Cost-Benefit Analysis of a System of Non-identical Units under Preventive Maintenance and Replacement Subject to Priority for Operation

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1 International Journal of Statistics and Systems ISSN Volume 11, Number 1 (216), pp esearch India ublications Cost-Benefit Analysis of a System of Non-identical Units under reventive Maintenance and eplacement Subject to riority for Operation Upma and S.C. Malik Department of Statistics, M.D. University, ohtak-1241, Haryana (India) upma82@yahoo.in and sc_malik@rediffmail.com Abstract The cost-benefit analysis of a system of non-identical units has been carried out by considering the concepts of preventive maintenance and priority for operation. The system starts its operation with the original unit (called Main Unit) and the duplicate unit is kept as spare in cold standby. Each unit has two modes-operative and complete failure. The repair activities are handled by a single server who visits the system immediately. The main unit undergoes for preventive maintenance with a constant rate. The duplicate unit is replaced by new one at its failure while main unit is repaired. riority is given to operation of the main unit over the duplicate unit. The random variable associated with failure, preventive maintenance, repair and replacement times are statistically independent. The time to failure of the units and the time to which unit goes for preventive maintenance are negative exponentially distributed whereas the distributions of preventive maintenance and replacement time of the unit are assumed as arbitrary with different probability density functions. The semi- Markov process and regenerative point technique are used to analyze the system model. Graphs for some important measures of system effectiveness are drawn to observe their behavior for arbitrary values of the parameters. Keywords: Non-identical units, reventive Maintenance, eplacement, riority for Operation and Cost-Benefit Analysis. Mathematics Subject Classification 21: 9B25, 6K1 Introduction The deterioration of repairable systems has been controlled by the engineers and scientists up to some extent by using proper repair mechanisms and better

2 38 Upma and S.C. Malik configuration of the components. The technique of redundancy and preventive maintenance has been considered as the best one to improve performance of such systems. In fact, cold standby redundancy has been used extensively by the researchers including Osaki and Asakura (197), Gopalan and Nagarwalla(1985), Kadian et al.(24) and Malik and Barak (213) while developing system models with preventive maintenance and repair. Further the idea of priority for operation to one of the units may be utilized in a system of non-identical units not only to reduce downtime but also to provide effective services to the costumers. Malik et al.(213) carried out cost benefit analysis of a cold standby system with priority for operation and preventive maintenance. In view of the above and practical applications of standby systems, here, a reliability model for a two unit cold standby system of non-identical units is developed in which main unit undergoes for preventive maintenance with a constant rate while the replacement of the duplicate unit is made by new one at its failure. However, repair of the main unit is done by the server at its failure. riority for operation to main unit is given over operation of the duplicate unit. The random variable associated with failure, preventive maintenance, repair and replacement are statistically independent. The distributions for time to failure of the units and the time to which unit goes for preventive maintenance are taken as negative exponential while that of preventive maintenance and replacement time of the unit are assumed as arbitrary with different probability density functions. The expressions for some important measures of system effectiveness are derived in study state using semi-markov process and regenerative point technique. The graphical behavior of the MTSF, availability and profit function has been examined for a particular case giving arbitrary values to various parameters and costs. Notation E : Set of regenerative states : Set of non-regenerative states λ / λ1 : Constant failure rate of the main unit/ duplicate unit α : The rate by which system undergoes for preventive maintenance α : The rate by which system undergoes for replacement θ : The rate by which system undergoes for repair β : reventive maintenance rate of main unit Mo/Do : Main/Duplicate unit is good and operative DCs : Duplicate unit is in cold standby mode MFUr /MFWr : The main unit is failed and under repair/waiting for repair DFUp/ : The duplicate unit is failed and under replacement/waiting for DFWp replacement Mm/ MWm : The main unit is under preventive maintenance/waiting for preventive maintenance MFU : The main unit is failed and under repair for repair continuously from previous state

3 Cost-Benefit Analysis of a System of Non-identical Units 39 DFU MM/ MWM g(t)/g(t) f(t)/f(t) r(t)/(t) q ij (t)/q ij (t) : The duplicate unit is failed and under replacement continuously from previous state : The main unit is under preventive maintenance/ waiting for preventive maintenance continuously from previous state : pdf/cdf of repair time of the unit : pdf/cdf of preventive maintenance time of the unit : pdf/cdf of replacement time of the unit : pdf /cdf of passage time from regenerative state Si to a regenerative state Sj q ij.kr (t)/q ij.kr (t) : pdf/cdf of direct transition time from regenerative state Si to a regenerative state Sj or to a failed state Sj visiting state Sk, Sr once in (, t] M i (t) : robability that the system up initially in state S i E is up at time t without visiting to any regenerative state. W i (t) m ij : robability that the server is busy in the state S i up to time t without making any transition to any other regenerative state or returning to the same state via one or more non-regenerative states. : Contribution to mean sojourn time ( i ) in state S i when system transits directly to state S j so that i m ij and m ij = j tdq ( t) q () ij ' ij & : Symbol for Laplace-Stieltjes convolution/laplace convolution / : Symbol for Laplace Transformation /Laplace Stieltjes Transformation Figure 1

4 4 Upma and S.C. Malik 1. Transition robabilities and Mean Sojourn Times Simple probabilistic considerations yield the following expressions for the non-zero elements pij Qij ( ) qij ( t) dt as (1) = = = 1 g ( 1) = r ( ) = 1 f ( 1) = (1 ( )) = r = f ( 1) = 1 f ( 1) = g ( 1) (1 r ( )) = (1 r ( )) = (1 r ( )) ( ) ( ) = 1 g ( 1) = = = = 1 (2) It can be easily verify that p 1 +p 2 = p 1 +p 15 = p 2 +p 24 = p 36 +p 37 +p 3 = = = = = 1 p 1 +p 13.5 =p 2 +p 23.4 =p 3 +p p 32.7 = 1 The mean sojourn times ( ) is in the state S i are (3) = m 1 + m 2, = m 1 +m 15, = m 2 +m 24 = m 1 +m 13.5, = m 2 +m 23.4, = m 3 +m m eliability and Mean Time to System Failure (MTSF) The i(t) be the cdf of first passage time from regenerative state Si to a failed state. egarding the failed state as absorbing state, we have the following recursive relations for i(t) : Ф (t) = Q 1 (t) S Ф 1 (t) + Q 2 (t) S Ф 2 (t) Ф 1 (t) = Q 1 (t) S Ф (t) + Q 15 (t) Ф 2 (t) = Q 2 (t) S Ф (t) + Q 24 (t) (4) Taking LST of above relations (4) and solving for (s), we have 1 ( s) = (5) s The reliability of the system model can be obtained by taking Inverse Laplace transform of (5). The mean time to system failure (MTSF) is given by 1 ( s) MTSF = lim =, (6) s s where N = p1 1 p 2 2 and D= 1 p1 p1 p2 p 2 (7)

5 Cost-Benefit Analysis of a System of Non-identical Units Steady State Availability Let A i (t) be the probability that the system is in up-state at instant t given that the system entered regenerative state S i at t =.The recursive relations for Ai () t are given as: A (t) = M (t) + q 1 (t) A 1 (t) + q 2 (t) A 2 (t) A 1 (t) = M 1 (t) + q 1 (t) A (t) + q 13.5 (t) A 3 (t) A 2 (t) = M 2 (t) + q 2 (t) A (t) + q 23.4 (t) A 3 (t) A 3 (t) = M 3 (t) + q 3 (t) A (t) + q 31.6 (t) A 1 (t) + q 32.7 (t) A 2 (t) (8) where _ ( M (t) = )t e, M 1 (t) = 1t e F() t, M 2 (t) = 1t ( e G() t, M 3 (t) = ) t e () t (9) Taking LT of above relations (8) and solving for A (s). The steady state availability is given by N A( ) lim sa( s ) = s D 1 1, where (1) N ((1 p p ) p p ) ( p (1 p p ) p p p ) and ( p (1 p p ) p p p ) ( p p p p ) ' ' 2( p2 (1 p13.5 p31.6 ) p1 p13.5 p32.7 ) ' 3( p1 p13.5 p2 p23.4 ) D ((1 p p ) p p ) ( p (1 p p ) p p p ) (11) (12) 4. Busy eriod Analysis for Server a) Due to epair Let Bi () t be the probability that the server is busy in repair the unit at an instant t given that the system entered regenerative state Si at t=.the recursive relations for B i () t are as follows: B () t = q 1 ( t) B 1 ( t) q 2 ( t) B 2 ( t ) B ( t) q ( t) B ( t) q ( t) B ( t ) B2 () t = W 2 ( t) q 2 ( t) B ( t) q 23.4 ( t) B 3 ( t ) B3 () t = q 3 ( t) B ( t) q 31.6 ( t) B 1 ( t) q 32.7 ( t) B 2 ( t ) (13) t 2 1 t where W ( t) e 1 G( t) ( e 1 1) G( t ) (14) Taking LT of above relations (13) and solving for is busy due to repair is given by B () s.the time for which server

6 42 Upma and S.C. Malik B sb s N 2 ( ) lim ( ) s D , where (15) N W () p p p p (1 p p ) and D 1 is already mentioned. (16) b) Due to eplacement p Let Bi () t be the probability that the server is busy in replacement the unit at an instant t given that the system entered regenerative state Si at t=.the recursive p i p p p ( ) 1( ) 1 ( ) 2( ) 2 ( ) p p p 1 ( ) 1( ) ( ) 13.5( ) 3 ( ) p p p 2 ( ) 2( ) ( ) 23.4( ) 3 ( ) p p p p relations for B () t are as follows: B t q t B t q t B t B t q t B t q t B t B t q t B t q t B t B ( t) W ( t) q ( t) B ( t) q ( t) B ( t) q ( t) B ( t ) (17) ( ) t ( ) t ( ) t 3 p where, W ( t) e ( t) ( e 1) ( t) ( e 1) ( t ) (18) Taking LT of above relations (17) and solving for B () s.the time for which server is busy due to replacement is given by p p N3 B ( ) limsb ( s), where (19) D s () N W p p p p and D 1 is already mentioned. (2) c) Due to reventive Maintenance Let Bi () t be the probability that the server is busy in preventive maintenance the unit at an instant t given that the system entered regenerative state Si at t=.the recursive relations for B i () t are as follows: ( ) 1( ) 1 ( ) 2( ) 2 ( ) 1 ( ) 1( ) 1( ) ( ) 13.5( ) 3 ( ) 2 ( ) 2( ) ( ) 23.4( ) 3 ( ) B t q t B t q t B t B t W t q t B t q t B t B t q t B t q t B t B ( t) q ( t) B ( t) q ( t) B ( t) q ( t) B ( t ) (21) where, 1t 1t 1 1 W ( t) e F( t) ( e 1) F( t ) (22) Taking LT of above relations (21) and solving for B () s.the time for which server is busy due to preventive maintenance is given by N B 4 ( ) limsb ( s), where (23) D s 1

7 Cost-Benefit Analysis of a System of Non-identical Units 43 N = W () p (1 p p ) p p p and D 1 is already mentioned. (24) Expected Number of epairs Let i () t be the expected number of repairs by the server in (, t] given that the system entered the regenerative state Si at t =. The recursive relations for i () t are given as: ( t) Q ( t) & ( t) Q ( t) & ( t) ( t) Q ( t) & ( t) Q ( t) & ( t) ( t) Q ( t) & (1 ( t)) Q ( t) & (1 ( t)) ( t) Q ( t) & ( t) Q ( t) & ( t) Q ( t) & ( t) (25) Taking LST of above relations (25) and solving for () s.the expected number of repairs per unit time by the server are giving by N5 ( ) lims ( s), where (26) D s 1 N5 p1 p13.5 p32.7 p2 (1 p13.5 p 31.6) and D 1 is already mentioned. (27) 6. Expected Number of eplacements Let pi () t be the expected number of replacements by the server in (, t] given that the system entered the regenerative state Si at t =. The recursive relations for pi () t are given as: p ( t) Q ( t) & p ( t) Q ( t) & p ( t) p ( t) Q ( t) & p ( t) Q ( t) & p ( t) p ( t) Q ( t) & p ( t) Q ( t) p ( t) p ( t) Q ( t) & (1 p ( t)) Q ( t) & (1 p ( t)) Q ( t) &(1 p ( t)) (28) Taking LST of above relations (28) and solving for p () s.the expected number of replacements per unit time by the server are giving by N6 p( ) limsp ( s), where (29) D s 1 N 6 p1 p13.5 p2 p 23.4 and D 1 is already mentioned. (3) 7. Expected Number of reventive Maintenances Let i () tbe the expected number of preventive maintenance by the server in (, t] given that the system entered the regenerative state S i at t =. The recursive relations

8 44 Upma and S.C. Malik for i () tare given as: ( t) Q ( t) & ( t) Q ( t) & ( t) ( t) Q ( t) &(1 ( t)) Q ( t) &(1 ( t)) ( t) Q ( t) & ( t) Q ( t) & ( t) ( t) Q ( t) & ( t) Q ( t) & ( t) Q ( t) & ( t) (31) Taking LST of above relations (31) and solving for () s.the expected number of preventive maintenances per unit time by the server are giving by N7 ( ) lim s ( s), where (32) s D 1 N 7 = 1( ) and D 1 is already mentioned. (33) 8. rofit Analysis The profit incurred to the system model in steady state can be obtained as p = KA K1B K2B K3B K4E K5Ep K6 Em Where = rofit of the system model K = evenue per unit up-time of the system K 1 = Cost per unit time for which server is busy due to repair K 2 = Cost per unit time for which server is busy due to replacement K 3 = Cost per unit time for which server is busy due to preventive maintenance K 4 = Cost per unit time repair K 5 = Cost per unit time replacement K 6 = Cost per unit time preventive maintenance 9. Graphical resentation of eliability Measures

9 Cost-Benefit Analysis of a System of Non-identical Units Conclusion t t t For the particular case gt= () e, f() t = e and rt () = e, the behavior of mean time to system failure (MTSF), availability and profit function has been observed for arbitrary values of the parameters as shown respectively in figures 2, 3 and 4. It is observed that these measures keep on increasing with the increase of repair rate (θ), replacement rate (α), preventive maintenance rate (β) while their values decline with the increase of failure rates and the rate by which unit undergoes for preventive maintenance. Hence the performance of a system of non-identical units can be improved by giving priority for operation to the main unit over duplicate unit provided the main unit does not undergo for preventive maintenance frequently. 11. eferences 1. Osaki, S. and Asakura, T.(197): A two-unit standby redundant systems with

10 46 Upma and S.C. Malik repair and preventive maintenance. Journal of Applied robability, Vol. 7(3), pp Gopalan, M.N. and Nagarwalla, H.E. (1985): Cost-benefit analysis of a oneserver two-unit cold standby system with repair and preventive maintenance. Microelectronics and eliability, Vol. 25(2), pp Kadian, M.S., Chander, S. and Grewal,A.S. (24): Stochastic analysis of non-identical units reliability models with priority and different modes of failure. Decision and Mathematical Science, Vol. 9(1-3), pp Malik, S.C. and Barak, Sudesh (213): eliability measures of a cold standby system with preventive maintenance and repair. International Journal of eliability, Quality and Safety Engineering, Vol. 2(6), (DOI: /S ). 5. Malik, S.C. and Nandal, Jyoti and Sureria, J.K. (213): eliability measures of a cold standby system with priority for operation and preventive maintenance. International Journal of Mathematical Archive, Vol. 4(3), pp