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