A Discrete Parametric Markov-Chain System Model of a Two ... · A Discrete Parametric Markov-Chain...

International Journal of Statistics and Systems

ISSN 0973-2675 Volume 12, Number 2 (2017), pp. 189-204

© Research India Publications

http://www.ripublication.com

A Discrete Parametric Markov-Chain System Model

of a Two Unit Standby System with Inspection and

Two Types of Repair

Rakesh Gupta¹* and Arti Tyagi²

¹ Department of Statistics, Ch. Charan Singh University, Meerut-250004 (India)

² Department of Statistics, Ch. Charan Singh University, Meerut-250004 (India)

*Corresponding authors

Abstract

The paper deals with cost benefit analysis of a two unit cold standby system

model with two possible modes of a unit- normal (N) and total failure (F). A failed

unit is inspected by an inspector to decide the kind of failure (major and minor)

and then the repair is performed accordingly by a repairman. The priority in repair

is given to the minor repair over the major repair. The failure time, inspection time

and both types of repair times are taken as independent random variables of

discrete nature having geometric distributions with different parameters.

Keywords: Regenerative point, reliability, MTSF, availability of system, busy

period of inspector and repairman, net expected profit.

1. INTRODUCTION

Two unit redundant system models have been widely analyzed including the authors

[2-5] by taking continuous distributions of failure and repair times as such type of

systems play the vital role in day to day life. But very few authors have analyzed

system models by taking discrete distributions of failure and repair times. Recently J.

Bhatti et al [1] analyzed a two identical unit cold standby system model with

inspection and two types of repair. They have considered geometric distributions of

failure time, inspection time and each type of repair time with different parameters. In

the above paper[1] we observed a lot of mistakes which reflects the mathematical

equations constructed for various characteristics as well as their mathematical

solutions. The few of major mistakes are highlighted below-

a) The authors have mentioned two types of repair namely minor and major but in

the analysis they have taken same repair rate for both types of repair. In view of

190 Rakesh Gupta and Arti Tyagi

this the consideration of minor and major repair is inconsequential. Moreover for

same repair rate the question of priority in the repair to minor categories has no

meaning.

b) In nomenclature section the authors have mentioned 1p as the probability that the

failed unit is inspected satisfactory whereas, the transition diagram given in the

paper indicates that 1p is the failure rate of an operating unit. Similarly,

according to the transition diagram 2p is inspection rate of a failed unit whereas

in nomenclature section the authors stated it as the probability of failure.

c) In state 1S one unit is under inspection and other is operative. But the authors

have not considered the possibility of failure of the operative unit in state 1S . In

view of this there will be five possible exists from state 1S whereas the authors

wrongly considered only two exists from 1S .

d) In state 1S if operative unit fails before the completion of inspection, then the

state i wiF ,F will be generated. This state is missing in the transition diagram

given by the authors.

e) The authors have considered states 7 Mw mrS F ,F and 8 mr MwS F ,F whereas for

identical units these two states must be the same.

In view of the above the transition diagram of the system model considered by J.

Bhatti et al is incorrect which results the wrong mathematics for evaluating the

various characteristics.

The purpose of the present paper is to analyze the system model consider by J. Bhatti

et al rectifying the above cumulative mistakes. The following economic related

measures of system effectiveness are obtained by using regenerative point technique-

i) Transition probabilities and mean sojourn times in various states.

ii) Reliability and mean time to system failure.

iii) Point-wise and steady-state availability of the system as well as expected up time

of the system during time (0, t-1).

iv) Expected busy period of inspector and repairman during time (0, t-1).

v) Net expected profit incurred by the system during a finite interval of time epochs

0, 1, 2, ……. (t-1) and steady-state.

2. MODEL DESCRIPTION AND ASSUMPTIONS

i. The system comprises of two identical units each having two modes- Normal (N)

and total failure (F). Initially one unit is operative and other is kept into cold

standby.

ii. A failed unit is inspected by an inspector to decide the category of failure (major

and minor) and then the repair is performed accordingly by a single repairman.

The minor repair gets the priority over the major repair.

iii. After inspection the unit goes for major repair or minor repair with fixed known

probabilities a and b (=1-a) respectively.

http://www.shabdkosh.com/translate/inconsequential/inconsequential-meaning-in-Hindi-English

A Discrete Parametric Markov-Chain System Model of a Two Unit Standby System… 191

iv. An inspector and a single repairman are always available with the system to do

their jobs.

v. After minor and major repairs a unit becomes as good as new.

vi. The time to failure, time to inspection and each category of repair time follow

geometric distributions with different parameters.

3. NOTATIONS AND STATES OF THE SYSTEM

a) Notations : x

1 1p q : P.m.f. of failure time of an operative unit; 1 1p q 1 . x

2 2p q : P.m.f. of inspection time of a failed unit; 2 2p q 1 x

i ir s : P.m.f. of repair time by repairman of minor and major categories

respectively for i=1, 2 and i ir s 1 .

ij ijq ,Q : P.m.f. and C.d.f. of one step or direct transition time from state iS tojS

ijp : Steady state transition probability from state iS to jS .

ij ijp Q

iZ t : Probability that the system sojourn in state iS up to epoch (t-1).

i : Mean sojourn time in state iS .

, h : Symbol and dummy variable used in geometric transform e. g.

t

ij ij ij

t 0

GT q t q h h q t

b) Symbols for the states of the system:

0 SN / N : Unit is in N-mode and operative/ into cold standby.

i wiF / F : Unit is in total failure (F) mode and under inspection/waits for

inspection.

mr wmrF / F : Unit is in total failure (F) mode and under minor repair/waits for

minor repair.

Mr wMrF / F : Unit is in total failure (F) mode and under major repair/waits for

major repair.

With the help of above symbols the possible states of the system are:

0 o SS N ,N , 1 i oS F ,N , 2 Mr oS F ,N , 3 mr oS F ,N

4 i wiS F ,F , 5 Mr iS F ,F , 6 mr iS F ,F , 7 Mr wMrS F ,F

8 mr wmrS F ,F , 9 mr wMrS F ,F


The transition diagram of the system model is shown in fig. 1.

Fig. 1

4. TRANSITION PROBABILITIES

Let ijQ t be the probability that the system transits from state iS to jS during time

interval (0, t) i.e., if ijT is the transition time from state iS to jS then

ij ijQ t P T t

By using simple probabilistic arguments we have

t 1

01 1Q t 1 q , t 11 2

12 1 2

1 2

aq pQ t 1 q q

1 q q

t 11 2

13 1 2

1 2

bq pQ t 1 q q

1 q q

, t 11 2

14 1 2

1 2

p qQ t 1 q q

1 q q

t 11 2

15 1 2

1 2

ap pQ t 1 q q

1 q q

, t 11 2

16 1 2

1 2

bp pQ t 1 q q

1 q q


t 11 2

20 1 2

1 2

q rQ t 1 q s

1 q s

, t 11 2

21 1 2

1 2

p rQ t 1 q s

1 q s

t 11 2

25 1 2

1 2

p sQ t 1 q s

1 q s

, t 11 1

30 1 1

1 1

q rQ t 1 q s

1 q s

t 11 1

31 1 1

1 1

p rQ t 1 q s

1 q s

, t 11 1

36 1 1

1 1

p sQ t 1 q s

1 q s

t 1

45 2Q t a 1 q , t 1

45 2Q t b 1 q

t 12 2

51 2 2

2 2

q rQ t 1 q s

1 q s

, t 12 2

52 2 2

2 2

ap rQ t 1 q s

1 q s

t 12 2

53 2 2

2 2

bp rQ t 1 q s

1 q s

, t 12 2

57 2 2

2 2

ap sQ t 1 q s

1 q s

t 12 2

59 2 2

2 2

bp sQ t 1 q s

1 q s

,

Q61

t( ) =r1q

2

1- s1q

2

1- s1q

2( )t+1é

ëêùûú

Q62

t( ) =ar

1p

2

1- s1q

2

1- s1q

2( )t+1é

ëêùûú,

Q63

t( ) =br

1p

2

1- s1q

2

1- s1q

2( )t+1é

ëêùûú

Q68

t( ) =bs

1p

2

1- s1q

2

1- s1q

2( )t+1é

ëêùûú,

Q69

t( ) =as

1p

2

1- s1q

2

1- s1q

2( )t+1é

ëêùûú

t 1

72 2Q t 1 s , t 1

83 92 1Q t Q t 1 s (1-26)

The steady state transition probabilities from state iS to jS can be obtained from

(1-26) by taking t , as follows:

01p 1 , 1 212

1 2

aq pp

1 q q

, 1 2

13

1 2

bq pp

1 q q

,

p14

=p

1q

2

1- q1q

2

1 215

1 2

ap pp

1 q q

, 1 2

16

1 2

bp pp

1 q q

, 1 2

20

1 2

q rp

1 q s

, 1 2

21

1 2

p rp

1 q s

1 225

1 2

p sp

1 q s

, 1 1

30

1 1

q rp

1 q s

, 1 1

31

1 1

p rp

1 q s

, 1 1

36

1 1

p sp

1 q s

45p a , 46p b , 2 2

51

2 2

q rp

1 q s

, 2 2

52

2 2

ap rp

1 q s

p53

=bp

2r2

1- q2s

2

, 2 257

2 2

ap sp

1 q s

, 2 2

59

2 2

bp sp

1 q s

, 1 2

61

1 2

r qp

1 s q


1 262

1 2

ar pp

1 s q

, 1 2

63

1 2

br pp

1 s q

,

p68

=bs

1p

2

1- s1q

2

,

p69

=as

1p

2

1- s1q

2

p

72= p

83= p

92= 1

We observe that the following relations hold-

01 72 83 92p p p p 1 , 12 13 14 15 16p p p p p 1 ,

20 21 25p p p 1

30 31 36p p p 1 , 45 46p p 1 ,

51 52 53 57 59p p p p p 1

61 62 63 68 69p p p p p 1 (27-33)

5. MEAN SOJOURN TIMES

Let iT be the sojourn time in state iS (i=0-9) theni mean sojourn time in state iS is

given by

i

t 1

P T t

In particular,

10

1

q

p , 1 2

1

1 2

q q

1 q q

, 1 2

2

1 2

q s

1 q s

, 1 1

3

1 1

q s

1 q s

24

2

q

p , 2 2

5

2 2

s q

1 s q

, 1 2

6

1 2

s q

1 s q

, 2

7

2

s

r

18 9

1

s

r (34-42)

6. METHODOLOGY FOR DEVELOPING EQUATIONS

In order to obtain various interesting measures of system effectiveness we developed

the recurrence relations for reliability, availability and busy period of repairman as

follows-

a) Reliability of the system-

Here we define iR t as the probability that the system does not fail up to epochs 0,

1, 2,.., (t-1) when it is initially started from up state iS . To determine it, we regard the

failed state 4S , 5S ,

6S , 7S ,

8S and 9S as absorbing state. Now, the expression for

iR t ; i=0, 1, 2, 3; we have the following set of convolution equations.

t 1

t

0 1 01 1

u 0

R t q q u R t 1 u


0 01 1Z t q t 1 R t 1

Similarly,

1 1 12 2 13 3R t Z t q t 1 R t 1 q t 1 R t 1

2 2 20 0 21 1R t Z t q t 1 R t 1 q t 1 R t 1

3 3 30 0 31 1R t Z t q t 1 R t 1 q t 1 R t 1 (43-46)

Where,

t t1 1 2Z t q q , t t

2 1 2Z t q r , t t3 1 1Z t q r

b) Availability of the system-

Let iA t be the probability that the system is up at epoch (t-1), when it initially

started from state iS . Then, by using simple probabilistic arguments, as in case of

reliability the following recurrence relations can be easily developed for iA t ; i=0 to

9.

0 0 01 1A t Z t q t 1 A t 1

1 1 12 2 13 3 14 4A t Z t q t 1 A t 1 q t 1 A t 1 q t 1 A t 1

15 5 16 6q t 1 A t 1 q t 1 A t 1



A

4t( ) = q

45t -1( )ãA

5t -1( ) + q

46t -1( )ãA

6t -1( )

5 51 1 52 2 53 3A t q t 1 A t 1 q t 1 A t 1 q t 1 A t 1

57 7 59 9q t 1 A t 1 q t 1 A t 1

6 61 1 62 2 63 3A t q t 1 A t 1 q t 1 A t 1 q t 1 A t 1

68 8 69 9q t 1 A t 1 q t 1 A t 1

7 72 2A t q t 1 A t 1

8 83 3A t q t 1 A t 1

9 92 2A t q t 1 A t 1 (47-56)

Where,

The values of iZ t ; i=0 to 3 are same as given in section 6(a).


c) Busy period of inspector

Let I

iB t be the probability that the inspector is busy in the inspection of a failed unit

at epoch t-1, when it initially started from state iS . Then, by using simple probabilistic

arguments, as in case of reliability the following recurrence relations can be easily

developed for B

i

I t( ); i=0 to 9.

I I

0 01 1B t q t 1 B t 1

I I I I

1 1 12 2 13 3 14 4B t Z t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

I I

15 5 16 6q t 1 B t 1 q t 1 B t 1

I I I I

2 20 0 21 1 25 5B t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

I I I I

3 30 0 31 1 36 6B t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

I I I

4 4 45 5 46 6B t Z t q t 1 B t 1 q t 1 B t 1

I I I I


I I

57 7 59 9q t 1 B t 1 q t 1 B t 1

I I I I


I I

68 8 69 9q t 1 B t 1 q t 1 B t 1

I I

7 72 2B t q t 1 B t 1

B

8

I t( ) = q83

t -1( )ãB3

I t -1( )

I I

9 92 2B t q t 1 B t 1 (57-66)

Where,

The values of 1Z t is same as given in section 6(a), t

4 2Z t q , t t

5 2 2Z t q s and

t t

6 1 2Z t s q .

d) Busy period of repairman-

Let B

i

mi t( ) and

B

i

Ma t( ) be the respective probabilities that the repairman busy in the

minor and major repair of a failed unit at epoch t-1, when it initially started from state

iS . Then, by using simple probabilistic arguments, as in case of reliability the

following recurrence relations can be easily developed for B

i

j t( ) ; i=0 to 9.

j j

0 01 1B t q t 1 B t 1

j j j j

1 12 2 13 3 14 4B t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

j j

15 5 16 6q t 1 B t 1 q t 1 B t 1


j j j j

2 2 20 0 21 1 25 5B t 1 Z t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

j j j j


j j j

4 45 5 46 6B t q t 1 B t 1 q t 1 B t 1

j j j j

5 5 51 1 52 2 53 3B t 1 Z t q t 1 B t 1 q t 1 B t 1 q t 1 B t 1

j j

57 7 59 9q t 1 B t 1 q t 1 B t 1

j j j j


j j

68 8 69 9q t 1 B t 1 q t 1 B t 1

j j

7 7 72 2B t 1 Z t q t 1 B t 1

B

8

j t( ) = dZ t( )+ q83

t -1( )ãB3

j t -1( )

j j

9 92 2B t Z t q t 1 B t 1 (67-76)

Where,

1 and 0 respectively for j=mi and Ma. The values of iZ t ; i=2, 3, 5, 6 are same as

given in section 6(a) and (b), and t

7 2Z t s , t

1Z t s .

7. ANALYSIS OF RELIABILITY AND MTSF

Taking geometric transforms of (43-46) and simplifying the resulting set of algebraic

equations for 0R h we get

1

01

N hR h

D h

(77)

Where,

2 * * 2 * * * * * 2 * * * 2 * * *

1 12 21 13 31 0 01 1 01 12 2 01 13 3N h 1 h q q h q q Z hq Z h q q Z h q q Z

2 * * 2 * * 3 * * * 3 * * *

1 12 21 13 31 01 12 20 01 13 30D h 1 h q q h q q h q q q h q q q

Collecting the coefficient of th from expression (77), we can get the reliability of the

system 0R t . The MTSF is given by-

1t

h 1 1t 1

N 1E T lim h R t 1

D 1

(78)

1 12 21 13 31 0 1 12 2 13 3N 1 1 p p p p p p

1 12 21 20 13 31 30D 1 1 p p p p p p


8. AVAILABILITY ANALSIS

On taking geometric transforms of (47-56) and simplifying the resulting equations for

we get

2

0

2

N hA h

D h

(79)

Where,

*

01

* * * * *

12 13 14 15 16

* * *

20 21 25

* * *

30 31 36

* *

45 46

2 * * * * *

51 52 53 57 59

* * * * *

61 62 63 68 69

7

1 hq 0 0 0 0 0 0 0 0

0 1 hq hq hq hq hq 0 0 0

hq hq 1 0 0 hq 0 0 0 0

hq hq 0 1 0 0 hq 0 0 0

0 0 0 0 1 hq hq 0 0 0N h

0 hq hq hq 0 1 0 hq 0 hq

0 hq hq hq 0 0 1 0 hq hq

0 0 hq

*

2

*

83

*

92

0 0 0 0 1 0 0

0 0 0 hq 0 0 0 0 1 0

0 0 hq 0 0 0 0 0 0 1

and

D2

h( ) =

Z0

* -hq01

* 0 0 0 0 0 0 0 0

Z1

* 1 -hq12

* -hq13

* -hq14

* -hq15

* -hq16

* 0 0 0

Z2

* -hq21

* 1 0 0 -hq25

* 0 0 0 0

Z3

* -hq31

* 0 1 0 0 -hq36

* 0 0 0

0 0 0 0 1 -hq45

* -hq46

* 0 0 0

0 -hq51

* -hq52

* -hq53

* 0 1 0 -hq57

* 0 -hq59

*

0 -hq61

* -hq62

* -hq63

* 0 0 1 0 -hq68

* -hq69

*

0 0 -hq72

* 0 0 0 0 1 0 0

0 0 0 -hq83

* 0 0 0 0 1 0

0 0 -hq92

* 0 0 0 0 0 0 1


The steady state availabilities of the system due to operation of unit -

2

0 0t h 1

2

N hA limA t lim 1 h

D h

But 2D h at h=1 is zero, therefore by applying L. Hospital rule, we get

2

0

2

N 1A

D 1

(80)

Where,

2 0 0 1 1 2 2 3 3N 1 u u u u

and

2 0 0 1 0 14 4 2 2 3 3 5 5 59 57 7D 1 u u p u u u p p

6 6 68 69u p p

where,

i iu U 0 and U

i

* h( ); i=0, 1, 2, 3, 5, 6 are the minors of 1st, 2nd, 3rd, 4th, 6th and 7th

elements lying in first column of the matrix corresponding to determinant D

2h( ).

Now the expected up time of the system due to operative unit upto epoch (t-1) is

given by

t 1

up 0

x 0

t A x

so that

0

up

A hh

1 h

(81)

7. BUSY PERIOD ANALSIS

a) Busy period of inspector-


B

0

I* h( )we get

B0

I* h( ) =N

3h( )

D2

h( ) (82)

Where,

* * * * * *3 01 1 1 14 4 5 5 6 6N h hq U Z hq Z U Z U Z

and 2D h is same as in availability analysis.

In the long run the respective probability that the inspector is busy in the inspection of

failed unit given by-


B0

I = limt®¥

Bo

I t( ) = limh®1

1- h( )N

4h( )

D2

h( )


B0

I = -N

41( )

¢D2

1( ) (83)

Where,

3 1 1 14 4 5 5 6 6N 1 u p u u

and 2D 1 is same as in availability analysis.

Now the expected busy period of inspector is in the inspection of a failed unit up to

epoch (t-1) is given by -

t 1

I I

b 0

x 0

t B x

So that,

I

0I

b

B hh

1 h

(84)

b) Busy period of repairman-


B

0

mi* h( ) and B

0

Ma* h( ) we get

B0

mi* h( ) =N

4h( )

D2

h( ) and

5Ma

0

2

N hB h

D h

(85-86)

Where,

* * * * * * * * * * *4 01 3 3 3 3 68 69 5 59N h hq U Z U Z hq hq Z U hq Z

* * * * * * *5 01 2 2 5 5 57 7N h hq U Z U Z hq Z

and 2D h is same as in availability analysis.

In the long run the respective probabilities that the repairman is busy in the minor

repair and major repair of a failed unit are given by-

4mi mi

0 ot h 1

2

N hB limB t lim 1 h

D h

5Ma Ma

0 ot h 1

2

N hB limB t lim 1 h

D h



4mi

0

2

N 1B

D 1

and

5Ma

0

2

N 1B

D 1

(87-88)

Where,

4 3 3 5 59 6 6 68 69N 1 u u p u p p

5 2 2 5 5 57 7N 1 u u p

and 2D 1 is same as in availability analysis.

Now the expected busy period of the repairman in minor repair and major repair of a

failed unit up to epoch (t-1) are respectively given by-

t 1

mi mi

b 0

x 0

t B x

, t 1

Ma Ma

b 0

x 0

t B x

So that,

mi

0mi

b

B hh

1 h

,

Ma

0Ma

b

B hh

1 h

(89-90)

8. PROFIT FUNCTION ANALYSIS

We are now in the position to obtain the net expected profit incurred up to epoch (t-1)

by considering the characteristics obtained in earlier section.

Let us consider,

0K revenue per-unit time by the system due to operative unit.

1K cost per-unit time when inspector is busy in the inspection of failed unit.

2K cost per-unit time when repairman is busy in the minor repair of a failed unit.

3K cost per-unit time when repairman is busy in the major repair of a failed unit.

Then, the net expected profit incurred up to epoch (t-1) given by

P t( ) = K

0m

upt( )- K

1m

b

I t( ) - K2m

b

mi t( )- K3m

b

Ma t( ) (91)

The expected profit per unit time in steady state is given by-

2

t h 1

P tP lim lim 1 h P h

t

I mi2 2 20 0 0

0 1 2h 1 h 1 h 1

A h B h B hK lim 1 h K lim 1 h K lim 1 h

1 h 1 h 1 h

-K3limh®1

1- h( )2 B

0

Ma* h( )1- h( )


I mi Ma

0 0 1 0 2 0 3 0K A K B K B K B (92)

9. GRAPHICAL REPRESENTATION

The curves for MTSF and profit function have been drown for different values of

failure parameters. Fig. 2 depicts the variation in MTSF with respect to inspection rate

2p for different values of minor repair rate 1r of a unit and failure rate 1p when

values of other parameters are kept fixed as 2r 0.02 and a 0.25 . From the curves

we conclude that expected life of the system increases as the values of 1r and 2p

increases and decrease with increase in 1p .

Behavior of MTSF with respect to 1p , 2p and 1r

Fig. 2

Similarly, Fig. 3 reveals the variations in profit (P) with respect to 2p for varying

values of 1r and 1p , when other parameters are kept fixed as 2r 0.02 , a 0.25 ,

0K 150 , 1K 150 , 2K 180 and 3K 220 . From the figure it is clearly observed

from the smooth curves, that the system is profitable if the value of parameter 2p is

greater than 0.0550, 0.0590 and 0.0690 respectively for 1r 0.10 , 0.15 and 0.20 for

fixed value of 1p 0.025 . From dotted curves, we conclude that system is profitable

60

80

100

120

140

160

180

0.05 0.07 0.09 0.11 0.13 0.15 0.17

MTSF

p₂

r₁=0.10,p₁=0.025 r₁=0.10,p₁=0.030

r₁=0.15,p₁=0.025 r₁=0.15,p₁=0.030

r₁=0.20,p₁=0.025 r₁=0.20,p₁=0.030


only if value of parameter 2p is greater than 0.0825, 0.0925 and 0.1200 respectively

for 1r 0.10 , 0.15 and 0.20 for fixed value of 1p 0.030 .

Behavior of Profit (P) with respect to 1p , 2p and 1r

Fig. 3

REFERENCES

[1] Bhatti, J., Chitkara, A and Bhardwaj, N. (2011) Profit analysis of two unit cold

standby system with two types of failure under inspection policy and discrete

distribution. International Journal of Scientific & Engineering Research,

12(2):1-7

[2] El-Said Khaled M. and M. S. EL-Sherbeny (2005) Profit Analysis of a Two-

Unit Cold Standby System with Preventive Maintenance and Random

Changes in Units. J. Mathematics and Statistics 1(1):71-77.

[3] Goel, L. R., Gupta, R. and Singh, S. K. (1985) Availability analysis of a two-

unit (dissimilar) parallel system with inspection and bivariate exponential life

times. Microelectron. Reliab. 25(1):77-80.

-40

-30

-20

-10

0

10

20

30

40

50

60

0.05 0.07 0.09 0.11 0.13 0.15 0.17

PROFIT

p₂

r₁=0.10,p₁=0.025 r₁=0.10,p₁=0.030

r₁=0.15,p₁=0.025 r₁=0.15,p₁=0.030

r₁=0.20,p₁=0.025 r₁=0.20,p₁=0.030


[4] Kouichi Adachi and Masanori Kodama (1980) Availability analysis of two-

unit warm standby system with inspection time. Micro electron. Reliab.

20(4):449-455.

[5] Mokaddis, G. S., M. S. El-Sherbeny, and Entesar Al-Esayeh (2008) A

dissimilar two-unit parallel system with common-cause failure and preventive

maintenance. The Journal of Mathematics & Computer Science, 19(2): 153–

167.

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