{"product_id":"phase-type-distributions-theory-and-application-hardback-9781848219458","title":"Phase Type Distributions; Theory and Application (Hardback) 9781848219458","description":"\u003cfont face=\"Georgia\"\u003e\r\n\u003cp\u003e\u003cfont size=\"6\"\u003ePhase Type Distributions\u003c\/font\u003e\u003cbr\u003e\r\n\u003cfont size=\"5\"\u003eTheory and Application\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\u003cp\u003e\u003cfont size=\"4\"\u003eAndrás Horváth (Author), Miklós Telek (Author), Miklós Telek (Author)\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e9781848219458, Wiley\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eHardback, published 14 November 2024\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e288 pages\u003cbr\u003e23.5 x 15.6 x 1.9 cm, 0.671 kg\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\r\n\r\n\r\n\u003cp align=\"justify\"\u003e\u003cstrong\u003e\u003cfont size=\"3\"\u003ePhase type distributions are widely applicable modeling and statistical tools for non-negative random quantities. They are built on Markov chains, which provide a simple, intuitive stochastic interpretation for their use. \u003ci\u003ePhase Type Distribution\u003c\/i\u003e starts from the Markov chain-based definition of phase type distributions and presents many interesting properties, which follow from the basic definition. \u003cbr\u003e\u003cbr\u003eAs a general family of non-negative distributions with nice analytical properties, phase type distributions can be used for approximating experimental distributions by fitting or by moments matching; and, for discrete event simulation of real word systems with stochastic timing, such as production systems, service operations, communication networks, etc. This book summarizes the up-to-date fitting, matching and simulation methods, and presents the limits of flexibility of phase type distributions of a given order. \u003cbr\u003e\u003cbr\u003eAdditionally, this book lists numerical examples that support the intuitive understanding of the analytical descriptions and software tools that handle phase type distributions.\u003c\/font\u003e\u003c\/strong\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003e\u003cp\u003eIntroduction xi\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 1 Mathematical Background 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1. Basic properties of random variables 1\u003c\/p\u003e \u003cp\u003e1.2. Moments of random variables and related quantities 2\u003c\/p\u003e \u003cp\u003e1.3. Laplace transformation 3\u003c\/p\u003e \u003cp\u003e1.4. z transform 4\u003c\/p\u003e \u003cp\u003e1.5. Matrix functions of quadratic matrices 5\u003c\/p\u003e \u003cp\u003e1.6. Matrix inverse 5\u003c\/p\u003e \u003cp\u003e1.7. Eigenvalues and the characteristic polynomial 5\u003c\/p\u003e \u003cp\u003e1.8. Spectral decomposition 6\u003c\/p\u003e \u003cp\u003e1.9. Ordinary differential equation of vector functions 10\u003c\/p\u003e \u003cp\u003e1.10. Exponential distribution 11\u003c\/p\u003e \u003cp\u003e1.11. Erlang distribution 12\u003c\/p\u003e \u003cp\u003e1.12. Discrete time Markov chain 12\u003c\/p\u003e \u003cp\u003e1.13. Continuous time Markov chain 14\u003c\/p\u003e \u003cp\u003e1.14. Kronecker algebra 15\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 2. Continuous Phase Type Distributions 19\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1. Definition and basic properties 19\u003c\/p\u003e \u003cp\u003e2.2 Stochastic meaning of (-A)-123\u003c\/p\u003e \u003cp\u003e2.3. Rational Laplace transform 24\u003c\/p\u003e \u003cp\u003e2.4. Decomposition of matrix exponential functions 26\u003c\/p\u003e \u003cp\u003e2.5. Similarity transformation 30\u003c\/p\u003e \u003cp\u003e2.5.1. Similarity transformation with identical sizes 30\u003c\/p\u003e \u003cp\u003e2.5.2. Similarity transformation with different sizes 31\u003c\/p\u003e \u003cp\u003e2.5.3. Full rank representation 32\u003c\/p\u003e \u003cp\u003e2.6. Closure properties 32\u003c\/p\u003e \u003cp\u003e2.7 Positive density on (0, 1) 34\u003c\/p\u003e \u003cp\u003e2.8. Eigenvalue structure 35\u003c\/p\u003e \u003cp\u003e2.9. Steepest increase property 39\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 3. Discrete Phase Type Distributions 43\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1. Definition and basic properties 43\u003c\/p\u003e \u003cp\u003e3.2 Stochastic meaning of (I-B)-1 45\u003c\/p\u003e \u003cp\u003e3.3. Rational z transform function 46\u003c\/p\u003e \u003cp\u003e3.4. Decomposition of the matrix geometric function 46\u003c\/p\u003e \u003cp\u003e3.5. Similarity transformations 48\u003c\/p\u003e \u003cp\u003e3.6. Closure properties 48\u003c\/p\u003e \u003cp\u003e3.7. Eigenvalue structure 49\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 4. Matrix Exponential and Matrix Geometric Distributions 51\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1. Matrix exponential distributions 51\u003c\/p\u003e \u003cp\u003e4.1.1. Non-negative matrix exponential subclasses 53\u003c\/p\u003e \u003cp\u003e4.2. Matrix geometric distributions 54\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 5. Classes and Representations of Continuous Phase Type Distributions 57\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1. Number of parameters 57\u003c\/p\u003e \u003cp\u003e5.2. Representations of continuous phase type distributions 58\u003c\/p\u003e \u003cp\u003e5.2.1. Matrix representation 58\u003c\/p\u003e \u003cp\u003e5.2.2. Markovian representation 58\u003c\/p\u003e \u003cp\u003e5.2.3. Laplace representation 59\u003c\/p\u003e \u003cp\u003e5.2.4. Moment representation 59\u003c\/p\u003e \u003cp\u003e5.3. Transformations between continuous phase type representations 60\u003c\/p\u003e \u003cp\u003e5.3.1. From matrix and Markovian representations to Laplace and moment representations 60\u003c\/p\u003e \u003cp\u003e5.3.2. From Laplace representation to moment representation 60\u003c\/p\u003e \u003cp\u003e5.3.3. From moment representation to matrix representation 60\u003c\/p\u003e \u003cp\u003e5.3.4. From Laplace representation to matrix representation 61\u003c\/p\u003e \u003cp\u003e5.3.5. From matrix representation to Markovian representation 61\u003c\/p\u003e \u003cp\u003e5.4. Properties of the matrix representation of PH(n) distributions 62\u003c\/p\u003e \u003cp\u003e5.4.1. 2n-1 versus n2 + n-1 parameters 62\u003c\/p\u003e \u003cp\u003e5.4.2. Different matrix representations 63\u003c\/p\u003e \u003cp\u003e5.5. Subclasses of continuous phase type distributions 65\u003c\/p\u003e \u003cp\u003e5.5.1. Subclasses with real eigenvalues 65\u003c\/p\u003e \u003cp\u003e5.5.2. Subclasses with complex eigenvalues 73\u003c\/p\u003e \u003cp\u003e5.6. Canonical Markovian representation 76\u003c\/p\u003e \u003cp\u003e5.7. Analysis of a non-Markovian representation 77\u003c\/p\u003e \u003cp\u003e5.7.1. Monocyclic representation 81\u003c\/p\u003e \u003cp\u003e5.7.2. Transformation to a Markovian representation 82\u003c\/p\u003e \u003cp\u003e5.8. Representation minimization 85\u003c\/p\u003e \u003cp\u003e5.8.1. Numerical issues 86\u003c\/p\u003e \u003cp\u003e5.9. Markovian representation minimization 86\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 6. Moment Matching 89\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1. Continuous phase type distributions with minimal squared coefficient of variation 89\u003c\/p\u003e \u003cp\u003e6.2. Moment bounds of continuous phase type distributions based on the steepest increase property 96\u003c\/p\u003e \u003cp\u003e6.3. Matrix exponential distributions with minimal squared coefficient of variation 99\u003c\/p\u003e \u003cp\u003e6.3.1. Matrix exponential distributions with complex eigenvalues 99\u003c\/p\u003e \u003cp\u003e6.3.2. Matrix exponential distributions with real eigenvalues 104\u003c\/p\u003e \u003cp\u003e6.4. Characteristics of moments of continuous phase type and matrix exponential distributions 107\u003c\/p\u003e \u003cp\u003e6.5. Matching 2n-1 moments with size n matrix representations 114\u003c\/p\u003e \u003cp\u003e6.5.1. Padé approximation for continuous distributions 114\u003c\/p\u003e \u003cp\u003e6.5.2. Padé approximation for discrete distributions 118\u003c\/p\u003e \u003cp\u003e6.6. Matching any three moments with minimal acyclic phase type distributions 120\u003c\/p\u003e \u003cp\u003e6.6.1. Moment bounds 121\u003c\/p\u003e \u003cp\u003e6.6.2. Explicit moment matching with minimal number of parameters 126\u003c\/p\u003e \u003cp\u003e6.6.3. Proof of theorem 6.28 130\u003c\/p\u003e \u003cp\u003e6.6.4. Proof of theorem 6.29 136\u003c\/p\u003e \u003cp\u003e6.7. Matching any valid odd number of moments with generalized hyper-Erlang distributions 137\u003c\/p\u003e \u003cp\u003e6.7.1. Moment matching procedure for generalized hyper-Erlang distributions 137\u003c\/p\u003e \u003cp\u003e6.7.2. Numerical examples 140\u003c\/p\u003e \u003cp\u003e6.8. Matching probability density function at zero 145\u003c\/p\u003e \u003cp\u003e6.8.1. Numerical examples 146\u003c\/p\u003e \u003cp\u003e6.8.2. Moment matching using the behavior around zero 146\u003c\/p\u003e \u003cp\u003e6.8.3. Improving matrix exponential approximation by matching the behavior around zero 148\u003c\/p\u003e \u003cp\u003e6.8.4. Improving matrix exponential approximation by approximating the behavior around zero 150\u003c\/p\u003e \u003cp\u003e6.9. Moment bounds of PH(2) distributions 151\u003c\/p\u003e \u003cp\u003e6.10. Moment bounds of PH(3) distributions 155\u003c\/p\u003e \u003cp\u003e6.10.1. The second and third normalized moments 156\u003c\/p\u003e \u003cp\u003e6.10.2. The fourth and fifth normalized moments 157\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 7. Distribution Fitting 165\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1. Distance measures 166\u003c\/p\u003e \u003cp\u003e7.2. Fitting based on maximum likelihood 168\u003c\/p\u003e \u003cp\u003e7.2.1. The expectation maximization algorithm 169\u003c\/p\u003e \u003cp\u003e7.2.2. Likelihood optimization for acyclic phase type distributions 176\u003c\/p\u003e \u003cp\u003e7.3. Fitting based on families of polynomials 179\u003c\/p\u003e \u003cp\u003e7.3.1. Bernstein polynomials and Bernstein expolynomials 179\u003c\/p\u003e \u003cp\u003e7.3.2. Application of Bernstein expolynomials to distribution fitting 182\u003c\/p\u003e \u003cp\u003e7.4. Fitting heavy-tailed distributions 185\u003c\/p\u003e \u003cp\u003e7.4.1. Fitting heavy-tailed distributions with monotone decreasing density function 186\u003c\/p\u003e \u003cp\u003e7.4.2. Fitting heavy-tailed distributions with possibly non-monotone density function 188\u003c\/p\u003e \u003cp\u003e7.5. Continuous versus discrete phase type distributions in practical fitting problems 190\u003c\/p\u003e \u003cp\u003e7.5.1. Limit behavior as the scale factor tends to zero 191\u003c\/p\u003e \u003cp\u003e7.5.2. The minimum squared coefficient of variation of scaled discrete phase type distributions 192\u003c\/p\u003e \u003cp\u003e7.5.3. The optimal scale factor in discrete phase type fitting 194\u003c\/p\u003e \u003cp\u003e7.5.4. Approximating non-Markovian models 198\u003c\/p\u003e \u003cp\u003e7.5.5. PH and scaled DPH approximation of continuous time models 203\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 8. Simulation 205\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1. Generating random samples based on the probabilistic interpretation 206\u003c\/p\u003e \u003cp\u003e8.1.1. Sub-classes of continuous phase type distributions 206\u003c\/p\u003e \u003cp\u003e8.1.2. The play algorithm 207\u003c\/p\u003e \u003cp\u003e8.2. Representation transformation to improve the Play method 209\u003c\/p\u003e \u003cp\u003e8.2.1. Formulating the optimization problem 210\u003c\/p\u003e \u003cp\u003e8.2.2. The solution for acyclic phase type distributions 211\u003c\/p\u003e \u003cp\u003e8.2.3. A heuristic representation optimization for general continuous phase type distributions 215\u003c\/p\u003e \u003cp\u003e8.3. An acceptance-rejection algorithm 216\u003c\/p\u003e \u003cp\u003e8.3.1. Generating random variables from matrix exponential distributions which have a Markovian generator 217\u003c\/p\u003e \u003cp\u003e8.3.2. Generating matrix exponential distributed random variables using feedback Erlang blocks 218\u003c\/p\u003e \u003cp\u003e8.4. Overview of simulation methods 223\u003c\/p\u003e \u003cp\u003e8.5. Numerical examples 224\u003c\/p\u003e \u003cp\u003e8.5.1. Optimizing acyclic phase type representations 224\u003c\/p\u003e \u003cp\u003e8.5.2. Optimizing general continuous phase type representations 225\u003c\/p\u003e \u003cp\u003e\u003cb\u003eChapter 9. Continuous Phase Type Distribution Based Random Variables 229\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1. Functions of continuous phase type distributions 229\u003c\/p\u003e \u003cp\u003e9.1.1. Bijective functions of PH distributions 230\u003c\/p\u003e \u003cp\u003e9.1.2. General functions of PH distributions 230\u003c\/p\u003e \u003cp\u003e9.1.3. The considered transformation functions 231\u003c\/p\u003e \u003cp\u003e9.2. Multivariate phase type distributions 235\u003c\/p\u003e \u003cp\u003e9.2.1. Subset-based multivariate phase type distributions 235\u003c\/p\u003e \u003cp\u003e9.2.2. Reward-based multivariate phase type distributions 236\u003c\/p\u003e \u003cp\u003e9.2.3. Linear projection based multivariate multivariate phase type distributions 238\u003c\/p\u003e \u003cp\u003eAppendices 241\u003c\/p\u003e \u003cp\u003eAppendix 1. Description of Related Software Tools 243\u003c\/p\u003e \u003cp\u003eAppendix 2. Acronyms 245\u003c\/p\u003e \u003cp\u003eReferences 247\u003c\/p\u003e \u003cp\u003eIndex 253\u003c\/p\u003e\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\u003cp\u003e\u003cfont size=\"3\"\u003eSubject Areas: Mathematics [\u003ca title=\"See our other books on Mathematics\" href=\"https:\/\/freshlyprintedbooks.co.uk\/search?q=%22Mathematics%20%5BPB%5D%22\"\u003ePB\u003c\/a\u003e]\u003c\/font\u003e\u003c\/p\u003e\r\n\r\n\r\n\u003c\/font\u003e","brand":"Wiley-ISTE","offers":[{"title":"Brand New","offer_id":52449406091544,"sku":"9781848219458","price":111.99,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0730\/2037\/5320\/files\/9781848219458.jpg?v=1785198348","url":"https:\/\/freshlyprintedbooks.co.uk\/products\/phase-type-distributions-theory-and-application-hardback-9781848219458","provider":"Freshly Printed Books","version":"1.0","type":"link"}