Matrix-analytic methods (MAM) were introduced by Professor Marcel Neuts and have been applied to a variety of stochastic models since. In order to provide a clear and deep understanding of MAM while showing their power, this book presents MAM concepts and explains the results using a number of worked-out examples.
This book’s approach will inform and kindle the interest of researchers attracted to this fertile field. To allow readers to practice and gain experience in the algorithmic and computational procedures of MAM, Introduction to Matrix Analytic Methods in Queues 1 provides a number of computational exercises. It also incorporates simulation as another tool for studying complex stochastic models, especially when the state space of the underlying stochastic models under analytic study grows exponentially.
The book’s detailed approach will make it more accessible for readers interested in learning about MAM in stochastic models.
Author(s): Srinivas R. Chakravarthy
Series: Mathematics and Statistics
Publisher: Wiley-ISTE
Year: 2022
Language: English
Pages: 368
City: London
Cover
Half-Title Page
Title Page
Copyright Page
Contents
List of Notations
Preface
Chapter 1. Introduction
1.1. Probability concepts
1.1.1. Random variables
1.1.2. Discrete probability functions
1.1.3. Probability generating function
1.1.4. Continuous probability functions
1.1.5. Laplace transform and Laplace-Stieltjes transform
1.1.6. Measures of a random variable
1.2. Renewal process
1.2.1. Renewal function
1.2.2. Terminating renewal process
1.2.3. Poisson process
1.3. Matrix analysis
1.3.1. Basics
1.3.2. Eigenvalues and eigenvectors
1.3.3. Partitioned matrices
1.3.4. Matrix differentiation
1.3.5. Exponential matrix
1.3.6. Kronecker products and Kronecker sums
1.3.7. Vectorization (or direct sums) of matrices
Chapter 2. Markov Chains
2.1. Discrete-time Markov chains (DTMC)
2.1.1. Basic concepts, key definitions and results
2.1.2. Computation of the steady-state probability vector of DTMC
2.1.3. Absorbing DTMC
2.1.4. Taboo probabilities in DTMC
2.2. Continuous-time Markov chain (CTMC)
2.2.1. Basic concepts, key definitions and results
2.2.2. Computation of exponential matrix
2.2.3. Computation of the limiting probabilities of CTMC
2.2.4. Computation of the mean first passage times
2.3. Semi-Markov and Markov renewal processes
Chapter 3. Discrete Phase Type Distributions
3.1. Discrete phase type (DPH) distribution
3.2. DPH renewal processes
3.3. Exercises
Chapter 4. Continuous Phase Type Distributions
4.1. Continuous phase type (CPH) distribution
4.2. CPH renewal process
4.3. Exercises
Chapter 5. Discrete-Batch Markovian Arrival Process
5.1. Discrete-batch Markovian arrival process
5.2. Counting process associated with the D-BMAP
5.3. Generation of D-MAP processes for numerical purposes
5.4. Exercises
Chapter 6. Continuous-Batch Markovian Arrival Process
6.1. Continuous-time batch Markovian arrival process (BMAP)
6.2. Counting processes associated with BMAP
6.3. Generation of MAP processes for numerical purposes
6.4. Exercises
Chapter 7. Matrix-Analytic Methods (Discrete-Time)
7.1. M/G/1-paradigm (scalar case)
7.2. M/G/1-paradigm (matrix case)
7.3. GI/M/1-paradigm (scalar case)
7.4. GI/M/1-paradigm (matrix case)
7.5. QBD process (scalar case)
7.6. QBD process (matrix case)
7.7. Exercises
Chapter 8. Matrix-Analytic Methods (Continuous-time)
8.1. M/G/1-type (scalar case)
8.2. M/G/1-type (matrix case)
8.3. GI/M/1-type (scalar case)
8.4. GI/M/1-type (matrix case)
8.5. QBD process (scalar case)
8.6. QBD process (matrix case)
8.7. Exercises
Chapter 9. Applications
9.1. Production and manufacturing
9.2. Service sectors
9.2.1. Healthcare
9.2.2. Artificial intelligence and the Internet of Things
9.2.3. Biological and medicine
9.2.4. Telecommunications
9.2.5. Supply chain
9.2.6. Consumer issues
References
Index
Summary of Volume 2
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