 Open Access
 Total Downloads : 208
 Authors : Mohit Kakkar , Ashok Chitkara, Jasdev Bhatti
 Paper ID : IJERTV2IS2458
 Volume & Issue : Volume 02, Issue 02 (February 2013)
 Published (First Online): 28022013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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.

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 .

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 [14] ,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

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

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
(0120)
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
(2124)
: 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)


ANALYSIS OF CHARACTERISTICS

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)
(2527)
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
(2829)
B8 W8 (t) q88.7 B8 (t) q80 B0 (t)
(3943)
0
0

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 upstate 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
In the steady state
(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)
(3034)
D1 is already specified.
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)
The steady state availability is
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 )
(3638)
(4750)
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)
D1 is already specified


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

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 figure2 and figure3 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

REFERENCES

Goyal,V.and Murari,K.cost anlysis of a two unit standby system with regular repairman and patience time , microelectron.reliab.vol.25 no.1,pp. 453 459,1984.

Tuteja,R.K., Taneja, G and Vashishta, Upasana, Cost benefit analysis of a system where operation and sometimes repair of mainunit depends on sub unit.Pure and Applied Mathematica Sciences, LIII (1 2), 4161,2001.

Singh, R. P., Taneja, G. and Siwach, B. S. Two unit warm standby system with accident and various types of repair. Journal of Decision and Mathematical Sciences, 56(13), pp7586,2000.

Vishal Sharma and Gaurav Varshney, Reliability analysis of an emergency shutdown system model in an industrial plant, RdE J. of Mathematical Sciences, Vol.1, pp.7182,2006.

Rakesh Gupta and Gaurav Varshney, A two non identical unit parallel system with Geometric failure and repair time distributions, IAPQR Trans. Vol. 31, No. 2, pp 127139,2007 .

Pradeep Chaudhary and Dharmendra Kumar, Stochastic analysis of a two unit cold standby system with different operative modes and different repair policies, Int. J. Agricult.Stat. Sciences, Vol. 3, No. 2, pp. 387394, (2007), .

Rakesh Gupta and Kailash Kumar, Cost benefit analysis of distillery plant system, Int. J.Agricult. Stat.
Sciences, Vol. 3, No. 2, pp. 541554 ,2007.

Amir Azaron ,Hideki Katagiri ,Kosuke Kato, Masatoshi Sakawa, Reliability Evaluation and Optimization Of DissimilarComponent ColdStandby Redundant Systems, Journal of the Operations Research Society of Japan , Vol. 48, No. 1, 718, (2005)

Attahiru Sule Alfa WL and Zhao, Y Q ,Stochastic analysis of a repairable system with three units and
repair facilities Microelectron. Reliab. , 38(4), (1998),585595, (1998).

Chao, A. and Hwang, W. D., Bayes estimation of reliability for special koutofm:G systems, IEEE Transactions on Reliability, R33, 246249,(1983).