Reliability Measures of a Series System with Weibull ... · increasing failure rate with passage of...

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)

http://www.amazon.in/Elsayed-A.-Elsayed/e/B001HNUPQI/ref=dp_byline_cont_book_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]


MTSF= Г1

𝛽1+1⁄

[𝜆1(𝛽1+1)𝛽1]1

𝛽1+1

Г1𝛽2+1⁄

[𝜆2(𝛽2+1)𝛽2]1

𝛽2+1


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


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


Fig.3: MTSF 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 →


λ=0.01

λ=0.02

λ=0.03

λ=0.04

λ=0.05




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

→


β=0.1

β=0.2

β=0.3

β=0.4

β=0.5




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 →


β=0.1

β=0.2

β=0.3

β=0.4

β=0.5


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

→


t=5

t=10

t=15

t=20

t=25


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


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:



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


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

→


λ=0.01

λ=0.02

λ=0.03

λ=0.04

λ=0.05



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


0

2

4

6

8

10

12

14

n 1 2 3 4 5 6 7 8 9

MTS

F →


λ=0.01

λ=0.02

λ=0.03

λ=0.04

λ=0.05



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


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

→


t=5

t=10

t=15

t=20

t=25


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.

REFERENCES

[1] Balagurusamy, E. (1984): Reliability Engineering, Tata McGraw Hill

Publishing Co. Ltd., India.

[2] Srinath, L.S. (1985): Concept in Reliability Engineering, Affiliated East-West

Press (P) Ltd.

[3] Rausand , M. and Hsyland, A. (2004): System Reliability Theory, John Wiley

& Sons, Inc., Hoboken, New Jersey,

[4] Navarro, J. and Spizzichino, F. (2010): Comparisons of series and parallel

systems with components sharing the same copula, Applied Stochastic Models

In Business and Industry, Vol. 26(6), pp. 775-791.

[5] Elsayed, A. (2012): Reliability Engineering, Wiley Series in Systems

Engineering and Management.

[6] Nandal, J., Chauhan, S.K. & Malik, S.C. (2015): Reliability and MTSF of a

Series and Parallel systems, International Journal of Statistics and Reliability

Engineering, Vol. 2(1), pp. 74-80.

[7] Chauhan, S.K. and Malik, S.C. (2016): Reliability Evaluation of a Series and

Parallel systems for Arbitrary Values of the Parameters, International Journal

of Statistics and Reliability Engineering, Vol. 3(1), pp.10-19,

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