Applications of Non-Parametric Recurrent Event Data Analysis in Reliability Engineering
Recurrent Event Data Analysis (RDA) can be used in various applied fields such as reliability, medicine, social sciences, economics, business and criminology. The non-parametric approach for modeling Recurrent Events Data Analysis is an easy methodology.
The non-parametric recurrent event data analysis method provides a nonparametric graphical estimate of the mean cumulative number or cost of recurrence per unit versus age. In the reliability field, non-parametric recurrent event data analysis has many applications. Some of the applications are: 
Non-parametric recurrent events data analysis is simple, informative, widely useful and requires minimal assumptions. In this article, we present some examples that illustrate the use of non-parametric recurrent events data analysis. For more details about the non-parametric approach for modeling recurrent events data and the MCF function, review the Reliability Basics article of this issue.
Example 1: Estimating Number of Failures
An oil refinery company maintains a piece of rotating equipment. The company wants to make projections about the expected cumulative number of failures for 100 similar rotating equipments after 4 years of operation. Failure and suspension data, in hours, for 22 equipments were collected. Every time an equipment fails, crews repair the equipment and put it back in service. The following table lists the collected data. F indicates a failure event, whereas E indicates the end of observation event (no failure at that time).
The data set is entered in Weibull++ using the Non-Parametric RDA sheet, which can be added into the Weibull++ project by selecting Add Specialized Folio then Add Non-Parametric RDA from the Project menu.
After calculating, the results can be obtained by clicking the Show Analysis Summary button (...) on the Main page of the control panel. The following is the MCF plot along with the 90% confidence bounds.
We can also observe the effectiveness of the maintenance activity by studying the shape of the plot. A smooth curve through the MCF plot has a derivative that decreases as the equipments age. That is, the repair rate decreases as each population ages. This is typical of products with manufacturing defects.
Looking at the above plot, the company can estimate the number of failures per equipment after 4 years of operation (35,040 hours). At t = 35040, MCF = 0.366; therefore, in 100 equipments, about 36.6 failures are estimated to happen after 4 years of operation. The 90% upper bound for the cumulative number of failures after 4 years is 0.569 x 100 = 56.9 failures. This estimation can be useful for predicting failure, estimating repair cost and planning for spare parts.
Example 2: Estimating Burn-in Period
An electronics company subjects its products to a burn-in program to reduce the occurrences for failures and the need for repairs in the field. Every produced unit is run, repaired upon failure and placed back into the test until the population's (instantaneous) recurrence rate decreases to a desired value rgoal. Note that this type of burn-in program is different from many burn-in programs in which failed units are disregarded and only the units that survive the burn-in are shipped to customers.
The company collected recurrent failures data and wants to determine the appropriate burn-in period tb that would lower the recurrence rate to a desired value rgoal equal to 1 failure per 50,000 cycles.
tb can be estimated from the sample MCF by moving a straight line with slope rgoal vertically until it becomes tangent to the smoothed MCF plot. The corresponding age at the tangent point determines the appropriate tb. 
The appropriate burn-in period, tb, is estimated to be 6744 cycles.
The process of drawing the smoothed MCF plot and the tangent line is not an automated feature of Weibull++. The above plot was constructed manually using the RS Draw feature in Weibull++.
Example 3: Different Shapes of MCF
As mentioned in Example 1, the shape of the MCF plot can reveal many things about the behavior of the recurrent events. The following plots are some example of the different MCF shapes and their interpretation.
The Constant Recurrence Rate:
When the smooth curve that goes through MCF plot increases monotonically, the systems are exhibiting what is usually referred to as useful life, which is the 'stable' period during which the failures occur at a constant rate.
The Increasing Recurrence Rate:
When the smooth curve that goes through the MCF plot is concave up (has a derivative that increases as the units age), the recurrence rate increases as the population ages. This is typical of products with wear out problems. It could also indicate that the maintenance effectiveness is degrading with time (because of inefficiencies in maintenance plans, degradation in maintenance tools, etc.).
The Decreasing Recurrence Rate:
When the smooth curve that goes through the MCF plot is concave down (has a derivative that decreases as the units age), the recurrence rate decreases as the population ages. This is typical of products with startup problems, which are usually attributed to manufacturing defects. It could also indicate that the maintenance effectiveness is improving with time (maintenance crews learning more about effective maintenance, etc.).
The "Bathtub" Recurrence Rate:
When the smooth curve that goes through the MCF plot is concave down, then increases monotonically then becomes concave up, the MCF plot is said to have a bathtub shape. This behavior is typical of systems that first experience infant mortality startup problems then exhibit what is usually referred to as useful life, which is the 'stable' period during which the failures occur at a constant rate. As the age increases, the systems start entering the wear out stage.
1. Nelson, Wayne, Recurrent Events Data Analysis for Product Repairs, Disease Recurrences, and Other Applications, ASA-SIAM, 2003.
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