Phase Type Distributions - András Horváth

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Résumé :

Introduction xi

Chapter 1 Mathematical Background 1

1.1. Basic properties of random variables 1

1.2. Moments of random variables and related quantities 2

1.3. Laplace transformation 3

1.4. z transform 4

1.5. Matrix functions of quadratic matrices 5

1.6. Matrix inverse 5

1.7. Eigenvalues and the characteristic polynomial 5

1.8. Spectral decomposition 6

1.9. Ordinary differential equation of vector functions 10

1.10. Exponential distribution 11

1.11. Erlang distribution 12

1.12. Discrete time Markov chain 12

1.13. Continuous time Markov chain 14

1.14. Kronecker algebra 15

Chapter 2. Continuous Phase Type Distributions 19

2.1. Definition and basic properties 19

2.2 Stochastic meaning of (-A)-123

2.3. Rational Laplace transform 24

2.4. Decomposition of matrix exponential functions 26

2.5. Similarity transformation 30

2.5.1. Similarity transformation with identical sizes 30

2.5.2. Similarity transformation with different sizes 31

2.5.3. Full rank representation 32

2.6. Closure properties 32

2.7 Positive density on (0, 1) 34

2.8. Eigenvalue structure 35

2.9. Steepest increase property 39

Chapter 3. Discrete Phase Type Distributions 43

3.1. Definition and basic properties 43

3.2 Stochastic meaning of (I-B)-1 45

3.3. Rational z transform function 46

3.4. Decomposition of the matrix geometric function 46

3.5. Similarity transformations 48

3.6. Closure properties 48

3.7. Eigenvalue structure 49

Chapter 4. Matrix Exponential and Matrix Geometric Distributions 51

4.1. Matrix exponential distributions 51

4.1.1. Non-negative matrix exponential subclasses 53

4.2. Matrix geometric distributions 54

Chapter 5. Classes and Representations of Continuous Phase Type Distributions 57

5.1. Number of parameters 57

5.2. Representations of continuous phase type distributions 58

5.2.1. Matrix representation 58

5.2.2. Markovian representation 58

5.2.3. Laplace representation 59

5.2.4. Moment representation 59

5.3. Transformations between continuous phase type representations 60

5.3.1. From matrix and Markovian representations to Laplace and moment representations 60

5.3.2. From Laplace representation to moment representation 60

5.3.3. From moment representation to matrix representation 60

5.3.4. From Laplace representation to matrix representation 61

5.3.5. From matrix representation to Markovian representation 61

5.4. Properties of the matrix representation of PH(n) distributions 62

5.4.1. 2n-1 versus n2 + n-1 parameters 62

5.4.2. Different matrix representations 63

5.5. Subclasses of continuous phase type distributions 65

5.5.1. Subclasses with real eigenvalues 65

5.5.2. Subclasses with complex eigenvalues 73

5.6. Canonical Markovian representation 76

5.7. Analysis of a non-Markovian representation 77

5.7.1. Monocyclic representation 81

5.7.2. Transformation to a Markovian representation 82

5.8. Representation minimization 85

5.8.1. Numerical issues 86

5.9. Markovian representation minimization 86

Chapter 6. Moment Matching 89

6.1. Continuous phase type distributions with minimal squared coefficient of variation 89

6.2. Moment bounds of continuous phase type distributions based on the steepest increase property 96

6.3. Matrix exponential distributions with minimal squared coefficient of variation 99

6.3.1. Matrix exponential ...