Introduction to Matrix Analytic Methods in Queues 1: Analytical and Simulation Approach - Basics

ISBN-13: 9781786307323 / Angielski / Twarda / 2022 / 368 str.

Chakravarthy

Spis treści

List of Notations ixPreface xiChapter 1 Introduction 11.1. Probability concepts 21.1.1. Random variables 51.1.2. Discrete probability functions 61.1.3. Probability generating function 71.1.4. Continuous probability functions 71.1.5. Laplace transform and Laplace-Stieltjes transform 91.1.6. Measures of a random variable 101.2. Renewal process 111.2.1. Renewal function 121.2.2. Terminating renewal process 151.2.3. Poisson process 161.3. Matrix analysis 181.3.1. Basics 181.3.2. Eigenvalues and eigenvectors 231.3.3. Partitioned matrices 271.3.4. Matrix differentiation 281.3.5. Exponential matrix 301.3.6. Kronecker products and Kronecker sums 321.3.7. Vectorization (or direct sums) of matrices 33Chapter 2 Markov Chains 352.1. Discrete-time Markov chains (DTMC) 362.1.1. Basic concepts, key definitions and results 362.1.2. Computation of the steady-state probability vector of DTMC 432.1.3. Absorbing DTMC 452.1.4. Taboo probabilities in DTMC 472.2. Continuous-time Markov chain (CTMC) 482.2.1. Basic concepts, key definitions and results 482.2.2. Computation of exponential matrix 522.2.3. Computation of the limiting probabilities of CTMC 572.2.4. Computation of the mean first passage times 582.3. Semi-Markov and Markov renewal processes 61Chapter 3 Discrete Phase Type Distributions 713.1. Discrete phase type (DPH) distribution 723.2. DPH renewal processes 923.3. Exercises 97Chapter 4 Continuous Phase Type Distributions 1014.1. Continuous phase type (CPH) distribution 1014.2. CPH renewal process 1204.3. Exercises 137Chapter 5 Discrete-Batch Markovian Arrival Process 1435.1. Discrete-batch Markovian arrival process (D-BMAP) 1445.2. Counting process associated with the D-BMAP 1525.3. Generation of D-MAP processes for numerical purposes 1625.4. Exercises 165Chapter 6 Continuous-Batch Markovian Arrival Process 1716.1. Continuous-time batch Markovian arrival process (BMAP) 1716.2. Counting processes associated with BMAP 1776.3. Generation of MAP processes for numerical purposes 1986.4. Exercises 206Chapter 7 Matrix-Analytic Methods (Discrete-Time) 2137.1. M/G/1-paradigm (scalar case) 2157.2. M/G/1-paradigm (matrix case) 2247.3. GI/M/1-paradigm (scalar case) 2447.4. GI/M/1-paradigm (matrix case) 2527.5. QBD process (scalar case) 2687.6. QBD process (matrix case) 2697.7. Exercises 278Chapter 8. Matrix-Analytic Methods (Continuous-time) 2918.1. M/G/1-type (scalar case) 2918.2. M/G/1-type (matrix case) 2958.3. GI/M/1-type (scalar case) 2978.4. GI/M/1-type (matrix case) 3008.5. QBD process (scalar case) 3048.6. QBD process (matrix case) 3058.7. Exercises 308Chapter 9. Applications 3219.1. Production and manufacturing 3229.2. Service sectors 3239.2.1. Healthcare 3249.2.2. Artificial Intelligence and the Internet of Things 3249.2.3. Biological and medicine 3259.2.4. Telecommunications 3259.2.5. Supply chain 3259.2.6. Consumer issues 326References 327Index 335Summary of Volume 2 339

O autorze

Srinivas R. Chakravarthy retired from Kettering University in Michigan, USA after serving as Professor of Mathematics, and as Professor and Head of Industrial and Manufacturing Engineering. He was bestowed the Distinguished Faculty (Kettering's Faculty and Alumni Honor Wall) award in 2015. He obtained his PhD under the supervision of Professor Marcel Neuts and is the co-founder of the International Conference Series on MAM in Stochastic Models. His research interests are in queues, inventory and reliability.