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Confidence limits on the failure rate and a rapid projection nomogram for the lognormal distribution
Microelectron. Reliab., Vol. 24, No. 1, pp. 101-124, 1984. Printed in Great Britain.
0026-2714/8453.00 + .00 Pergamon Press Ltd.
Confidence Limits on the Failure Rate and a Rapid Projection Nomogram for the Lognormal Distribution A . S. J O R D A N BELL
LABORATORIES, Murray
Hill,
NEW JERSEY United States (Received
for publication
07974, of A m e r i c a . 19th S e p t e m b e r
1983)
ABSTRACT
In order to predict the sample failure rate from a limited number of early malfunctions, we have developed the rapid lognormal projection nomogram (RLPN) which we present at 1, 5, and 10 years of service. In the core of each diagram is a family of curves showing the variation of failure rate as a function of sample median life and standard deviation. Superimposed are Arrhenius plots for a series of activation energies, all referenced with respect to a fixed time. We also display the standard deviation as a function of relative median life, using cumulative failure as a parameter. The RLPN provides graphical means to estimate the failure rate at the planned service life and operating temperature in an efficient manner from the time to failure at the aging temperature and sample size for known or assumed values of the standard deviation and activation energy. To obtain the confidence limits for the failure rate of the lognormal distribution at the 90% confidence level, we have derived an approximate formula that relates the bounds to the service life and the sample's size, median life, and standard deviation. Then, by an appropriate selection of the independent variables and parameters, the confidence inter,~als are displayed as a series of curve families. The applications of the diagrams are discussed by means of illustrative examples taken from the field of GaAs FETs. I. INTRODUCTION The widespread applicability of the lognormal distribution in describing the long-term reliability of semiconductor devices has been amply demonstrated.
In addition to components based on Si and
Ge, <~) devices involving III-V compounds such as GaP opto-isolators
Accelerated aging
experiments often provide an abundance of time to failure data which make it possible to estimate the characteristic parameters of the Iognorma| distribution, namely median life and standard deviation, by means of a relatively direct procedure, camAdditionally, testing at different temperatures permits the evaluation of the activation energy via the Arrhenius law for rate processes so that the failure rate prevalent in field operation can be forecast. However, some high quality devices may not fail in sufficient numbers during the time allotted to testing which would warrant an exhaustive reliability analysis. Indeed, even if all the data could be accumulated within 6 months, the device engineer would need an earlier indication of the long-term performance of his product.
In a recent paper we have developed the assurance test matrix plan
that allocates prior to testing a group of devices undergoing accelerated aging in such a fashion that the optimum information with regard to their maximum failure rate in field operation can be gained from the time to first failure. ~6) However, after a few malfunctions are detected the designer needs a 101
102
A . S . JoRDAN
convenient tool to project the failure rate from the meager data, and furthermore, it is desirable to estimate the uncertainty associated with this key quantity. One of the objectives of this paper is to construct a set of charts that is suitable for the efficient estimation of the failure rate. The suggested graphical technique will be referred to as the rapid lognormal projection nomogram (RLPN).
The other concern of this study is the evaluation of the
confidence limits on the failure rate at t i e 90% confidence level. To meet these objectives, first we briefly review the important statistical relationships characteristic of the lognormal distribution. Then, the charts for the R L P N are generated. Second, we outline the methods used to compute from sample data the confidence limits on the median life and standard deviation as a function of confidence level. Next, an approximate formula is derived that yields the bounds on the failure rate in relation to service life, sample size, median life and standard deviation. Third, by an appropriate selection of the independent variables and parameters, we display the confidence limits in a series of curve families. Finally, the application of the diagrams is discussed by means of an illustrative example taken from the field of GaAs FETs.
I!. The Rapid Lognormal Projection Nomogram I. Lognormal Concepts The cumulative failure function (CFF) for the Iognormal distribution is related to the time to failure, t, by the expression 1 I CFF - -~- erf ~ In t/tin ~/2s
(1)
where s and tm are the standard deviation and median life of the sample, respectively. (2)'(7~ The abbreviation erf designates the error function.
Usually, lognormal behavior of a given device is
demonstrated by the linearity on probability paper of the observed log t versus CFF, derived from aging studies. In general, CFF is given by CFF -
n------L--t N+I
(2)
Where N is the sample size and nt is the total number of devices that failed at or before t. If tm has been obtained at 2 or more temperatures, then the activation energy, Ea, can be determined by linear regression; for it is usually postulated that the Arrhenius law for rate processes is applicable to degradation phenomena. (l)'(2)'(s) Accordingly, assuming constant s, one has
Ea
lntm - A + - kT where A is a constant.
(3)
Clearly, tmo at the operating junction temperature, To, can be readily
calculated from Eq. (3). At To, the failure rate, f(t) -- defined as the logarithmic time derivative of the complement to Eq. (1) -- is of the form
Failure rates
103
~/2 exp [ - 2s 2 (ln t/tmo) 21 f(t) "~
(4)
l ts erfc " ~ s lnt/tmo
where t denotes the service life and erfc is the error function complement.
10
'°5F I
TIME TO FAILURE (HOURS) '100 t03 104
300
200
a= t 0 3 < L~ >-
LAJ rr I-< rt"
~< t00 L~ h,n~ ,,,
100 :E
U.J h-
t0
n~ ._J
O.
t0
t00
103
20
104
tm/t t00
I
J
103
104
I
t05
i
I
I
I
106
107
t08
t09
MEDIAN LiFE Fig. 1.
J 10 t0
(HOURS)
The rapid Iognormal projection nomogram at l year of service, a. Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
2. Construction of the Charts
We present the R L P N at 1, 5 and 10 years of service in Figs. 1, 2, and 3, respectively. The core of each diagram is the log failure rate versus log median life family of curves, based on Eq. (4). These plots are virtually the same as the ones.given in Ref. 2 and as before s is employed as the parameter. The solid segments of the lines indicate that the failure rate is also the highest rate attained at any time up to the designated service life. (2) Also shown is a horizontal grid of dashed lines which is a reciprocal absolute temperature scale corresponding to the range 20" and 300"C in
104
A . S . JORDAN
20°C steps. In addition, in the top right corner we have generated Arrhenius plots according to Eq. (3), taking arbitrarily t m - 10Sh at T - 573*K (300°C) and E a between 0.5 and 4 eV. Lastly, in the lower left corner the nomogram includes curves constructed from Eq. (l) in the form s versus tm/t using CFF as a parameter. The following sequence of operations is recommended in using the RLPNs: 1. In Region l select the appropriate service life (1, 5 or 10 years).
2. Plot the last observed time to failure at the
appropriate aging temperature (X). 3. Draw a line through X parallel with a selected El ray, based on experience with regard to the activation energy of like devices. 4. Terminate the line at the operating temperature and read from the graph the time to failure at To, to. 5. Knowing N and n t calculate C F F via Eq. (2) and then find the best matching curve in the lower corner (Region I1) in order to locate the tmcft o value corresponding to the anticipated s. 6. Having thus obtained tmo, the
10
105
TIME TO FAILURE (HOURS) 100 103 104
300
t04 200 I-14U') rr
10 3
UJ >IJ
100 100 ~or w n
UJ I-<[ rr L~
10
hi I'--
:D J
'1
0.1
co
20
0 10
I
100
103
104
tmlt I
100
,
L
I
L
103
104
105
106
107
108
109
101o
MEDIAN LIFE (HOURS)
Fig. 2.
The rapid lognormal projection nomogram at 5 years of service, a, Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
Failure rates
105
failure rate at To can be evaluated at the assumed s from the central family of curves (Region l l l ) . In general, the application of the nomogt'ams should be reserved for very reliable devices exhibiting only a few failures after long testing, in cases when employing maximum likelihood and least-square techniques to obtain tm and s are unsuitable, c2)
10
•~05
TIME TO FAILURE (HOURS) 100 103 104
500
t04_ 200
o
w
1,1 tw
©
l---
~- 1 0 0 ~
100
w cr w
w F-
10
0,t
rr t.J o..
0.1
20 1
10
100 tm/t
103
104
I
I
I
I
1
L
I
L
i
100
103
104
10 5
t0 6
10 7
10 8
10 9
1010
MEDIAN LIFE (HOURS) Fig. 3,
The rapid Iognormal projection nomogram at 10 years of service, a. Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
III. Confidence Limits on the Failure Rate. !. Derivation of the approximate Equations. For a systems designer the key property by means of which to assess the long-term performance of an electronic component is the failure rate.
Nonetheless, investigators usually refrain from
estimating confidence limits on f. Perhaps this is not surprising as exact bounds on f at a given confidence level are lacking in the case of the lognormal distribution.
,
106
A.S.
JORDAN
There is a need for confidence limits because estimated values of tm and s are only valid for a sample of size N and thus there must be a degree of assurance that any other sample d r a w n from the same population will not lead to drastically different parameters. An exact theorem which holds for the Iognormal distribution states that if lntm and s are the m a x i m u m likelihood ( M L H ) estimators of a random sample of size N taken from a population with unknown u ( - Intmp) and ~r (standard deviation of population) then the random variable ( l n t m - u) /(s/x/-N') obtained by random sampling is distributed according to Student's -- t probability density function with N - I degrees of f r e e d o m J 7) Another exact theorem states that if lntm and s are the M L H estimators of a random sample of size N chosen from a population with unknown u and a then the random variable (N-1)s2/a 2 is X2 -- distributed with N - I degrees of freedom. (7) The m a t h e m a t i c a l properties of these two distributions have been widely studied. (9) Based on these statistical principles, in a previous paper we presented convenient charts for the lower and upper limits of the (a) median life [tm~(tm) l/s] and (b) relative variance [(a/s) 2] as a function of confidence level for sample sizes ranging from 2 to well in excess of 100. (2) Both quantities tm and s enter into the expression for the failure rate (Eq. (4)).
Substituting
trapu or tmpl where the subscripts u and 1 denote the upper and lower limits, respectively, at the 90% confidence level (CL) into Eq. (4) we would offer assurance that no more than 10 out of 100 samples d r a w n from the same universal population would have median lines outside the confidence interval bounded by the limits. If then ~ru and o'l at the 90% C L were also introduced in Eq. (4) we would simultaneously g u a r a n t e e that less than 10 out of 100 samples would have standard deviations outside the a ~ - a l range. But what one wants is the failure rate which is at least for 90 samples, out of a total of 100, is confined to some calculated interval. Clearly, in pairing trap and a limits at the 90% C L to estimate bounds on f one in fact would be insisting that for one hypothetical sample both quantities should attain their "worst" level at the same time. To be sure there is only a very remote chance given by
for this to occur simultaneously. Therefore, in order not to confine the bounds on the failure rate in an unrealistic manner, it is suggested that the confidence limits on f at the 90% CL should be estimated from the parameters
trnp and a taken at the 67% CL. For, then
and there is a 100 x ( I - 0 . 1 ) % - 90% chance that both parameters as well as f fall within the interval. Consequently, the upper and lower confidence limits for the failure rate at the 90% C L become
Failure rates
107
_~ ,/~'exp•[- 12~ fu(t) -~ta.
(In t/tmpl) 2] (6)
1
erfc . ~ / ~
lnt/tmpl
and [
1..~._(lnt/tmpu)2]
fj(t) x]-~tal erfc ~
1
(7) lnt/ttmpu
where au, al, tmpu, and tmpl are the population values at the 67% CL. 2. Calculation and Organization of the Failure Rate Limit Charts.
In order to calculate the confidence interval for the failure rate at the 90% CL from Eqs. (6) and (7) it is necessary to determine the population estimates tm#, tmpu, al and ~u as a function of the sample parameters t m and s. In an earlier paper we have related the population and sample properties for the Iognormal distribution to sample size and CL. (2) Denoting the plotted results for the lower and upper limit of median life by r I and r u and like limits for the standard deviation by ~ t and ~ u , respectively, we have trope " tm ~'~ (N,0.67)
(8a)
tmpl - t m v~ (N,0.67)
(8b)
au - s-w/~u(N,0.67)
(8c)
al - s . ~ ( N , 0 . 6 7 )
(8d)
and
A convenient way to organize the error bounds on the failure rate is to fix the service life t at 1, 5, and 10 years. Subsequently, at a given t the sample's standard deviation is set. Thus a separate family of curves is generated for s - 0.5, 1, 1.5, 2, and 3. Within each family the remaining unspecified variables in Eqs. (6), (7), and (8) are the sample size N and tm. Accordingly, in analogy with the failure rate projection in Figs: 1 through 3, the upper and lower confidence limits on f will be presented as a function of sample tin, using N as the parameter ranging from 4 to 100• In Figs. 4, 5, and 6 the confidence limits on the lognormal failure rate at the 90% CL are displayed at 1, 5, and 10 years of service. Parts of a, b, c, d, and e of each figure provide charts for s -- 0.5, 1, 1.5, 2, and 3, respectively. At the center of each curve family is the nominal f given by Eq. (4) for the t and s corresponding to that figure. For obvious reasons only the upper limit is of practical significance in reliability engineering. The application of the charts is straightforward. First, one matches the service life and s as close as possible to the values given for one of the figures. Second, after the sample's t m at To is determined, as in Figs. 1 through 3, fu and fl appropriate for the actual sample size, together with f, are extracted from the graph.
108
A. S. JOKI)AI\
16
011 _. .
I
I
III
I
IO4
IO3
MEDIAN Fig. 4a.
Failure
I
rate at
III
I
\
IO5
LIFE
OF SAMPLE
I
I
I
I
IO7
IO6 (HOURS)
I year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 0.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
Failure
109
rates
1 YEAR
IO4
1
16 30
0.1 - 1 IO3
MEDIAN Fig. 4b.
Failure
LIFE
rate at 1 year of service as a function
depicts the upper and lower confidence standard
OF
deviation
IO7
IO6
IO5
IO4
SAMPLE of median life.
(HOURS) The family of curves
limits on the failure rate at the 90% CL.
The
of the sample is 1. On each curve the sample size is labelled.
The
central curve is the conventional
lognormal
failure rate for the sample.
110
A. s. JORDAN
I
I
Ill
I
I
I
II\
I
III
I
I
III
I
1 YEAR s=1.5
1
MEDIAN Fig. 4c
Failure
rate at
LIFE
OF SAMPLE
(HOURS)
1 year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
I
III
Failure rates
IO5 _
I
I
I
III
I
I
III
I
I
II\
I
II\
I
1 YEAR s=2 IO4
IO31
F ix Y
P 100 2 Id [y:
3 zlJ_
10
1
0.'
MEDIAN Fig. 4d.
Failure
LIFE
OF SAMPLE
(HOURS)
rate at I year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
If
A. S. JORDAN
II2
toy
I
I
I
III
I
I
III
I
I
I
III
I
I
II
1 YEAR s=3
IO4
10
1
0.4
MEDIAN Fig. 4e.
Failure
rate at
LIFE
OF SAMPLE
(HOURS)
1 year of service its a function of median \ife. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 3. On each curve the sample size is labelled.
central curve is the conventional lognormal failure rate for the sample.
I
II
FAILURE
RATE
(FITS)
A. S. JORDAN
114
60 100 100
1
0. 1 II 5
Fig. 5b.
I
I
III
I
I
Ill
I
lo5 IO4 MEDIAN LIFE OF SAMPLE Failure
rate at 5 years of service as a function
I
I
,
II
\
’
‘I
IO7
IO6 (HOURS)
of median life.
The family of curves
depicts the upper and lower confidence limits on the failure rate at the 9070 CL.
The
standard
The
deviation of the sample is 1.0. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
115
Failure rates
10%
’
I
III
I
I
I
III
I
I
III
I
Ill1
I
II
5 YEARS s= 1.5
0.1 IO5
IO6
MEDIAN Fig. SC.
Failure
IO7
LIFE
OF SAMPLE
rate at 5 years of service as a function
10'0
IO9
IO8
(HOURS)
of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
A.
’
s. JORDAN
’
“I
“I
’
“I
5 YEARS s=2
0.1
I
I
’ \ \\Y,
III
IO5
I
106 MEDIAN
Fig. 5d.
Failure
I
I
III
IO7 LIFE
I
I
I
\
I
108
OF SAMPLE
I
il
10'0 (HOURS)
rate at 5 years of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional lognormal failure rate for the sample.
117
Failure rates
105_
1
I
I
I(
I
I
I
II
I
I
5
I
II
I
I
I
III
YEARS s=3
P 2
100
W
a
3 ii! 10
MEDIAN Fig. 5e.
Failure
LIFE
OF SAMPLE
(HOURS)
rate at 5 years of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 3. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
II
A. S.
JORDAN
10! YEARS s =0.5 IO'
1
o?
IOC
IC
1
IO3
IO4 MEDIAN
Fig. 6a.
105
LIFE
OF SAMPLE
IO6
(HOURS)
Failure rate at 10 years of service as a function of median life. The family of curves depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 0.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
119
Failure rates
10
YEARS S=l
0.1 IO5
IO6 MEDIAN
Fig. 6b.
Failure
LIFE
depicts the upper and lower confidence standard central
OF SAMPLE
rate at 10 years of service as a function
deviation
(HOURS)
of median life.
of curves The
of the sample is 1. On each curve the sample size is labelled.
The
lognormal
on the failure
The family
rate at the 90% CL.
curve is the conventional
limits
lo'*
IO9
IO8
IO7
failure
rate for the sample.
A. S.
120
IO5
I
I
III
I
I
JORDAN
I
III
I
III
I
I
III
I
10 YEARS s= 1.5
404
IO3
F z
V
g
100
r
E 3
2
10
1
0.1 _ IO5 MEDIAN Fig. 6~.
LIFE OF SAMPLE
(HOURS)
Failure rate at 10 years of service as a function of median life. The family of curves depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
I
I
Failure rates
105_
I
I
III
I
I
III
I
I
III
10
I
I
I
III
YEARS s=2
IO4 r
s
10I3
ILL_ -
w
5 ml00 ii 2 z
LL
I
10 /
1
0.1
MEDIAN Fig. 6d.
LIFE
OF SAMPLE
(HOURS)
Failure rate at 10 years of service as a function of median life.
The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
II
A. S. JORDAN
122
IO5
10 YEARS
0.1 IO5
IO6 MEDIAN
Fig. 6~.
Failure
rate at 10 years
depicts
the upper
standard central
deviation
108 OF SAMPLE
IO7 LIFE
of s&vice
as a function
and lower confidence of the sample
curve is the conventional
is 3.
limits
of median
on the failure
On each curve
lognormal
failure
life.
IO9 (HOURS) The family
of curves
rate at the 90% CL.
the sample
size is labelled.
rate for the sample.
The The
10’0
Failure rates
123
IV. Application of tbe Charts and Discussion
Preliminary assessment of long-term device reliability can be readily accomplished by means of the rapid lognormal projection nomograms.
As an illustrative example, let us assume that in a
sample of 100 devices stress-aged at 180°C one device fails after 4000 hours.
From previous
experience on similar devices it is known that s ----- 3 and Ea -----0.86 eV. (5)'(1°) The question to be addressed is the projected failure rate at To - 100"C and 5 years of service. To proceed, first with the aid of the Arrhenius lines plotted in the top right corner (Region 1) in Fig. 2, a time to failure of 6.5 × 10~h at 100"C is predicted. Then, from the C F F curves in the bottom left corner (Region I) t m - 1500 x 6.5 x 103 ----.9.8 x 108h at CFF -
1% is determined.
Finally, the appropriate
failure rate curve (Region 11I) provides f - 10 FITs. Note that the order of using regions I and i i can be interchanged for the Arrhenius equation extrapolates t and tra identically if s is independent of temperature. In particular, one can first obtain t m at the aging temperature in region 11 and then continue to find tm at To in region I. One may also wish to determine the time when the second failure is likely to occur at the same aging temperature, assuming that the previous estimates are correct.
From region I! one gets at
C F F - 2% the ratio tm/t - 520 or t - 1.88 x 106h at 100"C. Then, operating in region 1, one has t - l1600h for the second failure at 1180°C. If, as is usual, aging at an additional temperature is also performed, one would like to know the time to first failure, for example, in a group of 20 devices tested at 220°C. Assuming that the previously assigned parameters are valid at the higher aging temperature, one finds from tm - 9.8 × 108h at to for CFF - 5%, t - 6.5 x 106h (Region II). Subsequently, one obtains t -- 6000h for the time of first failure at 220"C (Region I).
The
examples encountered herein may be considered representative of the reliability observed in the area of GaAs FETs protected by SijN4 .(l°) Naturally, many other practical applications may be envisioned in utilizing the nomograms. If, as above, for a sample of 100 the Iognormal parameters at To are tr~ - 9.8 x 108h and s - 3, then at 5 years of service the upper and lower confidence limits on f at the 90% CL are 30 and 4 FITs, respectively (Fig. 5e). Obviously, the lower bound is not relevant to the systems designer. In reality, one should exercise caution in making this prediction, for the knowledge of the parameters implies that the bulk of the components in the sample of 100 has failed.
In fact, the R L P N
estimates were based on a single failure. Therefore, a conservative upper limit on the failure rate may be 100-200 FITs which corresponds to a diminished apparent sample size of lx/]"~'- 10. Otherwise, the application of the error bounds on the failure rate to particular reliability problems is straightforward. Admittedly, the calculation of the confidence interval for the lognormal failure rate presented here is only approximate. A more accurate determination of the error bounds would require very
!24
A . S . JORDAN
extensive computer simulations employing the Monte Carlo technique. Nevertheless, the charts are quite reasonable on the whole as can be seen by noting the effect of the sundry parameters on the curves.
In particular, in the median life range of interest the confidence interval narrows with
sample size. Furthermore, as t m increases the failure rate limits broaden.
Finally, the confidence
band is only weakly sensitive to an increase in service life. ACKNOWLEDGEMENTS I am grateful to J. W. Nielsen for a critical reading of the manuscript.
REFERENCES [1]
See, for example, D. S. Peck and C. H. Zierdt, Jr., Proc. IEEE, 62, 185 (1974) and earlier references cited therein.
[2]
A.S. Jordan, Microelectronics and Reliability, 18, 267 (1978).
[3]
A~ S. Jordan, R. H. Peaker, R. H. Saul, H. J. Braun and H. H. Wade, Bell System Tech. Journal, 57, 2983 0978).
[4]
J.C. Irvin and A. Loya, Bell System Tech. Journal, 57, 2823 (1978).
[5]
A. S. Jordan, J. C. Irvin, and W. O. Schlosser, in "18th Annual Proceedings on Reliability Physics", p. 123 (1980).
[6]
A.S. Jordan and T. D. O'Sullivan, submitted to Microelectronics and Reliability.
[7]
J. Aitchison and J. A. C. Brown "The Lognormal Distribution", Cambridge University Press, Cambridge (1957).
[8]
Y.L. Maksoudian, "Probability and Statistics with Applications", International Textbook Co., Scranton, PA (1969).
[9]
H.
Fukui,
S~ H. Wemple,
J. C lrvin,
W.C. Niehaus,
J . C . M . Hwang,
H.M. Cox,
W. O. Schlosser and J. V. DiLorenzo, in. "lBth Annual Proceedings on Reliability Physics", p. 151 (1980).
0026-2714/8453.00 + .00 Pergamon Press Ltd.
Confidence Limits on the Failure Rate and a Rapid Projection Nomogram for the Lognormal Distribution A . S. J O R D A N BELL
LABORATORIES, Murray
Hill,
NEW JERSEY United States (Received
for publication
07974, of A m e r i c a . 19th S e p t e m b e r
1983)
ABSTRACT
In order to predict the sample failure rate from a limited number of early malfunctions, we have developed the rapid lognormal projection nomogram (RLPN) which we present at 1, 5, and 10 years of service. In the core of each diagram is a family of curves showing the variation of failure rate as a function of sample median life and standard deviation. Superimposed are Arrhenius plots for a series of activation energies, all referenced with respect to a fixed time. We also display the standard deviation as a function of relative median life, using cumulative failure as a parameter. The RLPN provides graphical means to estimate the failure rate at the planned service life and operating temperature in an efficient manner from the time to failure at the aging temperature and sample size for known or assumed values of the standard deviation and activation energy. To obtain the confidence limits for the failure rate of the lognormal distribution at the 90% confidence level, we have derived an approximate formula that relates the bounds to the service life and the sample's size, median life, and standard deviation. Then, by an appropriate selection of the independent variables and parameters, the confidence inter,~als are displayed as a series of curve families. The applications of the diagrams are discussed by means of illustrative examples taken from the field of GaAs FETs. I. INTRODUCTION The widespread applicability of the lognormal distribution in describing the long-term reliability of semiconductor devices has been amply demonstrated.
In addition to components based on Si and
Ge, <~) devices involving III-V compounds such as GaP opto-isolators
Accelerated aging
experiments often provide an abundance of time to failure data which make it possible to estimate the characteristic parameters of the Iognorma| distribution, namely median life and standard deviation, by means of a relatively direct procedure, camAdditionally, testing at different temperatures permits the evaluation of the activation energy via the Arrhenius law for rate processes so that the failure rate prevalent in field operation can be forecast. However, some high quality devices may not fail in sufficient numbers during the time allotted to testing which would warrant an exhaustive reliability analysis. Indeed, even if all the data could be accumulated within 6 months, the device engineer would need an earlier indication of the long-term performance of his product.
In a recent paper we have developed the assurance test matrix plan
that allocates prior to testing a group of devices undergoing accelerated aging in such a fashion that the optimum information with regard to their maximum failure rate in field operation can be gained from the time to first failure. ~6) However, after a few malfunctions are detected the designer needs a 101
102
A . S . JoRDAN
convenient tool to project the failure rate from the meager data, and furthermore, it is desirable to estimate the uncertainty associated with this key quantity. One of the objectives of this paper is to construct a set of charts that is suitable for the efficient estimation of the failure rate. The suggested graphical technique will be referred to as the rapid lognormal projection nomogram (RLPN).
The other concern of this study is the evaluation of the
confidence limits on the failure rate at t i e 90% confidence level. To meet these objectives, first we briefly review the important statistical relationships characteristic of the lognormal distribution. Then, the charts for the R L P N are generated. Second, we outline the methods used to compute from sample data the confidence limits on the median life and standard deviation as a function of confidence level. Next, an approximate formula is derived that yields the bounds on the failure rate in relation to service life, sample size, median life and standard deviation. Third, by an appropriate selection of the independent variables and parameters, we display the confidence limits in a series of curve families. Finally, the application of the diagrams is discussed by means of an illustrative example taken from the field of GaAs FETs.
I!. The Rapid Lognormal Projection Nomogram I. Lognormal Concepts The cumulative failure function (CFF) for the Iognormal distribution is related to the time to failure, t, by the expression 1 I CFF - -~- erf ~ In t/tin ~/2s
(1)
where s and tm are the standard deviation and median life of the sample, respectively. (2)'(7~ The abbreviation erf designates the error function.
Usually, lognormal behavior of a given device is
demonstrated by the linearity on probability paper of the observed log t versus CFF, derived from aging studies. In general, CFF is given by CFF -
n------L--t N+I
(2)
Where N is the sample size and nt is the total number of devices that failed at or before t. If tm has been obtained at 2 or more temperatures, then the activation energy, Ea, can be determined by linear regression; for it is usually postulated that the Arrhenius law for rate processes is applicable to degradation phenomena. (l)'(2)'(s) Accordingly, assuming constant s, one has
Ea
lntm - A + - kT where A is a constant.
(3)
Clearly, tmo at the operating junction temperature, To, can be readily
calculated from Eq. (3). At To, the failure rate, f(t) -- defined as the logarithmic time derivative of the complement to Eq. (1) -- is of the form
Failure rates
103
~/2 exp [ - 2s 2 (ln t/tmo) 21 f(t) "~
(4)
l ts erfc " ~ s lnt/tmo
where t denotes the service life and erfc is the error function complement.
10
'°5F I
TIME TO FAILURE (HOURS) '100 t03 104
300
200
a= t 0 3 < L~ >-
LAJ rr I-< rt"
~< t00 L~ h,n~ ,,,
100 :E
U.J h-
t0
n~ ._J
O.
t0
t00
103
20
104
tm/t t00
I
J
103
104
I
t05
i
I
I
I
106
107
t08
t09
MEDIAN LiFE Fig. 1.
J 10 t0
(HOURS)
The rapid Iognormal projection nomogram at l year of service, a. Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
2. Construction of the Charts
We present the R L P N at 1, 5 and 10 years of service in Figs. 1, 2, and 3, respectively. The core of each diagram is the log failure rate versus log median life family of curves, based on Eq. (4). These plots are virtually the same as the ones.given in Ref. 2 and as before s is employed as the parameter. The solid segments of the lines indicate that the failure rate is also the highest rate attained at any time up to the designated service life. (2) Also shown is a horizontal grid of dashed lines which is a reciprocal absolute temperature scale corresponding to the range 20" and 300"C in
104
A . S . JORDAN
20°C steps. In addition, in the top right corner we have generated Arrhenius plots according to Eq. (3), taking arbitrarily t m - 10Sh at T - 573*K (300°C) and E a between 0.5 and 4 eV. Lastly, in the lower left corner the nomogram includes curves constructed from Eq. (l) in the form s versus tm/t using CFF as a parameter. The following sequence of operations is recommended in using the RLPNs: 1. In Region l select the appropriate service life (1, 5 or 10 years).
2. Plot the last observed time to failure at the
appropriate aging temperature (X). 3. Draw a line through X parallel with a selected El ray, based on experience with regard to the activation energy of like devices. 4. Terminate the line at the operating temperature and read from the graph the time to failure at To, to. 5. Knowing N and n t calculate C F F via Eq. (2) and then find the best matching curve in the lower corner (Region I1) in order to locate the tmcft o value corresponding to the anticipated s. 6. Having thus obtained tmo, the
10
105
TIME TO FAILURE (HOURS) 100 103 104
300
t04 200 I-14U') rr
10 3
UJ >IJ
100 100 ~or w n
UJ I-<[ rr L~
10
hi I'--
:D J
'1
0.1
co
20
0 10
I
100
103
104
tmlt I
100
,
L
I
L
103
104
105
106
107
108
109
101o
MEDIAN LIFE (HOURS)
Fig. 2.
The rapid lognormal projection nomogram at 5 years of service, a, Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
Failure rates
105
failure rate at To can be evaluated at the assumed s from the central family of curves (Region l l l ) . In general, the application of the nomogt'ams should be reserved for very reliable devices exhibiting only a few failures after long testing, in cases when employing maximum likelihood and least-square techniques to obtain tm and s are unsuitable, c2)
10
•~05
TIME TO FAILURE (HOURS) 100 103 104
500
t04_ 200
o
w
1,1 tw
©
l---
~- 1 0 0 ~
100
w cr w
w F-
10
0,t
rr t.J o..
0.1
20 1
10
100 tm/t
103
104
I
I
I
I
1
L
I
L
i
100
103
104
10 5
t0 6
10 7
10 8
10 9
1010
MEDIAN LIFE (HOURS) Fig. 3,
The rapid Iognormal projection nomogram at 10 years of service, a. Failure rate as a function of median life. The label on each curve is the standard deviation of the sample. b. Standard deviation versus relative median life (bottom left-hand corner). parameter displayed is the CFF.
The
c. Junction temperature versus time to failure or
median life (top right-hand corner). Each straight-line represents a different activation energy.
III. Confidence Limits on the Failure Rate. !. Derivation of the approximate Equations. For a systems designer the key property by means of which to assess the long-term performance of an electronic component is the failure rate.
Nonetheless, investigators usually refrain from
estimating confidence limits on f. Perhaps this is not surprising as exact bounds on f at a given confidence level are lacking in the case of the lognormal distribution.
,
106
A.S.
JORDAN
There is a need for confidence limits because estimated values of tm and s are only valid for a sample of size N and thus there must be a degree of assurance that any other sample d r a w n from the same population will not lead to drastically different parameters. An exact theorem which holds for the Iognormal distribution states that if lntm and s are the m a x i m u m likelihood ( M L H ) estimators of a random sample of size N taken from a population with unknown u ( - Intmp) and ~r (standard deviation of population) then the random variable ( l n t m - u) /(s/x/-N') obtained by random sampling is distributed according to Student's -- t probability density function with N - I degrees of f r e e d o m J 7) Another exact theorem states that if lntm and s are the M L H estimators of a random sample of size N chosen from a population with unknown u and a then the random variable (N-1)s2/a 2 is X2 -- distributed with N - I degrees of freedom. (7) The m a t h e m a t i c a l properties of these two distributions have been widely studied. (9) Based on these statistical principles, in a previous paper we presented convenient charts for the lower and upper limits of the (a) median life [tm~(tm) l/s] and (b) relative variance [(a/s) 2] as a function of confidence level for sample sizes ranging from 2 to well in excess of 100. (2) Both quantities tm and s enter into the expression for the failure rate (Eq. (4)).
Substituting
trapu or tmpl where the subscripts u and 1 denote the upper and lower limits, respectively, at the 90% confidence level (CL) into Eq. (4) we would offer assurance that no more than 10 out of 100 samples d r a w n from the same universal population would have median lines outside the confidence interval bounded by the limits. If then ~ru and o'l at the 90% C L were also introduced in Eq. (4) we would simultaneously g u a r a n t e e that less than 10 out of 100 samples would have standard deviations outside the a ~ - a l range. But what one wants is the failure rate which is at least for 90 samples, out of a total of 100, is confined to some calculated interval. Clearly, in pairing trap and a limits at the 90% C L to estimate bounds on f one in fact would be insisting that for one hypothetical sample both quantities should attain their "worst" level at the same time. To be sure there is only a very remote chance given by
for this to occur simultaneously. Therefore, in order not to confine the bounds on the failure rate in an unrealistic manner, it is suggested that the confidence limits on f at the 90% CL should be estimated from the parameters
trnp and a taken at the 67% CL. For, then
and there is a 100 x ( I - 0 . 1 ) % - 90% chance that both parameters as well as f fall within the interval. Consequently, the upper and lower confidence limits for the failure rate at the 90% C L become
Failure rates
107
_~ ,/~'exp•[- 12~ fu(t) -~ta.
(In t/tmpl) 2] (6)
1
erfc . ~ / ~
lnt/tmpl
and [
1..~._(lnt/tmpu)2]
fj(t) x]-~tal erfc ~
1
(7) lnt/ttmpu
where au, al, tmpu, and tmpl are the population values at the 67% CL. 2. Calculation and Organization of the Failure Rate Limit Charts.
In order to calculate the confidence interval for the failure rate at the 90% CL from Eqs. (6) and (7) it is necessary to determine the population estimates tm#, tmpu, al and ~u as a function of the sample parameters t m and s. In an earlier paper we have related the population and sample properties for the Iognormal distribution to sample size and CL. (2) Denoting the plotted results for the lower and upper limit of median life by r I and r u and like limits for the standard deviation by ~ t and ~ u , respectively, we have trope " tm ~'~ (N,0.67)
(8a)
tmpl - t m v~ (N,0.67)
(8b)
au - s-w/~u(N,0.67)
(8c)
al - s . ~ ( N , 0 . 6 7 )
(8d)
and
A convenient way to organize the error bounds on the failure rate is to fix the service life t at 1, 5, and 10 years. Subsequently, at a given t the sample's standard deviation is set. Thus a separate family of curves is generated for s - 0.5, 1, 1.5, 2, and 3. Within each family the remaining unspecified variables in Eqs. (6), (7), and (8) are the sample size N and tm. Accordingly, in analogy with the failure rate projection in Figs: 1 through 3, the upper and lower confidence limits on f will be presented as a function of sample tin, using N as the parameter ranging from 4 to 100• In Figs. 4, 5, and 6 the confidence limits on the lognormal failure rate at the 90% CL are displayed at 1, 5, and 10 years of service. Parts of a, b, c, d, and e of each figure provide charts for s -- 0.5, 1, 1.5, 2, and 3, respectively. At the center of each curve family is the nominal f given by Eq. (4) for the t and s corresponding to that figure. For obvious reasons only the upper limit is of practical significance in reliability engineering. The application of the charts is straightforward. First, one matches the service life and s as close as possible to the values given for one of the figures. Second, after the sample's t m at To is determined, as in Figs. 1 through 3, fu and fl appropriate for the actual sample size, together with f, are extracted from the graph.
108
A. S. JOKI)AI\
16
011 _. .
I
I
III
I
IO4
IO3
MEDIAN Fig. 4a.
Failure
I
rate at
III
I
\
IO5
LIFE
OF SAMPLE
I
I
I
I
IO7
IO6 (HOURS)
I year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 0.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
Failure
109
rates
1 YEAR
IO4
1
16 30
0.1 - 1 IO3
MEDIAN Fig. 4b.
Failure
LIFE
rate at 1 year of service as a function
depicts the upper and lower confidence standard
OF
deviation
IO7
IO6
IO5
IO4
SAMPLE of median life.
(HOURS) The family of curves
limits on the failure rate at the 90% CL.
The
of the sample is 1. On each curve the sample size is labelled.
The
central curve is the conventional
lognormal
failure rate for the sample.
110
A. s. JORDAN
I
I
Ill
I
I
I
II\
I
III
I
I
III
I
1 YEAR s=1.5
1
MEDIAN Fig. 4c
Failure
rate at
LIFE
OF SAMPLE
(HOURS)
1 year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
I
III
Failure rates
IO5 _
I
I
I
III
I
I
III
I
I
II\
I
II\
I
1 YEAR s=2 IO4
IO31
F ix Y
P 100 2 Id [y:
3 zlJ_
10
1
0.'
MEDIAN Fig. 4d.
Failure
LIFE
OF SAMPLE
(HOURS)
rate at I year of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
If
A. S. JORDAN
II2
toy
I
I
I
III
I
I
III
I
I
I
III
I
I
II
1 YEAR s=3
IO4
10
1
0.4
MEDIAN Fig. 4e.
Failure
rate at
LIFE
OF SAMPLE
(HOURS)
1 year of service its a function of median \ife. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 3. On each curve the sample size is labelled.
central curve is the conventional lognormal failure rate for the sample.
I
II
FAILURE
RATE
(FITS)
A. S. JORDAN
114
60 100 100
1
0. 1 II 5
Fig. 5b.
I
I
III
I
I
Ill
I
lo5 IO4 MEDIAN LIFE OF SAMPLE Failure
rate at 5 years of service as a function
I
I
,
II
\
’
‘I
IO7
IO6 (HOURS)
of median life.
The family of curves
depicts the upper and lower confidence limits on the failure rate at the 9070 CL.
The
standard
The
deviation of the sample is 1.0. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
115
Failure rates
10%
’
I
III
I
I
I
III
I
I
III
I
Ill1
I
II
5 YEARS s= 1.5
0.1 IO5
IO6
MEDIAN Fig. SC.
Failure
IO7
LIFE
OF SAMPLE
rate at 5 years of service as a function
10'0
IO9
IO8
(HOURS)
of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
A.
’
s. JORDAN
’
“I
“I
’
“I
5 YEARS s=2
0.1
I
I
’ \ \\Y,
III
IO5
I
106 MEDIAN
Fig. 5d.
Failure
I
I
III
IO7 LIFE
I
I
I
\
I
108
OF SAMPLE
I
il
10'0 (HOURS)
rate at 5 years of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional lognormal failure rate for the sample.
117
Failure rates
105_
1
I
I
I(
I
I
I
II
I
I
5
I
II
I
I
I
III
YEARS s=3
P 2
100
W
a
3 ii! 10
MEDIAN Fig. 5e.
Failure
LIFE
OF SAMPLE
(HOURS)
rate at 5 years of service as a function of median life. The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 3. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
II
A. S.
JORDAN
10! YEARS s =0.5 IO'
1
o?
IOC
IC
1
IO3
IO4 MEDIAN
Fig. 6a.
105
LIFE
OF SAMPLE
IO6
(HOURS)
Failure rate at 10 years of service as a function of median life. The family of curves depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 0.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
119
Failure rates
10
YEARS S=l
0.1 IO5
IO6 MEDIAN
Fig. 6b.
Failure
LIFE
depicts the upper and lower confidence standard central
OF SAMPLE
rate at 10 years of service as a function
deviation
(HOURS)
of median life.
of curves The
of the sample is 1. On each curve the sample size is labelled.
The
lognormal
on the failure
The family
rate at the 90% CL.
curve is the conventional
limits
lo'*
IO9
IO8
IO7
failure
rate for the sample.
A. S.
120
IO5
I
I
III
I
I
JORDAN
I
III
I
III
I
I
III
I
10 YEARS s= 1.5
404
IO3
F z
V
g
100
r
E 3
2
10
1
0.1 _ IO5 MEDIAN Fig. 6~.
LIFE OF SAMPLE
(HOURS)
Failure rate at 10 years of service as a function of median life. The family of curves depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 1.5. On each curve the sample size is labelled.
central curve is the conventional
lognormal
failure rate for the sample.
I
I
Failure rates
105_
I
I
III
I
I
III
I
I
III
10
I
I
I
III
YEARS s=2
IO4 r
s
10I3
ILL_ -
w
5 ml00 ii 2 z
LL
I
10 /
1
0.1
MEDIAN Fig. 6d.
LIFE
OF SAMPLE
(HOURS)
Failure rate at 10 years of service as a function of median life.
The family of curves
depicts the upper and lower confidence limits on the failure rate at the 90% CL.
The
standard
The
deviation of the sample is 2. On each curve the sample size is labelled.
central curve is the conventional
lognormal failure rate for the sample.
I
II
A. S. JORDAN
122
IO5
10 YEARS
0.1 IO5
IO6 MEDIAN
Fig. 6~.
Failure
rate at 10 years
depicts
the upper
standard central
deviation
108 OF SAMPLE
IO7 LIFE
of s&vice
as a function
and lower confidence of the sample
curve is the conventional
is 3.
limits
of median
on the failure
On each curve
lognormal
failure
life.
IO9 (HOURS) The family
of curves
rate at the 90% CL.
the sample
size is labelled.
rate for the sample.
The The
10’0
Failure rates
123
IV. Application of tbe Charts and Discussion
Preliminary assessment of long-term device reliability can be readily accomplished by means of the rapid lognormal projection nomograms.
As an illustrative example, let us assume that in a
sample of 100 devices stress-aged at 180°C one device fails after 4000 hours.
From previous
experience on similar devices it is known that s ----- 3 and Ea -----0.86 eV. (5)'(1°) The question to be addressed is the projected failure rate at To - 100"C and 5 years of service. To proceed, first with the aid of the Arrhenius lines plotted in the top right corner (Region 1) in Fig. 2, a time to failure of 6.5 × 10~h at 100"C is predicted. Then, from the C F F curves in the bottom left corner (Region I) t m - 1500 x 6.5 x 103 ----.9.8 x 108h at CFF -
1% is determined.
Finally, the appropriate
failure rate curve (Region 11I) provides f - 10 FITs. Note that the order of using regions I and i i can be interchanged for the Arrhenius equation extrapolates t and tra identically if s is independent of temperature. In particular, one can first obtain t m at the aging temperature in region 11 and then continue to find tm at To in region I. One may also wish to determine the time when the second failure is likely to occur at the same aging temperature, assuming that the previous estimates are correct.
From region I! one gets at
C F F - 2% the ratio tm/t - 520 or t - 1.88 x 106h at 100"C. Then, operating in region 1, one has t - l1600h for the second failure at 1180°C. If, as is usual, aging at an additional temperature is also performed, one would like to know the time to first failure, for example, in a group of 20 devices tested at 220°C. Assuming that the previously assigned parameters are valid at the higher aging temperature, one finds from tm - 9.8 × 108h at to for CFF - 5%, t - 6.5 x 106h (Region II). Subsequently, one obtains t -- 6000h for the time of first failure at 220"C (Region I).
The
examples encountered herein may be considered representative of the reliability observed in the area of GaAs FETs protected by SijN4 .(l°) Naturally, many other practical applications may be envisioned in utilizing the nomograms. If, as above, for a sample of 100 the Iognormal parameters at To are tr~ - 9.8 x 108h and s - 3, then at 5 years of service the upper and lower confidence limits on f at the 90% CL are 30 and 4 FITs, respectively (Fig. 5e). Obviously, the lower bound is not relevant to the systems designer. In reality, one should exercise caution in making this prediction, for the knowledge of the parameters implies that the bulk of the components in the sample of 100 has failed.
In fact, the R L P N
estimates were based on a single failure. Therefore, a conservative upper limit on the failure rate may be 100-200 FITs which corresponds to a diminished apparent sample size of lx/]"~'- 10. Otherwise, the application of the error bounds on the failure rate to particular reliability problems is straightforward. Admittedly, the calculation of the confidence interval for the lognormal failure rate presented here is only approximate. A more accurate determination of the error bounds would require very
!24
A . S . JORDAN
extensive computer simulations employing the Monte Carlo technique. Nevertheless, the charts are quite reasonable on the whole as can be seen by noting the effect of the sundry parameters on the curves.
In particular, in the median life range of interest the confidence interval narrows with
sample size. Furthermore, as t m increases the failure rate limits broaden.
Finally, the confidence
band is only weakly sensitive to an increase in service life. ACKNOWLEDGEMENTS I am grateful to J. W. Nielsen for a critical reading of the manuscript.
REFERENCES [1]
See, for example, D. S. Peck and C. H. Zierdt, Jr., Proc. IEEE, 62, 185 (1974) and earlier references cited therein.
[2]
A.S. Jordan, Microelectronics and Reliability, 18, 267 (1978).
[3]
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