Trend Renewal Process

Another class of alternative models to the renewal process (RP) and NHPP are the trend renewal processes (TRP). This model is a generalisation of Berman’s gamma process [Lindqvist 2006] and works by generalising the following property of the NHPP. First the cumulative intensity function (CIF) corresponds to an intensity (λ). Then if T1, T2 is

an NHPP process (λ (t)) the time transformed stochastic process Λ (T1), Λ (T2) is HPP.

The TRP is defined by allowing the HPP to be any renewal process RP (F), with a specified distribution F for the inter-arrival times of this renewal process.

An example of this process is the replacement of a major part in a system (a tractor engine is used as an example); if the rest of the system is not subject to wear the RP would be a suitable model for the failure process, however if wear is present an increased replacement frequency could be expected. The TRP achieves this by accelerating the internal time of the renewal process which represents the cumulative wear. It can be seen that the TRP model has some similarities to the accelerated failure rate models [Lindqvist 2006].

Analysis of failure data associated with the operation of heterogeneous implementations must be approached with care. It can lead to an apparently decreasing failure rate, which can be counterintuitive due to the effects of wear and aging on the system. Proschan [Proschan 1963] demonstrated this fact statistically through using a result from Barlow et al. [Barlow et al. 1963] which implies that a mixture of exponential distributions has a decreasing failure rate. The connection between heterogeneity and the Poisson process was studied as early as 1920 [Greenwood et al. 1920] and it has been shown in biostatistics that neglecting individual heterogeneity may lead to severe bias in lifetime distributions through references in biostatistics literature by Aalen et al. [Aalen et al.1988], Hougaard et al. [Hougaard et al 1996] and Vaupel et al. [Vaupel et al. 1979]. Lindqvist [Lindqvist 2006] states that the presence of heterogeneity is often apparent from repairable systems data if there is a large variation in the number of events per system. In addition it is not really possible to distinguish

between heterogeneity and the dependence of the intensity on past events for a single process. Heterogeneity can be modelled by including an unobservable multiplicative constant in the conditional intensity of the process. For systems with a single type of event the conditional intensity γ(t) is replaced with aγ(t) where a is a random variable that represents the “frailty” of the system. Since a is unobservable one needs to review its distribution in order to derive the likelihood function from the observed data. Lindqvist et al [Lindqvist et al. 2003] introduces heterogeneity into the TRP and other processes and use a three dimensional cube based approach [Lindqvist 2006] to facilitate the presentation of maximum log likelihood values and parameter estimations. Several examples are shown which appear to support the conclusions of Proschan [Proschan 1963] and conclusions are drawn that there is no significant heterogeneity present in the stated examples, however a slight time trend with a p- value of 0.022 is detected

In many repairable systems one of the main aims is to detect trends in failure data which occur over time. These trends may be monotonic indicating an improving or deteriorating system, or a non-monotonic such as a bathtub curve or a cyclic trend. In this context there are two main types of trend testing available; graphical and statistical trend testing. Graphical testing normally entails using the plot of the failure pattern to identify any trends present. Examples of this method include the Nelson Aalen plot and the total time on test (TTT) plots, each of which identifies deviation in the intensity function corresponding to system changes. Statistical trend testing is biased towards detecting the null hypothesis for the HPP or renewal process. This test is designed to detect if the failure process is stationary rather than displaying a trend. There are several tests available for this analysis including the Laplace test. These tests are predominantly biased towards detecting if the failure process is an HPP. Additional tests are available to identify if the process is a renewal process, these tests include the modified Laplace test and Lewis Robinson test [Ascher et al 1984, Lindqvist 2006]

Having reviewed the most commonly deployed analysis methods it is logical to consider next the manner in which these methods can be applied in reliability analysis and performance assessment.

3.3 Lifetime Analysis Methods

One of the important topics in failure data analysis is to select and specify the most appropriate lifetime distribution that describes the times to failure of the system. There are two general approaches to fitting reliability distributions to failure data. The first method involves the derivation of an empirical reliability function directly from the data. The second method identifies and adopts an appropriate parametric distribution, such as Weibull, Gamma and the exponential lognormal which can be used within the process method to estimate the unknown parameters The second method is widely practised because of the ability to extrapolate data beyond the sample range and to apply more complex analysis methods to calculate properties such as hazard rates and mean time to failure (MTTF). There are several analysis methods supporting this approach which have applications in mechanical system reliability analysis, including those considered below.