probability analysis of a complex system working in a sugar mill with repair equipment failure and c

DOI: 10.15415/mjis.2012.11005

Mathematical Journal of Interdisciplinary Sciences

Vol. 1, No. 1, July 2012 pp. 57–66

©2012 by Chitkara University. All Rights

Reserved.

Probability Analysis of a Complex System Working in a Sugar Mill with Repair Equipment

Failure and Correlated Life Time

Mohit Kakkar

Department of Applied Sciences, Chitkara University, Punjab, India

Ashok K Chitkara

Chancellor, Chitkara University, Himachal Pradesh, India

Jasdev Bhatti

Department of Applied Sciences, Chitkara University, Punjab, India

Abstract

The aim of this paper is to present a reliability analysis of a complex (SJP) system in a sugar mill with the assumption that repair equipment may also fail during the repair. This paper considers the analysis of a three-unit system with one big unit and two small identical units of a SJP System in a sugar mill. Failure and Repair times of each unit 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

Complex systems have widely studied in literature of reliability theory as a large number of researchers are making lot of contribution in the field by incorporating some new ideas/concepts. Two/Three-unit

standby systems with working or failed stages have been discussed under various assumptions/situations by numerous researchers including [1-4], repair maintenance is one of the most important measures for increasing the reliability of the system, and they have assumed that repair equipment can never be failed, But in real situation repair equipment may also fail when repairman is busy in the repair of any one of the failed unit. They have also assumed that the failure and repair of units are independent of each other but in real life this is not so. To have an extensive study on such a practical situation including the above-mentioned important aspects, the sugar mills situated at Muzaffar Nagar in Uttar Pradesh were visited and the information was collected on the Sulphated Juice Pump(SJP) System Working therein, this system contains one big unit(B) and two small identical units(A)(one of them is in standby). Then, on the basis of the situation observed in this mill, we put a step towards this direction by analyzing the reliability and the profit of a 3-unit cold standby

Kakkar, M. Chitkara, A. K Bhatti, J.

58

Mathematical Journal of Interdisciplinary Sciences, Volume 1, Number 1, July 2012

SJP System, evaluating various measures of system effectiveness Taking these facts into consideration in this paper we investigate a complex system model assuming the possibility of repair equipment failure and correlated life time.

2. SYSTEM DESCRIPTION

Complex System consists of two small identical units of A and one big unit B both are connected in series, but A has two Sub unit connected in Parallel one of them is in standby mode, so both units A and B are initially operating but the operation of only one sub unit of A is sufficient for operating the system. There is single repair facility. When the repairman is called on to do the job, it takes a negligible time to reach at the system. When repair equipment fails during the repair of any failed unit, repairman starts the repair of repair-equipment first. The joint distribution of failure and repair times for each unit is taken to be bivariate exponential having density function. Each repaired unit works as good as new.

S3

S2 S1

S7S8

S5S0

X1

X1

X1

X2 X

Y2 Y2Y1

Y1

X2

X2

S4 S6

AO

AO,AO,AO,

AO

AO,

AO, AW,

AW,

AW,

AW

AW AW

BfrBfr

Afr

Afr,

MfrMfrMfr

Mfr

BW,BW,BO,

BOBO

BO,, BO

AS

AS AS

ω ω ω

ω

θ θ θ

θ

Figure1: Transition Diagram

3. NOTATIONS

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

A0 Unit A is in operative mode

B0 Unit B is in operative mode

Probability Analysis of a Complex

System Working in a Sugar Mill

59


Afr Unit A is in failure mode

Bfr Unit B is in failure mode

w Constant rate of failure of repair- equipment

θ Constant rate of repair-equipment’s repair

Aw unit A in failure mode but in waiting for repairman

Bw unit B in failure mode but in waiting for repairman

Mfr unit A in failure mode but in waiting for repairman

Bw unit B in failure mode but in waiting for repairman

Xi(i=1,2) random variables representing the failure times of A and B unit respectively for i=1,2

Yi(i=1,2) random variables representing the repair times of A and B unit respectively for i=1,2

fi(x,y) joint pdf of (xi,y

i);i=1,2

= − >

≤ <

− −α β α β α βα βi i i

x yi i i i i

i

r e I r xy X Y

r

i i( ) ( ( ); , , , ;

,

1 2 0

0 10

wheree I r xyr xy

ji i ii i i

j

j0 2

0

2( )( )

( !)α β

α β=

=

∞

∑

ki(Y/X) conditional pdf of Yi given X

i=x is given by

= β α βα βi

r x yi i ie I r xyi i i− −

0 2( ( )

gi(.) marginal pdf of Xi= α α

i ir xr e i i( ) ( )1 1− − −

hi(.): marginal pdf of Y

i= β β

i ir yr e i i( ) ( )1 1− − −

q Qij ij(.), (.) : pdf &cdf of transition time from regenerative states pdf &cdf of transition time from regenerative state S

i to S

j.

µi : Mean sojourn time in state Si.

⊕ Symbol of ordinary Convolution A t( )

⊕

B t A t u B u du

t

( ) ( ) ( )= −∫0

symbol of stieltjes convolution

A t( )

B t A t u dB u

t

( ) ( ) ( )= −∫0


60


3.1 Transition Probability and Sojourn Times

The steady state transition probability can be as follows

P01

=αφ

1 11( )−r P32

=1

P02

=αφ

2 21( )−rP

41=θφ

P10

=β

β ω φ( )

( )

1

11−

− + +r

rP

46=P

45.6= αθ φ1 11( )−+

r

P14

=ω

β ω φ( )1− + +rP

48= P

47.8=αθ φ

2 21( )−+

r

P15

=α

β ω φ2 21

1

( )

( )

−− + +

r

rP

51=βω β

2 2

2 2

1

1

( )

( )

−+ −

r

r

P17

=α

β ω φ1 11

1

( )

( )

−− + +

r

rP

56=

ωω β+ −2 21( )r

P20

=ββ ω

2 2

2 2

1

1

( )

( )

−− +

r

rP

71=βω β

1 1

1 1

1

1

( )

( )

−+ −

r

r

P23

=ω

β ω2 21( )− +rP

78=

ωω β+ −1 11( )r

P01

+P02

=1 P41

+ P47.8

+P45.6

=1

P10

+ P14

+P15

+P17

=1 P20

+P23

=1

P41

+P46

+P48

=1

P32

=1, P87

=1 , P65

=1

P71

+P78

=1

P56

+P51

=1 (1-26)

Mean sojourn times:

µφ0

1= µ

β ω φ1

1

1=

− + +( )r

µθ φ4

1=+ (27-29)



61


4. ANALYSIS OF CHARACTERISTICS

4.1 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

φ φ0 01 1 02( ) ( )t Q t Q= ⊗ +

φ φ φ1 10 0 14 4 15 17( ) ( ) ( )t Q t Q t Q Q= ⊗ + ⊗ + +

φ φ4 41 1 48 46( ) ( )t Q t Q Q= ⊗ + + (30-32)

Taking Laplace Stieltjes transforms of these relations and solving for φ0** ( )s ,

φ0** ( )

( )

( )s

N s

D s= (33)

Where

N P P P P P P P P= − + + + +µ µ µ0 14 41 1 01 02 41 2 01 14 021( ) ( ) ( )

D P P P P P P P= − − +( )1 14 41 01 10 02 41 10 (34-35)

4.2 Availability Analysis

Let A ti ( ) 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 availability A ti ( ) is seen to satisfy the following recursive relations

A t M t q A t q A t0 0 01 1 02 2( ) ( ) ( ) ( )= + ⊕ + ⊕

A t M q A t q A t q A t q A t q A1 1 10 0 14 4 15 5 17 7 1 18 7( ) ( ) ( ) ( ) ( ). .= + ⊕ + ⊕ + ⊕ + ⊕ + ⊕ 88 ( )t

A t q A t q A t2 20 0 23 3( ) ( ) ( )= ⊕ + ⊕

A t q A t3 32 2( ) ( )= ⊕

A t M q A t q A t q A t4 4 41 1 45 6 5 47 8 7( ) ( ) ( ) ( ). .= + ⊕ + ⊕ + ⊕

A t q A t q A t5 51 1 56 6( ) ( ) ( )= ⊕ + ⊕

A t q A t6 65 5( ) ( )= ⊕

A t q A t q A t7 71 1 78 8( ) ( ) ( )= ⊕ + ⊕

A t q A t8 87 7( ) ( )= ⊕ (36-44)


62


Now taking Laplace transform of these equations and solving them for A s0* ( ),

we get

A sN s

D s01

1

* ( )( )

( )= (45)

The steady state availability is

A sA sN

Ds0 0 01

1

= =→

lim( ( ))* (46)

Where

N P P P P P P P P P P

P1 0 23 71 18 7 56 65 78 51 45 6 14 15

78

1 1

1

= − − + +

−

µ ( )[ ( ) ( )

(. .

−− − + − −

+ −

P P P P P P P P P

P P P11 7 14 41 78 65 56 11 7 14 41

23 78 01

1

1. .) ( )]

( ) [(µµ µ µ µ1 4 14 56 65 1 4 14+ + +P P P P) ( )]

D P P P P P P P P P P P P P P1 0 78 10 20 51 1 78 01 20 51 2 78 51 14 47 8 021= − + − −µ µ µ ( ( . 110

18 7 51 78 11 7 3 51 11 7 14 41 01 10

71 14

1

)

) (. . .− − + − − −

−

P P P P P P P P P P

P P

µ

PP P P P P P P P P P P P47 8 4 01 51 14 20 78 51 6 5 20 01 78 15 45 6. .) ( ) ( )[( (µ µ µ+ + +PP

P P P P P P P P P P51

7 01 20 11 7 51 20 14 18 7 51 14 47 81 1

)]

[ ( ) ( ) (. . .+ − − + − −µ ))]

[ ( ) ]. .+ − +µ8 02 20 51 78 10 14 47 8 18 7P P P P P P P P

(47-48)4.3 Busy Period Analysis Of The Repairman

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 B

i(t)

B t q B t q B t0 01 1 02 2( ) ( ) ( )= ⊕ + ⊕

B t W q B t q B t q B t q B t q B1 1 10 0 14 4 15 5 17 7 1 18 7( ) ( ) ( ) ( ) ( ). .= + ⊕ + ⊕ + ⊕ + ⊕ + ⊕ 88 ( )t

B t W q B t q B t2 2 20 0 23 3( ) ( ) ( )= + ⊕ + ⊕

B t W q B t3 3 32 2( ) ( )= + ⊕

B t W q B t q B t q B t4 4 41 1 45 6 5 47 8 7( ) ( ) ( ) ( ). .= + ⊕ + ⊕ + ⊕

B t W q B t q B t5 5 51 1 56 6( ) ( ) ( )= + ⊕ + ⊕

B t W q B t6 6 65 5( ) ( )= + ⊕

B t q B t q B t7 71 1 78 8( ) ( ) ( )= ⊕ + ⊕

B t W q B t8 8 87 7( ) ( )= + ⊕ (49-57)



63


Taking Laplace transform of the equations of busy period analysis and solving them for B s0

* ( ) ,we get

B sN s

D s02

1

* ( )( )

( )= (58)

In the steady state

B sB sN

Ds0

00

2

1

= =→

lim( ( ))* (59)

Where

N P P P P P P P P P P

P2 0 23 71 18 7 56 65 78 51 45 6 14 15

78

1 1

1

= − − + +

−

µ ( )[ ( ) ( )

(. .

−− − + − −P P P P P P P P P11 7 14 41 78 65 56 11 7 14 411. .) ( )] (60)

D1 is already specified.

4.4 Expected Number of Visits by the Repairman

We defined as the expected number of visits by the repairman in (0,t],given that the system initially starts from regenerative state S

i

By probabilistic arguments we have the following recursive relations for V ti ( )

V t q V t q V t0 01 1 02 21 1( ) ( ( )) ( ( ))= ⊕ + + ⊕ +

V t q V t q V t q V t q V t q V t1 10 0 14 4 15 5 11 7 1 18 7 8( ) ( ) ( ) ( ) ( ) (. .= ⊕ + ⊕ + ⊕ + ⊕ + ⊕ ))

V t q V t q V t2 20 0 23 3( ) ( ) ( )= ⊕ + ⊕

V t q V t3 32 2( ) ( )= ⊕

V t q V t q V t q V t4 41 1 45 6 5 47 8 7( ) ( ) ( ) ( ). .= ⊕ + ⊕ + ⊕

V t q V t q V t5 51 1 56 6( ) ( ) ( )= ⊕ + ⊕

V t q V t6 65 5( ) ( )= ⊕

V t q V t q V t7 71 1 78 8( ) ( ) ( )= ⊕ + ⊕

V t q V t8 87 7( ) ( )= ⊕ (61-69)

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


64


And solving them for, V s0** ( ) we get

V sN s

D s03

1

** ( )( )

( )= (70)

In steady state

V sV sN

Ds0

00

3

1

= =→

lim( ( ))* (71)

where N P P P P P P P P

P P P3 0 23 47 8 2 02 51 18 7 02 41 48

4 02 48 78

1= − + + +

+

µ µ

µ

( ) ( )

( ). .

++ +µ8 47 8 021( ).P P

(72)

D1 is already specified

5. PROFIT ANALYSIS

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

P C A C B C V= − −0 0 1 0 2 0 (73)

Where

C0 =Revenue/unit up time 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 (α) of unit A for three different values of correlation coefficient, between X

and Y , while the other parameters are kept fixed as

α β β θ2 1 2 0

1 2

005 02 0 01 0 001 400

200 40

= = = = =

= =

. , . , . , . , ,

, ,

C

C C ωω= .004

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



65


0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

1000

800

600

400

-400

200

-200

PROFIT VS FAILURE RATE

FAILURE RATE OF UNIT A

r13=0.75

r12=0.50

r11=0.50

0

0.0101 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

1000

800

600

400

-400

200

-200

PROFIT VS FAILURE RATE


r13=0.75

r12=0.50

r11=0.50

0

Figure 3: Profit vs Failure Rate

8000

6000

4000

2000

0

MTSF VS FAILURE RATE


0.01 0.02

0.10.090.080.070.060.050.040.03

r11=0.25

r12=0.50

r13=0.75

MTSF

8000

6000

4000

2000

0

MTSF VS FAILURE RATE


0.01 0.02

0.10.090.080.070.060.050.040.03

r11=0.25

r12=0.50

r13=0.75

MTSF

Figure 2: MTSF vs Failure Rate

here we conclude that the high value of r tends to increase the expected life time of the system. From the fig.-3 it is clear that profit decreases linearly as


66


failure rate increases. Also for the fixed value of failure rate, the profit is higher for high correlation (r).

REFERENCES

[1] Amir Azaron ,Hideki Katagiri ,Kosuke Kato, Masatoshi Sakawa, (2005) “Reliability Evaluation and Optimization Of Dissimilar-Component Cold-Standby Redundant Systems”, Journal of the Operations Research Society of Japan , Vol. 48, No. 1, 71-8.

[2] Attahiru Sule Alfa WL and Zhao, Y Q (1998),“Stochastic analysis of a repairable system with three units and repair facilities” Microelectron. Reliab., 38(4), (1998), 585-595.

http://dx.doi.org/10.1016/S0026-2714(97)00204-7 [3] B. Parasher, and G.Taneja,. (2007) ,Reliability and profit evaluation of a standby system based on a

master-slave concept and two types of repair facilities‘. IEEE Trans. Reliabil., 56: pp.534-539. http://dx.doi.org/10.1109/TR.2007.903151[4] Chao, A. and Hwang, W. D.,(1983) “Bayes estimation of reliability for special k-out-of-m:G

systems”, IEEE Transactions on Reliability, R-33, 246-249. [5] G.Taneja, V.K.Tyagi, and P. Bhardwaj,. (2004) ‚Profit analysis of a single unit programmable logic

controller (PLC)‘, “Pure and Applied Mathemati-ka Sciences”, vol. LX no (1-2), pp 55-71, 2004.[6] G.Taneja, (2005)‚Reliability and profit analysis of a system with PLC used as hot standby’.

Proc.INCRESE Reliability Engineering Centre, IIT, Kharagpur India, pp: 455-464.(Conference proceedings).

[7] Goyal, A. and D.V. Gulshan Taneja, (2009). Singh analysis of a 2-unit cold standby system working in a sugar mill with operating and rest periods. Caledon. J. Eng., 5: 1-5.

[8] Khaled, M.E.S. and Mohammed, S.E.S,.(2005) ‚Profit evaluation of two unit cold standby system with preventive and random changes in units.‘ Journal of Mathematics and Statistics. vol 1, pp. 71-77. http://dx.doi.org/10.3844/jmssp.2005.71.77

[9] Pradeep Chaudhary and Dharmendra Kumar, (2007)“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. 387-394.

[10] Rakesh Gupta and Kailash Kumar, (2007) “Cost benefit analysis of distillery plant system”, Int. J. Agricult. Stat. Sciences, Vol. 3, No. 2, pp. 541-554.

[11] Vishal Sharma and Gaurav Varshney,(2006) “Reliability analysis of an emergency shutdown system model in an industrial plant”, RdE J. of Mathematical Sciences, Vol.1, pp.71-82.

Mohit Kakkar, is Professor and HOD of Mathematics, Department of Applied Sciences, Chitkara University, Punjab, India

Ashok K Chitkara, Chancellor, Chitkara University, Himachal Pradesh, India

Jasdev Bhatti, is Assistant Professor of Mathematics, Department of Applied Sciences, Chitkara University, Punjab, India

probability analysis of a complex system working in a sugar mill with repair equipment failure and c

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