The Exponential Repair Assumption: Practical Impacts
Charles M. Carter, ARINC Engineering Services
Anthony W. Malerich, Aspen Avionics
Key Words: Availability, Exponential distribution, Lognormal distribution, Mean Time Between Downing Events, Reliability
SUMMARY & CONCLUSIONS
For repairable systems, we assessed the impact of
assuming that the repair follows an exponential distribution
versus a lognormal distribution for systems with various ratios
of MTBF to MRT and various levels of redundancy. Real
repair times, as a rule, generally curve fit to the lognormal
distribution. We analyzed the system reliability parameters of
R(t), Ao, and MTBDE. We found that reliability was sensitive
to low values of the ratio MTBF to MRT, and to moderate and
low redundancies. In all cases that displayed a statistically
significant difference, the exponential repair assumption
inflated system reliability.
1 INTRODUCTION
We investigate the common simplifying assumption that
components repair according to an exponential density
function. In the study of repairable systems, one all too
frequently hears the phrase, “now we assume that repair of the
components is exponentially distributed”.
A statement
acknowledging that repair data is actually lognormally
distributed typically follows. Thus, the reliability community
recognizes that the exponential repair assumption is unrealistic
but still employs it, mainly because many problems in
repairable systems are not mathematically tractable without it.
There are some obvious questions that then follow this
supposition: “How close to reality is the assumption of
exponential repair?”, and “Does it really matter, and if so,
when?” So we seek to compare the impacts to RAM
parameters using the exponential repair distribution for
analysis when the lognormal distribution is known to be
appropriate.
The following anecdote of an automobile repair garage
illustrates some of the concerns inherent with assuming the
exponential distribution for repair, and the story is based on
the exponential distribution’s well-known characteristic of the
memory-less property. We have an auto repair shop called
Markov’s Garage. I take my car to Markov’s in the morning
on the way to work, and ask how long until the car will be
ready. The mechanic replies that the repair is exponentially
distributed with an expected value of four hours. So at the
lunch hour, I return to the shop, and ask if my car is ready; the
answer is no, not yet. So I ask how much longer it will be.
The mechanic answers, “Well, we’ve been working on it for
four hours, and since the repair is exponential, I estimate it
will be ready in, oh, about four more hours.” I return to work,
surprised. On my way home I stop at Markov’s anticipating
that I will pick up my car, but it is not ready. When I ask the
mechanic how much longer it should take, he tells me that
they’ve been working on it for eight hours and he expects it to
be ready at noon tomorrow. Now I am shocked. So I research
the exponential distribution. I realize that if the repair is truly
exponential, the memory-less property applies and the
expected time to completion is independent of work already
performed. We can see how this tale unravels, and eventually
I get my car. Most people would not accept such answers
from their mechanic, and yet exponential repair is frequently
assumed in the reliability community. Lets now turn our eye
to contrast exponential versus lognormal repairs, and
investigate their impact on Reliability (R(t)), Availability
(Ao), and Mean Time Between Downing Events (MTBDE).
2 METHODOLOGY
We would like to see the effects of the repair distribution
in reliability modeling. To this end, we create one Reliability
Block Diagram (RBD) topology of a repairable system that
contains four blocks, and another topology with two blocks.
Figure 1 - Screen shot of a sample RBD
The failure of each block is based on the exponential
distribution. The RAM outputs that we wish to examine are
the Reliability at time t, the steady-state Availability, and the
Mean Time Between Downing Events. We varied a number
of factors. First and foremost, of course, is the repair
distribution. The settings are exponential; lognormal with
standard deviation of 20% of the mean; and lognormal with
standard deviation of 50% of the mean. The next factor is the
ratio of the block’s Mean Time Between Failure (MTBF) to its
Mean Repair Time (MRT), and to cover the gamut in this area
the settings are 1, 10, 100, and 1000; so these components
range from shoddy to highly reliable. Another factor is the
degree of redundancy. The settings are high, moderate, low,
and none. Since we have four blocks in the RBD, high
corresponds to a 1-out-of-4 system, moderate to a 2-out-of-4
system and a 1-out-of-2 system, low to a 3-out-of-4 system,
and none to a system requiring all 4 blocks to be functioning
at once. The case of moderate redundancy is further broken
out because 1-out-of-2 systems are so common.
For
reliability, the time at which one reports the result may also be
considered as a factor.
We used simulation to solve each system. The software
platform chosen was Raptor. A discussion as to the simulation
parameters used to generate the solutions for Ao, MTBDE,
and R(t) is appropriate. For Ao, the focus is on steady-state
results. Hence, to properly implement aspects of simulation
theory, a few simulation trials of long duration are needed.
We conducted 10 trials of simulation length 50,000 hours.
One can also solve for MTBDE by this method, so the
MTBDE results were taken from these simulations as well.
Now to solve for reliability, one must simulate many trials of a
shorter duration. We used the results for MTBDE to find an
appropriate simulation length corresponding to the particular
system inputs, and conducted 10,000 simulation trials. For
more information relating to the selections of the simulation
parameters, we recommend Reference 1. Here we display a
matrix showing the various system parameters and simulation
settings.
MTBF:MRT
10:10
100:10
100:1
1000:1
Redundancy
High (1-out-of-4)
Moderate (2-of-4)
Moderate (1-of-2)
Low (3-out-of-4)
None (4-out-of-4)
Repair Dist
Exponential
LogN σ=0.2µ
LogN σ=0.5µ
Table 1 - Matrix of all the parameters varied
Simulations were conducted for all combinations of these
factors; so in all we had 60 distinct RBDs, and attempted 120
separate experiments.
Certain combinations of high
redundancy and high MTBF:MRT ratio required weeks of
computing time to yield statistically significant results, while a
few exceeded even that, and we did not complete them. As
will be seen in the next section, for those that could not be
completed, the factor of repair distribution is overwhelmed by
the dominance of redundancy and high component reliability,
and hence these final results are not necessary for our analysis.
3 RESULTS
The first simulations conducted were set up to determine
steady state availability and MTBDE. Each block diagram
was simulated to 50,000 hours, 10 times, and the mean and
standard deviation were calculated for each parameter. To
determine the impact of the exponential assumption, for each
combination of MTBF:MRT and redundancy level, the values
obtained when an exponential distribution was entered for
repair were compared to the values obtained when a lognormal distribution was utilized.
Statistical hypothesis testing was then implemented to
determine whether there was a difference in the values
obtained in the experiments. A null hypothesis, Ho, was
established so that the values for the mean were obtained from
the same population. First the t-statistic for the difference in
means from two populations was calculated using Equation 1.
t=
(y
1
s12
n
−y
+
1
2
)
s 22
n
(1)
2
Then, a t-distribution with (n1 + n2 – 2) degrees of freedom
was used to determine the probability of a Type 1 error. A
Type 1 error occurs when the null hypothesis is rejected while
it is actually true. Thus the probability in the right column of
the table below is the probably that you would be in error if
you rejected the assumption that the values were the same. A
low number (less than 0.01 if want to be 99% sure) means you
are fairly safe in assuming the numbers are from different
populations.
Table 2 shows the results for availability. The actual
values for Ao appear similar, and you cannot statistically
prove any differences.
MTBF:MRT, k of n
10:10, 4 of 4
10:10, 3 of 4
10:10, 2 of 4
10:10, 1 of 4
100:10, 4 of 4
100:10, 3 of 4
100:10, 2 of 4
100:10, 1 of 4
100:1, 4 of 4
100:1, 3 of 4
100:1, 2 of 4
100:1, 1 of 4
1000:1, 4 of 4
1000:1, 3 of 4
1000:1, 2 of 4
1000:1, 1 of 4
Availability
Expo
LN2
LN5
0.0624 0.0623 0.0624
0.3112 0.3150 0.3136
0.6863 0.6904 0.6893
0.9370 0.9377 0.9376
0.6831 0.6840 0.6822
0.9570 0.9569 0.9572
0.9975 0.9974 0.9973
0.9999 1.0000 1.0000
0.9606 0.9613 0.9613
0.9995 0.9994 0.9994
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
0.9961 0.9962 0.9961
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
1.0000 1.0000 1.0000
Ho: Same Mean
P(Type 1 Error) | Reject Ho
Expo-LN2
Expo-Ln5
0.9678
0.9870
0.1091
0.3152
0.0799
0.0981
0.5243
0.6048
0.8655
0.8525
0.9646
0.9282
0.5733
0.4276
0.7669
0.7484
0.2019
0.2902
0.0853
0.0696
0.5040
0.0216
N/A
N/A
0.7881
0.8860
0.2038
0.2392
N/A
N/A
N/A
N/A
Table 2
The column labeled Expo-LN2 shows the probability of a
Type 1 error when comparing the exponential results with the
100:10 MTBF:MRT, 4 of 4
1
0.8
R(t)
lognormal (20% stdev) results. Expo-LN5 refers to the
comparison of the exponential distribution to the lognormal
(50% stdev) distribution. The LN2-LN5 comparison was also
conducted but in every case, the results were extremely similar
so that column was not shown in order to improve table
readability.
Table 3 shows the results for MTBDE. The actual values
for MTBDE appear similar and you cannot statistically prove
they are different.
Expo
0.6
LN2
0.4
LN5
0.2
0
LN5
2.486
4.179
9.231
37.911
24.967
118.133
1303.087
>47498
25.231
829.431
>49999
>50000
261.363
>47499
>50000
>50000
Table 3
The next set of simulations was conducted to determine
reliability versus time for each case. Each block diagram was
simulated to approximately two times the steady state mean,
and 10,000 trials were conducted. R(t) was calculated at
various interval points.
The R(t) values at each time t were determined by
dividing the number of trials in which the system was still up
at time t by the number of trials. Thus a value for reliability is
a proportion of trials completed successfully. To compare
results from the exponentially repairing simulation with the
lognormal repair simulations, a hypothesis test was
implemented to test the difference in proportion from random
samples obtained from two populations. The test statistic z
was calculated using Equation 2.
z=
0
10
20
30
40
50
t
Figure 2
Figure 2 shows the results of a simulation in which four blocks
are in parallel, with the k of n set to 4-out-of-4. This is
essentially a series system configured in a parallel
arrangement. In this case, the MTBF was set to 100 hours and
the MRT was 10 hours. The three methods for defining repair
were each analyzed. The lines for R(t) fell right on top of
each other. This should be obvious since the system will fail
as soon as the first block fails and the selected repair
distribution will have no effect whatsoever.
It starts to get more interesting when the redundancy is
more complicated. Figure 3 shows the three plots with the
same MTBFs and MRTs, arranged in a parallel configuration
with 3-out-of-4 blocks required for the system to be up.
100:10 MTBF:MRT, 3 of 4
1
0.8
R(t)
MTBF:MRT, k of n
Expo
10:10, 4 of 4
2.509
10:10, 3 of 4
4.166
10:10, 2 of 4
9.180
10:10, 1 of 4
37.375
100:10, 4 of 4
25.276
100:10, 3 of 4
119.508
100:10, 2 of 4
1423.005
100:10, 1 of 4
>42498
100:1, 4 of 4
25.226
100:1, 3 of 4
910.161
100:1, 2 of 4
>44999
100:1, 1 of 4
>50000
1000:1, 4 of 4
258.750
1000:1, 3 of 4
>46666
1000:1, 2 of 4
>50000
1000:1, 1 of 4
>50000
MTBDE
LN2
2.465
4.176
9.328
37.871
25.199
117.855
1272.649
>38332
25.288
830.081
>44999
>50000
261.642
>47499
>50000
>50000
Ho: Same Mean
P(Type 1 Error) | Rej Ho
Expo-LN2 Expo-Ln5
0.154
0.418
0.729
0.660
0.115
0.489
0.459
0.395
0.844
0.376
0.655
0.717
0.163
0.305
N/A
N/A
0.835
0.987
0.079
0.036
N/A
N/A
N/A
N/A
0.712
0.730
N/A
N/A
N/A
N/A
N/A
N/A
Expo
0.6
LN2
0.4
LN5
0.2
0
0
50
100
150
200
t
(pˆ − pˆ )
1
2
⎛1
1⎞
pˆ (1 − pˆ )⎜ + ⎟
⎜n n ⎟
2⎠
⎝ 1
(2)
where
pˆ =
n pˆ + n pˆ
1 1
2 2
n1 + n2
(3)
Then the normal distribution was used to determine the
probability of a Type 1 error.
Figure 3
As Figure 3 and the associated Table 4 suggest, there are
differences now. At time 30, for example, R(t) was calculated
to be 0.8421 when the exponential repair distribution was used
and 0.8155 with the lognormal (LN2) repair distribution. The
hypothesis test backs this up by indicating an extremely low
probability of error if you reject the assumption that they are
the same.
100:10 MTBF:MRT, 3 of 4
Expo
0.9628
0.9026
0.8421
0.7829
0.7306
0.6791
0.6311
0.5870
0.5464
0.5078
0.4706
0.4340
0.4020
0.3734
0.3455
0.3229
0.3013
0.2779
0.2568
0.2391
R(t)
LN2
0.9519
0.8807
0.8155
0.7548
0.6975
0.6449
0.5965
0.5500
0.5054
0.4680
0.4308
0.3987
0.3686
0.3412
0.3153
0.2927
0.2699
0.2490
0.2282
0.2108
LN5
0.9544
0.8829
0.8120
0.7518
0.6970
0.6449
0.5969
0.5538
0.5154
0.4775
0.4401
0.4089
0.3757
0.3464
0.3221
0.2980
0.2768
0.2526
0.2323
0.2161
1
0.8
R(t)
time
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
10:10 MTBF:MRT, 2 of 4
Ho: Same Mean
P(Type 1 Error) | Rej
Expo-LN2
Expo-Ln5
1.366E-04 3.642E-03
6.298E-07 9.441E-06
5.944E-07 2.425E-08
2.440E-06 2.312E-07
2.227E-07 1.723E-07
3.187E-07 3.553E-07
5.041E-07 7.284E-07
1.278E-07 2.208E-06
6.416E-09 1.142E-05
1.805E-08 1.811E-05
1.553E-08 1.441E-05
4.122E-07 3.140E-04
1.218E-06 1.296E-04
2.023E-06 6.521E-05
5.629E-06 4.230E-04
3.725E-06 1.304E-04
8.870E-07 1.195E-04
3.503E-06 4.518E-05
2.379E-06 4.876E-05
1.649E-06 9.107E-05
Expo
0.6
LN2
0.4
LN5
0.2
0
0
15
20
Figure 5
Figure 5 shows the R(t) curves when the MTBF was set to 10
hours and the MRT was set to 10 hours. The values,
especially at the right edge of the graph, are way off. This is
confirmed in Table 5.
10:10 MTBF:MRT, 2 of 4
The plot for the 2-out-of-4 case with an MTBF of 100
hours and MRT of 10 hours is very similar to the 3-out-of-4
case. However, when the redundancy is very high, as in the 1of-4 case, the effect of the exponential assumption is reduced
again. Figure 4 shows that similar results are obtained
regardless of the repair distribution selected with high
redundancy.
100:10 MTBF:MRT, 1 of 4
1
0.8
Expo
0.4
LN2
LN5
R(t)
0.6
0.2
0
50000
10
t
Table 4
0
5
100000
t
Figure 4
A more obvious variable affecting the exponential assumption
is the ratio of MTBF to MRT. When a ratio of 100:1 or
1000:1 was used, the differences in values obtained for R(t)
were not statistically significant regardless of the redundancy
level chosen. The opposite was also clearly true. As the
MTBF:MRT ratio grew smaller, the differences became more
pronounced.
time
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Expo
0.9968
0.9810
0.9536
0.9179
0.8696
0.8185
0.7640
0.7132
0.6632
0.6113
0.5616
0.5139
0.4722
0.4363
0.4029
0.3736
0.3437
0.3171
0.2940
0.2734
R(t)
LN2
0.9971
0.9800
0.9438
0.8966
0.8367
0.7680
0.6950
0.6252
0.5537
0.4896
0.4385
0.3963
0.3596
0.3272
0.3003
0.2756
0.2517
0.2321
0.2160
0.1979
LN5
0.9978
0.9820
0.9452
0.8947
0.8310
0.7645
0.6986
0.6338
0.5747
0.5188
0.4720
0.4252
0.3861
0.3513
0.3181
0.2922
0.2644
0.2414
0.2204
0.2013
Ho: Same Mean
P(Type 1 Error) | Rej Ho
Expo-LN2
Expo-Ln5
7.005E-01 1.782E-01
6.091E-01 6.045E-01
1.683E-03 7.723E-03
2.083E-07 2.952E-08
4.973E-11 4.152E-14
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
0.000E+00 0.000E+00
Table 5
After observing the phenomenon of decreasing accuracy
with respect to moderate redundancy and low MTBF:MRT
ratios, we conducted a few additional experiments to assess
the impact of the exponential assumption on the most common
moderately redundant system used in the field, a 1-out-of-2
system. The same type of experiment was performed using
two blocks arranged in a parallel configuration with varying
MTBF:MRT ratios.
Similar results were obtained with a 1-of-2 configuration
as with the 2-of-4 configuration. At the 100:1 ratio, the effect
of an exponential assumption was minimal, as seen in Figure 6
and Table 6.
100:10 MTBF:MRT, 1 of 2
100:1 MTBF:MRT, 1 of 2
1
0.8
Expo
0.4
LN2
LN5
R(t)
0.6
0.2
0
0
5000
10000
t
Figure 6
100:1 MTBF:MRT, 1 of 2
time
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
8500
9000
9500
10000
Expo
0.9151
0.8278
0.7499
0.6768
0.6133
0.5596
0.5095
0.4608
0.4150
0.3760
0.3407
0.3075
0.2806
0.2549
0.2347
0.2123
0.1914
0.1732
0.1562
0.1432
R(t)
LN2
0.9092
0.8255
0.7489
0.6814
0.6161
0.5596
0.5049
0.4540
0.4086
0.3689
0.3337
0.3001
0.2728
0.2504
0.2271
0.2046
0.1856
0.1694
0.1557
0.1421
LN5
0.9046
0.8149
0.7422
0.6716
0.6106
0.5496
0.4953
0.4506
0.4081
0.3691
0.3305
0.3002
0.2722
0.2456
0.2219
0.2011
0.1837
0.1637
0.1494
0.1363
Ho: Same Mean
P(Type 1 Error) | Rej Ho
Expo-LN2
Expo-Ln5
0.1405
0.0096
0.6675
0.0169
0.8704
0.2104
0.4859
0.4318
0.6841
0.6950
1.0000
0.1547
0.5153
0.0446
0.3345
0.1475
0.3578
0.3213
0.2991
0.3126
0.2951
0.1265
0.2552
0.2612
0.2176
0.1835
0.4640
0.1290
0.2022
0.0310
0.1801
0.0502
0.2944
0.1625
0.4757
0.0729
0.9224
0.1825
0.8240
0.1604
Table 6
At a 100:10 ratio, the assumption is questionable once again.
100:10 MTBF:MRT, 1 of 2
1
0.8
Expo
0.4
LN2
LN5
R(t)
0.6
0.2
0
0
500
t
Figure 7
1000
time
50
100
150
200
250
300
350
400
450
500
550
600
650
700
750
800
850
900
950
1000
Expo
0.9340
0.8662
0.8007
0.7398
0.6843
0.6371
0.5870
0.5430
0.5044
0.4656
0.4309
0.3982
0.3715
0.3454
0.3174
0.2929
0.2714
0.2504
0.2327
0.2149
R(t)
LN2
0.9325
0.8647
0.7974
0.7327
0.6782
0.6251
0.5768
0.5321
0.4883
0.4513
0.4141
0.3844
0.3542
0.3296
0.3044
0.2829
0.2602
0.2426
0.2250
0.2074
LN5
0.9343
0.8614
0.7963
0.7370
0.6786
0.6270
0.5779
0.5317
0.4913
0.4522
0.4155
0.3851
0.3583
0.3323
0.3081
0.2841
0.2595
0.2390
0.2208
0.2017
Ho: Same Mean
P(Type 1 Error) | Rej Ho
Expo-LN2
Expo-Ln5
0.6709
0.9321
0.7559
0.3220
0.5603
0.4383
0.2546
0.6531
0.3546
0.3874
0.0786
0.1391
0.1437
0.1922
0.1221
0.1091
0.0228
0.0639
0.0424
0.0571
0.0162
0.0274
0.0456
0.0575
0.0110
0.0520
0.0181
0.0498
0.0470
0.1550
0.1184
0.1688
0.0730
0.0562
0.2006
0.0605
0.1949
0.0443
0.1938
0.0215
Table 7
4 DISCUSSION OF RESULTS
So, what conclusions can we draw? Well first off, it’s
safe to say that for steady state parameters like Availability
and MTBDE, the exponential assumption clearly had no
impact on the results obtained. This is no surprise however if
you think it through logically. Over the long run, each block
will have its own availability which is the ratio of its uptime to
total time. Regardless of the repair distribution selected, over
the long run, the total down time of a block will equal the
number of repair events times the MRT. The individual
block’s availability is independent of the repair distribution
selected. If the blocks operate independently, the system
availability can be calculated using standard equations and it is
independent of the repair distribution as well.
It is the short term results that are affected by the
exponential assumption. However, even in the short term,
there are many cases where it just doesn’t matter. If you have
a series system, and you are interested in reliability, it doesn’t
matter. If you have a highly redundant system, it is not
important. If you have a high MTBF:MRT ratio, it doesn’t
matter. But if you have moderate redundancy, and the
MTBF:MRT ratio is relatively low, then the assumption of
exponential repair can yield inaccurate results.
Over the years as reliability consultants we have
evaluated many systems, particularly in the aircraft and
petrochemical industries.
By far, the most common
redundancy configuration is the 1-out-of-2 system. Is it
appropriate to use an exponential repair distribution for
analysis of these systems? As usual, the answer is it depends.
It is common to see parallel electronics items, such as radios,
each with an MTBF of several thousand hours, and a short
repair time of a couple hours. The repair time is mainly used
to swap out one component with another.
Assuming
exponential repair will have little or no affect at all.
On the other hand, it is not uncommon in industry to have
large mechanical devices that fail once a year and can be down
for a week or more. A one week MRT would yield a
MTBF:MRT ratio of 52. At this point, the results are going to
be off, but it really depends on how critical the accuracy of
results must be. Even in the 100:1 ratio case, the exponential
results were inflated, but were generally accurate to the third
digit after the decimal place. The accuracy decreases in the
range when R(t) is below 0.5 but most systems are designed to
be much more reliable than that during the period of interest.
The 1-of-2 redundancy 100:10 MTBF:MRT values were off in
the second digit after the decimal. The other moderate
redundancy scenario we analyzed earlier, the 2-of-4 case, was
even less accurate.
Although we have never personally analyzed a system
containing components with an MTBF:MRT ratio of 1:1,
some of you may have. One thing we can say for sure is that
in this case the exponential assumption is very poor for short
term results.
Another point worth mentioning – in every case in which
the exponential assumption results differed from the lognormal result the exponential assumption overestimated the
reliability.
5 DIRECTIONS FOR FURTHER STUDY
For this paper, steady state availability was calculated for
each case. Availability over a fixed period of time that is short
(roughly anything less than 20 times the MTBF) was not
discussed. This parameter will likely be affected by the
exponential assumption in a similar manner as the reliability
parameter.
We have heard talk of, but found no references to, a
manner to compare k-out-of-n redundancies for various levels
of k and n. For example, would a 3-out-of-7 system match up
with our qualifier of moderate redundancy, or would it be
what we considered as highly redundant? Such results would
be of interest in and of themselves, and, since reliability
results are sensitive to low redundancies, a similar study to
ours for 1-out-of-3 and related systems would be particularly
relevant.
With regard to reliability, this paper intended to cover a
wide range of systems, so the RBDs chosen varied greatly.
Many systems are designed to be highly reliable and analysts
are concerned with detecting differences in values such as
0.9995 and 0.9999. Specific RBDs would need to be tested to
assess the affect of the exponential assumption for these
systems. One difficulty that would have to be overcome is
that long simulations are usually required to assess highly
reliable systems.
REFERENCES
1.
2.
3.
K.E. Murphy, C.M. Carter, L.H. Wolfe, “How Long
Should I Simulate, and for How Many Trials? A Practical
Guide to Reliability Simulations”, Proceedings Annual
Reliability & Maintainability Symposium, (Jan.) 2001, pp.
207-212.
M. Evans, N. Hastings, B. Peacock., Statistical
Distributions, Wiley-Interscience, 3rd Edition, 2000.
K. E. Murphy, C. M. Carter, S. O. Brown, “The
Exponential Distribution: the Good, the Bad and the
Ugly.
A Practical Guide to its Implementation”,
Proceedings Annual Reliability & Maintainability
Symposium, (Jan.) 2002, pp. 550-555.
BIOGRAPHIES
Charles M. Carter
ARINC Engineering Services, LLC
6565 Americas Parkway NE
Albuquerque, NM, 87110 USA
e-mail: [email protected]
Charles M. Carter has a B.S. in Aeronautical and Astronautical
Engineering from the University of Illinois and a M.S. in Systems
Engineering from the Air Force Institute of Technology. As an
officer in the USAF, Chuck was assigned to Vandenberg AFB, CA,
where he coordinated ground processing of Titan II and Titan IV
payloads and served on the launch crew. He was also assigned to
Kirtland AFB, NM, where he worked as a reliability engineer
evaluating space and electronics systems. He later worked for SAIC
at Johnson Space Center as a payload safety engineer, performing
safety analysis on Space Shuttle and International Space Station
payloads. Chuck is currently a principal engineer with ARINC and is
responsible for development of the Raptor simulation engine.
Anthony W. Malerich
Aspen Avionics
2305 Renard Place NE, Suite 110
Albuquerque, NM 87106 USA
e-mail: [email protected]
Anthony W. Malerich earned a Bachelor of Science in Printing
Management from Western Michigan University in 1991. He was
then recruited by the Peace Corps to teach Mathematics. After two
years in West Africa, Tony returned to the United States, where he
went to work for a printing ink manufacturer. However, the call of
Analysis, Abstract Algebra, and their ilk was too strong, so in 1999
Tony graduated from the University of New Mexico with an M.A. in
Pure Mathematics. In 2003, he joined ARINC’s Raptor Reliability
team, acting as a force to keep the team together and moving forward
at an impressive clip. Tony, feeling the need for increased risk, less
stability, and the chance to work with quaternionic coordinate
systems, took a position with the Aspen Avionics startup company in
2006.