Chapter 3: Introduction to Reliability Theory

Claver Diallo

OUTLINE
1. Part 1: Basic Reliability Models
1. 2. 3. 1. 2. 3. 4. 5. System Reliability function Probability distributions Reliability Block Diagram Serial and Parallel Structures Stand-by Structure k-out-of n Structure Complex structure

2. Part 2: Reliability of Structures

3. Part 3: Reliability Allocation 4. References
2 2 2 2 2 2 2 2

Chapter 3 - Part 1: Basic Reliability Models

SYSTEM
System: a collection of components or items performing a specific function.

4 4 4 4 4 4 4 4

STATE OF A SYSTEM
A system is considered to be in one of the two following states:
In operation (Up) Failed (Down)

Transition from one state to the other occurs according to a known or unknown probability function. de aF de aF delllliiiiaF de aF
5 5 5 5 5 5 5 5 noitca ecnanetniaM noitca ecnanetniaM noitca ecnanetniaM noitca ecnanetniaM riapeR riapeR riapeR riapeR e eruliaF

gn arepO gn arepO gniiiittttarepO gn arepO

SYSTEM LIFETIME
Lifetime is a measure of performance. Lifetime is a measure of performance. In general, lifetime is measured by the number In general, lifetime is measured by the of hours the system was in operation. number of hours the system was in operation. It can also be measured by the number of It can also be measured by the number of km or miles raced, number of pages copied, pages copied, km or miles raced, number wheel rotations, …etc. of wheel rotations, …etc. Lifetime (T) is a nonnegative random variable. Lifetime (T) is a nonnegative random variable.
6 6 6 6 6 6 6 6

LIFETIME PROBABILITY DENSITY FUNCTION f(x).dx : probability that the system’s lifetime is in the interval [x; x+dx], or the probability that the system fails between the instants x and x+dx.

0

x

x+dx

f ( x )dx = Prob{ x < lifetime ≤x + dx }

∫0

∞

f ( x )dx = 1
7 7 7 7 7 7 7 7

RELIABILITY FUNCTION
Reliability is defined as the probability that a system will perform its intended function satisfactorily for a specified period of time under specified operating conditions.

0 ≤ R (t ) ≤ 1

If T is a random variable representing the lifetime of the system with pdf f(t), then its reliability function R(t) is given by
R ( t ) = P {T > t } = 1 − P {T ≤ t} = 1 − F (t ) R(t ) =

∫t

∞

f (x )dx

Example: Find R(t) for f(t)= λe−λt .
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F(t): Cumulative distribution function associated with the lifetime t
F (t ) = Proba {T ≤ t } =

∫0

f ( x )dx

R(t): Reliability function
R(t ) = Proba {T > t } =

∫t

∞

f ( x )dx

R(t) is the probability that the system’s lifetime is higher than t, or probability that the device will fail after t.
R (t ) + F (t
)

= 1

Operating

Failed
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ESTIMATION OF RELIABILITY FUNCTIONS
Experiment: N identical components are put to test, under the same conditions. Assume that at the end of the test, we have:
Ns : # of components surviving the test Nf : # of components who failed during the test Ns(t) + Nf(t) = N ( no replacement )
Population size N

N

to

Time

01 01 01 01 01 01 01 01

F(t) and R(t)

N s (t ) R ( t ) = lim N →∞ N N f (t ) F ( t ) = lim N →∞ N N S (t ) N f (t ) R (t ) + F (t ) = + =1 N N N S (t ) : empirical reliability function N
R(t) N
R(t1) R(t2) t1 t2

F(t) N

t to

t to

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FAILURE OR HAZARD RATE FUNCTION
Let consider a system that has survived and reached instant t. h(t): instantaneous rate of failure at time t. Formally h(t) is the conditional probability of failure per unit of time given that the system has survived to time t. N
Ns(t) Nf(t)
Nf(t+∆) ∆

f (t ) h(t ) = R(t ) t ∆ time t+∆ ∆

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BATHTUB CURVE OR FAILURE RATE PLOT h(t) Burn-in (DFR) Useful Life (CFR) Wearout phase (IFR)
Maintenance action is carried out

Early failures Random failures

Wear out failures

time Causes: * Manufacturing defects; * Poor quality control; Reduced by: * Burn-in Testing; * Quality Control *Acceptance Testing Causes: * Environment; * Random Loads; * Human error; * « Acts of God »; * Chance events Reduced by: * Redundancy; * Excess strentgh Causes: * Fatigue; * Corrosion; * Aging; etc. Reduced by: * Derating; * Prev. Maintenance; * Technology

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Failure rate profile for:
Electronic device

Software

Mechanical system

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SUMMARY
The wearout or failure process is uniquely characterized (defined) once any of the following 4 functions is known: f(t), F(t), R(t) or h(t).
P { Lifetime > t } P { Lifetime ≤ t } Lifetime pdf → → → R (t ) = F (t ) =

∫t

∞ t

f ( x )dx
(

∫0 f

x )dx

dF ( t ) f (t ) = dt → h (t ) = f (t ) R (t )

Hazard rate function  R (t ) + F (t ) = 1    ∞    ∫0 f ( x )dx = 1  

51 51 51 51 51 51 51 51

f(t) is known:
F (t ) =

∫0 f

t

(

x dx →
)

R (t )

=

∫t

∞

f x dx
( )

→ h(t ) =

F(t) is known: f (t )

∫t

∞

f (t ) f ( x )dx

dF ( t ) = → R (t ) = 1 − F (t ) → dt

h (t )

dF ( t ) dt = 1 − F (t )

R(t) is known:
F (t ) = 1 − R (t ) → f (t ) = dF ( t ) dR ( t ) =− → h (t ) = dt dt − dR ( t ) dt = − 1 . dR ( t ) R (t ) R ( t ) dt

h(t) is known: dR ( t ) R (t ) dt d R (t ) − h ( t )d t = R (t ) h (t ) = − 1 . ln R ( t
)
t

= − ∫ h ( x )d x
0

t

R (t ) = e

− ∫ h ( x )d x
0

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f (t )

F (t)

R(t)

h(t ) f (t )

f (t ) F (t )

– dF (t) dt −dR(t) dt

∫

t

0

f ( x)dx
–

∫

∞

t

f ( x)dx

∫

∞

f ( x)dx dF (t )

t

1 − F (t )
–

[1 − F (t )]dt

R(t) h(t) 1 − R(t )

−dR(t ) R(t )dt

h(t ) ⋅ e ∫

− h( x) dx
0

t

1− e ∫

− h( x) dx
0

t

e∫

− h ( x )dx
0

t

–

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MEAN TIME TO FAILURE (MTTF)
Mean or expected value of the lifetime
State of the system

1 X1 0 Time X2 X3

MTTF ≈

X1 + X 2 + ... + Xn n

MTTF = ∫ tf (t)dt = ∫ R(t)dt
0 0
81 81 81 81 81 81 81 81

∞

∞

MEDIAN TIME TO FAILURE (TMED)
Median time to failure is defined by:

R(tmed ) = 0.5
The median divides the distribution into two halves, with 50% of the failures occurring before tmed and 50% of the failures occurring after the tmed. tmed may be preferred to MTTF when the distribution is highly skewed. (see fig. 2.2 page 27 in the textbook) Using the same structure of formulation, we can define the Design Life for a given reliability value V as the lifetime tv such as:

R(tv ) = V

Example: Q2.7 page 39
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CONDITIONAL RELIABILITY
Conditional reliability is defined for a system or a component that has been used for some time T0. Conditional reliability can be used to study a used system or determine the reliability of a system following a burn-in period.
R (T0 + t ) R ( t T0 ) = P {T > T0 + t T > T0 } = , R (T0 ) t,T0 ≥ 0.

Residual MTTF at T0:
RMTTF(T0 ) = ∫
∞ 0 ∞ 1 R ( t T0 )dt = ∫T0 R(t)dt, R (T0 )

T0 ≥ 0.

02 02 02 02 02 02 02 02

COMMON CONTINUOUS DISTRIBUTIONS
The Normal Distribution A r.v. x follows the normal distribution iff its pdf is given by f (x ) = 1 σ 2π e
1 x − µ −   2 σ     
2

E [x ] = µ

V [ x ] = σ2

The normal distribution with mean 0 and variance 1 is called the standard normal distribution. If the random variable x ~ N( µ,σ2) then the x − µ z = random variable ~ N(0,1). σ
12 12 12 12 12 12 12 12

The Exponential Distribution The r.v. x is said to be exponentially distributed with parameter λ>0 if its pdf is given by:

f x = λe
( )

−λx

E [ x ] = 1λ

V [x ] = 1

λ2

The exponential distribution is used to model failures or breakdowns that are completely random. λ= failures per unit of time. If the number of occurrences of an event has a Poisson distribution with parameter λ, then the distribution of the interval between occurrences is exponential with parameter λ.
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The exponential distribution is the only distribution with constant failure rate (CFR).
−λt    f ( t ) = λe    F ( t ) = 1 − e−λt      R ( t ) = e −λt  

h ( t ) = λ      MTTF = 1   λ 

Memoryless property:
R ( t T0 ) = P {T > T0 + t T > T0 } = R ( t + T0 ) = R(t ), R (T0 ) t,T0 ≥ 0.

The time to failure of a CFR component is not dependent on how long it has been operating. There is no aging or wearout effect. (old as good as new).
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42 42 42 42 42 42 42 42

RELIABILITY BOUNDS
Assume a constant failure rate that is bounded. Then we have: λ ≤ λ ≤λ
L U

Therefore if the failure rate is bounded, the reliability function can also be bounded. Even when the CFR model does not apply, it can be used to provide bounds on the system reliability because the exponential distribution is at the boundary between DFR and IFR distributions. If we know the value of MTTF for a DFR or IFR process then the following bounds can be used:
For a DFR process:
 −t M T T F e   R (t ) ≤   −1   M TTF .e    t if if t ≤ MTTF t > MTTF

e −λ U t ≤ R ( t ) ≤ e −λ L t

For an IFR process:

e  R (t ) ≥  0 

−t

M TTF

if if

t < M TT F t ≥ M TTF

1 R (t ) ≤  e

if
−W t

t ≤ MTTF t > MTTF

if

where 1 – w.MTTF = e-wt. w is computed for each value of t.

The Weibull distribution f (x ) = β x −γ θ θ

(

)

β −1

e

−

(

x−γ θ

)

β

x    β  with   θ   γ   

≥ θ = shape parameter = scale parameter = location param eter

The Weibull distr. is very flexible (can assume a wide variety of shapes). Useful distribution for modelling the lifetime of mechanical systems.
62 62 62 62 62 62 62 62

1  1 +  E x = γ + θΓ     β
[ ]

2 1    2  V x = θ  Γ  1 +  − Γ 1 +      β  β   
[ ]

2 

Γ(x + 1) = x Γ(x ) Γ(α) =

∫0

+∞

x α −1e−x dx

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WEIBULL DISTRIBUTION t−γ β   β t − γ β −1 −( θ )  f (t ) =  e  θ θ    t−γ β  −(  F (t ) = 1 − e θ ) t > γ   

(

)

t−γ β   −(  R (t ) = e θ )     β t − γ   h (t ) =  θ θ  

(

)

β −1

h(t)

β1

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FIELD DATA ANALYSIS

Statistical Approaches Accelerated Life-testing Reliability Databases Bayesian inference Histograms Lifetime Analysis Software Experfit; Weibull++; Stat:fit; Relex; … etc. Distributions a priori Exponential; Gamma; Weibull; Log-normal; Normal, … etc Lifetime probability density function f(t) Failure rate r(t) Reliability function R(t)

MTTF

Data

Field Data

Distribution function F(t)

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REFERENCES
Richard E. Barlow, Frank Proschan (1996). Mathematical Theory of Reliability, SIAM, Mathematical statistics. C.E. Ebeling (1997). An Introduction to Reliability and Maintainability Engineering, McGraw-Hill, New-York, NY. B.S. Blanchard, D. Verma, E.L. Peterson (1995). Maintainability: a key to effective serviceability and maintenance management, Wiley, New-York, NY.
03 03 03 03 03 03 03 03

Chapter 3 - Part 2: Reliability of Structures

OUTLINE
1. 2. 3. 4. 5. 6. 7. 8.

Reliability Block Diagram Serial Structure Parallel Structure Combined Series-Parallel Structures k-out-of-n Structures Standby Structure Complex Structures Reliability of repairable systems
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1. RELIABILITY BLOCK DIAGRAM (RBD)
A Reliability Block Diagram (RBD) is used to depict the relationship between the functioning of a system and the functioning of its components. It reads from left to right. A RBD does not necessarily represent how the components are physically connected in the system. Example: successful presentation
Class Computer Presentation file Laptop

Network

Projector

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2. SERIAL STRUCTURE (CONFIGURATION)
Presentation file Class Computer Network Projector

n independent components If one of the n components fails, the whole system fails. The system will operate iff all the components work.
Rs (t ) =

∏ Ri (t ) i =1

n

Rs (t ) : Reliability of the system Ri (t ) : Reliability of component i

Rs (t ) < min{Ri (t )}
43 43 43 43 43 43 43 43

EXAMPLE
A system is made of 6 components in series. Their failure rates are constant and given below. Find the reliability function of the system. What is the reliability for a mission of length 300 hours? λ (failure per hour)
Component 1 Component 2 Component 3 Component 4 Component 5 Component 6 0.005 0.005 0.05 0.005 0.05 0.05
53 53 53 53 53 53 53 53

SERIAL CONFIGURATION
Lifetime:
Ts = min {Ti }
1≤i ≤n

Failure rate: hs (t ) =

∑ hi (t ) i =1

n

hs (t ) : failure rate of the system hi (t ) : failure rate of component i

63 63 63 63 63 63 63 63

COMPETING FAILURE MODES
Failure modes: are the ways, or modes, in which something might fail.
Failure Modes: Chemical, Electrical, Physical, Mechanical, Structural, Thermal Processing, etc. (see page 169 textbook)

For many products (complex systems), there are many failure modes that can cause failure. For non-repairable systems, we can consider that there can only be one failure mode for each failed product.
Failure modes "compete" as to which will be the first to cause the failure of the system. This can be viewed as a series system reliability model, with each failure mode composing a block of the series system.
Corrosion Buckling Fatigue Wear

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FAILURE ON DEMAND
Some systems that operate on a cyclical basis may fail on demand. (e.g., AC-Heating system, light bulbs)
Idle Operating On Demand

If the system is CFR and there is probability p of failure on demand, then it will have an effective constant failure rate on a unit of time basis: tI tO 1 λeff = ⋅ λI + ⋅ λO + ⋅p tI + tO tI + tO tI + tO λI : average failure rate while idle λO : average failure rate while operating tI : average length of the idle time period per cycle tO : average length of the operating time period per cycle

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3. PARALLEL STRUCTURE (CONFIGURATION)
a.k.a redundant configuration n independent components The system works as long as at least one of the n components works. A parallel system is failed iff all components are failed. 1
Fs (t ) =

∏ Fi (t ) i =1

n

2

i

Fs (t ) : unreliability of the system Fi (t ) : unreliability of component i
93 93 93 93 93 93 93 93 n PARALLEL CONFIGURATION
Reliability of the parallel structure
Rs (t ) = 1 − ∏ [1 − Ri (t )] i =1 n

Lifetime
Ts = Max{Ti }
1≤i ≤n

Note: if

Ri (t ) = e−λt

then n 1 1 MTTF = ∑ λ i =1 i
04 04 04 04 04 04 04 04

4. COMBINED SERIES-PARALLEL SYSTEMS
Low-level redundancy system:
Each component comprising the system may have one or more parallel components.

High-level redundancy system:
The entire system is placed in parallel with one or more identical systems.

A component is at a lower level compared to the system which is at a higher level. Low-level redundancy system is always better than its corresponding high-level redundancy system.
14 14 14 14 14 14 14 14

5. K-OUT-OF-N PARALLEL STRUCTURE n independent components The system works as long as at least k out of the n components work. 1 R1(t ) = R2(t ) = ... = R(t ) then If 2 n n  
  [R(t )]j [1 − R(t )]n − j j ∑  j =k 

Rs (t ) =

k/n

i n

Note: A series structure is an n-out-of-n structure A parallel structure is an 1-out-of-n structure
24 24 24 24 24 24 24 24

K-OUT-OF-N PARALLEL STRUCTURE

Note:
A series structure is an n-out-of-n structure A parallel structure is an 1-out-of-n structure

If

Ri (t ) = e−λt

then n 1 2 k/n MTTF(k ,n )

1 1 = ∑ λ i =k i

i n
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VOTING REDUNDANCY – MAJORITY VOTING SYSTEM
In a voting redundancy system, n parallel signals are channeled through a decision-making device that provides the required output as long as at least a predetermined number k of the n signals are in agreement. Majority voting system 1 2 3
Sensors Alarm
2/ 3

E

If R1 (t ) = R2 (t ) = R3 (t ) = p(t ) , then
44 44 44 44 44 44 44 44

Rs (t ) = [3p 2 (t ) − 2p 3 (t )] ⋅ RE (t )

1

K-OUT-OF-N PARALLEL STRUCTURE

2 kn i n

An n-component system that works (or is good) iff at least k of the n components work (or are good) is called a k-out-of n:G system; An n-component system that fails iff at least k of the n components fail is called a k-out-of n:F system; A k-out-of n:G is equivalent to an (n-k+1)-out-of n:F system.
54 54 54 54 54 54 54 54

6. STANDBY STRUCTURE
SS

1 2

One component is operating and n-1 reserve n components are waiting to be switched on; The sensing and switching unit (SS) monitors the operation of the active component. Whenever a failure is detected, a reserve component is switched into active operation. 3 types of stand-by:
Cold stand-by: reserve component has zero failure rate (cannot fail while in stand-by) Warm stand-by: reserve component has a failure rate comprised between 0 and the failure rate of the active component. Hot stand-by: reserve component has same failure rate as active component (active redundant component).
64 64 64 64 64 64 64 64

COLD STANDBY WITH PERFECT SWITCH
SS Sensing and switching is perfect (instantaneous and failure free) Denote by λi the failure rate of component i if λ1 ≠ λ2 ≠... ≠ λn then:

1 2 n

Rs ( t ) =

if λ1 = λ2 =... = λn = λ then:
Rs ( t ) = e −λt ∑ n −1

( λt )k

k =0

k!

General Case (h1(t)= h2(t)=... = hn (t)= h(t) ):
74 74 74 74 74 74 74 74

[ − ln R(t ) ] Rs ( t ) = R(t ) ⋅ ∑ k! k =0

n −1

k

NUMBER OF SPARE PARTS REQUIRED
A spare part is modeled as a reserve component. Once the system reliability Rs(t,n) is known, we can determine the number of (n-1) of spare parts required to guarantee a pre-determined reliability R* for a mission duration t.
SS

1 2 n t, f(t), R*, n=n0

RS(t,n)

n=n+1

No

? Rs(t,n)>= R*
Yes

m*=n-1
84 84 84 84 84 84 84 84

NUMBER OF SPARE PARTS REQUIRED
Exponential distribution λ =0.02 ; i.i.d. components; without repair
1

8 Spares 7 SP
0.75

6 SP

Reliability R(T)

5 SP

0.5

4 SP

0.25

3 SP 2 SP

0 0 50 100 150 Mission Length T 200 250

1 Spare
300

94 94 94 94 94 94 94 94

7. COMPLEX STRUCTURE
Several methods can be used:
Parallel and Series Reductions; Pivotal Decomposition; Generation of Minimal cuts – Minimal Paths; Inclusion – Exclusion Method; Sum-of-Disjoint-Product (SDP) Method; Etc.

We will use the Parallel –Series Reductions and the Pivotal Decomposition method also known as the Shannon’s Decomposition.
05 05 05 05 05 05 05 05

EXAMPLE OF PARALLEL SERIES REDUCTION
4

2

5

3

6

1

7

10

8

11

9

15 15 15 15 15 15 15 15

PIVOTAL DECOMPOSITION
The Pivotal Decomposition method is based on the concept of conditional probability.
Pr(system works) = Pr(component i works) × Pr(system works | component i works) + Pr(component i fails) × Pr(system works | component i fails)

The efficiency of this method depends on the ease of evaluating the conditional probabilities. This means that the selection of the component to be decomposed plays an important role in the efficiency of this method. It involves choosing a “key” component and then calculating the reliability of the system twice:
Once as if the key component failed (R=0) Once as if the key component is working (R=1)

these two probabilities are then combined to obtain the reliability of the system, since at any given time the key component will be failed or operating.
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EXAMPLE:

THE BRIDGE STRUCTURE

1 3 2

4

5

R ( t ) = R3 (t ) ⋅ { 1 − [ 1 − R1(t ) ] ⋅ [ 1 − R2 (t ) ]} { 1 − [ 1 − R4 (t ) ] ⋅ [ 1 − R5 (t ) ]} + [ 1 − R3 (t ) ] ⋅ {1 − [ 1 − R1(t )R4 (t ) ] ⋅ [ 1 − R2 (t )R5 (t ) ]}

35 35 35 35 35 35 35 35

8. RELIABILITY OF REPAIRABLE SYSTEMS
Redundancy is defined as the use of additional components beyond the number actually required for satisfactory operation of a system for the purpose of improving its reliability.
A series system has no redundancy A parallel system has redundancy. Similarly k-out-of-n, parallelseries, series-parallel, and standby systems have redundancy.

For a system with redundancy, if an active component fails, other components can take the load and ensure that the system remains operational during the repair. It is therefore possible to define the concept of reliability of repairable systems for systems that have some level of redundancy.
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PARALLEL SYSTEM

WITH 2 I.I.D. COMPONENTS
1 2

1-2λ∆t λ∆t

2λ∆t λ∆t

0 µ∆t µ∆ t

1-λ∆tλ∆t µ∆t µ∆ t 1 λ∆t λ∆ t 2 1

R(t ) = P0 (t ) + P1 (t ) = 1 − P2 (t )

S1eS 2t − S2eS1t R(t ) = S1 − S2 −(3λ + µ) + λ2 + 6λµ + µ2 S1 = 2 −(3λ + µ) − λ2 + 6λµ + µ2 S2 = 2 3λ + µ MTTF = 2λ2

Improvement =

55 55 55 55 55 55 55 55

µ 2λ2

rrrrrrrriiiiiiiiaperrrrrrrr ttttttttuohttttttttiiiiiiiiW ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W
1-2λ∆t λ∆t 2λ∆t λ∆t 1-λ∆t λ∆t 0 1 λ∆t λ∆ t 2 1

rrrrrrrriiiiiiiiaperrrrrrrr httttttttiiiiiiiiW ape h W ape h W ape h W ape h W ape h W ape h W ape h W

R(t ) = 2e −λt − e −2λt 3 MTTF = 2λ

STANDBY SYSTEM WITH 2 I.I.D. COMPONENTS
1
SS

R(t)=e-λ t R(t)=e-λ t

1-λ∆t λ∆t

λ∆t λ∆ t

0 µ∆t µ∆ t

1-λ∆tλ∆t µ∆t µ∆ t 1 λ∆t λ∆ t 1

2

S1e S2t − S 2e S1t R(t ) = S1 − S 2 −(2λ + µ) + 4λµ + µ2 S1 = 2 −(2λ + µ) − 4λµ + µ2 S2 = 2 2λ + µ MTTF = λ2

R(t ) = P0 (t ) + P1 (t ) = 1 − P2 (t )

65 65 65 65 65 65 65 65

µ Improvement = 2 λ

tcefrep si tinu SS ehT tcefrep si tinu SS ehT tcefrep si tinu SS ehT tcefrep si tinu SS ehT

2

rrrrrrrriiiiiiiiaperrrrrrrr ttttttttuohttttttttiiiiiiiiW ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W ape uoh W
1-λ∆t λ∆t 0 λ∆t λ∆ t 1-λ∆t λ∆t 1 λ∆t λ∆ t 2

rrrrrrrriiiiiiiiaperrrrrrrr httttttttiiiiiiiiW ape h W ape h W ape h W ape h W ape h W ape h W ape h W

R(t ) = e−λt (1 + λt ) 2 λ

MTTF =

REFERENCES
B.S. Blanchard, D. Verma, E.L. Peterson (1995). Maintainability: a key to effective serviceability and maintenance management, Wiley, New-York, NY. C.E. Ebeling (1997). An Introduction to Reliability and Maintainability Engineering, McGraw-Hill, New-York, NY. W. Kuo, M.J. Zuo (2003). Optimal Reliability Modeling: principles and applications, Wiley, New-York, NY.J. B
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Chapter 3 - Part 3: Reliability Allocation

1. DEFINITION
Once the system reliability goals have been defined, reliability must then be allocated to the components and possibly subcomponents in a manner that will support these goals. Ideally, reliability allocation should be accomplished in a least-cost manner.
M in st: Z =

∑ C i (xi ) i =1

n

M ax st:

Z = Rs ( Ri + x i )

Rs ( Ri + x i ) ≥ R * 0 ≤ Ri + x i ≤ UB i < 1 where: ∑ C i ( x i ) ≤ Budget i =1

n

0 ≤ Ri + x i ≤ U B i < 1

Rs : is the system reliability R* : is the system reliability goal Ri : is the current reliability of component i UBi : is the upper bound on the current reliability of component i xi : is the increase in reliability of component i

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Ci ( xi ) : is the cost for increasing the reliability of component i by xi

2. METHODS
Several methods have been developed for reliability allocation such as ARINC, AGREE, etc. For this class, we will use two methods: the general optimization model describe in the previous slide, and the method to be presented next (reliability growth by improvement of one component at a time)

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3. RELIABILITY GROWTH BY IMPROVEMENT OF ONE COMPONENT AT A TIME Series structure Parallel structure
R s (t ) =

∏ R i (t ) j =1

n

R s (t ) = 1 −

∏ [ 1 − R i (t ) ] j =1

n

Select component k such as:

R k (t ) = M in { R i (t ) }
1≤ i ≤ n

Select component k such as: R k (t ) = M ax { R i (t ) }
1≤ i ≤ n

(the least reliable component)

(the most reliable component)

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sa hcus k nenopmoc ce eS sa hcus k nenopmoc ce eS ::::sa hcus k ttttnenopmoc ttttcelllleS sa hcus k nenopmoc ce eS

esac areneG ::::esac llllareneG esac areneG esac areneG

R s (t ) = Ψ ( R 1 (t ), R 2 (t ), ....., R n (t ))
∂ R s (t ) ∂ R s (t ) = M ax 1 ≤ i ≤ n ∂ R i (t ) ∂ R k (t )

{

}

REFERENCES
C.E. Ebeling (1997). An Introduction to Reliability and Maintainability Engineering, McGraw-Hill, New-York, NY. B.S. Blanchard, D. Verma, E.L. Peterson (1995). Maintainability: a key to effective serviceability and maintenance management, Wiley, NewYork, NY.

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