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R-WV-T-6-22-73 AD
TECHNICAL LIBRARY
RELIABILITY DATA ANALYSIS MODEL
R.RACICOT
BENET WEAPONS LABORATORY WATERVLIET ARSENAL
WATERVLIET, N.Y. 12189
JULY 1973
TECHNICAL REPORT
AMCMS No. 611102.11.85300.01
DA Project No. 1T061102B14A
Pron No. B9-2-67162-01-M2-M7
APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED
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Destroy this report when it is no longer needed. Do not return it
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DISCLAIMER
The findings in this report are not to be construed as an official
Department of the Army position unless so designated by other authorized
documents.
Unclassified Security Classification
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Watervliet Arsenal Watervliet, N.Y. 12189
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RELIABILITY DATA ANALYSIS MODEL
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U. S. Army Weapons Command
13. ABSTRACT
A computer model has been prepared to assist in the analysis of reliability test data. Essentially, the model computes point estimates and confidence limits for mission reliability of components, subsystems, and systems from component failure data. The main features of the model are: 1. Performs goodness-of-fit tests to determine the best fit probability distribution of component failure times, 2. Computes maximum likelihood estimates of distribution parameters, 3. Computes point estimates of reliability for the renewal nonconstant failure rate case, and 4. Computes lower confidence limits for component, subsystem, and system reliability for the constant failure rate case.
DD FORM I MOV •• 1473 RF.-LACE4 DO FORM 1471. I JAN 44. WHICH IS
oasOLKTS FOR ARMY USB.
^cSrU^ftjA,
Unclassified Security Classification
KEY WO*DS
Reliability
Statistics
Probability
Distribution Selection
Computer Model
Failure Rate
Bayesian Statistics
Hnrlassifif-d Security Classification
R.WV-T.6-22-73
RELIABILITY DATA ANALYSIS MODEL
R.RACICOT
BENET WEAPONS LABORATORY WATERVLIET ARSENAL
WATERVLIET, N.Y 12189
JULY 1973
TECHNICAL REPORT
AMCMS No. 611102.11.85300.01
DA Project No. 1T061102B14A
Pron No. B9-2-67162-01-M2-M7
APPROVED FOR PUBLIC RELEASE DISTRIBUTION UNLIMITED
RELIABILITY 'MTA ANALYSIS ''OPEL
ABSTRACT Cross-Reference
Pat a
A computer model has been Prepared to assist in the analysis o* reliabilitv test data. Essentially, the model confutes noint estimates and confidence limits for mission reliability of comnonents, sub- systems, and systems from component failure data. The main features of the model arc: 1. Performs ^oodness-of-fit test-s to deter- mine the best fit probability distribution of component failure times, 2. Commutes maximum likelihood estimates of distri- bution parameters, 3. Comnutes noint esti- mates cf reliabilitv for the renewal non- constant failure rate case, and 4. Confutes lower confidence limits for comnonent, subsystem, and system reliability for the constant failure rate case.
peliability
Statistics
">robabilitv
Distribution Selection
Computer Model
Failure Rate
Bavesian Statistics
FOREWORD
This project was supported in part by the Project Manager's
Office for Close Support Weapon Systems and by USASASA for the Future
Rifle System.
The author would like to thank R. Hagjjerty ^d R^ Scanlon of the
Benet Weapons Laboratory for preparing a number of the comnuter sub-
routines used in the model.
TABLE OF CONTENTS
Introduction 5
Purpose of Reliability Corn-nut er Model fi
Tyne of Data Considered for Analysis r,
Type of Information Generated 7
Usual Assumptions for Data Analysis and Their Limitations ^
Main Features and Status of the Present Model \/\
Analytical Methods 15
Computation of Component Mission Reliability 1C
(a) Component Reliability Based on Age 17
(b) Mission Reliability of a Component Within a System jo
(c) Numerical Solution for the Renewal Rate 9f>
(d) Numerical Solution for Mission Reliability ?R
(e) Average Mission Reliability ->0
Computation of System Mission Reliability 30.
Probability Distribution Selection 37
(a) Candidate Distributions 32
(b) Least Squares Fit of Data 35
(c) Computation of the Standard Error 3A
(d) Kolmogorov-Smimov Goodness of Fit 30
Maximum Likelihood Estimates of Parameters 4]
Bayesian and Classical Confidence Intervals for Component Reliability 45
Bayesian Confidence Intervals for System Reliability 5]
(a) Constant Failure Rate Components in Series r^j
(b) Non-Constant Failure Rate Components in Series 53
Computer Program 53
References 55
Appendix I-Input Format for Programs DISTSEL and RELIAB 57
(a) Overall System and Subsystem Data Cards 57
(b) Element or Component Data Cards 57
(c) Sample Problem 53
Appendix II-Output Data for Sample Problem 60
DD Form 1473
figures
1. Component Failure Rate vs Component Age 18
2. Component Failure Rate vs System Age for the Case of Ideal Repair 21
3. Component Renewal Rate vs System Age 25
4. Component Reliability vs System Age 25
Tables
I. Average System Reliability for Equal Components Having a Weibull Distribution of failure Times \n
RELIABILITY DATA ANALYSIS MODEL
INTRODUCTION
Current Army regulations 111 reouire the snecification of ouanti-
fied reliability goals in the develonment of new weanon systens.
rhese reliability goals must also be verified bv tests nrior to final
acceptance and fielding of the systen. Reliability, bv definition,
is a nrobabilistic quantitv. In addition, the amount of testing that
can be conducted is limited by cost and time considerations and conse-
quently reliability assessment must be conducted within a statistical
framework. To assist in this assessment effort a statistical data
analysis computer model has been nrenared and is described in this
report. Essentially, the model computes noint estimates and confi-
dence limits for reliability of components, subsystems and system
from component test data.
The snecific reliability indnx considered is the mission relia-
bility defined as the probability that a given system will success-
fully perform its intended function without failure for a specified
period of time under a given set of conditions. Time can be given
in terms of clock time, rounds, cycles, miles, etc. The svstem mav be
required to nerform a number of missions over its exnected life.
Army Regulation AR70S-50, "Army Materiel "el i abil itv and ''attain- ability," Headquaters Dent, of the Army, Washington, D.C., 8 January 1968.
PURPOSE OF THE RELIABILITY COMPUTER MODEL
The reliability model is intended to serve a number of useful
purposes. First, reliability analyses for the realistic situations
encountered in weapon system testing and for all but the simplest
assumptions made for computation entail rather complicated mathemati-
cal formulation and tedious computation. The computer model conse-
quently provides a systematic means of incorporating more complex
analyses with less restrictive assumptions into a routine data analy-
sis procedure. Second, the model provides greater speed in obtaining
results of data analyses. Third, the mathematical procedures and for-
mulations incorporated into the model provide a means of standardizing
reliability assessment among design agencies, test agencies and the
user. Finally, the model provides a basis for determining the type
and amount of test data required to perform a particular kind of
analysis. This provides insight into planning of tests and data
collection procedures required.
TYPE OF DATA CONSIDERED FOR ANALYSIS
The type of data considered for analysis is failure and suspen-
sion data on a component or a number of components making up a sub-
system or system. Definition of failure depends on the particular
requirements and the mission profile of a givep system and can
include part breakage, out-of-tolerance performance, incipient
failure, safety hazard, etc. Suspended data is obtained when a compo-
nent is removed from the test without failure, the test is terminated
without failure or when the failure of a component is attributable to
the failure of another component or cause.
Since failure rates and resulting mission reliabilities are
generally transient, all data is assumed collected as a function of
system age. Snecifically, the age given in terms such as rounds, time
or miles on individual romnororts within a systor at failure or sus-
pension is required. In pen^ral if is not enough to record onlv the
total test time and number of failures except in the case of constant
failure rate.
TYPE OF INFORMATION nr. NEGATED
The tvpe of information generated bv the model includes results
of analyses for each individual component within a subsvstem, for
each subsystem making UP a given svstem and for the entire system.
Component information includes results of distribution selection
procedures, Kolmogorov-Smimov goodness-of-fit tests, maximum likeli-
hood estimates of distribution parameters, noint estimates of the
mission reliability as a function of svstem age ror the non-constant
failure rate case and confidence limits on average mission reliahilitv
over the system life for the constant failure rate case. Subsystem
and system information generated includes no^nt estimates and confi-
dence intervals on mission reliability for the constant failure rate
case.
A more complete description of required input data for the model
and the resulting output information will be presented in subseouent
sections dealing with the computer program.
USUAL ASSUMPTIONS FOR DATA ANALYSIS AND THEIR LIMITATIONS
It is worthwhile at this point to review the usual assumptions
made in performing a reliability assessment for a given system and
the desirability of relaxing these assumptions for some of the realis-
tic cases encountered in weapon system testing. Undoubtedly, the most
common assumption made in reliability data analysis is the assumption
of constant failure rate for components and systems. The most signi-
ficant implication of this assumption is that the probability of
failure of a component or system is independent of its age. Components
are assumed to fail at completely random points in time. This, of
course, does not permit the full treatment of the cases of early
failures and wear-out failures common to mechanical components.
There are a number of reasons why the constant failure rate
assumption is so often made. First, the constant failure rate is
often assumed primarily to simplify analysis and computation. Second,
straightforward techniques for determining confidence intervals on
component reliability from test data are readilv available for con-
stant failure rate. Third, testing procedures are greatly simplified
since only the total number of failures and total test time are re-
quired for data analysis. Also, the amount of data required for each
component is not as critical as in those cases where theoretical
8
distribution selection must be considered. Finally, for system relia-
bility verification tests, individual component data is not required
witb all failures being treated as system failures. This is true
onlv if all components have enua] test times.
Assuming a constant failure rate can lead to three tynes of
errors; the first is in the computation of reliability for fixed '1TRr
(mean time between failures), the second is in the computation of
confidence limits, and the third is in the estimation of 'TB^ when
susnension or censored data has been generated. The magnitude of the
error in computing reliability for fixeH ^(TRT7 cannot be determined ;n
general. For illustrative purposes, however, consider ^ hypothetical
case in which a system is made up of coual components in series.
Assume further that each component has a Weibull distribution and an
*1TBF (mean time between failure) enual to twice the expected system
life. The number of missions over system life is assumed to be ISO.
The magnitude of the error in assuming constant failure rate can now
be determined for this particular case. Table I lists the results of
average system reliability for different numbers of components in the
system and for various values of the Weibull shape narameter R. FOT
a shape parameter 8 equal to 1.0 the Weibull distribution reduces to
the exponential which describes the distribution for constant failure
rate. For values of R greater than 1.0 the failure rate is an increas-
ing function of time characteristic of wear-out phenomena. The greater
the value of R the more peaked the distribution is about the mean.
As can be seen from Table I, large differences can be obtained when
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assuming exponential or constant failure rate for Weihul1 components
that in actuality have 3 values greater than 1.0, ^or example, a svs-
tem of SO equal components, each component having a Weihul1 share para-
meter equal to 3.0 would have an averarc svsten reliability of 0.072.
Assuming a constant failure rate op the other hand would yield a con-
nuted reliability of 0.846 which is significantly different than the
true value for this case.
Reliability for the Weihul 1 distribution withf,>l was computed
using renewal theory where, in this rase, a component is replaced or
renewed upon failure. The mathematical formulation for the renewal
case is described in the Analytical Methods section of this report.
The second source of error in assuming constant failure rate is in
the computation of confidence limits. Consider as an examnle nroblem
a test sample o^ failure times given as 2000, 2200, 1800, 1000 and
2100. For this nroblem the '1TBF is 2000 and the standard deviation is
158. Lower confidence limits on the true '1TBF can readily be computed
assuming different underlying distributions of failure times. ror
examnle, the lower 00° confidenced moans for the exponential (constant
failure rate)and the normal (increasing failure rate) distributions
are 1250 and 1758 respective^'. It is common in reliability verifica-
tion tests to rcnuire that the lower confidenced "ITBF (or reliability)
exceed a given fixed value for accentance. If the requirement in the
above problem were 1500, the assumption of the exponential distribution
would lead to rejection and the assumption of the normal distribution
would lead to acceptance. Although the above is an oversimplification,
11
it does indicate the possible error in assuming the wrong distribution
in determining confidence limits.
The third source of error in assuming constant failure rate lies
in the unrealistic handling of suspension data in estimating population
parameters. Consider as an example problem a test sample of failure
times given as 3800, 3900, 4100 and 4200 and a samnle of suspension
times given as 3500, 4000, 4000 and 4500. In this example, eight dif-
ferent components were tested but onlv four failed. The point esti-
mates of MTBF assuming exponential and Weibull distributions are 8000
and 4130 respectively. As can be seen the '1TBF for constant failure
rate is nearly twice the value for the nonconstant failure rate Weibull
distribution.
In the first two sources of error discussed above the computed
reliability is generally conservative for components and systems that
wear out. Reliability and lower confidence limits are conservative in
that lower than true values are computed. In the third source of error
the resulting reliability is generally nonconservative. The unrealis-
tic handling of suspension data seems to be the most significant
source of error in assuming constant failure rate for testing of
mechanical systems.
A second assumption or requirement which is often made in relia-
bility analysis is that the data sample be complete. Consequently
only the failure times in a given test are considered for analysis.
The main reason for this assumption is that statistical methods that
treat suspended data along with the failure data are in many cases
12
restrictive, complicated or not available. It is clear, however, that
susnended data, particularly data involving suspension times that are
greater than some or all of the failure times should contribute signifi-
cant information toward determininn the unknown population parameters.
This was observed in the above discussion of the constant failure rate
assumption. Suspepded data can result in manv ways. In weanon svstem
testing, for example, it cap arise from the removal of a comnonent from
test without failure, comnletiop of a system test nrior to failure or
through failure resulting directly from the failure o*" another compo-
nent or cause such as accident.
A third assurmtiop which is often made in determining confiHence
limits on system reliability from svstem tests is that the svstem can
be treated as if it were a single comnonent with no differentiation
being made as to which component within the svstem actually fails.
All failures are treated as svstem failures. A necessary underlving
assumption for this approach is that either all components have con-
stant failure rate or all components have failed a number of times so
that steady state conditions prevail. In this instance confidence
intervals on system reliability are readilv derived usin^ the theory
applicable to the constant failure rate case. A limitation o*" this
assumption is that results of individual component or subsvstem tests
cannot be included with the system test results. Mso, if components
are redesigned during the course of a test, which is often the case in
large weapon system testing, total test time on the redesigned compo-
nents are not the same as the total system test time. The usual
13
methods of analyzing system test data consequently do not annly in
this case.
Finally, large sample theory is often assumed in computing confi-
dence intervals since methods to handle small samples may not exist
or are too difficult to use.
MAIN FEATURES AND STATUS OP THE PRESENT MODEL
The reliability data analysis model presented in this renort con-
tains a number of general solutions to overcome many of the limitations
of the usual assumptions discussed in the previous section. The main
features of the model are summarized as follows:
a. Performs goodness-of-fit test to determine the best fit
probability distribution of component failure times. The theoretical
distributions considered are the exponential, normal, lognormal,
Weibull and gamma.
b. Computes maximum likelihood estimates of population para-
meters for general theoretical distribution of failure times.
c. Can handle suspended data which results when a component is
tested without failure.
d. Computes point estimates of reliability for the renewal case;
that is, for the non-constant failure rate case.
e. Computes lower confidence limits on component, subsystem
and system reliability for the constant failure rate case.
14
The present version of the reliability data analysis computer
model does not contain all o^ the features ultimately planned for in
the final version. Mnst significant o*" the limitations is the assumn-
tion of constant failure rate in determining confidence intervals
although point estimtes n* reliability are determined fnr the general
non-constant failure rate case. In addition components are assumed to
he in series for system reliabi]ity computation and no provision is
presently made for nreventive maintenance narts replacement of comno
nents. The analytical methods section o* this report descrihes the
techniques to be employed in removing these limitations with work on
future versions of the model presently being undertaken.
The remainder of this report contains a discussion of the analyti-
cal methods used for commutation followed by a presentation of the
computer program with example input and outout information.
ANALYTICAL METHODS
The purpose of this section is to briefly outline the mathematical
and statistical methods used or planned in the reliahility data
analysis model. A complete discussion of probability and statistical
theory will not be presented in this report and the reader is referred
to the references cited for more complete discussions. Much of the
theory presented is straightforward and readily found in the litera-
ture. However, some of the statistical and computational methods used
are the result of research efforts at the Watervliet Arsenal and will
IS
be the subject of forthcoming reports and publications.
There are a number of good texts on the general subiect of reli-
ability. The texts found particularly useful to this writer are those
by Lloyd and Linow [2], Barlow and Proschan [3], ^nedenko, Belyayev
and Solowev [4] and Pieruschka [5].
Computation of Component Mission Reliability.
Although one of the computational goals in th<* model is system
reliability it is necessary to develon the analytical methods for
computing system reliability by considering the individual components
or elements making up the svstem. This is narticularlv true in the
case of non-constant failure rate components.
At the present time there is considerable confusion among design
engineers on the definition of component reliability when the compo-
nent is part of a system. The confusion lies primarily on the time
reference used in computing reliability. Many texts on reliability
consider time in terms of component age. However, for system reliabi-
lity the system age is the important time reference. Since components
that fail within a system are replaced or renewed at random noints in
2 Lloyd, D.K., and Linow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1962).
3 Barlow, R.E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley ft Sons, New York (1965).
4 Cnedenko, B.V., Belyayev, Yu.K., and Solovyev, A.D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1969).
5Pieruschka, E., "Principles of Reliability," Prentice-Hall, Englewood Cliffs, New Jersey (1963).
16
time, comnonent apes are generally not known a ~riori as a function of
system age. In this instance renewal theorv must be emnloved t-o
determine the transient mission reliability as a function of system
age. This asnect of the nroblem will be considered in the following
sections.
(a) Component Reliability Baser! on Component Ape.
Consider first a single con-<-nen* '/here the time reference
is component age. Let
f(t) = Probability densitv distribution o*" coT->on°nt failure times or tine between Fiilur^q. t
F(t) = / f(t)dt = cumulative distribution r>mofion. 0
• Probabilitv that the component will *ai 1 in the time interval (0,t).
R(t) • Reliability defined as the probability that the comno- nent will not fail in the time interval fO,t) where the comnonent is assumed new at tine o.
= 1 - P(t)
Aft) = Conditional failure rate where X(t)dt is the nrohabilitv of failure in the time interval (t,t+dt) "iven that the comnonent has survived to time t.
The conditional failure rate > ft") defined above is a useful and
descrintive quantity in reliability theory. It can be determined fron
the probability distribution of failure tines as follows:
Aft) = fft)/fl-^ft))
= fft)/Rft) m
Figure 1 shows descrintivelv the failuro rate for three tvmcal cases
in reliability; decreasing, constant and increasing failure rate.
17
Component Failure Rate
1 MTBF"
Increasing railure Rate flVear Out ^ail ires) /
Constnnt \ (Independent of A.rre')
necrensin" fRarlv railures )
0 Component Ar;e
Figure 1. Component failure Rate vs Component Ape
The decreasing failure rate implies that the longer a component survives
the lower its probability of failure. This is typical of early failures
in which components containing manufacturing or material flaws tend to
fail early. The increasing failure rate is tynical of components that
wear out where the longer a component survives the higher the probabi-
lity of failure. Mechanical components in which early failures have
been eliminated generally have increasing failure rates caused by such
phenomena as fatigue, corrosion, erosion and abrasion. The constant
failure rate is typical of components which have a probability of
failure independent of its age. This can result, for example, from
a situation in which excessive loads can cause failure where load
fluctuations are nurely random in time such as wind loads.
18
Examples:
1. Exponential Distribution
For this case
f(t) = Xe"Xt
F(t) = l-e'Xt
Rft) = e"Xt
A(t) = > = constant f2)
The exponential distribution describes the constant failure rate cas**.
2. Weibull Distribution
p R-l B f(t) = — - e
F(t) - 1 - e-(t/n)B
R(t) = e"(t/n^
B t6-1 A(t) = w ^ (3)
in which B and n, are distribution or population parameters. Note that
for the Weibull distribution B values less than, equal to and greater
than unity yield failure rates which are decreasing, constant and
increasing respectively. This characteristic makes the Weibull distri.
bution a useful one in reliability analysis.
(b) Mission Reliability of a Component Within a System [3,
Consider next the situation of a component within a system.
The auantity of interest here is the reliability of the component for
10
3 Barlow, R. E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley F, Sons, New York (1965).
4 flnedenko, B. V,, Belyayev, Yu.K., and Solovyev, A. D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1069).
Pieruschka, E., "Princinles of Reliability," Prentice-Hall, Ent»lewood Cliffs, New Jersey (1963).
20
a mission time interval of (t, t • T) wh^re t is the system aj>e and T
is the mission length. Prior to time t the component could have
failed and been replaced one or more times. Figure 2 denicts the
conditional failure rate Aft) of a component within a svstem showing
failure noints for an increasing failure rate.
Component failure Rate
nailures with Subsequent Penewals
System Are
Figure 2. Component Failure Rate vs Svstem Age For The Case Of Ideal Repair
In general the failure times of components within a system are not
known in advance as depicted in Figure 2 and consenuently must be
treated probabilistically. This is accomplished by defining another
quantity h(t) called the renewal rate or unconditional failure rate
which is an ensemble average of the failure rate over the population
of all systems. The renewal rate is defined such that
h(t)dt = unconditional probability of component failure in system time interval (t,t+dt).
21
The renewal rate for a given component is a function of the
underlying failure distribution given by the following equation:
h(t) = f(t) + fZ h(x)f(t-x)dx. 0
Derivation of this equation is presented in references ]?.], [3] and
elsewhere. Logically, equation (4) can be derived from the theorems
of total and conditional probabilities [6] alonp with the definitions
of the terms in equation (4). Multiplying both sides of equation (4)
by dt and considering the integral as a sum we have
f(t)dt • Probability that the original component fails
in the time interval (t, t + dt).
h(x)dx • Probability that a failure and subsequent renewal
occurred in time interval (x, x • dx).
f(t-x)dt • Conditional probability that a component which
was renewed at time x fails in the time interval
(t, t + dt).
h(x)f(t-x)dxdt • Unconditional probability of failure in time
interval (t, t • dt) for a component which could
fail at time x.
f(t)dt +
/ h(x)f(t-x)dxdt • Total probability of failure which is the sum of 0
all possible conditions which could lead to
failure in the time interval (t, t + dt),
= h(t)dt by definition.
The mission reliability can now be determined from the renewal
22
2 Lloyd, D.K., and Linow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1D62).
3 Barlow, R. E., and Proschan, F., "Mathematical Theory of Reliability," John Wiley 5 Sons, New York (1965).
Papoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).
23
rate h(t) using the relation from reference f4]
R(t,T) = 1-F(t+T) + / [l-r(t+T-x)lhCx)dx (5) 0
in which
R(t,r) = Mission reliability at svstem time t for a
mission length T,
= Probability of no failure in (t, t+x).
l-F(t+r) = Probability that the original component in the
system has not failed at time t+T.
l-F(t+T-x) • Probability that a component which failed at
time x has not failed at time t+T.
h(x)dx • Probability of failure at time x.
Typical examples of the renewal rate and mission reliability of an
increasing failure rate component are deoicted in Figures 3 and 4. As
can be seen the renewal rate increases with svstem age until about a
system time enual to the MTBF (mean time between failure) of the
component. The renewal rate then approaches a constant value enual to
the reciprocal of the MTBF. The reliability OP the other hand de-
creases from a value of 1.0 at t=0 and approaches a constant value.
The asymptotic values of h(t) and R(t,T) can be derived directlv from
equations (4) and (5) by passing t to the limit infinity. These
values are given by the relations
h(») = l/'TTBF
1 R(«yr) - jjjrr/ [1-F(x)|dx.
T (6)
4Hnedenko, B.V., Belyayev, Yu.K., and Solovyev, A.D., "Mathematical Methods of Reliability Theory," Academic Press, New York (1969).
24
Component Renewal Rate
<1TB^
0 ' Svsten Are
Figure 3. Component Renewal Rate vs System Ape.
1.0
Comnonent Reliability
0 Svstexii A^e
Figure 4. Component Reliability vs System Age.
25
For the exponential distribution (constant failure rate) the
renewal rate and reliability are both constant throughout system life
-AT and are equal to X and e respectively.
It should be noted at this point that for a high reliability sys-
tem (say .85 or greater) the MTBF of the comnonents comnrising the
system should be of the same order of magnitude as the expected or
required system life. Consequently, the components within a system
are essentially exercised within the transient nortion of their lives
thus indicating the general importance of considering renewal theorv
in determining component reliability for high reliabilitv systems,
(c) Numerical Solution for the Renewal Rate.
A number of numerical techniques were investigated to
solve the integral equation (4) for the renewal rate h(t). The most
computationally efficient solution investigated thus far was through
the use of finite difference methods [7], Finite difference yields a
direct solution for the renewal rate. Other more general technioues
investigated which give comnlete solutions to the renewal problem
(e.g. probability distribution of total number of failures or time to
nth failure) involved the use of orthogonal expansions of Lanlace
transforms. Two sets of orthogonal functions considered were trigono-
metric and Laguerre nolynomials. The finite difference solution,
although limited to solution of the renewal rate, generally required
less computer time for given accuracy. The solution is also quite
general in that it can be used to determine renewal rate for most
theoretical distributions of failure time applicable to reliability
7 McKelvey, R. W., "An Analysis of Approximate Methods for Predholm's Integral Equation of the First Kind," December 1956, ADr>5f)530.
26
theory.
In solving equation (4) using finite differences, the density
f(t) and the renewal rate h(t) are discretized over a fixed time inter-
val generally the system life. In this case f(t) and h(t) are written
as
f(t) = f(kAt) = fk
h(t) = h(kAt) = hk
t = kAt where k=n,l,...,n.
The integral equation (4) can then he written as
t A k_1 / f(t-x)h(x)dx • _ I [f(kAt-iAx)h(jAx) + 0 2 i=n
•f(kAt-(i*l)Ax)hC(i*l)Ax)]
Ax v
j = l [f. .h. • f. . .h. .] J k-i j k-i-1 i + lJ
for k = 1,... ,n
0 for k = o.
(7)
Substituting into equation (4) then gives
j=0
for k • l,...,n (8)
= f for k = o.
Equation (8) renresents n+1 equations with n+1 unknowns which can be
readily solved using Gaussian elimination.
27
A difficulty is encountered in some situations where f(0) • «
such as in the case of the Weibull distribution with shape narameter
8<1. In this instance f(0) is fixed at a finite value such that the
actual area under f(t) in the interval (0, At) (i.e. F(At)) is made
equal to the finite difference area (f(0) + f(At))At/2.
(d) Numerical Solution for Mission Reliability.
Once the renewal rate is determined the transient
mission reliability can then be computed from equation (5):
R(t,T) » l-F(t+x) + fZ p-F(t+T-x)lh(x)dx. n
Numerically, this equation is difficult to solve since the integral
must be evaluated over the whole interval (0,t) for each different
value of t. By making use of the renewal equation an equivalent
equation for R(t,r) can be derived which is commitationally more
efficient. Integrating the renewal equation (4) from 0 to T gives
H(T) » F(T) + / H(T-x)f(x)dx (9) 0 j
where H(T) = / h(t)dt. 0
Making the substitution T=t+x in equation (9) gives the following
equation:
t+T H(t+T) » F(t+T) + / H(t+T-x)f(x)dx. (10)
0
Integrating by parts and using the fact that H(0) = n and F(0) = n
then gives
2«
t+T H(t+T) = F(t+T) + / P(x)h(t*T-x)dx
0 t+T
= F(t+T) + / T7(t+T-x)hfx)dx. fll) n
From equation (11) the following equation can he readilv derived:
t t + T
/ fl-F(t+T-x)]h(x)dx = r(t*x] - J ri-^ft*T-x)]hfx)dx n t
(12) t
where the definition H(t) = / h(x)dx was used. o
Substituting equation (12) into equation (S) for mission reliability
finally yields
t+T R(t,T) = 1 - / ri-T7(t+x-x)lh(x)dx (13)
t
Note that in this equation the integral is evaluated onlv over the
interval (t, t+T) and that the value of F(v) is reouired onlv over
the interval (0,x). This simplifies the numerical solution for mission
reliability.
(e) Average Mission Reliability.
Interval reliability as discussed in the previous
section is a transient function of system age. Weapon svstem renuire-
ments, however, generally specify one value of mission reliability for
the system. This snecified value could represent the lowest reliability
to be experienced by the system or an average value over system life.
In the computer model average system reliability is comnuted using the
relation
1 ? R(T) - - I R.(T) (14)
n i.i i
in which
R-CT) = Reliability for the ith mission
n • Expected number of missions over system life.
Computation of System Mission Reliability.
In general, system reliability can be computed directly from
the component reliabilities using an appropriate reliability model
[2,8], For example, for the series reliability model, system reliabili-
ty Rs(t) is determined from
n
Rs(t) « TTRi(t) (is) i-1
in which
R^(t) • Reliability of the ith component
n = Total number of components.
A series model is applicable whenever the failure of a single
component within a system results in a failure of the system. This
model has been initially chosen for the data analvsis computer model.
In the more general case where redundancy and load sharing are
inherent in the system, reliability models are more complicated but
csn be derived in most given situations [2,81.
30
2 Lloyd, D. K., and Linow, M., "Reliability: Manajremrnt, Methods, and Mathematics," Prentice Hall, flnj>lewood Cliffs, New Jersey (lOri?).
g Bazovsky, I., "Reliability Theory and Practice," Prentice-Flail, Englewood Cliffs, New Jersey (10^2).
31
Probability Distribution Selection.
One of the more difficult and questionable aspects of data
analysis is the selection of a theoretical distribution applicable to
a given set of data. This is particularly true for small data sarnies
where information is ultimately required at the tails of the distri-
bution. In the case of component failure data, theoretical considera-
tions, prior history and experience play a large part in distribution
selection. For example, for particular failure modes such as fatigue
the lognormal and Weibull distributions have been found to yield Rood
characterizations of the data [9, 10],
The data analysis computer model includes a distribution fitting
program to assist in the selection of a best-fit theoretical distri-
bution for use in reliability computation. Essentially, this program
computes the standard error for a number of different candidate
distributions. The standard error is a measure of the deviation of
the data from the theoretical cumulative distributions.
The Kolmogorov-Smimov statistical goodness-of-fit test is also
used to determine which of the theoretical distributions can be
rejected for given confidence level. Final selection of the distri-
bution can then be made based on the computer results, theoretical
considerations and/or personal experience and judgment.
(a) Candidate Distributions.
Following are the candidate distributions considered
for characterizing component failure times. In order to standardize
terminology used to describe the distribution parameters, the terms
32
Dolan, T. J., "Basic Concents of Fatirnj* Damage in Metals," 'fetal Fatif»ue, McGraw-Hill, New York (1950), Chanter 3, np. 3 -67.
' Freudenthal, A. M., and Shinozuka, "'., "Structural Safety Under Conditions of Ultimate Load Failure and ^atipue," Columbia Univcrsitv, Mew York, N.Y., October 1P61, WADD Technical ^e^ort fil-177.
S3
scale, shape and location parameters are used for all distributions
and are defined below.
(1) Exponential
f(t) = ~e't/n (16)
where n = MTBF = scale parameter
= 1/X
X = constant failure rate.
(2) Normal 1 t-U 2
f(t) - e 2 ° (17) /2if a
where u = *fi*BF • scale parameter
o • Standard deviation • shane parameter.
(3) Weibull
6 t-y e-i -(^)B
f(t) = (ft rS-J e n (18)
where u • >1TBF = y +nr(l+l/8) 2 2 2
a = Variance = rf [r(l*2/B)-r (1*1/6)]
H = Scale parameter
3 • Shape parameter
y • Location parameter.
(4) Lognormal
(t-y)/2T 8
1 2 1 - rsT(*n(t>Y)-n)
f(t) * e 2P (19)
34
where V = '1TBF » y+e 2 ->
a = Variance = e (e -1)
p • Average {£n t} = Scale parameter
B = Standard Deviation f?,n tl • Shape parameter
Y = Location parameter.
(5) r.amma
f(t) . Ii^!ll e"^ C20) T(B)p3
where u = lTBP = Y*BP
2 2 a = Variance « Bp
p = Scale parameter
B = Shane narameter
Y = Location parameter.
More detailed discussions of these distributions can be found
in the literature [6,11],
(b) Least Squares Fit of Data [12].
For each distribution other than Hamma a least
squares fit of the data to the distribution is made. Henerally, the
theoretical distribution is linearized as far as nossible to simnlify
computation. In the case of the Camma distribution maximum likelihood
estimates of oarameters are used rather than solving the more difficult
nonlinear nroblem.
The cumulative distribution function £(t) is used in fittintj the
35
Papoulis, A., "Probability, Random Variables, and Stochastic Pro- cesses," McGraw-Hill, New York (1965).
nireson, W. G., Editor, "Reliability Handbook" McGraw-Hill, New York (1966), Chapter 4.
12 Draper, N. R., and Smith, H., "Annlied Regression Analysis," John Wiley 5 Sons, New York (1966).
.36
data with the median ranks being used as the true values of the cumula-
tive distribution associated with each ordered data point [13]. The
median ranks for a complete data sample are computed using the relation
i-0.3 MR. = . (21)
.1 n-0.4
where n = Data sample size
i = l,...,n
= Order numbers for the data where the data is
sorted in increasing order.
When suspended data has been generated along with failure data,
the median ranks are determined for the failure data onlv with the
suspension times being used to modifv the median ranks ri3]. In this
instance the failure and suspension data are sorted in increasing
order. The sample size n in equation (21) is now the total number of
failure and suspension data items. The order number i associated with
each failure item is determined as follows; initiallv i is set equal
to zero. An increment is then computed which is to be added to i to
give the order number for the first failure item using the general
relation
n • 1 - (Previous Order Number) New Increment =
1+(Number of Items rollowing the Present Suspension Set).
C22)
If there are no suspension data prior to the first failure, then the
increment is 1.0 and the order number for the first failure item is
1.0. This procedure is repeated for each subsequent failure item
1 ^ Johnson, L.G.t "Theory and Technique of Variation Research," Hlsevier, New York (1964), Chapter 8. _
using equation (22) in each case. The median ranks are then computed
using equation (21).
The median ranks are considered to be the true values of ^(t)
associated with the ordered failure times t.. The distribution para-
meters for a given theoretical form of F(t) are thon determined by
minimizing the sum of the squares of the error between the theoretical
distribution evaluated at the failure times and the median ranks or
between the transformed distribution and median ranks.
Consider as an example the Weibull distribution. In this case
t-Y B
F(t) = 1 - e "HH . (23)
Rearranging equation (23) and then taking double logs gives
£n£n(l/(l-F(t))) » B(An(t-y)-inn). (24)
For fixed Y, this equation is linear in £n(t-Y). The error between
the transformed theoretical distribution and the data is then given by
e. » 6(P-n(ti-Y) - £nn) - lnta(l/(l-»1R )) (25)
where t. • ith failure time l
MR. = ith median rank. l
The oarameters 8, o. and Y are then found which minimize the total
2 r 2 square error c • 2,ej* A similar approach is used for other distri-
butions.
(c) Computation of the Standard Error [12],
In general, the standard error in the fitting of
12Draper, N.R., and Smith, H., "Applied Regression Analysis" John Wiley g Sons, New York (1966).
38
a theoretical distribution to data is not the same as the error dis-
cussed in the previous section. The standard error is a measure of
the difference between the untransformed theoretical distribution and
the median ranks. The standard error is defined bv the following
equation:
( I Crct.)-MR.)2)1/2 i = l * T
s.e. = Standard Error = —— (26) n - n - 1
where F(t.) = Theoretical distribution evaluated at the failure 1 tines tj.
»|R. a Median ranks
n = Number o^ failure noints
p - Number of nonulation parameters estimated in determining p(t).
The smaller the s.e. the closer the data fits the theoretical distri-
bution.
(d) Kolmogorov-Smimov Goodness of Fit N4"|.
A non-parametric distribution has been derived bv
Kolmogorov [15] and Smirnov f16] for a particular statistic d which
is a measure of the fluctuation of sample data about the theoretical
distribution from which the sample is drawn. The statistic d is de-
fined as the maximum absolute deviation between the theoretical and
observed cumulative nrobability distributions. The theoretical distri
bution is fixed by specifying both the functional form and the para-
meters^
Siegel, S., "Nonparametric Statistics for the Behavioral Sciences," McGraw-Hill, New York (1956), Chapter 4.
30
Kolmogorov, A., "Confidence Limits for an Unknown Distribution Function," Annals Mathematical Statistics, Vol. 12, (1941), nn. 461- 463.
16Smimov, N. V., "Table for Estimating the Hoodness of Pit of Empirical Distributions," Annals Mathematical Statistics, Vol. 19, (1948), pv. 279-281.
40
In application of the K-S statistic, a Riven set of data is
hypothesized to have heen drawn from a Riven theoretical distribution.
The statistic d is then determined from the data and compared to the
K-S distribution of d. That is, for given significance level a or
confidence level (1-ct) the theoretical value d is determined from
K-S tables and compared to the computed d. If d>d , the hypothesis
is rejected. If d<d then there is no sufficient reason to reject the
given theoretical distribution and the hypothesis may be accepted.
The statistic d is determined for the candidate distributions
in the computer model to provide bases for reiecting distributions and
to help indicate the best-fit distribution.
Maximum Likelihood Estimates of Parameters [17],
Maximum likelihood estimates of population parameters are
determined for each component within a svstem for which failure data
has been generated. There are a number of reasons for generating
maximum likelihood estimators as part of the reliability model. First,
maximum likelihood yields estimates of parameters which have a number
of desirable attributes. For example, if an efficient estimator for
small samples exists (i.e. one with minimum variance), then maximum
likelihood provides such an estimator. In general then minimum con-
fidence intervals can be derived in this case. Maximum likelihood
estimators are also consistent in that thev anoroach the true para-
meter values as sample size increases. In addition, maximum likeli-
hood estimators are asymptotically normal as sample size increases
Mood, A.M., and Craybill, r.A., "Introduction to the Theory of Statistics," McGraw-Hill, New York (1063), Chapter 8.
41
for most cases. This simplifies determination of confidence intervals
for large samples.
A second reason for determining maximum likelihood estimates is
that they can be computed for relatively general underlying probability
distributions through the use of routine mathematical procedures.
Finally, the numerical procedures planned to be used in the computer
model to determine confidence intervals require the maximum likelihood
estimates of parameters to improve computational efficiency.
The basic idea behind maximum likelihood estimation is relatively
straightforward. One assumes first that a sample of nf failures and
n suspensions have been generated from a given theoretical distribu-
tion with parameters c^ * (a. ,a2,... ,ct ). The failure and suspension
data are designated as t^. (i=l,... ,nf) and t .(i=l,...,n 1 respectively.
The joint probability distribution of the random sample of failure and
suspension times can be written as
L(t;o/) » gt(tfl,..*,tfn »tsl»---»tsn ;i) (2?)
where L(t;a) » Defined as the likelihood function
g. « Joint distribution of the sample outcome
t = Vector of the sample data values
a • Vector of parameters for given underlying distributions of failure times.
It is next assumed that each failure and suspension time is a
statistically independent outcome. Equation(27) can then be written
in terms of the failure density f(t) and the cumulative distribution
42
F(t) as follows [6]
ns
L(t;o) - C | | f(tf.;op [1-F(t .;o)] i«l i=»l
(28)
where C • Normalizing constant such that the area under L(t;a) is unity
f(t;a) • Theoretical density distribution of failure times.
F(t;o) • Cumulative distribution function
= J* f(x;o)dx n
[1-F(t;ct)] = Probability that the failure time is greater than t. This term is used to represent the nrobabilitv of obtaining the suspension times.
As can be seen, the likelihood function L is defined as the
multivariate probability distribution of the random failure and sus-
pension times. Solving for the parameters c^ such that L is maximized
consequently gives the highest probability density for the given sample
outcome t. The parameter values which maximize L are denoted as a
and are called the maximum likelihood estimates of cu
In practice in L given by equation (29) is maximized rather than
L to simplify computation. This can be done since the maximum of
any positive function and its log are equivalent.
Jin L(t_;a) = In C * I In f (t_. ;a) i-1 "
"s • I An[l-F(t :o)] (29)
j-1 *">
Panoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).
43
Consider the Weibull distribution as an example. For this case
F(t;ci) = 1 - e n
where a • (n,B,Y).
The likelihood function can then be written as
nf
An L * An C + £ r£nB-£nn.+ (B-l)(£n(t -Y) i=l fl
*£/*" "s V"7* - Ann) - (—••—D ] ~ f (—~~) (30)
n i^i *
Solving for n,B and y which maximize An L yields the maximum likeli-
hood estimators for the Weibull parameters.
Likelihood functions such as given by equation (30) are maximized
in the computer model using Rosenbrock's algorithm ("18] which is a
general solution of the unconstrained minimization problem for non-
linear functions. Minimizing the negative of An L maximizes the
function.
Once point estimates of the failure distribution parameters are
generated for a given component, the mission reliability can be com-
puted using the methods previously presented.
1 8 Rosenbrock, H.H., "Automatic Method for Findinp the Greatest or Least Value of a Function," Computer .Journal, Vol. 3, (1061), pj). 175-184.
44
Bayesian and Classical Confidence Intervals for Component
Reliability.
Classical methods of confidencing component mission reli-
ability for small samples are generally available onlv for the constant
? failure rate case. In this instance the x distribution represents
the inferencing distribution for the "fTBF from which confidenced reli-
ability can be derived [2]. Bayesian methods will consequently be used in
the computer model for the non-constant failure rate components since
this method provides a svstematic means o^ determining confidence inter-
vals for more general problems. A complete discussion of Bayesian
statistics will not be presented in this report and the reader is
referred to the books by Lindley [19] for a more complete discussion
of this topic.
Bayesian inferencing is based, as the name implies, on Bayes"
probability theorem which is essentially a conditional probability
statement:
Bayes' Theorem:
f(X.x) = f(y;x)f(x) = f(x;y)f(y) (SI)
in which
f(y»x) • Joint probability density of random variables x and y.
f(y;x) • Conditional probability of y given x
f(x;y) = Conditional probability of x given y
2 Lloyd, D. K., and Lipow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1962).
19 Lindley, D.V., "Introduction to Probability and Statistics from a Bayesian Viewpoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge (1965).
46
f(x)» f(y) • Marginal distribution of x and y respectively.
From equation (.31)
f(y) f(y;x) = f(x;y) *— " (37)
f(x)
In the Bayesian aonroach thn unknown mrameters of the distrihu-
tion of failure times are considered to be random variables. The
sample outcome t from a given test are also random variables which
are dependent on the distribution narameters. Bayes' theorem, eouation
(32), can be written in this instance as
f(a) f(a;x) = f(x;a) (33)
f(x)
in which
f (a;x) = Density distribution of narameters given the test sample x.
f (x_;a) = Density distribution of the samnle outcome given the narameters m.
f(a),f(x) = Marginal densities of rx_ and x_ respectively.
In equation (33) the distribution f(x;ot) is iust the likelihood
function L(x_;ot) by definition. The distribution f(oi) is called the
prior of a and f (rt;x) is called the posteriori distribution of r^
since it is determined after the samnle x_ is determined and is con-
sequently conditioned by the test results.
The prior distribution f(a) generally reflects nrior knowledge
47
or prior data on the population parameters a before the test results
x have been generated. For the case of weapon system components there
is at the present time little or no prior information and consequently
f(a) must reflect in some way our prior ignorance of the values of a.
Choosing the prior distribution to reflect decrees of ignorance is one
of the more difficult and questionable aspects of Bayesian statistics.
Much research work is presently being conducted on the question of
prior distribution selection. Priors which represent maximum ignorance
for some of the simpler problems have been found [19,70]. It has also
been found, however, that a uniform nrior on the parameters yields
confidence intervals that are generally close to exact in a classical
frequency sense for a number of reliability indices. The uniform
distribution assigns equal probability to all values of a random
variable over a given range. The robustness of uniform priors is
discussed somewhat in reference [19]. Uniform priors have consequently
been used in our reliability statistics work and have also been found
to be generally robust.
The distribution f(x) in eauation (33) is considered to be a
constant and is determined so that the area under f(o;x_) is unity.
Equation (33) can now be rewritten in the following form:
f(a;x) - C L(xjo) (34)
where f (a;x) • Posteriori distribution of ot_ given x_.
C = Constant such that area under f (nyx) is unity. The constant C contains the constant terms f(x) and f(ct).
4*
19 Lindley, D. V., "Introduction to Probability and Statistics rrom a Bayesian Viewpoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge fl9f,S).
20 Jaynes, E. T., "Prior Probabilities," IEEE Transactions on Svste^s Science and Cybernetics, Vol. SSC-4, No. 3, Scot. 1^8, no. 227-241
49
L(x;a) • Likelihood function.
In computing confidenced reTiability our interest is not on the
•parameters themselves but rather on a function of the parameters.
From probability theory [6] the cumulative distribution of any function
z of the random variables a is determined by the relation
F_(z;x) = /// f(a;x)dn f35)
in which
F (z;xj = Posteriori distribution of Z
D « Domain of Z_ such that Z(o)<z
f(a;x) • Posteriori distribution of a i»iven by equation (34).
Once the posteriori distribution ^(z;x) is known confidence
intervals can be constructed on z for any given confidence level CL
by determining the apnronriate Percentage points directs from F(z;50.
For example, a lower confidence limit is determined by solving the
following equation for z. :
Fz(zL;x) « 1 - CL f.36)
in which
Zt * Lower confidence limit.
CL = Confidence level.
Since component mission reliability is a direct function of the
failure distribution and hence of the parameters, confidence intervals
Panoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York ("1965).
SO
on mission reliability can be determined using the above formulation.
Solution of equations (34), (35) and (36) for confidence limits can
be accomplished numerically using computer routines.
Bayesian Confidence Intervals for Svstem Reliability.
(a) Constant Failure Rate Comnonents In Series.
Confidence intervals on component reliability for con-
2 stant failure rate can be determined using the x distribution f2].
For a system of components in series, however, classical confidence
intervals can be readily determined only if all components have eoual
test times. In this instance the svstem can be treated as if it were
a component with all failures being counted as system failures regard-
2 less of which components fail. The x distribution can again be used
to determine confidence limits.
For series components with unequal test times, however, classical
methods do not generallv apply. For this case Bayesian statistics, as
discussed in the previous section, can be employed to determine confi-
dence intervals.
Consider the series svstem
R = | R. (37) i=l 1
in which R = Svstem reliability, s
n = Number of components, c
R. = Reliability of the ith component, l
2 Lloyd, D. K., and Lioow, M., "Reliability: Management, Methods, and Mathematics," Prentice-Hall, Englewood Cliffs, New Jersey (1062),
For constant failure rate, R. = e i and the system reliability
becomes
nc -A.T -A T
R = ||e = e S (38) 5 i=l
where n c x = y x.
i=l x
The system failure rate A is enunl to the sum of the individual s
component failure rates. The posteriori probability density of the
individual component failure rates can be determined usin.c? equation
(34) by letting ot = A . Since A is equal to the sum of independent — i s
random variables, the distribution for A is equal to the convolution s
of the individual component distributions [6]. Performing the required
convolution is generally tedious and it is at this point that a simpli-
fying assumption is made. Using the Central Limit Theorem for sums of
random variables [6] it is assumed that the oosteriori distribution for
As is Gaussian. Consequently, all that is required is the mean and
variance of A to define the oosteriori distribution. These are deter- s
mined using the relations
nc E(A J = I E(A.) (30)
s i=l
nc Var(As) = I VarCXj) M")
i=l
Where E(*) • Expected value
Var(*) = Variance
Papoulis, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (T>65).
52
Confidence intervals can now be constructed for X and hence for system
reliability using equations (39) and (40).
The accuracy of making the Gaussian assumption was checked and it
was found that a system with about ten components and one to three fail-
ures experienced ner comnonent yielded relatively accurate confidence
intervals in comparison to true values. In the example problem consi-
dered test times for each component were assumed equal which permitted
the comoutation of classical confidence limits.
(b) Non-constant Failure Rate Components In Series.
For the series svstem eouation (37) applies:
"c
R - | j R s i-1 1
Taking logs of both sides of this eouation yields
nc In R = 7 InR.
i-1
Equation (41) represents a sum of independent random variables
and consequently the posteriori distribution of InR can be determined
using the Gaussian assumption as discussed in the previous section.
Confidence intervals can then he constructed from this distribution.
COMPUTER PROGRAM
Two computer programs have been prepared to perform the compu-
tations required for reliability data analysis. One program called
53
DISTSEL performs goodness-of-fit computations and is used to assist in
distribution selection given component failure and suspension data.
Included in this program are computations of the standard error and
Kolmogorov-Smirnov statistic. At the nresent time, the theoretical
distributions that can be handled are the exponential, two and three
parameter Weibull, two parameter lognormal, two parameter famma and
the normal.
The second computer program called RRLIAB Performs corn-nutations
required to determine reliability and confidence limits for components,
subsystems and system given component data. The component data in-
cludes failure and suspension times and the associated theoretical
distributions to be assumed.
The fortran listings of the computer programs are lengthv;
DISTSEL and RELIAB contain 17 and 25 subroutines respectively. These
listings are consequently not contained in this report but can be
made available upon request. The innut data format for both programs
is presented in Appendix I and output data for a sample problem are
given in Appendix II.
As a final note, it should be emphasized that the computer models
are flexible. Input and output data formats can readily be changed
for particular situations, perhaps to render the programs compatible
with a given computer data file system. Improved versions of the
programs will also be generated in the future.
54
REFERENCES
1. Army Regulation AR705-5D, "Army ''ateriel Reliability and Maintain- ability," Headquaters Dent, of the Army, Washington, D. C.f 8 January 1968.
2. LLOYD, D. K., and LIPOW, M., "Reliability: Management, Methods, and Mathematics," Prentice-Mall, Englewood Cliffs, New Jersey (1962).
3. BARLOW, R. E., and PROSCHAN, F., "Mathematical Theorv of Reli- ability," John Wiley F, Sons, New York (1965).
4. GNEDENKO, B. V., BELYAYEV, Yu.K., and SOLOVYEV, A. n., "Mathema- tical Methods of Reliabilitv Theory," Academic Press, New York (1969).
5. PIERUSCHKA, E., "Principles of Reliability," Prentice-Hall, Engle- wood Cliffs, New Jersev (1963).
6. PAPOULIS, A., "Probability, Random Variables, and Stochastic Processes," McGraw-Hill, New York (1965).
7. McKELVEY, R. W., "An Analysis of Approximate Methods for Fredholm's Integral Equation of the First Kind," December 1956, AD650530.
8. BAZOVSKY, I., "Reliability Theory and Practice," Prentice-Hall, Englewood Cliffs, New Jersey (1067).
9. DOLAN, T. J., "Basic Concents of fatigue Damage in Metals," Metal Fatigue, McGraw-Hill, New York (1959), Chanter 3, np. 39-67.
10. FREUDENTHAL, A. M., and SHINOZUKA, M., "Structural Safety Under Conditions of Ultimate Load Failure and Fatigue," Columbia Univer- sity, New York, N.Y., October 1961, WADD Technical Report 61-177.
11. IRESON, W. G., Editor, "Reliability Handbook," McGraw-Hill, New York (1966), Chapter 4.
12. DRAPER, N. R., and SMITH, H., "Applied Regression Analysis," John Wiley T, Sons, New York (1966).
13. JOHNSON, L. C, "Theory and Technique of Variation Research," Elsevier, New York (1964), Chapter 8.
55
14. SIEGEL, S., "Nonparametric Statistics for the Behavioral Sciences," McGraw-Hill, New York (1956), Chanter 4.
15. KOLMOGOROV, A., "Confidence Limits for an Unknown Distribution Function," Annals Mathematical Statistics, Vol. 12, (1941), op. 461-463.
16. SMIRNOV, N. V., "Table for Estimating the Goodness of Fit of Empirical Distributions," Annals Mathematical Statistics, Vol. 19, (1948), pp. 279-281.
17. MOOD, A. M., and GRAYBILL, F. A., "Introduction to the Theory of Statistics," McGraw-Hill, New York (1963), Chanter 8.
18. ROSENBROCK, H. H., "Automatic Method for Finding the Greatest or Least Value of a Function," Comnuter Journal, Vol. 3, (l°r,n) f pp. 175-184.
19. LINDLEY, D. V., "Introduction to Probabilitv and Statistics from a Bayesian Viewnoint," Part 1: Probability and Part 2: Inference, Cambridge University Press, Cambridge (1965).
20. JAYNES, E. T., "Prior Probabilities," IEEE Transactions on Svstems Science and Cybernetics, Vol. SSC-4, No. 3, Sent. 1968, nn. 227-241,
56
APPENDIX I
INPUT FORMAT FOR PROGRAMS DISTSEL AND RELIAB
The following listings give the card by card description of the
innut data and format required by the reliability data analysis
programs.
(a) Overall System And Subsystem Data Cards
Card No.
Column No. Description Format
1
2
4 to 4 +NSIJB
1-80
1-10 11-20 21-30
1-40
1-80
Overall system identification.
Number of missions over svstem life. Specified system life. Desired confidence level for relia- bility.
Number of components per subsystem for which data is supplied. UP to four subsystems can be handled.
Alphanumeric
110 FlO.o F10.0
4110
Subsystem identification. NSUB is the number of subsystems where one card is used to describe each subsystem. Alphanumeric
(b) Element or Component Data Cards
The following data cards are required for each element
starting with the elements of the first subsystem down to the elements
of the last subsystem:
57
Card No.
Column No.
Description format
1 2-10
11-80
2 to Tlj
2-11
12-80
Theoretical distribution code number*, Element identification code number. Description of element.
Code number for determining if data on card is a failure time, a suspen- sion time or last card ^or current element**. n.r is the number of failures and n is the number of suspensions. Failure nr susnensiop time. If this is the last card then the remaining columns are blank. Description o*- the failure or suspen- sion.
II 19
Alohanumeri c
II
^10.0
Alnhanuneri r
Distribution Code Number: 1 - Exponential 2 - Two parameter Weibul1 3 - Three parameter Weibul1 4 - Two parameter lognormal 5 - Three narameter lo^nonnal 6 - Two parameter gamma 7 - Three parameter gamma 8 - Normal
Failure or Suspension Code Number: 1 - Suspension 2 - Failure 3 - End of data
(c) Sample Problem
The following is a listing of data innut for a hypothetical reli
ability data analysis problem:
58
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS 10 2000.0 0.70 3 1 1 0
SAMPLE PROBLEM SUBSYSTEM A SAMPLE PROBLEM SUBSYSTEM B SAMPLE PROBLEM SUBSYSTEM C 1100000001 SAMPLE PROBLEM ELEMENT M 2 1200.0 SPACE FOR rMLURE DESCRIPTION 2 000.0 SPACE FOR FAILURE DESCRIPTION 2 1500.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FMLURE DESCRIPTION 2 1000.0 SPACE FOR FAILUF-H DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1100.0 SPACE FOR FAILURE DESCRIPTION 2 850.0 3 1100000002
SPACE F0" PAIUPE DESCRIPTION
SAMPLE PROBLEM ELEMENT A? 2 1500.0 SPACE ^OR FAILURE DESCRIPTION 1 8000.0 3 1100000003
SPACE FOR SUSPENSION DESCRIPTION
SAMPLE PROBLEM ELEMENT A3 1 0000.0 3 2200000001
SPACE FOR SUSPENSION DESCRIPTION
SAMPLE PROBLEM ELEl*ENT Bl 2 1200.0 SPACE ^OR FAILURE DESCRIPTION 2 000.0 SPACE FOP. FAILURE DESCRIPTION 2 1500.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FAILURE DESCRIPTION 2 1000.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1100.0 SPACE FOR FAILURE DESCRIPTION 2 850.0 3 4300000001
SPACE FOR FAILURE DESCRIPTION
SAMPLE PROBLEM ELEMENT Cl 2 1200.0 SPACE FOR FAILURE DESCRIPTION 2 000.0 SPACE FOR FAILURE DESCRIPTION 2 1500.0 SPACE ^OR FAILURE DESCRIPTION 2 1300.0 SPACE FOR FAILURE DESCRIPTION 2 1350.0 SPACE FOR FAILURE DESCRIPTION 2 1000.0 SPACE FOR FAILURE DESCRIPTION 2 1300.0 SPACE "OR FAILURE DESCRIPTION 2 1100.0 SPACE ^OR FAILURE DESCRIPTION 2 850.0 3
SPACE FOR FAILURE DESCRIPTION
'•ODEL
5P
APPENDIX II
OUTPUT DATA TT)R SAMPLE PROBLEM
The following pages contain output data renerated bv p-rorrams
DISTSEL and RELIAB for the sample innut data presented in Appendix I
60
RESULTS FROM PROGRAM DISTSEL
FAILED SPECIMENS AND MEDIAN RANKS...
850.0 0.07447 SPACE FOR FAILURE DESCRIPTION 900.0 0.18085 SPACE norc FAILURE DESCRIPTION 1000.0 0.28723 SPACE FOR FAILURE DESCRIPTION 1100.0 0.39362 SPACE noi> FAILURE DESCRIPTION 1200.0 0.50000 SPACE FOR FAILURE DESCRIPTION 1300.0 0.60638 SPACE TOR FAILURE DESCRIPTION 1300.0 0.71277 SPACE "OR FAILURE DESCRIPTION 1350.0 0.81915 SPACE FOR FAILURE DESCRIPTION 1500.0 0.92553 SPACE FOR FAILURE DESCRIPTION
61
STANDARD ERROR 0^ ESTIMATE...
DISTRIBUTION
EXPONENTIAL NORMAL LOC-NORMAL WEIBULL-2 PARM. WEIBULL-3 PARM. GAMMA-2 PARM.
S.E.
0.280808E on 0.566338E-01 0.657507E-01 0.530806E-01 0.642205E-01 0.83200OE-O1
DISTRIBUTION SELECTED... WEIBULL-2 PARAMETER
K-S STATISTIC ( 0.141) IS LESS THAN TABLE VALUE ( 0.388) THEREFORE MAY ACCEPT HYPOTHETICAL DISTRIBUTION AT 10 PERCENT LEVEL
62
)DflTA
1500 J_
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973
100000000 SAMPLE PROBLEM ELEMENT Al
EXPONENTIAL DISTRIBUTION...LEAST SQUARES ESTIMATE MEAN LIFE = HUB. 173
1339 __
1195 __
X X
1067 __
952 __
850 __
-+- •+• -t- -t- -+- .001 01 .05
f>7,
.10 .20 .HO .60 .80 ."90
CUMULATIVE DISTRIBUTION
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973
DATA
isoo _i_
lOOOOOOOO SAMPLE PROBLEM ELEMENT Al
NORMAL DISTRIBUTION...LEAST SQUARES ESTIMATES MEAN LIFE
STANDARD DEVIATION =»2H5.0730
1166.665
137Q __
1240 __
1110 __
980 __
650 __
.001 .20
M
HO .60 .80 .90 .95
CUMULATIVE DISTRIBUTION
SAMPLE PROBLEM TO DEMONSTRATE RELIRBILITT DATA ANALTSIS MODEL MARCH 1973
DATA
1SQO _L
lOOOOOOOO SAMPLE PROBLEM ELEMENT HI
LOG NORMAL DISTRIBUTION...LEAST SQUARES ESTIMATES
STANDARD DEVIATION =2S9.UU38
MEAN LIFE = 1175.337
1195 __
.001
65
.40 .60 .BO .30 .95
CUMULATIVE DISTRIBUTION 99
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973
DRTfl
1500
1UU0O00OU SAMPLE PROBLEM ELEMENT Al
2-PARM. HEIBULL...LEAST SQUARES ESTIMATES SHAPE = 5.587
SCALE = 1259.OD
1339 __
1195 __
1067 __
9SE _.
850 __
.001 .01 1— .05
66
•+- -+- •+• -i—
.10 .20
CUMULATIVE .40 .60 .80 .90
DISTRIBUTION
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973
.DflTfl
903
100000000 SAMPLE PROBLEM ELEMENT Al
3-PARM. HEIBULL...LEAST SQUARES ESTIMATES SHAPE = 2.494
SCALE = 648.866
LOCATION = 597.253
700 __
543 __
421 --
326 __
253 __
H .001
•+• -+- -+- •+• •+ •+• 01 .05
67
.10 .20 .40 .60 .80 .90
CUMULATIVE DISTRIBUTION
SAMPLE PROBLEM TO DEMONSTRATE RELIABILITY DATA ANALYSIS MODEL MARCH 1973
lOOOOOOOO SAMPLE PROBLEM ELEMENT Al
GAMMA DISTRIBUTION...2-PARAMETER MAXIMUM LIKELIHOOD ESTIMATES ALPHA = 30.71U
BETA = 37.98U
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