1. What It Models

Quantity: The failure rate, or hazard, $h (t)$ is the rate at which units that have survived to time $t$ fail in the next instant. It is a conditional rate, so the denominator counts only survivors, and it is related to the reliability function by $R (t) = exp [- H (t)]$ with $H (t) = \int h d t$ (NIST).
Bathtub curve: Plotted against age, the failure rate of many products falls during an early-failure period, stays roughly constant through the intrinsic period and rises in wear-out (NIST; TI). Early failures last weeks to a few months and are attributed to manufacturing defects, the intrinsic period is the useful life, and wear-out is degradation (NIST; TI).
Shapes: A Weibull hazard with shape $γ$ below 1 falls, with $γ$ = 1 it is the constant-rate exponential model, and above 1 it rises (NIST; the reading of the exponent is mine). A full bathtub is then a sum of three hazards, which I construct as a model and the sources do not give as a formula (background knowledge).
Constant-rate model: In the useful life the rate is a constant $λ$ , the life is exponential, $R = e^{- λ t}$ , the mean life is $1/ λ$ , and only the exponential has a constant rate (NIST).
FIT: The semiconductor industry quotes the constant rate in FIT, failures per $1 0^{9}$ device-hours, and the mean life as $1 0^{9} / FIT$ hours. A FIT is a rate and not an absolute number (TI).
Population and repairable systems: The hazard describes non-repairable populations. For repairable systems the quantity on the vertical axis is the repair rate, or rate of occurrence of failures, the derivative of the mean cumulative number of failures (NIST).
Scope: This page defines the failure rate, its units and the bathtub curve. The distributions (Weibull, lognormal) and the acceleration models (Arrhenius and others) have their own pages in this folder, and the mechanism pages under Stress Types say why and where each mechanism acts.

2. Usage

From test to field rate: An accelerated test at a stress temperature is converted to the use temperature with an acceleration factor, the equivalent device-hours are $N t A F$ , and the upper limit of the failure rate at a chosen confidence level follows from the chi-square value for $2 r + 2$ degrees of freedom (JESD85 as quoted by TI and Analog Devices). Industry typically uses 60 per cent confidence, and 90 per cent is also reported (TI; Analog Devices).
Reporting: A FIT figure is meaningful only with its activation energy, use temperature, confidence level, sample size, test hours and failure count, because each changes the number. The examples in the sources use 0.7 $eV$ , 55 $^{\circ} C$ use and 125 $^{\circ} C$ stress (TI; Analog Devices).
Zero failures: Modern qualification is typically accepted on zero failures (TI). The result is an upper bound that falls with sample size, and small samples give artificially high FIT rates (Analog Devices).
Mission profile: A product that spends time at several temperatures has a mean rate equal to the time-weighted sum of the rates at each temperature, which is not the rate at the average temperature (TI; Q6).
Converting: $λ_{F I T}$ times $8.76 \times 1 0^{- 6}$ gives the yearly failure probability in the small-rate limit (computed), so 100 FIT is 876 $ppm$ per year; the early-failure period is quoted as a cumulative fraction in $ppm$ (TI).
Locating a product on the curve: Divide the failures in an interval by the survivors and the interval length to estimate the hazard, and compare successive intervals; falling values mean early failures, flat values the useful life and rising values wear-out (NIST). Whether a given product follows the curve is a question for its own field data: a review of electronics field data and experiments argues that the bathtub model does not reliably predict hazard rates (IEEE Access), and the hard-drive data change shape from one product generation to the next (Backblaze).
Link to other pages: The distribution to fit to the intervals is chosen from the Weibull and lognormal pages, the temperature scaling from the acceleration-model pages, and the stress and place of failure from the mechanism pages under Stress Types.

Reliability Functions and the Hazard Rate

Name	Formula	Related Formulae	Variables/Constants
Reliability function, hazard rate and cumulative hazard	$R (t) = 1 - F (t) = exp [- H (t)]$ $h (t) = \frac{f ( t )}{R ( t )} = - \frac{d l n R ( t )}{d t}$	$H (t) = \int_{0}^{t} h (s) d s = - ln R (t)$ $AFR (T_{1}, T_{2}) = \frac{H ( T _{2} ) - H ( T _{1} )}{T _{2} - T _{1}}$	$R (t) : p r o babi l i t y t ha t a u ni t s u r v i v es t o t im e t, F (t) : c u m u l a t i v e d i s t r ib u t i o n f u n c t i o n o f t h e f ai l u r e t im e, f (t) : i t s d e n s i t y, h (t) : ha z a r d (f ai l u r e) r a t e, t h e in s t an t an eo u s r a t e o f f ai l u r e o f t h e s u r v i v or s, so a co n d i t i o na l r a t e p er u ni t t im e, H (t) : c u m u l a t i v e ha z a r d$ $A F R : a v er a g e f ai l u r e r a t e o v er t h e in t er v a l f r o m T_{1} t o T_{2}, w i t h A F R (T) = H (T) / T f r o m z er o (N I S T 8.1.2.3), t h e p a g e p r in t s F (t) = 1 - e^{H (t)}, w hi c h l a c k s t h e min u s s i g n in t h e e x p o n e n t (m y r e a d in g)$ $(co m p u t e d : f or a co n s t an t ha z a r d λ, H = λ t an d t h e a v er a g e r a t e o v er an y in t er v a l i s λ; f or a W e ib u l l o f s ha p e γ, A F R (T) = H (T) / T = h (T) / γ (m y d er i v a t i o n), so t h e a v er a g e o v er 0 t o T i s tw i ce t h e in s t an t an eo u s r a t e a t T w h e n γ = 0.5)$
Empirical hazard from population data	$\hat{h} = \frac{r}{N _{s} Δ t}$	$\hat{h}_{13} \approx \frac{r _{13}}{N _{12} \times 720 h}$	$\hat{h} : es t ima t e o f t h e ha z a r d a v er a g e d o v er an in t er v a l$ $r : u ni t s t ha t f ai l in t h e in t er v a l$ $N_{s} : u ni t s s t i l l w or k in g a t i t s s t a r t$ $Δ t : in t er v a l l e n g t h, in t h e N I S T e x am pl e N_{12} u ni t s s u r v i v e t o t h e s t a r t o f m o n t h 13 an d r_{13} f ai l d u r in g t ha t m o n t h o f 720 h, co v er in g t h e a g es 8640 t o 9360 h (N I S T 8.1.2.4), f ai l u r es a r e co u n t e d o n l y am o n g t h e s u r v i v or s$ $(co m p u t e d w i t h a ss u m e d co u n t s : N_{s} = 10000 an d r = 2 in 720 h g i v e 2.78 \times 1 0^{- 7} p er h, w hi c h i s 278 FIT)$
Exponential model, mean life and median	$R (t) = e^{- λ t}, h (t) = λ$	$MTTF = \frac{1}{λ}$ $t_{50} = \frac{l n 2}{λ}$ $σ^{2} = \frac{1}{λ ^{2}}$	$λ : co n s t an t f ai l u r e r a t e$ $M T T F : m e an t im e t o f ai l$ $t_{50} : m e d ian l i f e$ $σ^{2} : v a r ian ce (N I S T 8.1.6.1), t h e e x p o n e n t ia l i s t h e o n l y d i s t r ib u t i o n w i t h a co n s t an t f ai l u r e r a t e, H (t) = λ t, a W e ib u l l w i t h s ha p e 1 r e d u ces t o i t w i t h t h e sc a l e α = 1/ λ = M T T F (N I S T 8.1.6.2), t h e N I S T p a g e u ses M T T F f or t h e m e an an d d oes n o t u se M T B F$ $(co m p u t e d : t h e s u r v i v in g f r a c t i o n a t t = M T T F i s e^{- 1} = 36.8 p er ce n t, an d t h e m e d ian i s 0.693 o f t h e M T T F)$

The Bathtub Curve

Name	Formula	Related Formulae	Variables/Constants
Weibull hazard for the three regions of the bathtub	$h (t) = \frac{γ}{α} (\frac{t}{α})^{γ - 1}$ $R (t) = exp [- (\frac{t}{α})^{γ}]$	$\overset{ˉ}{t} = α Γ (1 + \frac{1}{γ})$ $γ < 1, γ = 1, γ > 1$	$γ : s ha p e$ $α : sc a l e (t h e c ha r a c t er i s t i c l i f e, a t w hi c h R = e^{- 1})$ $Γ : g amma f u n c t i o n$ $\overset{ˉ}{t} : m e an l i f e, t h e ha z a r d f a l l s w i t h t im e f or γ b e l o w 1 (e a r l y f ai l u r es), i s co n s t an t f or γ = 1 (in t r in s i c f ai l u r es) an d r i ses f or γ ab o v e 1 (w e a r - o u t), t h e e x p o n e n t γ - 1 b e in g r e a d f r o m t h e f or m u l a (N I S T 8.1.6.2, m y r e a d in g), r e g i o n s o f t h e c u r v e (N I S T 8.1.2.4; T I) : e a r l y f ai l u r e, a l so in f an t m or t a l i t y, l a s t in g w ee k s t o a f e w m o n t h s, a hi g h b u t r a p i d l y f a l l in g r a t e, e x p r esse d in ppm an d a tt r ib u t e d t o man u f a c t u r in g d e f ec t s (T I); in t r in s i c or s t ab l e f ai l u r e, r o ug h l y co n s t an t f or m os t o f t h e u se f u l l i f e, e x p r esse d a s λ in FIT (T I); w e a r - o u t, a r i s in g r a t e f r o m d e g r a d a t i o n, w h ose o n se t s h o u l d co m e a f t er t h e t a r g e t u se f u l l i f e (T I)$ $(co m p u t e d : a u ni t s u r v i v es t o t = α w i t h p r o babi l i t y e^{- 1} = 36.8 p er ce n t f or an y γ)$
Bathtub curve as a sum of three hazards	$h (t) = h_{in f} (t) + λ_{0} + h_{w} (t)$	$R (t) = exp [- H_{in f} (t) - λ_{0} t - H_{w} (t)]$	$h_{in f} : d ecr e a s in g e a r l y - f ai l u r e ha z a r d, f or e x am pl e a W e ib u l l w i t h γ b e l o w 1$ $λ_{0} : co n s t an t in t r in s i c r a t e$ $h_{w} : in cr e a s in g w e a r - o u t ha z a r d, f or e x am pl e a W e ib u l l w i t h γ ab o v e 1$ $H : t h e ma t c hin g c u m u l a t i v e ha z a r d s, in d e p e n d e n t c a u ses o f f ai l u r e a dd t h e i r ha z a r d s, w hi c h i s t h e v er t i c a l s u m o f t h e c u r v es (ma s k e d c a u ses v i g n e tt e), t h e t h r ee - t er m f or m f or a ba t h t u b i s m y co n s t r u c t i o n (ba c k g r o u n d k n o w l e d g e), I E E E A ccess (2021) : t h e ba t h t u b m o d e l i s a r g u e d n o t t o p r e d i c t ha z a r d r a t es r e l iab l y f or e l ec t r o ni c co m p o n e n t s, p r o d u c t s an d sy s t e m s, an d a p r eco n ce i v e d m o d e l o f t h e ha z a r d r a t e o v er t im e s h o u l d n o t b e u se d, B a c k b l a z e (2025) : t h e f ai l u r e r a t e o f ha r d d r i v es b y a g e w a s c l oses t t o a ba t h t u b in t h e 2013 d a t a, w i t h a p e ak o f 13.73 p er ce n t a y e a r a t ab o u t 3 y e a r s 3 m o n t h s, an d n e a r l y f l a t f or m os t o f t h e l i f e in t h e 2021 an d 2025 d a t a, w i t h a s t ee p r i se n e a r t h e e n d, a t ab o u t 10 y e a r s w i t h a p e ak o f 4.25 p er ce n t in 2025, N o w l an an d H e a p (a s r e p or t e d) : 4 p er ce n t o f t h e U ni t e d A i r l in es i t e m s f o l l o w e d t h e ba t h t u b p a tt er n$ $(co m p u t e d f or an a ss u m e d i l l u s t r a t i o n : e a r l y f ai l u r es w i t h γ = 0.5 an d α = 1.1 \times 1 0^{10} h, c h ose n so t ha t H (1000 h) = 300 ppm, pl u s 10 FIT an d w e a r - o u t w i t h γ = 4 an d α = 3 \times 1 0^{5} h; t h e t o t a l ha z a r d i s 1510 FIT a t 10 h, 160 FIT a t 1000 h, a minim u m o f 47 FIT a t 22000 h (2.5 yr), 93 FIT a t 50000 h an d 519 FIT a t 100000 h; t h e e a r l y - f ai l u r e t er m i s s t i l l 15 FIT a t 100000 h, ab o v e t h e co n s t an t 10 FIT, so t h e f l a t r e g i o n i s n o t f l a t)$
Series system of independent parts	$R_{sy s} (t) = \prod_{i} R_{i} (t)$ $λ_{sy s} = \sum_{i} λ_{i}$	$α_{sy s} = (\sum_{i} α_{i}^{- γ})^{- 1/ γ}$ $MTTF_{sy s} = \frac{1}{λ _{sy s}}$	$R_{i} an d λ_{i} : r e l iabi l i t y an d co n s t an t f ai l u r e r a t e o f p a r t i, a ser i es sy s t e m f ai l s w h e n i t s f i r s t p a r t f ai l s$ $α_{sy s} : sy s t e m sc a l e i f a l l p a r t s a r e W e ib u l l w i t h t h e s am e s ha p e γ, t h e sy s t e m ha z a r d i s t h e s u m o f t h e co m p o n e n t ha z a r d s (ma s k e d c a u ses v i g n e tt e) an d t h e p r o d u c t f or m an d t h e co mm o n - s ha p e sc a l e f o l l o w f r o m R = e^{- H} (m y d er i v a t i o n)$ $(co m p u t e d w i t h a ss u m e d p a r t s : 100, 50 an d 20 FIT a dd t o 170 FIT, an M T T F o f 5.88 \times 1 0^{6} h; t h r ee p a r t s w i t h γ = 2 an d α = 1 \times 1 0^{5}, 2 \times 1 0^{5} an d 3 \times 1 0^{5} h g i v e α_{sy s} = 8.57 \times 1 0^{4} h)$

FIT, Confidence Limits and Acceleration

Name	Formula	Related Formulae	Variables/Constants
FIT, MTTF and the probability of failure in a year	$λ_{F I T} = 1 0^{9} λ$ $MTTF = \frac{1 0 ^{9}}{λ _{F I T}} h$	$P_{y r} = 1 - exp (- λ t_{y r}) \approx λ t_{y r}$ $t_{y r} = 8760 h$	$λ : f ai l u r e r a t e p er h o u r$ $λ_{F I T} : t h e s am e in F I T, f ai l u r es p er 1 0^{9} d e v i ce - h o u r s$ $M T T F : m e an t im e t o f ai l f or a co n s t an t r a t e (T I : M T B F = M T T F in t ha t c a se)$ $P_{y r} : p r o babi l i t y t ha t a u ni t f ai l s in a y e a r o f co n t in u o u s u se$ $t_{y r} : h o u r s in a y e a r (8760 a ss u m e d)$ $T I : a F I T i s a r a t e an d n o t an ab so l u t e n u mb er, so i t ha s t o b e a g g r e g a t e d o v er t h e l i f e o f t h e p r o d u c t, an d 60 p er ce n t i s t h e u s u a l co n f i d e n ce l e v e l in in d u s t r y$ $(co m p u t e d : 1, 10, 100 an d 1000 F I T g i v e an M T T F o f 1.0 \times 1 0^{9}, 1.0 \times 1 0^{8}, 1.0 \times 1 0^{7} an d 1.0 \times 1 0^{6} h, an d a y e a r l y f ai l u r e p r o babi l i t y o f 8.76, 87.6, 876 an d 8722 ppm; 1 p er ce n t p er 1000 h i s 10000 F I T)$
Upper confidence limit of the failure rate from a stress test	$λ_{U C L} = \frac{χ ^{2} ( C L , 2 r + 2 )}{2 N t A F} \times 1 0^{9} [FIT]$ $A F = exp [\frac{E _{a}}{k _{B}} (\frac{1}{T _{u se}} - \frac{1}{T _{s t r ess}})]$	$χ^{2} (C L, 2) = - 2 ln (1 - C L)$ $N_{min} = \frac{χ ^{2} ( C L , 2 )}{2 t A F λ _{t a r g e t}}$	$λ_{U C L} : u pp er l imi t o f t h e f ai l u r e r a t e a t t h e co n f i d e n ce l e v e l C L, χ^{2} (C L, 2 r + 2) : c hi - s q u a r e v a l u e f or 2 r + 2 d e g r ees o f f r ee d o m$ $r : n u mb er o f f ai l u r es$ $N : n u mb er o f u ni t s in t h e t es t$ $t : t es t d u r a t i o n in h o u r s$ $A F : a cce l er a t i o n f a c t or f r o m t h e t es t t o t h e u se t e m p er a t u r e, N t A F : e q u i v a l e n t d e v i ce - h o u r s$ $E_{a} : a c t i v a t i o n e n er g y$ $k_{B} : B o l t z mann co n s t an t 8.617 \times 1 0^{- 5} eV/K (T I u ses 8.6 \times 1 0^{- 5})$ $T_{u se} an d T_{s t r ess} : u se an d s t r ess t e m p er a t u r es in K$ $λ_{t a r g e t} : t a r g e t r a t e p er h o u r, t h e f or m f o l l o w s J E S D 85 a s q u o t e d b y T I an d A na l o g D e v i ces, t h e c l ose d f or m f or z er o f ai l u r es f o l l o w s f r o m t h e c hi - s q u a r e d i s t r ib u t i o n w i t h 2 d e g r ees o f f r ee d o m (m y d er i v a t i o n), T I e x am pl e : E_{a} = 0.7 eV, 125^{\circ} C an d 1000 h, 231 u ni t s, n o f ai l u r es, 60 p er ce n t, u se a t 55^{\circ} C, g i v in g 50.9 F I T w i t h A F ab o u t 78.6, an d a t ab l e o f 12.8, 3.3 an d 1.6 F I T f or 922, 3600 an d 7500 u ni t s (127, 32, 8.2 an d 3.9 F I T a t 90 p er ce n t f or 231, 922, 3600 an d 7500 u ni t s), A na l o g D e v i ces : A F o f 258 f or 150^{\circ} C, 128 f or 135^{\circ} C, 77 f or 125^{\circ} C an d 8 f or 85^{\circ} C, e a c h t o 55^{\circ} C, an d i t w a r n s t ha t t h e m e t h o d g i v es a r t i f i c ia l l y hi g h F I T r a t es f or s ma l l s am pl es$ $(co m p u t e d w i t h E_{a} = 0.7 eV an d k_{B} = 8.617 \times 1 0^{- 5} eV/K : A F = 259, 128.0, 77.65 an d 7.95 f or t h e f o u r s t r ess t e m p er a t u r es; χ^{2} (0.60, 2) = 1.833 an d χ^{2} (0.90, 2) = 4.605; 231 u ni t s f or 1000 h g i v e 17.9 \times 1 0^{6} e q u i v a l e n t d e v i ce - h o u r s an d 51.1 F I T, w i t h 50.5 F I T f or A F = 78.6; t h e o t h er s am pl e s i z es g i v e 12.8, 3.28 an d 1.57 F I T a t 60 p er ce n t an d 128, 32.2, 8.24 an d 3.95 F I T a t 90 p er ce n t; o n e an d tw o f ai l u r es r ai se t h e 231 - u ni t l imi t t o 113 an d 173 F I T; a t a r g e t o f 10 F I T n ee d s 1180 u ni t s a t 60 p er ce n t or 2966 u ni t s a t 90 p er ce n t)$
Mission profile and effective acceleration	$A F_{e f f} = \frac{1}{\sum _{i} a _{i} / A F _{i}}$ $λ_{mi ss i o n} = \sum_{i} a_{i} λ_{i}$	$\sum_{i} a_{i} = 1$	$a_{i} : f r a c t i o n o f t h e mi ss i o n t im e s p e n t a t t e m p er a t u r e i$ $A F_{i} : a cce l er a t i o n f a c t or f r o m t h e t es t t o t ha t t e m p er a t u r e$ $λ_{i} : f ai l u r e r a t e e v a l u a t e d a t t ha t t e m p er a t u r e$ $A F_{e f f} : e f f ec t i v e a cce l er a t i o n f or t h e w h o l e p r o f i l e (T I E q . 3 a s r eco n s t r u c t e d in t h e s u mma r y), T I e x am pl e : a p r o f i l e g i v in g 1.90 F I T in t o t a l, a g ain s t 19.88 F I T i f t h e w or s t c a se o f 85^{\circ} C h e l d f or t h e w h o l e l i f e, w hi c h a ppl i es t o o n l y 2 p er ce n t o f i t, g i v in g an M T T F o f ab o u t 5.26 \times 1 0^{8} h (T I)$ $(co m p u t e d : 2 p er ce n t o f t h e t im e a t 85^{\circ} C a dd s 0.398 F I T t o t h e 19.88 F I T r a t e (T I); f or an a ss u m e d p r o f i l e o f 98 p er ce n t a t 55^{\circ} C an d 2 p er ce n t a t 85^{\circ} C t h e m e an r a t e i s 1.139 t im es t h e r a t e a t 55^{\circ} C, b ec a u se t h e r a t e a t 85^{\circ} C i s 7.95 t im es hi g h er, w h er e a s t h e a v er a g e t e m p er a t u r e o f 55.6^{\circ} C w o u l d g i v e o n l y 1.046 t im es)$

Important Questions

Ques: Do electronic devices follow the bathtub curve, and is its flat region really flat?

NIST calls the bathtub curve an empirical shape for many mechanical and electronic products and applies it to repairable systems through the repair rate. An IEEE Access assessment finds that the model does not reliably predict hazard rates in electronics, Backblaze hard-drive data are nearly flat for most of the life in 2021 and 2025, and only 4 per cent of the United Airlines items of Nowlan and Heap followed it. These populations differ, so they need not conflict with NIST (my reading). In my assumed illustration the total hazard has a minimum of 47 $FIT$ at 22000 $h$ and the early term is still 15 $FIT$ at 100000 $h$ (computed), so the flat region is only approximate. I regard the curve as a model of three competing causes and not a law, and a constant FIT as valid only where it was measured (my reading). Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1 (pages 8.1.2.3 failure rate, 8.1.2.4 bathtub curve, 8.1.2.5 repair rate, 8.1.6.1 exponential and 8.1.6.2 Weibull distributions) (fetch-tool summaries; the page on the relations of the reliability functions was not available); ‘An Assessment of Validity of the Bathtub Model Hazard Rate Trends in Electronics’, IEEE Access, January 2021 (abstract only, read in the DOAJ record, which does not list the authors); Backblaze Drive Stats Team, ‘Are Hard Drives Getting Better? Let’s Revisit the Bathtub Curve’, 15 October 2025 (fetch-tool summary; the author notes that bulk purchasing, a different decommissioning process and changing drive standards can skew the data); F. S. Nowlan and H. F. Heap, ‘Reliability-Centered Maintenance’, United Airlines for the US Department of Defense, 29 December 1978, AD-A066579, Exhibit 2.13, as reported in ‘The Six Failure Patterns: Nowlan and Heap Explained’, Reliable Media (September 2026) (secondary source; the report itself was not read).

Ques: Is the failure rate of 231 devices 50.9 FIT, and is the acceleration factor 77, 77.65 or 78.6?

TI gives 50.9 FIT for 231 units, 1000 $h$ at 125 $^{\circ} C$ , no failures, 60 per cent confidence and use at 55 $^{\circ} C$ , with an acceleration factor of about 78.6. Analog Devices lists 77. With $k_{B}$ = $8.617 \times 1 0^{- 5}$ $eV/K$ and 273.15 $K$ I get 77.65 and 51.1 FIT (computed). The factor 78.6 follows only from $k_{B}$ = $8.6 \times 1 0^{- 5}$ with 273 $K$ , which gives 50.5 FIT, while 50.9 FIT needs 77.9, which is the exact constant with 273 $K$ (computed). So TI appears to mix constants, which is my reading. The differences are 1.2 per cent and do not matter for a bound. I regard 77.65 as the correct factor and 51 FIT as the result. Equivalent device-hours are $17.9 \times 1 0^{6}$ . Reference: Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary); Analog Devices, ‘Details of Reliability Calculations’ (wafer-fabrication data page) (fetch-tool summary; chi-square values not listed).

Ques: Why does the 90 per cent confidence limit come out 2.5 times the 60 per cent one?

For zero failures the chi-square value with 2 degrees of freedom is $- 2 ln (1 - C L)$ , which is 1.833 at 60 per cent and 4.605 at 90 per cent (computed), a ratio of 2.51. TI’s 231-unit values of 50.9 and 127 FIT give 2.50, and the other rows agree. With a failure the value for 4 degrees of freedom raises the limit from 51 to 113 FIT, and for two failures to 173 FIT (computed). So the confidence level is a choice that changes the number by a factor, not a property of the device. Analog Devices warns that small samples give artificially high rates, which is expected since the limit scales as $1/ N$ . A target of 10 FIT needs 1180 units at 60 per cent and 2966 units at 90 per cent (computed). I regard the level as something to state in every FIT figure. Reference: Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary); Analog Devices, ‘Details of Reliability Calculations’ (wafer-fabrication data page) (fetch-tool summary; chi-square values not listed).

Ques: What does a FIT value mean for a single unit: does 100 FIT mean it lasts 1141 years?

For a constant rate the mean life is $1 0^{9} / FIT$ hours, which is $1.0 \times 1 0^{7}$ $h$ or 1141 years at 100 FIT (computed). That does not mean that one unit lasts that long. The yearly failure probability is 876 $ppm$ (computed), and the fraction surviving to the mean life is $e^{- 1}$ = 36.8 per cent, with the median at 0.693 of it (computed). The figure holds only in the useful life, before wear-out. TI states that a FIT is a rate and not an absolute number, and that it needs to be aggregated over the life of the product. I regard the yearly probability, or the cumulative fraction over the mission, as the number to quote to a customer, and the mean life as a way of restating the rate (my reading). Reference: Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1 (pages 8.1.2.3 failure rate, 8.1.2.4 bathtub curve, 8.1.2.5 repair rate, 8.1.6.1 exponential and 8.1.6.2 Weibull distributions) (fetch-tool summaries; the page on the relations of the reliability functions was not available).

Ques: Is the mean life MTTF or MTBF, and is the failure rate the same as the repair rate?

The NIST page on the exponential distribution uses only MTTF, the mean time to fail, for the mean $1/ λ$ and does not use MTBF. TI uses both and states that for a constant failure rate MTBF equals MTTF. I regard them as the same number for a constant rate and as different in general, since MTBF for a repairable item includes the time to repair, which I take from background knowledge. NIST also separates the population failure rate $h (t)$ , a conditional rate for survivors, from the repair rate, the derivative of the mean cumulative number of failures for repairable systems. NIST does not say when the two are equal, so I leave that open. Reference: NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1 (pages 8.1.2.3 failure rate, 8.1.2.4 bathtub curve, 8.1.2.5 repair rate, 8.1.6.1 exponential and 8.1.6.2 Weibull distributions) (fetch-tool summaries; the page on the relations of the reliability functions was not available); Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary).

Ques: Why is the average temperature the wrong temperature for a mission profile?

The rate depends exponentially on the inverse temperature, so hot hours count more than cool ones. TI weights the rate at each temperature by the fraction of time and sums, and gives 1.90 FIT for its profile, against 19.88 FIT if 85 $^{\circ} C$ held throughout, which applies to 2 per cent of the life. For my assumed profile of 98 per cent at 55 $^{\circ} C$ and 2 per cent at 85 $^{\circ} C$ the rate at 85 $^{\circ} C$ is 7.95 times that at 55 $^{\circ} C$ , so the mean rate is 1.139 times, while the average temperature of 55.6 $^{\circ} C$ gives 1.046 times (computed). Using the average underestimates the excess by a factor of about 3 (computed: 0.139 against 0.046). I regard the time-weighted sum of rates as correct for a thermally activated mechanism with a fixed activation energy. Reference: Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary).

Ques: Why is early failure quoted in ppm and the useful life in FIT?

TI expresses early life in $ppm$ and the useful life as a constant rate in FIT. The reason is that the early hazard changes quickly, so no single rate describes it, while the cumulative fraction that fails by a given age is well defined. For a Weibull with shape $γ$ the average rate over 0 to $T$ is $h (T) / γ$ (computed from $H / T$ ), so with $γ$ = 0.5 it is twice the rate at $T$ . In my illustration the fraction failed by 1000 $h$ is 300 $ppm$ , the average rate over that time is 300 FIT and the instantaneous rate at its end is 150 FIT (computed, assumed parameters). I regard the cumulative fraction as the right quantity for the early period, and the instantaneous hazard as the right one for the flat region (my reading). Reference: Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary); NIST/SEMATECH e-Handbook of Statistical Methods, Section 8.1 (pages 8.1.2.3 failure rate, 8.1.2.4 bathtub curve, 8.1.2.5 repair rate, 8.1.6.1 exponential and 8.1.6.2 Weibull distributions) (fetch-tool summaries; the page on the relations of the reliability functions was not available).

Ques: Is 1000 hours at 125 degrees Celsius the same as ten years at 55 degrees Celsius?

Analog Devices describes its baseline as 1000 $h$ at 125 $^{\circ} C$ translated to 10 years at 55 $^{\circ} C$ , and lists an acceleration factor of 77. That gives 77000 $h$ , or 8.79 years, and 77.65 gives 8.86 years (computed). Ten years, or 87600 $h$ , needs a factor of 87.6 or 1128 $h$ of test (computed). I regard 8.8 to 9 years as the figure and ten years as a rounded claim, which is my reading. The equivalence also holds only for a mechanism with an activation energy of 0.7 $eV$ and a constant rate. It says nothing about wear-out beyond the test time, since TI requires that wear-out should begin after the target life, and a 1000 $h$ test cannot show that. Mechanisms with other activation energies are scaled by their own factors. Reference: Analog Devices, ‘Details of Reliability Calculations’ (wafer-fabrication data page) (fetch-tool summary; chi-square values not listed); Texas Instruments application report SPRABY3, FIT rate and MTTF calculation for semiconductor devices (year not read; Equations 2 and 3 were garbled in the summary and were reconstructed there) (fetch-tool summary).

Quartz 5

Explorer

Failure Rate and the Bathtub Curve