A new reliability growth model: its mathematical comparison to the Duane model

Microelectronics Reliability 40 (2000) 533±539

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A new reliability growth model: its mathematical comparison to the Duane model John Donovan a,*, Eamonn Murphy b a Institute of Technology, School of Engineering, Sligo, Ireland National Centre for Quality Management, University of Limerick, Limerick, Ireland

b

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 undergoing development testing. This paper proposes a new reliability growth model derived from variance stabilisation transformation 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. This paper 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 later failures, compared to early failures for the Duane model, it has the further advantage of leading to reduced test times 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 proposed for monitoring and tracking reliability improvement during product development. Duane's model [1] has remained one of the primary graphical models and was initially developed as an empirical model based on observations by Duane over a number of projects. This model represents a relationship between the cumulative MTBF …y† and the cumulative test time (T ) such that y ˆ a1 T b1 : ln a1 and b1 represent the intercept and Duane slope, respectively, as seen from the following log±log model.

* Corresponding author. Tel.: +353-71-55231; fax: +35371-55390. E-mail address: [email protected] (J. Donovan).

ln…y † ˆ 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 Nonhomogeneous Poisson Process with a Weibull intensity function. Donovan and Murphy [3] more recently, have formulated a new reliability growth model which was derived from variance stabilisation theory for regression 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.

0026-2714/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 6 - 2 7 1 4 ( 9 9 ) 0 0 2 3 5 - 8

534

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

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 continued in the new model. As each failure occurs in accordance with a Poisson process, the cumulative time can be viewed as a ``count'' of the number of hours during which a number of failures have occurred, or in other words, a count of the time in which a speci®c cumulative MTBF has been reached. Variance stabilisation transformation theory suggests that for a Poisson process, the transformation equals the square root of the count. Therefore, in the case of cumulative time, the appropriate transformation for the x-axis is the square 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 MTBF is plotted directly without transformation. The model therefore becomes p y ˆ a2 ‡ b2 T …2† where y and T represent the cumulative MTBF and cumulative test time respectively, and a2 and b2 represent the intercept and slope, respectively. This model bears a resemblance to the Duane model when the Duane slope …b1 † is 0.5, but has a number of advantages 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 in Ref. [3]. These advantages can be summarised as follows: . In the Duane model, early failures have a high in¯uence on the resulting model as measured by the Cook's distance …Di † statistic which combines both the leverage and in¯uence of each failure. If the Duane model is used to observe growth during testing, then the resulting graph is overly a€ected by those failures occurring early in time. This does not occur in the new model. . In the Duane model, failures occurring towards the latter part of the test tend to cluster together due to the nature of the ln(Cumulative Time ). This arises as there is little di€erence between the natural logs of large values of cumulative time. The clustering e€ect is avoided in the new model as the log scales are avoided. . Simulation has shown that the latter failures have the greatest in¯uence and leverage on the new model. The early failures have little in¯uence or leverage on this model.

. The simplicity of the new model is evident as there is no requirement to transform the y-axis resulting in the further advantage of reading the cumulative MTBF directly from the graph. The new model thereby 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 provide a better ®t to the data. This is not too great a hindrance 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 the earliest failures. It is unrealistic to believe that after possibly 100,000 h of testing, the earliest failure remains the most in¯uential. This de®ciency is largely overcome in the Duane model by omitting the earliest failures [4,5]. This becomes unnecessary with the new model, which has the added advantage of plotting all the 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 of how well the model ®ts the data. It represents the percentage variability in y explained by x. It is important to remember, however, that the R 2 of respective models provides an inappropriate comparison if the model y-axes di€er. As the new model has completely di€erent axes from the Duane model, the method chosen to compare both models is from Hamilton [6]. This requires ®nd-

Fig. 1. Di€erence in the R2 of both models …R2 Diff † related to the Duane slope.

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

^ 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. In à the Duane model, this means calculating eln…Cum:MTBF† for the various failures. One then evaluates the coecient of determination between these predicted values and the y values (i.e. the observed cumulative MTBF). It has been decided to call this as R2 to distinguish it from the R2 described earlier. In this respect, R2 is a measure of the closeness of the predicted to the actual. The better model is considered to be the model with the highest R2 : The di€erence in the R2 values for the Duane and new model arising from 6200 simulated datasets is shown in Fig. 1. A positive di€erence indicates that the Duane model provides a better ®t to the data, while a negative di€erence indicates that the new model provides a better ®t. The further the Duane slope deviates from 0.5, the more likely will be the di€erence in the respective R2 of both models. The Duane model is better at higher Duane slopes, while the new model is better for the vast range of slopes typically observed during 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 ˆ a ‡ bT g

…3†

In the speci®c form identi®ed by variance stabilisation transformations, where g is represented by 0.5, this model becomes: ySq ˆ a2 ‡ b2 T 0:5

yDu ˆ a1 T b1

n ÿ X
2 y^i ÿ y

R2 ˆ

iˆ1

n X  2 …yi ÿ y† iˆ1

Therefore, for R2 Du , n  X

R2 Du ˆ

y^ Dui ÿ y

iˆ1 n ÿ X

yi ÿ y

2 …6†


2

iˆ1

where yi represents the cumulative MTBF for each failure i from 1 to n. y^ Du represents the regression line of y on yDu as illustrated in Fig. 2, while y^Dui represents the predicted cumulative MTBF for each failure i. y is the mean of the observed cumulative MTBF and can be evaluated as: n 1X Ti y ˆ n iˆ1 i

y^Du is represented as the straight line: y^Du ˆ a3 ‡ b3 yDu

…7†

Combining Eqs. (7) and (5), yields: y^Du ˆ a3 ‡ b3 a1 T b1

…8†

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

…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 predicted by the new model and Duane model, respectively. The notation Sq is used for the new model as this model contains the Square Root of cumulative time. The Duane model is represented by:

R2 Du ˆ

iˆ1

a3 ‡ b3 a1 Ti 1 ÿ y b

n ÿ X

yi ÿ y

2


2

iˆ1

…5†

The method by which the models are evaluated is to compare the R2 for the Duane and new model, respectively: R2 Du ˆ Coefficient of Determination between yDu and y R2 Sq ˆ Coefficient of Determination between ySq and y where y is the observed cumulative MTBF. The equation for coecient of determination [7] is:

535

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

…9†

536

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

  n X 0:5 yi b2 T0:5 ÿ b T 2 i

Using Eq. (6), R2 Sq can be presented as:

R2 Sq ˆ

n  2 X y^Sqi ÿ y iˆ1 n ÿ X

yi ÿ y

ˆ) b4 ˆ …10†


2

iˆ1

In order to reduce this equation further, the values of a4 and b4 must be determined. Let us start with the slope b4 : From regression analysis theory [7], the slope of a regression line is: n X

Slope ˆ

Sxy ˆ iˆ1n X Sxx  2 …x i ÿ x†

ySqi ÿ y Sq

…12†

2

  n X 0:5 yi a2 ‡ b2 T0:5 ÿ a ÿ b T 2 2 i iˆ1 n  X

a2 ‡

iˆ1

b2 T0:5 i

ÿ a2 ÿ b2 T

iˆ1 n  X

0:5 T 0:5 i ÿT

2

…14†

Therefore, the intercept a4 of the regression line y^ Sq is: a4 ˆ y ÿ b4 y Sq

As y Sq ˆ a2 ‡b2 T 0:5 , by substitution for ySq and y Sq ,

b4 ˆ

  n X 0:5 yi T 0:5 i ÿT

Intercept ˆ y ‡ Slope…x †

  n X yi ySqi ÿ y Sq

iˆ1

where Ti represents the cumulative test time to failure i. When one considers the new growth model in Eq. (4), the slope b2 can be represented as:

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

In the case of y^Sq , the slope b4 is therefore:

iˆ1 n  X

…13†

iˆ1

iˆ1

iˆ1

b4 ˆ

2

iˆ1 ˆX  2 n 0:5 b2 T0:5 i ÿT

b2 ˆ

 yi …x i ÿ x†

0:5 b2 T0:5 i ÿ b2 T

  n X 0:5 yi T0:5 i ÿT

iˆ1

where y^Sq represents the regression line of y on ySq : As shown in Fig. 3, y^Sq ˆ a4 ‡ b4 ySq : Substituting from Eq. (4) yields: ÿ
y^ Sq ˆ a4 ‡ b4 a2 ‡ b2 T 0:5 …11†

iˆ1 n  X

0:5

2

  ˆ) a4 ˆ y ÿ y Sq ˆ y ÿ a2 ‡ b2 T 0:5

…15†

Similarly, the intercept of the new growth model, a2 , is: a2 ˆ y ÿ b2 T 0:5 ˆ) y ˆ a2 ‡ b2 T 0:5 , which by substituting into Eq. (15) yields:   a4 ˆ a2 ‡ b2 T 0:5 ÿ a2 ‡ b2 T 0:5 ˆ 0 Since y^Sq ˆ a4 ‡ b4 ySq , substituting the values of a4 ˆ 0 and b4 ˆ 1 yields: y^Sq ˆ ySq ˆ a2 ‡ b2 T 0:5

Fig. 3. The y^Sq regression line of y on ySq :

…16†

This shows that when there is no transformation of the y-axis, R2 ˆ R2 : In the case of the new model, R2 represents the coecient of determination between y and T 0:5 By combining Eqs. (10) and (16), the value of R2 Sq yields:

J. Donovan, E. Murphy / Microelectronics Reliability 40 (2000) 533±539 n  X iˆ1

R2 Sq ˆ

 a2 ‡ b2 T0:5 i ÿy n ÿ X

yi ÿ y

2 …17†


2

537

better at higher Duane slopes, while the new model is better for the vast range of slopes normally observed during a reliability growth programme.

iˆ1 2 Eqs. (9) and (17), for R2 Du and RSq , respectively, are in a form that can be easily compared. The di€erence in 2 2 the R2 …R2 Diff ˆRDu ÿ RSq † of the two models is:

R2 Diff ˆ

2 X 2 n  n  X b  a3 ‡ b3 a1 T i 1 ÿ y ÿ a2 ‡ b2 T 0:5 ÿ y i iˆ1

n ÿ X

iˆ1

yi ÿ y


2

iˆ1

…18† This di€erence is a stationary point when dR2 Diff =dT ˆ 0: Now

dR2 Diff ˆ dT

d

n  X iˆ1

2 X 2 n  b  a3 ‡ b3 a1 Ti 1 ÿ y ÿ a2 ‡ b2 T0:5 i ÿy dT

iˆ1

ˆ0   n  X dR2 b b ÿ1 Diff ˆ2 a3 ‡ b3 a1 Ti 1 ÿ y b1 b3 a1 T i 1 ÿ k dT iˆ1 ÿ n  X
ÿ0:5  ÿ2 ÿk a2 ‡ b2 T0:5 i ÿ y 0:5b2 T i iˆ1

ˆ0

ˆ)

n  X iˆ1

ˆ

b a3 ‡ b3 a1 Ti 1 ÿ y

n  X iˆ1

  b ÿ1 b1 b3 a1 T i 1 ÿ k

ÿ
ÿ0:5  a2 ‡ b2 T0:5 ÿk i ÿ y 0:5b2 T i

 where k represents a constant arising from dy=dT: 2 By inspection, RDiff is a stationary point when the Duane slope …b1 † is 0.5. It follows that: a3 ˆ a2

and

b b3 ˆ 2 a1

By substituting these expressions for a3 and b3 into 2 Eq. (18), the actual di€erence between R2 Du and RSq is zero when the Duane slope …b1 † is 0.5. This is consistent with the simulation results presented in Fig. 1. The simulation also shows that at values of Duane slope less than 0.5, the new model provides a more e€ective ®t to the data. The further the Duane slope deviates from 0.5, the more likely will be the di€erence in the respective R2 of both models. The Duane model is

5. Comparison of instantaneous failure rates The instantaneous MTBF re¯ects the actual MTBF at a particular time t, if testing terminates and no further improvements are made to the product. The Duane model is frequently used to extrapolate to a particular cumulative test time T, at which a certain instantaneous MTBF is achieved [8,9]. This is used in development testing to predict the test time required to achieve a speci®ed MTBF. The instantaneous MTBF of both models is developed for such a purpose. The cumulative MTBF for the Duane model is represented as: yDu ˆ

T n…T †

…19†

where n(T ) represents the expected number of failures by time T. Combining Eqs. (5) and (19), and expressing it in terms of expected number of failures n(T ), then yields n…T † ˆ

T 1ÿb1 a1

…20†

Di€erentiating with respect to T, we obtain the instantaneous failure rate for the Duane model, lDu : ÿ
1 ÿ b1 T ÿb1 dn…T † ˆ lDu ˆ …21† dT a1 Rearranging Eq. (21), we can observe that: lDu ˆ

1 ÿ b1 1 ÿ b1 ˆ a1 T b yDu

…22†

In developing the instantaneous failure rate of the new model, the same procedure is conducted as described in Eqs. (19)±(22). The new model is represented by: ySq ˆ a2 ‡ b2 T 0:5 n…T † ˆ

T a2 ‡ b2 T 0:5

…23†

Di€erentiating with respect to T, we obtain the instantaneous failure rate for the new model, lSq :
ÿ dn…T † 0:5 2a2 ‡ b2 T 0:5 …24† ˆ lSq ˆ ÿ
2 dT a2 ‡ b T 0:5 2

Although Eq. (24) appears more complicated than

Eq. (22), it can reasonably be approximated to: …25†

which is quite similar in structure to Eq. (22), when the Duane slope is 0.5. Assuming a constant failure rate, the instantaneous MTBF …yDuInst † of the Duane model becomes:


2 a2 ‡ b2 T 0:5 ÿ
0:5 2a2 ‡ b2 T 0:5

…27†

"ÿ ˆ) TDu ˆ

#1=b1
1 ÿ b1 yinst a1

Natural log of cumulative MTBF

The aim of a reliability growth test program may be to continue testing till a cumulative time T, by which time a speci®ed instantaneous MTBF, yinst will be achieved. 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 instantaneous MTBF is called TDu and TSq for the Duane and new model, respectively. …28†

ˆ) TSq ˆ

 1 ÿ b2 yinst ‡ 4a2 b2 4 16b2 pp 2 ÿ b2 yinst yinst ‡ 8a2

Cumulative MTBF

while

…29†

6. Worked example A reliability growth example, comprising of 10 failures, is shown in Table 1. This illustrates the reliability growth technique and the associated calculations.

Table 1 Reliability growth data

As the Duane model is in¯uenced by the early failures, this has the e€ect of extending the time TDu to achieve a speci®ed instantaneous MTBF over that time required by the new model, TSq : Therefore, while the new model provides a better ®t to the data for reliability growth situations when the Duane slope is less than 0.5, it also has the additional advantage of envisaging less test time to achieve a speci®ed value of instantaneous MTBF.

6.934 6.884 7.489 7.515 7.319 7.754 7.815 7.956 8.142 8.625

ÿ

ySqInst ˆ

1027 977 1789 1836 1508 2330 2478 2854 3434 5568

Similarly, the instantaneous MTBF …ySqInst † of the new growth model becomes:

6.934 7.578 8.588 8.902 8.928 9.545 9.761 10.036 10.339 10.927

…26†

1027 1954 5366 7344 7540 13,981 17,344 22,830 30,907 55,684

a1 T b1 1 ÿ b1

Natural log of cumulative time

yDuInst ˆ

1 2 3 4 5 6 7 8 9 10

0:5 ySq

Cumulative failure time

lSq ˆ

32.047 44.204 73.253 85.697 86.833 118.241 131.697 151.096 175.804 235.975

Square root of cumulative time

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

Failure

538

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

Fig. 4. Duane plot of data from Table 1.

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 calculated as: ln…yDu † ˆ 3:82 ‡ 0:417 ln…T †

539

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

proven to be mathematically equivalent in ®tting the data when the Duane slope equals 0.5. When extrapolation 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 instantaneous MTBF.

Regression analysis for the new model results in the equation: ySq ˆ ÿ22:95 ‡ 21:175T 0:5 2 The R2 Du and RSq values are calculated by Eqs. (6) and (10), respectively. These reveal a R2 Du of 92.4% and a R2 of 94.4%, indicating that the new model provides Sq a better ®t to the observed data. If one wishes to continue testing until an instantaneous MTBF …yInst † of 10,000 h is achieved, then the required test time can be calculated for each of the models by Eqs. (28) and (29). This results in a TSq ˆ 55,750 h for the new model, while the Duane model provides a TDu ˆ 112,700 h. In summary, the new model required only an additional 66 test hours to achieve an instantaneous MTBF of 10,000 h, while the Duane model envisages a further 57,000 h of testing.

7. Summary The new reliability growth model represents an advancement on the empirically based Duane model for reliability growth situations. The Duane model is unduly in¯uenced by the early failures, while the new model, more realistically, is in¯uenced by late failures. Both models bears certain similarities and have been

References [1] Duane JT. Learning curve approach to reliability monitoring. IEEE Trans Aerospace 1964;AS-2:553±66. [2] Crow LH. Reliability analysis for complex, repairable systems. 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 maintainability engineering. New York: McGraw-Hill, 1997. p. 345±7.