Reliability Assessment of Repairable Systems Using Simple Regression Models

The statistical nature of failures in repairable systems does not have a behaviour similar to non-repairable systems. The statistical models developed for the study of the reliability of repairable systems mostly based on the application of stochastic processes. However, there is a group of prediction models for reliability based on time series analysis. Below are the results and conclusions of the application of simple regression models in the escalators Avante model (TNE), in order to assess their potential use by maintenance organizations. KeywordsRepairable system, Reliability models, Simple regression, Escalators.


Introduction
Much of the developed statistical models for the study of the reliability of repairable systems based on the application of stochastic processes in which a failure of an element is a random variable, and once repaired the next failure is another random variable, which may or may not have equal probability density function. The most commonly used stochastic models are Markov chains, Homogeneous Poisson Processes (HPP), Non-Homogeneous Poisson Processes (NHPP) and Renewal Processes (RP). Most of the Markov chain methods based on two-state models (Kumamoto et al., 1980;Simpson and Kelly, 2002;Xiao and Li, 2008;Zheng et al., 2006), and to a lesser extent, models with multi-state Markov chains have developed.
The HPP model can only applied if it demonstrated that the Time Between Failure (TBF) data of the repairable system are independent, stationary and with exponential distribution (Tan et al., 2008;Wu et al., 2011). The NHPP model applies if the repairable system TBF data shown to be independent and non-stationary, being the Power Law Process (PLP) the most extended (Crow, RP models apply if TBF data from the serviceable system is shown to be independent, stationary, and with any distribution. The mathematical solution for this integral equation with convolution is complex and must particularized to each distribution presented by the TBF (Baxter et al., 1982;Andronov, 2014;Maghsoodloo and Helvaci, 2014).
The difficulty lies in choosing the most appropriate model for the data of the repairable systems under study, and although the scientific community has reached some meeting points regarding the applicability of the HPP, NHPP and RP models. To date, there is no consensus in relation to which imperfect models (or others) are the most appropriate to try to model those repairable systems that do not can apply to the HPP, NHPP or RP models.
The intensive development over decades of statistical models for the reliability of repairable systems denotes the great complexity of the treatment and modelling of the real data obtained in the systems in operation. Most of these models based on stochastic processes, but not all. There are statistical models of reliability that not based on stochastic processes: • Models of differential equations, Lloyd-Lipow (design phase), Aroef, IBM, etc. Most developed in the 60s. • Cumulative damage shock models, for example the Kijima model (Kijima and Nakagawa, 1991). They are suitable models for mechanical components. • Monte Carlo models, (Kaminskiy and Krivtsov, 1998).
• Prediction models based on time series analysis, (Liang, 2011). This paper classified within this group of models.
Reliability prediction models through time series analysis start from a data-oriented approach, which does not require a priori model specification. This time series technique has the flexibility to fit an appropriate empirical model, which is an adaptation of the data structure itself.
Therefore, the stochastic nature of the time series can be model with greater precision, although it is necessary to use self-correlation, partial correlation and spectral analysis tools to examine the underlying properties of the data, such as the existence of no seasonality, trend, etc.
Stand out as reliability prediction models based on time series analysis: • Regression models. This paper classified within this group of models.
Reliability regression models are oriented to construct a statistical model that describes the impact of one or more quantitative factors on a dependent variable. It is about finding the model that best fits the data, without needing to understand and explain the origin of the behaviour of the TBF. The regression methods commonly applied to reliability models are: simple, multiple, logistics, negative binomial, nonlinear, polynomial, Poisson and Cox proportional hazards.
In this paper, 27 simple regression models are tested and evaluated on the failure data of 40 escalators to model their reliability. The main advantage of this type of tests with respect to other models, is that the tests are simpler, require less technical resources and data processing times, quickly obtaining the presentation of results.

The System Tested
The escalator defined as a motorized ladder, inclined and in continuous movement, used to raise or lower people on which the transport surface, for example stairs that remain horizontal. The escalators made up of repairable components that, if they fail, replaced by other useful spare parts, the electromechanical components being predominant.
The escalators object of this test are of the TNE model and installed in a subway system in 2005, their large building blocks corresponding to those represented in Figure 1.
Regarding the type and treatment of the data, there is continuous and complete data for a finite population without sampling. They are quantitative data, uncensored and truncated by time, validated at source and without screening. No failure record is deleted, even if the distribution appears initially abnormal or outside the expected system values.
The 40 escalators of the TNE series under study have the same technical and constructive design, as well as the same operational context. The failures records correspond to the period of 2005-2014. The accumulated operating hours for each escalator exceed 67,000 hours.

Selected Models and Formulation
The models chosen for reliability estimation of the 40 escalators are simple regression (least squares method). It has chosen to test these models, knowing the high volume of data to be processed and taking into account the available resources, in order to assess their ability to adjust to the data and level of acceptance. For each escalator, the regression model that best adapted to the data was selected, that is, the one with a higher of R 2 determination coefficient adjusted in a range from zero to 100%. The goodnessof-fit test integrated within the ANOVA (analysis of variance) model by decomposing the variability of the dependent variable Y into a sum of squares model of the error or residues.
Of particular interest in this analysis is the test F and its associated P-value to test the statistical significance of the adjusted model. A small P-value (less than 0.05 at a significance level of 5%) indicates that a statistical relationship of the specified form exists between Y and X.
The formulation of simple regression models to better acceptance testing goodness of fit tests on 27 models of the 40 escalators attached. Linear, Square root of x, Square of y, square of x,

Tests and Numerical Examples
The total number of failure records of the 40 escalators in the study period is 8,837. There is an important and detailed database, which ensures that the results obtained in statistical tests have an adequate degree of integrity, see summary in Table 1. For each escalator, 27 simple regression models have tested, presenting the results in Table 2 of the model that has obtained a better fit to the failure data.  (t)] that best fits is linear and has no trend to failure (constant). There are 3 escalators with no trend of failures that have the best model as the square of y, square of x. For the 5 escalators with the growing trend in the failures, the best accepted regression simple model is the square of x. For the 2 escalators with the decreasing trend in failures, the best adjusted simple regression model is the square root of x. Figure 2 shows the linear simple regression model without trend to failures for the escalator number 6.       After more than 67,000 hours of operation of each escalator a large majority that does not have a trend to failures, but there is a significant dispersion in the value of E [N(t)]. This result does not seem to be the expected one in the case of escalators with the same technical and constructive design, as well as the same operational context. For example, escalator 38 accumulates 63 failures, while escalator 5 has 389 failures.
In all escalators observed that TBFs have the accumulation of several consecutive failures during short time periods, which precede and precede long time periods without accumulated failures. This widespread phenomenon in repairable systems called repetitive failures and they accumulate during operating life.
Repetitive failures have their origin in different causes of difficult diagnosis, such as the correct repair of complex repairable systems (Hatton, 1999;Karanikas, 2013). Repetitive failures have given rise to multiple models based on imperfect stochastic processes. The

Discussions and Limitation
The tests carried out for the estimation of the reliability by means of simple regression models (least squares method) of E[N(t)], have achieved a wide acceptance in the tests of goodness of fit, consuming low resources in the treatment of the data, the performance of the tests and the obtaining of results.
In the application to 40 escalators, it is observed that 33 escalators do not show a trend in E[N(t)], 5 escalators have a growing trend and 3 escalators decreasing trend. The most striking result is the range of values of E[N(t)] values of each escalator at the end of the test, exceeding 67,000 operating hours, which is 63 to 389 failures.
Among the limitations of this study, it is highlighted that for each escalator, more than one simple regression model has been accepted in the tests of goodness of fit, between 3 to 6 models, although the model that has been presented for each escalator a bigger fit to TBF data.
The systematic occurrence of repetitive failures observed in all escalators, which condition and cause some escalators accumulate many more failures as others, over long periods of operation.

Conclusions
Reliability models based on time series analysis are an alternative to models based on stochastic processes, when limited consumption of human and technical resources is required in the processing of data and tests, as well as obtaining results quickly.
These models, not having to adjust to an a priori model, obtain a high degree of acceptance in the goodness of fit tests, even with TBF with errant behaviours or complex explanation, as a contrast to the high levels of rejection that are usually obtained by testing models based on stochastic processes.
Reliability models based on time series analysis help limited to explain the behaviour of TBF, only to represent them with the least error the data, but they are a tool that due to its ease of compression and rapidity in obtaining results, they are very attractive to maintenance managers.