| Introduction | p. 1 |
| Aim and Scope of the Book | p. 1 |
| Brief Overview | p. 5 |
| Failure Rate and Mean Remaining Lifetime | p. 9 |
| Failure Rate Basics | p. 10 |
| Mean Remaining Lifetime Basics | p. 13 |
| Lifetime Distributions and Their Failure Rates | p. 19 |
| Exponential Distribution | p. 19 |
| Gamma Distribution | p. 20 |
| Exponential Distribution with a Resilience Parameter | p. 22 |
| Weibull Distribution | p. 23 |
| Pareto Distribution | p. 24 |
| Lognormal Distribution | p. 25 |
| Truncated Normal Distribution | p. 26 |
| Inverse Gaussian Distribution | p. 27 |
| Gompertz and Makeham-Gompertz Distributions | p. 27 |
| Shape of the Failure Rate and the MRL Function | p. 28 |
| Some Definitions and Notation | p. 28 |
| Glaser's Approach | p. 30 |
| Limiting Behaviour of the Failure Rate and the MRL Function | p. 36 |
| Reversed Failure Rate | p. 39 |
| Definitions | p. 39 |
| Waiting Time | p. 42 |
| Chapter Summary | p. 43 |
| More on Exponential Representation | p. 45 |
| Exponential Representation in Random Environment | p. 45 |
| Conditional Exponential Representation | p. 45 |
| Unconditional Exponential Representation | p. 47 |
| Examples | p. 48 |
| Bivariate Failure Rates and Exponential Representation | p. 52 |
| Bivariate Failure Rates | p. 52 |
| Exponential Representation of Bivariate Distributions | p. 54 |
| Competing Risks and Bivariate Ageing | p. 59 |
| Exponential Representation for Competing Risks | p. 59 |
| Ageing in Competing Risks Setting | p. 60 |
| Chapter Summary | p. 65 |
| Point Processes and Minimal Repair | p. 67 |
| Introduction - Imperfect Repair | p. 67 |
| Characterization of Point Processes | p. 70 |
| Point Processes for Repairable Systems | p. 72 |
| Poisson Process | p. 72 |
| Renewal Process | p. 73 |
| Geometric Process | p. 76 |
| Modulated Renewal-type Processes | p. 79 |
| Minimal Repair | p. 81 |
| Definition and Interpretation | p. 81 |
| Information-based Minimal Repair | p. 83 |
| Brown-Proschan Model | p. 84 |
| Performance Quality of Repairable Systems | p. 85 |
| Perfect Restoration of Quality | p. 86 |
| Imperfect Restoration of Quality | p. 88 |
| Minimal Repair in Heterogeneous Populations | p. 89 |
| Chapter Summary | p. 92 |
| Virtual Age and Imperfect Repair | p. 93 |
| Introduction - Virtual Age | p. 93 |
| Virtual Age for Non-repairable Objects | p. 95 |
| Statistical Virtual Age | p. 95 |
| Recalculated Virtual Age | p. 98 |
| Information-based Virtual Age | p. 102 |
| Virtual Age in a Series System | p. 105 |
| Age Reduction Models for Repairable Systems | p. 107 |
| G-renewal Process | p. 107 |
| 'Sliding' Along the Failure Rate Curve | p. 109 |
| Ageing and Monotonicity Properties | p. 115 |
| Renewal Equations | p. 123 |
| Failure Rate Reduction Models | p. 125 |
| Imperfect Repair via Direct Degradation | p. 127 |
| Chapter Summary | p. 130 |
| Mixture Failure Rate Modelling | p. 133 |
| Introduction - Random Failure Rate | p. 133 |
| Failure Rate of Discrete Mixtures | p. 138 |
| Conditional Characteristics and Simplest Models | p. 139 |
| Additive Model | p. 141 |
| Multiplicative Model | p. 143 |
| Laplace Transform and Inverse Problem | p. 144 |
| Mixture Failure Rate Ordering | p. 149 |
| Comparison with Unconditional Characteristic | p. 149 |
| Likelihood Ordering of Mixing Distributions | p. 152 |
| Mixing Distributions with Different Variances | p. 157 |
| Bounds for the Mixture Failure Rate | p. 159 |
| Further Examples and Applications | p. 163 |
| Shocks in Heterogeneous Populations | p. 163 |
| Random Scales and Random Usage | p. 164 |
| Random Change Point | p. 165 |
| MRL of Mixtures | p. 167 |
| Chapter Summary | p. 168 |
| Limiting Behaviour of Mixture Failure Rates | p. 171 |
| Introduction | p. 171 |
| Discrete Mixtures | p. 172 |
| Survival Models | p. 175 |
| Main Asymptotic Results | p. 177 |
| Specific Models | p. 179 |
| Multiplicative Model | p. 179 |
| Accelerated Life Model | p. 182 |
| Proportional Hazards and Other Possible Models | p. 183 |
| Asymptotic Mixture Failure Rates for Multivariate Frailty | p. 184 |
| Introduction | p. 184 |
| Competing Risks for Mixtures | p. 185 |
| Limiting Behaviour for Competing Risks | p. 187 |
| Bivariate Frailty Model | p. 189 |
| Sketches of the Proofs | p. 192 |
| Chapter Summary | p. 196 |
| 'Constructing' the Failure Rate | p. 197 |
| Terminating Poisson and Renewal Processes | p. 197 |
| Weaker Criteria of Failure | p. 201 |
| Fatal and Non-fatal Shocks | p. 201 |
| Fatal and Non-fatal Failures | p. 205 |
| Failure Rate for Spatial Survival | p. 207 |
| Obstacles with Fixed Coordinates | p. 207 |
| Crossing the Line Process | p. 210 |
| Multiple Availability on Demand | p. 213 |
| Introduction | p. 213 |
| Simple Criterion of Failure | p. 215 |
| Two Consecutive Non-serviced Demands | p. 218 |
| Other Weaker Criteria of Failure | p. 221 |
| Acceptable Risk and Thinning of the Poisson Process | p. 222 |
| Chapter Summary | p. 223 |
| Failure Rate of Software | p. 225 |
| Introduction | p. 225 |
| Several Empirical Models for Software Reliability | p. 226 |
| The Jelinski-Moranda Model | p. 227 |
| The Moranda Model | p. 228 |
| The Schick and Wolverton Model | p. 229 |
| Models Based on the Number of Failures | p. 230 |
| Time-dependant Operational Profile | p. 231 |
| General Setting | p. 231 |
| Special Cases | p. 233 |
| Chapter Summary | p. 235 |
| Demographic and Biological Applications | p. 237 |
| Introduction | p. 237 |
| Unobserved Overall Resource | p. 242 |
| Mortality Model with Anti-ageing | p. 246 |
| Mortality Rate and Lifesaving | p. 250 |
| The Strehler-Mildvan Model and Generalizations | p. 252 |
| 'Quality-of-life Transformation' | p. 253 |
| Stochastic Ordering for Mortality Rates | p. 255 |
| Specific Population Modelling | p. 256 |
| Definitions of Life Expectancy | p. 260 |
| Comparison of Life Expectancies | p. 263 |
| Further Inequalities | p. 265 |
| Tail of Longevity | p. 268 |
| Chapter Summary | p. 273 |
| References | p. 275 |
| Index | p. 287 |
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