A Two-unit System with inspection before failure Essay Example

conditions. A system is considered to hold failed under three conditions: One is due to when it becomes wholly inoperable. Second is due to when it is still operable but is no longer able to execute its intended map satisfactorily and 3rd is due to when the serious impairment has made it undependable and insecure for continued usage, So this needs its immediate remotion for the system for fix or replacing. Numerous dependability theoretical accounts for standby with different fix mechanism have been proposed by the research worker including Mishra and Balagurusamy [ 1976 ] , Chiang and Niu [ 1981 ] , Gopalan and Naidu [ 1982 ] , Goel et.al [ 1985 ] , Tuteja and malik [ 1994 ] taking some certain premises. To increase the dependability and handiness of the system, we take a normal review of two indistinguishable unit before failure. In this paper, an operative unit is inspected after a certain period of its operation and so it is distinct whether unit can run further or needs certain care. Besides it is assumed that operative unit is non inspected if another unit is failed and when a working unit is fails it goes under review of the maintenance man and he decides whether the unit is repairable or non. If a unit is non repairable it is replaced by a new one. In this paper system will be analyzed to find assorted dependability steps by utilizing mathematical tools MTSF/MTBF, Markov concatenation, Markov procedure, reclamation procedure etc.

Description of system and Premise: 

In this paper, an operative unit is inspected after a certain period of its operation and it is

distinct whether unit can run further or needs certain care.

• The system consists of two indistinguishable units - Initially one unit is operative and 2nd unit is kept as cold standby.
• System is considered in Up-state if one unit is working and in down province if no unit is working.
• Each unit of the system has two modes-normal operative or failed.
• Here a everyday review is conducted to analyze the operating unit after a certain fixed period.
• It is assumed that operative unit is non inspected if another unit is failed.
• Besides it is assumed that when a working unit is fails it goes under review of the maintenance man and he decides whether the unit is repairable or not.If a unit is non repairable, it is replaced by a new one.
• Inspection clip is excessively little to travel for care of 2nd unit.
• A unit under care would non neglect.
• A repaired and replaced unit is every bit good as new.
• All the random variable are independent.

Notations

Tocopherol: Set of regenerative provinces

A : Set of non-regenerative provinces

Nitrogen: Routine review

Nitrogen: Everyday review uninterrupted

Oxygen: Unit of measurement is in operative province

_Second: Unit of measurement is in cold standby province

Changeless failure rate of a unitg ( T ) , G ( T ) : pdf and cdf of fix clip of a failed unit

Nitrogen_mCare of unit

F_R: Failed unit under fix

F_I: Failed unit under review

F_I:Failed unit under review

F_WisconsinFailed unit waiting for review

M ( T ) Care rate

I ( T ) , I ( T ) pdf and cdf of review clip of normal unit

H ( T ) , H ( T ) pdf and cdf of fix

clip of a failed unit before review

©Symbol for Laplace whirl

®symbol for Laplace Stieltjes Convolution

The system can be in any of the undermentioned provinces with regard of the above symbols:

Second₀= ( O, N_s) Second₄= ( N_I, F_{Badger state})

Second₁= ( N_I,O ) S₅= ( N_m, F_Wisconsin)

Second₂= ( F_I, O ) S₆= ( F_R,O )

Second₃= ( N_m, O ) S₇= ( F_I, F_{Badger state})

Second₈= ( F_R, F_Wisconsin)

Passage Probabilities

The era of entry into provinces { S₀, S₁, S₂, S_3,Second₅, S₆, S₈} are regenerative provinces. The passage chances from the provinces S_Ito S_Jare given by Q_ijand in the provinces provinces p_ijdenotes the passage chance from provinces S_Ito S_Jare given under

P₀₁= ?/ ( ?+? ) P₂₆= qh^*( ? )

P₀₂= ?/ ( ?+? ) P₂₀= pH^*( ? )

P₁₀= a i* ( ? ) P₂₇= { 1- H^*( ? ) }

P₁₃= B i* ( ? ) P₃₀= m^*( ? )

P₁₄= { 1- I^*( ? ) } P₃₅= { 1- m^*( ? ) }

P₅₂= 1 P₆₀= g^*( ? )

P₈₂= 1p₆₈= { 1- g^*( ? ) }

P₁^{( 4 )}₂=a { 1- I^*( ? ) } P₁^{( 4 )}₅=b { 1- I^*( ? ) }

P₂^{( 7 )}₂=p { 1- H^*( ? ) } P₂^{( 7 )}₈=q { 1- H^*( ? ) }

It can be easy verified that

P₀₁+p₀₂=1p₅₂=1

P₁₀+p₁₃+p₁₄=1p₈₂=1

P₂₀₊P₂₆+p₂₇=1p₁^{( 4 )}₂+ P₁^{( 4 )}₅=p₁₄

P₃₀+p₃₅=1p₆₀+p₆₈=1

P₁^{( 4 )}₂+ P₁^{( 4 )}₅+p₁₃+p₁₀=1p₂₀₊P₂₆+ P₂^{( 7 )}₂+ P₂^{( 7 )}₈=1

P₂^{( 7 )}₂+ P₂^{( 7 )}₈=p₂₇P₃₅=1-p₃₀

P₆₈=1-p₆₀

Mean Sojourn Times

Mean Sojourn Times may be defined by µ_I=

So that in steady province we have following dealingss

µ₀=1/ ( ?+? ) µ₁= { 1- I^*( ? ) } / ?

µ₂= [ 1- H^*( ? ) ] /?µ₃= [ 1- m^*( ? ) ] /?

µ₆= [

1- g^*( ? ) ] /?

The unconditioned mean clip taken by the system to pass through from any provinces S_Ito S_Jis mathematically given by mij =tdQij ( T ) =-q_ij^*( s )^’/_{at s=0}

So that

m₀₁=?/ ( ?+? )²m₂₀=-ph^*’( ? )

m₀₂= ?/ ( ?+? )²m₂₆=-qh^*’( ? )

m₁₀= -ai^*’( ? ) m₂₇= [ { 1- H^*( ? ) } /? ] +h^*’( ? )

m₁₃= -bi^*’( ? ) m₃₀= -m^*’( ? )

m₁₄= [ { 1- I^*( ? ) } /? ] +i^*’( ? ) m₃₅= [ { 1- m^*( ? ) } /? ] +m^*’( ? )

m₆₀=-g^*’( ? ) m₆₈= [ { 1- g^*( ? ) } /? ] +g^*’( ? )

It can be easy verified that

m₀₁+ m₀₂= µ₀m₁₀+ m₁₃+m₁₄= µ₁

m₂₀+ m₂₆+ m₂₇= µ₂m₃₀+ m₃₅= µ₃

m₆₀+ m₆₈= µ₆

Average Time to System Failure

The average clip to system failure is given by the equations

?₀( T ) = Q₀₁( T ) ® ?₁( T ) + Q₀₂( T ) ® ?₂( T )

?₁( T ) = Q₁₀( T ) ® ?₀( T ) + Q₁₃( T ) ® ?₃( T ) + Q₁₄

?₂( T ) = Q₂₀( T ) ® ?₀( T ) + Q₂₆( T ) ® ?₆( T ) + Q₂₇( T )

?₃( T ) = Q₃₀( T ) ® ?₀( T ) + Q₃₅( T )

?₆( T ) = Q₆₀( T ) ® ?₀( T ) + Q₆₈( T )

Solving above equation by taking Laplace Stieltjes transmutations and work outing for ?₀^**( s ) , we get

?₀^**( s ) =

Where

N ( s ) = - Q₀₁Q₁₄- Q₀₁Q₁₃Q₃₅- Q₀₂Q₂₇- Q₀₂Q₂₆Q₆₈

D ( s ) = -1+ Q₀₁Q₁₀+ Q₀₁Q₁₃Q₃₀+ Q₀₂Q₂₀+ Q₀₂Q₂₆Q₆₀

MTSF = ?₀=[ { 1-?₀^**( s ) } /s ]

= { D^?( 0 ) -N^?( 0 ) } /D ( 0 )

Where

Calciferol^?( 0 ) -N^?( 0 ) = - [ µ₀+ µ₁P₀₁+ µ₂P₀₂+ µ₃P₀₁P₁₃+ µ₆P₀₂P₂₆]

D ( 0 ) = - [ P₀₁P₁₄+p₀₁P₁₃P₃₅+p₀₂P₂₇+p₀₂P₂₆P₆₈]

Handiness of the system

The point wise handiness a‚?_I( T ) of the system is given by

a‚?₀( T ) = I?₀( T ) + Q₀₁( T ) © a‚?₁( T ) + Q₀₂( T ) © a‚?₂( T )

a‚?₁( T ) = I?₁( T ) + Q₁₀( T ) © a‚?₀( T ) + Q₁^{( 4 )}₂( T ) © a‚?₂( T ) +q₁₃( T ) © a‚?₃( T )

a‚?₂( T ) = I?₂( T ) + Q₂₀( T ) © a‚?₀( T ) + Q₂₆( T ) © a‚?₆( T ) +q₂^{( 7 )}₂© a‚?₂( T ) +q₂^{( 7 )}₈© a‚?₈( T )

a‚?₃( T ) = I?₃( T ) +q₃₀( T ) © a‚?₀( T ) + Q₃₅( T ) © a‚?₅( T )

a‚?₅( T ) = Q₅₂( T ) © a‚?₂( T )

a‚?₆( T ) = I?₆( T ) +q₆₀( T ) © a‚?₀( T ) + Q₆₈( T ) © a‚?₈( T )

a‚?₈( T ) = Q₈₂( T ) © a‚?₂( T )

Now taking Laplace transform of these equations and work outing them for a‚?₀^*( s ) , we get

a‚?₀^*( T ) =

The steady provinces handiness is given by

a‚?₀^**=

Where

Nitrogen₁( 0 ) = µ₁P₀₁[ P₂₆+1 ] + µ₃P₀₁P₁₃[ P₂₆P₆₈+p₂^{( 7 )}₈]

and

Calciferol₁( 0 ) =0I?₀( T ) =?₀( T ) I?₁( T ) =?₁( T ) I?₂( T ) =?₂( T ) I?₃( T ) =?₃( T ) I?₆( T ) =?₆( T )

Calciferol₁^,( 0 ) = - { ( m₂⁽

7 )₂+m₂₆P₆₈+m₆₈P₂₆+ m₂^{( 7 )}₈) ( 1-p₀₁P₁₀-p₀₁P₁₃P₃₀)

+ ( 1-p₂₇-p₂₆P₆₈) ( m₀₁P₁₀+m₁₀P₀₁+m₀₁P₁₃P₃₀+m₁₃P₀₁P₃₀+m₃₀P₀₁P₁₃)

+ ( m₀₁P₁^{( 4 )}₂+m₁^{( 4 )}₂P₀₁+m₀₁P₁₃P₃₅+m₁₃P₀₁P₃₅+m₃₅P₀₁P₁₃+m₀₂) ( P₂₀+p₂₆P₆₀)

+ ( m₂₀+m₂₆P₆₀+m₆₀P₂₆) ( P₀₁P₁^{( 4 )}₂+p₀₁P₁₃P₃₅+p₀₂) }

Inspection Time Before Failure-

Let I_Iis the review clip get downing from a regenerative provinces S_Iat t=0 is given by

O?₀( T ) = Q₀₁( T ) © O?₁( T ) + Q₀₂( T ) © O?₂( T )

O?₁( T ) = U₁( T ) + Q₁₀( T ) © O?₀( T ) + Q₁^{( 4 )}₂( T ) © O?₂( T ) +q₁₃( T ) © O?₃( T )

O?₂( T ) = Q₂₀( T ) © O?₀( T ) + Q₂₆( T ) © O?₆( T ) +q₂^{( 7 )}₂© O?₂( T ) +q₂^{( 7 )}₈© O?₈( T )

O?₃( T ) = Q₃₀( T ) © O?₀( T ) + Q₃₅( T ) © O?₅( T )

O?₅( T ) = Q₅₂( T ) © O?₂( T )

O?₆( T ) = Q₆₀( T ) © O?₀( T ) + Q₆₈( T ) © O?₈( T )

O?₈( T ) = Q₈₂( T ) © O?₂( T )

Where

Nitrogen₂( 0 ) = U₁P₀₁( -1+p₂₇+p₂₆P₆₈)

and

Uracil₁=+ ?

The review clip is given by

O?₀^*( T ) =

O?₀^**=

Calciferol₁^?( 0 ) is already defined

Inspection Time After Failure-

Let A¬_Iis the review clip get downing from a regenerative provinces S_Iat t=0 is given by

A¬₀( T ) = Q₀₁( T ) © A¬₁( T ) + Q₀₂( T ) © A¬₂( T )

A¬₁( T ) = Q₁₀( T ) © A¬₀( T ) + Q₁^{( 4 )}₂( T ) © A¬₂( T ) +q₁₃( T ) © A¬₃( T )

A¬₂( T ) = V₂+ Q₂₀( T ) © A¬₀( T

) + Q₂₆( T ) © A¬₆( T ) +q₂^{( 7 )}₂© A¬₂( T ) +q₂^{( 7 )}₈© A¬₈( T )

A¬₃( T ) = Q₃₀( T ) © A¬₀( T ) + Q₃₅( T ) © A¬₅( T )

A¬₅( T ) = Q₅₂( T ) © A¬₂( T )

A¬₆( T ) = Q₆₀( T ) © A¬₀( T ) + Q₆₈( T ) © A¬₈( T )

A¬₈( T ) = Q₈₂( T ) © A¬₂( T )

Where

Nitrogen₃( 0 ) = -V₂( P₀₁P₁^{( 4 )}₂+p₀₁P₁₃P₃₅+p₀₂)

and

The review clip is given by

A¬₀^*( T ) =

A¬₀^**=( sA¬₀^*( s ) ) =<

Calciferol₁^?( 0 ) is already defined

Care Time

Let K_Iis the Maintenance clip get downing from a regenerative provinces S_Iat t=0 is given by

K₀( T ) = Q₀₁( T ) © K₁( T ) + Q₀₂( T ) © K₂( T )

K₁( T ) = Q₁₀( T ) © K₀( T ) + Q₁^{( 4 )}₂( T ) © K₂( T ) +q₁₃( T ) © K₃( T )

K₂( T ) = Q₂₀( T ) © K₀( T ) + Q₂₆( T ) © K₆( T ) +q₂^{( 7 )}₂© K₂( T ) +q₂^{( 7 )}₈© K₈( T )

K₃( T ) =W₃+ Q₃₀( T ) © K₀( T ) + Q₃₅( T ) © K₅( T )

K₅( T ) = W₅+q₅₂( T ) © K₂( T )

K₆( T ) = Q₆₀( T ) © K₀( T ) + Q₆₈( T ) © K₈( T )

K₈( T ) = Q₈₂( T ) © K₂( T )

The Maintenance clip is given by

K₀^*( T ) =

K₀^**=( sK₀^*( s ) ) =

Where

Nitrogen₄( 0 ) = P₀₁P₁₃( W₃+W₅P₃₅) ( P₂₆P₆₈+p₂^{( 7 )}₈)

Where Calciferol₁^?( 0

) is already defined

Repair Time

Let R_Iis the Repair clip get downing from a regenerative provinces S_Iat t=0 is given by

Roentgen₀( T ) = Q₀₁( T ) © R₁( T ) + Q₀₂( T ) © R₂( T )

Roentgen₁( T ) = Q₁₀( T ) © R₀( T ) + Q₁^{( 4 )}₂( T ) © R₂( T ) +q₁₃( T ) © R₃( T )

Roentgen₂( T ) = Q₂₀( T ) © R₀( T ) + Q₂₆( T ) © R₆( T ) +q₂^{( 7 )}₂© Roentgen₂( T ) +q₂^{( 7 )}₈© Roentgen₈( T )

Roentgen₃( T ) = Q₃₀( T ) © R₀( T ) + Q₃₅( T ) © R₅( T )

Roentgen₅( T ) = Q₅₂( T ) © R₂( T )

Roentgen₆( T ) = Ten₆+q₆₀( T ) © R₀( T ) + Q₆₈( T ) © R₈( T )

Roentgen₈( T ) = Ten₈+q₈₂( T ) © R₂( T )

The Repair clip is given by

Roentgen₀^*( T ) =

Roentgen₀^**=

Where

Nitrogen₅( 0 ) = - [ { ( X₆+X₈P₆₈) P₂₆+X₈P₂^{( 7 )}₈} { P₀₁P₁^{( 4 )}₂+p₀₁P₁₃P₃₅+p₀₂} ]

Where

Ten₆=

Particular instances:

If we take repair rate and review clip as negative binomial distributions as

I ( T ) =H ( T ) = ?

Then we get,

P₀₁=?/ ?+?µ₀=1/?+?

P₀₂=?/?+?µ₁=1/?+?

P₁₀=a?/?+?µ₂=1/?+?

P₁₃= b?/?+?µ₃= 1/?+?

P₁₄= ?/?+?µ₆= 1/?+?

P₂₀= p?/?+?m₀₁=?/ ( ?+? )²

P₂₆= q?/?+?m₀₂=?/ ( ?+? )²

P₂₇= ?/?+?m₁₀=a?/ ( ?+? )²

P₃₀= ?/?+?m₁₃=b?/ ( ?+? )²

P₃₅= ?/?+?m₁₄=?/ ( ?+? )²

P₅₂= 1m₂₀=p?/ ( ?+? )²

P₈₂= 1m₂₆=q?/ ( ?+? )²

P₆₀= ?/?+?m₂₇=?/ ( ?+? )²

P₆₈= ?/?+?m₃₀=?/ ( ?+? )²

P₁^{( 4 )}₂=a?/?+?m₃₅=?/ ( ?+? )²

P₁^{( 4 )}₅=b?/?+?m₆₀=?/ ( ?+? )²

P₂^{( 7 )}₂=p?/?+?m₆₈=?/ ( ?+? )²

P₂^{( 7 )}₈=q?/?+?m₁^{( 4 )}₂=a [ ( 1/?²) - ( 1/ ( ?+? )²) ]

m₁^{( 4 )}₅=b

[ ( 1/?²) - ( 1/ ( ?+? )²) ]

m₂^{( 7 )}₂=p [ ( 1/?²) - ( 1/ ( ?+? )²) ]

m₂^{( 7 )}₈=q [ ( 1/?²) - ( 1/ ( ?+? )²) ]

Mentions: 

1. Nakagawa, T. and Osaki, S. ( 1975 ) . Stochastic behavior of two unit analogues redundant system with preventative care, Microlectron Reliab. 14, p. 457-461.
2. Jui-Hsiang Chian and John Yuan, ( 2001 ) , optimum care policy for a deteriorating production system under review,
3. Tuteja, R.K. , and Gulshan, T. `` cost – benefit analysis of a two- server-two unit warm standby system with different type of failure '' , Microelectron. Reliab. , 1992, VOL.32, p1353-1359.
4. Brown, M. , Proschan, F. ( 1983 ) “Imperfect repair” , Journal of Applied Probability,20: 851–859.
5. Jiang, R. and Jardine, A.K.S. ( 2005 ) “Two Optimization Models of the Optimum Inspection Problem” , Journal of the Operational Research Society,56: 1176–1183.
6. LEWIS E. E. Introduction to Reliability Engineering. John Wiley and Sons. Co. 1987.
7. H. Wang. A study of care policies of deteriorating systems. European Journal of Operational Research, 139 ( 3 ) :469–489, 2002
8. Barlow R.E. , Proschan F. Mathematical Theory of Reliability. SIAM, 1996.