07 - analysis of stochastic models in manufacturing systems pertaining to repair machine failure

7/27/2019 07 - Analysis of Stochastic Models in Manufacturing Systems Pertaining to Repair Machine Failure

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7Analysis of StochasticModels inManufacturing

Systems Pertaining

to RepairMachine Failure

7.1 Introduction7.2 System Description and Assumptions7.3 Notation and States of the System

7.4 Model ATransition Probabilities and Sojourn Times Analysis ofReliability and Mean Time to System Failure AvailabilityAnalysis Busy-Period Analysis Expected Number of RepairsDuring (0, t) Particular Case

7.5 Model BTransition Probabilities and Sojourn Times Analysis ofReliability and Mean Time to System Failure Availability

Analysis Busy-Period Analysis Expected Number of RepairsDuring (0, t) Particular Case Expected Number of RepairsDuring (0, t)

7.6 Model CTransition Probabilities and Sojourn Times Analysis ofReliability and Mean Time to System Failure AvailabilityAnalysis Busy-Period Analysis

7.7 Profit Analysis7.8 Graphical Study of System Behaviour

This chapter deals with three stochastic models A, B, and C, each consisting of two nonidentical units

in standby network. One unit is named as the priority unit (p-unit) and the other as the nonpriority or

ordinary unit (o-unit). In each model, the p

-unit gets priority in operation over the o

-unit. A single

server is available to repair a failed unit and a failed repair machine (R.M.). The R.M. is required to do

the repair of a failed unit. In models A

and C

, the o

-unit gets priority in repair over the p

-unit, whereas

Rakesh Gupta

Ch. Charan Singh University

Alka Chaudhary

Meerut College

2001 by CRC Press LLC


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in model-

B

the priority in repair is also given to the p

-unit over the o

-unit. In each model it is assumed

that the R.M. may also fail during its working and then the preference in repair is given to R.M. over

any of the units. In models A

and B

, the failure and repair times of each unit are assumed to be

uncorrelated independent random variables (r.vs.), whereas in model C

these two r.vs. are assumed to

be correlated having bivariate exponential distribution. In each model we have obtained various economic

measures of system effectiveness by using the regenerative point technique.

7.1 Introduction

Two-unit standby systems have been widely studied in the literature of reliability due to their frequent

and significant use in modern business and industry. Various authors including [1, 3, 810, 1723, 25]

have studied two-unit standby systems with different sets of assumptions and obtained various charac-

teristics of interest by using the theories of semi-Markov process, regenerative process, Markov-renewal

process and supplementary variable technique. They have given equal priority to both the units in respect

of operation and repair. But realistic situations may arise when it is necessary to give priority to the mainunit in respect of operation and repair as compared to the ordinary (standby) unit. A very good example

of this situation is that of a system consisting of two units, one power supply and the other generator.

The priority is obviously given to the power through power station rather than generator. The generator

will be used only when the power supply through power station is discontinued. Further, due to costly

operation of the generator, the priority in repair may be given to power station rather than the generator.

Keeping the above concept in view, Nakagawa and Osaki [24] have studied the behavior of a two-unit

(priority and ordinary) standby system with two modes of each unit normal and total failure. Goel et al

.

[2] have obtained the cost function in respect of a two-unit priority standby system with imperfect

switching device. They have assumed general distributions of failure and repair times of each unit.

Recently, Gupta and Goel [11] investigated a two-unit priority standby system model under the assump-

tion that whenever an operative unit fails, a delay occurs in locating the repairman and having himavailable to repair a failed unit/system. Some other authors including [1215] have also investigated two-

unit priority standby system models under different sets of assumptions. The common assumption in

all the above models is that a single repairman is considered and the preference with respect to operation

and repair is given to priority (

p

) unit over the ordinary (

o

) unit. However, situations may also arise

when one is to provide preference to priority (

p

) unit only in operation and not in repair. Regarding the

repair, either the preference may be given to o

-unit over the p

-unit or the repair discipline may be first

come first serve (FCFS). So, more recently Gupta et al

.

[16] investigated a two nonidentical unit cold

standby system model assuming that the preference in operation is given to the first unit (

p

-unit) while

in repair the preference is given to the second unit (

o

-unit). The system model under this study can be

visualised by a very simple example: Suppose in a two-unit cold standby system model two nonidenticalunits are an air conditioner (A.C.) and an air cooler. Obviously the preference in operation will be given

to the A.C. and air cooler will get the preference in repair as the repair of A.C. is costly and time-consuming.

The case of standby redundant system is not seen in the literature of reliability when the preference in

operation is given top

-unit but in repair the policy is FCFS.

All the above discussed authors have analysed the system models under the assumptions that the

machine/device used for repairing a failed unit remains good forever. In real situations this assumption

is not always practicable as the repair machine (R.M.) may also have a specified reliability and can fail

during the repair process of a failed unit. For example, in the case of nuclear reactors, marine equipments,

etc., the robots are used for the repair of such type of systems. It is evident that a robot, a machine, may

fail while performing its intended task. In this case obviously the repairman first repairs the repair machine

and then takes up the failed unit for repair.In this chapter we discuss three system models,A

, B

, and C

, each consisting of two nonidentical units

named as p

-unit and o

-unit. It is assumed that in each model the p

-unit gets priority in operation as

only one unit is sufficient to do the required job. A repair machine (R.M.) is required to do the repair



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of a failed unit which can also fail during its operation. Further, a single repairman is available to repair

a failed unit as well as a failed R.M. and in each model the priority is given to R.M. over any of the failed

units. Regarding the repair of failed units, it is assumed in model B

that the p

-unit gets preference in

repair over the o

-unit, whereas in modelsA

and C

the priority in repair is given to the o

-unit rather than

to the p

-unit. In models A

and B

, the basic assumption is that the failure and repair times are taken

uncorrelated independent r.vs. However, a common experience of system engineers and managers reveals

that in many system models there exists some sort of correlation between failure and repair times. It is

observed that in most of the system models an early (late) failure leads to early (delayed) repair. The

concept of linear relationship is the main point of consideration. Therefore, taking this concept in view,

in model C

, the joint distribution of failure and repair times is assumed to be bivariate exponential

(B.V.E) of the form suggested by Paulson (0 r

1). The p.d.f. of the B.V.E. is

(1)

where is the modified Bessel function of type

and order Zero. Some authors

including [47,16] have already analysed system models by using the above mentioned concept.

Using regenerative point technique in the Markov renewal process, the following reliability

characteristics of interest to system designers and operation managers have been obtained for models

A

, B

, and C

.

(i) reliability of the system and mean time to system failure (MTSF);

(ii) pointwise and steady state availabilities of the system;

(iii) the probability that the repairman is busy at an epoch and in steady state;

(iv) expected number of repairs by the repairman in (0, t

) and in steady state; and

(v) expected profit incurred by the system in (0, t

) and in steady state.

Some of the above characteristics have also been studied and compared through graphs and important

conclusions have been drawn in order to select the most suitable model under the given conditions.

7.2 System Description and Assumptions

(i) The system is comprised of two nonidentical units and a repair machine (R.M.). The units are

named as priority (

p

) unit and ordinary (

o

) unit. The operation of only one unit is sufficient to

do the job.

(ii) In each model thep

-unit gets priority in operation over the o

-unit. The o

-unit operates only when

p

-unit has failed. So, initially the p-unit is operative and o

-unit is kept as cold standby which

cannot fail during its standby state.

(iii) Each unit of the system has two modes normal (

N

) and total failure (

F

). A switching device is

used to put the standby unit into operation and its functioning is always perfect and instantaneous.

(iv) A single repairman is available with the system to repair a failed unit and failed R.M. In modelsA

and

C

, the o

-unit gets priority in repair over thep

-unit, whereas in model B

, the priority in repair is given

to thep

-unit over the o

-unit. Further, the R.M. gets the preference in repair over both the units.

(v) The R.M. repairs a failed unit and it can also fail during the repair of a unit. In such a situation

the repair of the failed unit is discontinued and the repairman starts the repair of the R.M. as a

single repairman is available. Each repaired unit and R.M. work as good as new.

(vi) The R.M. is good initially and it cannot fail until it begins functioning.(vii) In models A

and B

, the failure times and repair times of a unit and R.M. are assumed to be

independent and uncorrelated r.vs., whereas in model C

the failure and repair times of the units

are correlated r.vs.

f x y,( ) 1 r( )exyI0 2 r y( )

x y 0, 0 r 1, , ,

I0 z( ) k0

Z2( )2 K

K!( )2--------------------



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(i) Model A

Failed states:

The epoch of transition from S2 to S4 is nonregenerative while all the other entrance epochs into the

states are regenerative. The transition diagram of the system model along with failure/repair rates or

repair time cdf is shown in Figure 7.1.

(ii) Model B

Failed states:

FIGURE 7.1 Transition diagram of system model with failure/repair rates or repair time cdf.

Up states : S0N10,N2S

RMg , S1

F1r,N20

RMo , S2

F1w,N20

RMr

S3F1w,F2r

RMo , S4

F1w,F2w

RMr

Up states: S0N10,N2S

RMg , S1

F1r,N20

RMo , S2

F1w,N20

RMr

S5N10,F2r

RMo , S6

N10,F2w

RMr

S3F1r,F2w

RMo , S4

F1w,F2w

RMr



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The epochs of transition entrance into the states S3 from S1 and S4 from S2, S6 are nonregenerative.

The transition diagram of the system model along with failure/repair rates or repair time cdf is shown

in Figure 7.2.(iii) Model C

The epochs of entrance from S0 to S1, S1 to S3, S5 to S3, S2 to S6, and S6 to S3 are nonregenerative while

all the other entrance epochs into the states are regenerative. The transition diagram of the system model

along with failure/repair times or failure/repair rates is shown inFigure 7.3.

FIGURE 7.2 Transition diagram of the system model with failure/repair rates or repair time cdf.

Up states : SoN10,N2S

RMg , S1

F1r,N20

RMo

S2F1w,N20

RMr , S5

F1r,N20

RMo

Failed States: S3F1w,N2r

RMo , S4

F1w,F2w

RMr

S6F1w,F2w

RMr , S7

F1w,F2r

RMo



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7.4 Model A

Transition Probabilities and Sojourn Times

Let T0 (0), T1, T2, ... denote the epochs at which the system enters any state SiE, and letXn be thestate visited at epoch Tn , i.e., just after the transition at Tn. Then {Xn, Tn} is a Markov renewal process

with state space E. If

Then the transition probability matrix (t.p.m.) is given by

(i) By simple probabilistic reasoning the nonzero elements ofQ (Qij(t)) may be obtained as follows:

FIGURE 7.3 Transition diagram of the system model with failure/repair times or failure/repair rates.

Qij t( ) P Xn+1 Sj,Tn+1 Tn t Xn Si[ ]

P ij( ) Qij ( )[ ] Q ( )



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For the system to reach state S1 from S0 on or before time t, we suppose that the system transits from S0to S1 during (u, udu), ut. The probability of this event is 1e

udu. Since u varies from o to t, therefore,

Similarly,

(18)

To derive an expression for , we suppose that the system transits from state S2 to S4 during the

time interval (u, udu), ut; the probability of this event is (u)du. Further, suppose that

the system passes from state S4 to S3 during the interval (v, vdv) in (u, t); the probability of this event

is dH(v)/ (u). Thus,

(9)

(ii) The steady-state transition probabilities are given by

Q01 t( ) 1e1u

ud 1 e

1t

0

t

Q10 t( ) e2 ( )u G1 u( )d0

t

Q12 t( ) e2 ( )u G1 u( ) ud

0

t

Q13 t( ) 2 e2 ( )u G1 u( ) ud

0

t

Q21 t( ) e2u H u( )d

0

t

Q24 t( ) 2 e2u H u( ) ud

0

t

Q31 t( ) eu

G2 u( )d0

t

Q34 t( ) eu

G2 u( ) ud0

t

Q43 t( ) H u( ) H t( )d0

t

Q234( )

t( )2e

2u H

H

Q234( )

t( ) 2e2u H u( ) u H v( )H( ) u( )d

u

t

d0

t

2 H v( ) e2u u (by change of order of integration)d

0

v

d0

t

1 e2v( ) H v( )d

0

t

pij Qijt lim t( )



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Therefore,

(1017)

It is easily verified that

(1820)

(iii) Mean sojourn time i in state Si is defined as the expected time for which the system stays in stateSi before transiting to any other state. To calculate the mean sojourn time 0 in state S0. We observe

that so long as the system is in state S0, there is no transition to S1. Hence ifT0 denotes the sojourn

time in S0, then

(21)

Similarly,

(2225)

(iv) We define mij as the mean sojourn time by the system in state Si when the system is to transit to

regenerative state Sj, i.e.,

Therefore,

(2634)

p01 p43 1

p10 G1 2( ), p12 1 G1 2( )[ ] 2( )

p13 2 1 G1 2( )[ ] 2( )p21 H 2( ), p23

4( )p24 1 H

2( )

p31 G ( ), p34 1 G2 ( )

p10 p12 p13 1

p21 p234( )

p24( ) 1

p31 p34 1

0 p T0 t( ) t e1t t 11dd

1 1 G1 2( )[ ] 2( ), 2 1 H 2( )[ ]2

3 1 G2

( )[ ], 4 H t( ) td

mij t Qd ij t( ) tq ij t( ) td

m01 1 te1t t 11.d

m10 te2+( )t

G1 t( ).dm12 te

2+( )t G1 t( ) t.d

m13 2 te2+( )t G1 t( ) t.d

m21 te2t Hd t( ).

m234( )

t 1 e2t

( ) Hd t( ).

m31

tet

G2

d t( ).

m34 tet

G2 t( ) t.d

m43 t H t( ).d



11/50


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For brevity, we have omitted the argument s from , and . Computing the above

matrix equation for , we get

(5)

where

and

Taking the Inverse Laplace Transform (ILT) of (7.5), we can get the reliability of the system when

initially it starts from S0.

The mean time to system failure (MTSF) can be obtained using the well-known formula

(6)

To determineN1(0) and D1(0) we first obtain (0), using the result

Therefore

Thus, using and the above results, we get

and

Availability Analysis

Let and be the probabilities that the system is up (operative) at epoch tdue top-unit and

o-unit, respectively, when initially it starts from state SiE.

By simple probabilistic laws, is the sum of the following mutually exclusive contigencies:

(i) The system continues to be up in state S0 until epoch t. The probability of this event is .

(ii) The system transits to S1 from S0 in (u, udu), utand then starting from S1 it is observed to

be up at epoch t, with probability . Therefore,

qij s( ) Z1 s( ) Ri s( )R0 s( )

R0 s( ) N1 s( )D1 s( )

N1 s( ) 1 q12q21( )Z0 q01 Z1 Z2q12( )

D1 s( ) 1 q12q21 q01q10

E T0( ) R0 t( ) td R0 s( )s 0lim N1 0( )D1 0( )

Z1

Zi s( )s 0lim Zi t( ) td

Z0 0( ) 0, Z1 0( ) 1, Z2 0( ) 2

qij 0( ) pij

N1 0( ) 0 1 p12p21( )0 1 p122

D1 0( ) 1 p10 p12p21

Aip

t( ) Ai0

t( )

A0p

t( )

e 1t

Aip

t u( )

A0p

t( ) e 1t q01 u( ) u A1

pt u( )d

0

t

Z0 t( ) q01 t( )A1

pt( )



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Similarly,

(15)

Taking LT of relations (7.17.5) and solving the resulting set of algebraic equations for , we get

(7)

where

(8)

and

(9)

Similarly, the recurrence relations among pointwise availabilities , i.e., when system is up due to

o-unit, can also be obtained and are as follows:

(1014)

where

Taking LT of the relations (7.107.14) and solving for , we get

(15)

where

(16)

Now to obtain the steady-state probabilities that the system will be operative due top-unit and o-unit,

we proceed as follows:

A1p

t( ) q10 t( )A0p

t( ) q12 t( )A2p

t( ) q13 t( )A3p

t( )

A2p

t( ) q21 t( )A1p

t( ) q 23( )4( )

t( )A3p

t( )

A3p

t( ) q31 t( )A1p

t( ) q34 t( )A4p

t( )

A4p

t( ) q43 t( )A3p

t( )

A0p

s( )

A0p

s( ) N2 s( )D2 s( )

N2 s( ) 1( q12 q21( q234( )

q31 ) q13q31 q34q43 1 q12q21( ) ]Z0

D2 s( ) 1 q12 q21( q234( ) q31 ) q13q31 q34q43 1 q12q21( )

q01 q10 1 q34 q43( )

Ai0

t( )

A00

t( ) q01 t( )A10

t( )A1

0t( ) Z1 t( ) q 10 t( )A0

0t( ) q12 t( )A1

0t( ) q13 t( )A3

0t( )

A2t( )

t( ) Z2 t( ) q 21 t( )A10

t( ) q234( )

t( )A30

t( )

A30

t( ) q31 t( )A10

t( ) q34 t( )A40

t( )

A40

t( ) q43 t( )A30

t( )

Z1 t( ) e2 ( )t

G1 t( ) and Z2 t( ) e2t

H t( )A0

0s( )

A00

N3 s( )D2 s( )

N3 s( ) q01 1 q34q43( ) Z1 q12Z2( )

Zi 0( ) Zi t( ) td i i 0 1 3, ,( )



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Using the result , we have

Therefore, the steady-state probability that the system will be operative due to p-unit is given by

(17)

where

To obtain , we collect the coefficients of in for various values ofi and

jas follows:

(i) Coefficient ofm01 p10 (1 p34)

p31p10(ii) Coefficient ofm10 1p34p31

(iii) Coefficient ofm12 p21p23p31p34p21 p121(1 p34) p23p31

p31(iv) Coefficient ofm13 p31(v) Coefficient ofm21 p12p12p34

p31p12(vi) Coefficient of

(vii) Coefficient ofm31 p12 p13(viii) Coefficient ofm34 1 p12p21p10

p12 p13(ix) Coefficient ofm43 p34 (1p12p21)p10p34

p34(1 p10p12p21)p34(p12 p13)

Thus,

(18)

Similarly, the steady-state probability that the system will be operative due to o-unit is given by

(19)

where

and has already been defined by equation (18).

qij 0( ) pij

D2 0( ) 0

A0p

N2 t( )D2 t( )t lim

s N2 s( )D2 s( )s 0lim

N2 0( )D2 0( )

N2 0( ) p31p100

D2 0( ) qij 0( ) mij( ) D2 0( )

m234( )

p31p12p23

4( )

p234( )

p234( )

D2 0( ) P31 p100 1 p124 ( ) p13 p12p234( )

( ) 3 p344( )

A 0o

N3 t( )D2 t( ) N3 0( )D2 0( )t lim

N3 0( ) p31 1 p122( )

D2 0( )



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Busy-Period Analysis

Let , , and be the respective probabilities that the repairman is busy in the repair

of p-unit, o-unit, and RM when system initially starts from state Si E. Using elementary

probabilistic arguments in respect to the above definition of , we have the following recursive

relations:

(15)

where

For an illustration, is the sum of the following mutually exclusive contingencies:

(i) The repairman remains busy in state S1 continuously up to time t. The probability of this event

is

(ii) The system transits from state Si to Sj(j 0, 2, 3) during time (u, udu), u tand then starting

from state Sj the repairman may be observed to be busy with p-unit at epoch t. The probability

of this contingency is

Taking LT of relations (15) and solving the resulting set of algebraic equations for we get

(6)

where

and D2(s) is the same as given by equation (3.9).

Similarly, the recursive relations in and may be developed as follows:

(711)

Bip

t( ) Bio

t( ) Bim

t( )

Bip

t( )

B0p

t( ) q01 t( )B1p

t( )

B1p

t( ) Z1 t( ) q 10 t( )B0p

t( ) q12 t( )B2p

t( ) q13 t( )B3p

t( )

B2p

t( ) q21 t( )B1p

t( ) q234( )

t( )B3p

t( )

B3p

t( ) q31 t( )B1p

t( ) q34 t( )B4p

t( )

B4p

t( ) q43 t( )B3p

t( )

Z1 t( ) e2+( )t G1 t( )

B1p

t( )

e2 ( )t G1 t( )

qij u( ) u Bjp

t u( ) qij t( )Bjp

t( )d0

t

B0p

s( )

B0p

s( ) N4 s( )D2 s( )

N4 s( ) q01 1 q34 q43( )Z1

Bio

t( ) Bim

t( )

B00

t( ) q01 t( )B10

t( )

B10

t( ) q10 t( )B00

t( ) q12 t( )B20

t( ) q13 t( )B30

t( )

B20

t( ) q21 t( )B10

t( ) q234( )

t( )B30

t( )

B30

t( ) Z3 t( ) q 31 t( )B10

t( ) q34 t( )B40

t( )

B40

t( ) q43 t( )B30

t( )



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Expected Number of Repairs During (0, t)

Let , , and be the expected number of repairs of thep-unit, o-unit, and RM, respec-

tively, during (0, t)| E0Si. Here, using the above definition, the recursive relations in are as follows:

As an illustration, is the sum of the following contingencies:

(i) The system transits from S1 to S0 in the interval (u, udu), u thaving completed one repair

and then starting from S0 at epoch u we may count the expected number of repairs during time(t u). The probability of this event is

(ii) The system transits from S1 to Sj (j 2, 3) during time (u, udu), u tand then starting from

Sj at epoch u we may count the expected number of repairs during time (tu). The probability

of this contingency is

Taking Laplace-Stieltjes transform (LST) of the above equations (15) and solving for we have

(6)

where

and D3(s) can be written simply on replacing and , respectively, by and in D2(s) given by

equations (3.87).

Similarly, the recurrence relations in and can be obtained to get the expected number

of repairs ofo-unit and RM, respectively, and are as follows:

(711)

Nip

t( ) Nio

t( ) Nim

t( )Ni

pt( )

N0p t( ) Q01 t( ) N1p t( )

N1p

t( ) Q10 t( ) 1 N0p

t( )[ ] Q12 t( ) N2p

t( ) Q13 t( ) N3p

t( )

N2p

t( ) Q21 t( )$$N1p

t( ) Q234( )

t( ) N3p

t( )

N3p

t( ) Q31 t( )$$N1p

t( ) Q34 t( ) N4p

t( )

N4p

t( ) Q43 t( )$$N3p

t( )

N1p

t( )

Q10 u( ) 1 N0p

t u( )[ ] Q10 t( ) 1 N0p

t( )[ ]d0

t

Qij u( )Njp

t u( ) Qij t( ) Njp

t( )d0

t

N 0p

s( )

N 0p

s( ) N7 s( )D3 s( )

N7 s( ) Q01Q10 1 Q34Q43( )

qij qijk( )

Qij Q ijk( )

Nio

t( ) Nim

t( )

N0o

t( ) Q01 t( ) N1o

t( )

N1o

t( ) Q10 t( ) N0o

t( ) Q12 t( ) N2o

t( ) Q13 t( ) N3o

t( )

N2o

t( ) Q21 t( ) N1o

t( ) Q234( )

t( ) N3o

t( )

N3o t( ) Q31 t( ) 1 N 1o t( )[ ] Q34 t( ) N4o t( )

N4o

t( ) Q43 t( ) N3o

t( )



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and

(1216)

Taking LST of the above relations (711) and (1216) and solving for and , we get

and

where,

and

Now let , and , be the expected number of repairs ofp-unit, o-unit, and RM in steady

state, respectively. Then their expressions can be obtained as follows:

Similarly,

Since

and

therefore the value comes out to be same as that of given by equation (3.18).

Particular Case

In this section, we consider a case when the repair time distributions are also negative exponential, i.e.,

N0m

t( ) Q01 t( ) N1m

t( )

N1m

t( ) Q10 t( ) N0m

t( ) Q12 t( ) N2m

t( ) Q13 t( ) N3m

t( )

N2m t( ) Q21 t( ) 1 N 1m t( )[ ] Q234( ) t( ) N3m t( )

N3m

t( ) Q31 t( ) N1m

t( ) Q34 t( ) N4m

t( )

N4m

t( ) Q43 t( ) 1 N 3m

t( )[ ]

N 0o

s( ) N 0m

s( )

N 0o

s( ) N8 s( )D3 s( )

N 0m

s( ) N9 s( )D3 s( )

N8 s( ) Q01Q31 Q13 Q12Q234( )

( )

N9 s( ) Q01 Q12Q21 1 Q34Q43( ) Q 34Q43 Q13 Q12Q234( )

( )[ ]

N0p

, N0o

N0m

N0p

N0p

t( )t lim N0

p s( )s 0lim

P31P10D3 0( )

N0o

p31 p13 p12p234( )

( )D3 0( )

N0m

p12p21p31 p34 p13 p( 12p234( )

[ ]D3 0( )

qij 0( ) Qij 0( ) pij, qijk( )

0( ) Qijk( )

0( ) pijk( )

qij 0( ) Q ij 0( ) mij , qijk( )

0( ) Qijk( )

0( ) mijk( )

D3 0( ) D2 0( )

g1 t( ) 1e1t , g2 t( ) 2e

2t

h t( ) e t



19/50

Then in results (1.111.17) and (1.221.25) we have the following changes:

7.5 Model B


(i) By simple probabilistic arguments as discussed in Section 7.4 the nonzero elements ofQ (Qij(t))

are as follows:

(114)

p10 1 1 2 ( ), p12 1 2 ( )

p13 2 1 2 ( ), p21 2( ), p234( )

2 2( )

p31 2 2( ) , p34 2( ), 1 1 1 2 ( )2 1 2( ), 3 1 2( ), 4 1

Q01 t( ) 1e1us

0

t

ud

Q10 t( ) e2+( )

0

t

G1 u( )d

Q12 t( ) e2+( )u G1 u( ) ud

0

t

Q13 t( ) 2 e2+( )u G1 u( ) ud

0

t

Q21 t( ) e2u

0

t

dH u( )

Q24 t( ) 2 e2u

0

t

H u( ) ud

Q34 t( ) eu

0

t

G1 u( ) ud

Q35 t( ) eu

0

t

Gd 1 u( )

Q43 t( ) Hd0

t

u( ) H t( )

Q50 t( ) e1+( )u G2d u( )0

t

Q53 t( ) 1 e1+( )u G2 u( ) ud

0

t

Q56 t( ) e1+( )u G2 u( ) ud

0

t

Q64 t( ) 1 e1u

0

t

H u( ) ud

Q65 t( ) e1u

0

t

Hd u( )



20/50

(ii) To derive an expression for , we suppose that the system transits from state S1 to S4 during

the time interval (u, u du), u t, the probability of this event is . Further,

suppose that the system passes from state S3 to S4 during the interval (v, v dv) in (u, t); the

probability of this event is

Thus,

Similarly,

(1518)

(iii) Using the result

the steady-state transition probabilities are as follows:

(1933)

Q143

t( )2e

1 ( )u G1 u( ) ud

e v u( ) v Gd v( )G1 u( )

Q143

t( ) 2e2 ( )u G1 u( ) u e

v u( )v G1d v( )G1 u( )

0

t

d0

t

2 ev

G1 v( ) v e2u ud

0

v

d0

t

ev

1 e2v

[ ]G1 v( ) vd0

t

Q153

t( ) e v 1 e2v

[ ] Gd 1 v( )0

t

Q234

t( ) 1 e2v

[ ] Hd v( )0

t

Q634

t( ) 1 e1v

[ ] Hd v( )0

t

pij Qij t( )t lim

p01 p43 1

p10 G1 2 ( )

p12 1 G

1 2 ( )[ ] 2 ( )p13 2 1 G1 2 ( )[ ] 2 ( )

p143( )

1 G1 ( ) 1 G1 2 ( )[ ] 2 ( )

p153( )

G1 ( ) G1 2 ( )

p21 H 2( ), p24 p23

4( )1 H 2( )

p34 1 G1 ( ) , p35 G1 ( )

p50 G2 1 ( )

p53 1 1 G

2 1 ( )[ ] 1 ( )p56 1 G2 1 ( )[ ] 1 ( )

p64 p634( )

1 H 1( ), p65 H 1( )



21/50

It is easily verified that

(3438)

(iv) By similar probabilistic arguments as in Section 7.4, the mean sojourn times in various states are

as follows:

(3945)

(v) mij, the mean sojourn time by the system in state Si when the system is to transit to regenerative

state Sj is given by

therefore,

p10 p12 p13 p143( )

p153( )

( ) 1, p21 p24 p234( )( ) 1

p34 p35 1, p50 p53 p56 1

p64 p634( )( ) p65 1

0 e1t td 11

1 e2 ( )t G1 t( ) td

2 e2t

H t( ) td3 e

t G1 t( ) td

4 H t( ) td

5 e2 ( )t G2 t( ) td

6 e1t H t( ) td

mij t Qd ij t( ) tq ij t( ) td

m01 1 te1t td 11

m10

te2+( )t G

1

t( )d

m12 te2+( )t G1 t( ) td

m13 2 te2+( )t G1 t( ) td

m143( )

tet

1 e 2t

( )G1 t( ) td

m153( )

tet

1 e 2t

( ) G1d t( )

m21 te2t Hd t( )

m24 2 te2t H t( ) td

m234

t 1 e2t[ ] Hd t( )



22/50

(4664)

The following relations among mijs and is are observed:

Analysis of Reliability and Mean Time to System Failure

To determine Ri(t), we assume that the failed states S3 and S4 of the system as absorbing. By probabilistic

arguments as used in Section 7.4, we observe that the following recursive relations hold good:

(13)

where

m34 tet G1 t( ) td

m35 tet Gd 1 t( )

m43 t Hd t( )

m50 te1+( )t G2d t( )

m53 1 te1+( )t G2 t( ) td

m56 te1+( )t G2 t( ) td

m64 1 te1t H t( ) td

m634

t 1 e1t

[ ] Hd t( )m65 te

1t Hd t( )

m01 0, m43 4

m10 m12 m143( )

m153( )

3

m21

m23

4( )

4

m34 m35 3

m50 m53 m56 5

m634( )

m65 4

R0 t( ) Z0 t( ) q 01 t( )R1 t( )

R1 t( ) Z1 t( ) q 10 t( )R0 t( ) q12 t( )R2 t( )

R2 t( ) Z2 t( ) q 21R1 t( )

Z0 t( ) e1t

Z1 t( ) e2+( )t G1 t( )

Z2 t( ) e2t H t( )



23/50


24/50

where

Taking LT of relations (7.17.7) and solving the resulting set of algebraic equations for , we get

(8)

where

and

Similarly, the recursive relations among pointwise availability (i.e., when system is up due to

o-unit) can also be obtained and are as follows:

(915)

where

and

Z0 t( ) e1t

Z5 t( ) ea1 ( )t

G2 t( )

Z6 t( ) e1t H t( )

A0p

s( )

A0p

s( ) N2 s( )D2 s( )

N2 s( ) 1 q12q21( ) 1 q34q43( ) 1 q56q65( ) q35 q53 q56q634( )

( )[ ]Z0

q01 q12q234( )

q35 q143( )

q43q35 q153( )

1 q34q43( ) [ ] Z5 q56Z6( )

D2 s( ) 1 q12q21 q10q01( ) 1 q34q43( ) 1 q56q65( ) q35 q53 q56q634( )

( )[ ]

q01q50 q12q234( )

q143( )

q43( )q35 q153( )

1 q34q43( )[ ]

Aio

t( )

A0o

t( ) q01 t( ) A1o

t( )

A1o

t( ) Z1 t( ) q10 t( ) Aoo

t( ) q12 t( ) A2o

t( ) q143( )

t( ) A4o

t( ) q153( )

t( ) A5o

t( )

A2o

t( ) Z2 t( ) q21 t( ) A1o

t( ) q234( )

t( ) A3o

t( )

A3o

t( ) q34 t( ) A4o

t( ) q35 t( ) A5o

t( )

A4o t( ) q43 t( ) A3o t( )

A5o

t( ) q50 t( ) A0o

t( ) q53 t( ) A3o

t( ) q56 t( ) A6o

t( )

A6o

t( ) q634( )

t( ) A3o

t( ) q65 t( ) A5o

t( )

Z1 t( ) e2+( ) t G1 t( )

Z2 t( ) e2t H t( )



25/50

Taking LT of the above relations and solving for , we get

(16)

where

and D2(s) has already been defined.

To obtain the steady-state probabilities that the system will be operative due to p-unit and o-unit, we

proceed as in Section 7.4 and find that

(17)

(18)

where

and

(19)


Using the same probabilistic arguments as in Section 7.4, we find that the probabilities satisfy

the following recursive relations:

(17)

where andZ1(t) has already been defined.

A00

s( )

A00

s( ) N3 s( )D2 s( )

N3 s( ) q01Z1 q01q12Z2( ) 1 q34q43( ) 1 q56q65( ) q35 q53 q56q634( )

( )[ ]

A0p

N2 0( )D2 0( )

A00

N3 0( )D2 0( )

N2 0( ) p35 1 p12p21( )p500 p143( )

p153( )

p12p234( )

( ) 5 p566( )[ ]

N3 0( ) p35p50 1 p122( )

D2 0( ) p35p50 1 p12p21( )0 3 p124 [ ]

1 p10 p12p21( ) 1 p56p65( ) p153( )p50[ ] 3 p344( )

p143( )p35p504 p35 1 p10 p 12p21( ) 5 p566( )

Bip

t( )

B0p

t( ) q01 t( ) B1p

t( )

B1p

t( ) Z1 t( ) q13 t( ) Z3 t( ) q10 t( )B0p

t( ) q12 t( ) B2p

t( )

q143( )

t( ) B4p

t( ) q153( )

t( ) B5p

t( )

B2p

t( ) q21 t( ) B1p

t( ) q234( )

B3p

t( )

B3p

t( ) Z3 t( ) q34 t( ) B4p

t( ) q35 t( ) B5p

t( )

B4p

t( ) q43 t( ) B3p

t( )

B5

pt

( )q

50t

( ) B

0

pt

( )q

53t

( ) B

3

pt

( )q

56t

( ) B

6

pt

( )

B6p

t( ) q634( )

t( ) B3p( )

t( ) q65 t( ) B5p

t( )

Z3 t( ) et

G1 t( )



26/50


27/50

and

The steady-state probabilities , and that the repairman is busy with p-unit, o-unit, and

RM, respectively, can be easily obtained, and are as follows:

(2527)

where

and

Expected Number of Repairs during (0, t)

Similar probabilistic arguments as in Section 7.4 yield the recurrence relations for , andas follows:

(17)

N6 s( ) q01q12 Z2 q24Z4( ) 1 q34q43( ) 1 q56q65( ) q35 q53 q56q634( )

( )[ ]

q01q12q234( )

q34 1 q56q65( ) q35q56q64[ ]Z4 q35q56Z6{ } q01q14

3( )1 q 35q53 q56q65 q64q43 q63

4( )( )q35q56[ ]Z4 q35q56q43Z6{ }

q01q153( )

q34q53 q 56q64 q64q43 q633( )

( )q34q56[ ]Z4 q56 1 q 34q43( )Z6{ }

B0p

, B0o

Bom

B0p

N4 0( )D2 0( )

B0o

N5 0( )

D2 0( )

B0m

N6 0( )D2 0( )

N4 0( ) p35p50 1 p133( ) 1 p56p65( ) p143( )

p12p2314( )

( )3

p153( )

p53 p56p634( )

( )3

N5 0( ) p35 p143( )

p153( )

p12p234( )

( )5

N6 0( ) p12p35p50 2 p244( ) p12p234( )

p34 p35p56 p56p65( )[

p143( )

1 p35 p53 p56p65( ) p153( )

p53p34 p56p64( ) ]4

p35p56 p12p23 p143( )

p153( )

( )6

Nip t( ), Nio t( )Ni

mt( )

N0p

t( ) Q01 t( ) N1p

t( )

N1p

t( ) Q01 t( ) 1 N0p

t( )[ ] Q12 t( ) N2p

t( ) Q143( )

t( ) N4p

t( )Q153( )

t( ) 1 N 5p

t( )[ ]

N2p

t( ) Q21 t( ) N1p

t( ) Q234( )

t( ) N3p

t( )

N3p

t( ) Q34 t( ) N4p

t( ) Q35 t( ) 1 N 5p

t( )[ ]

N4p

t( ) Q43 t( ) N3p

t( )

N5p

t( ) Q50 t( ) N0p

t( ) Q53 t( ) N3p

t( ) Q56 t( ) N6p

t( )

N6p

t( ) Q634( )

t( ) N3p

t( ) Q65 t( ) N5p

t( )



28/50

Similarly,

(814)

and

(1520)

Taking LST of the above equations (17), (814), and (1520) and solving them for ,

and , we get

(2123)

where

N0o

t( ) Q01 t( ) N1o

t( )

N1o

t( ) Q10 t( ) N0o

t( ) Q12 t( ) N2o

t( ) Q143( )

t( ) N4o

t( ) Q153( )

t( ) N5o

t( )

N2o

t( ) Q21 t( ) N1o

t( ) Q234( )

t( ) N3o

t( )

N30

t( ) Q34 t( ) N4p

t( ) Q35 t( ) N5o

t( )

N4o

t( ) Q43 t( ) N3o

t( )

N5o

t( ) Q50 t( ) 1 N0o

t( )[ ] Q53 t( ) N3o

t( ) Q56 t( ) N6o

t( )

N6o

t( ) Q634( )

t( ) N3o

t( ) Q65 t( ) N5o

t( )

N0m t( ) Q01 t( )$$ N1m t( )

N1m

t( ) Q10 t( )$$ N0m

t( ) Q12 t( )$$ N2m

t( ) Q143( )

t( )$$ N4m

t( ) Q153( )

t( )$$ N5m

t( )

N2m

t( ) Q21 t( )$$ 1 N1m

t( )[ ] Q234( )

t( )$$ 1 N3m

t( )[ ]

N3m

t( ) Q34 t( )$$ N4m

t( ) Q35 t( )$$ N5m

t( )

N4m

t( ) Q43 t( )$$ 1 N3m

t( )[ ]

N5m

t( ) Q50 t( )$$ N0m

t( ) Q53 t( )$$ N3m

t( ) Q56 t( )$$ N6m

t( )

N6m

t( ) Q634( )

t( )$$ 1 N3m

t( )[ ] Q65 t( )$$ 1 N5m

t( )[ ]

N 0p

s( ), N 0o

s( )N 0

ms( )

N 0p

s( ) N7 s( )D3 s( )

N 00

s( ) N8 s( )D3 s( )

N 0m

s( ) N9 s( )D3 s( )

N7 s( ) 1 Q34Q43( ) 1 Q56Q65( ) Q35 Q53 Q56Q634( )

( )[ ] Q10 Q153( )

( )

Q35Q01 1 Q56Q65( ) Q12Q234( )

Q143( )

Q43( ) Q153( )

Q53 Q56Q634( )

( )[ ]

N8 s( ) Q01Q50 Q12Q234( )

Q 35 Q143( )

Q 43Q35 Q153( )

1 Q34 Q43( ) [ ]

N9 s( ) Q01Q12 Q21 Q234( )

( ) 1 Q34Q43( ) 1 Q65Q65( ) Q35 Q53 Q56Q634( )

( )[ ]

Q01Q12Q234( )

Q34Q43 1 Q56 Q65( ) Q35Q56 Q634( )

Q65( )[ ]

Q 01Q 143( )

Q 43 1 Q 35 Q 53 Q 56Q 65( )

Q 35Q 56 Q 634( )

Q 65( )

[ ]

Q01Q153( )

Q43Q53Q34 Q56 1 Q34Q43( ) Q634( )

Q65( )[ ]



29/50

and D3(s) can be written simply on replacing in D2(s) by .

In steady state the expected number of repairs ofp-unit, o-unit, and RM per unit time are, respectively,

as follows:

and

where

and

Particular Case

As in Section 7.4, we consider a case when the repair time distributions are also negative exponential, i.e.,

then in the result (1.201.33), (1.401.45), we have the following changes:

qij Qij

N0p N7 0( )D2 0( )

N0o

N8 0( )D2 0( )

N0m

N9 0( )D2 0( )

N7 p35 p10 p153( )

( )p50 1 p56p65( ) p143( )

p12p234( )

( ) p153( )

p53 p56p634( )

( ) [ ]

N8 p35p50 p143( )

p153( )

p12p234( )

( )

N9 p12p35p50 p35p56 p143( )

p153( )

p12p234( )

( ) p12p234( )p34 1 p56 p65( )

p143( ) 1 p35 p53 p56 p65( ) p153( )p53p34

g1 t( ) 1e1t( ), g2 t( ) 2e

2t( ), h t( ) e t( )

p10 1 1 2 ( ), p12 1 2 ( ), p13 2 1 2 ( )

p143( )

1 1 1 ( ) 1 2 ( ) , p153( )

1 1 ( ) 1 1 2 ( )

p21 2( ), p24 p234( )( ) 2 2( ), p34 1( )

p35 1 1( ), p50 2 2 1 ( ), p53 1 2 1 ( )

p56 2 1 ( ), p64 p634( )( ) 1 1( ), p65 1( )

1 1 1 2 ( ), 2 1 2( ), 3 1 1( ) 4 12, 5 1 2 1 ( ), 6 1 1( )



30/50

7.6 Model C


(i) By definition and simple probabilistic arguments as in earlier sections, the direct of one-stepunconditional transition probabilities can be obtained as follows:

(19)

(ii) By the definition ofQij|x, the direct or one-step conditional transition probabilities from the states

where the repair of a unit depends upon its failure time are as follows:

Probability that the system transits from state S1 to S0 in time t given that thep-unit entered into F-mode after an operation of timex.

[From state S1 the system does not transit to state S3 up to time u and transit into S0 during

(u, udu)]

(On changing the limits of integration)

Q01 t( ) 1 1 r1( )e1 1 r1( ) ud

0

t

1 e 1 1 r1( )t

Q25 t( ) eu

ue2 1 r2( )ud

0

t

1 e 2 1 r2( ){ }t[ ] 2 1 r2( ){ } Q26 t( ) 21 r2 1 e

2 1 r2( ){ }t[ ] 2 1 r2( ){ }

Q47 t( ) 1 et

Q63 t( )

Q50 t( ) 1 1 e1 2 1 r2( ) { }t[ ] 1 2 1 r2( ) { }

Q52 t( ) 1 e1 2 1 r2( ) { }t

[ ] 1 2 1 r2( ) { }

Q53 t( ) 2 1 r2( ) 1 e1 2 1 r2( ) { }t

[ ] 1 2 1 r2( ) { }

Q74 t( ) 1 e 2( )t

[ ] 2( )

Q75 t( ) 2 1 e 2( )t

[ ] 2( )

Q10|x t( )

0

tP

e 2 1 r2( ){ }

1e1u 1r1x( ) |0 2 11r1xu( ) ud0

t

1e1r1x 11r1x

j

j!( )2-------------------- e

1 2 1 r2( ) { }u

uj

ud0

t

j0

Q12|x t( ) e 2 1 r2( ){ }

u

u 1e1y 1r1x( ) |0 2 11r1xy( ) ydu

t

d0t

e 2 1 r2( ){ }

u

u 1e1r1x 11r1x( )

j

j!( )2------------------------- e

1y yj

ydu

t

j0

d0

t

1e1r1x 11r1x( )

j

j!( )2------------------------- e

1y yj

e 2 1 r2( ){ }

u

ud0

y

yd0t

j0

1e1r1x

2 1 r2( )-------------------------------------

11r1x( )j

j!( )2------------------------- e

1y yj

1 e 2 1 r2( ){ }y

[ ] yd0

t

j0



31/50


32/50


33/50

Obviously,

(3637)

(vi) To obtain steady-state transition probabilities through nonregenerative state(s), we observe that

or

or

Similarly,

therefore,

The following relations are seen to be satisfied:

(3841)

(vii) The unconditional mean sojourn times in states Si (i 0, 2, 4, 5, 6, 7) are given by

Similarly,

(4246)

p10 p12 p13 1

p34 p35 1

Qijk( )

t( ) Qik t( ) Qkj t( )

Qijk( )

s( )s 0lim Qik s( )

s 0lim Qkj s( )

pijk( )

pik pkj

pijk ,l( )

pikpklplj

p001( )

p01p10 p10

p021( )

p01p12 p12

p041,3( )

p01p13p34 p13p34

p051,3( )

p01p13p35 p13p35

p246,3( )

p26p63p34 p26p34

p25

6,3( )p

26

p63

p35

p26

p35

p543( )

p53p34

p553( )

p53p35

p001( )

p021( )

p041,3( )

p051,3( )

1

p246,3( )

p25 p256,3( )

1

p50 p52 p543( )

p553( )

1

p74 p75 1

0 e1 1 r1( )t td 11 1 r1( )

2 1 2 1 r2( ){ }

4 6 1 5 1 1 2 1 r2( ) { }

7 1 2 ( )



34/50

The conditional mean sojourn times in states S1 and S3 are as follows:

(47)

Similarly,

(48)

From the conditional mean sojourn times in states S1 and S3, the unconditional mean sojourn times

are given by

(49)

(50)

Analysis of Reliability and Mean Time to System Failure

Assuring the failed states S3, S4, S6, and S7, as absorbing and enjoying the probailistic arguments, the

reliability of the system Ri(t) satisfies the following relations:

(13)

where

1 x e 2 1 r2( ){ }

t

1e1u 1r1x( ) |0 2 11r1xu( ) udt

td

1e1r1x 11r1x( )

j

j!( )2------------------------- e

1u uj

e 2 1 r2( ){ }

t

td0

u

udj0

1

2 1 r2( )------------------------------------- 1 2e

1r1x 1 1( )

3 x1--- 1 2e

2r2x 1 2( )

1 1 ! xq1 x( ) xd

2 1( r2{ }1

1 1e1r1x 1 1( ){ }1 1 r1( )e

1 1 r1( )x xd

2 1( r2{ }1

11 1 r1( )1 r11

----------------------------

3 3 xq2 x( ) xd

11 1e

1r1x 1 1( ){ }2 1 r2( )e2 1 r2( )x dx

1

12 1 r2( )1 r22

----------------------------

R0 t( ) Z0 t( ) q01 t( ) Z1 t( ) q001( )

t( ) R0 t( ) q021( )

t( ) R2 t( )

R2 t( ) Z2 t( ) q25 t( ) R5 t( )

R5 t( ) Z5 t( ) q50 t( ) R0 t( ) q52 t( ) R2 t( )

Z0 t( ) e1 1 r1( )t

Z1 t( ) e 2 1 r2( )t{

1e1y 1r1x( ) |0 2 11r1xy( ) yd

t

q1 x( ) xd

Z2 t( ) e 2 1 r2( ){ } t

Z5 t( ) e1 2 1 r2( ) { }



35/50

Taking Laplace Transform of relations (13) and solving the resulting set of algebraic equations for

we get

(4)

where

and

Taking inverse LT of (4), we can get the reliability of the system when it starts initially from state S0.

The mean time to system failure can be obtained using the formula

(5)

To obtain the RHS of (5), we note that

and

Therefore,

(6)

Availability Analysis

According to the definition of the elementary probabilistic arguments yield the following recursive

relations:

(15)

R0 s( )

R0 s( ) N1 s( )D1 s( )

N1 s( ) Z0 q01Z1( ) 1 q25q52( ) Z2 q25Z5( )q021( )

D1 s( ) 1 q001( )

( ) 1 q25q52( ) q021( )

q25q50

E T0( ) R0 t( ) td R0 s( )s 0lim

N1 0( )D1 0( )--------------

Zis 0lim s( ) Zi t( ) td i

qij s( )s 0lim pij, qijk( ) s( )

s 0lim pijk( )

p001( )

p10, p021( )

p12, p01 1

E T0( )

0 1( ) 1 p25p52( ) 2 p255( )p12

1 p10( ) 1 p25p52( ) p12p25p50----------------------------------------------------------------------------------------------------

Aip

t( ),

A0p

t( ) Z0 t( ) q001( )

t( ) A0p

t( ) q021( )

A2p

t( ) q041,3( )

t( ) A4p

t( ) q051,3( )

t( ) A5p

t( )

A2p

t( ) q246,3( )

t( ) A4p

t( ) q25 t( ) q256,3( )

t( ){ } A5p

t( )

A4p

t( ) q47 t( ) A7p

t( )A5

pt( ) q50 t( ) A0

pt( ) q52 t( ) A2

pt( ) q54

3( )t( ) A4

pt( ) q55

3( )t( ) A5

pt( )

A7p

t( ) q74 t( ) A4p

t( ) q75 t( ) A5p

t( )



36/50

Similarly, the relations for are:

(610)

Taking Laplace Transform of relations (15) and (610) and simplifying the resulting set of equations

for and , we get

(11)

and

(12)

where

and

In the long run, the probabilities that the system will be up (operative) due to p-unit and o-unit,

respectively, are given by

(13)

(14)

Using the results

Aio

t( )

A0o

t( ) q01 t( ) Z1 t( ) q001( )

t( ) A0o

t( ) q021( )

t( ) A2o

t( ) q041,3( )

t( ) A4o

t( )

q051,3( )

t( ) A5o

t( )

A2o

t( ) z2 t( ) q246,3

t( ) A4o

t( ) q25 t( ) q256,3( )

t( ){ } A5o

t( )

A4o

t( ) q47 t( ) A7o

t( )

A50

t( ) Z5 t( ) q50 t( ) A0o

t( ) q52 t( ) A2o

t( ) q543( )

t( ) A4o

t( ) q553( )

A5o

t( )

A70

t( ) q74 t( ) A4o

t( ) q75 t( ) A5o

t( )

A0p

s( ) A0o

s( )

A0p

s( ) N2 s( )D2 s( )

A0o

s( ) N3 s( )D2 s( )

N2 s( ) J1Z0, N3 s( ) q01J1Z1 J2Z2 J5Z5

D2 s( ) 1 q001( )

( )J1 q50J5

J1 1 q47q74( )(1 q553( ) q25q52 q256, 3( ) q52) q47q75 q543( ) q246, 3( ) q52( )

J2 1 q553( )

( ) 1 q47q74( ) q47q75q543( )

{ }q021( )

q051,3( )

q52 1 q47q74( ) q041, 3( )

q47q75q52

J5 q021( )

q25 q021( )

q256, 3( )

q051,3( )

( ) 1 q47q74( ) q041, 3( )

q021( )

q256, 3( )

( )q47q75

A0p

sS 0lim A0

ps( ) N2 0( )D2 0( )

A0o

sS 0lim A0

os( ) N2 0( )D2 0( )

Zi 0( ) Zi t( ) td iqij 0( ) pij, qij

k ,l( ) 0( ) pijk, l( )

pijk( )

pjkpkj and pijk ,l( )

pikpklplj



37/50


38/50

Thus in view of the results (1820), the expression (17) becomes

(21)


We have already defined , , and as probabilities that the repairman is busy withp-unit,

o-unit, and RM at time t, respectively. When the system initially starts from regenerative state Si using

the usual probabilistic reasoning for we have the following relations:

(15)

The relations in are as follows:

(610)

For we have

(1115)

Taking LT of relations (15), (610), and (1115) and simplifying the resulting sets of equations for

, and we get

(16)

(17)

D2 0( ) 0 1 p133 ( )1 2 p263 p266 ( )2

4 7( ) 3 5 p533( )4

Bip

t( ) Bio

t( ) Bim

t( )

Bip

t( ),

B0p

t( ) q01 t( ) Z1 t( ) q001( )

t( ) B0p

t( ) q021( )

t( ) B2p

t( ) q041,3( )

t( ) B4p

t( ) q051,3( )

t( ) B5p

t( )

B2p

t( ) q246 3,( )

t( ) B4p

t( ) q25 t( ) q 256,3( )

t( ){ } B5p

t( )

B4p

t( ) q47 t( ) B7p

t( )B5

pt( ) Z5 t( ) q50 t( ) B0

pt( ) q52 t( ) B2

pt( ) q54

3( )t( ) B4

pt( ) q55

3( )t( ) B5

pt( )

B7p

t( ) q74 t( ) B4p

t( ) q75 t( ) B5p

t( )

B10

t( )

B00

t( ) q01 t( ) q13 t( ) Z3 t( ) q001( )

t( ) B00

t( ) q02 t( ) B20

t( )

q 041,3( )

t( ) B40

t( ) q051,3( )

t( ) B50

t( )

B20

t( ) q26 t( ) q63 t( ) Z3 t( ) q 246, 3( )

t( ) B40

t( ) q25 t( ) q 256,3( )

t( ){ } B50

t( )B4

0t( ) q47 t( ) B7

0t( )

B50

t( ) q50 t( ) B00

t( ) q52 t( ) B20

t( ) q543( )

t( ) B40

t( ) q553( )

t( ) B50

t( )

B70

t( ) Z7 t( ) q 74 t( ) B40

t( ) q75 t( ) B50

t( )

Bim

t( )

B0m

t( ) q001( )

t( ) B0m

t( ) q021( )

t( ) B2m

t( ) q041,3( )

t( ) B4m

t( ) q051,3( )

t( ) B5m

t( )

B2m

t( ) Z2 t( ) q 26 t( ) Z6 t( ) q 246, 3( )

t( ) B4m

t( ) q25 t( ) q 256,3( )

t( ){ } B5m

t( )B4

mt( ) Z4 t( ) q 47 t( ) B7

mt( )

B5m

t( ) q50 t( ) B0m

t( ) q52 t( ) B2m

t( ) q543( )

t( ) B4m

t( )

B7m

t( ) q74 t( ) B4m

t( ) q75 t( ) B5m

t( )

B0p

s( ), B0o

s( ) B0m

s( ),

B0p

s( ) N4 s( )D2 s( )

B0o

s( ) N5 s( )D2 s( )



39/50

and

(18)

where

and

Now using the result and various relations among pij, the above three probabilities in

steady state are given by

and

(1921)

where

and has already been defined.

B0m

s( ) N6 s( )D2 s( )

N4 s( ) 1 q47q74( ) 1 q553( )

q25q52 q256,3( )

q52( ) q47q75 q543( )

q246,3( )

q52( )[ ]q01Z1

[ q021( )

q25 q021( )

q256,3( )

q051,3( )

( ) 1 q47q74( ) q041,3( )

q021( )

q246,3( )

( )q47q75 ]Z5

N5 s( ) 1 q47q74( ) 1( q553( )

q25q52 q256,3( )

q52 ) q47q75 (q543( )

q246,3( )

q52)[ ]q01q13

Z3

Z3q26

q63

q2

1( ) 1 q55

3( ) ( ) q05

1,3( ) q52[ ] Z3

q26 q63 q47 [q52 q04

1,3( ) q75 q05

1,3( ) q74)

q021( )

(q543( )

q75 q74 1 q553( )

( ) )]

+q47Z7[q02

1( ) (q24

6,3( ) (1 q55sup 3( )

) q54

3( ) q25

q54

3( ) q25

6,3( ) ]

q041,3( )

1 q553( )

q52 q25q52q25

6,3( ) ( ) q051,3( ) q543( ) q52 q246,3( ) ( )[ ]

N6 S( ) Z4[q2

1( ) (q24

6,3( )1( q55

3( ) )q54

3( ) q25 q54

3( ) q25

6,3( ) )

q041,3( )

(1( q553( )

q52 q25

q52 q256,3( )

) q051,3( )

q541,3( ) ( q52 q256,3 ( ) ) ]

Z2

q 26Z6

( )[ q21( ) 1( q55m3( ) q051,3( ) q52( )

q47 q52 q41,3( ) q75 q 051,3( ) q74( ) q021( ) q543( ) q75 q 74

1( q553( ) ( )( )]

qij

0( ) pij

B0p

N4 0( )D2 0( )

B0o

N5 0( )D2 0( )

B0m

N6 0( )D2 0( )

N4 s( ) p75p501 p12 p13( ) p755N5 0( ) p13 p26( )3 p12p75 1 p53( ) p13p75p52[ ]

p347 p12p26 1 p12( )p26p35p53 p13 p25 p53 p13p52( ) [ ]

N6 0( ) 2 p266( ) p12p75 1 p53( ) p13p75p52[ ]

p354 p12p26 1 p12( )p26p35p53 p13 p25 p53 p13p52( ) [ ]

D2

0( )



40/50

Expected Number of Repairs during (0, t)

We have already defined as the expected number of repairs of thep-unit during (0, t)|E0 si. For

this model the recurrence relations among , i 0, 2, 4, 5, 7 are as follows:

(15)

Similarly, the recursive relations in and can be obtained to obtain the expected number

of repairs ofo-unit and RM, respectively. They are as follows:

(711)

and

(1216)

Taking LST of the above equations (711) and (1216) and solving, then and can be

easily obtained.

In steady state the expected number of repairs ofp-unit, o-unit, and RM per unit are, respectively,

and

(1719)

where

Nip

t( )Ni

pt( )

N0p

t( ) Q001

( ) t( ) 1 N0p

t( )[ ] Q021

( ) t( ) N2p

t( ) Q041,3

( ) t( ) N4p

t( ) Q051,3

( ) t( ) N5p

t( ) N2

pt( ) Q24

6 3,( )t( ) N4

pt( ) Q25 t( ) Q 25

6,3( )t( )[ ] N5

pt( )

N4p

t( ) Q47 t( ) N7p

t( )

N5p

t( ) Q50 t( ) 1 N 0p

t( )[ ] Q52 t( ) N2p

t( ) Q543( )

t( ) N4p

t( ) Q553( )

t( ) N5p

t( )

N7p

t( ) Q74 t( ) N4p

t( ) Q75 t( ) N5p

t( )

N10

t( ) Nim

t( )

N00

t( ) Q001( )

t( ) N00

t( ) Q021( )

t( ) N20

t( ) Q041,3( )

t( ) N40

t( ) Q051, 3( )

t( ) 1 N50

t( )[ ]

N20

t( ) Q246, 3( )

t( ) N40

t( ) Q25 t( ) Q256,3( )

t( )[ ] 1 N 50

t( )[ ]N40

t( ) Q17 t( ) N60

t( )

N50

t( ) Q50 t( ) N00

t( ) Q52 t( ) N20

t( ) Q543( )

t( ) N40

t( ) Q553( )

t( ) N50

t( )

N70

t( ) Q64 t( ) N40

t( ) Q75 t( ) 1 N50

t( )

N0m

t( ) Q001( )

t( ) N0m

t( ) Q021( )

t( ) N0m

t( ) Q041, 3( )

t( ) N4m

t( )

Q051, 3( ) t( ) N5m t( )

N2m

t( ) Q246,3( )

t( ) N4m

t( ) Q25 t( ) Q256,3( )

t( )[ ] 1 N5m

t( )[ ]

N4m

t( ) Q47 t( ) 1 N7m

t( )[ ]

N5m

t( ) Q50 t( ) N0m

t( ) Q52 t( ) N2m

t( ) Q543( )

t( ) N4m

t( ) Q553( )

t( ) N5m

t( )

N7m

t( ) Q74 t( ) N4m

t( ) Q75 t( ) N5m

t( )

N 00

s( ) N 0m

s( )

N0p

N7 0( )D2(0)

N00

N8 0( )D2(0)

N0m

N9 0( )D2 0( )

N7 0( ) p75p50/D2 0( )

N8 0( ) p13 p36( )p35 p12p75 1 p53( ) p13p75p52[ ]

p34p75 p12p26 1 p12( )p26p35p53 p13 p25 p53 p13p52( ) [ ]



41/50

and

7.7 Profit Analysis

The two profit functions P1(t) and P2(t) can be found easily for each of the three models A, B, and C

with the help of the characteristics obtained in the earlier sections. The net expected total profit (gain)

incurred during (0, t) are

(1)

and

(2)

where K0 and K1 are the revenues per-unit up-time due top-unit and o-unit, respectively: K2, K3, and K4are the amounts paid to the repairman per-unit of time when he is busy in repairing the p-unit, o-unit,

and RM, respectively: K5, K6, and K7 are the per-unit repair costs of the p-unit, o-unit, and RM, respec-

tively. Also the mean up-times of the system due to the operation ofp-unit and o-unit during (0, t) are

given by

(3)

and

(4)

so that

(5)

and

(6)

Further, , and are the expected busy periods of the repairman with the p-unit,

o-unit, and RM, respectively, in (0, t) and are given by

(7)

(8)

N9 0( ) p25 p26p35( ) p12p75 1 p53( ) p13p75p52[ ]

p34 p13 p12p26 1 p12( )p26p35p53 p25 p53 p13p52( ) [ ]

P1 t( ) expected total revenue in (0,t) expected total expenditure during 0, t( )

K0upp

K1up0

t( ) K3b0

t( ) K4bm

()

P2 t( ) K0upp

t( ) K1upo

t( ) K5N0p

t( ) K6N0o

t( ) K7N0m

t( )

upp t( ) A0

p u( ) ud0t

upo

t( ) A00

u( ) ud0t

upp s( ) A0p s( )s

upo

s( ) A0o

s( )s

bp

t( ), bo

t( ) bm

t( )

bp

t( ) B0p

u( ) ud0

t

bo

t( ) B0p

u( ) ud0

t



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and

(9)

so that

(10)

(11)

and

(12)

Now the expected total profit per unit time in steady state is given by

On using (5, 6, 1012) we have

(13)

Similarly,

(14)

The values of and can be substituted in (13) and (14) from Sections

7.4, 7.5, and 7.6 for each of the three models A, B, and C, respectively.

7.8 Graphical Study of System Behaviour

For a more concrete study of system behaviour of models A and B, we plot curves inFigures 7.4to7.7

for the profit functions P1 and P2 obtained in earlier section w.r.t. 1 for different values of1 while the

other parameters are kept fixed as 2 0.05, 2 0.075, 0.025, 0.075, K0 250, K1 100,

K2 75, K3 25, K4 50, K5 100, K6 50, and K7 75.

The comparison of profit functions P1 and P2 for modelA is shown in Figure 7.4. From the figure it

is clear that both the profit functions decrease uniformly as the failure rate parameter ofp-unit (1)

increases. Also with the increase in the value of repair rate parameter ofp-unit (1), the profit functions

P1 and P2 increase. Further, it is observed that the function P2 provides the higher profit as compared tothe function P1 irrespective of the values of 1 and 1.

Figure 7.5 provides the comparison of profit functions P1 and P2 for model B. Here also the same

trends are observed for P1 and P2 as inFigure 7.4. One of the important features in this figure is that the

function P1 carries loss for 1 0.08 at 1 0.28.

bm

t( ) B0m

u( ) ud0

t

bp

s( ) B0p

s( )s

bo

s( ) B0o

s( )s

bm

s( ) B0m

s( )s

P1 P1 t( )tt lim s

2P1 s( )

s lim

K0 s2up

ps( ) K1

s 0lim s

2up

os( ) K2

s 0lim s

2b

ps( )

s 0lim K3

s 0lim s

2b

0s( ) K4

s 0lim b

ms( )

P1 K0 s 0lim s A0

ps( ) K1 s A0

os( ) K2 s B0

p

s 0lim

s 0lim s( )

K3 s B0o

s( )s 0lim K4

s 0lim s B0

ms( )

K0A0p

K1A0o

K2B0p

K3B0o

K4B0m

P2 K0A0p

K1A0o

K5N0p

K6N00

K7N0m

A0p

,A0o, B0

p, B0

o, B0

m,N0

p,N0

0, N0

m



43/50

FIGURE 7.4 Comparison of profit functions P1 and P2 for modelA.

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FIGURE 7.5 Comparison of profit functions P1 and P2 for model B.

2001byCRC

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FIGURE 7.6 Comparison of modelsA and B with respect to profit functions P1.

2001byCRC

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FIGURE 7.7 Comparative impression of function P2 for modelsA and B.

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48/50

FIGURE 7.9 Behaviours of profit functions P1 and P2 for model Cw.r.t. 1 for different values (0, 0.25, 0.50, 0.75) of correlati

2001byCRC

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50/50

[15] Gupta, R. and A. Chaudhary, Stochastic analysis of a priority unit standby system with repair

machine failure. Int. Jr. System Science, 26, 24352440 (1995).

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failure and repairs.Microelectron. Reliab. 37(5), 845849 (1997).

[17] Jaiswal, N.K. and J.V. Krishna, Analysis of two-dissimilar-unit standby redundant system with

administrative delay in repair. Int. Jr. Systems Science, 11 (4), 495511 (1980).

[18] Kapur, P.K. and K.R. Kapoor, Interval reliability of a two-unit standby redundant system.Micro-

electron. Reliab., 23(1), 167168 (1983).

[19] Kumar, A., D. Ray and M. Agarwal, Probabilistic analysis of a two-unit standby redundant system

with repair efficiency and imperfect switch over. Int. Jr. System Science, 9(7), 731742 (1978).

[20] Mahmoud, M.I. and M.A.W. Mahmoud, Stochastic behaviour of a two-unit standby redundant

system with imperfect switch over and preventive maintenance. Microelectron. Reliab., 23(1),

153156 (1983).

[21] Mahmoud, M.A.W. and M.A. Esmail, Probabilistic analysis of a two-unit warm standby system

subject to hardware and human error failures. Microelectron. Reliab., 36(10), 15651568 (1996).

[22] Murari, K. and V. Goyal, Comparison of two-unit cold standby reliability models with three typesof repair facilities.Microelectron. Reliab., 24(1), 3549 (1984).

[23] Naidu, R.S. and M.N. Gopalan, On the stochastic behaviour of a 1-server 2-unit system subject to

arbitrary failure, random inspection and two failure modes.Microelectron. Reliab., 24(3), 375378

(1984).

[24] Nakagawa, T. and S. Osaki, Stochastic behaviour of a two-unit priority standby redundant system

with repair.Microelectron. Reliab., 14(2), 309313 (1975).

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07 - analysis of stochastic models in manufacturing systems pertaining to repair machine failure

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