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Markov Models and Reliability |
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1 Preliminaries |
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1.1 Introduction |
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1.2 Why Markov Models? |
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1.3 Poisson Processes and Queues |
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1.4 Probability Densities, Failure Rates, and MTBFs |
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1.5 Prior and Conditional Probabilities |
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2 Markov Model Fundamentals |
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2.1 What Is A Markov Model? |
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2.2 A Simple Markov Model for a Two-Unit System |
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2.3 Matrix Notation |
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2.4 Delayed Repair of Total Failures |
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2.5 Transient Analysis |
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2.6 Discussion |
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3 Considerations for More Complex Systems |
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3.1 Modeling Infrequent Periodic Repairs for First-Order States |
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3.2 Simplifying Higher-Order Models By State Aggregation |
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3.3 Iterative Solution of Steady-State Models |
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3.4 Modeling Periodic Repairs for Higher-Order States |
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3.5 Exact Solution of Complex Models with Infrequent Periodic Repair |
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3.6 Exact Solution With Multiple Distinct Periodic Repair Intervals |
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3.7 Model Truncation and Completeness |
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3.8 Models with Variable Transition Rates |
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4 Examples and Applications |
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4.1 Active/Backup System with Internal/External Fault Monitoring |
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4.2 Markov Models Of Dual-Redundant Systems |
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4.3 Redundant Systems With A Common Threat |
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4.4 Latent Failures of Threat Protection |
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4.5 Complete Markov Models and Reliability |
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4.6 Markov Models with Boolean Transition Logic |
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4.7 Dual Failures with General Densities |
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4.8 Time and Distance for Dual Engines |
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4.9 Required Order Factors |
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5 Formal Considerations |
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5.1 Two Properties of Markov Models |
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5.2 Open-Loop and Closed-Loop Markov Models |
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5.3 Asymptotic Rate of an Open Loop Markov Model |
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5.4 Complete Solutions of Linear Systems |
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5.5 Iterative Solutions of Homogeneous Linear Systems |
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5.6 Markov Models and Fault Trees |
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5.7 Evaluating Probabilities of Boolean Events |
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5.8 Negative Faults |
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5.9 Lowest-Order Transient Response |
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6 Non-Markovian Repairs |
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6.1 Periodic and Continuous Repair Models |
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6.2 Hierarchical Repair |
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6.3 Mixed Periodic Repairs |
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6.4 Series-Parallel Systems |
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6.5 Average Product of Sawtooth Functions |
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6.6 Reliability with Periodic Repairs |
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6.7 Failure Rates and Normalized Probabilities |
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7 Quantitative Reliability Analysis |
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7.1 Normalized Average Probability |
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7.2 The Arsenal Companion |
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7.3 Probability for Regulatory Requirements |
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7.4 Latencies and Periodic Repairs |
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7.5 Tiling Product of Matrices |
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7.6 Mean Rate Matrix and Diagonalization |
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7.7 Probabilities with Variable Failure Rates |
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7.8 Latent Protection and Uncertain Threat |
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7.9 The Distribution of Mission Length |
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Appendices |
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Appendix A: Reliability Models with Aging Components |
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A.1 Weibull Analysis |
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A.2 Lightning with Aging Protection |
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A.3 Markov Models With Variable Transition Rates |
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A.4 Timed Markov Models |
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A.5 Markov Models with Aging Components |
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A.6 Age Distributions in Continuous Markov Models |
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Appendix B: Quantifying Latent Risk |
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Appendix C: Probability Calculations Using Excel |
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Appendix D: Advisory Circular 25.1309-1B |
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