Fleet Analysis

Fleet analysis is similar to the repairable systems analysis described previously. The main difference is that a fleet of systems is considered and the models are applied to the fleet failures rather than to the system failures. In other words, repairable system analysis models the number of system failures versus system time; whereas fleet analysis models the number of fleet failures versus fleet time.

The main motivation for fleet analysis is to enable the application of the Crow Extended model for fielded data. In many cases, reliability improvements might be necessary on systems that are already in the field. These types of reliability improvements are essentially delayed fixes (BD modes) as described in the Crow Extended chapter of this on-line guide.

Recall from the Crow Extended chapter that in order to make projections using the Crow Extended model, the $\beta$ of the combined A and BD modes should be equal to 1. Since the failure intensity in a fielded system might be changing over time (e.g. increasing if the system wears out), this assumption might be violated. In such a scenario, the Crow Extended model cannot be used. However, if a fleet of systems is considered and the number of fleet failures versus fleet time is modeled, the failures might become random. This is because there is a mixture of systems within a fleet, new and old, and when the failures of this mixture of systems are viewed from a cumulative fleet time point of view, they may be random. Figures 13.5 and 13.6 illustrate this concept. Figure 13.5 shows the number of failures over system age. It can be clearly seen that as the systems age, the intensity of the failures increases (wearout). The superposition system line, which brings the failures from the different systems under a single timeline, also illustrates this observation. On the other hand, if you take the same four systems and combine their failures from a fleet perspective, and consider fleet failures over cumulative fleet hours, then the failures seem to be random. Figure 13.6 illustrates this concept in the System Operation plot when you consider the Cum. Time Line. In this case, the $\beta$ of the fleet will be equal to 1 and the Crow Extended model can be used for quantifying the effects of future reliability improvements on the fleet.

Methodology for Fleet Analysis

Figures 13.5 and 13.6 illustrate that the difference between repairable systems data analysis and fleet analysis is the way that the data set is treated. In fleet analysis, the time-to-failure data from each system is "stacked" to a cumulative timeline. For example, consider the two systems in Table 13.2.

As this example demonstrates, the accumulated timeline is determined based on the order of the systems. So if you consider the data in Table 13.2 by taking System 2 first, the accumulated timeline would be: 4, 9, 13, 18, 22, with an end time of 25. Therefore, the order in which the systems are considered is somewhat important. However, in the next step of the analysis the data from the accumulated timeline will be grouped into time intervals, effectively eliminating the importance of the order of the systems. Keep in mind that this will NOT always be true. This is true only when the order of the systems was random to begin with. If there is some logic/pattern in the order of the systems, then it will remain even if the cumulative timeline is converted to grouped data. For example, consider a system that wears out with age. This means that more failures will be observed as this system ages and these failures will occur more frequently. Within a fleet of such systems, there will be new and old systems in operation. If the data set collected is considered from the newest to the oldest system, then even if the data points are grouped, the pattern of fewer failures at the beginning and more failures at later time intervals will still be present. If the objective of the analysis is to determine the difference between newer and older systems, then that order for the data will be acceptable. However, if the objective of the analysis is to determine the reliability of the fleet, then the systems should be randomly ordered.

Data Analysis for Fleet Analysis

Once the accumulated timeline has been generated, it is then converted into grouped data. To accomplish this, a group interval is required. The group interval length should be chosen so that it is representative of the data. Also note that the intervals do not have to be of equal length. Once the data points have been grouped, the parameters can be obtained using maximum likelihood estimation as described in the Crow Discrete Reliability Growth Model section of this on-line reference. The data in Table 13.2 can be grouped into 5 hr intervals. This interval length is sufficiently large to insure that there are failures within each interval. The grouped data set is given in Table 13.3.

The Crow-AMSAA model for Grouped Failure Times is used for the data in Table 13.3 and the parameters of the model are solved by satisfying the following maximum likelihood equations as described in the Crow Discrete Reliability Growth Model section of this on-line reference.

Fielded Systems Example 4

Table 13.4 presents data for a fleet of 27 systems. A cycle is a complete history from overhaul to overhaul. The failure history for the last completed cycle for each system is recorded. This is a random sample of data from the fleet. These systems are in the order in which they were selected. Suppose the intervals to group the current data are 10000, 20000, 30000, 40000 and the final interval is defined by the termination time. Conduct the fleet analysis.

System	Cycle Time $T_{j}$	Number of failures $N_{j}$	Failure Time $X_{ij}$
1	1396	1	1396
2	4497	1	4497
3	525	1	525
4	1232	1	1232
5	227	1	227
6	135	1	135
7	19	1	19
8	812	1	812
9	2024	1	2024
10	943	2	316, 943
11	60	1	60
12	4234	2	4233, 4234
13	2527	2	1877, 2527
14	2105	2	2074, 2105
15	5079	1	5079
16	577	2	546, 577
17	4085	2	453, 4085
18	1023	1	1023
19	161	1	161
20	4767	2	36, 4767
21	6228	3	3795, 4375, 6228
22	68	1	68
23	1830	1	1830
24	1241	1	1241
25	2573	2	871, 2573
26	3556	1	3556
27	186	1	186
Total	52110	37

Solution to Fielded Systems Example 4

For the system data in Table 13.4, the data can be grouped into 10000, 20000, 30000, 4000 and 52110 time intervals. Table 13.5 gives the grouped data.

Based on the above time intervals, the maximum likelihood estimates of $\widehat{\lambda }$ and $\widehat{\beta }$ for this data set are then given by:

Failures in Interval	Interval End Time
1	5
1	10
1	15
1	20
1	25

Time	Observed Failures
10000	8
20000	16
30000	22
40000	27
52110	37