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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.
Preface
Introduction
Probability concepts
Random variables
Discrete probability functions
Probability generating function
Continuous probability functions
Laplace transform and Laplace-Stieltjes transform
Measures of a random variable
Renewal process
Renewal function
Terminating renewal process
Poisson process
Matrix analysis
Basics
Eigenvalues and eigenvectors
Partitioned matrices
Matrix differentiation
Exponential matrix
Kronecker products and Kronecker sums
Vectorization (or direct sums) of matrices
Markov Chains
Discrete-time Markov chains (DTMC)
Basic concepts, key definitions and results
Computation of the steady-state probability vector of DTMC
Absorbing DTMC
Taboo probabilities in DTMC
Continuous-time Markov chain (CTMC)
Basic concepts, key definitions and results
Computation of exponential matrix
Computation of the limiting probabilities of CTMC
Computation of the mean first passage times
Semi-Markov and Markov renewal processes
Discrete Phase Type Distributions
Discrete phase type (DPH) distribution
DPH renewal processes
Exercises
Continuous Phase Type Distributions
Continuous phase type (CPH) distribution
CPH renewal process
Exercises
Discrete-Batch Markovian Arrival Process
Discrete-batch Markovian arrival process
Counting process associated with the D-BMAP
Generation of D-MAP processes for numerical purposes
Exercises
Continuous-Batch Markovian Arrival Process
Continuous-time batch Markovian arrival process (BMAP)
Counting processes associated with BMAP
Generation of MAP processes for numerical purposes
Exercises
Matrix-Analytic Methods (Discrete-Time)
M/G/1-paradigm (scalar case)
M/G/1-paradigm (matrix case)
GI/M/1-paradigm (scalar case)
GI/M/1-paradigm (matrix case)
QBD process (scalar case)
QBD process (matrix case)
Exercises
Matrix-Analytic Methods (Continuous-time)
M/G/1-type (scalar case)
M/G/1-type (matrix case)
GI/M/1-type (scalar case)
GI/M/1-type (matrix case)
QBD process (scalar case)
QBD process (matrix case)
Exercises
Applications
Production and manufacturing
Service sectors
Healthcare
Artificial intelligence and the Internet of Things
Biological and medicine
Telecommunications
Supply chain
Consumer issues
References
Index