Recurrence Data from Repairable Systems

The RELIABILITY Procedure

Recurrence Data from Repairable Systems

When a repairable system fails, it is repaired and placed back in service. As a repairable system ages, it accumulates a history of repairs and costs of repairs. The mean cumulative function (MCF) M(t) is defined as the population mean of the cumulative number (or cost) of repairs up until time t. You can use the RELIABILITY procedure to compute and plot nonparametric estimates and plots of the MCF for the number of repairs or the cost of repairs. The Nelson (1995) confidence limits for the MCF are also computed and plotted. You can compute and plot estimates of the difference of two MCFs and confidence intervals. This is useful for comparing the repair performance of two systems. See "Analysis of Recurrence Data on Repairs" and "Comparison of Two Samples of Repair Data" for examples of the analysis of recurrence data from repairable systems.

Refer to Nelson (1995), Nelson (1988), Doganaksoy and Nelson (1991), and Nelson and Doganaksoy (1989) for discussions and examples of repairable systems analysis.

Formulas for the MCF estimator $\hat{M}(t)$ and the variance of the estimator Var $(\hat{M}(t))$ are given in Nelson (1995). Table 30.48 shows a set of artificial repair data from Nelson (1988). For each system, the data consist of the system and cost for each repair. If you want to compute the MCF for the number of repairs, rather than cost of repairs, then you should set the cost equal to 1 for each repair. A plus sign (+) in place of a cost indicates that the age is a censoring time. The repair history of each system ends with a censoring time.

Table 30.48: System Repair Histories for Artificial Data

Unit	(Age in Months, Cost in $100)
6	(5,$3)	(12,$1)	(12,+)
5	(16,+)
4	(2,$1)	(8,$1)	(16,$2)	(20,+)
3	(18,$3)	(29,+)
2	(8,$2)	(14,$1)	(26,$1)	(33,+)
1	(19,$2)	(39,$2)	(42,+)

Table 30.49 illustrates the calculation of the MCF estimate from the data in Table 30.48. The RELIABILITY procedure uses the following rules for computing the MCF estimates.

Order all events (repairs and censoring) by age from smallest to largest.
- If the event ages of the same or different systems are equal, the corresponding data are sorted from the largest repair cost to the smallest. Censoring events always sort as smaller than repair events with equal ages.
- When event ages and values of more than one system coincide, the corresponding data are sorted from the largest system identifier to the smallest. The system IDs can be numeric or character, but they are always sorted in ASCII order.
Compute the number of systems I in service at the current age as the number in service at the last repair time minus the number of censored units in the intervening times.
For each repair, compute the mean cost as the cost of the current repair divided by the number in service I.
Compute the MCF for each repair as the previous MCF plus the mean cost for the current repair.

Table 30.49: Calculation of MCF for Artificial Data

		Number I in	Mean
Event	(Age,Cost)	Service	Cost	MCF
1	(2,$1)	6	$1/6=0.17	0.17
2	(5,$3)	6	$3/6=0.50	0.67
3	(8,$2)	6	$2/6=0.33	1.00
4	(8,$1)	6	$1/6=0.17	1.17
5	(12,$1)	6	$1/6=0.17	1.33
6	(12,+)	5
7	(14,$1)	5	$1/5=0.20	1.53
8	(16,$2)	5	$2/5=0.40	1.93
9	(16,+)	4
10	(18,$3)	4	$3/4=0.75	2.68
11	(19,$2)	4	$2/4=0.50	3.18
12	(20,+)	3
13	(26,$1)	3	$1/3=0.33	3.52
14	(29,+)	2
15	(33,+)	1
16	(39,$2)	1	$2/1=2.00	5.52
17	(42,+)	0

The variance of the estimator of the MCF Var $(\hat{M}(t))$ is computed as in Nelson (1995). If the VARMETHOD2 option is specified, the method of Lawless and Nadeau (1995) is used to compute the variance of the estimator of the MCF. This method is recommended if the number of systems or events is large. Approximate two-sided $\gammax 100\%$ confidence limits for M(t) are computed as

$M_{L}(t) = \hat{M}(t) - K_{\gamma}\sqrt{{Var}(\hat{M}(t))}$

$M_{U}(t) = \hat{M}(t) + K_{\gamma}\sqrt{{Var}(\hat{M}(t))}$

where $K_{\gamma}$ represents the $100(1+\gamma)/2$ percentile of the standard normal distribution.

Figure 30.29 displays the tabular output produced by the RELIABILITY procedure for the artificial data. The first table in Figure 30.29 displays the input data set, the number of observations used in the analysis, the number of systems (units), and the number of repair events. The second table displays the system age, MCF estimate, standard error, approximate confidence limits, and system ID for each event.

The RELIABILITY Procedure

Repair Data Summary
Input Data Set	WORK.MCFART
Observations Used	17
Number of Units	6
Number of Events	11

Repair Data Analysis
Age	Sample MCF	Standard Error	95% Confidence Limits		Unit ID
Age	Sample MCF	Standard Error	Lower	Upper	Unit ID
2.00	0.167	0.167	-0.160	0.493	sys4
5.00	0.667	0.494	-0.302	1.636	sys6
8.00	1.000	0.516	-0.012	2.012	sys2
8.00	1.167	0.543	0.103	2.230	sys4
12.00	1.333	0.667	0.027	2.640	sys6
12.00	.	.	.	.	sys6
14.00	1.533	0.764	0.035	3.032	sys2
16.00	1.933	0.951	0.069	3.797	sys4
16.00	.	.	.	.	sys5
18.00	2.683	0.913	0.894	4.473	sys3
19.00	3.183	0.641	1.926	4.440	sys1
20.00	.	.	.	.	sys4
26.00	3.517	0.679	2.185	4.848	sys2
29.00	.	.	.	.	sys3
33.00	.	.	.	.	sys2
39.00	5.517	0.679	4.185	6.848	sys1
42.00	.	.	.	.	sys1

Figure 30.29: PROC RELIABILITY Output for the Artificial Data

Estimates of the difference between two MCFs MDIFF(t) = M₁(t)-M₂(t) and the variance of the estimator are computed as in Doganaksoy and Nelson (1991). Confidence limits for the MCF difference function are computed in the same way as for the MCF, using the variance of the MCF difference function estimator.

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