Items related to Applied Probability Models
Synopsis
Intended for a course in Probability Models at the undergraduate or graduate level, this book is designed for those who will actually use probability and is designed to fit diverse audiences (business students, applied engineering students, and biology students). The course focuses on applications of probability through the presentation of models rather than theory alone. In this practical and interesting book, author Do Le (Paul) Minh provides accessible coverage for a course in probability models. Minh motivates the material with interesting application problems relating to medicine, business, and engineering, many of which are based on real studies and applications. Throughout the book, he thoughtfully integrates the use of computers and spreadsheets to solve problems.
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Review
1. REVIEW OF PROBABILITY THEORY Experiments and Outcomes / Sample Space and Events / Probabilities Defined on Events / Probability Models / Probabilities of Combined Events / Conditional Probabilities / Conditional Arguments / Independent Events / Random Variables / Discrete Random Variables and Their Expected Values / Bernoulli Random Variables / Geometric Distribution / Binomial Distribution / Negative Binomial Distribution / Continuous and Mixed Random Variables / Continuous Random Variables and Their Expected Values / Uniform Distribution / Normal Distribution / Conditional Arguments for a Random Variable / Conditional Expectations / Random Sum / Summary / Problems 2. DISCRETE-TIME MARKOV CHAINS Stochastic Processes / Independent and Identically Distributed Random Variables / Markov Chains of Order i / First-Order Markov Chains / Matrix Notation for Markov Chains / Transition Diagrams / The Importance of First-Order Markov Chains / Time-Homogeneity / Seasonal Chains / The Rat in the Maze / Empirical Transition Matrices / Population-Homogeneity / Enlarging the State Space / Using Supplementary Variables / Refining the State Space / What Do a Little Rat and a Big Company Have in Common? / Lumpable States / Discrete-Time Chains In a Continuous-Time Chain / The Sojourn Times / Summary / Problems 3. TRANSIENT AND LIMITING RESULTS Paths / k-Step Transition Probabilities / Path Analysis / Matrix Multiplication Method / Chapman-Kolmogorov Equation / Initial Conditions / Transient Results / Validating the Model / Limiting Distributions / Expected Numbers of Visits / Summary / Problems 4. CLASSIFICATION OF FINITE CHAINS Classification of Finite Chains / Transient and Recurrent States / Reachability Between States / Communication of Two States / Classes / Transient and Recurrent Classes / Canonical Forms / Absorbing Chains / Irreducible Chains / Periodic States / Periodic Classes / Finite, Periodic Chains / Summary / Problems 5. FINITE ABSORBING CHAINS Absorbing Chains / First-Step Analysis / Expected Number of Visits to a Transient State / Expected Absorption Time / Absorbing Probabilities / Limiting Distributions / Summary / Problems 6. FINITE NON-ABSORBING CHAINS Non-Absorbing Chains / The Taxicab Example / Limiting Probabilities / Long-term Visiting Rates / Expected First Reaching Times / Expected Return Times / Regenerative Property / Periodic Chains / Finite Reducible Chains / Transient Behavior / Class-Absorption Probabilities / Long-Term Visiting Rate / Limiting Distribution / Summary / Problems 7. INFINITE CHAINS Infinite Random Walks / Infinite Chains / Infinite Absorbing Chains / Infinite Irreducible Chains / Positive-Recurrence / Null-Recurrence / Summary / Problems 8. POISSON STREAMS OF EVENTS Continuous-Time Chains / Markovian Streams / Regular Streams / Poisson Streams / Instantaneous and Average Rates / Rates / Remaining Lives / Exponential and Hyperexponential Distributions / Lives / The Shorter Remaining Life / Poisson (Counting) Process / Erlang Distributions / Random Streams / Superposition of Two Poisson Streams / Decomposition of a Poisson Stream / Compound Poisson Process / Summary / Problems 9. RENEWAL STREAMS OF EVENTS Renewal Streams and Renewal (Counting) Processes / Elementary Renewal Theorem / Renewal Rates / Renewal-Reward Processes / Gradual-Reward Processes / Regenerative Processes / Remaining Lives / Lattice Random Variables / Spread-Out Random Variables / Limiting Distributions / Summary / Problems 10. SEMI-MARKOV CHAINS Semi-Markov Chains / The Rat Maze Example / A Marketing Example / Holding Times / Sojourn Times / Classification of States / Absorbing Semi-Markov Chains / Irreducible Semi-Markov Chains / Expected First Reaching Times / Expected Return Times / Visiting Rates / Limiting Results / Summary / Problems 11. CONTINUOUS-TIME MARKOV CHAINS Continuous-Time Markov Chain / Memoryless Property / Transition Rates / Transition Diagrams / The Q-Matrix / Equally-Spaced Discrete-Time Chains / Why This Model? / Competitive Chains / The Machine-Repair Example / The Balance Equations / Birth-and-Death Processes / Summary / Problems 12. MARKOVIAN QUEUES Queuing Theory / Queuing Systems / Kendall Notation / Performance Measures / Little's Result / Markovian Queues / M/M/k/N Queues / M/M/k Queues / M/M/k Queues with Feedback / M[G]/M/1 Queues / M/M/2 Queues with Different Service Rates / M/M/1 Queues with N-Policy / M[G]/M/1 Queues / Queues in Series with Blocking / Queues in Series without Blocking / Acyclic Open Jackson Networks / General Open Jackson Networks / Summary / Problems 13. GENERAL SINGLE-SERVER QUEUES G/G/1 Queues / Busy Cycles / Waiting Times and Idle Times / Model Approximation: M/G/1 Queues / Analytical Approximations / Numerical Methods / Simulation / Random Number Generators / Random Variate Generators / Inverse Transform Technique for the Discrete Distributions / Inverse Transform Technique for the Continuous Distributions / Acceptance-Rejection Technique for the Discrete Distributions / Acceptance-Rejection Technique for the Continuous Distributions / Generating Specific Distributions / Simulating E^2/ E^2/1 Queues / Simulation and Analytical Methods / Summary / Problems
Synopsis
Intended for a course in Probability Models at the undergraduate or graduate level, this book is designed for those who will actually use probability and is designed to fit diverse audiences (business students, applied engineering students, and biology students). The course focuses on applications of probability through the presentation of models rather than theory alone. In this practical work, the author provides accessible coverage for a course in probability models. The author motivates the material with application problems relating to medicine, business, and engineering, many of which are based on real studies and applications. Throughout the book, he integrates the use of computers and spreadsheets to solve problems.
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Applied Probability Models
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