# Probability Analysis Of A Cold Standby Unit System With Slow Switching And Correlated Appearance And Disappearance Of Repairman

DOI : 10.17577/IJERTV2IS2458

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#### Probability Analysis Of A Cold Standby Unit System With Slow Switching And Correlated Appearance And Disappearance Of Repairman

Mohit Kakkar1, Ashok Chitkara2, Jasdev Bhatti3

1Department of Applied Sciences,Chitkara University,Punjab,India 2Chancellor, Chitkara University, Himachal Pradesh, India 3Department of Applied Sciences, Chitkara University, Punjab,India

Abstract

The aim of this paper is to present a reliability analysis of a cold standby unit system with the assumption that the switching is not instantaneous

.There is only one repair facility. Appearance and Disappearance of repairman are assumed to be correlated. Using regenerative point technique various reliability characteristics are obtained which are useful to system designers and industrial managers. Graphical behaviors of MTSF and profit function have also been studied.

Keywords: Transition Probabilities, MTSF, Availability, Busy Period, Profit Function.

1. INTRODUCTION

Two identical unit cold standby systems have widely studied in literature of reliability theory, repair maintenance is one of the most important

measures for increasing the reliability of the

appearance and disappearance time of repairman taken as correlated random variables having their joint distribution as bivariate exponential .

2. SYSTEM DESCRIPTION

System consists of two identical units , initially both units are not operative but the only one of them is sufficient for operating the system, other one is in cold standby. There is single repair facility. When one unit fails another unit takes the charge but switching is not instantaneous. The joint distribution of appearance and disappearance of repairman is taken to be bivariate exponential having density function. Each repaired unit works as good as new.

Afr Aw

a

Afr Aw

a

S7

S6 Afr

Ao

a

Y

system. Many authors have studied various system models under different repair policies [1-4] ,they have assumed that appearance and disappearance times of repairman are uncorrelated random variable. But in real situation rest period of the repairman depends on workload on the repairman. They have also assumed that the switching is instantaneous but in real life this is not so. Taking these facts into consideration in this paper we investigate a two unit cold standby system model assuming the possibility of slow switching and

S2 So

Ao X Ao

As As

r Y a

Y

S4 S5

Aw Aw

A

A

Asb

o

r r

S1

Afr,

Asb

a

Afr,

Asb

a

As Asb

a

As Asb

a

S3

S8 Ao

Afr

a

Y

Fig.1 : Transition Diagram

Fig.1 : Transition Diagram

3. NOTATIONS

For defining the states of the system we assume the following symbols:

A0 : Unit A is in operative mode

AS: Unit A is in standby mode

1. #### Transition Probability and Sojourn Times

The steady state transition probability can be as follows

P01= P18.67=

Afr :

Unit A is in failure mode

(1 r )

( )( )

Asb : Unit A(stand by) is being switched

: Constant rate of repair of unit A

: Constant rate of failure of unit A

P02=

P16=

(1 r )

(1 r )

P10.6=

P10.6=

( )( )

( )

: Constant rate of switching

(1 r )

Aw: Unit A in failure mode but in waiting for repairman

P13=

P32.6= 2 2

1 2 (1 r2 )

X: Random variables representing the disappearance of repair man

P20=

(1 r )

(1 r )

P40=

2

2 1 (1 r1 )

Y: Random variables representing the appearance of repair man

P24=

(1 r )

P41.5=

1 (1 r1 )

2 1 (1 r1 )

fi(x,y) Joint pdf of (xi,yi);i=1,2

(1 r )ei xi y I (2 ( r xy ); X ,Y ,

P01+P02=1 P18.67+P10.6+P10.3=1

P16+P13=1 P20+P24=1

i i i

0 ri 1,

0 i i i

P41+P48.5=1 P80+P88.7=1

( r xy) j

(01-20)

whereI0 (2

i i ri xy )

j 0

i i i

( j!)2

#### Mean sojourn times:

ki (Y/X): Conditional pdf of Yi given Xi=x is given by

r x y

0

1

(1 r)

1

= ie

i i i

I0 (2 (i i ri xy )

2 (1 r)

i i

i i

gi(.): Marginal pdf of Xi= (1 r )ei (1ri ) x

hi(.): Marginal pdf of

1

1

1

qij (.),

i :

Yi= (1 r )ei (1ri ) y

i i

i i

pdf &cdf of transition time from regenerative states pdf &cdf of transition time from regenerative state Si to Sj.

Mean sojourn time in stateSi.

8 6

(21-24)

: Symbol of ordinary Convolution

A(t)

t

B(t) A(t u)B(u)du

0

: symbol of stieltjes convolution

t

0

0

A(t) B(t) A(t u)dB(u)

4. ANALYSIS OF CHARACTERISTICS

1. #### MTSF (Mean Time to System Failure)

To determine the MTSF of the system, we regard the failed state of the system as absorbing state, by probabilistic arguments, we get

0 Q01 Q02 2 (t)

2 Q24

Q20

0

(t)

0

0

Taking Laplace Stieltjes transforms of these relations and solving for ** (s) ,

** (s) N (s)

Let Bi(t) be the probability that the repairman is busy at instant t, given that the system entered regenerative state I at t=0.By probabilistic arguments we have the following recursive relations for Bi(t)

0 D(s)

B0 q01 B1(t) q02 B2 (t)

(25-27)

B W (t) q

B (t) q

B (t) q

B (t)

Where

1 1 10.6 0

18.67 8

10.3 0

N 1 2

P02

B2 q20 B0 (t) q24 B4 (t)

B4 q41 B1(t) q48.5 B8 (t)

D 1 P10 P01 P20 P02 P30 P03 P40 P04

(28-29)

B8 W8 (t) q88.7 B8 (t) q80 B0 (t)

(39-43)

0

0

2. #### Availability Analysis

Taking Laplace transform of the equations of busy

period analysis and solving them for

B* (s) ,we get

Let Ai (t) be the probability that the system is in up-state at instant t given that the system entered regenerative state i at t=0.using the arguments of

the theory of a regenerative process the point wise

B* (s) N2 (s)

0

0

D1 (s)

availability Ai (t) is seen to satisfy the following

(44)

recursive relations

B (sB* (s)) N2

A M (t) q A (t) q

A (t)

0 lim 0 D

0 0 01 1

02 2

s0 1

A1 q10.6 A0 (t) q18.67 A8 (t) q10.3 A0 (t)

A2 M2 (t) q20 A0 (t) q24 A4 (t)

Where N2 1(P01P80 P02P24P80 )

(45)

A q A (t) q A (t)

8 (P01P18.67 P02P24P48.5 P02P24P41P18.67 )

4 41 1

48.5 8

(46)

A8 q88.7 A8 (t) q80 A0 (t)

(30-34)

#### 4.4 Expected Number of Visits by the

Now taking Laplace transform of these equations

#### Repairman

and solving them for

A* (s), We get

We defined as the expected numer of visits by

* N1 (s)

0

0

the repairman in (0,t],given that the system initially

1

1

A0 (s) D (s)

(35)

starts from regenerative state Si

By probabilistic arguments we have the following recursive relations for Vi (t)

V (t) q

(1V (t)) q

V (t)

N 0 01

1 02 2

A

A

(sA* (s))

(sA* (s))

1

1

0 lim 0 D

V1 (t) q10.6 V0 (t) q18.67 V8 (t) q10.3 V0 (t)

s0 1

Where

V2 (t) q20 V0 (t) q24 V4 (t)

V4 (t) q41 (1 V1(t)) q48.5 (1 V8 (t))

N1 P80(0 2 P02)]

D1 P80[0 1 (P01 P02P41P24 ) 2 P02 4 P02P24 ]

8 (P01P18.67 P02P24P48.5 P02P24P41P18.67 )

(36-38)

(47-50)

0

0

Taking Laplace stieltjes transform of the equations of expected number of visits

#### 4.3 Busy Period Analysis of The Repairman

And solving them forV ** (s) , we get

V

V

0

0

** (s)

N3 (s)

D1 (s)

(51)

increases. Also for the fixed value of failure rate, the profit is higher for high correlation (r).From the fig.-4 it is clear that profit decreases linearly as disappearance rate of repairman increases. Also for the fixed value of disappearance rate, the profit is

In steady state higher for high correlation (r).

V (sV * (s)) N3

0 lim 0 D

Where

s0 1

(52)

N3 0 (1 P24)P80 2 (P02 P41P88.7 P02P48.5 P88.7 )

(4 )P02P24P80 8 (1 P24P02)

(53)

5. PROFIT ANALYSIS

The expected total profit incurred to the system in steady state is given by

P C0 A0 C1B0 C2V0

Fig. 2:MTSF vs Failure Rate

Fig. 2:MTSF vs Failure Rate

(54)

Where

C0 =Revenue/unit uptime of the system

C1 =Cost/unit time for which repairman is busy

C2 =Cost/visit for the repairman

6. CONCLUSION

For a more clear view of the system characteristics w.r.t. the various parameters involved, we plot curves for MTSF and profit function in figure-2 and figure-3 w.r.t the failure parameter ( ) of unit A for three different values of correlation coefficient (r1 =0.25, r2 =0.50, r3

=0.75), between X and Y , while the other

parameters are kept fixed as

.005, .02, 0.001,C0 400,

C1 200,C2 40, .004

Fig. 3:Profit vs Failure Rate

Fig. 3:Profit vs Failure Rate

From the fig.-2 it is observed that MTSF decreases as failure rate increases irrespective of other parameters. the curves also indicates that for the same value of failure rate,MTSF is higher for higher values of correlation coefficient(r),so here we conclude that the high value of r between appearance and disappearance tends to increase the expected life time of the system.From the fig.-3 it is clear that profit decreases linearly as failure rate

Fig. 4:Profit vs Disappear Rate

Fig. 4:Profit vs Disappear Rate

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