Probability and Statistics with
Reliability, Queuing, and
Computer Science Applications, Second Edition, John Willey & Sons
Contents
1. Introduction
- 1.1 Motivation
- 1.2 Probability Models
- 1.3 Sample Space
- 1.4 Events
- 1.5 Algebra of Events
- 1.6 Graphical Methods of Representing Events
- 1.7 Probability Axioms
- 1.8 Combinatorial Problems
- 1.8.1 Ordered Samples of Size k, with Replacement
- 1.8.2 Ordered Samples of Size k, without Replacement
- 1.8.3 Unordered Samples of Size k, without Replacement
- 1.9 Conditional Probability
- 1.10 Independence of Events
- 1.11 Bayes' Rule
- 1.12 Bernoulli Trials
2 Discrete Random Variables
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2.1 Introduction
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2.2 Random Variables and Their Event Spaces
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2.3 The Probability Mass Function
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2.4 Distribution Functions
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2.5 Special Discrete Distributions
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2.5.1 The Bernoulli pmf
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2.5.2 The Binomial pmf
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2.5.3 The Geometric pmf
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2.5.4 The Negative Binomial pmf
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2.5.5 The Poisson pmf
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2.5.6 The Hypergeometric pmf
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2.5.7 The Discrete Uniform pmf
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2.5.8 Constant Random Variable
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2.5.9 Indicator Random Variable
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2.6 Analysis of Program MAX
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2.7 The Probability Generating Function
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2.8 Discrete Random Vectors
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2.9 Independent Random Variables
3 Continuous Random Variables
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3.1 Introduction
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3.2 The Exponential Distribution
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3.3 The Reliability and Failure Rate
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3.4 Some Important Distributions
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3.4.1 Hypoexponential Distribution
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3.4.2 Erlang and Gamma Distribution
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3.4.3 Hyperexponential Distribution
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3.4.4 Weibull Distribution
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3.4.5 Log-Logistic Distribution
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3.4.6 Normal or Gaussian Distribution
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3.4.7 The Uniform or Rectangular Distribution
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3.4.8 Pareto Distribution
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3.4.9 Defective Distribution
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3.5 Functions of a Random Variable
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3.6 Jointly Distributed Random Variables
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3.7 Order Statistics
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3.8 Distribution of Sums
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3.9 Functions of Normal Random Variables
4 Expectation
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4.1 Introduction
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4.2 Moments
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4.3 Expectation Based on Multiple Random Variables
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4.4 Transform Methods
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4.5 Moments and Transforms of Some Distributions
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4.5.1 Discrete Uniform Distribution
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4.5.2 Bernoulli pmf
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4.5.3 Binomial Distribution
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4.5.4 Geometric Distribution
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4.5.5 Poisson pmf
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4.5.6 Continuous Uniform Distribution
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4.5.7 Exponential Distribution
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4.5.8 Gamma Distribution
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4.5.9 Hypoexponential Distribution
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4.5.10 Hyperexponential Distribution
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4.5.11 Weibull Distribution
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4.5.12 Log-logistic Distribution
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4.5.13 Pareto Distribution
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4.5.14 The Normal Distribution
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4.6 Computation of Mean Time to Failure
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4.6.1 Series System
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4.6.2 Parallel System
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4.6.3 Standby Redundancy
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4.6.4 TMR and TMR/Simplex Systems
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4.6.5 The k-out-of-n System
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4.6.6 The Hybrid k-out-of-n System
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4.7 Inequalities and Limit Theorems
5 Conditional Distribution and Expectation
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5.1 Introduction
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5.2 Mixture Distributions
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5.3 Conditional Expectation
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5.4 Imperfect Fault Coverage and Reliability
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5.5 Random Sums
6 Stochastic Processes
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6.1 Introduction
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6.2 Classification of Stochastic Processes
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6.3 The Bernoulli Process
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6.4 The Poisson Process
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6.5 Renewal Processes
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6.6 Availability Analysis
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6.7 Random Incidence
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6.8 Renewal Model of Program Behavior
7 Discrete-Time Markov Chains
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7.1 Introduction
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7.2 Computation of n-step Transition Probabilities
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7.3 State Classification and Limiting Probabilities
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7.4 Distribution of Times Between State Changes
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7.5 Markov Modulated Bernoulli Process
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7.6 Irreducible Finite Chains with Aperiodic States
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7.6.1 Memory Interference in Multiprocessor Systems
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7.6.2 Models of Program Memory Referencing Behavior
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7.6.2.1 The Independent Reference Model
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7.6.2.2 Performance Analysis of Cache Memories
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7.6.2.3 The LRU Stack Model
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7.6.3 Slotted Aloha Model
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7.6.4 Performance Analysis of an ATM Multiplexer
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7.7 The M/G/1 Queuing System
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7.8 Discrete-Time Birth--Death Processes
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7.9 Finite Markov Chains with Absorbing States
8 Continuous-Time Markov Chains
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8.1 Introduction
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8.2 The Birth--Death Process
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8.2.1 The M/ M/1 Queue
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8.2.2 The M/M/m Queue
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8.2.3 Finite State Space
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8.2.3.1 Machine Repairman Model
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8.2.3.2 Wireless Handoff Performance Model
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8.3 Other Special Cases of the Birth--Death Model
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8.3.1 The Pure Birth Process
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8.3.2 Pure Death Processes
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8.3.2.1 Death Process with a Constant Rate
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8.3.2.2 Death Process with a Linear Rate
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8.4 Non-Birth--Death Processes
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8.4.1 Availability Models
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8.4.2 Performance Models
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8.4.2.1 Markov Modulated Poisson Process (MMPP)
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8.4.2.2 The Counting Process
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8.4.2.3 The MMPP/M/1 Queue
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8.4.3 Performance and Availability Combined
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8.5 Markov Chains with Absorbing States
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8.6 Solution Techniques
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8.6.1 Methods for Steady-State Analysis
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8.6.1.1 Power Method
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8.6.1.2 Successive Overrelaxation (SOR)
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8.6.2 Methods for Transient Analysis
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8.6.2.1 Fully Symbolic Method
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8.6.2.2 Numerical Methods
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8.7 Automated Generation
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8.7.1 Petri Nets
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8.7.2 Stochastic Petri Nets
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8.7.3 Generalized Stochastic Petri Nets
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8.7.4 Stochastic Reward Nets
9 Networks of Queues
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9.1 Introduction
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9.2 Open Queuing Networks
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9.3 Closed Queuing Networks
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9.4 General Service Distribution and Multiple Job Types
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9.5 Non-product-form Networks
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9.6 Computing Response Time Distribution
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9.6.1 Response Time Distribution in Open Networks
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9.6.1.1 Response Time Blocks
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9.6.2 Response Time Distribution in Closed Networks
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9.7 Summary
10 Statistical Inference
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10.1 Introduction
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10.2 Parameter Estimation
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10.2.1 The Method of Moments
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10.2.2 Maximum-Likelihood Estimation
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10.2.3 Confidence Intervals
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10.2.3.1 Sampling from the Normal Distribution
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10.2.3.2 Sampling from the Exponential Distribution
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10.2.3.3 Sampling from the Weibull Distribution
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10.2.3.4 Sampling from the Bernoulli Distribution
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10.2.4 Estimation related to Markov Chains
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10.2.4.1 Discrete-Time Markov Chains
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10.2.4.2 Estimating Parameters of an M/M/1 Queue
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10.2.4.3 Estimation of Availability
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10.2.4.4 Estimation for a Semi-Markov Process
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10.2.5 Estimation with Dependent Samples
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10.3 Hypothesis Testing
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10.3.1 Tests on the Population Mean
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10.3.2 Hypotheses Concerning Two Means
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10.3.3 Hypotheses Concerning Variances
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10.3.4 Goodness-of-fit Tests
11 Regression and Analysis of Variance
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11.1 Introduction
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11.2 Least-squares Curve Fitting
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11.3 The Coefficients of Determination
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11.4 Confidence Intervals in Linear Regression
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11.5 Trend Detection and Slope Estimation
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11.5.1 Mann--Kendall Test
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11.5.2 Sen's Slope Estimator
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11.6 Correlation Analysis
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11.7 Simple Nonlinear Regression
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11.8 Higher-dimensional Least-squares Fit
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11.9 Analysis of Variance
A Bibliography
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A.1 Theory
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A.1.1 Probability Theory
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A.1.2 Stochastic Processes
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A.1.3 Queuing Theory
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A.1.4 Reliability Theory
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A.1.5 Statistics
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A.2 Applications
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A.2.1 Computer Performance Evaluation
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A.2.2 Communications
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A.2.3 Analysis of Algorithms
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A.2.4 Simulation
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A.2.5 Computer-Communication Networks
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A.2.6 Operations Research
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A.2.7 Fault-Tolerant Computing
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A.2.8 Software Reliability
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A.2.9 Numerical Solutions
B Properties of Distributions
C Statistical Tables
D Laplace Transforms
E Program Performance Analysis
Author Index
Subject Index