تجزیه و تحلیل سیستم اصلاح پذیر در حضور متغیرهای کمکی و اثرات تصادفی

This paper aims to model the failure pattern of repairable systems in presence of explained and unexplained heterogeneity. The failure pattern of each system is described by a Power Law Process. Part of the heterogeneity among the patterns is explained through the use of a covariate, and the residual unexplained heterogeneity (random effects) is modeled via a joint probability distribution on the PLP parameters. The proposed approach is applied to a real set of failure time data of powertrain systems mounted on 33 buses employed in urban and suburban routes. Moreover, the joint probability distribution on the PLP parameters estimated from the data is used as an informative prior to make Bayesian inference on the future failure process of a generic system belonging to the same population and employed in an urban or suburban route under randomly chosen working conditions.

Most of the papers on the reliability analysis of repairable systems deal with the homogeneous case although homogeneous populations and homogeneous operating conditions can be hardly found in practice [1]. Indeed, in many cases a substantial heterogeneity is observed among the failure patterns of apparently identical repairable systems, due to the presence of differences among units characteristics and/or differences in operating and environmental conditions. Part of such heterogeneity can be frequently explained by the presence of assignable causes (often called explanatory variables or covariates). The residual part (often called the unobserved heterogeneity or random effects) remains rather unexplained, since it is caused by unassignable (unidentified or not recorded) factors. Refs. [1], [2], [3], [4], [5], [6] and [7] considered, among others, heterogeneity in non-homogeneous Poisson processes, whereas Lindqvist et al. [8] discussed heterogeneity in the more general case of trend-renewal processes. An interesting example of the joint presence of explained heterogeneity and random effects is provided by the powertrain system of a fleet of buses employed in urban and suburban routes of Naples, Italy. Routes differ among themselves mainly in the traffic conditions: suburban service is characterized by noticeable thinner traffic, with a more regular use of the engine and a less frequent use of the gearbox with respect to the urban service, so that service type is expected to have some significant effect on the powertrain reliability. However, the large variability observed in the failure patterns (depicted in Fig. 1 and Fig. 2, for urban and suburban service, respectively) is only partially explained by service type. Other factors which potentially affect the reliability of the system (inherent characteristics of each system and/or not recorded unit specific service characteristics, such as the mean slope of the roads, the mean payload, etc.) leave unexplained part of the existent heterogeneity.This paper introduces and studies a method to analyze the failure pattern of the powertrain systems in presence of identified and unidentified sources of heterogeneity. In particular, the failure pattern of each individual system is described by a Power Law Process (PLP) [9] and [10], that is, the non-homogeneous Poisson process with failure intensity equation(1) View the MathML sourceλ(t)=βα(tα)β−1,α,β>0 Turn MathJax on The heterogeneity among the patterns is then modeled through the combined use of: (a) a joint probability distribution g(α,β) on the PLP parameters α and β that models the random effects, and (b) a dummy covariate x, that defines service type (urban or suburban service) and acts only on the distribution of the scale parameter α, via the covariate parameter q. The proposed joint distribution g(α,β) is indexed by five hyperparameters θ=(a,b, c,d,y) in order to allow the first two marginal moments of α and β and the cross moment E{αβ} to be assessed each independently of the others. The hyperparameters θ and the covariate parameter q, are estimated via two different maximum likelihood procedures: (a) the herein-called 2-step procedure, which first estimates the PLP parameters α and β of each bus and then estimates θ and q by treating the obtained ML estimates of α and β as they were the true (unit specific) values of the considered unknown parameters; and (b) the herein-called 1-step procedure, which estimates θ and q directly from the observed failure data. The hypotheses of presence of explained heterogeneity and random effects against the null hypothesis of common parameters across all the systems are checked via the likelihood ratio testing procedure. Based on the ML estimates of θ and q, a joint distribution on the PLP parameters of a generic powertrain system belonging to the same population and employed in an urban or suburban route under randomly chosen working conditions is formulated. This informative distribution is then used in a Bayesian framework to make inference on the failure process of a future system, in conjunction with its early failure data. Finally, to make the potential users more confident in the effectiveness of the proposed approach, a comparison between the results obtained adopting the proposed informative Bayesian procedure and those obtained without using prior information closes the work.

The model proposed in this paper showed to be able to describe the heterogeneity in the intensity function of the powertrain system of a fleet of 33 buses employed in urban or suburban service. Part of the existing heterogeneity is explained by a covariate representing service type, while the remaining part of the heterogeneity, due to different working conditions, is modeled through a joint probability distribution on the parameters which index the adopted stochastic process, namely the PLP. The proposed regression model allows one to estimate the powertrain reliability under new future operating conditions, different from the actual ones and consisting, for example, of suburban lines with heavy traffic. Due to the strong correlation between the estimated PLP parameters, it was possible to formulate an informative joint distribution on the PLP parameters for a generic powertrain system belonging to the same population and employed in an urban or suburban route under randomly chosen working conditions. Using this informative distribution in conjunction with the early failures of a future system, allows one to obtain point estimates of the PLP parameters closer to the true values and smaller coefficients of variation, when compared to those obtained by using the non-informative prior density.