication networks: circuit
switched transmission, packet transmission, multiple access channels, and the
M/M/1 queue. Aspects of parallel computation are covered as well including,
basics of job allocation, rollback-based parallel simulation, assorted priority
queueing models that might be used in performance models of various com-
puter architectures, and asymptotic coupling of processors. These applications
are thoroughly analysed using the tools developed in the first half of the book.

Large Deviations for
Performance Analysis
Queues, Communication and Computing
Adam Shwartz
Technion — Israel Institute of Technology
Alan Weiss
Nokia Bell Labs
Appendix by Robert J. Vanderbei
Princeton University
Firstpublishedin1995
byCRCPress
Thiseditionfirstpublishedin2018byRoutledge
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v
Tableofcontents
0. WhatthisBookIs,andWhatItIsNot .............................1
0.1. WhattoDowiththisBook ...................................5
0.2. AbouttheFormatoftheBook ................................5
0.3. Acknowledgments ..........................................6
1. LargeDeviationsofRandomVariables ............................9
1.1. HeuristicsandMotivation ................................... 9
1.2. I.I.D.RandomVariables ....................................14
1.3. Examples—I.I.D.RandomVariables .........................19
1.4. I.I.D.RandomVectors ..................................... 23
1.5. EndNotes ................................................ 25
2. GeneralPrinciples ...............................................29
2.1. TheLargeDeviationsPrinciple ............................. 29
2.2. Varadhan’sIntegralLemma .................................33
2.3. TheContractionPrinciple .................................. 36
2.4. EmpiricalMeasures: Sanov’sTheorem ...................... 37
3. RandomWalks,BranchingProcesses .............................45
3.1. TheBallotTheorem ........................................46
3.2. BranchingRandomWalks ..................................48
4. PoissonandRelatedProcesses ................................... 53
4.1. TheOne-DimensionalCase .................................53
4.2. JumpMarkovProcesses ....................................56
4.3. MartingalesandMarkovProcesses ..........................61
5. LargeDeviationsforProcesses ...................................69
5.1. Kurtz’sTheorem .......................................... 75
5.2. PropertiesoftheRateFunction ..............................85
5.3. TheLowerBound ........................................105
5.4. TheUpperBound: Orientation .............................114
5.5. ProofoftheUpperBound .................................117
6. Freidlin-WentzellTheory .......................................129
6.1. TheExitProblem .........................................132
6.2. BeyondtheExitProblem ..................................157
6.3. Discontinuities ...........................................160
6.4. ConvergenceofInvariantMeasures ........................ 164
vi
7. ApplicationsandExtensions ....................................169
7.1. EmpiricalDistributionsofFiniteMarkovProcesses ..........169
7.2. SimpleJumpProcesses ................................... 175
7.3. TheFreeM/M/1Process ..................................179
7.4. MeaningoftheTwistedDistribution ........................183
7.5. EndNotes ...............................................189
8. BoundaryTheory ..............................................191
8.1. TheRateFunctions .......................................195
8.2. PropertiesoftheRateFunction ............................ 201
8.3. ProofoftheUpperBound .................................215
8.4. ConstantCoefficientProcesses .............................220
8.5. TheLowerBound ........................................230
8.6. EndNotes ...............................................242
Applications ........................................................243
9. AllocatingIndependentSubtasks ................................245
9.1. UsefulNotions ...........................................246
9.2. Analysis .................................................249
9.3. EndNotes ...............................................253
10. ParallelAlgorithms: Rollback ..................................255
10.1. RollbackAlgorithms ......................................255
10.2. AnalysisofaRollbackTree ............................... 258
11. TheM/M/1Queue ..............................................261
11.1. TheModel ...............................................261
11.2. HeuristicCalculations .....................................262
11.3. MostProbableBehavior ...................................267
11.4. ReflectionMap ...........................................272
11.5. TheExitProblemandSteadyState .........................274
11.6. TheProbabilityofHittingaPoint ..........................277
11.7. TransientBehavior ....................................... 279
11.8. ApproachtoSteadyState ..................................281
11.9. FurtherExtensions ........................................283
11.10. EndNotes ...............................................286
vii
12. Erlang’sModel .................................................289
12.1. ScalingandPreliminaryCalculations .......................290
12.2. StartingwithanEmptySystem ............................ 292
12.3. StartingwithaFullSysteminLightTraffic ..................300
12.4. Justification ..............................................302
12.5. Erlang’sModel: GeneralStartingPoint .....................305
12.6. LargeDeviationsTheory ..................................308
12.7. ExtensionstoErlang’sModel ..............................313
12.8. TransientBehaviorofTrunkReservation ....................318
12.9. EndNotes ...............................................323
13. TheAnick-Mitra-SondhiModel .................................325
13.1. TheSimpleSourceModel .................................328
13.2. BufferStatistics ..........................................330
13.3. SmallBuffer .............................................341
13.4. LargeBuffer .............................................345
13.5. ConsequencesoftheSolution ..............................348
13.6. Justification ..............................................350
13.7. ControlSchemes .........................................361
13.8. MultipleClasses ..........................................378
13.9. EndNotes ...............................................386
14. Aloha ..........................................................387
14.1. TheI.D.ModelandHeuristics .............................387
14.2. RelatedModels .......................................... 394
14.3. BasicAnalysis ...........................................397
14.4. LargeDeviationsofAloha .................................398
14.5. Justification ..............................................400
14.6. AParadox—Resolved .................................... 405
14.7. SlottedAlohaModels .....................................409
14.8. EndNotes ...............................................417
15. PriorityQueues ................................................419
15.1. PreemptivePriorityQueue ................................ 421
15.2. MostProbableBehavior—PP ..............................422
15.3. TheVariationalProblem—PP ..............................423
15.4. ProbabilisticQuestions—PP ...............................436
15.5. Justification—PP .........................................438
15.6. ServetheLongestQueue ..................................443
viii
15.7. MostProbableBehavior—SL ..............................444
15.8. TheMainResult—SL .....................................445
15.9. Justification—SL .........................................446
15.10. JointheShortestQueue ...................................454
15.11. EndNotes ...............................................458
16. TheFlatto-Hahn-Wrightmodel .................................459
16.1. MostProbableBehavior ...................................462
16.2. FormalLargeDeviationsCalculations ......................463
16.3. JustificationoftheCalculation .............................467
16.4. EndNotes ...............................................470
A. AnalysisandProbability ........................................471
A.1. Topology,MetricSpaces,andFunctions ....................471
A.2. OrdinaryDifferentialEquations ............................482
A.3. ProbabilityandIntegration ................................ 483
A.4. Radon-NikodymDerivatives ...............................493
A.5. StochasticProcesses ......................................495
B. Discrete-SpaceMarkovProcesses ...............................499
B.1. GeneratorsandTransitionSemigroups ......................499
B.2. TheMarkovProcess ......................................501
B.3. Birth-DeathProcesses .....................................503
B.4. Martingales ..............................................505
C. CalculusofVariations ..........................................515
C.1. HeuristicsoftheCalculusofVariations .....................515
C.2. CalculusofVariationsandLargeDeviations .................518
C.3. One-DimensionalTricks .................................. 520
C.4. ResultsfromtheCalculusofVariations .....................525
D. LargeDeviationsTechniques ....................................527
D.1. TheGa¨rtner-EllisTheorem ................................527
D.2. SubadditivityArguments ..................................533
D.3. ExponentialApproximations ...............................533
D.4. LevelII ..................................................534
References ..........................................................539
Index ...............................................................551
Chapter 0
What this Book Is, and What It Is Not
Thefieldofcommunicationandcomputernetworksisbustlingwithactivity. One
oftheactiveareasfallsundertherubric“performance.” Researchersanddevel-
opment engineers tackle systems that are huge, complex and fast; think of the
telephone network in the United States. The resulting models are, for the most
part, discrete-event, continuous time stochastic processes, technically known as
jump Markov processes. The objective is to analyze the behavior of these sys-
tems,withthegoalofdesigningsystemsthatprovidebetterservice. “Better”may
mean faster, less prone to error and breakdown, more efficient, or improved by
manyothercriteria.
Until quite recently, the tools brought to bear on these problems were appro-
priate for small, simplesystems. Some of these methods take into account only
averagebehavior(orperhapsvariances). Butthisisoftennotenough,astheper-
formanceofmanysystemsislimitedbyeventswithasmallprobabilityofoccur-
ring,butwithconsequencesthataresevere. Clearly,newtoolsareneeded. Com-
putersimulationisonerelativelynewtool. Butthismethod,forallitspower,is
limitedinthatitusuallydoesnotproviderulesofthumbfordesign,maynotgive
estimatesonthesensitivityofresultstovariousparameters,andcanbeextremely
costly in terms of both computer time and programming (especially debugging)
time. Analyticmethodsclearlyretainsomeadvantages. Thisbookisaboutafairly
newanalyticmethodcalledlargedeviations.
Large deviations is a mathematical theory that is very active at present. The
theorydealswithrareevents,andisasymptoticinnature;itisthusanaturalcan-
didateforanalyzingrareeventsinlargesystems. Thetheoryoflarge deviations
includesasetoftechniquesforturninghardprobabilityproblemsthatconcerna
class of rare events into analytic problems in the calculus of variations. It also
provides a nice qualitative theory for understanding rare events. As an asymp-
totictechnique,itseffectivenessresidesintherelativesimplicitywithwhichone
mayanalyzesystemswhosesizemaybegrowingwiththeasymptoticparameter,
or whose “conditioning” may be getting worse. The theory is often useful even
when simulation or other numerical techniques become increasingly difficult as
theparametertendstoitslimit.
However, the theory is noted for being technically (mathematically) very de-
manding,andsolvingaprobleminthecalculusofvariationsisnottypicallyanen-
gineer’sdream. Althoughthetheoryisbeingincreasinglyusedforanalyzingrare
eventsinlargesystems,thisisdonebyarelativelysmallnumberofresearchers.
Webelievethatthereasonforthisstateofaffairsisthatthetheoryisnoteasilyac-
cessibletonon-mathematicians,andthefinalresultsseemtorequireanadditional
1
2 Chapter0. WHATTHISBOOKIS,ANDWHATITISNOT
translationtoengineeringlingo. Hence
Largedeviationsisuseful.
Largedeviationsisformidablytechnical.
What’sastudenttodo?
Hereiniscontainedonepointofviewonwhat’stodo. Wedevelopthetheory
oflargedeviationsfromthebeginning(independent,identicallydistributed(i.i.d.)
randomvariables)throughrecentresultsonthetheoryforprocesseswithbound-
aries, keeping to a very narrow path: continuous-time, discrete-state processes.
Bydevelopingonlywhatweneedfortheapplicationswepresent,wetrytokeep
thetheorytoamanageablelevel,bothintermsoflengthandintermsofdifficulty.
Wemakenoparticularclaimtooriginalityofthetheorypresentedherein,except
forthematerialconcerningboundaries,whichisthesubjectofChapter8. Mostof
thetrailblazingworkofFreidlinandWentzell[FW],andofDonskerandVaradhan
[DV1–DV4]goesfurtherthanwedo. Also,othershavesubsequentlytreatedthe
generaltheorymuchmorethoroughly;e.g.Ellis[Ell],Wentzell[Wen],Deuschel
andStroock[DeS],DemboandZeitouni[DZ],andtherecentworkofDupuisand
Ellis[DE2]. Wehave,however,formulatedacomplete,self-containedsetoftheo-
remsandproofsforjumpMarkovprocesses. Sinceourscopeislimitedtoaclass
of relatively simple processes, the theory is much more accessible, and less de-
mandingmathematically. Toenhancethepedagogicalvalueofthiswork,wehave
attemptedtoconveyasmuchintuitionaswecould,andtokeepthestylefriendly.
In addition, we present for the first time a complete theory for processes with a
flatboundary,andforsomeprocessesinarandomenvironment. Thelevelofthe
bookissomewhatuneven,asindicatedinthedependencechartFigure0.1. Thisis
purposeful—webelievethataneophytewouldnotwanttoreadthedifficultchap-
ters,andthatanexpertdoesn’twantasmuchhandholdingasabeginner.
Webelievethatourapplicationsareimportantenoughtorequirenoapologies.
AsMark Kacsaid, “Theoremscomeand go,but anapplication isforever.” Our
applicationscoverlargeareasofthetheoryofcommunicationnetworks: circuit-
switchedtransmission(Chapter12),packettransmission(Chapter13),multiple-
accesschannels(Chapter14),andthe M/M/1queue(Chapter11). Wecoveras-
pects of parallel computationin a much more spottyfashion: basicsof job allo-
cation(Chapter9),rollback-basedsimulation(Chapter10),andassortedpriority
queuingmodels(Chapters15and16)thatmaybeusedinperformancemodelsof
variouscomputerarchitectures.
The key word in the phrase “our applications” is “our.” We present only our
ownresultsconcerningtheapplications. Wedonotsynthesizeexistingtheoryex-
ceptinournarrowfashionforjumpprocesses. Weignorepossibleimprovements
inordertoremainwithintherealmofthoselargedeviationsboundsthatweactu-
allyuse. Forexample,Anantharam’sbeautifulresultsontheG/G/1queue[An]
arecertainlyrelevanttothesubjectsweaddress,buthistechniquesaredifferent.
WedonotobtainthebestresultsknownforjumpMarkovprocesses. Itiscertainly
arguablewhetherthisisawisechoice. However,wewantedtopresentaconsis-