System Reliability

Random State Variables

1 2

1 2

( ), ( ), , ( ) are stochatically independent

binary random variables at time t

Pr ( ) 1 ( ) where 1,2, ,

Pr ( ) 1 ( )

where

( ) ( ), ( ), , (

n

i i

S

n

X t X t X t

X t p t i n

t p t

t X t X t X t

X

X

)

System Reliability/Availability

1 2

( ) 0 Pr ( ) 0 1 Pr ( ) 1

( )

Similarly, ( ) ( )

It can be shown that when the components are indep.

( ) ( ), ( ), , ( ) ( )

i i i

i

S

S n

E X t X t X t

p t

p t E t

p t h p t p t p t h t

X

p

Series Structure

1

1

1 1

( )

( )

( ) ( )

min ( )

n

ii

n

ii

n n

i ii i

ii

t t

h t E t E t

E t p t

h t p t

X X

p X X

X

p

Series Structure

A series structure is at most as reliable as the least reliable component. For a series structure of order n with the same components, its reliability is

10

( ) ( )

For example, 10, ( ) 0.95

( ) 0.95 0.60

nS

S

p t p t

n p t

p t

Parallel Structure

11

1

1 1

( ) 1 [1 ( )]

1 1 ( )

1 1 ( ) ( )

n n

i iii

n

ii

nn

i ii i

t t t

h t E t E t

p t p t

X X X

p X X

k-out-of-n Structure

1

1

1

1 if ( )

( ( ))

0 if ( )

let ( ) ( )

( ) ( ) 1, 2, ,

( ) Pr ( ) ( ) 1 ( )

n

ii

n

ii

n

ii

i

nn jj

Sj k

X t k

t

X t k

Y t X t

p t p t i n

np t Y t k p t p t

j

X

Non-repairable Series Structures

01 1

0 01

1

1 2

( ) ( ) exp ( )

exp ( ) exp ( )

( ) ( )

1If ( ) then

n n t

S i ii i

nt t

i Si

n

S ii

i in

R t R t r u du

r u du r u du

r t r t

r t MTTF

Non-repairable Parallel Structures

1 2 1 2

1 2 1 2

1 2 1

1

( )

01 2 1 2

( )1 2 1 2

(

( ) 1 1 ( )

For two-component system with constant failure rates

( )

1 1 1( )

( ) ( )( )

( )

n

S ii

t t tS

S

t t tS

S t tS

R t R t

R t e e e

MTTF R t dt

R t e e er t

R t e e e

2 )t

This example illustrates that even if the individual components of a systemhave constant failure rates, the system itself may have a time-variant failurerate.

r(t)

Non-repairable 2oo3 Structures

1 2 2 3 3 1 1 2 3

1 2 2 3 3 1 1 2 3

Structure Function

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )

System Reliability

( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( )

If all three components have the common constant failS

X t X t X t X t X t X t X t X t X t

R t R t R t R t R t R t R t R t R t R t

X t

2 3

0

ure rate

( ) 3 2

3 1 2 1 5 1( )

2 3 6

t tS

S

R t e e

MTTF R t dt

2ln

A System with n Components in Parallel

• Unreliability

• Reliability

n

iiFF

1

n

iiRFR

1

)1(11

A System with n Components in Series

• Reliability

• Unreliability

n

iiRR

1

n

iiFRF

1

)1(11

Upper Bound of Unreliability for Systems with n

Components in Series

n

ll

nj

n

i

i

ji

n

ii FFFFF

1

1

2

1

11

)1(

n

iiF

1

Reactor

PIA PICAlarm

atP > PA

PressureSwitch

PressureFeed

SolenoidValve

Figure 11-5 A chemical reactor with an alarm and inlet feed solenoid. The alarm and feed shutdown systems are linked in parallel.

C o m p o n e n t

F a i l u r e R a t e( F a u l t s / y r )

R e l i a b i l i t y

tetR )(U n r e l i a b i l i t y

F = 1 - R

P r e s s u r e S w i t c h # 1 0 . 1 4 0 . 8 7 0 . 1 3A l a r m I n d i c a t o r 0 . 0 4 4 0 . 9 6 0 . 0 4P r e s s u r e S w i t c h # 2 0 . 1 4 0 . 8 7 0 . 1 3S o l e n o i d V a l v e 0 . 4 2 0 . 6 6 0 . 3 4

Alarm System

• The components are in series

56.51

180.0ln

165.0835.011

835.0)96.0)(87.0(2

1

MTTF

R

RF

RRi

i

Faults/year

years

Shutdown System

• The components are also in series:

80.11

555.0ln

426.0574.011

574.0)66.0)(87.0(2

1

MTTF

R

RF

RRi

i

The Overall Reactor System

• The alarm and shutdown systems are in parallel:

7.131

073.0ln

930.0070.011

070.0)426.0)(165.0(2

1

MTTF

R

FR

FFj

j

Non-repairable k-out-of-n Structures

0

1 1

0

System reliability

( ) (1 )

Mean time to failure

(1 )

let

1(1 )

1 ( 1)!( )! =

!

nj t t n j

j k

nj t t n j

j k

t

nj n j

j k

n

j k

nR t e e

j

nMTTF e e dt

j

v e

nMTTF v v dv

j

n j n j

j n

1 1n

j k j

Structure Function of a Fault TreeState variables of basic events

1 if the th basic event occurs at time ( )

0 otherwise

where, 1,2, , , and is the total number of

basic events in a fault tree

The structure function

i

i tY t

i n n

1 2

of the fault tree is

( ) ( ), ( ), , ( )

1 if the top event occurs at time

0 otherwise

nt Y t Y t Y t

t

Y

System Unreliability

The probability that the basic event i occurs at time t

( ) Pr ( ) 1 ( )

The probability that the top event occurs at time t

( ) Pr ( ) 1 ( )

The probability that component i in a function

i i i

o

q t Y t E Y t

Q t t E t

Y Y

1 2

1 2

ing state is

( ) 1 ( )

System unreliability

( ) 1 ( ) 1 1 ( ),1 ( ), ,1 ( )

= ( ), ( ), , ( ) ( )

i i

o n

n

p t q t

Q t h t h q t q t q t

g q t q t q t g t

p

q

Fault Trees with a Single AND-gate

1

1

1 1

Structure function of the fault tree

( ) ( )

Since the basic events are assumed to be indep

( ) ( ) ( )

( ) ( )

n

ii

n

o ii

n n

i ii i

t Y t

Q t E t E Y t

E Y t q t

Y

Y

Fault Trees with a Single OR-gate

11

1

1 1

Structure function of the fault tree

( ) ( ) 1 1 ( )

Since the basic events are assumed to be indep

( ) ( ) 1 (1 ( ))

1 (1 ( ) ) 1 (1 ( ))

n n

i iii

n

o ii

n n

i ii i

t Y t Y t

Q t E t E Y t

E Y t q t

Y

Y

Approximate Formula for System Unreliability

1 2

j

o

Consider a fault tree with k MCSs

, , ,

The probability that the minimal cut parallel

structure j fails at time t:

Q ( ) ( )

If all minimal cut parallel structure are independent,

Q ( ) Q

j

k

ii K

K K K

t q t

t

j j

11

o j j1 1

( ) 1 1 Q ( )

Since the same basic event may occur in several cut sets,

the minimal cut parallel structure could be dependent. Thus,

Q ( ) 1 1 Q ( ) Q ( )

If all ( ) 's a

k k

jj

k k

j j

i

t t

t t t

q t

o j j1 1

re very small,

Q ( ) 1 1 Q ( ) Q ( )k k

j j

t t t

Exact System Reliability

• Structure Function

• Pivotal Decomposition

• Minimal Cut (Path) Sets

• Inclusion-Exclusion Principle

Reliability Computation Based on Structure Function

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

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

2

2S

X X X X X X X X X X X X X X X X

p p p p p p p p p p p p p p p p p

X

Reliability Computation Based on Pivotal Decomposition

1

1

1

1

1

1

(1 )

= (1 )

= (1 )

j j

j j

j j

ny yj j

j

S

ny yj j

j

ny yj j

j

X X

p E

E X X

p p

y

y

y

X y

X

y

y

Reliability Computation Based on Minimal Cut or Path Sets

1 1

1 1

jj

jj

pk

i ij i Pi K j

pk

S i ij i Pi K j

X X

p p p

X

Unreliability Computation Based on Inclusion-Exclusion Principle

1

k1

1 2j=1

1

1 2 3

1j=1

Let denote the event that the components in are all in a failed state.

Pr

Pr

= Pr Pr ( 1) Pr

= - - 1

where, Pr

j

j j

j j ii K

k

o jj

kj i j k

i j

k

k

j

E K

E Q q

Q E

E E E E E E

W W W W

W E

k

2

1 2

; Pr ; ;

Pr

i ji j

k k

W E E

W E E E

Example

1 2 3 41,2 , 4,5 , 1,3,5 , 2,3,4 K K K K

Example

1 2 3 4

1 1 2 4 5 1 3 5 2 3 4

1 2 4 5 1 3 5 2 3 4

2 1 2 1 3 1 4 2 3

2 4 3 4

1 2 4 5 1 2 3 5 1 2 3 4 1 3 4 5

Pr Pr Pr Pr

=

Pr Pr Pr Pr

+ Pr Pr

=

oQ W W W W

W B B B B B B B B B B

q q q q q q q q q q

W E E E E E E E E

E E E E

q q q q q q q q q q q q q q q q q

2 3 4 5 1 2 3 4 5

3 1 2 3 4 5

4 1 2 3 4 5

4

q q q q q q q q

W q q q q q

W q q q q q

Upper and Lower Bounds of System Unreliability

1

1 2

1 2 3

1 1 1

1

( 1) ( 1) ( 1)

1,2, ,

o

o

o

jj j i

o ii

Q W

W W Q

Q W W W

Q W

j k

Redundant Structure and Standby Units

Active Redundancy

The redundancy obtained by replacing the important unit with two or more units operating in parallel.

Passive Redundancy

The reserve units can also be kept in standby in such a way that the first of them is activated when the original unit fails, the second is activated when the first reserve unit fails, and so on. If the reserve units carry no load in the waiting period before activation, the redundancy is called passive. In the waiting period, such a unit is said to be in cold standby.

Partly-Loaded Redundancy

The standby units carry a weak load.

Cold Standby, Passive Redundancy, Perfect

Switching, No Repairs

Life Time of Standby System

The mean time to system failure

n

iiTT

1

n

iis MTTFMTTF

1

Exact Distribution of Lifetime

If the lifetimes of the n components are independent and exponentially distributed with the same failure rate λ. It can be shown that T is gamma distributed with parameters n and λ. The survivor (reliability) function is

tn

k

k

s ek

ttR

1

0 !

)()(

Approximate Distribution of Lifetime

Assume that the lifetimes are independent and identically distributed with mean time to failure μ and standard deviation σ. According to Lindeberg-Levy’s central limit theorem, T will be asymptotically normally distributed with mean nμ and variance nσ^2.

1 1

1

( ) Pr 1 Pr

=1 Pr

where denotes the distribution function of the

standard normal distribution (0,1).

n n

S i ii i

n

ii

R t T t T t

T nt n n t

n n n

N

Cold Standby, Imperfect Switching, No Repairs

2-Unit System

• A standby system with an active unit (unit 1) and a unit in cold standby. The active unit is under surveillance by a switch, which activates the standby unit when the active unit fails.

• Let be the failure rate of unit 1 and unit 2 respectively; Let (1-p) be the probability that the switching is successful.

21,

Two Disjoint Ways of Survival

1. Unit 1 does not fail in (0, t], i.e.

2. Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 is activated at time τ and does not fail in the time interval (τ,t].

tT 1

Probabilities of Two Disjoint Events

• Event 1:

• Event 2: tetT 1

1Pr

depetTt t 12

10

)(2 )1(Pr

Unit 1 failsSwitching successful

Unit 2 working afterwards

System Reliability

)()1(

)( 121

21

1

21

ttts ee

petR

ts etptR

)1(1)(

21

Mean Time to Failure

210

1)1(

1)(

pdttRMTTF ss

Partly-Loaded Redundancy, Imperfect Switching, No

Repairs

Two-Unit System

Same as before except unit 2 carries a certain load before it is activated. Let denote the failure rate of unit 2 while in partly-loaded standby.

0

Two Disjoint Ways of Survival

1. Unit 1 does not fail in (0, t], i.e.

2. Unit 1 fails in the time interval (τ, τ+dτ], where 0<τ<t. The switch is able to activate unit 2. Unit 2 does not fail in (0, τ], is activated at time τ and does not fail in the time interval (τ,t].

tT 1

Probabilities of Two Disjoint Events

• Event 1:

• Event 2: tetT 1

1Pr

deepetTt t 102

10

)(2 )1(Pr

Unit 1 failsat τSwitching

successful

Unit 2 still working after τ Unit 2 working

in (0, τ]

System Reliability

][)1(

)(

0

)(

210

1

210

1021 ttts ee

petR

tts tepetR 21

1

021

)1()(

0

Mean Time to Failure

)()1(

1

)(

012

1

1

0

p

dttRMTTF ss