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B.KwiatuszewskaSarneckaMODELSOFRELIABILITYANDAVAILABILITYIMPROVEMENTOFSERIESANDPARALLELSYSTEMSRELATEDTOTHEIROPERATIONPROCESSES

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MODELS OF RELIABILITY AND AVAILABILITY IMPROVEMENT OF SERIES AND PARALLEL SYSTEMS RELATED TO THEIR OPERATION PROCESSES

B. Kwiatuszewska-Sarnecka,

Gdynia Maritime University, Gdynia, Poland

e-mail: [email protected]

ABSTRACT Integrated general models of approximate approaches of complex multi-state series and parallel systems, linking their reliability and availability improvement models and their operation processes models caused changing reliability and safety structures and components reliability characteristics in different operation states, are constructed. These joint models are applied to determining improved reliability and availability characteristics of the considered multi-state series and parallel systems related to their varying in time operation processes. The conditional reliability characteristics of the multi-state systems with hot, cold single reservation of component and the conditional reliability characteristics of the multi-state systems with reduced rate of departure by a factor of system components are defined. 1 INTRODUCTION

Taking Most real transportation systems are very complex. Large numbers of components and subsystems and their operating complexity cause that the evaluation and optimization of their reliability improvement and availability improvement is complicated. A convenient tool for solving this problem is semi-markov modeling of the systems operation processes combining with three methods of reliability and availability improvement proposed in this paper. Therefore, the common usage of the systems reliability and availability improvement models and the semi-markov model for the systems operation process modeling in order to construct the joint general system reliability and availability improvement model related to its operation process is proposed.

. 2 RELIABILITY IMPROVEMENT OF MULTI-STATE SYSTEM COMPONENT IN

VARIABLE OPERATION CONDITIONS

We assume that the reliability of a single system component can be improved by using hot or cold reserve of this component or by replacing this component by an improved component with the reduced rates of departure from the reliability state subset },...1,{ zuu + by a factor

),(u ,1)(0 ubi (2)


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i = 1,2,...,n, u = 1,2,,z, ,,...,2,1 =b then (a) the reliability function of multi-state system component with a single hot reservation at the operational state ,bz ,,...,2,1 =b is respectively given by ],)],([,,)]1,([,1[)],([ )()1()()1()()1( bibibi ztRtRtR K= t (,), ,,...,2,1 =b (3) where

1])],([[1)],([ 2)()()1( == bibi utFutR , t < 0, 2)()()1( ])],([[1)],([ bi

bi utFutR = ])(2exp[])(exp[2 )()( tutu bibi = , t 0, (4)

,0)()( >ubi i = 1,2,...,n, u = 1,2,,z, ,,...,2,1 =b (b) the reliability function of multi-state system component with single cold reservation at the operational state ,bz ,,...,2,1 =b is respectively given by ])],([,,)]1,([,1[)],([ )()2()()2()()2( bi

bi

bi ztRtRtR K= , t (,), ,,...,2,1 =b (5)

where ])],([)],([[1)],([ )()()()2( bi

bi

bi utFutFutR = 1= , t < 0,

])],([)],([[1)],([ )()()()2( bib

ib

i utFutFutR = ])(exp[]])(1[ )()( tutu bibi += , t 0, (6) (c) the exponential reliability function of multi-state system component with the reduced rate of departure by a factor ),(u ,,...,2,1 zu = at the operational state ,bz ,,...,2,1 =b is respectively given by ])],([,,)]1,([,1[)],([ )()3()()3()()3( bi

bi

bi ztRtRtR K= t (,), ,,...,2,1 =b (7)

where 1)],([ )()3( =bi utR for t < 0, ])()(exp[)],([ )()()3( tuuutR bibi = for t 0, (8)

3 ASYMPTOTIC APPROACH TO EVALUATION OF RELIABILITY IMPROVEMENT

OF LARGE MULTI-STATE SYSTEMS IN VARIABLE OPERATION CONDITIONS Technical Main results concerning asymptotic approach to multi-state system reliability improvement with ageing components in fixed operation conditions are comprehensively detailed. In order to combine the results on the reliability improvement of multi-state systems related to their operation processes and the results concerning limit reliability functions of the multi-state systems, and to obtain results of asymptotic approach to evaluation of the multi-state system reliability improvement in variable operation conditions (Koowrocki K. & Kwiatuszewska-Sarnecka B. (2008)), we use the following definition. A reliability function

),,()],,(),...,1,(,1[),( = tzttt where

,]),([),( )(1

bv

bb utput = = ,,...,2,1 zu =

is called a limit reliability function of a multi-state system in its operation process with reliability function )],,(),...,1,(),0,([),( ztttt nnnn RRRR =

where )(

1]),([),( b

v

bnbn utput = RR , ,,...,2,1 zu =

where


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bp = )(lim tpbt

= ,

1=

v

lll

bb

M

M

,,...,2,1 vb =

are the limit values of the transient probabilities at the particular operation states )(tpb = P(Z(t) = bz ) , ),,0 +ua bn ),,()()( ub bn ,,...,2,1 zu = ,,...,2,1 vb =

such that for )()]([ buCt , ,,...,2,1 zu = ,,...,2,1 vb = )()()()( )],([)]),()(([lim bbbn

bnnn

utuubtua =+ R . Hence, the following approximate formulae are valid

)],,(),...,1,(),0,([),( ztttt nnnn RRRR = where

,)],)(

)((),( )(

1)(

)(bv

b bn

bn

bn uuaubt

put [=

R ),,( t .,...,2,1 zu = The following propositions are concerned with the homogeneous exponential systems i.e. the systems which components have at operational states ,bz ,,...,2,1 vb = exponential reliability functions. Proposition 3.1 If components of the homogeneous multi-state series system at the operational state bz have exponential reliability functions and the system have: (a) a single hot reservation of system components and

)()( ua bn = ,)(1

)( nub )()( ub bn = 0, u = 12,..., ,z, ,,...,2,1 vb =

then )],,(,),1,(,1[),( )1()1()1( zttt K= t (-,), (9) where

1),()1( =ut for t < 0, ]exp[),( 21

)1( tputv

bb = = for t 0, (10)

is its unconditional multi-state limit reliability function. Hence, the following exact formulae is valid ),()1( tnR = [1, )1,()1( tnR ,..., ),()1( ztnR ], (11) where

1),()1( =utnR for ,0


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1),()2( =ut for t < 0, ]exp[),( 21

)2( tputv

bb = = for t 0, u = 1,2,...,z (14)

is its unconditional multi-state limit reliability function. Hence the following approximate formulae is valid ),()2( tnR = [1, )1,()2( tnR ,..., ),()2( ztnR ], (15) where

1),()2( =utnR for ,0


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]]exp[exp[1),(1

)2( tputv

bb = = , t (-,), u = 1,2,...,z, (27)

is its unconditional multi-state limit reliability function, hence, the following approximate formulae is valid )],(),...,1,(,1[),( )2()2()2( zttt nnn RRR = , (28) where

),()2( utnR 1 ))]]()()((exp[exp[ )()()(1

ubutup bnbbv

bb = , t (-,), ,,...,2,1 zu = (29)

(c) components with the reduced rate of departure by a factor ),(u ,,...,2,1 zu = and )()( ua bn = )()(

1)( uub , )(

)( ub bn = nuublog

)()(1

)( , u = 1,2,...,z, ,,...,2,1 vb = then )],,(,),1,(,1[),( )3()3()3( zttt K= t (-,), (30) where

]]exp[exp[1),(1

)3( tputv

bb = = , t (-,), u = 1,2,...,z, (31)

is its unconditional multi-state limit reliability function hence, the following approximate formulae is valid ),()3( tnR = [1, )1,()3( tnR ,..., ),()3( ztnR ], (32) where

),()3( utnR 1 )]]log)()((exp[exp[ )()(1

ntuup bbv

bb = , t (-,), .,...,2,1 zu = (33)

4 AVAILABILITY OF MULTI-STATE SERIES AND PARALLEL SYSTEMS IN

VARIABLE OPERATION CONDITIONS There is presented a combination of reliability, availability improvement models of multi-state renewal systems only with non-ignored time of renovation in a model of variable in time operation processes. On the basis of those joined models, with assumption, that systems improved conditional reliability functions dependent on variable in time operation states are the same as improved limit reliability functions of the exponential non-renewal multi-state series and parallel systems, improved availability characteristics of the systems are determined. 4.1 Multi-state systems with non-ignored time of renovation in variable operation conditions We assume similarly as in non renewal systems considered in Point 3 that the changes of the process Z(t) states have an influence on the multi-state system reliability structure. The main characteristics of multi-state renewal system with hot, cold single reservation of system components and system with improved components reliability related to their operation process can be approximately determined by taking account described systems operation process properties.

Proposition 4.1 If components of the multi-state renewal system with non-ignored time of renovation at the operational states ,bz ,,...,2,1 vb = have exponential reliability functions and the time of the system renovation has the mean value )()]([ bo r and the standard deviation )(2 )]([ bo r , then:


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i) the distribution function of the time )()( rS kN , ,3,2,1=k until the Nth systems renovation, for sufficiently large N, has approximately normal distribution

)))()(()),()((( 22)()( rrNrrNN ok

ok ++

i.e.,

== ),()()(

rtFkN

))(( )( trSP kN


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),))()((

)()()(

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

22)(

0)(

0

0)(

)()1,0(

rrrr

rtrtrrN

F

ok

k

ok

kN

++

+++ ,...2,1=N , },,...,2,1{ zr

viii) the expected value and the variance of the number ),()( rtN k , ,3,2,1=k of exceeding the reliability critical state r of this system up to the moment ,0, tt for sufficiently large t, are approximately respectively given by

)()()(

),( )(0

)(

rrrt

rtHo

k

k

+

+ , ),()(

rtDk

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

)( 22)(3)(

0 rrrr

rto

k

ok

+++ },,...,2,1{ zr

ix) the availability coefficient of the system at the moment t is given by the formula

)()()(),( )(

)()(

rrrrtK

ok

kk

+ , ,0t },,...,2,1{ zr x) the availability coefficient of the system in the time interval ,0),, >+< tt is given by the formula

,),()()(

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

dtrtrrrtKk

no

kk R ,0t ,0> },,...,2,1{ zr

where for ,ru = and )()( rk and )()( rk , ,3,2,1=k are given by: - for a homogeneous series system (a) with a single hot reservation of system components

nr

pr bv

bb )(2

)( )(1

)1(

=, },,...,2,1{ zr (34)

=2)1( )]([ r +

,)]([),( 2)1()1(2 rrtdt n F },,...,2,1{ zr where )()1( r is given by the formula (34)

),,(1),( )1()1( rtrt nn RF = ),,( t and ),()1( rtnR is given by the formulae (11)-(12) for ,ru = },...,2,1{ zr , (b) with a single cold reservation of system components

nr

prb

v

bb 2)(

)()(1

)2(

=, },,...,2,1{ zr (35)

=2)2( )]([ r +


),,(1),( )2()2( rtrt nn RF = ),,( t and ),()2( rtnR is given by the formulae (15)-(16) for ,ru = },...,2,1{ zr , (c) with the reduced rate of departure by a factor ),(u ,,...,2,1 zu = of its components

nrrpr b

v

bb )()(

1)( )(1

)3(

= , },,...,2,1{ zr (36)

=2)3( )]([ r +


),,(1),( )3()3( rtrt nn RF = ),,( t


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and ),()3( rtnR is given by the formulae (19)-(20) for ,ru = },...,2,1{ zr , - for a homogeneous parallel system (a) with a single hot reservation of system components

)()1( r )),(/2log)(/( )()(1

rnrCp bbv

bb += },,...,2,1{ zr (37)

where 5772.0C is Eulers constant =2)1( )]([ r +

,)]([),( 2)1()1(2 rrtdt n F },,...,2,1{ zr

where )()1( r is given by the formula (37) ),,(1),( )1()1( rtrt nn RF = ),,( t

and ),()1( rtnR is given by the formulae (23)-(24) for ,ru = },...,2,1{ zr , (b) with a single cold reservation of system components

)()2( r )),()(log()(/( )()()(1

rbrnrCp bnbbv

bb + = },,...,2,1{ zr (38)

where 5772.0C is Eulers constant and =2)]([ r +

,)]([),( 22 rrtdt n F },,...,2,1{ zr

where )()2( r is given by the formula (38) ),,(1),( )2()2( rtrt nn RF = ),,( t

and ),()2( rtnR is given by the formulae (25)-(26) for ,ru = },...,2,1{ zr , (c) with the reduced rate of departure by a factor ),(u ,,...,2,1 zu = of its components )()3( r ))()(/(( )(

1rrCp b

v

bb = ))),()(/(log

)( rrn b + },,...,2,1{ zr (39) where 5772.0C is Eulers constant

=2)3( )]([ r +


),,(1),( )3()3( rtrt nn RF = ),,( t and ),()3( rtnR is given by the formulae (32)-(33) for ,ru = },...,2,1{ zr . 5 EFFECTS OF COMPARISON ON RELIABILITY AND AVAILABILITY

IMPROVEMENT In order to determine the value of the coefficient )(r , },,...,2,1{ zr by which it is necessary to reduce (multiply by) the failure rates of departure of the reliability states subsets of basic components of the non renewal system or renewal with non-ignored time of renovation in order to receive the system having the mean value of the number of exceeding the critical reliability state during the time t as the mean value of the number of exceeding the critical reliability state during the time t of the system with either hot or cold single reserve of basic components we can solve either the equation )()1( r )()3( r= , ,0t },,...,2,1{ zr (40) or respectively the equation )()2( r )()3( r= , ,0t },,...,2,1{ zr (41)


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and if the system is the system with not ignored time of renovation in order to receive the system having the same availability as the availability of the system with either hot or cold single reserve of basic components we can solve either the equation )()1( rK )()3( rK= , ,0t },,...,2,1{ zr (42) or respectively the equation )()2( rK )()3( rK= , ,0t }.,...,2,1{ zr (43) Proposition 5.1 If components of the multi-state non renewal system or renewal system with non ignored time of renovation at the operational states ,bz ,,...,2,1 vb = have exponential reliability functions then the coefficient )(r , },,...,2,1{ zr is in the form - for a homogeneous series system in the first case by (40) , after considering (34) and (36), we get

nr

2)( = , },,...,2,1{ zr in the second case by (41), after considering (35) and (36), we get

nr

2)( = , },,...,2,1{ zr - for a homogeneous parallel system in the first case by (40) , after considering (36) and (38), we get

nCnCr

2loglog)( +

+= , },,...,2,1{ zr in the second case by (41), after considering (37) and (38), we get

nrbrCnCr b

nb )()(loglog)( )()( +

+= , },,...,2,1{ zr

where 5772.0C is Eulers constant and nrbr

rbrb

nb

bn

b

=)()(

)]()(exp[)()(

)()(

, }.,...,2,1{ zr 6 CONCLUSION

In the paper the multi-state approach to the improvement of systems reliability, and availability for series and parallel systems has been presented. Constructed in this paper the final integrated, general analytical models of complex systems reliability and availability improvement related to their operation processes are very important for the further optimization of system operation costs. 6 PREFERENCES

- Barlow, R. E. & Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Probability Models. Holt Rinehart and Winston, Inc., New York.

- Gniedenko, B. W. (1943). Sur la distribution limite du terme maximum dune serie aleatoire. Ann. of Math. 44, 432453.

- Koowrocki, K. (2004). Reliability of Large Systems. Elsevier, Amsterdam - Boston - Heidelberg - London - New York - Oxford - Paris - San Diego - San Francisco - Singapore - Sydney - Tokyo.

- Koowrocki K. & Kwiatuszewska-Sarnecka B. (2008). Safety and Reliability Optimization of Complex Industrial Systems and Processes. WP5 - Task 5.1. Models of safety, reliability


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and availability improvement and optimization of complex industrial systems related to their operation processes. English 31.05.2009. Poland-Singapore Joint Project. MSHE Decision No. 63/N-Singapore/2007/0.

- Soszyska, J. (2007). Systems reliability analysis In variable operation conditions. PhD Thesis, Gdynia Maritime University-System Research Institute Warsaw, (in Polish).

Acknowledgements The paper describes part of the work in the Poland-Singapore Joint Research Project titled Safety and Reliability of Complex Industrial Systems and Processes supported by grants from the Polands Ministry of Science and Higher Education (MSHE grant No. 63/N-Singapore/2007/0) and the Agency for Science, Technology and Research of Singapore (A*STAR SERC grant No. 072 1340050)

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