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TitleMarkov Chain Models โ{128}{148} Rarity and Exponentiality [electronic resource] / edited by Julian Keilson
ImprintNew York, NY : Springer New York, 1979
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Descript XIV, 184 p. online resource


in failure time distributions for systems modeled by finite chains. This introductory chapter attempts to provide an overยญ view of the material and ideas covered. The presentation is loose and fragmentary, and should be read lightly initially. Subsequent perusal from time to time may help tie the matยญ erial together and provide a unity less readily obtainable otherwise. The detailed presentation begins in Chapter 1, and some readers may prefer to begin there directly. ยงO.l. Time-Reversibility and Spectral Representation. Continuous time chains may be discussed in terms of discrete time chains by a uniformizing procedure (ยง2.l) that simplifies and unifies the theory and enables results for discrete and continuous time to be discussed simultaneously. Thus if N(t) is any finite Markov chain in continuous time governed by transition rates vmn one may write for pet) = [Pmn(t)] โ{128}ข P[N(t) = n I N(O) = m] pet) = exp [-vt(I - a )] (0.1.1) v where v > Max r v ' and mn m n law ̃ 1 - v-I * Hence N(t) where is governed r vmn Nk = NK(t) n K(t) is a Poisson process of rate v indep- by a ' and v dent of N โ{128}ข k Time-reversibility (ยง1.3, ยง2.4, ยง2.S) is important for many reasons. A) The only broad class of tractable chains suitable for stochastic models is the time-reversible class


0. Introduction and Summary -- 1. Discrete Time Markov Chains; Reversibility in Time -- ยง1.00. Introduction -- ยง1.0. Notation, Transition Laws -- ยง1.1. Irreducibility, Aperiodicity, Ergodicity; Stationary Chains -- ยง1.2. Approach to Ergodicity; Spectral Structure, Perron-Romanovsky-Frobenius Theorem -- ยง1.3. Time-Reversible Chains -- 2. Markov Chains in Continuous Time; Uniformization; Reversibility -- ยง2.00. Introduction -- ยง2.0. Notation, Transition Laws; A Review -- ยง2.1. Uniformizable Chains โ{128}{148} A Bridge Between Discrete and Continuous Time Chains -- ยง2.2. Advantages and Prevalence of Uniformizable Chains -- ยง2.3. Ergodicity for Continuous Time Chains -- ยง2.4. Reversibility for Ergodic Markov Chains in Continuous Time -- ยง2.5. Prevalence of Time-Reversibility -- 3. More on Time-Reversibility; Potential Coefficients; Process Modification -- ยง3.00. Introduction -- ยง3.1. The Advantages of Time-Reversibility -- ยง3.2. The Spectral Representation -- ยง3.3. Potentials; Spectral Representation -- ยง3.4. More General Time-Reversible Chains -- ยง3.5. Process Modifications Preserving Reversibility -- ยง3.6. Replacement Processes -- 4. Potential Theory, Replacement, and Compensation -- ยง4.00. Introduction -- ยง4.1. The Green Potential -- ยง4.2. The Ergodic Distribution for a Replacement Process -- ยง4.3. The Compensation Method -- ยง4.4. Notation for the Homogeneous Random Walk -- ยง4.5. The Compensation Method Applied to the Homogeneous Random Walk Modified by Boundaries -- ยง4.6. Advantages of the Compensation Method. An Illustrative Example -- ยง4.7. Exploitation of the Structure of the Green Potential for the Homogeneous Random Walk -- ยง4.8. Similar Situations -- 5. Passage Time Densities in Birth-Death Processes; Distribution Structure -- ยง5.00. Introduction -- ยง5.1. Passage Time Densities for Birth-Death Processes -- ยง5.2. Passage Time Moments for a Birth-Death Process -- ยง5.3. PF?, Complete Monotonicity, Log-Concavity and Log-Convexity -- ยง5.4. Complete Monotonicity and Log-Convexity -- ยง5.5. Complete Monotonicity in Time-Reversible Processes -- ยง5.6. Some Useful Inequalities for the Families CM and PF? -- ยง5.7. Log-Concavity and Strong Unimodality for Lattice Distributions -- ยง5.8. Preservation of Log-Concavity and Log-Convexity under Tail Summation and Integration -- ยง5.9. Relation of CM and PF? to IFR and DFR Classes in Reliability -- 6. Passage Times and Exit Times for More General Chains -- ยง6.00. Introduction -- ยง6.1. Passage Time Densities to a Set of States -- ยง6.2. Mean Passage Times to a Set via the Green Potential -- ยง6.3. Ruin Probabilities via the Green Potential -- ยง6.4. Ergodic Flow Rates in a Chain -- ยง6.5. Ergodic Exit Times, Ergodic Sojourn Times, and Quasi-Stationary Exit Times -- ยง6.6. The Quasi-Stationary Exit Time. A Limit Theorem -- ยง6.7. The Connection Between Exit Times and Sojourn Times. A Renewal Theorem -- ยง6.8. A Comparison of the Mean Ergodic Exit Time and Mean Ergodic Sojourn Time for Arbitrary Chains -- ยง6.9. Stochastic Ordering of Exit Times of Interest for Time-Reversible Chains -- ยง6.10. Superiority of the Exit Time as System Failure Time; Jitter -- 7. The Fundamental Matrix, and Allied Topics -- ยง7.00. Introduction -- ยง7.1. The Fundamental Matrix for Ergodic Chains -- ยง7.2. The Structure of the Fundamental Matrix for Time-Reversible Chains -- ยง7.3. Mean Failure Times and Ruin Probabilities for Systems with Independent Markov Components and More General Chains -- ยง7.4. Covariance and Spectral Density Structure for Time-Reversible Processes -- ยง7.5. A Central Limit Theorem -- ยง7.6. Regeneration Times and Passage Times-Their Relation For Arbitrary Chains -- ยง7.7. Passage to a Set with Two States -- 8. Rarity and Exponentiality -- ยง8.0. Introduction -- ยง8.1. Passage Time Density Structure for Finite Ergodic Chains; the Exponential Approximation -- ยง8.2. A Limit Theorem for Ergodic Regenerative Processes -- ยง8.3. Prototype Behavior: Birth-Death Processes; Strongly Stable Systems -- ยง8.4. Limiting Behavior of the Ergodic and Quasi-stationary Exit Time Densities and Sojourn Time Densities for Birth-Death Processes -- ยง8.5. Limit Behavior of Other Exit Times for More General Chains -- ยง8.6. Strongly Stable Chains, Jitter; Estimation of the Failure Time Needed for the Exponential Approximation -- ยง8.7. A Measure of Exponentiality in the Completely Monotone Class of Densities -- ยง8.8. An Error Bound for Departure from Exponentiality in the Completely Monotone Class -- ยง8.9. The Exponential Approximation for Time-Reversible Systems -- ยง8.10. A Relaxation Time of Interest -- 9. Stochastic Monotonicity -- ยง9.00. Introduction -- ยง9.1. Monotone Markov Matrices and Monotone Chains -- ยง9.2. Some Monotone Chains in Discrete Time -- ยง9.3. Monotone Chains in Continuous Time -- ยง9.4. Other Monotone Processes in Continuous Time -- References

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