Reliability HotWire

Issue 109, March 2010

Hot Topics

Crow Extended Fleet Analysis

RGA offers two data types that have been designed to analyze data from systems operating in the field: Repairable and Fleet. This example illustrates the use of the Fleet data type. This type of data is intended to be used for analyzing randomly selected systems from the field. The results are returned in terms of total number of fleet failures vs. total fleet operation time (as opposed to the Repairable data type, which returns results in terms of individual system time). The main advantage of the Fleet data type is the ability to apply the Crow Extended model to data obtained from in-service systems that are being used in the field. With this analysis, you can evaluate the improvement (i.e. the "jump" in the MTBF) that could be achieved by rolling out a fix (or set of fixes) for all units operating in the field. In this article we will illustrate this application.


When applying the Crow Extended model for fielded system analysis (with both Repairable and Fleet data types), the underlying assumption is that beta = 1. Beta determines the behavior of the rate at which failures are being observed (i.e. the behavior of the failure intensity). A beta greater than 1 would be an indication that the system/fleet may be wearing out. A beta smaller than 1 may show that the system/fleet is improving over time. A beta of 1 would indicate failures are observed at a constant rate and the system is not improving nor wearing out. This is the underlying assumption associated with the Crow Extended model when the data set contains A modes (indicating a fix will not be applied) and BD modes (indicating a delayed fix will be applied after the termination time). This assumption implies that the system is in a relatively steady state as it is being used; it is neither wearing out nor exhibiting reliability growth. The failure intensity of a fielded system might be changing over time (e.g. increasing if the system exhibits wearout) and this would violate the assumption of beta = 1 that is required for the Crow Extended model. However, if you consider the data from a fleet perspective (where the cumulative number of fleet failures versus cumulative fleet time is modeled), the rate at which fleet failures are observed may be constant. Therefore, the Fleet data type can be used even when the underlying assumption of beta = 1 does not hold true for repairable systems analysis and you wish to apply the Crow Extended model to analyze the improvement achieved by fixing the failure modes with the BD mode classification. The following example demonstrates this potential application.

Fleet Analysis Example
The purpose of the analysis is to project the improvement in a system based on the implementation of planned fixes for some failure modes. Assume failure data has been collected from 22 fielded systems. The start time for all systems is time = 0. The end times for the systems are shown in Table 1 and the failures of the observed systems are recorded in Table 2.

Table 1: System End Times

System ID End Time
1 1396
2 4497
3 2132
4 3698
5 2514
6 2024
7 5822
8 5371
9 4877
10 2527
11 1980
12 5079
13 1023
14 3163
15 4767
16 6228
17 2156
18 5630
19 1841
20 5852
21 3556

Table 2: System Failures

System ID Event Time to Event Classification Mode
1 F 1396 BD 1
2 F 4497 BD 2
3 F 2132 BD 3
4 F 3698 BD 4
5 F 2514 BD 5
6 F 2024 BD 3
7 F 2100 BD 3
7 F 5822 A  
8 F 5371 A  
9 F 4233 BD 2
9 F 4234 BD 6
9 F 4877 BD 5
10 F 1877 BD 1
10 F 2527 A  
11 F 1420 BD 7
11 F 1980 BD 8
12 F 5079 BD 9
13 F 1023 BD 2
14 F 3163 BD 8
15 F 4767 BD 5
16 F 3795 BD 1
16 F 4375 A  
16 F 6228 BD 7
17 F 2156 A  
18 F 5630 A  
19 F 1841 BD 10
20 F 3385 BD 4
20 F 5852 BD 10
21 F 3556 A  
22 F 3956 A  
22 F 5425 BD 6

Initially, the Repairable data type with the Power Law model is used to analyze the data. The parameter results are shown below:

Figure 1: Results with Power Law model and repairable data type

The estimate for beta, 2.3829, is greater than 1, but the confidence bounds on beta also must be checked to see if they include beta = 1. The confidence bounds on beta can be obtained from the Quick Calculation Pad, using the icon shown in Figure 2.

Figure 2: Parameter calculation in Quick Calculation Pad

For this analysis, the lower one-sided confidence bound on beta at a 90% confidence level is:

BetaLower = 1.4907

The lower confidence on beta does not include 1. This violates the assumption associated with the Crow Extended model and the model cannot be applied to this data set.

However, this assumption may not be violated if the data set is considered from a fleet perspective. Note that the data can be easily transferred to a Fleet data type by clicking Transfer to new Data Folio on the control panel.

When analyzing this data as a fleet, an interval must be selected for grouping the data. In this example, a constant interval of 8000 hours was chosen. This provides a sufficient number of groups for the analysis (11). The goodness-of-fit tests also return favorable results. The results in Figure 3 are obtained when analyzing the data using the Crow Extended model.

Figure 3: Results with Crow Extended model and fleet data type

This value is very close to 1. Just to verify, we can also see that the lower confidence bound at a 90% confidence level on beta includes 1.

BetaLower = 0.8302

Given this, there is no evidence we are violating the assumption that beta = 1. Therefore, the analysis can proceed with the application of the Crow Extended model. A conservative effectiveness factor (EF) for the BD modes was chosen, such that EF = 0.4. This represents the expected reduction in the failure intensity associated with each BD mode from applying the corrective actions. In other words, 60% of each BD mode’s failure intensity will remain in the system after the corrective actions have been implemented. Figure 4 shows the resulting Growth Potential MTBF plot.

Figure 4: Growth Potential MTBF plot for fleet analysis data

The demonstrated MTBF for the fleet is equal to 2631 hours. Therefore, based on the current configuration of the system in the fleet, a failure within the fleet is expected to occur once for every 2631 hours of fleet operation. Given the proposed corrective actions, the fleet MTBF is expected to jump to 3387 hours. This jump in the fleet’s MTBF, a 28% improvement over the demonstrated MTBF, is the projected improvement based on the 10 proposed corrective actions (i.e. fixes for the 10 BD modes in the data set).


In this article, we showed an example where, even though the beta = 1 assumption was violated when looking at the individual systems (repairable system analysis), we could still proceed with a fleet data analysis. By doing that, we were able to estimate the MTBF improvement that could be achieved in the fleet by rolling out a set of fixes for all units operating in the field.

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