N0460284100

K. Uma Maheswari et al Int. Journal of Engineering Research and Applications www.ijera.com

ISSN : 2248-9622, Vol. 4, Issue 6( Version 2), June 2014, pp.84-100

www.ijera.com 84 | P a g e

Probabilistic Behaviour of A Redundant Complex System with

Imperfect Switching, Environmental Common Cause and Human

Error Effects under Head-Of-Line Repair Discipline

K. Uma Maheswari#1

, A. Mallikarjuna Reddy#1

and R. Bhuvana Vijaya#2

#1

Department of Mathematics, Sri Krishnadevaraya University, Ananthapuramu-515003, A.P., India #2

Department of Mathematics, J.N.T.U., Ananthapuramu-515001, A.P., India

ABSTRACT Series-parallel systems are made up of combination several series & parallel configuration to obtain system

reliability down into homogeneous subsystems. This is very simple to analyse a simple series-parallel system

without any error effects, But very difficult to analyse a complex system with various error effects. This paper

we presented mathematical analysis of a Redundant Complex System with Imperfect switching, Environmental

and Common Cause and Human error effects under Head of Line Repair Discipline.

Keywords: Imperfect switching, Reliability, Availability, MTTF, State Transition diagram.

I. INTRODUCTION Kontoleon & Konoleon [2] had considered a

system subject to partial and catastrophic failures to

be repaired at a single service station and also

assumed exponential distribution for repair. Gupta

and Mittal [3] considered a standby redundant system

and two types of failures to be repaired at a single

service station with general repair time distribution

under pre-emptive resume repair Discipline

incorporating environmental effects. Further, Gupta

and Agarwal [1] developed a model with two types of

failure under general repair time distribution and

under different repair discipline. Consequently

Agarwal, Mittal et. al., [4] have solved the model of a

Parallel redundant complex system with two types of

failures with Environmental effect under preemptive

Resume repair Discipline. Mittal, Gupta et. al., [1,3]

have assumed constant repair rate due to

environmental failure and perfect switching over

device of a standby redundant system under pre-

emptive resume and repeat discipline respectively.

But it is not always possible that the switch is perfect.

It may have some probability of failure. So the

authors have initiated the study keeping in mind the

practical aspect of imperfect switch. Also repair due

to environmental effects is considered to follow

general time distribution.

In this paper, we consider a complex system

consisting of three independent, repairable

subsystems A, B & C. Subsystem B is comprised of

two identical units B1 & B2. The subsystem C is in

standby redundancy to be switched into operation

when both the identical units B1 & B2 of subsystem B

fails, through an imperfect switching over device. For

the smooth operation of the system, the subsystems B

or C are of vital importance. The failure of the

subsystem C results into non-operative state of the

system. At the time of installation, the units of

subsystem B have the same failure rate but due to

adverse environmental effects, the failure rate of the

subsystem C increases by the time, it is switched into

operation. Subsystem A has minor failure and

subsystem C has major and minor failures. Minor

failure reduces the efficiency of the system causing a

degraded state while major failure results into a non-

operable state of the system. The whole complex

system can also be in a total failure state due to

Human error or Environmental effects like

temperature, humidity etc. The repair rate due to

Human error or Environmental failure is considered

general. Laplace transforms of the time dependent

probabilities of the system being in various states

have been obtained by employing the techniques

using supplementary variable under Head-of line

Repair Discipline. The general solution of the

problem is used to evaluate the ergodic behaviour

and some particular case. At the end of the chapter

some numerical illustrations are given and various

reliability parameters have been calculated. The flow

of states is depicted in the state transition diagram.

RESEARCH ARTICLE OPEN ACCESS




The following symbols are used in state transition diagram:

: Operable : Degarded

: Imperfect : : Failed

switch

Fig. 1. STATE TRANSITION DIAGRAM

II. NOTATIONS: 1 : Minor failure rate of the operating units B1

& B2 of subsystem B.

2(1<2): Minor failure rate of the

operating unit of subsystem C

1m : Minor failure of subsystem A.

2m : Major failure of subsystem B & C.

x: Elapsed repair time for both units of

subsystem B.

y: Elapsed minor repair time for the

subsystem A, B & C.

z: Elapsed major repair time for the

subsystem B & C.

v: Elapsed repair time for environmental

failure.

h: Elapsed repair time for human error

failure.

B(x) : Repair rate of subsystem B

)(1

ym : Repair rate of minor failure of

subsystems A, B & C.

)(2

zm : Repair rate of major failure of

subsystems B & C.

: Failure due to environmental effects.

: Failure due to human error.

: Failure due to common cause.

(v) : Repair rate due to environmental

effects.

(h) : Repair rate due to Human error.

(c) : Repair rate due to common cause

b, R1 Probability of the successful operation

of the switching over device

and constant repair rate of

switching over device.

III. ASSUMPTIONS (1) Initially, at time t = 0 the system is in operable

state, i.e., it operates in its normal efficiency

state

(2) The subsystem B consists of two identical units

B1 & B2.

(3) Switching device is imperfect.

(4) When the system starts functioning the

subsystems A & B will operate and C is in stand

by.

(5) When the system starts functioning both the

subsystems B & C have the same failure rate 1

as minor failure, but as the time passes due to the

adverse environmental effects, the failure rate of

the stand by unit i.e., subsystems C increase to 2

which is the minor failure of subsystem C by the

time it is needed to operate.

(6) Subsystem A has no major failures.

(7) During the degraded state of the system due to

minor failure in subsystems A, B & C, major

failures may also occur in subsystems B & C.

(8) All the failures are distributed exponentially.

(9) All the repair follows general time distribution.




(10) The system fails completely due to

environmental effects or due to major failures of

the subsystems B & C or due to the failure

caused by Human error.

(11) Repair of the subsystem C is taken only in failed

state.

(12) During the repair of the subsystem C in failed

state, the failed units of subsystem B are also

repaired.

STATE PROBABILITIES DESCRIPTION:

(i) )(0, tPa : Probability that the system is in

operable state at the time t, where a=0, 0, 1,

2.

(ii) ),(0,3 txP : Probability that the system is the

failed state and is under repair with elapsed

repair time in the interval (x, x+).

(iii) ),(1, tyP ma : Probability that the system is in

degraded state and is under repair with

elapsed repair time lying in the interval (y,

y+) where a = 0,1,2.

(iv) ),(2, tzP ma : Probability that the system is

the failed state and is under repair with

elapsed repair time in the interval (z, z+)

where a = 0,1,2.

(v) ),( tvPE : Probability that the system is in

failed state due to Environmental failure and

is under repair with elapsed Repair time in

the interval (v, v+).

(vi) )(tPr : Probability that the system is in

failed state due to failure of the switch in

switching the subsystem C when subsystem

B completely fails at any time t.

(vii) ),( thPh : Probability of the system is in

failed state due to Human error and is under

repair with elapsed repair time in the

interval (h, h+).

(viii) ),( tuPv : Probability of by the system is in

failed state due to common cause error and

is under repair with elapsed repair time in

the interval (u, u+).

IV. FORMULATION OF THE

MATHEMATICAL MODEL: Using continuity arguments and probability

considerations, we obtain the following difference

differential equations governing the stochastic

behaviour of the complex system, which is discrete is

space and continuous in time:

0

00

,2

0

0

,0

0

,0

0

0,30,01

)(),(

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

)(),()(),(

)(),()(

22

2211

12

xdutuP

xdhthPdzztzPdxvtvP

dzztzPdyytyP

dxxtxPtPdt

d

C

HmmE

mmmm

Bmm

(1)

0

00

,2

0

0

,1

0

,1

0

0,30,11

)(),(

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

)(),()(),(

)(),()(

22

2211

12

xdutuP

xdhthPdzztzPdxvtvP

dxztzPdyytyP

dxxtxPtPdt

d

C

HmmE

mmmm

Bmm

(2)




0

,21

0,110,010,22

)(),()(

)()()(

11

21

dyytyPtPR

tPbtPbtPdt

d

mmr

mm

(3)

0

,1

0,1

0

,00,010,0

1

),(

)(),()()(

)1()(2

1

1

1

dytyP

tPdytyPtPbtP

btPRdt

d

m

m

r

(4)

0),()(2 0,3

txPx

xtB (5)

0),()(112 ,01

tyPy

ytmmm (6)

0),()(112 ,11

tyPy

ytmmm (7)

0),()(112 ,22

tyPy

ytmmm (8)

0),()(22 ,0

tzPz

ztmm (9)

0),()(22 ,1

tzPz

ztmm (10)

0),()(222 ,2

tzPz

ztmm (11)

0),()(2

thPh

htH (12)

0),()(2

tvPv

vtE (13)

0),()(2

tuPu

utC (14)

BOUNDARY CONDITIONS

0

,220,220,3 ),()(),0(1

dytyPtPtP m (15)

)(),0( 0,0,0 11tPtP mm (16)

)(),0( 0,1,1 11tPtP mm (17)

0

1,11

0

,010,2,2 )(),(),()(),0(1121

tPRdytyPbdytyPbtPtP rmmmm (18)

0

,00,0,0 ),()(),0(1222

dytyPtPtP mmmm (19)




0

,10,1,1 ),()(),0(1222

dytyPtPtP mmmm (20)

0

,20,2,2 ),()(),0(1222

dytyPtPtP mmmm (21)

0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dytyPdytyP

dytyPtPtPtPtP

mm

mE

(22)

0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dytyPdytyP

dytyPtPtPtPtP

mm

mC

(23)

)(),0( 0,2 tPtPH (24)

INITIAL CONDITIONS

P0,0(0) = 1 and all other states probabilities are zero at time t = 0.

from 6

0),()(112 ,01

tyPy

ytmmm

Taking Laplace transforms on both sides

)(),(

),(

12

1

1

1

,0

,0

yy

SsyP

syPy

mm

m

m

Integrating ‘y’ b/w the limits ‘0’ to ‘y’

y

mm dyyys

mm esPsyP 0112

11

)(][

,0,0 ),0(),(

from equation (16)

)(),0( 0,0,0 11tPtP mm


y

mm yyys

mm esPsyP 0112

11

)(][

0,0,0 )(),(

On simplification

)()(),( 10,0,0 11sVsPsyP mm (25)

where

1

1

1

2

21)(1

)(m

mm

s

sssV

from equation (7)

0),()(112 ,11

tyPy

ytmmm





)(),(

),(

12

1

1

1

,1

,1

yssyP

syPy

mm

m

m

Integrating ‘y’ between the limits ‘0’ to ‘y’

y

mm dyyys

mm esPsyP 0112

11

)(][

,1,1 ),0(),(

from equation 16

)(),0( 0,1,0 11tPtP mm


)(),0( 0,1,1 11sPsP mm

y

mm dyyys

mm esPsyP 0112

11

)(][

0,1,1 )(),(

)()(),( 10,1,1 11sVsPsyP mm (26)

From equation 4

0

,1

0,1

0

,00,0

11

),(

)(),()(

)1()(2

1

1

dytyP

tPdytyPtP

btPRdt

d

m

m

r


0

,1

0,1

0

,00,0

1

1

),(

)(),()(

2

)1()(

1

1

dysyP

sPdysyPsP

Rs

bsP

m

m

r

)]()(1()()(1[(2

)1(0,010,01

1

1

11sPsVsPsV

Rs

bmm

]))[(1[(2

)1(101

1

1

1PPsV

Rs

bm

])[( 102 PPsV (27)

where )(1[(2

)1()( 1

1

12 1

sVRs

bsV m

from equation 8

0),()(112 ,22

tyPy

ytmmm


)(),(

),(

12

1

1

2

,2

,2

yssyP

syPy

mm

m

m

Integrating ‘y’ between the units ‘0’ to ‘y’

y

mm dyyys

mm esPsyP 0122

11

)(][

,2,2 ),0(),(

from equation 18




0

1,11

0

,010,2,2 )(),(),()(),0(1111

tPRdytyPbdytyPbtPtP rmmmm


)]()()[()()(),( 0,10,050,24,2 1sPsPsVsPsVsyP m

where

2

2

3

2

11)(1

)(m

mm

s

sssV (28)

1

)()( 34 msVsV

)()()( 211115 sVRsVmbsV

From equation (10)

0),()(22 ,1

tzPz

ztmm


)(),(

),(

2

2

2

,1

,1

zsszP

szPy

m

m

m

Integrating ‘z’ between the limits ‘0’ to ‘z’

z

m dzzsz

mm esPszP 02

22

)(

,1,1 ),0(),(

From equation 19

0

,10,1,1 ),()(),0(1222

dytyPtPtP mmmm


z

m dzzsz

mmmm edysyPsPszP 02

1222

)(

0

,10,1,1 ]),()([),(

)()(

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

0,17

0,161,1 122

sPsV

sPsVsVszP mmm

(29)

where s

sssV

m )(1)( 2

6

)()](1[)( 617 12sVsVsV mm

From equation (12)

0),()(222 ,2

tzPz

tmm


)(2),(

),(

2

2

2

,2

,2

zsszP

szPy

m

m

m

Integrating ‘z’ between the limits ‘0’ to ‘z’

z

m dzzsz

mm esPszP 02

22

)(

,2,2 ),0(),(

From equation 21.




0

,20,2,2 ),()(),0(2222

dytyPtPtP mmmm


z

m dzzsz

mmmm edysyPsPszP 02

1222

)(2

0

,20,2,2 ),()(),(

)()()(

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

0,085

0,1580,284

2

22

sPsVsV

sPsVsVsPsVsV

m

mm

)()()()()( 0,00,11029 sPsPsVsPsV (30)

where s

sssV

m )(21)( 2

8

)())(1()( 849 2sVsVsV m

)()()( 5810 2sVsVsV m

From equation (12)

0),()(2

thPh

htH


)(2),(

),(

hsshP

shPh

H

H

Integrating ‘h’ between the limits ‘0’ to ‘h’

h

dhhsh

HH esPshP 0

)(2

),0(),(

From equation 22

)(),0( 0,2 tPtPH


)(),0( 0,2 tPtPH

After simplification

)()(),( 0,211 sPsVshPH (31)

where s

sssV h )(21)(11

from equation 13

0),()(2

tvPv

vtE


)(2),(

),(

vssvP

svPv

E

E

Integrating ‘v’ between the limits ‘0’ to ‘v’

v

dvvsv

EE esPsvP 0

)(2

),0(),(

From equation 14




0),()(2

tuPu

utC


)(),(

),(

ussuP

suPv

C

C

Integrating ‘u’ between the limits ‘0’ to ‘u’

u

duusu

CC esPsuP 0

)(2

),0(),(

From equation 22

0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dytyPdytyP

dytyPtPtPtPtP

mm

mE


0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dysyPdysyP

dysyPsPsPsPsP

mm

mE


)())()](1[

)()()]()(1[

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

0,2124

0,11251

0,01251

1

1

sPsVsV

sPsVsVsV

sPsVsVsVsP

m

mE

)]()()[()()()( 0,10,0140,213 sPsPsVsPsVsPE (32)

where

s

sssV

)(21)(12

)()](1[)( 12413 sVsVsV

)()]()(1[)( 125114 1sVsVsVsV m

From equation 23

0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dytyPdytyP

dytyPtPtPtPtP

mm

mC


0

,2

0

,1

0

,00,20,10,0

),(),(

),()()()(),0(

11

1

dysyPdysyP

dysyPsPsPsPsP

mm

mC





)()()](1[

)()()]()(1[

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

0,2124

0,11251

0,01251

1

1

sPsVsV

sPsVsVsV

sPsVsVsVsP

m

mC

)]()()[()()()( 0,10,0160,215 sPsPsVasPsVsPC (33)

where

s

sssV

)(21)(12

)()](1[)( 12415 sVsVsV

)()]()(1[)( 125116 1sVsVsVsV m

From equation 5

0),()(2 0,3

txPx

xtB


)(2),(

),(

0,3

0,3

xssxP

sxPx

B

Integrating ‘x’ between the limits ‘0’ to ‘x’

x

B dxxsx

esPsxP 0

)(2

0,30,3 ),0(),(

From equation 14

0

,220,220,3 ),()(),0(1

dytyPtPtP m


0

,220,220,3 ),()(),0(1

dysyPsPsP m

Substituting the value of P3,0(0,s) in the above equation we get

)]()()[()()(),( 0,00,1190,2180,3 sPsPsVsPsVsxP (34)

where

s

sssV B )(21)(17

)()](1[)( 174218 sVsVsV

)()()( 175219 sVsVsV

From equation 3

0

,21

0,110,010,22

)(),()(

)()()(

11

21

dyytyPtPR

tPbtPbtPdt

d

mmr

mm


0

,21

0,110,010,22

)(),()(

)()()(

11

21

dyysyPsPR

sPbsPbsPS

mmr

mm




Substituting the values of )(sPr , ),(,2 1 symP in the above equation we get

)]()()[()( 0,10,0220,2 sPsPsVsP (35)

)()()()( 2521120 121 mmm sSsVsVRbsV

)(

)()(

2

221

1211

12

mmmm

mm

sS

ssV

)(

)()(

21

2022

sV

sVsV

from equation 2

0

00

,2

0

0

,1

0

,1

0

0,30,11

)(),(

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

)(),()(),(

)(),()(

22

2211

12

xdutuP

xdhthPdzztzPdxvtvP

dzztzPdyytyP

dxxtxPtPdt

d

C

HmmE

mmmm

Bmm


0 0 0

,2

00

,1

0

,1

0

0,30,11

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

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

)(),()(

22

2211

12

dxusuPdxhshPdzzszP

dxvsvPdzzszPdyysyP

dxxsxPsPS

CHm

Emmmm

Bmm

m

Substituting the values of

),(0.3 sxP , ),(1.1 syP m , ),(

2.1 szP m , ),(2.2 szP m , ),( svPE , ),( shPH , ),( suPC

In the above equation we get

)()()()()( 0,0270,2260,1 sPsVsPsVsP (36)

where

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

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

22 44

44223

sSsSsVsSsV

sSsVsSsVsV

mm

B

)()()()](

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

22

11

85

1515224

sSsVsSsV

sVsSsVsVsSsVsV

mm

mmB

)()](1[)()()()(21212 11124125 sSsVssmmsVSsV mmmmm

)(

)()(

25

2326

sV

sVsV

)(

)()(

25

2427

sV

sVsV

From equation 1




0

00

,2

0

0

,0

0

,0

0

0,30,01

)(),(

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

)(),()(),(

)(),()(

22

2211

12

xdutuP

xdhthPdzztzPdxvtvP

dzztzPdyytyP

dxxtxPtPdt

d

C

HmmE

mmmm

Bmm


0 0 0

,2

00

,0

0

,0

0

0,30,01

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

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

)(),()(

22

2211

12

dxusuPdxhshPdzzszP

dxvsvPdzzszPdyysyP

dxxsxPsPS

CHm

Emmmm

Bmm

m

Substituting the values of

),(0.3 sxP , ),(1.0 syP m , ),(

2.0 szP m , ),(2.2 szP m , ),( svPE , ),( shPH , ),( suPC in the above

equation we get

1)()()()()()( 0,1240,2230,025 sPsVsPsVsPsP (37)

From equation 34

)]()()[(),( 0,10,0220,2 sPsPsVsxP

Substitute the value of )(0,1 sP in this equation we get

)(P)()()(P )()()()()( 0,027220,226220,0220,2 ssVsVssVsVsPsVsP

)()()(1

)](1)[()( 0,0

2622

27220,2 sP

sVsV

sVsVsP

)()(

)()( 0,0

28

290,2 sP

sV

sVsP (38)

where )()(1)( 262228 sVsVsV

)](1)[()( 272229 sVsVsV

From equation 36

)()()()()( 0,0270,2260,1 sPsVsPsVsP

Substitute the vale of )(0,2 sP in this equation we get

)()()(1

)()()()( 0,0

2226

2722260,1 sP

sVsV

sVsVsVsP

)()(

)()( 0,0

28

300,1 sP

sV

sVsP

where

)()()()( 27222630 sVsVsVsV

From equation 37

1)()()()()()( 0,1240,2230,025 sPsVsPsVsPsP




Substitute the value of )(),( 0,10,2 sPsP in this equation we get

)(

)()(

31

280,0

sV

sVsP (39)

where )()()()()(P)()( 302429230,02531 sVsVsVsVssVsV

Substituting the value )(P 0,0 s in equation (37, 38) we get

)(

)()(

31

300,1

sV

sVsP (40)

)(

)()(

31

290,2

sV

sVsP (41)

Evaluation of operational Availability and Non-Availability

The Laplace transforms of the probability that the system is in operable up and down state at time ‘t’ can be

evaluated as follows.

)()]([1

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

)()()()()()()(

0,24

0,151m0,051m

,2,1,00,20,10,0

11

111

sPsV

sPsVsVsPsVsV

sPsPsPsPsPsPsP mmmUP

(42)

)()()()()()()()()(222 ,2,1,00,3 sPsPsPsPsPsPsPsPsP CHrEmmmdawn

)()](()()()()([

))()()](()()()()([

0,2181513119

0,10,0191614107

sPsVsVsVsVsV

sPsPsVsVsVsVsV

(43)

ERGODIC BEHAVIOUR

Using Abel’s lemma is Laplace transform,

viz., )()(0

)(0

sayftft

ltssf

s

lt

, provided that the limit on the R.H.S exists, the time independent

up and down state probabilities are as follows.

)0(

)0()]0([1

)0(

)0()]0()0([1

)0(

)0()]0()0([1

)()()()()()()(

31

294

31

3051m

31

2851m

,2,1,00,20,10,0

11

111

V

VV

V

VVV

V

VVV

sPsPsPsPsPsPsP mmmUP

(44)

)()()()()()()()()(222 ,2,1,00,3 sPsPsPsPsPsPsPsPsP CHrEmmmdown

)0(

)0()]0()0()0()0()0([

)0(

)0(

)0(

)0()]0()0()0()0()0([

31

29181513119

31

30

31

28191614107

V

VVVVVV

V

V

V

VVVVVV

(45)

PARTICULAR CASE:- When repair follows exponential distribution setting

i

ii

ssS

)( and

ssS )( where i = B, m1, m2 in results (25) to (34) (39) (40) (41) one may get the various probabilities as

follows:




)(

)()(

31

280,0

sg

sgsP

)(

)()(

31

300,1

sg

sgsP

)(

)()(

31

290,2

sg

sgsP

)]()()[()()()( 0,10,0190,2180,3 sPsPsgsPsgsP

)()()( 0,01,0 11sPsgsP mm

)()()( 0,11,1 11sPsgsP mm

)]()()[()( 0,10,02 sPsPsgsPr

)]()()[()()()( 0,10,050,24,2 1sPsPsgsPsgsP m

)()()( 0,07,0 2sPsgsP m

)()()( 0,17,1 2sPsgsP m

)]()()[()()()( 0,00,1100,29,2 2sPsPsgsPsgsP m

)()()( 0,211 sPsgsPH

)]()()[()()()( 0,10,0140,213 sPsPsgsPsgsPE

)]()()[()()()( 0,10,0160,215 sPsPsgsPsgsPC

where 1

115 )()(12

mmSsg

)](1[2

)1()( 1

1

12 1

sgRs

bsg m

1

23 )()(12

mmSsg

134 )()( msgsg

)()()( 211115 sgRsgmbsg

1

6 )()(2

mssg

)()](1[)( 617 12sgsgsg mm

)()(

2

2

8

m

m

ss

ssg

)()](1[)( 849 2sgsgsg m

)()]()( 8510 2sgsgsg m

)()(11

h

h

ss

ssg

)()(12

ss

ssg

)()](1[)( 12413 sgsgsg

)()]()(1[)( 125114 1sgsgsgsg m

)()](1[)( 12415 sgsgsg




)()]()(1[)( 125116 1sgsgsgsg m

)()(17

B

B

ss

ssg

)()](1[)( 174218 sgsgsg

)()()( 175219 sgsgsg

1

2521120 )()()()(21

mm ssgsgRbsg

1

2

221

)(

)()(

1211

12

mmmm

mm

s

ssg

)(

)()(

21

2022

sg

sgsg

1

1

4

1

4

1

4

1

4223

)(

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

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

222

s

ssgssg

ssgssgsg

mmm

BB

1

8

1

51

1

51

1

5224

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

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

2221

1

mmmm

mBB

ssgssgsg

ssgsgssgsg

1

1

1

24225

)()](1[)(

)()()(

2212111

12

mmmmmmm

mm

ssgs

sgssg

)(

)()(

25

2326

sg

sgsg

)(

)()(

25

2427

sg

sgsg

)()(1)( 262228 sgsgsg

)](1)[()( 272229 sgsgsg

)()()()( 27222630 sgsgsgsg

)()()()()()()( 30242923282531 sgsgsgsgsgsgsg

UP and Down state probabilities

The Laplace transforms of up and down state probabilities are as follows.

)()](1[

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

)()()()()()()(

0,24

0,1510,051m

,2,1,00,20,10,0

11

111

sPsg

sPsgsgsPsgsg

sPsPsPsPsPsPsP

m

mmmUP

(44)

)(][

)(][

)(][

)()()()()()()()()(

0,2181513119

0,119164107

0,0191614107

,2,1,00,3 222

sPggggg

sPggggg

sPggggg

sPsPsPsPsPsPsPsPsP CHrEmmmdown

RELIABILITY: Laplace transforms of the reliability of the system given as follows:




))((1

1)( 111 111

DsBs

b

Bs

b

Cs

b

CsASsR

mmm

Taking inverse Laplace transforms of the above equation we get

))((

)()())(()(

)())((1)(

1

11

111

1

1111

111

DBBA

be

AC

b

AC

me

BDBA

b

AB

be

AC

b

ADAB

b

AB

b

ACetR

mDt

mCtmBt

mmmAt

M.T.T.F Mean time to system failure is the expected time to operate the system successfully which is given as

follows:

0

)(.. dttRFTMT

))(()()(1

111 1111

ABAD

b

AC

b

AC

m

AB

mb

A

mm

))((

1

)()(

1

))((

1

1

111

1

111

DADB

b

D

CA

b

ACCBABD

b

AB

b

B

m

mmm

where 121 mmA , 121 mmB

12mC , 22mD

Numerical Illustrations:

Reliability Analysis:

01.01 , 02.02 , 011.01m , 015.0

2m , = 0.01, = 0.02, = 0.03, b = 0.96 and for

different values of t in the equation (60) one may obtain the reliability of the system as given in fig 2. The

reliability of the system decreases slowly as the time period increases. It also depicts the reliability of the system

for a long time period.

M.T.T.F

1 = 0.01, 2 = 0.02, 01.01m , b = 0.96, = 0.02 and taking different values of

2m in the equation (61)

one may obtain the variations of M.T.T.F. of the system against the environmental failure rate shown in figure

3.

The variations in M.T.T.F w.r.to Environmental failure rate as the major failure rate of the subsystem A

increases. The series of curves Represents that MTTF decreases as the environmental failure increases

apparently.




0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

Time (t)

Re

lia

bil

ity

Fig. 5.2.

Fig. 2 Fig. 3

Fig. 5.4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.02 0.04 0.06 0.08 0.1 0.12Environmental Failure Rate

M.T

.T.F

.

Series1

Series2

Series3

Fig. 5.5.

Series 1 2 3

2.02m 4.0

2m 6.0

2m

Fig. 4 Series 1 2 3

2.01m

4.02m

6.03m

Fig.5

Fig. 5.6.

Fig.6

V. DISCUSSION In this paper we presented mathematical models

of imperfect switching, environmental, common

cause and human error effects, so probabilistic

behaviour considering various values of coefficient

of human error, common cause, environmental

effects respectively.

VI. CONCLUSIONS : The Reliability, MTTF curves are plotted in figures (2),

(3). From these graphs we observe that

i) The Reliability of the system decreases slowly as

the time period increases.

ii) The MTTF decreases as the environmental

failure increases

REFERENCES [1] Gupta, P.P. and Agarwal, S.C. : A parallel

redundant complex system with two types of

failure under different repair discipline, IEEE

Trans. Reliability, Vol R-32, 1983.

[2] Kontoleon, J.M. and Kontoleon, N., 1974.

Reliability Analysis of a system subjected to

partial & Catastrophic Failures, IEEE Trans.

On Rel., Vol. R-23, pp 277-278.

[3] Mittal, S.K., Gupta, Ritu and C.M. Batra,

2007. Probabilistic behavior of a Redundant

Complex System, With Imperfect Switching

and Environmental Effects under Head of Line

Repair Discipline, Bulletin of pure and

Applied Sciences, Vol. 6E(1), 2007, pp. no.43-

58.

[4] Mittal S.K., Agarwal, S.C., and Kumar Sachin,

2004. Operational Behaviour of a parallel

redundant complex system with two types of

failures with environmental effect under Pre-

Emptive Resume Repair Discipline, Bulletin of

Pure and Applied Sciences, Vol. 23E (No.2),

pp.253-263.

N0460284100

Documents

environmental failure

system reliability

system subject

standby redundant system

total failure state

minor failure rate

different repair discipline

probability of failure