Performance, Reliability, and Availability Evaluation of Computational Systems, Volume I: Performance and Background

841 Pages·2023·22.377 MB·English

by  Paulo Romero, Martins Maciel

Performance,
Reliability, and
Availability Evaluation
of Computational
Systems, Volume 1
This textbook intends to be a comprehensive and substantially self-contained two-volume
book covering performance, reliability, and availability evaluation subjects. The volumes
focus on computing systems, although the methods may also be applied to other systems.
The first volume covers Chapter 1 to Chapter 14, whose subtitle is “Performance Model-
ing and Background”. The second volume encompasses Chapter 15 to Chapter 25 and has
the subtitle “Reliability and Availability Modeling, Measuring and Workload, and Lifetime
Data Analysis”.
This text is helpful for computer performance professionals for supporting planning, de-
sign, configuring, and tuning the performance, reliability, and availability of computing
systems. Such professionals may use these volumes to get acquainted with specific sub-
jects by looking at the particular chapters. Many examples in the textbook on computing
systems will help them understand the concepts covered in each chapter. The text may also
be helpful for the instructor who teaches performance, reliability, and availability evalua-
tion subjects. Many possible threads could be configured according to the interest of the
audience and the duration of the course. Chapter 1 presents a good number of possible
courses programs that could be organized using this text.
Volume 1 is composed of the first two parts, besides Chapter 1. Part I gives the knowledge
required for the subsequent parts of the text. This part includes six chapters. It covers an
introduction to probability, descriptive statistics and exploratory data analysis, random
variables, moments, covariance, some helpful discrete and continuous random variables,
Taylor series, inference methods, distribution fitting, regression, interpolation, data scal-
ing, distance measures, and some clustering methods. Part II presents methods for perfor-
mance evaluation modeling, such as operational analysis, Discrete-Time Markov Chains
(DTMC), and Continuous Time Markov Chains (CTMC), Markovian queues, Stochastic
Petri nets (SPN), and discrete event simulation.

Performance,
Reliability, and
Availability Evaluation
of Computational
Systems, Volume 1
Performance and Background
Paulo Romero
Martins Maciel
First edition published 2023
by CRC Press
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ISBN: 978-1-032-29537-4 (hbk)
ISBN: 978-1-032-30639-1 (pbk)
ISBN: 978-1-003-30601-6 (ebk)
DOI: 10.1201/9781003306016
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Dedication
TotheOneandTriuneGod,theHolyMysterythatisWhollyLove.

Contents
Preface.....................................................................................................................xiii
Acknowledgement...................................................................................................xv
Chapter1 Introduction.....................................................................................1
1.1 AnOverview...........................................................................1
1.2 AGlanceatEvaluationPlanning............................................9
PART I Fundamental Concepts
Chapter2 IntroductiontoProbability............................................................25
2.1 SetsandAlgebraofSets.......................................................25
2.2 Probability.............................................................................29
2.3 ConditionalProbability.........................................................34
2.4 Independence........................................................................39
2.5 Bayes’RuleandtheLawofTotalProbability......................40
2.6 Counting...............................................................................45
2.6.1 N-Permutation.........................................................45
2.6.2 KoutofNPermutationwithReplacement.............47
2.6.3 KoutofNPermutationwithoutReplacement.......48
2.6.4 KoutofNCombinationwithoutReplacement......48
2.6.5 KoutofNCombinationwithReplacement...........49
Chapter3 ExploratoryDataAnalysis............................................................55
3.1 DiagramsandPlots...............................................................55
3.2 StatisticsofCentralTendency.............................................62
3.3 MeasuresofDispersion........................................................66
3.4 StatisticsofShape(AsymmetryandKurtosis).....................68
3.5 Outliers.................................................................................70
Chapter4 IntroductiontoRandomVariables.................................................79
4.1 DiscreteRandomVariables...................................................79
4.2 ContinuousRandomVariables.............................................90
4.3 Moments...............................................................................98
4.4 JointDistributions...............................................................110
4.4.1 JointDiscreteRandomVariables..........................111
4.4.2 JointContinuousRandomVariables.....................115
4.4.3 Convolution...........................................................120
4.4.4 Expect.andVar.ofProd.ofRand.Variab............124
4.4.5 Expect.andVar.ofSumsofRand.Variab............128
4.5 SummaryofPropertiesofExpectationandVariance.........131
4.6 Covariance,Correlation,andIndependence.......................132
vii
viii Contents
Chapter5 SomeImportantRandomVariables.............................................139
5.1 SomeDiscreteRandomVariables......................................139
5.1.1 Bernoulli...............................................................139
5.1.2 Geometric............................................................141
5.1.3 Binomial..............................................................144
5.1.4 NegativeBinomial...............................................147
5.1.5 Hypergeometric...................................................150
5.1.6 Poisson.................................................................152
5.2 SomeContinuousRandomVariables.................................156
5.2.1 Uniform................................................................156
5.2.2 Triangular.............................................................158
5.2.3 Normal.................................................................160
5.2.4 Chi-Square...........................................................165
5.2.5 Student’st............................................................168
5.2.6 FDistributions.....................................................171
5.2.7 Exponential..........................................................174
5.2.8 Gamma.................................................................178
5.2.9 Phase-Type...........................................................181
5.2.10 Erlang...................................................................181
5.2.11 Hypoexponential..................................................185
5.2.12 Hyperexponential.................................................190
5.2.13 Cox.......................................................................193
5.2.14 Weibull.................................................................197
5.3 FunctionsofaRandomVariable.........................................202
5.4 TaylorSeries.......................................................................207
Chapter6 StatisticalInferenceandDataFitting..........................................219
6.1 ParametricConfidenceIntervalforMean...........................219
6.1.1 ConfidenceIntervalwhenVarianceisKnown......221
6.1.2 ConfidenceIntervalwhenVarianceisUnknown..225
6.2 ParametricConfidenceIntervalforSD2andSD.................228
6.3 ParametricConfidenceIntervalforProportion...................231
6.3.1 ParametricConfid.Interv.for pbasedonb(n,k).231
6.3.2 ParametricConfid.Interv.for pbasedonN(µ,σ)235
6.4 ParametricConfidenceIntervalforDifference...................237
6.4.1 ConfidenceIntervalforPairedComparison.........237
6.4.2 Conf.Interv.forNon-Corresp.Measurements.....241
6.5 Bootstrap.............................................................................246
6.5.1 BasicBootstrap.....................................................246
6.5.2 Bootstrap-t............................................................249
6.5.3 Semi-ParametricBootstrap...................................252
6.6 GoodnessofFit...................................................................255
6.6.1 Probability–ProbabilityPlotMethod....................255
6.6.2 χ2Method.............................................................258
6.6.3 Kolmogorov-SmirnovMethod..............................262
Contents ix
6.7 DataFitting.........................................................................268
6.7.1 LinearRegression.................................................268
6.7.2 PolynomialRegression.........................................276
6.7.3 ExponentialRegression........................................279
6.7.4 Lagrange’sPolynomial.........................................282
Chapter7 DataScaling,Distances,andClustering......................................295
7.1 DataScaling........................................................................295
7.2 DistanceandSimilarityMeasures......................................313
7.3 ClusterDistances................................................................318
7.4 Clustering:anintroduction.................................................323
7.5 K-Means.............................................................................330
7.6 K-MedoidandK-Median...................................................337
7.7 HierarchicalClustering.......................................................341
PART II Performance Modeling
Chapter8 OperationalAnalysis...................................................................365
8.1 UtilizationLaw...................................................................367
8.2 ForcedFlowLaw................................................................368
8.3 DemandLaw.......................................................................369
8.4 Little’sLaw.........................................................................372
8.5 GeneralResponseTimeLaw..............................................376
8.6 InteractiveResponseTimeLaw..........................................378
8.7 BottleneckAnalysisandBounds........................................382
Chapter9 DiscreteTimeMarkovChain......................................................393
9.1 StochasticProcesses...........................................................393
9.2 Chapman-KolmogorovEquation........................................398
9.3 TransientDistribution.........................................................404
9.4 SteadyStateDistribution....................................................407
9.5 ClassificationofStates,MRTandMFPT...........................408
9.6 HoldingTime(SojournTimeorResidenceTime).............421
9.7 MeanTimetoAbsorption...................................................423
9.8 SomeApplications..............................................................425
Chapter10 ContinuousTimeMarkovChain.................................................439
10.1 RateMatrix.........................................................................439
10.2 Chapman-KolmogorovEquation........................................442
10.3 HoldingTimes....................................................................448
10.4 StationaryAnalysis.............................................................450
10.4.1 GaussElimination.................................................452
10.4.2 Gauss-SeidelMethod............................................458
10.5 TransientAnalysis..............................................................463
10.5.1 IntervalSubdivision..............................................464