The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 1

ISSN 2229-5518

Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults

Rajeev Kumar and Pooja Bhatia

ABSTRACTThe paper deals with a model developed for a single centrifuge system working in Thermal Power Plant, Panipat (Haryana) India, which has alternate periods of operation and rest. The system may have minor, neglected and major faults. It is assumed that the occurrence of a minor fault leads to degradation of the system whereas occ urrence of a major fault leads to failure of the system. The neglected faults that are in the system are generally neglected for repair during operation of the system until the system goes to rest or complete failure and the system has to be stopped on occurrence of minor fault for repair. Various measures of system effectiveness are obtained regarding the reliability and cost analysis of the system is carried out and the conclusions on the basis of the graphical studies are given.

KEY W ORDScentrifuge system, mean time to system failure, neglected faults, Markov process, profit, regenerative point technique.

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1 INTRODUCTION

N the field of reliability modeling, several different types of systems considering various aspects such as types of failure (faults) repairs, inspection polices, modes of operations, switching etc. have been analyzed by several researchers
including [1],[2],[3],[4],[5],[6], [7].
In many practical situations, for instance in thermal
power plant for oil purification, milk plants for making butter,
laboratories, blood fractionation, wine clarification, etc.
centrifuge systems are used and act as the main systems or
sub-systems. In these situations the reliability and cost of
centrifuge systems play a very important and crucial role.
It was observed, while collecting real data on faults/
failures and repairs on a centrifuge system working in Thermal
Power plant, Panipat (Haryana) which undergoes periodic rest
(normally after eight hours), that a minor fault leads to
degradation of the system whereas a major fault leads to
complete failure of the system. Some faults such as vibration,
abnormal sound, etc are generally neglected for repair during
the operation of the system until system goes to rest or to
complete failure. Sometimes these neglected faults also lead to
complete failure of the system. Further the system has to be
stopped on occurrence of minor fault. The cost and
maintenance analysis of centrifuge system considering the
aspects of periodic rest period, neglected faults and stoppage
on occurrence of minor faults has not been reported in the
literature of reliability so far. However, the reliability and
availability analyses of a centrifuge system considering minor,
ignored and major faults has been carried out by [8], [9].
Keeping above in view, the present paper deals with a
single unit centrifuge system considering major, minor and
neglected faults wherein a minor fault degrades the system
whereas a major fault leads to complete failure of the system. The neglected fault is taken as the fault that may be neglected for repair during the operation of the system until system goes to rest or to complete failure. The system undergoes periodic rest. During the rest period or complete failure, the repairman first inspect whether the fault is repairable or non repairable and accordingly carry out repair or replacement of the faulty components. Sometimes when minor fault occurs, the system has to be stopped. Various measures of system effectiveness, such as mean sojourn times, mean time to system failure, expected uptime, busy period of repairman and expected profit are obtained using Markov processes and regenerative point technique. Various conclusions regarding the reliability and cost of the system on the basis of graphical analyses are drawn.

2 OTHER ASSUMPTIONS

1. Faults are self- announcing.

2. There is single repairman that reaches the system in

negligible time, whenever called for repair.

3. The system is as good as new after each repair /

replacement.

4. Switching is perfect and instantaneous.

5. The time distributions of various faults, rest, stoppage

and restart of the system are exponential whereas
other time distributions are general.

3 NOTATIONS

λ123 Rate of occurrence of major/minor/ neglected faults
a/b Probability that the fault is non repairable /
repairable, b = 1- a

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 2

ISSN 2229-5518

p/q Probability that the neglected fault lead to/

don’t lead to complete failure, q=1-p

1 Rate at which the unit goes to rest

2 Rate at which the unit is stopped

β Rate at which the unit restarts after rest
i1(t)/i2(t)/i3(t) p.d.f. of time to inspection of the unit at
failed/stopped/rest state
I1(t)/I2(t)/I3(t) c.d.f. of time to inspection of the unit at
failed/stopped/rest state
g1(t)/g2(t)/g3(t) p.d.f. of time to repair the unit at failed /
stopped/rest state
Operative State Failed State Degraded State
Rest State Stopped State

Fig.1 State Transition Diagram

5 TRANSITION PROBABILITIES AND MEAN SOJOURN TIMES

The transition probabilities are
G1(t)/G2(t)/G3(t) c.d.f. of time to repair the unit at failed /
stopped/rest state

dQ01 (t)

e ( 1 2 3 1 ) t dt

( 1 2 3 1 ) t

dQ02 (t)

e ( 1 2 3 1 ) t dt

( 1 2 3 1 ) t

h1(t)/h2(t)/h3(t) p.d.f. of time to replacement of the unit at failed/stopped/rest state

dQ03 (t)

3e dt

dQ04 (t) e dt

H1(t)/H2(t)/H3(t) c.d.f. of time to replacement of the unit at

dQ15 (t) ai1 (t)dt

2 ( t )

dQ16 (t) bi1 (t)dt

failed/stopped/rest state
dQ27 (t)

2 e dt

dQ31 (t) pk(t)dt

( t )

k(t)/K(t) p.d.f./c.d.f. of time to delay in repair of the
neglected fault

3.1 STATES OF THE SYSTEM

O/R Operative / Rest state
Or/On/ Od Operative state under inspection/ neglected fault/degradation
Ri /Rr/Rrp Rest state under inspection/repair/
replacement

dQ34 (t) qk(t)dt


dQ (t) ai (t)e- ( t ) dt

dQ50 (t) h1 (t)dt dQ78 (t) ai3 (t)dt dQ80 (t) h3 (t)dt dQ10,4 (t) h2 (t)dt

dQ40 (t) e I2 (t)dt

dQ (t) bi (t)e ( t ) dt




dQ60 (t) g2 (t)dt dQ79 (t) bi3 (t)dt dQ90 (t) g1 (t)dt dQ11, 4 (t) g3 (t)dt

lim Q** (s)
Fi/ Fr / Frp Failed state under inspection/ repair/
The non-zero elements pij are
pij

s 0 ij

replacement

4 THE MODEL

A diagram showing the various states of transition of the system is shown in Fig. 1. The epochs of entry in to state 0, 1, 2,
p01

p03

1

1 2 3 1

3

1 2 3 1

p02

p04

2

1 2 3 1

1

1 2 3 1

3, 4, 5, 6, 7, 8, 9, 10, 11 are regenerative point and thus all the states are regenerative states.
p15

ai* (0)

*


p bi* (0)

*


p31


p40 p4,11 p60 p79 p90

p11, 4
pk (0)


1 i * ( ) ai * ( ) g* (0) bi* (0) g* (0)

g * (0)

p34


p4,10 p50 p78 p80 p10, 4

qk (0)

bi * ( )



h * (0) ai* (0) h * (0) h * (0)
By these transition probabilities, it can be verified that p01 + p02 + p03 + p04 = p15 + p16 = p34 + p31 = p4,10 +
p4,11 + p40 = p78 + p79 = 1, p27 = p50 = p60 = p80 = p90
= p10,4 = p11,4 = 1
The mean sojourn time in the regenerative state i(µi) is defined as the time of stay in that state before transition to any other state then we have

1

0 =

1 2 3 1

* /

1 = 1



1

(0) 2=

2

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 3

ISSN 2229-5518




*/ *

* / For graphical analysis the following particular cases are

3 = k (0) 4 = 1 i2 ( ) 5=

h1 (0)
considered:


= g*/ (0) µ =

i* / (0) =

* (0)

g1 (t)

1 ( t )

1

g2 (t)

2 ( t )

2

g3 (t)

3 ( t )

3




= g*/ (0) =

* (0) 11 =

* (0)



k(t) e

( t )

3 ( t )


h1 (t) 1e

1 ( t )

( t )


h2 (t) 2 e

2 ( t )

( t )

The unconditional mean time taken by the system to transit for any regenerative state j, when it is counted from

h3 (t)

i3 (t)

3e

3 ( t )

3

i1 (t) 1e
i2 (t) 2 e
epoch of entrance into that state i, is mathematically stated as- mij = td Qij(t)
Various graphs are drawn for the MTSF and the profit (P)
for the different values of the rates of occurrence of major,


Thus,

m01 + m02 + m03 + m04 = 0 m15 + m16 = 1


m27 = 2 m34 + m31 = 3


m40 + m4,10 + m4,11 = 4 m50 = 5

m60 = 6 m78 + m79 = 7
m80 = 8 m90 = 9
m10,4 = µ10 m11,4 = µ11

6 Other Measures of System Effectiveness

Using probabilistic arguments for regenerative processes, various recursive relations are obtained and are solved to derive important measures of the system effectiveness that are as given below:

Mean time to system failure

(TO) = µ0 + µ2p02

Expected up time of the system

(AO) = N1/ D1

Busy period of repair man (Inspection time only)

(Bi) = N2/ D1

Busy period of repair man (Repair time only)

( Br)= N3/ D1

Busy period of repair man (Replacement time only)

(Brp)= N4/ D1

where
N1 = µ0 + µ2p02 + µ3p03
N2= p40 [p02p27 µ7+ (p01+p03p31) µ1 + (p03p34+p04) µ 4]
N3= p40[p02µ2+p16 µ6(p01+p03p31)+p02p27p79 µ9]
+ p4,11µ11(p03p34+p04)
N4= p40[p02p27p78 µ8+ p15µ5(p01+p03p31)]+ µ10p4,10(p03p34+p04)
D1= p40 0 + p03 µ3 + p02 2 + p79µ9 + p78µ8 + µ7) +
(p01+p03p31) (µ1+p15 µ5+ p16µ6)] +
11p4,11+ µ10p4,10 + µ4) (1-p01-p02- p03p31)

7 PROFIT ANALYSIS




The expected profit incurred of the system is
P = C0 A0 C1 Bi C2 Br C3Brp-C
where
C0 = revenue per unit uptime of the system
C1 = cost per unit time of inspection
C2 = cost per unit time of repair
C3 = cost per unit time of replacement
C = cost of installation of the unit

8 GRAPHICAL ANALYSES

minor and neglected faults (λ1, λ2, λ3), repair rates (β1, β2, β3), replacement rates (γ1, γ2, γ3), inspection rates (α1, α2, α3) and

rates of rest, stoppage and delay for repair ( 1, 2,δ) of the unit.
Fig. 2 and Fig.3 give the graphs between MTSF (To) and
the rate of occurrence of major and neglected faults (λ1 & λ3), respectively for different values of rate of occurrence of minor faults (λ2) and rate at which the system has to be stopped (η2). The graph reveals that the MTSF deceases with increase in the values of rates of occurrence of major, minor and neglected faults, respectively. Further it may also be concluded from the Fig.3 that the delay in the repair of neglected faults result into decrease in the values of MTSF.

Fig. 2

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 4

ISSN 2229-5518


Fig.3

The graph in Fig. 4 shows the pattern of profit with respect to rates of occurrence of minor faults for different values of major faults (λ2 & λ1). The curves in the graph indicate that the profit of the system decreases with the increase in the values of the rates of occurrence of minor faults for different values of major faults.
Fig.4
The curves in the Fig.5 show the behavior of the profit with respect to rate of occurrence of neglected faults (λ3) of the system for the different values of rate of delay in repair of neglected faults (δ). It is evident from the graph that profit decreases with the increase in the rate due to occurrence of neglected faults. From the Fig. 5 it may also be observed that for δ =2, the profit is > or = or < 0 according as λ3 is < or = or
>.879. Hence the system is profitable to the company whenever λ3 ≤ .879. Similarly, for δ =3 and δ =4 respectively the profit is > or = or < 0 according as λ3 is < or = or > .825 and .8 respectively. Thus, in these cases, the system is profitable to the company whenever λ3 ≤ .825 and .8 respectively.
Fig. 5

The graph in Fig. 6 shows the pattern of profit with respect to the rates of occurrence of neglected faults for different values of rate of system has to be stopped (λ32). The curves in the graph indicate that the profit of the system decreases with the increase in the values of the rate of occurrence of neglected faults as well as rate with which system is stopped.
Fig. 6
The curves in the Fig.7 show the behavior of the profit with respect to the revenue per unit up time (CO) of the system for the different values of rate of occurrence of neglected faults (λ3) due to neglected faults. It is evident from the graph that profit increases with the increase in revenue up time of the system for fixed value of the rate of occurrence of neglected faults. From the Fig.7 it may also be observed that for λ3 =
0.001, the profit is > or = or < 0 according as CO is > or = or < Rs.20664.21. Hence the system is profitable to the company whenever CO ≥ Rs. 20664.21. Similarly, for λ3 = 0.201 and λ3 =
0.401 respectively the profit is > or = or < 0 according as CO is > or = or < Rs.24093.56 and Rs.28257.06 respectively. Thus, in these cases, the system is profitable to the company whenever CO ≥ Rs. 24093.56 and Rs. 28257.06 respectively.

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The research paper published by IJSER journal is about Economic and Reliability Analysis of a Centrifuge System with Rest period, Neglected Faults and Stoppage on Minor Faults 5

ISSN 2229-5518

M.D.University,Rohtak(INDIA)

Ms. Pooja Bhatia Research Scholar M.D.University,Rohtak(INDIA)

Fig. 7

REFERENCES

[1] Goel, L.R., Sharma, G. and Gupta, R., “Reliability analysis of a system with prevention maintenance and two types of repairing,” Microelectronics Reliability, 26, pp. 429-433 1986.

[2] Gopalan, M.N. and Murlidhar, N. N.,”Cost analysis of a one unit repairable system subject to on-line prevention maintenance and/or repair,” Microelectronics Reliability,

31(2/3), pp.233-228 1991.

[3] Goyal, A., Taneja, G. and Singh, D.V., “Reliability modeling and analysis of a sulphated juice pump system comprising three identical units with two types of working capacity and rest period,” Pure and Applied Mathematical Sciences, 71(1-2), pp.133-143 2010.

[4] Gupta, M.L. and Kumar, A., “On profit consideration of a maintenance system with minor repair,” Microelectronics Reliability, 23, pp.437-439 1983.

[5] Kumar, R. and Bhatia, P. “Availability analysis of a reliability model on a centrifuge system with rest period, neglected faults and stoppage on minor faults,” Proceedings of international conference on advances in modeling, optimization and computing 2011.

[6] Kumar, R. and Bhatia, P., “Reliability and availability analysis of a single unit centrifuge system considering various faults,” International conference on science and engineering (ICSE11), pp.536-539 2011.

[7] Murari, K. and Goyal, V.,” Reliability system with two types of repair facilities,” Microelectronics Reliability, 23(6), pp.1015-1025 1983.

[8] R.C. Garg and A. Kumar, “A complex system with two types of failure and repair”, IEEE Trans. Reliability., 26, pp.299-300,1977.

[9] Taneja, G., Tayagi, V. K. and Bhardwaj, P.,” Profit analysis of a single unit programmable logic controller,” Pure and Applied Mathematical Sciences, LX(1-2), pp.55-71,2004.

Dr. Rajeev Kumar

Associate Professor

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