A new reliability growth model: its mathematicalcomparison to the Duane model

John Donovana,*, Eamonn Murphyb

aInstitute of Technology, School of Engineering, Sligo, IrelandbNational Centre for Quality Management, University of Limerick, Limerick, Ireland

Received 8 July 1999; received in revised form 18 October 1999

Abstract

The Duane reliability growth model has been traditionally used to model electronic systems undergoingdevelopment testing. This paper proposes a new reliability growth model derived from variance stabilisationtransformation theory which surpasses the Duane model in typical reliability growth situations. This new model is

simpler to plot and ®ts the data more closely than the Duane model whenever the Duane slope is less than 0.5. Thispaper explores the mathematical relationships between these two models; and shows that at a Duane slope of 0.5,both models are mathematically equivalent in their capacity to ®t the observed data. The instantaneous MTBF of

the new model is also developed and compared to that of Duane. As the new model is in¯uenced by the laterfailures, compared to early failures for the Duane model, it has the further advantage of leading to reduced testtimes for achieving a speci®ed instantaneous MTBF. As the reliability of electronic systems increases, this has

positive implications for testing. # 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Many reliability growth models have been proposedfor monitoring and tracking reliability improvement

during product development. Duane's model [1] hasremained one of the primary graphical models and wasinitially developed as an empirical model based on ob-servations by Duane over a number of projects. This

model represents a relationship between the cumulativeMTBF �y� and the cumulative test time (T ) such thaty � a1T b1 : ln a1 and b1 represent the intercept and

Duane slope, respectively, as seen from the followinglog±log model.

ln�y� � ln a1 � b1 ln�T� �1�

Data following this relationship, when plotted on log±

log paper, falls on a straight line. Crow [2] showed

that this empirical model is essentially a Nonhomoge-

neous Poisson Process with a Weibull intensity func-

tion. Donovan and Murphy [3] more recently, have

formulated a new reliability growth model which was

derived from variance stabilisation theory for re-

gression analysis problems. This model represents an

improvement in Duane's model for reliability growth

situations and yet has certain mathematical similarities,

especially when the Duane slope equals 0.5. These

improvements and mathematical similarities are

explored in this paper.

Microelectronics Reliability 40 (2000) 533±539

0026-2714/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved.

PII: S0026-2714(99 )00235-8

www.elsevier.com/locate/microrel

* Corresponding author. Tel.: +353-71-55231; fax: +353-

71-55390.

E-mail address: donovan.john@itsligo.ie (J. Donovan).

2. The new reliability growth model

The precedent of plotting of cumulative MTBF on

the y-axis and cumulative time on the x-axis is contin-ued in the new model. As each failure occurs in ac-cordance with a Poisson process, the cumulative time

can be viewed as a ``count'' of the number of hoursduring which a number of failures have occurred, or inother words, a count of the time in which a speci®c

cumulative MTBF has been reached. Variance stabilis-ation transformation theory suggests that for a Poissonprocess, the transformation equals the square root ofthe count. Therefore, in the case of cumulative time,

the appropriate transformation for the x-axis is thesquare root of the cumulative time.An advantage of this model is that there is no need

to transform the y-axis so that the cumulative MTBFis plotted directly without transformation. The modeltherefore becomes

y � a2 � b2����Tp

�2�where y and T represent the cumulative MTBF andcumulative test time respectively, and a2 and b2 rep-resent the intercept and slope, respectively.

This model bears a resemblance to the Duane modelwhen the Duane slope �b1� is 0.5, but has a number ofadvantages when used for reliability growth plotting.

3. Advantages of the new model over Duane's model

Many of the advantages of the new model over

Duane's model have been simulated and presented inRef. [3]. These advantages can be summarised as fol-lows:

. In the Duane model, early failures have a high in¯u-ence on the resulting model as measured by theCook's distance �Di� statistic which combines both

the leverage and in¯uence of each failure. If theDuane model is used to observe growth during test-ing, then the resulting graph is overly a�ected by

those failures occurring early in time. This does notoccur in the new model.

. In the Duane model, failures occurring towards thelatter part of the test tend to cluster together due to

the nature of the ln(Cumulative Time ). This arises asthere is little di�erence between the natural logs oflarge values of cumulative time. The clustering e�ect

is avoided in the new model as the log scales areavoided.

. Simulation has shown that the latter failures have

the greatest in¯uence and leverage on the newmodel. The early failures have little in¯uence orleverage on this model.

. The simplicity of the new model is evident as thereis no requirement to transform the y-axis resulting

in the further advantage of reading the cumulativeMTBF directly from the graph. The new modelthereby produces a graphical display that is easier to

plot, interpret and visualise.. Simulation has shown that new model provides a

better ®t to the data when the Duane slope is less

than 0.5. Above 0.5, the Duane model tends to pro-vide a better ®t to the data. This is not too great ahindrance as many reliability growth programs have

Duane slopes less than 0.5.

From an estimation and extrapolation point of view,the new model adopts the most pragmatic approach

with failures. It is in¯uenced by the latest failures,which should realistically be more important than theearliest failures. It is unrealistic to believe that afterpossibly 100,000 h of testing, the earliest failure

remains the most in¯uential. This de®ciency is largelyovercome in the Duane model by omitting the earliestfailures [4,5]. This becomes unnecessary with the new

model, which has the added advantage of plotting allthe failure data.

4. Model comparisons on di�erent scales

The simplest measure of model adequacy is the

Coe�cient of Determination (R 2), and is a measure ofhow well the model ®ts the data. It represents the per-centage variability in y explained by x. It is importantto remember, however, that the R 2 of respective

models provides an inappropriate comparison if themodel y-axes di�er.As the new model has completely di�erent axes

from the Duane model, the method chosen to compareboth models is from Hamilton [6]. This requires ®nd-

Fig. 1. Di�erence in the R�2 of both models �R�2Diff � related to

the Duane slope.

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539534

ing the coe�cient of determination between y and y:In the case of a transformed y, this implies to ®rst

apply the inverse transformation to obtain y values. Inthe Duane model, this means calculating eln�Cum:M ÃTBF�

for the various failures. One then evaluates the coe�-

cient of determination between these predicted valuesand the y values (i.e. the observed cumulative MTBF).It has been decided to call this as R�2 to distinguish it

from the R2 described earlier. In this respect, R�2 is ameasure of the closeness of the predicted to the actual.The better model is considered to be the model with

the highest R�2:The di�erence in the R�2 values for the Duane and

new model arising from 6200 simulated datasets isshown in Fig. 1. A positive di�erence indicates that the

Duane model provides a better ®t to the data, while anegative di�erence indicates that the new model pro-vides a better ®t. The further the Duane slope deviates

from 0.5, the more likely will be the di�erence in therespective R�2 of both models. The Duane model isbetter at higher Duane slopes, while the new model is

better for the vast range of slopes typically observedduring a reliability growth programme.The equality of both models at a Duane slope of 0.5

is proven below.The new model in its general form is represented as:

y � a� bT g �3�

In the speci®c form identi®ed by variance stabilisationtransformations, where g is represented by 0.5, thismodel becomes:

ySq � a2 � b2T0:5 �4�

where ySq represents the predicted cumulative MTBF

of the new model at cumulative test time T.To distinguish between the two models, ySq and yDu

are used to signify the cumulative MTBF as predictedby the new model and Duane model, respectively. The

notation Sq is used for the new model as this modelcontains the Square Root of cumulative time.The Duane model is represented by:

yDu � a1T b1 �5�

The method by which the models are evaluated is to

compare the R�2 for the Duane and new model, re-spectively:

R�2Du � Coefficient of Determination between yDu and y

R�2Sq � Coefficient of Determination between ySq and y

where y is the observed cumulative MTBF.The equation for coe�cient of determination [7] is:

R2 �

Xni�1

ÿyi ÿ �y

�2Xni�1�yi ÿ �y�2

Therefore, for R�2Du,

R�2Du �

Xni�1

�yDuiÿ �y

�2Xni�1

ÿyi ÿ �y

�2 �6�

where yi represents the cumulative MTBF for each

failure i from 1 to n.yDu represents the regression line of y on yDu as illus-

trated in Fig. 2, while yDuirepresents the predicted

cumulative MTBF for each failure i. �y is the mean of

the observed cumulative MTBF and can be evaluatedas:

�y � 1

n

Xni�1

Ti

i

yDu is represented as the straight line:

yDu � a3 � b3yDu �7�Combining Eqs. (7) and (5), yields:

yDu � a3 � b3a1Tb1 �8�

Combining Eqs. (8) and (6) yields:

R�2Du �

Xni�1

�a3 � b3a1T

b1i ÿ �y

�2Xni�1

ÿyi ÿ �y

�2 �9�

Fig. 2. The yDu regression line of y on yDu:

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539 535

Using Eq. (6), R�2Sq can be presented as:

R�2Sq �

Xni�1

�ySqiÿ �y

�2Xni�1

ÿyi ÿ �y

�2 �10�

where ySq represents the regression line of y on ySq:As shown in Fig. 3, ySq � a4 � b4ySq: Substituting

from Eq. (4) yields:

ySq � a4 � b4ÿa2 � b2T

0:5�

�11�

In order to reduce this equation further, the values ofa4 and b4 must be determined. Let us start with the

slope b4: From regression analysis theory [7], the slopeof a regression line is:

Slope � Sxy

Sxx�

Xni�1

yi�x i ÿ �x�Xni�1�x i ÿ �x�2

In the case of ySq, the slope b4 is therefore:

b4 �

Xni�1

yi�ySqi ÿ �ySq

�Xni�1

�ySqi ÿ �ySq

�2 �12�

As �ySq�a2�b2T 0:5, by substitution for ySq and �ySq,

b4 �

Xni�1

yi�a2 � b2T

0:5i ÿ a2 ÿ b2T 0:5

�Xni�1

�a2 � b2T

0:5i ÿ a2 ÿ b2T 0:5

�2

�) b4 �

Xni�1

yi�b2T

0:5i ÿ b2T 0:5

�Xni�1

�b2T

0:5i ÿ b2T 0:5

�2

�

Xni�1

yi�T0:5i ÿ T 0:5

�Xni�1

b2�T0:5i ÿ T 0:5

�2 �13�

where Ti represents the cumulative test time to failurei.

When one considers the new growth model in Eq.(4), the slope b2 can be represented as:

b2 �

Xni�1

yi�T 0:5

i ÿ T 0:5

�Xni�1

�T 0:5

i ÿ T 0:5

�2 �14�

Combining Eqs. (13) and (14), yields b4 � b2=b2 � 1:From regression theory [7],

Intercept � �y� Slope� �x�

Therefore, the intercept a4 of the regression line ySq is:

a4 � �yÿ b4 �ySq

�) a4 � �yÿ �ySq � �yÿ�a2 � b2T 0:5

��15�

Similarly, the intercept of the new growth model, a2,is:

a2 � �yÿ b2T 0:5

�) �y�a2�b2T 0:5, which by substituting into Eq. (15)yields:

a4 � a2 � b2T 0:5 ÿ�a2 � b2T 0:5

�� 0

Since ySq � a4 � b4ySq, substituting the values of a4 �0 and b4 � 1 yields:

ySq � ySq � a2 � b2T0:5 �16�

This shows that when there is no transformation of they-axis, R�2 � R2: In the case of the new model, R2 rep-resents the coe�cient of determination between y and

T 0:5

By combining Eqs. (10) and (16), the value of R�2Sq

yields:Fig. 3. The ySq regression line of y on ySq:

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539536

R�2Sq �

Xni�1

�a2 � b2T

0:5i ÿ �y

�2Xni�1

ÿyi ÿ �y

�2 �17�

Eqs. (9) and (17), for R�2Du and R�2Sq, respectively, are in

a form that can be easily compared. The di�erence inthe R�2 �R�2Diff�R�2DuÿR�2Sq� of the two models is:

R�2Diff �

Xni�1

�a3 � b3a1T

b1i ÿ �y

�2ÿXni�1

�a2 � b2T

0:5i ÿ �y

�2Xni�1

ÿyi ÿ �y

�2�18�

This di�erence is a stationary point when dR�2Diff=dT �0: Now

dR�2Diff

dT�

dXni�1

�a3 � b3a1T

b1i ÿ �y

�2ÿXni�1

�a2 � b2T

0:5i ÿ �y

�2dT

� 0

dR�2Diff

dT� 2

Xni�1

�a3 � b3a1T

b1i ÿ �y

��b1b3a1T

b1ÿ1i ÿ k

�

ÿ 2Xni�1

�a2 � b2T

0:5i ÿ �y

�ÿ0:5b2T

ÿ0:5i ÿ k

�� 0

�)Xni�1

�a3 � b3a1T

b1i ÿ �y

��b1b3a1T

b1ÿ1i ÿ k

��Xni�1

�a2 � b2T

0:5i ÿ �y

�ÿ0:5b2T

ÿ0:5i ÿ k

�where k represents a constant arising from d�y=dT:By inspection, R�2Diff is a stationary point when the

Duane slope �b1� is 0.5. It follows that:

a3 � a2 and b3 �b2a1

By substituting these expressions for a3 and b3 intoEq. (18), the actual di�erence between R�2Du and R�2Sq iszero when the Duane slope �b1� is 0.5. This is consist-

ent with the simulation results presented in Fig. 1. Thesimulation also shows that at values of Duane slopeless than 0.5, the new model provides a more e�ective

®t to the data. The further the Duane slope deviatesfrom 0.5, the more likely will be the di�erence in therespective R�2 of both models. The Duane model is

better at higher Duane slopes, while the new model isbetter for the vast range of slopes normally observed

during a reliability growth programme.

5. Comparison of instantaneous failure rates

The instantaneous MTBF re¯ects the actual MTBFat a particular time t, if testing terminates and nofurther improvements are made to the product. The

Duane model is frequently used to extrapolate to aparticular cumulative test time T, at which a certain in-stantaneous MTBF is achieved [8,9]. This is used in

development testing to predict the test time required toachieve a speci®ed MTBF. The instantaneous MTBFof both models is developed for such a purpose.

The cumulative MTBF for the Duane model is rep-resented as:

yDu � T

n�T� �19�

where n(T ) represents the expected number of failuresby time T.Combining Eqs. (5) and (19), and expressing it in

terms of expected number of failures n(T ), then yields

n�T� � T 1ÿb1

a1�20�

Di�erentiating with respect to T, we obtain the instan-taneous failure rate for the Duane model, lDu:

dn�T�dT� lDu �

ÿ1ÿ b1

�T ÿb1

a1�21�

Rearranging Eq. (21), we can observe that:

lDu � 1ÿ b1a1T b �

1ÿ b1yDu

�22�

In developing the instantaneous failure rate of the newmodel, the same procedure is conducted as describedin Eqs. (19)±(22). The new model is represented by:

ySq � a2 � b2T0:5

n�T� � T

a2 � b2T 0:5�23�

Di�erentiating with respect to T, we obtain the instan-

taneous failure rate for the new model, lSq:

dn�T�dT� lSq � 0:5

ÿ2a2 � b2T

0:5�ÿ

a2 � b2T 0:5�2 �24�

Although Eq. (24) appears more complicated than

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539 537

Eq. (22), it can reasonably be approximated to:

lSq � 0:5

ySq

�25�

which is quite similar in structure to Eq. (22), whenthe Duane slope is 0.5.

Assuming a constant failure rate, the instantaneousMTBF �yDuInst

� of the Duane model becomes:

yDuInst� a1T b1

1ÿ b1�26�

Similarly, the instantaneous MTBF �ySqInst� of the new

growth model becomes:

ySqInst�

ÿa2 � b2T

0:5�2

0:5ÿ2a2 � b2T 0:5

� �27�

The aim of a reliability growth test program may be tocontinue testing till a cumulative time T, by which

time a speci®ed instantaneous MTBF, yinst will beachieved.Rearranging Eqs. (26) and (27), one observes the

e�ect that cumulative test time T has on both models.The cumulative time T to achieve this instantaneousMTBF is called TDu and TSq for the Duane and new

model, respectively.

�) TDu �"ÿ

1ÿ b1�yinst

a1

#1=b1

�28�

while

�) TSq � 1

16b42

�ÿ b2yinst � 4a2b2

ÿ b2��������yinst

p ���������������������yinst � 8a2

p �2�29�

As the Duane model is in¯uenced by the early failures,this has the e�ect of extending the time TDu to achievea speci®ed instantaneous MTBF over that time

required by the new model, TSq: Therefore, while thenew model provides a better ®t to the data for re-liability growth situations when the Duane slope is less

than 0.5, it also has the additional advantage of envi-saging less test time to achieve a speci®ed value of in-stantaneous MTBF.

6. Worked example

A reliability growth example, comprising of 10 fail-ures, is shown in Table 1. This illustrates the reliabilitygrowth technique and the associated calculations. T

able

1

Reliabilitygrowth

data

Failure

Cumulativefailure

time

CumulativeMTBF

NaturallogofcumulativeMTBF

Naturallogofcumulativetime

Square

rootofcumulativetime

11027

1027

6.934

6.934

32.047

21954

977

6.884

7.578

44.204

35366

1789

7.489

8.588

73.253

47344

1836

7.515

8.902

85.697

57540

1508

7.319

8.928

86.833

613,981

2330

7.754

9.545

118.241

717,344

2478

7.815

9.761

131.697

822,830

2854

7.956

10.036

151.096

930,907

3434

8.142

10.339

175.804

10

55,684

5568

8.625

10.927

235.975

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539538

The resulting Duane plot and new model plot are

presented in Figs. 4 and 5, respectively.The Duane model in the form of Eq. (1) is calcu-

lated as:

ln�yDu � � 3:82� 0:417 ln�T�

Regression analysis for the new model results in the

equation:

ySq � ÿ22:95� 21:175T 0:5

The R�2Du and R�2Sq values are calculated by Eqs. (6) and

(10), respectively. These reveal a R�2Du of 92.4% and aR�2Sq of 94.4%, indicating that the new model providesa better ®t to the observed data.

If one wishes to continue testing until an instan-taneous MTBF �yInst� of 10,000 h is achieved, then therequired test time can be calculated for each of themodels by Eqs. (28) and (29). This results in a TSq �55,750 h for the new model, while the Duane modelprovides a TDu � 112,700 h. In summary, the newmodel required only an additional 66 test hours to

achieve an instantaneous MTBF of 10,000 h, while theDuane model envisages a further 57,000 h of testing.

7. Summary

The new reliability growth model represents an

advancement on the empirically based Duane modelfor reliability growth situations. The Duane model isunduly in¯uenced by the early failures, while the newmodel, more realistically, is in¯uenced by late failures.

Both models bears certain similarities and have been

proven to be mathematically equivalent in ®tting thedata when the Duane slope equals 0.5. When extrapol-ation to a speci®ed instantaneous MTBF is required,the new model has the further advantage of requiring

less cumulative test time to achieve the speci®ed instan-taneous MTBF.

References

[1] Duane JT. Learning curve approach to reliability moni-

toring. IEEE Trans Aerospace 1964;AS-2:553±66.

[2] Crow LH. Reliability analysis for complex, repairable sys-

tems. In: Proschan F, Ser¯ing RJ, editors. Reliability and

Biometry: Statistical Analysis of Lifetimes. Philadelphia:

SIAM, 1974. p. 379±410.

[3] Donovan J, Murphy E. Reliability growth Ð a new

graphical model. Quality and Reliability Engineering

International 1999;15(3):167±74.

[4] Mead PH. Reliability growth of electronic equipment.

Microelectron Reliab 1975;14:439±43.

[5] Mondro MJ. Practical guidelines for conducting a

Reliability Growth Program. In: Proceedings of the

Institute of Environmental Sciences. 1993. p. 110±6.

[6] Hamilton LC. In: Regression with graphics: a second

course in applied statistics. Paci®c Grove: Brooks/Cole,

1992. p. 181.

[7] Montgomery DC, Runger GC. In: Applied statistics and

probability for engineers. New York: Wiley, 1994. p. 471±

522.

[8] Clarke JM, Cougan WP. RPM Ð a recent real life case

history. In: Proceedings of the Annual Reliability and

Maintainability Symposium. 1978. p. 279±85.

[9] Ebeling CE. In: An introduction to reliability and main-

tainability engineering. New York: McGraw-Hill, 1997. p.

345±7.

Fig. 4. Duane plot of data from Table 1.Fig. 5. New model plot of data from Table 1.

J. Donovan, E. Murphy /Microelectronics Reliability 40 (2000) 533±539 539