Yusuf I., Gaawa R.I. Volume, December 206 Probabilisic Models for Reliabiliy Analysis of a Sysem wih Three Consecuive Sages of Deerioraion Ibrahim Yusuf Deparmen of Mahemaical Sciences, Bayero Universiy, Kano, Nigeria iyusuf.mh@buk.edu.ng Ramau Idris Gaawa Deparmen of Mahemaics, Norh Wes Universiy, Kano, Nigeria rigaawa@yahoo.com Absrac In his paper we presen availabiliy and mean ime o failure esimaion of a sysem where he deerioraion raes follow he Weibull disribuion. The paper presens modeling and evaluaion of availabiliy and mean ime o sysem failure (MTSF) of a consecuive hree sage deerioraing sysem. The sysem has hree possible modes: working wih full capaciy, deerioraion and failure mode. The hree sages of deerioraion are minor, medium and major deerioraions. Minor and major mainenance are allowed a minor and medium deerioraion saes and replacemen a sysem failure. Explici expressions for he availabiliy and mean ime o failure of he sysem are obained analyically. Graphs have been ploed o deermine he behavior of availabiliy and mean ime o sysem failure wih respec o ime for differen values of deerioraion, mainenance and replacemen raes. Also, high values of he shape parameer decreases mean ime o sysem failure and availabiliy. The sysem is analyzed using differenial difference equaions. Keywords: lis, keywords, ener, here I. Inroducion In pracical engineering applicaions, mos repairable sysems are deerioraive ha sysem failure ofen canno be as good as new, i is more reasonable for hese deerioraing repairable sysems o assume ha he successive working imes of he sysem afer repair will become shorer and shorer while he consecuive repair imes of he sysem afer failure will become longer and longer. Mos of hese sysems are subjeced o random deerioraion which can resul in unexpeced failures and disasrous effec on he sysem availabiliy and he prospec of he economy. Therefore i is imporan o find a way o slow down he deerioraion rae, and o prolong he equipmen s life span. Mainenance policies are vial in he analysis of deerioraion and deerioraing sysems as hey help in improving reliabiliy and availabiliy of he sysems. Mainenance models can assume minor mainenance, major mainenance before sysem failure, perfec repair (as good as new), minimal repair (as bad as old), imperfec repair and replacemen a sysem failure. Several models on deerioraing sysems under differen condiions have been sudied by several researchers such as Bérenguer (2008), Frangopol e al (2004), Lam and Zhang (200), Nicolai e al (2007), Rani and Sukumari (204), Vinayak and Dharmaraja (202), Yuan e al (202), Yuan and Xu (20). Analysis of reliabiliy and availabiliy model for deerioraing sysem have been sudied under differen condiions such as Liu e al. (20) who invesigaed reliabiliy analysis of a deerioraing sysem wih delayed vacaion of repairman, Tuan e al (20).A Reliabiliy-based Opporunisic Predicive Mainenance Model for k-ou-of-n Deerioraing Sysems, Xiao e al (20) proposed he Bayesian reliabiliy esimaion for deerioraing sysems wih limied samples using he maximum enropy approach, Yusuf e al (202). Presens modelling he reliabiliy and availabiliy characerisics of a sysem wih hree sages of deerioraion, Zhang and Wang (2007) 0
Yusuf I., Gaawa R.I. Volume, December 206 deal wih he sudy of deerioraing cold sandby repairable sysem wih prioriy in use. This paper considers a sysem wih hree consecuive sages of deerioraion before failure and derived is corresponding mahemaical models. Furhermore, we sudy mean ime o sysem failure and availabiliy using differenial difference equaion mehod. The focus of our analysis is primarily o capure he effec of minor mainenance, minor deerioraion and shape parameer on mean ime o sysem failure and availabiliy. The organizaion of he paper is as follows. Secion 2 conains a descripion of he sysem under sudy. Secion presens formulaions of he models. The resuls of our numerical simulaions are presened and discussed in secion 4. Finally, we make some concluding remarks in Secion 5. II. Descripion and Saes of he Sysem In his paper, one uni sysem is considered. I is assumed ha he sysem mos pass hrough hree consecuive sages of deerioraion which are minor, medium and major deerioraion before failure. The uni is considered o be non repairable. A early sae of he sysem life, he operaing uni is exposed o minor deerioraion wih rae and his deerioraion is recified hrough minor mainenance which rever he uni o is earlies posiion before deerioraion. If no mainained, he uni is allowed o coninue operaing under he condiion of minor deerioraion which laer changes o medium deerioraion wih rae 2. A his sage, he srengh of he uni is sill srong ha i can recify o early sae wih rae 2. However, he sysem can move o major deerioraion sage wih rae where he srengh of he uni has decreases o he exen ha i canno be revered o is early sae, neiher ha of minor nor medium deerioraion sages. Here he uni is allowed o coninue operaion unil i fails wih parameer and he sysem is 4 immediaely replaced by wih a new one wih rae. Deerioraing raes follow Weibull disribuion f, k,2,, is he shape parameer. k 2 0 2 2 4 4 Figure : Transiion diagram of he sysem Table : Saes of he sysem Sae Descripion S0 Iniial sae, he sysem is operaive. S The sysem is in minor deerioraion mode and is under online minor mainenance, and is operaive. S2 The sysem is in medium deerioraion mode and is under online major mainenance and is operaive. S The sysem is in major deerioraion mode and is operaive. S4 The sysem is inoperaive.
Yusuf I., Gaawa R.I. Volume, December 206 III. Formulaion of he Models In order o analyze he sysem availabiliy of he sysem, we define P i () o be he probabiliy ha he sysem a 0 is in sae S i. Also le P () be he row vecor of hese probabiliies a ime. The iniial condiion for his problem is: P(0) [ p 0, p 0, p 0, p 0, p (0)],0,0,0,0 0 2 4 We obain he following differenial difference equaions from Figure : 0 p 0( ) p ( ) 2 p 2( ) p 4( ) d 2 p ( ) p 0( ) d 2 2 p 2( ) 2 p ( ) d 4 p ( ) p 2( ) d 4 p 4( ) 4 p ( ) () d This can be wrien in he marix form as P TP, (2) where 2 0 2 0 0 0 T 0 2 2 0 0 0 0 4 0 0 0 0 4 The seady-sae availabiliy (he proporion of ime he sysem is in a funcioning condiion or equivalenly, he sum of he probabiliies of operaional saes) is given by N AV ( ) p0 p p2 p D () where 2 2 N 4 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 4 2 2 4 2 4 4 D 2 4 4 2 2 4 2 4 2 Following Trivedi (2002), Wang and Kuo (2000), Wang e al. (2006) o develop he explici for MTSF. The procedures require deleing rows and columns of absorbing saes of marix T and ake he ranspose o produce a new marix, say M. The expeced ime o reach an absorbing sae is obained from 2
Mean ime o sysem failure Yusuf I., Gaawa R.I. Volume, December 206 E T 0 P0 P P absorbing M where he iniial condiions are given by P(0) [ p 0, p 0, p 0, p 0 ],0,0,0 and 0 2 0 0 2 2 0 M 0 0 0 0 4 The explici expression for is given by MTSF 2 2 (4) 2 2 4 2 2 4 2 2 4 2 MTSF 24 (5) IV. Discussion Numerical examples are presened o demonsrae he impac of repair and failure raes on seadysae availabiliy and ne profi of he sysem based on given values of he parameers. For he purpose of numerical example, he following ses of parameer values are used: 0., 0., 2 0.5, 0.4, 2 0.5, 0.6, 4 0.2, 0.9, 0 0 0 25 =0. 20 =0.4 =0.5 5 0 0 2 4 5 6 7 8 9 0 Figure 2: Mean ime o sysem failure agains for differen values of (0.,0.4,0.5)
Mean ime o sysem failure Availabiliy Yusuf I., Gaawa R.I. Volume, December 206 0.94 0.92 0.9 =0. =0.4 =0.5 0.88 0.86 0 2 4 5 6 7 8 9 0 Figure : Availabiliy agains for differen values of (0.,0.4,0.5) 25 20 =0.4 =0.5 =0.6 5 0 5 0 2 4 5 6 7 8 9 0 Figure 4: Mean ime o sysem failure agains for differen values of (0.4,0.5,0.6) 4
Mean ime o sysem failure Availabiliy Yusuf I., Gaawa R.I. Volume, December 206 0.95 =0.4 =0.5 =0.6 0.9 0.85 0.8 0.75 0 2 4 5 6 7 8 9 0 Figure 5: Availabiliy agains ime for differen values of (0.4,0.5,0.6) 40 0 =.2 =. =.4 20 0 0 0 2 4 5 6 7 8 9 0 Figure 6: Mean ime o sysem failure agains for differen values of (0.4,0.5,0.6) 5
Availabiliy Yusuf I., Gaawa R.I. Volume, December 206 0.95 0.9 =.2 =. =.4 0.85 0.8 0.75 0.7 0.65 0 2 4 5 6 7 8 9 0 Figure 7: Availabiliy agains ime for differen values of (.2,.,.4) Numerical resuls of mean ime o sysem failure and availabiliy wih respec o ime are depiced in Figures 2 and for differen values of minor mainenance raes. In hese Figure he mean ime o sysem failure and availabiliy increases as ime increases when increases from 0. o 0.5. This sensiiviy analysis implies ha minor mainenance o he sysem should be invoked o increase he life span of he sysem. On he oher hand, simulaions in Figures 4 and 5, depics he impac of ime on mean ime o sysem failure and availabiliy for differen values of minor deerioraion. From hese Figures, he mean ime o sysem failure and availabiliy decreases as ime increases for differen values of. The above sensiiviy analysis depiced he effec of minor mainenance and deerioraion raes on mean ime o sysem failure and availabiliy. I can be observe ha minor mainenance played a significan role in increasing he mean ime o sysem failure and availabiliy whereas minor deerioraion slow down he mean ime o sysem failure. From simulaions depiced in Figures 6 and 7, i is eviden ha he choice of he shape parameer influences he ime aken for he sysem o reach he failure sae. The higher he value of his shape parameer, he less he values of mean ime o sysem failure and availabiliy. V. Conclusion This paper sudied a one uni sysem wih hree consecuive sages of deerioraion before failure. Explici expressions for he mean ime o sysem failure and availabiliy are derived. The numerical simulaions presened in Figures 2 7 provide a descripion of he effec of mainenance and deerioraion raes on mean ime o sysem failure. On he basis of he numerical resuls obained for paricular cases, i is suggesed ha he sysem availabiliy can be improved significanly by: Adding more cold sandby unis. Increasing he mainenance rae. Exchange he sysem a major deerioraion wih new one before failure. 6
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