The Mathematical Methods in Reliability conferences serve as a forum for discussing fundamental issues on mathematical modeling in reliability theory and its applications. It is a forum that brings together mathematicians, probabilists, statisticians, and computer scientists with a central focus upon reliability.

The University of Strathclyde hosted the fifth in the series of conferences in Glasgow in 2007. Previous conferences were held in Bucharest, Romania, in Bordeaux, France, in Trondheim, Norway, and in Sante Fe, New Mexico, USA.

This book contains a selection of papers originally presented at the conference and now made available to a wider audience in revised form. The book has been organized into a number of sections that represent different themes from the meeting, and important current research areas within the overall area.

1. Graphical Modeling and Bayesian Networks

Graphical methods are becoming increasing popular for modeling and supporting the computation of the reliability of complex systems. The papers within this section address a number of challenges currently facing these methods. Langseth provides a brief review of the state of the art of Bayesian Networks in relation to reliability and then focuses on the current challenges of modeling continuous variables within this framework. Hanea and Kurowicka extend the theory for non-parametric continuous Bayesian Networks to include ordinal discrete random variables, where dependence is measured through rank correlations. Donat, Bouillaut and Leray develop a Bayesian Network approach to capture reliability that is changing dynamically. Jonczy and Haenni develop a method using propositional directed acyclic graphs to represent the structure function and hence facilitate the computation of the reliability of networks.

2. Repairable Systems Modeling

One of the fundamental problems in reliability is to find adequate models for failure and repair processes. Simple renewal models provide familiar examples to students of probability and reliability, and provide the basic building blocks for many commercial simulation packages. However, such models do not come near to describing the complex interactions between failure and repair. The paper of Kahle looks at the way (possibly) incomplete repair interacts with the failure process through a Kijima type process. Often the overall failure repair process in real systems follows a homogeneous Poisson process, and Kahle shows that maintenance schedules can be constructed to generate this type of output. Volf looks at models where degradation is modeled through a number of shocks or some other random process, and considers how one can choose optimal repair policies that stabilize the equipment hazard rate. Finally, Daneshkhah and Bedford show how Gaussian emulators can be used to perform computations of availability. A major problem in practice is to understand how sensitive availability calculations are to parameter choices, and emulators provide the potential to perform such calculations on complicated systems to a fair degree of accuracy and in a computationally efficient manner.

3. Competing Risk

Competing risks arise in reliability and maintenance analysis through the ways in which data is censored. Rather than getting “pure” failure data we usually have a messy mixture of data, for there may be many different reasons for taking equipment offline and bringing it back to “as new”, or at least in an improved state. A competing risk model is used to model the times at which such failure causes would be realized, taking into account possible interdependencies between them. There has been a growing interest in competing risk modeling over the last 10–15 years, and the papers presented here demonstrate this. Dewan looks at the interrelationship between various kinds of independence assumptions in competing risk modeling. Sankaran and Ansa consider the problem in which the failure cause is sometimes masked, and additional testing might be required to find the true failure cause. The final two papers of this section move from IID models of Competing Risk to take a point process perspective. Lindqvist surveys a number of recent papers on this topic and discussing the benefits of moving to this wider framework. Finally Dijoux, Doyen and Gaudoin generalize the “usual” independent competing model theory for IID and show that in the point process generalization one can properly formulate and solve the corresponding identifiability issues.

4. Mixture Failure Rate Modeling

Mixture models provide a means of analyzing reliability problems where there exist, for example, multiple failure modes or heterogeneous populations. It will not always be possible to observe all factors influencing the time to event occurrence, hence a random effect, called a frailty, can be included in the model. A frailty is an unobserved proportionality factor that modifies the hazard function of an item or group of items. Frailty models can be classed as univariate, when there is a single survival endpoint, or multivariate, when there are multiple survival endpoints such as under competing risks or recurrent event processes. There is much interest in modeling mixtures and frailty in survival analysis. We include two papers in this area. Finkelstein and Esaulova derive the asymptotic properties of a bivariate competing risks model, where the lifetime of each component is indexed by a frailty parameter and, under the assumption of conditional independence of the components, the correlated frailty model is considered. The other paper, due to Badá and Berrade, aims to give insights into the properties of the reversed hazard rate, defined as the ratio of the density to the distribution function, and the mean inactivity time in the context of mixtures of distributions.

5. Signature

The signature of a system refers to a vector where the i-th element is the probability that the system fails upon the realization of i components. The Samaniego representation of the failure time of a system distinguishes the properties of the system through the signature from the probability distributions on the lifetime of the components. Such a representation is effective for comparing the reliability of different systems. This section of papers is concerned with developments of Samaniego representation. Rychlik develops bounds for the distributions and moments of coherent system lifetimes. Triantafyllou and Koutras develop methods to facilitate the calculation of the signature of a system through generating functions. Hollander and Samaniego develop a new signature based metric for comparing the reliability of systems. An important generalization of the concept of independence is that of exchangeability. This assumption is key to Bayesian and subjectivist modeling approaches. The paper of Spizzichino considers symmetry properties arising as a result of exchangeability and discusses generalizations to non-exchangeable systems.

6. Relations among Aging and Stochastic Dependence

Aging properties have always played an important role in reliability theory, with a multiplicity of concepts available to describe subtle differences in aging behavior. A particularly interesting development is to place such aging concepts in a multivariate context, and consider how multiple components (or multiple failure modes) interact. The paper of Spizzichino and Suter looks at aging and dependence for generalizations of the Marshall-Olkin model. Their work develops closure results for survival copulas in certain classes with specified aging properties. Belzunce, Mulero and Ruiz develop new variants on multivariate increasing failure rate (IFR) and decreasing mean residual life (DMRL) notions. Some of the basic properties and relationships between these definitions are given.

7. Theoretical Advances in Modeling, Inference and Computation

This collection of papers is concerned with developments in modeling, inference and computation for reliability assessment. Ruggeri and Soyer develop hidden Markov modeling approaches and self exciting point process models to address the issue of imperfect reliability development of software. Huseby extends the use of matroid theory to directed network graphs and derives results to facilitate the calculation of the structure function. Coolen and Coolen-Schrijner extend nonparametric predictive inference techniques to address k-out-of-m systems.

8. Recent Advances in Recurrent Event Modeling and Inference

Recurrent event processes correspond to those processes where repeated events are generated over time. In reliability and maintenance, recurrent event processes may correspond to failure events of repaired systems, processes for detection and removal of software faults, filing of warranty claims for products and so forth. Common objectives for recurrent event analysis includes describing the individual event processes, characterizing variation across processes, determining the relationship of external factors on the pattern of event occurrence and modeling multi-state event data. Model classes include Poisson, renewal and intensity-based for which a variety of parametric, semi-parametric and non-parametric inference is being developed. There has been growing interest in recurrent event analysis and modeling in reliability, medical and related fields as the papers presented here demonstrate. Adekpedjou, Quiton and Peña consider the problem of detecting outlying inter-event times and examine the impact of an informative monitoring period in terms of loss of statistical efficiency. Mercier and Roussignol study and compute the first-order derivatives for some functional of a piece-wise deterministic Markov process, used to describe the time-evolution of a system, to support sensitivity analysis in dynamic reliability. Lisnianski considers a multi-state system with a range of performance levels which are observed together with the times at which the system makes a transition in performance state and provides a method for estimating the transition intensities under the assumption that the underlying model is Markovian. Finally, van der Weide, van Noortwijk and Suyono present new results in renewal theory with costs that can be discounted according to any discount function which is non-increasing and monotonic over time.

Acknowledgments

The organization of the conference was made possible by the hard work of a number of different people working at Strathclyde:

Anisah Abdullah, Babakalli Alkali, Samaneh Balali, Tim Bedford, Richard Burnham, Daosheng Cheng, Alireza Daneshkhah, Gavin Hardman, Kenneth Hutchison, Alison Kerr, Haiying Nan, John Quigley, Matthew Revie, Caroline Sisi, Lesley Walls, Bram Wisse.

The conference itself was sponsored by the University of Strathclyde, Glasgow City Council and Scottish Power, whom we thank for their contributions to the event.