0. Introduction and Summary.- 1. Discrete Time Markov Chains; Reversibility in Time.- §1.00. Introduction.- §1.0. Notation, Transition Laws.- §1.1. Irreducibility, Aperiodicity, Ergodicity; Stationary Chains.- §1.2. Approach to Ergodicity; Spectral Structure, Perron-Romanovsky-Frobenius Theorem.- §1.3. Time-Reversible Chains.- 2. Markov Chains in Continuous Time; Uniformization; Reversibility.- §2.00. Introduction.- §2.0. Notation, Transition Laws; A Review.- §2.1. Uniformizable Chains - A Bridge Between Discrete and Continuous Time Chains.- §2.2. Advantages and Prevalence of Uniformizable Chains.- §2.3. Ergodicity for Continuous Time Chains.- §2.4. Reversibility for Ergodic Markov Chains in Continuous Time.- §2.5. Prevalence of Time-Reversibility.- 3. More on Time-Reversibility; Potential Coefficients; Process Modification.- §3.00. Introduction.- §3.1. The Advantages of Time-Reversibility.- §3.2. The Spectral Representation.- §3.3. Potentials; Spectral Representation.- §3.4. More General Time-Reversible Chains.- §3.5. Process Modifications Preserving Reversibility.- §3.6. Replacement Processes.- 4. Potential Theory, Replacement, and Compensation.- §4.00. Introduction.- §4.1. The Green Potential.- §4.2. The Ergodic Distribution for a Replacement Process.- §4.3. The Compensation Method.- §4.4. Notation for the Homogeneous Random Walk.- §4.5. The Compensation Method Applied to the Homogeneous Random Walk Modified by Boundaries.- §4.6. Advantages of the Compensation Method. An Illustrative Example.- §4.7. Exploitation of the Structure of the Green Potential for the Homogeneous Random Walk.- §4.8. Similar Situations.- 5. Passage Time Densities in Birth-Death Processes; Distribution Structure.- §5.00. Introduction.- §5.1. Passage Time Densities for Birth-Death Processes.- §5.2. Passage Time Moments for a Birth-Death Process.- §5.3. PF?, Complete Monotonicity, Log-Concavity and Log-Convexity.- §5.4. Complete Monotonicity and Log-Convexity.- §5.5. Complete Monotonicity in Time-Reversible Processes.- §5.6. Some Useful Inequalities for the Families CM and PF?.- §5.7. Log-Concavity and Strong Unimodality for Lattice Distributions.- §5.8. Preservation of Log-Concavity and Log-Convexity under Tail Summation and Integration.- §5.9. Relation of CM and PF? to IFR and DFR Classes in Reliability.- 6. Passage Times and Exit Times for More General Chains.- §6.00. Introduction.- §6.1. Passage Time Densities to a Set of States.- §6.2. Mean Passage Times to a Set via the Green Potential.- §6.3. Ruin Probabilities via the Green Potential.- §6.4. Ergodic Flow Rates in a Chain.- §6.5. Ergodic Exit Times, Ergodic Sojourn Times, and Quasi-Stationary Exit Times.- §6.6. The Quasi-Stationary Exit Time. A Limit Theorem.- §6.7. The Connection Between Exit Times and Sojourn Times. A Renewal Theorem.- §6.8. A Comparison of the Mean Ergodic Exit Time and Mean Ergodic Sojourn Time for Arbitrary Chains.- §6.9. Stochastic Ordering of Exit Times of Interest for Time-Reversible Chains.- §6.10. Superiority of the Exit Time as System Failure Time; Jitter.- 7. The Fundamental Matrix, and Allied Topics.- §7.00. Introduction.- §7.1. The Fundamental Matrix for Ergodic Chains.- §7.2. The Structure of the Fundamental Matrix for Time-Reversible Chains.- §7.3. Mean Failure Times and Ruin Probabilities for Systems with Independent Markov Components and More General Chains.- §7.4. Covariance and Spectral Density Structure for Time-Reversible Processes.- §7.5. A Central Limit Theorem.- §7.6. Regeneration Times and Passage Times-Their Relation For Arbitrary Chains.- §7.7. Passage to a Set with Two States.- 8. Rarity and Exponentiality.- §8.0. Introduction.- §8.1. Passage Time Density Structure for Finite Ergodic Chains; the Exponential Approximation.- §8.2. A Limit Theorem for Ergodic Regenerative Processes.- §8.3. Prototype Behavior: Birth-Death Processes; Strongly Stable Systems.- §8.4. Limiting Behavior of the Ergod
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