Issue 59, January 2006

Parametric Recurrence Data Analysis and the GRP Model

Issue 57 introduced the concept of recurrent event data and presented a non-parametric approach to analyze such data. Weibull++ 7 also offers a parametric approach that is performed using the Parametric RDA Specialized Folio. This type of folio can capture the trend, estimate the rate and predict the total number of recurrences. The failure and repair data of a repairable system can be treated as one type of recurrence data. Past and current repairs may affect the future failure process. For most recurrent events, time (distance, cycles, etc.) is a key factor. With the time, the recurrence rate may keep constant, increase or decrease. For other recurrent events, not only the time, but also the number of events can affect the recurrence process, e.g. the debugging process in software development.

The parametric analysis approach utilizes the General Renewal Process (GRP) model [1]. In this model, the repair time is assumed to be negligible so that the processes can be viewed as point processes. This model provides a way to describe the rate of occurrence of events over time such as in the case of data obtained from a repairable system. This model is particularly useful in modeling the failure behavior of a specific system and understanding the effects of the repairs on the age of that system. For example, consider a system that is repaired after a failure, where the repair does not bring the system to an as-good-as-new or an as-bad-as-old condition. In other words, the system is partially rejuvenated after the repair. Traditionally in as-bad-as-old repairs, also known as minimal repairs, the failure data from such a system would have been modeled using a homogeneous or non-homogeneous Poisson process (NHPP). On rare occasions, a Weibull distribution has been used as well, in cases where the system is almost as-good-as-new after the repair, also known as Perfect Renewal Process (PRP). However, for the intermediate states after the repair, there has not been a commercially available model, even though many models have been proposed in literature. In Weibull++ 7, the GRP model provides the capability of modeling such systems with partial renewal (general repair or imperfect repair/maintenance) and allows for a variety of predictions such as reliability, expected failures, etc.

The GRP Model for Parametric Recurrence Data Analysis
In this model, the concept of virtual age is introduced. Denote by t₁,t₂,...,t_n the successive failure times and let x₁,x₂,...,x₃ represent the time between failures. Assume that after each event, actions are taken to improve the system performance. Let q be the action effectiveness factor. There are two GRP models.

Type I:

Type II:

where v_i is the virtual age of the system right after the i^th repair. The Type I model assumes that the i^th repair cannot remove the damage incurred before the i^th failure. It can only reduce the additional age x_i to qx_i. The Type II model assumes that at the i^th repair, the virtual age has been accumulated to v_i-1 + x_i . The i^th repair will remove the cumulative damage from both current and previous failures by reducing the virtual age to q(v_i-1 + x_i ).

The power law function is used to model the rate of recurrence, which is:

The conditional pdf is:

The MLE method is used to estimate the model parameters. The log likelihood function is [1]:

where n is the total number of events during the entire observation period. T is the stop time of the observation. T = t_n if the observation stops right after the last event.

Confidence Bounds for Parametric Recurrence Data Analysis

In general, in order to obtain the virtual age, the exact occurrence time of each event (failure) should be available as described by the Type I and Type II models. However, the times are unknown until the corresponding events occur. For this reason, there are no closed-form expressions for total failure number and failure intensity, which are functions of failure times and virtual age. Therefore, in Weibull++, Monte Carlo simulation is used to predict values of virtual time, failure number, MTBF and failure rate. The approximate confidence bounds obtained from simulation are provided. The uncertainty of model parameters is also considered in the bounds. Further details will be discussed in future issues. For more information about confidence bounds for parametric recurrence data analysis, click here.

Example

The following table gives the failure times of the air-condition unit of an aircraft [2]. The observation is ended at the time of the last failure.

The purpose of the analysis is to estimate the GRP model parameters using the Type I virtual age option and obtain failure number and instantaneous failure intensity estimates at t = 1800 of accumulate time in operation.

Enter the data set into a Parametric RDA Specialized Folio in Weibull++. Choose 3 under Parameters and Type I under Settings. Keep the default simulation settings.

After clicking Calculate, the estimated parameters can be obtained as follows: β=1.1976, λ=4.94E-03 and q=0.1344.

The failure number and instantaneous failure intensity are given in the following plots.

Using the QCP, the failure number and instantaneous failure intensity can be estimated as follows:

References:

1- Mettas, A, and Zhao, W., "Modeling and Analysis of Repairable Systems with General Repair," 2005 Proceedings Annual Reliability and Maintainability Symposium, Alexandria, Virginia, 2005.

2- Cox, F. R., and Lewis, P.A. W. , The Statistical Analysis of Series of Events, London: Methuen, 1966.

Copyright 2006 ReliaSoft Corporation, ALL RIGHTS RESERVED