Continuous-Time Markov Chains: An Applications-Oriented ApproachContinuous time parameter Markov chains have been useful for modeling various random phenomena occurring in queueing theory, genetics, demography, epidemiology, and competing populations. This is the first book about those aspects of the theory of continuous time Markov chains which are useful in applications to such areas. It studies continuous time Markov chains through the transition function and corresponding q-matrix, rather than sample paths. An extensive discussion of birth and death processes, including the Stieltjes moment problem, and the Karlin-McGregor method of solution of the birth and death processes and multidimensional population processes is included, and there is an extensive bibliography. Virtually all of this material is appearing in book form for the first time. |
Contents
| 1 | |
| 8 | |
Resolvent Functions and Their Properties | 21 |
The FunctionalAnalytic Setting for Transition Functions and Resolvents | 33 |
Feller Transition Functions | 42 |
Kendalls Representation of Reversible Transition Functions | 48 |
Appendix | 54 |
CHAPTER 2 | 65 |
Determination of Invariant Measures from the QMatrix | 191 |
CHAPTER 6 | 204 |
Ordinary Ergodicity | 212 |
The CroftKingman Lemmas | 224 |
CHAPTER 7 | 233 |
Exponential Families of Transition Functions | 243 |
Dual Processes | 251 |
CHAPTER 8 | 261 |
Examples of ContinuousTime Markov Chains | 89 |
Birth and Death Processes | 96 |
Continuous Time Parameter Markov Branching Processes | 113 |
CHAPTER 4 | 120 |
NonuniquenessConstruction of QFunctions Other Than the Minimal | 138 |
UniquenessThe NonConservative Case | 148 |
CHAPTER 5 | 155 |
Subinvariant and Invariant Measures | 166 |
Classification Based on the QMatrix | 184 |
The Stieltjes Moment Problem | 273 |
The KarlinMcGregor Method of Solution | 280 |
Total Positivity of the Birth and Death Transition Function | 288 |
Extinction Times and Probability of Extinction for Upwardly SkipFree | 298 |
Multidimensional Population Processes | 307 |
TwoDimensional Competition Processes | 317 |
Birth Death and Migration Processes | 330 |
| 347 | |
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Continuous-Time Markov Chains: An Applications-Oriented Approach William J. Anderson No preview available - 2011 |
Common terms and phrases
assume backward equations birth and death bounded C₁ Chapman-Kolmogorov equation communicating class components componentwise condition continuous-time Markov chain Convergence Theorem death process define Definition denote distribution E₁ E₂ ergodic example exists exponential fact Feller fi(t fij(t finite forward equations Fubini's theorem given Hence holds honest i₁ irreducible j₁ jump chain l₁ Laplace transform Lemma linear m₁ Markov property matrix measure minimal Q-function minimal solution Monotone Convergence Theorem non-negative solution Note numbers orthogonal orthogonal polynomials P₁ P₁(t P₁j(t parameter Pi(t Pij(t polynomials Proposition 3.1 Q is conservative q-matrix q-matrix Q random variables resolvent equation result Reuter right-hand side row coordination family semigroup sequence space statements are equivalent Stieltjes moment problem Suppose t₁ transition function transition probabilities vector weakly symmetric zero λ Σ λη μη Σ Σ