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Reliability exploration of microcomputer systems using the Weibull distribution
Microelectron.Reliab.,Vol. 32, No. 3, pp. 337-340, 1992. Printed in Great Britain.
0026-2714/9255.00+ .00 © 1991 PergamonPress plc
RELIABILITY EXPLORATION OF MICROCOMPUTER SYSTEMS USING THE WEIBULL DISTRIBUTION IRENEUSZ J. Jr~WIAK Institute of Engineering Cybernetics, Technical University of Wroctaw, Janiszewskiego Str. 11-17, 50-372 Wroclaw, Poland
(Receivedfor publication 30 October 1990) Almtraet--The Weibull model is utilized for reliability exploration of microcomputer systems. The usefulness of the model for reliability exploration of technical systems is shown.
1. INTRODUCTION In making reliability analyses of technical systems it is usually assumed that reliability is a function of time only (equipment age or exploitation time) [1-8]. Analysing the experimental data about the random sample it is assumed that the failure time of a random variable has an assigned distribution in the relation of time (e.g. the Weibull distribution) [1-3, 8-12]. The efficient (failure) time of the device is the device work time until failure. The distribution parameter estimation is made using graphical or analytical methods by checking whether the assumed model (the r a n d o m variable T distribution) fits empirical data. If so, the hazard function form h(t) can be assigned [10-12]. The function h (t) can be used for the instantaneous rate of failure prediction at time t. If, for example, the device failure time has a Weibull distribution with parameters 2 and p, it can be assumed that the hazard function takes the form
h(t) = 2p(2/y'-1, t > 0.
croprocessor I N T E L 8080, 32 kB RAM, two floppy disks, power supply, monitor display, keyboard, IO controller) was examined. The data are shown in Table 1. The table also contains information about the state of the system at the end of the experiment (U, system failed at time t~; S, system working at the end of the experiment; t~ is the work-time to the end of the experiment). Failure is the state of the system in which it cannot fulfil its functions. Input data from the examination of m = 10 systems are shown in Table 1 with the following notation: ti is the ith system failure time (or suspension time for systems which have not failed at the end of the experiment), SS means system state at the end of the experiment (U means failed system, S means good system) and R(ti) is the reliability function.
(1)
Moreover,
f(t) h(t) = R(t)
R'(t) R(t)
(2)
where f(t) is the probability density function and R(t) is the reliability function [1-3]. The reliability function R(t) is given by
R(t)=exp(-flh(u)du)=exp(-(2t)e).
3. ESTIMATION OF THE MODEL PARAMETERS WITH THE GRAPHICAL METHOD The values of the Weibull distribution parameters 2 and p are estimated with a graphical method. The values of the reliability function R(t~) at the efficient times and the values of In t i and l n ( - l n R(t~)) were calculated (Table 1) according to the graphical method of Weibull distribution parameter estimation. The values of the efficient (failure) time t~logarithm and the values of l n ( - l n R(ti)) are shown in Fig. I, using the following notation:
(3)
Particularly for p = l, the Weibull distribution is the exponential distribution and for p = 2 the Weibull distribution is the Rayleigh distribution [1-3]. The Weibull distribution is accepted as the baseline distribution. The application of the Weibull distribution with parameters 2 and p to reliability exploration of microcomputer systems is presented in this paper. 2. FORMULATION OF THE PROBLEM
To check the usefulness of the Weibull distribution (model) to the microcomputer system, reliability (mi337
wi = In 6,
ui = I n ( - l n R(ti)).
(4)
Table 1. Data of rn = 10 microcomputer systems and the reliability function
i
t~(h)
R(t,)
SS
In t~
I n ( - In R(t,))
1 2 3 4 5 6 7 8 9 10
2018 2197 2256 2064 2534 2512 1664 1631 1498 2685
0.6 0.4 0.3 0.5 0.1 0.2 0.7 0.8 0.9 0.0
U U U S U S U U U S
7.6099 7.6948 7.7213 7.6329 7.8376 7.8288 7.4170 7.3969 7.2917 7.8954
-0.67173 -0.08742 0.18563 -0.36651 0.83403 0.47588 - 1.03093 - 1.49994 -2.25036 --
IREtCEUSZJ. J6~:WIAK
338
..f
I --
0
cr
L
I
L
7.2
7.3
7.4
_5 = L
o
g -2
--
I
t
7.5 7.6 ~7~ 7.8 ~ ° W=knt
7.9
/
Fig. 1. Graphical interpretation of data concerning the microcomputer system efficient times. The coefficients of the regression line [10, 11] are given by
Equating both derivatives to 0 and solving the system of equations:
u = Aw + B
(5)
1041.2110A + 136.8608B + 64.0728 = 0
where A and B are regression line coefficients. The regression line is also drawn in Fig. 1. In order to estimate the regression line coefficients A and B the minimum sum of square deviations was used according to the suggestion made in Refs [10, I 1, 13]. So the minimum of the following expression is found:
18B + 136.8608A + 8.8228 --- 0,
S = ~ (ul- A w i - B) 2
(10)
the values of A and B were found: A =4.997,
B --- -37.776,
(11)
for which S has its minimum: &2S = 1041.2110 > 0 OA 2
(6)
(12)
i=l
where S is the sum of square deviations of the u coordinate from the regression line at points of failure and m is the number of systems examined. Substituting the values of u; and w; from Table 1 into the expression (6), using notation (4) for each failure, the regression for the sum of square deviations for the considered data can be found:
0zS OB----i = 18 > 0
(13)
&2S -
-
t3A • OB
- 136.8608 > 0.
(14)
The values of the Weibull distribution parameters 2 and p were worked out by substitution of the calculated values of A and B:
S = (0.83403 - A • 7.8376 - B) 2
p = A = 4.997
(15)
+ (0.47588 - A ' 7.8288 - B) 2 2 =exp(B)=4.926x
+ ( 0 . 1 8 5 6 3 - A • 7 . 7 2 1 3 - B) 2 + ( - 0 . 0 8 7 4 2 - A . 7.6948 - B) 2
10-4=0.0004926.
(16)
The value of the K o l m o g o r o v statistic [2, 3] is equal to 0.32 for the confidence region ~t _< 0.4.
+ ( - 0 . 3 6 6 5 1 - A • 7 . 6 3 2 4 - B): + ( - 0 . 6 7 1 7 3 - A • 7.6099 -- B) 2 +(-
4. E S T I M A T I O N O F T H E M O D E L P A R A M E T E R S WITH THE MAXIMUM LIKELIHOOD METHOD
1.03093 -- A • 7.4170 - B) 2
+(1.49994 - A • 7.3969 - B) 2 + ( - 2 . 2 5 0 3 6 - A • 7.2917 - B) 2.
(7)
After calculating the sums of squares, expression (7) reduces to the following form:
The Weibull distribution parameters 2 and p are estimated with the maximum likelihood method for the full likelihood [10, 11, 14-16]:
L = f l h(tj)" f l R(tk)
S = 520.6055A 2 + 9B 2 + 136.8608AB + 64.0728A + 8.8228B + 9.9003.
j=l
(8)
The partial derivatives of S with respect to A and B are: 0S
--
OA
(17)
k=l
where L is the full likelihood, n is the number of failed systems (n = 7) and m is the number of systems examined (m = 10). The formula for the full likelihood for the accepted model can be found to be:
= 1041.2110A + 136.8608B + 64.0728
0S 0--~ = 18B + 136.8608A + 8.8228.
L = f i ~p(2tj) p-I" f i exp(--(Atk)r). j~I
(9)
(18)
k=l
As suggested in Refs [10, 11, 15, 16] the logarithms
Reliability exploration of microcomputers using the Weibull distribution of equation (18) were taken:
The value of the Kolmogorov statistic [2, 3] is equal to 0.89 for the confidence region ct ~< 0.4.
lnL =n lnp +np ln2 + (p - 1). ~ In t j - ~. (2tk)p j=,
(19)
k=l
and its maximum was found. The partial derivatives of In L with respect to 2 and p were found: a lnL ~2
np 2
~ tkp(2tk)p- 1
(20)
k=l
and lnL - =-n+nln2+ @ P
~ l n t j - - ~ (2tk)Pln(2tk). k=~
j=~
(21) Such values of 2 and p are found, for which In L achieves its maximum [11, 15, 16], so the right hand sides of equations (20) and (21) are equated to zero and the system is solved with respect to 2 and p: tkp(2tk)P- J = 0
,~
(22)
k=l
and
n P
+ n In 2 + ~ In t j - ~ (2t~)r ln(2tk) = 0. (23) j=1
A
(24)
k=l
Thus 2P -
-
n -
(25)
(tky' k=l
Transforming equation (23) we obtain: n
- + n l n 2 + ~ In t j = 2 ' l n 2 . j=l
~ (tk)r k=,
+ 2 ?. ~ (tk) pIn tk.
(26)
k=l
When we use (25) in (26) we obtain: - +
In tj = m /='
Comparing the values of parameters 2 and p estimated with the graphical method [equations (15) and (16)] with the values of 2 and p estimated with the maximum likelihood method [equations (28) and (29)] it was found that the values obtained do not differ significantly. So, it can be said that both the methods are effective. The maximum likelihood method is more credible, because it considers information about systems which have not failed until the end of the experiment. So the values of parameters estimated with the maximum likelihood method are accepted for further calculations. Substituting the estimated values of 2 = 0.0004168 and p = 4.913 and the efficient time of the devices from Table 1 into equation (19) we obtain: In L = - 56.2475.
(30)
Application of a random variable distribution (in this case the Weibull distribution) to the quantitative representation of a physical process needs checking if the distribution fits the data using one of the statistical tests (X: statistic or Kolmogorov statistic) [2, 3, 8]. REFERENCES
~ (tk)r.
:p2 p-l"
P
5. CONCLUSIONS
k=~
From equation (22) we obtain:
P
339
"
(tk) p In t k.
(27)
Y(tkY' k=, k=l
Substituting the values of In tj and In tk from Table 1 into equation (27), the value of parameter p is calculated, for which the equation is true. The following value of p is found from the program: p = 4.913.
(28)
Substituting the value of p and the values of tk (from Table 1) into equation (25) we obtain: 2 = 4.168 x 10 -4 = 0.0004168.
(29)
I. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. John Wiley, New York (1965). 2. D. Bobrowski, Mathematical Models and Methods in Reliability Theory. WNT, Warsaw (1985). 3. B. W. Gniedenko, J. K. Bielajev and A. D. Sotoviev, Mathematical Methods in Reliability. WNT, Warsaw (1966). 4. I. J. J62wiak, Zuverlaessigkeit einer ausgewaehlten Klasse von Systemen mit hierarchischer Struktur. Nachrichtentech. Elektron., Berlin 35(5), 188-189 (1985). 5. I. J. J62wiak, Zeit haveriefreier Arbeit eines digitalen Systems mit Zuverlaessigkeitsreserve. Nachrichtentech. Elektron., Berlin 40(2), 72-73 (1990). 6. I. J. J6~wiak, Funktionelles Zuverlaessigkeits Modell eines eine Aufgabenmenge realisierenden digitalen Systems. Nachrichtentech. Elektron., Berlin 40(3), 95-97 (1990). 7. I. J. J6~'wiak and U. Kosmol, Stochastische Modelierung von fehlertoleranten Systemen. Nachrichtentech. Elektron., Berlin 39(4), 152-153 (1989). 8. J. M. Juran and F. M. Gryna, Quality Planning and Analysis. McGraw-Hill, New York (1970). 9. D. R. Cox and D. V. Hinkley, Theoretical Statistics. Chapman & Hall, London (1974). 10. D. R. Cox and D. Oakes, Analysis of Survival Data. Chapman & Hall, London (1983). 11. J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data. John Wiley, New York (1980). 12. N. R. Mann, R. E. Schafer and N. D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York (1974). 13. I. J. J62wiak and I. Mazurkiewicz, The graphical method of evaluation of the Weibull proportional hazards model parameters. Polish Acad. Sci. J. Z E M (submitted). 14. D. R. Cox, Partial likelihood. Biometrika 62(2), 269-276 (1975).
340
IRENEUSZJ. J6~WIAK
15. A. K. S. Jardine and M. Anderson, Use of concomitant variables for reliability estimation. Maintenance Management Int. 5, 135-140 (1985).
16. I. J. J6~ciak, Weibull proportional hazards model for concomitant variables dependent on time. Polish Acad. Sci. J. Z E M (submitted).
0026-2714/9255.00+ .00 © 1991 PergamonPress plc
RELIABILITY EXPLORATION OF MICROCOMPUTER SYSTEMS USING THE WEIBULL DISTRIBUTION IRENEUSZ J. Jr~WIAK Institute of Engineering Cybernetics, Technical University of Wroctaw, Janiszewskiego Str. 11-17, 50-372 Wroclaw, Poland
(Receivedfor publication 30 October 1990) Almtraet--The Weibull model is utilized for reliability exploration of microcomputer systems. The usefulness of the model for reliability exploration of technical systems is shown.
1. INTRODUCTION In making reliability analyses of technical systems it is usually assumed that reliability is a function of time only (equipment age or exploitation time) [1-8]. Analysing the experimental data about the random sample it is assumed that the failure time of a random variable has an assigned distribution in the relation of time (e.g. the Weibull distribution) [1-3, 8-12]. The efficient (failure) time of the device is the device work time until failure. The distribution parameter estimation is made using graphical or analytical methods by checking whether the assumed model (the r a n d o m variable T distribution) fits empirical data. If so, the hazard function form h(t) can be assigned [10-12]. The function h (t) can be used for the instantaneous rate of failure prediction at time t. If, for example, the device failure time has a Weibull distribution with parameters 2 and p, it can be assumed that the hazard function takes the form
h(t) = 2p(2/y'-1, t > 0.
croprocessor I N T E L 8080, 32 kB RAM, two floppy disks, power supply, monitor display, keyboard, IO controller) was examined. The data are shown in Table 1. The table also contains information about the state of the system at the end of the experiment (U, system failed at time t~; S, system working at the end of the experiment; t~ is the work-time to the end of the experiment). Failure is the state of the system in which it cannot fulfil its functions. Input data from the examination of m = 10 systems are shown in Table 1 with the following notation: ti is the ith system failure time (or suspension time for systems which have not failed at the end of the experiment), SS means system state at the end of the experiment (U means failed system, S means good system) and R(ti) is the reliability function.
(1)
Moreover,
f(t) h(t) = R(t)
R'(t) R(t)
(2)
where f(t) is the probability density function and R(t) is the reliability function [1-3]. The reliability function R(t) is given by
R(t)=exp(-flh(u)du)=exp(-(2t)e).
3. ESTIMATION OF THE MODEL PARAMETERS WITH THE GRAPHICAL METHOD The values of the Weibull distribution parameters 2 and p are estimated with a graphical method. The values of the reliability function R(t~) at the efficient times and the values of In t i and l n ( - l n R(t~)) were calculated (Table 1) according to the graphical method of Weibull distribution parameter estimation. The values of the efficient (failure) time t~logarithm and the values of l n ( - l n R(ti)) are shown in Fig. I, using the following notation:
(3)
Particularly for p = l, the Weibull distribution is the exponential distribution and for p = 2 the Weibull distribution is the Rayleigh distribution [1-3]. The Weibull distribution is accepted as the baseline distribution. The application of the Weibull distribution with parameters 2 and p to reliability exploration of microcomputer systems is presented in this paper. 2. FORMULATION OF THE PROBLEM
To check the usefulness of the Weibull distribution (model) to the microcomputer system, reliability (mi337
wi = In 6,
ui = I n ( - l n R(ti)).
(4)
Table 1. Data of rn = 10 microcomputer systems and the reliability function
i
t~(h)
R(t,)
SS
In t~
I n ( - In R(t,))
1 2 3 4 5 6 7 8 9 10
2018 2197 2256 2064 2534 2512 1664 1631 1498 2685
0.6 0.4 0.3 0.5 0.1 0.2 0.7 0.8 0.9 0.0
U U U S U S U U U S
7.6099 7.6948 7.7213 7.6329 7.8376 7.8288 7.4170 7.3969 7.2917 7.8954
-0.67173 -0.08742 0.18563 -0.36651 0.83403 0.47588 - 1.03093 - 1.49994 -2.25036 --
IREtCEUSZJ. J6~:WIAK
338
..f
I --
0
cr
L
I
L
7.2
7.3
7.4
_5 = L
o
g -2
--
I
t
7.5 7.6 ~7~ 7.8 ~ ° W=knt
7.9
/
Fig. 1. Graphical interpretation of data concerning the microcomputer system efficient times. The coefficients of the regression line [10, 11] are given by
Equating both derivatives to 0 and solving the system of equations:
u = Aw + B
(5)
1041.2110A + 136.8608B + 64.0728 = 0
where A and B are regression line coefficients. The regression line is also drawn in Fig. 1. In order to estimate the regression line coefficients A and B the minimum sum of square deviations was used according to the suggestion made in Refs [10, I 1, 13]. So the minimum of the following expression is found:
18B + 136.8608A + 8.8228 --- 0,
S = ~ (ul- A w i - B) 2
(10)
the values of A and B were found: A =4.997,
B --- -37.776,
(11)
for which S has its minimum: &2S = 1041.2110 > 0 OA 2
(6)
(12)
i=l
where S is the sum of square deviations of the u coordinate from the regression line at points of failure and m is the number of systems examined. Substituting the values of u; and w; from Table 1 into the expression (6), using notation (4) for each failure, the regression for the sum of square deviations for the considered data can be found:
0zS OB----i = 18 > 0
(13)
&2S -
-
t3A • OB
- 136.8608 > 0.
(14)
The values of the Weibull distribution parameters 2 and p were worked out by substitution of the calculated values of A and B:
S = (0.83403 - A • 7.8376 - B) 2
p = A = 4.997
(15)
+ (0.47588 - A ' 7.8288 - B) 2 2 =exp(B)=4.926x
+ ( 0 . 1 8 5 6 3 - A • 7 . 7 2 1 3 - B) 2 + ( - 0 . 0 8 7 4 2 - A . 7.6948 - B) 2
10-4=0.0004926.
(16)
The value of the K o l m o g o r o v statistic [2, 3] is equal to 0.32 for the confidence region ~t _< 0.4.
+ ( - 0 . 3 6 6 5 1 - A • 7 . 6 3 2 4 - B): + ( - 0 . 6 7 1 7 3 - A • 7.6099 -- B) 2 +(-
4. E S T I M A T I O N O F T H E M O D E L P A R A M E T E R S WITH THE MAXIMUM LIKELIHOOD METHOD
1.03093 -- A • 7.4170 - B) 2
+(1.49994 - A • 7.3969 - B) 2 + ( - 2 . 2 5 0 3 6 - A • 7.2917 - B) 2.
(7)
After calculating the sums of squares, expression (7) reduces to the following form:
The Weibull distribution parameters 2 and p are estimated with the maximum likelihood method for the full likelihood [10, 11, 14-16]:
L = f l h(tj)" f l R(tk)
S = 520.6055A 2 + 9B 2 + 136.8608AB + 64.0728A + 8.8228B + 9.9003.
j=l
(8)
The partial derivatives of S with respect to A and B are: 0S
--
OA
(17)
k=l
where L is the full likelihood, n is the number of failed systems (n = 7) and m is the number of systems examined (m = 10). The formula for the full likelihood for the accepted model can be found to be:
= 1041.2110A + 136.8608B + 64.0728
0S 0--~ = 18B + 136.8608A + 8.8228.
L = f i ~p(2tj) p-I" f i exp(--(Atk)r). j~I
(9)
(18)
k=l
As suggested in Refs [10, 11, 15, 16] the logarithms
Reliability exploration of microcomputers using the Weibull distribution of equation (18) were taken:
The value of the Kolmogorov statistic [2, 3] is equal to 0.89 for the confidence region ct ~< 0.4.
lnL =n lnp +np ln2 + (p - 1). ~ In t j - ~. (2tk)p j=,
(19)
k=l
and its maximum was found. The partial derivatives of In L with respect to 2 and p were found: a lnL ~2
np 2
~ tkp(2tk)p- 1
(20)
k=l
and lnL - =-n+nln2+ @ P
~ l n t j - - ~ (2tk)Pln(2tk). k=~
j=~
(21) Such values of 2 and p are found, for which In L achieves its maximum [11, 15, 16], so the right hand sides of equations (20) and (21) are equated to zero and the system is solved with respect to 2 and p: tkp(2tk)P- J = 0
,~
(22)
k=l
and
n P
+ n In 2 + ~ In t j - ~ (2t~)r ln(2tk) = 0. (23) j=1
A
(24)
k=l
Thus 2P -
-
n -
(25)
(tky' k=l
Transforming equation (23) we obtain: n
- + n l n 2 + ~ In t j = 2 ' l n 2 . j=l
~ (tk)r k=,
+ 2 ?. ~ (tk) pIn tk.
(26)
k=l
When we use (25) in (26) we obtain: - +
In tj = m /='
Comparing the values of parameters 2 and p estimated with the graphical method [equations (15) and (16)] with the values of 2 and p estimated with the maximum likelihood method [equations (28) and (29)] it was found that the values obtained do not differ significantly. So, it can be said that both the methods are effective. The maximum likelihood method is more credible, because it considers information about systems which have not failed until the end of the experiment. So the values of parameters estimated with the maximum likelihood method are accepted for further calculations. Substituting the estimated values of 2 = 0.0004168 and p = 4.913 and the efficient time of the devices from Table 1 into equation (19) we obtain: In L = - 56.2475.
(30)
Application of a random variable distribution (in this case the Weibull distribution) to the quantitative representation of a physical process needs checking if the distribution fits the data using one of the statistical tests (X: statistic or Kolmogorov statistic) [2, 3, 8]. REFERENCES
~ (tk)r.
:p2 p-l"
P
5. CONCLUSIONS
k=~
From equation (22) we obtain:
P
339
"
(tk) p In t k.
(27)
Y(tkY' k=, k=l
Substituting the values of In tj and In tk from Table 1 into equation (27), the value of parameter p is calculated, for which the equation is true. The following value of p is found from the program: p = 4.913.
(28)
Substituting the value of p and the values of tk (from Table 1) into equation (25) we obtain: 2 = 4.168 x 10 -4 = 0.0004168.
(29)
I. R. E. Barlow and F. Proschan, Mathematical Theory of Reliability. John Wiley, New York (1965). 2. D. Bobrowski, Mathematical Models and Methods in Reliability Theory. WNT, Warsaw (1985). 3. B. W. Gniedenko, J. K. Bielajev and A. D. Sotoviev, Mathematical Methods in Reliability. WNT, Warsaw (1966). 4. I. J. J62wiak, Zuverlaessigkeit einer ausgewaehlten Klasse von Systemen mit hierarchischer Struktur. Nachrichtentech. Elektron., Berlin 35(5), 188-189 (1985). 5. I. J. J62wiak, Zeit haveriefreier Arbeit eines digitalen Systems mit Zuverlaessigkeitsreserve. Nachrichtentech. Elektron., Berlin 40(2), 72-73 (1990). 6. I. J. J6~wiak, Funktionelles Zuverlaessigkeits Modell eines eine Aufgabenmenge realisierenden digitalen Systems. Nachrichtentech. Elektron., Berlin 40(3), 95-97 (1990). 7. I. J. J6~'wiak and U. Kosmol, Stochastische Modelierung von fehlertoleranten Systemen. Nachrichtentech. Elektron., Berlin 39(4), 152-153 (1989). 8. J. M. Juran and F. M. Gryna, Quality Planning and Analysis. McGraw-Hill, New York (1970). 9. D. R. Cox and D. V. Hinkley, Theoretical Statistics. Chapman & Hall, London (1974). 10. D. R. Cox and D. Oakes, Analysis of Survival Data. Chapman & Hall, London (1983). 11. J. D. Kalbfleisch and R. L. Prentice, The Statistical Analysis of Failure Time Data. John Wiley, New York (1980). 12. N. R. Mann, R. E. Schafer and N. D. Singpurwalla, Methods for Statistical Analysis of Reliability and Life Data. John Wiley, New York (1974). 13. I. J. J62wiak and I. Mazurkiewicz, The graphical method of evaluation of the Weibull proportional hazards model parameters. Polish Acad. Sci. J. Z E M (submitted). 14. D. R. Cox, Partial likelihood. Biometrika 62(2), 269-276 (1975).
340
IRENEUSZJ. J6~WIAK
15. A. K. S. Jardine and M. Anderson, Use of concomitant variables for reliability estimation. Maintenance Management Int. 5, 135-140 (1985).
16. I. J. J6~ciak, Weibull proportional hazards model for concomitant variables dependent on time. Polish Acad. Sci. J. Z E M (submitted).