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This paper presents a maintenance strategy model of a system with a gamma deterioration process, regular inspection times, and intervention delay. Interventions are deemed necessary when the deterioration process is greater than an intervention level at inspection time. The objective is to determine the long-term mean costs of different maintenance strategies and to optimize these costs with respect to specific parameters, such as the intervention level or the inspection interval. Semi-regeneration properties at the inspection times and associated Markov renewal techniques are used in order to compute the long-term mean costs. This model is applied to the maintenance of railway tracks.

This paper discusses the intervention scheduling of a railway track, based on the observation of two dependent randomly increasing deterioration indicators. These two indicators are modeled through a bivariate Gamma process constructed by trivariate reduction. Empirical and maximum likelihood estimators are given for the process parameters and tested on simulated data. An EM algorithm is used to compute the maximum likelihood estimators. A bivariate Gamma process is then tted to real data of railway track deterioration. Intervention scheduling is de ned, ensuring that the railway track remains of good quality with a high probability. The results are compared to those based on both indicators taken separately, and also on one single indicator. The policy based on the joint information is proved to be safer than the other ones, which shows the potential of the bivariate model.

BackgroundThe aim of this article is to propose several methods that allow to investigate how and whether the shape of the hazard ratio after an intermediate event depends on the waiting time to occurrence of this event and/or the sojourn time in this state.MethodsA simple multi-state model, the illness-death model, is used as a framework to investigate the occurrence of this intermediate event. Several approaches are shown and their advantages and disadvantages are discussed. All these approaches are based on Cox regression. As different time-scales are used, these models go beyond Markov models. Different estimation methods for the transition hazards are presented. Additionally, time-varying covariates are included into the model using an approach based on fractional polynomials. The different methods of this article are then applied to a dataset consisting of four studies conducted by the German Breast Cancer Study Group (GBSG). The occurrence of the first isolated locoregional recurrence (ILRR) is studied. The results contribute to the debate on the role of the ILRR with respect to the course of the breast cancer disease and the resulting prognosis.ResultsWe have investigated different modelling strategies for the transition hazard after ILRR or in general after an intermediate event. Including time-dependent structures altered the resulting hazard functions considerably and it was shown that this time-dependent structure has to be taken into account in the case of our breast cancer dataset. The results indicate that an early recurrence increases the risk of death. A late ILRR increases the hazard function much less and after the successful removal of the second tumour the risk of death is almost the same as before the recurrence. With respect to distant disease, the appearance of the ILRR only slightly increases the risk of death if the recurrence was treated successfully.ConclusionsIt is important to realize that there are several modelling strategies for the intermediate event and that each of these strategies has restrictions and may lead to different results. Especially in the medical literature considering breast cancer development, the time-dependency is often neglected in the statistical analyses. We show that the time-varying variables cannot be neglected in the case of ILRR and that fractional polynomials are a useful tool for finding the functional form of these time-varying variables.

For any railway infrastructure manager, track maintenance is one of the most important tasks to perform. This mission requires a regular follow-up of geometric track quality and monitoring of intervention execution. Among other tools, TIMON, a computer application using track geometry measurements and maintenance operation data, is currently in use in France. This application illustrates the evolution of track quality indicators.We present a possible extension of this tool: a systematic analysis of the available data by applying a stochastic model for the track geometry deterioration is proposed. The use of environmental variables in order to identify the significance of their influence on track degradation is introduced.The stochastic model has two advantages: on the one hand it enables one to take into account the intrinsic variability of the degradation phenomenon and on the other, significant environmental variables are included into the model. The work presented here aims to classify sections of the high-speed railway network with respect to similar degradation behaviour. Two different graphical representations that could ease decision-making are proposed.