Cost Benefit Analysis of a Two Unit Model with Priority to Preventive Maintenance of the Unit over Other Activity Done By the Server

INTRODUCTION

After the Second World War numbers of systems or units were found fail during their operation time, before operation time as well as between the operation times. Researchers were found lot of reason for their failure as well as suggested to improve their performance during such situations. In literature, the stochastic behavior of cold standby system has been widely discussed by many researchers including, Osaki and Nakagawa [1971] discussed a two-unit standby redundant system with standby failure. Nakagawa and Osaki [1975] analyzed stochastic behavior of a two-unit priority standby redundant system with repair. Subramanian et al. [1976] explored reliability of a repairable system with standby failure. Gopalan and Nagarwalla [1985] evaluated cost benefit analysis of one server two unit cold standby system with repair and age replacement. Gupta and Goel [1989] studied profit analysis of two-unit priority standby system with administrative delay in repair. Reliability and availability analysis of a system with standby and common cause failures have been explained by Dhillon [1992]. Lam [1997] studied a maintenance model for two–unit redundant system. Malik [2009] discussed reliability modelling and cost-benefit analysis of a system – A case study. Dhankhar and Malik [2011] studied cost- benefit analysis of system reliability models with server failure during inspection and repair. Bhardwaj and Kaur [2014] analyzed reliability and profit of a redundant system with possible renewal of standby subject to inspection. Recently, Grewal et al. [2017] obtained economic analysis of a system having duplicate cold standby unit with priority to repair of original unit. Rohila and Kumar [2018] analyzed cost benefit of industry having duplicate cold standby unit with different failure rate. Rohila [2019] studied the cost-benefit analysis of a two unit system model with the concept with server facility-FCFS. The present paper is to analyze a cold standby system by giving priority to preventive maintenance over repair. Two identical units are taken having two modes- operative and complete failure. There is a single server who visits the system immediately for conducting maintenance and repair. Server conducts preventive maintenance of the unit after a maximum operation time ‗t‘. However, repair of the unit is done at its complete failure. Priority is given to preventive maintenance of one unit over repair of the other unit. The random variables associated with failure time, completion of maximum operation time, preventive maintenance and repair times are statistically independent. The failure time of the unit and the time by which unit undergoes for preventive maintenance and repair times are taken as arbitrary. Several measures of system effectiveness such as transition probabilities mean

E0 : Set of regenerative states. O/Cs : The unit is operative/cold standby. 0 : The 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 time of a failed unit. Pm/WPm: The unit is under preventive maintenance/waiting for preventive maintenance. PM/FUR: The unit is under preventive maintenance/under repair continuously from previous state. FUr/Fwr : The failed unit is under repair/waiting for repair. mij : The unconditional mean time taken by the system to transit from any regenerative state Si when it (time) is counted from epoch of entrance in to that state Sj. Mathematically, it can be written as



. i : The mean Sojourn time in state Si which is given by



0)()(jijimdttTPtE

, where T denotes the time to system failure. Wi(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. / : Symbol for Laplace Stieltjes transform/ Laplace transform. (desh) : Used to represent alternative result.

1. The system has of two identical units which may fail directly from normal mode. 2. Initially one unit is operative and the other is kept as spare in cold standby. 3. There is a single server who visits immediately to the system. 4. The preventive maintenance of the unit is carried out after a maximum operation time. 5. Repair of the unit is done at its failure. 6. The server cannot leave the system while performing jobs. 7. The unit works as a new after repair and preventive maintenance. 8. The switch over is instantaneous and perfect. 9. Priority is given to preventive maintenance over repair. The failure time of the unit and the rate by which unit undergoes for preventive maintenance follow negative exponential distributions. 10. The distributions for preventive maintenance and repair times of the units are taken as arbitrary with different probability density functions. 11. All random variable are statistically independent.

The differential transition probabilities considerations yield the following expressions for non-zero elements It can be easily verified that The mean sojourn time )(iin the regenerative state Si are defined as the time of stay in that state before transition to any other state. If T denotes the sojourn time in the regenerative state Si, then The unconditional mean time taken by the system to transit for any regenerative state Sj when it (time) is counted from the epoch of entrance into the state Si is mathematically, states as

Let )(tibe the c.d.f. of first passage time from the regenerative state Si to a failed state. Regarding the failed state as absorbing state, we have the following recursive relations for )(ti; Taking Laplace–Stieltjes transform of above relations (5) and solving for)(~0s. We have The reliability of the system model can be obtained by taking Laplace inverse transform of (6). The mean time to system failure (MTSF) is given by

AVAILABILITY ANALYSIS

Let )(tAibe 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 )(tAiare given as Here, )(tMiis the probability that the system is up initially in state ESi is up at time t without visiting to any other regenerative state, we have Taking L.T. of above relations (8) and (9) and solving for )(*0sA, The steady state availability is given by where

given that system entered state Si at t=0. The recursive relations for )(tBpi are as follows: where )(1tWand )(4tWbe the probability that the server is busy in state S1 and S4 due to preventive maintenance 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 Taking Laplace transform of above relations (11 and 12) and solving for )(*0tBp. The time for which server is busy due to preventive maintenance is given by Where and Dhas already mentioned in relation (10).

Let )(tBRi be the probability that the server is busy in repair of the unit at an instant ‗t‘ given that system entered state Si at t=0. The recursive relations for )(tBRi are as follows: without making any transition to any other regenerative state or returning to the same via one or more non-regenerative state and so Taking Laplace transform of above relations (14 and 15) and solving for)(*0tBR. We obtained. The time for which server is busy due to repair is given by Where and Dhas already mentioned in relation (10).

Let )(tRPibe the expected number of preventive maintenance of unit by the server in (0,t] given that the system entered the regenerative state Si at t=0.

Taking L.S.T of relations (17) and, solving for)(~0tRp. The expected number of preventive maintenances per unit time are respectively of given by Where

andDhas already defined in relation (10).

Let )(tRRi be the expected number of repairs of unit by the server in (0,t] given that the system entered the regenerative state Si at t=0. The recursive relations for )(tRRiis given as Taking L.S.T of relations (19) and, solving for )(~0tRR.The expected numbers of repairs per unit time are respectively given by Where and Dhas already defined in relation (10).

The profit incurred to the system model in steady state can be obtained as 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 done by the server K5= Total installation cost of the system

Let us take tetg)(andtetf)(, then the following results are obtained: Availability Busy period due to preventive maintenance Busy period due to Repair Expected Number of visits by the server for conducting preventive maintenance Expected Number of visits by the server for doing repair

First Curve of the figure 2 corresponding to L2

(α0=5, λ=0.01 and =3) represents that MTSF having increasing trend when the preventive maintenance rate ‗α‘ is increasing between the range 5 to 14, third curve of this figure at its

minimum range .38 to .56 corresponding to L4

(α0=5, λ=0.02 and =3). Similarly effect of the other parameters reflects from this figure. The graphical behaviour of mean time to system failure corresponding to preventive maintenance rate α, represents that L3(α0=7, λ=.01, =2.5) having increasing treand but at its lowest range, and the other parameters have very small impact on the MTSF with increasing trend.

Figure 3 represents the graphical behavior of availability of the system with respect to the preventive maintenance rate α. The idea of preventive maintenance of the unit after a specific period of time can enhance the availability of the system. The curve L1 (α0=5, λ=0.01 and =2.5) coincide with the second curve corresponding to L2 (α0=5, λ=0.01 and =3) indicate that the effect of the parameter  does not affect availability of the system at high level. The fourth curve L4 (α0=5, λ=0.02 and =3) corresponding to the parameter λ also having negligible impact on the availability of the system. Only α0 is the parameter, which can affect on the availability of the system as shown in the curve L3.

Graphical behaviour of the profit of the system with respect to preventive maintenance of the unit α indicated in the figure 4, i.e profit of the system having increasing treand when the preventive maintenance rate α increases from 5 to 14.

Second curve of the figure having its maximum range (2159 to 3142). When the parameter α0 increases from 5 to 7 then profit of the system at its lowest range but in increasing pettren between the range (1365 to 2401). Profit of any system is depends upon the availability of the sysem as well as MTSF of the system. Figure 4 represents the graphical behaviour of the profit of the system corresponding to preventive maintenance rate α. The curves L1, L2, and L4 are coincide to each other corresponding to α0=5, λ=0.01 and =2.5, =3 and λ=.02 respectively. And the curve L3(α0=5, λ=0.01 and =2.5) at its lowest range as indicated in the graph.

The graphs for mean time to system failure, availability and profit function have been drawn with respect to preventive maintenance rate giving particular values to the parameters and costs as shown respectively in figures 2, 3 and 4. It is observed that the values of these reliability measures go on increasing with the increase of preventive maintenance and repair rates. However, their values decline with the increase of maximum

1. Osaki, S. and Nakagawa, T. (1971): On a two-unit standby redundant system with standby failure. Operations Research, 19(2), pp. 510-523. 2. Nakagawa, T. and Osaki, S. (1975): Stochastic behavior of a two-unit priority standby redundant system with repair, Microelectronics Reliability, 14(3), pp. 309 – 313.

Research, 24(1) pp. 169-176. 4. Gopalan, M. N. and Naidu, R. S. (1982): Cost-benefit analysis of a one-server system subject to inspection, Microelectronics Reliability, Vol. 22, pp. 699 – 705. 5. Naidu, R. S. and Gopalan, M. N. (1984): Cost-benefit analysis of one-server two-unit system subject to arbitrary failure, inspection and repair. Journal of Reliability Engg, Vol. 22, pp. 8-11. 6. Gupta, R. and Goel, L. R. (1989): Profit analysis of two-unit priority standby system with administrative delay in repair, International Journal of System Sciences, Vol. 20, pp. 1703 – 1712. 7. Lam Y. F. [1997]; a maintenance model for two-unit redundant system, Microelectronics Reliability, 37(3) pp. 497-504. 8. Malik S. C. [2009]: reliability modeling and cost-benefit analysis of a system – A case study, Journal of Mathematics and System Sciences, 5(2) pp. 26-38. 9. Dhankhar, A .K. and Malik, S.C. (2011): Cost- benefit analysis of system reliability models with server failure during inspection and repair. International Journal of Statistics & Analysis, 1(3), pp. 265-278. 10. Bhardwaj, R. K. and Kaur, Komaldeep (2014): Reliability and profit analysis of a redundant system with possible renewal of standby subject to inspection, International Journal of Statistics and Reliability Engineering, 1(1), pp. 36-46. 11. Ajit Grewal, Indu Rohilla and Jitender Kumar (2017): Economic analysis of a system having duplicate cold standby unit with priority to repair of original unit, International Journal of Latest Research in Engineering and Technology (IJLRET), 3(12), pp. 98-103. 12. Indu Rohilla (2019): Cost benefit analysis of a two unit system model with the concept with service facility-FCFS, Journal of Advance and Scholarly Research in Allied Education (JASRAE), 16(9), June, pp. 467-472.

Singhania University, Pechari Beri, Jhunjhunu, Rajasthan indurohilla24@gmail.com