Comparative study of two reliability models on a two -unit hot standby system with unannounced failures

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1 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) Comparative study of two reliability models on a two -unit hot standby system with unannounced failures Anita Taneja Associate Prof. epartment of Mathematics, IEC College of Engg, Greater oida ABSTRACT The present paper deals with the comparative study of two reliability models on a two-unit hot standby system wherein failures are unannounced. Initially, both the units are loaded but one of them is operative and the other is hot standby i.e. the total control of the machine is on the operative unit. Hot standby unit is hot in the sense that the latest information is transformed to it by the operative unit. The hot standby unit may also get failed. In the first model, the failure of a unit is not self announcing but it is revealed only when both the units get failed i.e. on the failure of the system whereas in the second model, the units are observed with regard to failures i.e. the failures are observable meaning thereby the failure of unit is detected as soon as it fails. Inspection is carried out for the failed unit to see as to whether it is repairable or irreparable and then it is repaired or replaced accordingly. Various measures of the system effectiveness are obtained by making use of semi-markov processes and regenerative point technique. Study through graphs is also made. Introduction A large number of research papers in the field of reliability developed various reliability models for standby systems including Tuteja and Taneja (992), Tuteja et al. (200), Parashar and Taneja (2007), Zuhair and Rizwan (2007), Goyal et.al (2009), Rizwan et al. (200), Mathew et al. (20), Kumar and Kapoor (202), Zhang et al. (202), Singh and Taneja [203]. So, a lot of work has been done in the field of reliability by large number of researchers considering two-unit systems similar or dissimilar. In most of the studies, it has been assumed that failures are self announcing. However, there may be situations where, in hot standby systems, we may not come to know whether the operative unit has got failed or not till the complete failure of both the units. Taking the aspect that failures may not be self announcing, the present paper deals with the comparative study of two reliability models on a two-unit hot standby system wherein failures are unannounced. Initially, both the units are loaded but one of them is operative and the other is hot standby i.e. the total control of the machine is on the operative unit. Hot standby unit is hot in the sense that the latest information is transformed to it by the operative unit. The hot standby unit may also get failed. Two models have been discussed. In the first model, the failure of a unit is not self announcing but it is revealed only when both the units get failed i.e. on the failure of the system whereas in the second model, the units are observed with regard to failures i.e. the failures are observable meaning thereby the failure of unit is detected as soon as it fails. Inspection is carried out for the failed unit to see as to whether it is repairable or irreparable and then it is repaired or replaced accordingly. Various measures of the system effectiveness are obtained by making use of semi-markov processes and regenerative point technique. Study through graphs is also made. 92

2 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) otations p p 2 g(t), G(t) g e(t), G e(t) h (t), H (t) h e(t), H e(t) h 2(t), H 2(t) Constant failure rate of operative unit Constant failure rate of hot stand by unit. probability that unit is repairable. probability that unit is irreparable p.d.f. and c.d.f. of repair time of ordinary repairman p.d.f. and c.d.f. of repair time of expert repairman p.d.f. and c.d.f. of time to inspection for detecting reparability of a failed unit p.d.f. and c.d.f. of inspection time of expert repairman p.d.f. and c.d.f. of replacement time Symbols for the states of the system are o hs F ui F uei F ur F re F rep F UI/F UIe F Ur/F Re F REP operative unit hot standby unit Failed unit under inspection by ordinary repairman Failed unit under inspection by expert repairman Failed unit under repair by ordinary repairman Failed unit under repair by expert repairman Failed unit under replacement Inspection of the failed unit is continuing by the ordinary/expert from the previous state Repair of the failed unit continuing by the ordinary/expert from the previous state Replacement of the failed unit continuing by the ordinary repairman from the previous state Model The idea of inspection and replacement for a two-unit hot standby system is used in model, where failure of unit is not self announcing but it is revealed only when both the units get failed i.e. on the failure of the system. Initially both the units are loaded but one of them is operative and the other is hot standby the total control of the machines is on the operative unit. Hot standby unit is hot in the sense that the latest information is transformed to it by the operative unit. The hot standby unit may also get failed in standby unit. The ordinary repairman replaces the unit on finding it irreparable. 93

3 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) The state transition diagram for the model is shown as in Fig.. The following measures of the system effectiveness are obtained by making use of semi-markov processes and regenerative point technique: p 2 h (t) 0 (+)e (+)t e t p h (t) 2 3 (o, hs) (Fu i, w i ) (Fu r, w i ) (F rep, w i ) g(t) p h (t) p 2 h (t) (F rep, o) 6 p 2 h (t) 4 (o, F ui ) 7 (F UI, w i ) p h (t) g(t) (F REP, w i ) (F ur, o) (F UR, w i ) g(t) Up state Failed state Regeneration point Fig. Mean time to system failure (MTSF), T 0 = 0 **(s) Lim s0 s Where = + 2 and = ( + ) Availability analysis, A 0 = / = 0[p 45p 50 + p 46p 60] p p 46 = (p 45p 50 + p 46p 60) 0 + ( + p 45p 50 + p 46p 60) + {p 2(p 45p 50 94

4 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) + p 46p 60) + p 42 (7) + p 45} 2 + {p 3(p 45p 50 + p 46p 60) + p 43 (7) + p 46} 3 (Repair Time Only) B 0 = 2 / where 2 = 2[( p 43 (7) p 45 p 54 (8) p 46 p 64 (9))p 2 + p 42 (7) p 3] + 2p 45 (Inspection Time Only) BI 0 = 3 / Where 3 = [p 45 p 50 + p 46 p 60 + ] 4 (Replacement Time Only) BR 0 = 4 / Where 4 = [p 46 + p 43 (7) + p 3(p 45p 50 + p 46 p 60)] 3 Expected number of visits by ordinary repairman V 0 = 5 / Where 5 = p 45 p 50 + p 46 p 60 Expected number of replacements Profit analysis R 0 = 6 / Where 6 = p 3[p 45 p 50 + p 46 p 60 + p 46] The expected total profit incurred to the system in steady-state is given by P 4 = C 0A 0 C B 0 C 2BI 0 C 3BR 0 C 4V 0 C 5R 0 Where C 0 = revenue per unit up time of the system C = cost per unit time for which the repairman is busy under repair C 2 = cost per unit time for which the repairman is busy under inspection C 3 = cost per unit time for which the repairman is busy under replacement C 4 = cost per visit of ordinary repairman C 5 = cost per unit time for replacement 95

5 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) Particular case For graphical interpretation the following particular case is considered : g(t) = e t h (t) = e t h 2(t) = e t p 0 = p 2 = p p 3 = p 2 p 24 = p 34 = p 45 = p p 46 = p 2 p 47 (2) = p p 47 (3) = p 2 : p 50 = p 54 (8) = p 60 = : p 64 (9) = : 0 = 2 ( ) =, 3 =, 4 =, 5 =, 6 = we can find profit for this particular case. Model 2 In this model, we analyse a two-unit hot standby system. It is assumed that failures are observable i.e. failure of a unit is detected as soon as it fails. On the failure of a unit, it goes under inspection which is carried out by an ordinary repairman to detect whether the unit is repairable or not. Two reliability models have been studied. In the first model, it is assumed that when the ordinary repairman declares that unit is not repairable, it is replaced with a new one. The unit is replaced if the ordinary repairman declares that unit is not repairable state transition diagram is shown as in Fig. 2 (F ui, o) 0 + g(t) p h (t) 2 (o, hs) g(t) (F ur, o) 7 g(t) (F ur, w i ) 5 p 2 h (t) (F UR, w i ) p h (t) 4 p 2 h (t) (F UI, w i ) (F rep, w i ) (F rep,o) (F REP, w i ) 96

6 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) Up state Failed state Regeneration point Fig. 2 Following measures of the system effectiveness are obtained by making use of semi-markov processes and regenerative point technique. Mean time to system failure T 0 = / where = p p 3 3 = (p 2p 20 + p 3p 30) Availability analysis A 0 = where = (p 2 p 20 + p 3p 30) p p 3 3 = (p 2p 20 + p 3p 30) 0 + k + (p 2 + p 7 (4)) 7 + (p 3 + p 8 (4)) 8 (Repair Time Only) B 0 = 2 where 2 = {p 2 + p 7 (4)} 7 (Inspection Time Only) BI 0 = 3 where 3 = (Replacement Time Only) BR 0 = 4 where 4 = (p 3 + p 8 (4)) 8 97

7 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) Expected number of visits by the repairman V 0 = 5 Expected number of replacements where 5 = p 2 p 2 (5) p 3 p 3 (6) p 7 (4) p 8 (4) R 0 = 6 where 6 = p 3 + p 8 (4) Profit analysis The expected total profit incurred to the system in steady-state is given by P 5 = C 0A 0 C B 0 C 2BI 0 C 3BR 0 C 4V 0 C 5R 0 where C 0 = revenue per unit up time of the system C = cost per unit time for which the repairman is busy under repair C 2 = cost per unit time for which the repairman is busy under inspection C 3 = cost per unit time for which the repairman is busy under replacement C 4 = cost per visit of ordinary repairman C 5 = cost per unit time for replacement Particular case For graphical interpretation the following particular case is considered: g(t) = e t h (t) = e t h 2(t) = e t : p 0 = p 2 = p p 3 = p 2 : p 4 = p 7 (4) = p p 8 (4) = p 2 p 20 = p 2 (5) = p 25 = : p 30 = p 3 (6) = p 36 = : p 7 = p 8 = 98

8 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) 0 = = 2 = : 3 = 7 = 8 = : K = Using the above equations, we can find profit of the particular system. Comparison Between Model and Model 2 Fig. 3 shows the behavior of the difference between P P 2 with respect to cost (C 4) for different values of C 0 where P and P 2 are the profits for Model and Model 2 respectively. The following conclusions are drawn: 99

9 IJREAS VOLUME 5, ISSUE 8 (August, 205) (ISS ) International Journal of Research in Engineering and Applied Sciences (IMPACT FACTOR 5.98) If C 0 = 000, then P P 2 > or = or < 0 according as C 4 < or = or > 62. So, Model is better or worse than Model 2 if C 4 < or > 62. Both the Models are equally good if C 4 = 6.6. For C 0 = 4000, C 0 = 7000 or C 0 = 0000 the difference becomes negative at a very high cost (C 4). Therefore, Model 2 is better than Model till C 4 becomes very high References:. Goyal Amit, Taneja G and Singh V, Analysis of a 2-unit cold standby system working in a sugar mill with operating and rest periods, Caledonian Journal of Engineering, (0): Kumar R and Kapoor S, Cost-benefit analysis of a reliability model for a base transceiver system considering Hardware/Software Faults and Congestion of calls, International Journal of Science and Technology, 202 4(6): Mathew AG, Rizwan SM, Majumder MC, Ramachandran KP and Taneja G, Reliability analysis of an identical two-unit parallel CC plant system operative with full installed capacity, International Journal of Performability Engineering, 207(2): Parashar B and Taneja G, Reliability and profit evaluation of a PLC hot standby system based on a master slave concept and two types of repair facilities,ieee Trans Reliab, (3): Rizwan SM, Khurana V and Taneja G,Reliability analysis of a hot standby industrial system, International Journal of modeling and Simulation, (3): R. K. Tuteja and G. Taneja : Cast benefit analysis of two server two unit warm standby system with different types of failure, Microelectron, Reliab.,32, (992) 7. Singh alip and Taneja G, Reliability analysis of a power generating system through gas and steam turbines with scheduled inspection, Aryabhatta Journal of Mathematics & Informatics, 203Vol. 5, o. 2: Tuteja R. K. and Taneja, G. and Malik, Archana, Reliability and profit analysis of a two- unit cold standby system with partial failure and two types of repairman, Journal of ecision and Mathematical Sciences, 5-6 (-3) (200). 9. Zuhair Al Hemyari and Rizwan S.M., Reliability analysis of a two-unit system, Proc. IEEE International Conference on Industrial Engineering and Engineering Management, Singapore, 2007, Zhang Zhisheng, Gao W, Zhou Y and Zhang Zhiqiang, Reliability Modelling and Maintenance Optimization of the iesel System in Locomotives, Maintenance and Reliability, 202 4(4):