Riassunto
This book was written for an introductory one-semester or two-quarter course in stochastic processes and their applications. The reader is assumed to have a basic knowledge of analysis and linear algebra at an undergraduate level. Stochastic models are applied in many fields such as engineering systems, physics, biology, operations research, business, economics, psychology, and linguistics. Stochastic modeling is one of the promising kinds of modeling in applied probability theory. This book is intended to introduce basic stochastic processes: Poisson pro- cesses, renewal processes, discrete-time Markov chains, continuous-time Markov chains, and Markov-renewal processes. These basic processes are introduced from the viewpoint of elementary mathematics without going into rigorous treatments. This book also introduces applied stochastic system modeling such as reliability and queueing modeling. Chapters 1 and 2 deal with probability theory, which is basic and prerequisite to the following chapters. Many important concepts of probabilities, random variables, and probability distributions are introduced. Chapter 3 develops the Poisson process, which is one of the basic and im- portant stochastic processes. Chapter 4 presents the renewal process. Renewal- theoretic arguments are then used to analyze applied stochastic models. Chapter 5 develops discrete-time Markov chains. Following Chapter 5, Chapter 6 deals with continuous-time Markov chains. Continuous-time Markov chains have im- portant applications to queueing models as seen in Chapter 9. A one-semester course or two-quarter course consists of a brief review of Chapters 1 and 2, fol- lowed in order by Chapters 3 through 6.
Contenuti
1 Probability Theory.- 1.1 Introduction.- 1.2 Sample Spaces and Events.- 1.3 Probabilities.- 1.4 Combinatorial Analysis.- 1.5 Problems 1.- 2 Random Variables and Distributions.- 2.1 Introduction.- 2.2 Random Variables and Distributions.- 2.3 Discrete Distributions.- 2.4 Continuous Distributions.- 2.5 Multivariate Distributions.- 2.6 Limit Theorems.- 2.7 Problems 2.- 3 Poisson Processes.- 3.1 Stochastic Processes.- 3.2 The Poisson Process.- 3.3 Interarrival Time Distributions.- 3.4 Conditional Waiting Time Distributions.- 3.5 Nonhomogeneous Poisson Processes.- 3.6 Problems 3.- 4 Renewal Processes.- 4.1 Introduction.- 4.2 Renewal Functions.- 4.3 Limit Theorems.- 4.4 Delayed and Stationary Renewal Processes.- 4.5 Problems 4.- 5 Discrete-Time Markov Chains.- 5.1 Introduction.- 5.2 Chapman-Kolmogorov Equation.- 5.3 State Classification.- 5.4 Limiting Probabilities.- 5.5 Finite-State Markov Chains.- 5.6 Problems 5.- 6 Continuous-Time Markov Chains.- 6.1 Introduction.- 6.2 Pure Birth Processes.- 6.3 Pure Death Processes.- 6.4 Birth and Death Processes.- 6.5 Finite-State Markov Chains.- 6.6 Problems 6.- 7 Markov Renewal Processes.- 7.1 Introduction.- 7.2 Markov Renewal Processes.- 7.3 Stationary Probabilities.- 7.4 Alternating Renewal Processes.- 7.5 Problems 7.- 8 Reliability Models.- 8.1 Introduction.- 8.2 Lifetime Distributions and Failure Rates.- 8.3 Availability Theory.- 8.4 Replacement Models.- 8.4.1 Age Replacement Models.- 8.4.2 Block Replacement Models.- 8.5 Ordering Models.- 8.5.1 Model I.- 8.5.2 Model II.- 8.6 Problems 8.- 9 Queueing Models.- 9.1 Introduction.- 9.2 Single Server Queueing Models.- 9.2.1 M/M/1/? Queueing Models.- 9.2.2 M/M/1/N Queueing Models.- 9.3 Multiple Server Queueing Models.- 9.3.1 M/M/c/?Queueing Models.- 9.3.2 M/M/c/c Queueing Models.- 9.3.3 M/M/?/? Queueing Models.- 9.4 Queues with a Finite Population.- 9.4.1 M/M/1/K/K Queueing Models.- 9.4.2 M/M/c/K/K Queueing Models.- 9.4.3 M/M/c/c/c Queueing Models.- 9.5 Problems 9.- A Laplace-Stieltjes Transforms.- A.1 Laplace-Stieltjes Transforms.- A.2 Properties of Laplace-Stieltjes Transforms.- A.3 Applications to Distributions.- A.4 Applications to Differential Equations.- A.5 Applications to Renewal Functions.- B Answers to Selected Problems.- C The Bibliography.
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