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 and the
variance of the estimator Var 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 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 twosided confidence
limits for M(t) are computed as
where represents the 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 
Lower 
Upper 
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) = M_{1}(t)M_{2}(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.
Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.