3-Unit System Comprising Two Types of Units with First Come First Served Repair Pattern Except When Both Types of Units are Waiting for Repair

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1 Journal of Mathematics and Statistics 6 (3): 36-30, 00 ISSN Science Publications 3-Unit System Comrising Two Tyes of Units with First Come First Served Reair Pattern Excet When Both Tyes of Units are Waiting for Reair Amit Goyal, Gulshan Taneja and 3 D.V. Singh Deatment of Mathematics, G.G.G.I. Dinarur, Ambala, Haryana, India Deatment of Mathematics, M.D. University, Rohtak, Haryana, India 3 Deatment of Mathematics, N.I. Technology, Kurukshetra, Haryana, India Abstract: Problem statement: In this study we investigated the robabilistic analysis of a three unit system working in a sugar Mill, wherein one unit is big and the other two units are small and identical, is examined. Uon failure of the big unit, both small units are made oerative and unit is undertaken for reair immediately. Priority for oeration is given to the big unit. Priority for reair is given to big unit if both tyes (i.e., big and small) of units are in queue to get the reair done. System is able to work with full caacity only if big unit or both small units are in good condition. If only one small unit is oerable, the system works at reduced caacity. The system under consideration goes to rest during the non-seasonal eriod. Aroach: System was analyzed by making use of semi-markov rocesses and regenerative oint technique and various measures of the system effectiveness are obtained including the rofit incurred to the system. Results: Grahs had been lotted to deict the behavior of the rofit with resect to failure rate for different values of reair rate and with resect to revenue er unit u time for which system is working at full caacity for different values of cost for PM/CM. Conclusion: The rofit increases with the decrease in the values of the failure rate and with the increase in the values of revenue er unit u time. It has lower (higher) values on increasing the values of cost (reair rate). Cut-off oints obtained for failure rate hel decide about having the system with failure rate lesser than that the value at cut-off oint. Also, the rice of the roduct should be fixed in such a way so as to get the revenue greater than the value at cut-off oint. Key words: 3-unit system, two tyes of units, oerating and rest eriods, regenerative oint technique, reliability, measure of system effectiveness, cost benefit analysis INTRODUCTION Two/three-unit standby systems with two stages, that is, working or, have been widely discussed under various assumtions/situations by a large number of researchers including (Mokaddis et al., 997; Tuteja et al., 00; Goel et al., 986; Taneja and Nanda, 003; Li et al., 998; Dhillon, 99; Taneja, 00; Rizwan, 007; Murari and Maruthachalam, 98). There may be one more stage i.e., stage of Rest eriod of the system which also exists in case of some seasonal systems like system working in a sugar mill. Goyal et al. (009) studied reliability and rofit analysis of a two-unit standby system working in a sugar mill with oerating and rest eriods. Thereafter, Goyal et al. (00) did reliability modeling and analysis of a sulhated juice um system comrising three identical units with two tyes of working caacity and rest eriod on the basis of information gathered visiting some sugar mills in Haryana (India). It was observed that the systems used in the mills remain in the functional mode seasonally (i.e., November to Aril in the resent study) and go to Rest in the non-seasonal eriod (i.e., May to October). When the Rest eriod is about to comlete, the PM/CM is carried out to make the system ready for oeration. They considered that the riority for reair as well as oeration is given to the big unit. However, from the economic oint of view, it may not be good all the times to give such riority. Thus, the resent study analyses the reliability and economic asect of such a system considering FCFS (first come first served) attern for reair in all the cases excet when both tyes of units are waiting for reair. When different tyes of units (i.e., one big and one small) are waiting for reair, the riority for reair is given to big unit. It is also assumed on the basis of information obtained from the mills that the system is brought to study at reduced caacity when the system is about to go to the rest eriod due to shortage of sugarcane near the end of season. Corresonding Author: Amit Goya, Deartment of Mathematics, G.G.G.I. Dinarur, Ambala, Haryana, India 36

2 MATERIALS AND METHODS In this study, the robabilistic analysis of the system is analyzed by making use of semi-markov rocesses and regenerative oint technique and have obtained various measures of system effectiveness such as Mean time to system failure, Availability at full caacity and reduced caacity, PM/CM analysis, busy eriod analysis of reairman during working and rest eriods, Rest eriod analysis and rofit function. For other assumtions, (Goyal et al., 00) may be referred to. Notations: : Constant failure rate of the big unit when oerative : Constant failure rate of the small unit when oerative g (t),g (t):.d.f. and c.d.f. of reair time for big unit during working eriod g (t),g (t):.d.f. and c.d.f. of reair time for big unit during rest eriod h (t),h (t):.d.f. and c.d.f. of reair time for small unit during working eriod h (t),h (t):.d.f. and c.d.f. of reair time for small unit during rest eriod q ij (t),q ij (t):.d.f. and c.d.f. of first assage time from a regenerative state i to a regenerative state j or to a state j without visiting any other regenerative state in (0,t] Φ i (t): c.d.f. of first assage time from a regenerative state i to a state j : Rate of going to rest α : Rate of change from full caacity to reduced caacity α : Rate of change from reduced caacity to full caacity,, 3, 4, : Rate when rest eriod is about to start β : Rate of going for PM/CM β : Rate of doing PM/CM β 3 : Rate of starting of the working eriod after the comletion of PM/CM Symbols for the states of system: β 0 : S 0 : B s : S s : Big unit is oerative Small unit is oerative Big unit as cold standby Small unit as cold standby J. Math. & Stat., 6 (3): 36-30, F Br : Reair of big unit under reairman when S r : Reair of small unit under reairman when F Brr : Reair of big unit under reairman when in rest state S rr : Reair of small unit under reairman when in rest state F Brus : Reair of big unit under susension when S rus : Reair of small unit under susension when F BR : Big unit is under reair by reairman from revious state B ro : Big unit ready to become oerative as the working eriod is about to start S rs : Small unit ready to become standby as the working eriod is about to start S wr : Small unit waiting for reair when r B : Big unit under rest r S : Small unit under rest PM/CM: Preventive Maintenance/Corrective Maintenance The diagram showing the various states of transition of the system is shown in Fig.. Transition robabilities and mean sojourn times: The eoch of entries into the states and 7 are regenerative oints and thus these are regenerative states. States 9 and 9 are states. States 0,, 4,, 6, 7,, and 3 are the states where the system is at rest. States, 4, 8 and 0 are states where rest eriod is about to start. States -8 and are states where system is at reduced caacity. The non-zero element lim ij s 0q ij *(s) = is given by: = ; = ; = g * ( + ) [ g *( +) 3 = + 3 [ g *( + ) = + [ g * ( + ) 4 = + = 36 = 47 =, = ; 68 = g *( + 3 ) + (9) 69 = [ g* ( + 3)] = 6,3 + 3 = [ g* ( + )] = = (0) , 3 +

3 J. Math. & Stat., 6 (3): 36-30, 00 P 8, =, = 4, =,6 = 6,7 = P 7,0 = ;,0 = h * ( + 4 ),4 = [ h * ( + 4)] + 4 = [ h * ( + )] 4, , = h * ( + ) = [ h * ( + )] = (9) 3,9 3,6 + = [ h * ( + )] 3,0 + 8, = 0, =,3 =,4 = 3, = 4, = = h * ( + ), = P [ h * ( + )] (9),6,9 + = [ h * ( + )] = = (6),7, + Fig. : State transition diagram Mean Sojourn times (µ i ) in the regenerative state i is 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 i, then: µ i = E(T) = Pr(T>y) The unconditional mean time taken by the system to transit for any regenerative state j, when it is counted 38

4 from the eoch of entrance in to the state i, is mathematically stated as: J. Math. & Stat., 6 (3): 36-30, 00 * ij = ij = ij o m tdq (t) q (0) Measures of the system effectiveness: Following measures of the system effectiveness are obtained using semi-markov rocesses, regenerative oint technique, recursive relations and Lalace/Stieltjes transforms: Mean time to system failure (Including rest eriod) Steady state availability at full caacity (A 0 ) Availability Analysis at Reduced Caacity (AR 0 ) Analysis of the exected rest eriod of the system (R 0 ) Busy eriod analysis for reairing the units during working eriod for Big Unit (B 0 ) Busy eriod analysis for reairing the units during working eriod for Small Unit (BS 0 ) Busy eriod analysis for reairing the units during rest eriod for Big Unit (BR 0 ) Busy eriod analysis for reairing the units during rest eriod for Small Unit (BRS 0 ) PM/CM Analysis of the system Cost-benefit analysis: In steady state, the exected rofit er unit time incurred to the system is given by: Profit (P ) = C 0 A 0 + C AR 0 C B 0 C 3 BS 0 C 4 P 0 C BR 0 C 6 BRS 0 C7R 0 Fig. : Profit versus revenue er unit u time for different values of cost er unit time for PM/CM Fig. 3: Profit versus failure rate ( ) for different values of reair rate (α) Where: C 0 = Revenue er unit u time for which system is working at full caacity C = Revenue er unit u time for which system is working at reduced caacity C = Cost er unit time for which the reairman is busy to reair big unit C 3 = Cost er unit time for which the reairman is busy to reair small unit C 4 = Cost er unit time for PM/CM C = Cost er unit time for which the reairman is busy to reair big unit at rest eriod C 6 = Cost er unit time for which the reairman is busy to reair small unit at rest eriod C 7 = Cost er unit time for which the system remains at rest RESULTS The following articular case is considered to obtain various results: g (t) = αe -αt ; g (t) = βe -βt ; h (t) = e-t ; h (t) = µ e-µt 39 For taking the values of β = 0.00; β = 0.; β 3 = 00; β = 0.; = = = 3 = 4 = = 0.00; α = α = 00; C = 00; C = 800; C 3 = 400; C = 800; C 6 = 000; C 7 = 30 the grah for rofit with resect to failure rate for different values of reair rate is lotted in Fig.. Taking the values β = 0.00; β = 0.; β 3 = 00; β = 0.; = = = 3 = 4 = = 0.00; α = α = 00; C 0 = 000; C = 00; C = 800; C 3 = 400; C 4 = C = 800; C 6 = 000; C 7 = 30 the grah for rofit with resect to failure rate for different values of reair rate is lotted in Fig. 3. DISCUSSION Figure deicts that the rofit increases with the increase in the values of C 0. But C 0 has the lower values on increasing the values of C 4. The following can also be interreted from Fig. : For C 4 = 600, the value of the rofit is > or = or<0 according to whether C 0 >or = or<08.. So, in this case the system is rofitable only if C 0 >08.

5 J. Math. & Stat., 6 (3): 36-30, 00 For C 4 = 800, the value of the rofit is > or = or<0 according to whether C 0 >or = or< So, in this case the system is rofitable only if C 0 > For C 4 = 3000, the value of the rofit is >or = or<0 according to whether C 0 >or = or< So, in this case the system is rofitable only if C 0 >078.0 Figure 3 deicts that the rofit decreases with the increase in the values of. But has the higher values on increasing the values of α. The following can also be interreted from Fig. 3: For α =, the value of the rofit is > or = or <0 according to whether the failure rate ( ) is <or = or> So, in this case the system is rofitable only if the <0.348 For α = 3, the value of the rofit is >or = or <0 according to whether the failure rate ( ) is <or = or > So, in this case the system is rofitable only if the <0.496 For α = 4, the value of the rofit is >or = or<0 according to whether the failure rate ( ) is <or = or > So, in this case the system is rofitable only if the <0.666 CONCLUSION It is concluded that the cut-off oints for failure rate can be obtained which hel decide that that system should be had/urchased which has the failure rate lesser than the value at cut-off oint. Also, the cut-off oint for the revenue er unit u time reveals that the rice for the roduct being manufactured by the user of the system should be fixed in such a way that the revenue er unit u time comes out to be greater than the value at cut-off oint so that a ositive rofit is attained. Recommendation/suggestion for the users of such system: The user of such system should adot the model discussed and imlement it for deciding about fixing of various costs like ayment made to the reairman, the sales rices of the roduct and maximum accetable failure rate of the system. Goel, L.R., G.C. Sharma and P. Guta, 986. Reliability analysis of a system with reventive maintenance insection and two tye of reair. Microelect. Reliab., 6: htt://cat.inist.fr/?amodele=affichen&csidt=79 68 Goyal, A., G. Taneja and D.V. Singh, 009. Reliability and rofit evaluation of a -unit cold standby system working in a sugar mill with oerating and rest eriods. Caledon. J. Eng. Oman, : -. Goyal, A., G. Taneja and D.V. Singh, 00. Reliability modeling and analysis of a sulhated juice um system comrising three identical units with two tyes of working caacity and rest eriod. Pure Alied Math. Sci., 7: Li, W., A.S. Alfa and Y.Q. Zhao, 998. Stochastic analysis of a reairable system with three units and reair facilities. Microelect. Reliab., 38: 8-9. DOI: 0.06/S006-74(97) Murari, K. and C. Maruthachalam, 98. Two-unit arallel system with eriods of working and rest. IEEE Trans. Reliab., 30: 9-9. DOI: 0.09/TR Mokaddis, G.S., S.W. Labib and A.M. Ahmed, 997. Analysis of a two-unit warm standby system subject to degradation. Microelect. Reliab., 37: DOI: 0.06/S006-74(96) Rizwan, S.M., 007. Reliability analysis of a two unit system with two reairmen. Caledon. J. Eng. Oman, 3: - Taneja, G. and J. Nanda, 003. Probabilistic analysis of a two-unit cold standby system with resume and reeat reair olices. Pure Alied Math. Sci., 7: htt://direct.bl.uk/bld/placeorder.do?uin= &ETOC=RN&from=searchengine Taneja, G., 00. Reliability and rofit analysis of a system with PLC used as hot standby. Proceeding of International Conference on Reliability and system Engineering, Dec. -3, Reliability Engineering Centre, Kharagur, India, : Tuteja, R.K., G. Taneja and A. Malik, 00. Reliability and rofit analysis of two-unit cold standby system with artial failure and two tyes of reairman. J. Dec. Math. Sci., -6: REFERENCES Dhillon, B.S., 99. Stochastic modeling of k-out-of-n units family of system. Int. J. Syst. Sci., 3: DOI: 0.080/