reliability measures of a series system with weibull ... · increasing failure rate with passage of...
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International Journal of Statistics and Systems
ISSN 0973-2675 Volume 11, Number 2 (2016), pp. 173-186
© Research India Publications
http://www.ripublication.com
Reliability Measures of a Series System with Weibull
Failure Laws
S.K. Chauhan and S.C. Malik
Department of Statistics, M.D. University, Rohtak – 124001 (Haryana)
Email: [email protected] & [email protected]
Abstract
The Weibull distribution is widely used in reliability and life data analysis due
to its versatility. And, this distribution has been considered as a popular life
time distribution which describes modeling phenomena with monotonic failure
rates of components. Depending on the values of the parameters it can be used
to model a variety of life behaviors. In this paper, reliability measures such as
reliability and mean time to system failure (MTSF) of a series system of ‘n’
identical components by considering Weibull failure laws are obtained. The
results for these measures are also evaluated for the special case of Weibull
distribution i.e. by assuming Rayleigh failure laws. The behavior of MTSF and
reliability has been observed graphically for arbitrary values of the parameters
related to number of components, failure rates and operating time.
Keywords: Series System, Reliability, MTSF, Weibull Failure Laws
1. INTRODUCTION It has commonly known that performance of operating systems depends entirely on
the configurations of their components. The system may have simple or complex
structure of the components. And, accordingly several configurations of the
components have been evolved as a result of research in the field of reliability
engineering. The series systems are one of them being used in many systems like
wheat harvesting system where a tractor, wagon and combine are connected in series.
In a series system, the components are arranged in such a way that the successive
operation of the system depends on the proper operation of all the components.
Therefore, reliability of such systems has become a matter of concern for the
engineers and researchers in order to identify the factors which can be used to
improve their performance. There are several systems in which components have
174 S.K. Chauhan and S.C. Malik
monotonic failure rates. For example, the hazard rate of rotating shafts, valves and
cams are of non linear nature due to aging and working stress. In such systems, the
component’s life time distributed by cumulative damage and thus they have
increasing failure rate with passage of time. Balagurusamy (1984) and Srinath(1985)
determine reliability measures of a series system for Exponential distribution. Elsayed(2012) developed reliability measure of some system configurations using
Exponential, Rayleigh and Weibull distributions. Navarro and Spizzichino(2010)
made a Comparison of series and parallel systems with components sharing the same
copula. Recently, Nandal et al. (2015) evaluated the Reliability and Mean time to
System Failure (MTSF) of a Series System with exponential failure laws.
The Weibull distribution is widely used in reliability and life data analysis due to its
versatility. And, this distribution has been considered as a popular life time
distribution which describes modeling phenomena with monotonic failure rates of
components. Depending on the values of the parameters it can be used to model a
variety of life behaviors. In this paper, reliability measures such as reliability and
mean time to system failure (MTSF) of a series system of ‘n’ identical components by
considering Weibull failure laws are obtained. The results for these measures are also
evaluated for the special case of Weibull distribution i.e. by assuming Rayleigh failure
laws. The behavior of MTSF and reliability has been observed graphically for
arbitrary values of the parameters related to number of components, failure rates and
operating time.
2. NOTATIONS
R(t) = Reliability of the system, 𝑅𝑖(𝑡) = Reliability of the 𝑖𝑡ℎ component
h(t)= Instantaneous failure rate of the system,
ℎ𝑖(𝑡) = Instantaneous failure rate of 𝑖𝑡ℎ component, λ = Constant failure rate
T = Life time of the system, 𝑇𝑖= Life time of the 𝑖𝑡ℎ component.
3. SYSTEM DESCRIPTION
Here, a series system of ‘n’ components is considered which can fail at the failure of
any one of the components. The state transition diagram is shown in Fig. 1
Fig:1 A series system of ‘n’ components.
The reliability of the system is given by
R(t) = Pr[T>t] = Pr[min(𝑇1, 𝑇2,…….., 𝑇𝑛)>t] = Pr[𝑇1>t, 𝑇2>t,……….,𝑇𝑛>t]
=∏ Pr [𝑇𝑖 > 𝑡]𝑛𝑖=1 =∏ 𝑅𝑖(𝑡)𝑛
𝑖=1 (1)
Reliability Measures of a Series System with Weibull Failure Laws 175
The mean time to system failure is given by
MTSF=∫ ∏ 𝑅𝑖(𝑡)𝑛𝑖=1 𝑑𝑡
∞
0 (2)
4. RELIABILITY MEASURES OF A SERIES SYSTEM WITH WEIBULL
DISTRIBUTION:
Suppose failure rate of all components are governed by the Weibull failure law i.e.
ℎ𝑖(𝑡) = 𝜆𝑖𝑡𝛽𝑖
Then, the components reliability is given by
𝑅𝑖(𝑡) = 𝑒− ∫ ℎ𝑖(𝑢)𝑑𝑢𝑡
0 = 𝑒− ∫ 𝜆𝑖𝑢𝛽𝑖𝑑𝑢𝑡
0 = 𝑒−𝜆𝑖
𝑡𝛽𝑖+1
𝛽𝑖+1
Therefore, the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑅𝑖(𝑡) = ∏ 𝑒−𝜆𝑖
𝑡𝛽𝑖+1
𝛽𝑖+1 = 𝑛𝑖=1 𝑒
− ∑ 𝜆𝑖𝑡𝛽𝑖+1
𝛽𝑖+1𝑛𝑖=1 𝑛
𝑖=1
And,
MTSF = ∫ 𝑒− ∑ 𝜆𝑖
𝑡𝛽𝑖+1
𝛽𝑖+1𝑛𝑖=1 dt
∞
0 = ∏
Г1𝛽𝑖+1⁄
[𝜆𝑖(𝛽𝑖+1)𝛽𝑖]
1𝛽𝑖+1
𝑛𝑖=1
For identical components we can have
ℎ𝑖(𝑡) = 𝜆𝑡𝛽
Then the system reliability is given by
𝑅𝑠(𝑡) = 𝑒−𝑛𝜆
𝑡𝛽+1
𝛽+1 and MTSF= ∫ 𝑒−𝑛𝜆
𝑡𝛽+1
𝛽+1 𝑑𝑡 = Г
1
𝛽+1
[𝑛𝜆(𝛽+1)𝛽]1
𝛽+1
∞
0
Illustrations
1. For a single component, the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑅𝑖(𝑡)1𝑖=1 = ∏ 𝑒
−𝜆𝑖𝑡𝛽𝑖+1
𝛽𝑖+1 =1𝑖=1 𝑒
−𝜆1𝑡𝛽1+1
𝛽1+1 and MTSF= Г1
𝛽1+1⁄
[𝜆1(𝛽1+1)𝛽1]1
𝛽1+1
For identical components, we can have
ℎ𝑖(𝑡) = 𝜆𝑡𝛽 Then the system reliability is given by
𝑅𝑠(𝑡) = 𝑒−𝑛𝜆
𝑡𝛽+1
𝛽+1 and MTSF= Г
1
𝛽+1
[𝜆(𝛽+1)𝛽]1
𝛽+1
2. Suppose system has two components, then the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑒−𝜆𝑖
𝑡𝛽𝑖+1
𝛽𝑖+1 = 𝑒− ∑ 𝜆𝑖
𝑡𝛽𝑖+1
𝛽𝑖+12𝑖=1 =2
𝑖=1 𝑒−[𝜆1
𝑡𝛽1+1
𝛽1+1+𝜆2
𝑡𝛽2+1
𝛽2+1]
176 S.K. Chauhan and S.C. Malik
MTSF= Г1
𝛽1+1⁄
[𝜆1(𝛽1+1)𝛽1]1
𝛽1+1
Г1𝛽2+1⁄
[𝜆2(𝛽2+1)𝛽2]1
𝛽2+1
For identical components, we can have
ℎ𝑖(𝑡) = 𝜆𝑡𝛽 Then the system reliability is given by
𝑅𝑠(𝑡) = 𝑒−2𝜆
𝑡𝛽+1
𝛽+1 and MTSF= ∫ 𝑅𝑠∞
0(𝑡)𝑑𝑡 = ∫ 𝑒
−2𝜆𝑡𝛽+1
𝛽+1∞
0𝑑𝑡 =
Г1
𝛽+1
[2𝜆(𝛽+1)𝛽]1
𝛽+1
In a similar way we can obtain reliability and MTSF of a system having three or more
components connected in series.
5. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE
PARAMETERS
Reliability and mean time to system failure (MTSF) of the system has been obtained
for arbitrary values of the parameters associated with number of components(n),
failure rate (λ) , operating time of the component (t) and shape parameter (β). The
results are shown numerically and graphically as:
Table 1: Reliability Vs No. of Components (n)
No. of
Components
n
Reliability
λ=0.01,
t=10,
β=0.1
λ=0.02,
t=10, β=0.1
λ=0.03,
t=10, β=0.1
λ=0.04,
t=10, β=0.1
λ=0.05,
t=10, β=0.1
1 0.891859 0.795412 0.709395 0.6326797 0.564261
2 0.795412 0.63268 0.503241 0.4002835 0.31839
3 0.709395 0.503241 0.356996 0.2532513 0.179655
4 0.63268 0.400284 0.253251 0.1602269 0.101372
5 0.564261 0.31839 0.179655 0.1013723 0.0572
6 0.503241 0.253251 0.127446 0.0641362 0.032276
7 0.44882 0.201439 0.09041 0.0405777 0.018212
8 0.400284 0.160227 0.064136 0.0256727 0.010276
9 0.356996 0.127446 0.045498 0.0162426 0.005799
10 0.31839 0.101372 0.032276 0.0102763 0.003272
Reliability Measures of a Series System with Weibull Failure Laws 177
Fig.2: Reliability Vs No. of Components (n)
Table 2: MTSF Vs No. of Components (n)
No. of
Compo
nents n
MTSF
λ=0.01,
t=10, β=0.1
λ=0.02,
t=10, β=0.1
λ=0.03,
t=10, β=0.1
λ=0.04,
t=10, β=0.1
λ=0.05,
t=10, β=0.1
1 69.2305736 36.8667 25.500655 19.63228 16.02768
2 36.8667028 19.63228 13.5796227 10.45459 8.535069
3 25.500655 13.57962 9.39300903 7.231428 5.903697
4 19.6322766 10.45459 7.23142806 5.567284 4.545099
5 16.0276796 8.535069 5.90369698 4.545099 3.710593
6 13.5796227 7.231428 5.00197028 3.850884 3.14384
7 11.8039398 6.28584 4.34790843 3.347339 2.732749
8 10.4545907 5.567284 3.85088401 2.964693 2.420359
9 9.39300903 5.00197 3.45985696 2.663652 2.17459
10 8.5350687 4.545099 3.14383993 2.420359 1.975967
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n 1 2 3 4 5 6 7 8 9
Re
liab
ility
→
No. of Components →
λ=0.01
λ=0.02
λ=0.03
λ=0.04
λ=0.05
178 S.K. Chauhan and S.C. Malik
Fig.3: MTSF Vs No. of Components (n)
Table 3: Reliability Vs No. of Components (n)
No. of
Components
n
Reliability
β=0.1,
λ=0.01,t=10
β=0.2,
λ=0.01,t=10
β=0.3,
λ=0.01,t=10
β=0.4,
λ=0.01,t=10
β=0.5,
λ=0.01,t=10
1 0.891859 0.876276 0.857716 0.8357544 0.809921
2 0.795412 0.767859 0.735678 0.6984855 0.655972
3 0.709395 0.672856 0.631003 0.5837623 0.531286
4 0.63268 0.589608 0.541221 0.4878819 0.430299
5 0.564261 0.516659 0.464214 0.4077495 0.348509
6 0.503241 0.452736 0.398164 0.3407784 0.282264
7 0.44882 0.396721 0.341512 0.2848071 0.228612
8 0.400284 0.347637 0.292921 0.2380288 0.185158
9 0.356996 0.304626 0.251243 0.1989336 0.149963
10 0.31839 0.266937 0.215495 0.1662596 0.121458
0
10
20
30
40
50
60
70
80
n 1 2 3 4 5 6 7 8 9
MTS
F →
No. of Components →
λ=0.01
λ=0.02
λ=0.03
λ=0.04
λ=0.05
Reliability Measures of a Series System with Weibull Failure Laws 179
Fig.4: Reliability Vs No. of Components (n)
Table 4: MTSF Vs No. of Components (n)
No. of
Compo
nents n
MTSF
β=0.1,
λ=0.01,t=10
β=0.2,
λ=0.01,t=10
β=0.3,
λ=0.01,t=10
β=0.4,
λ=0.01,t=10
β=0.5,
λ=0.01,t=1
0
1 69.2305736 50.82565 39.0466108 31.09362 25.48514
2 36.8667028 28.52493 22.909827 18.95177 16.05463
3 25.500655 20.34613 16.7713103 14.18634 12.25198
4 19.6322766 16.00908 13.441888 11.55123 10.11378
5 16.0276796 13.29254 11.3217643 9.849336 8.715795
6 13.5796227 11.41888 9.84023475 8.646671 7.718261
7 11.8039398 10.04233 8.73992948 7.745149 6.964474
8 10.4545907 8.984791 7.88676206 7.040556 6.371284
9 9.39300903 8.144808 7.2036175 6.472461 5.890136
10 8.5350687 7.460185 6.64282138 6.003237 5.490606
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n 1 2 3 4 5 6 7 8 9
Re
liab
ility
→
No. of Components →
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
180 S.K. Chauhan and S.C. Malik
Fig.5: MTSF Vs No. of Components (n)
Table 5: Reliability Vs No. of Components (n)
No. of
Componen
ts
n
Reliability
t=5,
λ=0.01,
β=0.1
t=10,
λ=0.01,
β=0.1
t=15,
λ=0.01,
β=0.1
t=20,
λ=0.01,
β=0.1
t=25,
λ=0.01,
β=0.1
1 0.948009 0.891859 0.836294 0.7824509 0.73083
2 0.89872 0.795412 0.699387 0.6122294 0.534112
3 0.851994 0.709395 0.584893 0.4790394 0.390345
4 0.807698 0.63268 0.489142 0.3748248 0.285276
5 0.765705 0.564261 0.409067 0.293282 0.208488
6 0.725894 0.503241 0.3421 0.2294787 0.152369
7 0.688154 0.44882 0.286096 0.1795558 0.111356
8 0.652376 0.400284 0.23926 0.1404936 0.081382
9 0.618458 0.356996 0.200092 0.1099294 0.059476
10 0.586304 0.31839 0.167336 0.0860143 0.043467
0
10
20
30
40
50
60
70
80
n 1 2 3 4 5 6 7 8 9
MTS
F →
No. of Components →
β=0.1
β=0.2
β=0.3
β=0.4
β=0.5
Reliability Measures of a Series System with Weibull Failure Laws 181
Fig.6: Reliability Vs No. of Components and Time
6. RELIABILITY MEASURES FOR A SPECIAL CASE (RAYLEIGH
DISTRIBUTION) OF WEIBULL DISTRIBUTION:
The Rayleigh distribution has extensively been used in life testing experiments,
reliability analysis, communication engineering, clinical studies and applied statistics.
This distribution is a special case of Weibull distribution with the shape parameter
β=1.
When components are governed by Rayleigh failure laws, the component reliability is
given by
𝑅𝑖(𝑡) = 𝑒− ∫ ℎ𝑖(𝑢)𝑑𝑢𝑡
0 = 𝑒− ∫ 𝜆𝑖𝑢𝑑𝑢𝑡
0 = 𝑒−𝜆𝑖𝑡2
2 , where ℎ𝑖(𝑡) = 𝜆𝑖𝑡
Therefore, the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑅𝑖(𝑡) = ∏ 𝑒−𝜆𝑖𝑡2
2 𝑛
𝑖=1= 𝑒− ∑
−𝜆𝑖𝑡2
2𝑛𝑖=1
𝑛
𝑖=1
And, MTSF =∫ 𝑅(𝑡)𝑑𝑡∞
𝑡=0 = ∫ 𝑒− ∑
−𝜆𝑖𝑡2
2𝑑𝑡𝑛
𝑖=1∞
𝑡=0 = √
𝛱
2 ∑ 𝜆𝑖𝑛𝑖=1
For identical components we can have
𝜆𝑖𝑡 = 𝜆𝑡
The system reliability is given by
𝑅𝑠(𝑡) = 𝑒−𝑛𝜆𝑡2
2 and MTSF= ∫ 𝑅(𝑡)𝑑𝑡∞
𝑡=0= ∫ 𝑒
−𝑛𝜆𝑡2
2 𝑑𝑡∞
𝑡=0 = √
𝛱
2𝑛𝜆
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n 1 2 3 4 5 6 7 8 9
Re
liab
ility
→
No. of Components →
t=5
t=10
t=15
t=20
t=25
182 S.K. Chauhan and S.C. Malik
Illustrations:
1. For a single component the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑒−𝜆𝑖𝑡2
2 = 1𝑖=1 𝑒− ∑
−𝜆𝑖𝑡2
21𝑖=1
And MTSF= ∫ 𝑅(𝑡)𝑑𝑡∞
𝑡=0 = ∫ 𝑒− ∑
−𝜆𝑖𝑡2
2𝑑𝑡1
𝑖=1∞
𝑡=0 = √
𝛱
2 ∑ 𝜆𝑖1𝑖=1
= √𝛱
2𝜆1
For identical components, we can have
𝜆𝑖𝑡 = 𝜆𝑡 Then the system reliability is given by
𝑅𝑠(𝑡) = 𝑒−𝜆𝑡2
2 and MTSF= ∫ 𝑒−𝜆𝑡2
2 𝑑𝑡∞
𝑡=0 = √
𝛱
2𝜆
2. Suppose system has two components, then the system reliability is given by
𝑅𝑠(𝑡) = ∏ 𝑅𝑖(𝑡)2𝑖=1 = 𝑒− ∑
−𝜆𝑖𝑡2
22𝑖=1
And MTSF= ∫ 𝑅(𝑡)𝑑𝑡∞
𝑡=0 = ∫ 𝑒− ∑
−𝜆𝑖𝑡2
2𝑑𝑡2
𝑖=1∞
𝑡=0 = √
𝛱
2 ∑ 𝜆𝑖2𝑖=1
= √𝛱
2(𝜆1+𝜆2)
For identical component, we can have
𝜆𝑖𝑡 = 𝜆𝑡
Then the system reliability is given by
𝑅𝑠(𝑡) = 𝑒−𝜆𝑡2 and MTSF= ∫ 𝑒−𝜆𝑡2
𝑑𝑡∞
𝑡=0 =
1
2√
𝛱
𝜆
In a similar way we can obtain reliability and MTSF of a system having three or more
components connected in series.
7. RELIABILITY MEASURES FOR ARBITRARY VALUES OF THE
PARAMETERS
Reliability and mean time to system failure (MTSF) of the system has been obtained
for arbitrary values of the parameters associated with number of components(n),
failure rate (λ) and operating time of the component (t) The results are shown
numerically and graphically as:
Reliability Measures of a Series System with Weibull Failure Laws 183
Table 6: Reliability Vs No. of Components (n)
Number of
Components
n
Reliability
λ=0.01,t=10 λ=0.02,t=10 λ=0.03,t=10 λ=0.04, t=10 λ=0.05, t=10
1 0.606531 0.367879 0.2231302 0.135335283 0.08208499862
2 0.367879 0.135335 0.0497871 0.018315639 0.00673794700
3 0.22313 0.049787 0.0111090 0.002478752 0.00055308437
4 0.135335 0.018316 0.0024788 0.000335463 0.00004539993
5 0.082085 0.006738 0.0005531 0.000045400 0.00000372665
6 0.049787 0.002479 0.0001234 0.000006144 0.00000030590
7 0.030197 0.000912 0.0000275 0.000000832 0.00000002511
8 0.018316 0.000335 0.0000061 0.000000113 0.00000000206
9 0.011109 0.000123 0.0000014 0.000000015 0.00000000017
10 0.006738 0.000045 0.0000003 0.000000002 0.00000000001
Fig.7: Reliability Vs No. of Components (n)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
n 1 2 3 4 5 6 7 8 9
Re
liab
ility
→
No. of Components →
λ=0.01
λ=0.02
λ=0.03
λ=0.04
λ=0.05
184 S.K. Chauhan and S.C. Malik
Table 7: MTSF Vs No. of Components (n)
No. of
Components
n
MTSF
λ=0.01,t=10 λ=0.02,t=10
λ=0.03,t=1
0
λ=0.04,t=1
0
λ=0.05,t=1
0
1 12.5331 8.8622 7.236 6.2645 5.6049
2 8.8622 6.2665 5.1166 4.4311 3.9633
3 7.23601 5.1166 4.1777 3.16806 3.23604
4 6.2665 4.4311 3.618 3.1332 2.8024
5 5.6049 3.9633 3.236 2.8024 2.5066
6 5.1166 3.618 2.954 2.5583 2.2882
7 4.737 3.3496 2.7349 2.3685 2.1184
8 4.4311 3.1332 2.5583 2.2155 1.9817
9 4.1777 2.95408 2.412 2.0888 1.8683
10 3.9633 2.8024 2.2882 1.9816 1.7724
Fig.8: MTSF Vs No. of Components (n)
0
2
4
6
8
10
12
14
n 1 2 3 4 5 6 7 8 9
MTS
F →
No. of Components →
λ=0.01
λ=0.02
λ=0.03
λ=0.04
λ=0.05
Reliability Measures of a Series System with Weibull Failure Laws 185
Table 8: Reliability Vs No. of Components (n)
No. of
Components
n
Reliability
t=5, λ=0.01
t=10,
λ=0.01
t=15,
λ=0.01 t=20, λ=0.01 t=25, λ=0.01
1 0.882497 0.606531 0.3246525 0.135335283 0.043936933623
2 0.778801 0.367879 0.1053992 0.018315639 0.001930454136
3 0.687289 0.22313 0.0342181 0.002478752 0.000084818235
4 0.606531 0.135335 0.0111090 0.000335463 0.000003726653
5 0.535261 0.082085 0.0036066 0.000045400 0.000000163738
6 0.472367 0.049787 0.0011709 0.000006144 0.000000007194
7 0.416862 0.030197 0.0003801 0.000000832 0.000000000316
8 0.367879 0.018316 0.0001234 0.000000113 0.000000000014
9 0.324652 0.011109 0.0000401 0.000000015 0.000000000001
10 0.286505 0.006738 0.0000130 0.000000002 0.000000000000
Fig.9: Reliability Vs No. of Components (n)
8. DISCUSSION OF THE RESULTS
The results obtained for arbitrary values of the parameters indicate that reliability and
mean time to system failure of a series system of 10 identical components keep on
decreasing with the increase of the number of components and their failure rates.
However, the effect of number of components and their failure rates on reliability of
the system is much more in case components governed by Rayleigh failure laws then
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n 1 2 3 4 5 6 7 8 9
Re
liab
ility
→
No. of Components →
t=5
t=10
t=15
t=20
t=25
186 S.K. Chauhan and S.C. Malik
that of Weibull failure laws. In case of mean time to system failure, the effect is much
more when components follow Weibull failure laws rather than Rayleigh failure laws.
The reliability of the system goes on decreasing with the increase of operating time
irrespective of distributions governed by failure time of the components. The effect of
operating time on reliability is much more in case components follow Rayleigh failure
laws as compare to Weibull failure laws. However, there is no effect of operating time
on mean time to system failure (MTSF).
The results obtained for some more particular values of the shape parameter β (0.1,
0.2 ,0.3, 0.4 and 0.5) indicate that reliability and mean time to system failure of a
series system of 10 identical components decline with the increase of the value of β.
The results are shown numerically and graphically in respective tables and figures.
CONCLUSION
In present study, we conclude that the reliability and MTSF keep on decreasing with
the increase the number of components, failure rates and operating time of the
component. It is suggested that least number of component should be used in a series
system for better performance. However, the performance of such systems can be
improved by utilizing components which follow Weibull failure laws.
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