Ijer 2014 412

International Journal of Engineering Research Vo lu me No.3, Issue No.4, pp :241-244

(ISSN : 2319-6890) 01 April 2014

A Standby System with P riority to Repair over Preventive Maintenance S.C. Malik, Sudesh K. Barak Department of Statistics, M.D. University Rohtak (Haryana) sc_malik@rediffmail.com, ms_barak@rediffmail.com Abstract: A cold standby system of two identical units is analyzed stochastically under priority in repair disciplines.

Unit has a constant failure from normal mode. There is a single server who visits the system immediately to carry out repair activities. Server conducts preventive maintenance of the unit after a pre-specific time‘t’ of operation. However, repair of the unit is done by the server at its failure. Priority is given to repair of one unit over preventive maintenance of the other unit. The ra ndom variables associated with different rates are statistically independent. The failure time and time by which unit goes for preventive maintenance follow exponential distribution while the distributions of maintenance and repair times are taken as arbitrary with different probability density functions. The expressions for several reliability measures of the system are derived using semi-Markov process and regenerative point technique. Graphs are drawn to depict the behavior of MTSF, availability and profit function for particular values of various parameters and costs.

Key words: Cold standby system / Maximu m operation time / Preventive maintenance / Repair / Priority / and Stochastic Analysis.

Introduction The performance of operating systems can be improved by providing redundancy as well as by conducting maintenance and repair of the failed co mponents. And, cold standby systems are one of the most important structures in reliability engineering. Therefore, a lot of research work has been done on reliability modeling of cold standby systems considering different aspects on failure and repair policies . Gopalan and Naidu (1982), Cao and Wu (1989), Lam (1997) and Yadavalli et al. (2004) developed reliability models of a cold standby system with single repair facility.

Further, continued operation and ageing of operable systems reduce their performance, reliability and safety.

Therefore, preventive maintenance of such systems can be conducted after a pre-specific time (called maximu m operation time) in order to slow the deterioration process. Recently, Malik (2013) investigated a reliability model of a computer system with cold standby redundancy and preventive maintenance.

Also, sometimes it becomes necessary to give priority in repair disciplines of one unit over the other in order to reduce down time of the system. Chhillar et al. (2013) analyzed a parallel system with priority to repair over maintenance subject to random shocks.

In view of the above practical situations in mind, here a reliability model for a t wo-unit cold standby system is developed considering the aspect of priority in repair discip lines. The units are identical in nature wh ich may fail d irectly fro m normal mode. There is a single server who visits the system immediately for conducting maintenance and repair. Server conducts

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preventive maintenance of the unit after a maximu m operation time‘t’. Ho wever, repair of the unit is done at its complete failure. Priority is given to repair of one unit over maintenance of the other unit. The random variables associated with failure time, co mp letion of maximu m operation time, preventive maintenance and repair times are statistically independent. The failure t ime and time by which unit goes for preventive maintenance follow exponential d istribution while the distributions for maintenance and repair times are taken as arbitrary. Several measures of system effectiveness such as transition probabilities, mean sojourn times, mean time to system failure (MTSF), availab ility, busy period of the server due to maintenance and repair, expected number of maintenances and repairs of the unit and profit function are obtained using semi-Markov process and regenerative point technique. Graphs are drawn to depict the behavior of MTSF, availability and profit function for part icular values of various parameters and costs.

Methodol ogy The semi-Markov and regenerative point processes are adopted to obtain reliability measures of the system model. A brief description as follows: Semi-Markov Process It is a continuous-time stochastic process in which transition from one state to another is governed by the transition probabilit ies of a Markov process and time interval between two successive transitions is a random variab le whose distribution may depend upon the state fro m which transition takes place as well as on the state to which next transitions take place.

Regenerati ve Process It is a stochastic process with time points at which, fro m a probabilistic point of view, the process restarts itself.

These time points may themselves be determined by the evolution of the process. i.e., the process {X(t), t ≥ 0} is a regenerative process if there exist time points 0 ≤ T 0 < T1 < T2 < ... such that the post-Tk process {X(Tk+t ) : t ≥ 0} has the same distribution as the post-T process {X(T0 + t) : t ≥ 0} and is independent of the pre-Tk process {X(t) : 0 ≤ t < Tk }. These time points are called regenerative points. Regenerative stochastic process was defined by Smith (1955) and has been crucial in the analysis of the complex system.

Notati ons and Assumptions E : Set o f regenerative states.

O/Cs : The unit is operative/cold standby 0 : The constant rate by which unit undergoes for preventive maintenance  : Constant failure rate of the unit. f(t)/F(t) : pdf /cdf of preventive maintenance time g(t)/ G(t) : pdf /cdf of repair t ime of a failed unit Pm/WP m /PM :The unit is under preventive

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International Journal of Engineering Research Vo lu me No.3, Issue No.4, pp :241-244 maintenance/waiting for preventive Maintenance / under preventive maintenance continuously from previous state.

FUr/Fwr/FUR :The failed unit is under repair/waiting for repair/under repair continuously from prev ious state. q ij (t) / Qij (t) :pdf and cdf of direct transition time fro m a regenerative state Si to a regenerative state Sj without visiting any other regenerative state q ij.k (t) / Qij.k (t) :pdf and cdf of first passage time fro m a regenerative state Si to a regenerative state Sj or to a failed state Sj visiting state k once in (0,t ]. mij : The contribution to mean sojourn time in regenerative state Si when system is to make transition in to regenerative state Sj .

Mathematically, it can be written as 

mij  E (T ij )   td [Qij (t )]  qij (0) , 0

where T ij is the transition time

fro m state

Si to Sj ; Si , Sj ε E.

i

: The mean Sojourn time in state S i this is given by

(ISSN : 2319-6890) 01 April 2014

pij  Qij ()   qij (t ) dt

0  * , p02  , p10  f (0   ) , 0   0   0 p13  [1  f * ( 0   )] 0    p16  [1  f * ( 0   )] , p20  g* (0   ) , 0    p25  [1  g * ( 0   )] 0   0 p24  [1  g * ( 0   )] , 0   p31  p41  p52  p61  1 0 p11.3  [1  f * ( 0   )] 0   ,

p01 

p21.4 



i  E (T i )   P(Ti  t )dt   mij , where j

0

Ti

is the sojourn time in state Si .

W i (t)

: Probability that the server is busy in the state Si up to time ‘t’ without making any transition to any other regenerative state or returning to the same state via one or more regenerative states. / :Sy mbol for Laplace Stieltjes convolution/Laplace convolution / : Sy mbol for Laplace Stelt jes transform (LST)/ Laplace transform (LT) The possible transition diagram of the system model is shown in fig. 1.

State Transition Diagram O CS

S0 

O Pm



S1

f(t)

f(t)  

FUR FWr

g(t)

WPm Fur

 0  

[1  g * ( 0   )] ,

[1  g * ( 0   )]

(2)

It can be easily verified that

p01  p02  p10  p13  p16  p20  p24  p25  1 The mean sojourn times

(3)

( i ) is the state Si are

1 1 [1  f * ( 0   )] , , 1  0   0   1 2  [1  g * ( 0   )] 0   0   1 1 ' 1'  , 2  and  6   ( 0     )  

0 

(4)

Reliability and Mean Ti me to System Failure (MTS F) Let

i (t ) be

the cdf of first passage time fro m the

regenerative state Si to a failed state. Regarding the failed state as absorbing state, we have the following recursive relations for

0 (t )  Q01 (t )  1 (t )  Q02 (t )  2 (t ) 1 (t )  Q10 (t )  0 (t )  Q13 (t )  Q16 (t ) 2 (t )  Q20 (t )  0 (t )  Q24 (t )  Q25 (t )

(5)

Taking LST of relations (5) and solving for 0 ( s ) . We have FUR WPm

R* ( s ) 

S4

Fig 1.

Transition Probabilities and Mean Sojourn Ti mes Simp le probabilistic considerations yield the expressions for non-zero elements

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0  

i (t ) :

S6

5



S2

p22.5 

0

S5 g(t)

O Fur

S3

g(t)

g(t) 

PM WP m

(1)

following

~ 1  0 ( s ) s

(6)

The reliability of the system model can be obtained by taking inverse Laplace transformation of (6). The mean time to system failure (MTSF) is given by

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International Journal of Engineering Research Vo lu me No.3, Issue No.4, pp :241-244

~ 1  0 (s) 0  1 p01  2 p02 MTSF  lim  s 0 s 1  p01 p10  p02 p20

(7)

Steady State Availability

Ai (t ) be

Let

the probability that the system is in up-

state at instant‘t’ given that the system entered regenerative state Si at t=0. The recursive relations for

A2 (t )  M 2 (t )  q20 (t )  A0 (t )  q21.4 (t )  A1 (t )  q22.5 (t )  A2 (t ) A6 (t )  q61 (t )  A1 (t ) (8) state

Si  E

the probability that the system is up initially in is up at time t without visiting to any other

regenerative state, we have M 0 (t )  e( 0  )t , M 1 (t )

M 2 (t )  e

 ( 0   ) t

e

 ( 0   ) t

BR2 (t )  W2 (t )  q20 (t )  B0R (t )  q21.4 (t )  B1R (t )  q22.5 (t )  B2R (t ) B6R (t )  W6 (t )  q61 (t )  B1R (t )

Wi (t ) be the probability that the server is busy in state

Si due to preventive maintenance and repair up to time ‘t’ without making any transition to any other regenerative state or returning to the same via one or more non-regenerative state and so

W1 (t )  {e( 0  )t  (0e( 0  )t 1)}F (t ) W2 (t )  {e (0  )t  ( 0e (0  )t  1)  (e (0  )t  1)}G(t ) , W6 (t )  G(t ) taking LT of relat ions (11). So lving for

F (t ) ,

G (t )

(9)

* Taking LT of relat ions (8) and solving for A0 ( s) , the steady

state availability is given by

N A0 ()  lim sA0* ( s)  , where s 0 D N  0{(1  p22.5 )[(1  p11.3 )  p16 p61 ]}

(10)

 1{ p01 (1  p22.5 )  p02 p21.4 }  2{ p02 p61  p02 (1  p11.3 )} D  0 {(1  p11.3 )(1  p22.5 )  p61 p16 (1  p22.5 )}  1{ p01 (1  p22.5 )  p02 p21.4 }  2 { p02 p61 p16  p02 (1  p11.3 )}  6 { p02 p21.4 p16  p01 (1  p22.5 )} Busy Period Anal ysis for Server p Let Bi (t ) and

B1R (t )  q10 (t )  B0R (t )  q11.3 (t )  B1R (t )  q16 (t )  B6R (t )

where

A1 (t )  M 1 (t )  q10 (t )  A0 (t )  q11.3 (t )  A1 (t )  q16 (t )  A6 (t )

M i (t ) is

B0R (t )  q01 (t )  B1R (t )  q02 (t )  B2R (t )

(11)

Ai (t ) are given as

A0 (t )  M 0 (t )  q01 (t )  A1 (t )  q02 (t )  A2 (t )

where

(ISSN : 2319-6890) 01 April 2014

BiR (t ) be the probability that the server is busy

due to preventive maintenance and repair of the unit at an instant ‘t’ given that system entered state Si at t=0.

B0* p (t ) and B0*R (t ) , the

time for wh ich server is busy due to preventive maintenance and repair respectively are given by

N1p , s 0 D NR B0R (t )  lim sB0* R (t )  2 , s 0 D B0p (t )  lim sB0* p (t ) 

p

Where N1

and (12)

(t )  W1*  p01 (1  p22.5 )  p02 p21.4  ,

N 2R (t )  W2*{ p02 (1  p11.3  p16 p61 )}  W6*{ p16 ( p01 p22.5  p01  p02  p21.4 )} and D has already mentioned.

Expected Number of Repairs and Preventi ve Maintenances of the Unit Let

RiR (t ) and RiP (t ) be

the expected number of

repairs and preventive maintenances of the unit by server in time interval (0,t] g iven that the system entered the regenerative state R

Si at t=0. The recursive relations for Ri as

(t ) and RiP (t ) are given

R0R (t )  Q01 (t )  R1R (t )  Q02 (t )  R2R (t )

B0P (t )  q01 (t )  B1p (t )  q02 (t )  B2p (t )

R1R (t )  Q10 (t )  R0R (t )  Q11.3 (t )  R1R (t )  Q16 (t )  R6R (t )

B1P (t )  W1 (t )  q10 (t )  B0p (t )  q11.3 (t )  B1p (t )  q16 (t )  B6p (t )

R2R (t )  Q20 (t )  [1  R0R (t )]  Q22.5 (t )  [1  R2R (t )]  Q21.4 (t )  [1  R1R (t )]

B2P (t )  q20 (t )  B0p (t )  q22.5 (t )  B2p (t )  q21.4 (t )  B1p (t )

R6R (t )  Q61 (t )  [1  R1R (t )]

B6P (t )  q61 (t )  B1p (t )

and

and

R0p (t )  Q01 (t )  R1p (t )  Q02 (t )  R2p (t ) R1p (t )  Q10 (t )  [1  R0p (t )]  Q11.3 (t )  [1  R1p (t )]  Q16 (t )  R6p (t ) R2p (t )  Q20 (t )  R0p (t )  Q22.5 (t )  R2p (t )  Q21.4 (t )  R1p (t ) R6p (t )  Q61 (t )  R1p (t ) (13)

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International Journal of Engineering Research Vo lu me No.3, Issue No.4, pp :241-244 Taking LST of relations (13) and solving for

~ ~ R0R (t ) and R0p (t )

. The expected number of repairs and preventive maintenances per unit time are respectively given by

NR ~ R0R ()  lim sR0R ( s )  3 s 0 D NP ~ and R0P ()  lim sR0P ( s )  4 s 0 D

(ISSN : 2319-6890) 01 April 2014 iv.

Yadavalli, V.S.S., Chanderasekhar and Natarajan, R. (2004): A study on two-unit standby system with Erlangian repair time. AsiaPacific Journal of Operational Research,Vol. 21(3), pp. 271-277.

(14) , where

N 3R  p02 (1  p11.3  p16 p61 )  p16 p61[ p01 (1  p22.5 )  N 4P  ( p10  p11.3 ){ p01 (1  p22.5 )  p02 p21.4 }

v.

S.C. Malik (2013): Reliability modeling of a computer system with preventive maintenance and priority subject to maximum operation and repair times. International Journal System Assurance Engineering p02 p21.4Management ] DOI/10.1007/S13198-013-0144-y/ vi.

S.K. Chillar, A.K. Barak and S.C. Malik (2013): Analysis of a parallel system with priority to repair over maintenance subject to random shocks. International Journal of Computer Science Issues vol.

No.2, May 2013, pp. 298-303.

and D has already defined.

Profit Anal ysis The profit incurred to the system model in steady state can be obtained as

1

(15)

K0 = Revenue per unit up-time of the system K1 = Cost per unit time for which server is busy due preventive maintenance K2 = Cost per unit time for which server is busy due to repair K3 = Cost per unit time repair K4 = Cost per unit time preventive maintenance K5 = Total installation cost of the system

0.8

MTSF

P  K 0 A0  K1B0P  K 2 B0R  K 3 R0R  K 4 R0P  K 5

 513  5225  5125

1.2

0.6 0.4 0.2

 7125

0 5

Conclusion The trends of some important reliability measures have been observed graphically fo r the particular case

7

8

9

10

11

12

13

14

g (t )  e  t 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

 513

 5125

 5225

5

6

7

8

9

10

 7125

11

12

13

Fig. 3: Preventive maintenance rate() 3500

 5125

3000

14

 513

2500

Profit

i.

Gopalan, M.N. and Naidu, N.S.(1982): Stochastic behavior of a two-unit repairable system subject to inspection. Microelectron Reliab. Vol. 22, pp.717-722. ii.

Cao, Jinhua and Wu, Yan Hong (1989): Reliability of twounit cold standby system with replaceable repair facility.

Microelectronics & Reliability, Vol. 29(2), pp. 145-150. iii.

Lam, Y.F. (1997): A maintenance model for two-unit redundant system. Microelectronics & Reliability, Vol. 37(3), pp. 497504.

Availability

 t

and f (t )  e by taking numerical values of various parameters and costs as shown in figures 2, 3 and 4. It is found that mean time to system failure (MTSF), availability and profit function go on increasing with the increase of preventive maintenance and repair rates. However, their values decline with the increase of failure rate and the rate by which unit under goes for preventive maintenance. The study also reveals that effect of the rate α0 is more on these measures as compared to the other rates. Hence, a cold standby system of two identical units in which priority is given to repair of one unit over preventive maintenance of the other unit can only be made reliab le and profitable to use by repairing the unit in a time less than the time taken for p reventive maintenance of the unit. The beauty of the present study is to make a suggestion that performance of a cold standby system can be improved by conducting preventive maintenance after a maximu m operation time rather than to spent more money on repair after failure. The application of the present study can be visualized in a water pu mp system.

References

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6

Fig. 2: Preventive maintenance rate()

2000

1500

 5225

1000

 7125

500 0 5

6

7

8

9

10

11

12

13

14

Fig.4: Preventive maintenance rate()

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