Evaluation of Reliability and Availability Characteristics of 2-out of -3 Standby System under a Perfect Repair Condition

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1 Americn Journl of Mthemtics nd ttistics, (5): -9 DOI:.59/j.jms.5. Evlution of Reliility nd Avilility Chrcteristics of -out of - tndy ystem under erfect Repir Condition Irhim Yusuf *, Nfiu Hussini Deprtment of Mthemticl ciences, Fculty of cience, Byero University, Kno Astrct Mny uthors hve studied the effectiveness of redundnt system under two or three types of filure under the ssumption tht such filures re repirle. Little ttention is pid on whether such repir ction cn restore the system operting condition to s good s new (perfect repir) nd the effect of such perfect repir on the system performnce. In this study, vrious mesures of system effectiveness such s men time to system filure (MTF), stedy stte vilility, usy period nd profit function of -out-of- repirle system with perfect repir re nlyzed using Kolmogorov s forwrd eqution method. ome prticulr cses hve een discussed grphiclly. The results hve indicted tht perfect repir ction plys vitl role on system performnce. imultions results show tht perfect repir is importnt prticulrly in incresing men time to system filure, vilility nd system performnce s whole. Keywords Men Time To ystem Filure (MTF), Avilility, rofit, Redundncy, erfect Repir, Kolmogorov s Eqution Method, -Out-Of- tndy ystem. Introduction Redundncy is technique used to improve system reliility nd vilility. Reliility is vitl for proper utiliztion nd mintennce of ny system. It involves technique for incres ing system effectiveness through reducing filure frequency nd mintennce costminimizti on. One of the forms of redundncy is the k-out-of-n system which finds wide ppliction in industril system. Redundncy, repir ction (perfect), nd preventive mintennce re some of the wys y which the reliility of system cn e improved. K-out-of-n systems re often encountered industril pplictions. Electronics industry, telecommuniction network systems, power genertor nd trnsmission systems re the common exmples ofk-out-ofn systems. In generl, there re three types in stndy, i.e. cold, hot nd wrm stndy. A lot of ppers deling with the reliility nd vilility of k-out-of-n systems (stndy systems) hve een pulished.[] discussed cost nlysis of k-out of-n: G system with ctive stndy redundncy components nd repir fcility,[] studied the cost nlysis of two unit cold stndy redundnt system with two types of unit filure involving common cuse filure,[] dels with * Corresponding uthor: irhimyusif@yhoo.com (Irhim Yusuf) ulished online t Copyright cientific & Acdemic ulishing. All Rights Reserved cost nlysis of two unit cold stndy redundnt system with two types of filure nd preventive mintennce,[] studied profit nlysis of two unit cold stndy system with preventive mintennce,[6] dels with stochstic nlysis of redundnt system involving common humn filure,[5] dels with the evlution of reliility nd vil ility chrcteristics of different system.[7] studied cost nlysis of series systems with cold stndy components nd repirle service sttion, Bhtti et l[8], considered two identicl unit cold stndy systems with single repirmn. Ku mr et l[9] nlyzed cost enefit nlysis of two unit system in which units work in prllel nd ecome degrded fter repir, Gupt el l[] dels with stochstic nlysis of two non identicl unit stndy model. The system consist priority unit (p) nd ordinry unit (o). A single repir fcility ppers nd disppers from the system rndomly with constnt rtes, Mlik et l[] nly zed two reli ility models for system of non identicl units originl nd duplicte using regenertive point technique., Mhmoud nd Moshref[] del with the study of the stochstic nlysis of two unit cold stndy system considering hrdwre filure, humn error filure nd preventive mintennce, Yusuf nd Bl[], studied stochstic two models of two unit prllel system. In model I, the system cn e norml, deteriortion (slow, mild or fst deteriortion), filure wheres in model II, the system cn either e in norml of filure modes. Using liner first order liner d ifferentil equtions, vrious mesures of system effectiveness such s men time to system filure (MTF) nd vilility re

2 5 Americn Journl of Mthemtics nd ttistics, (5): -9 otined to see the effect of deteriortion on such mesures, Kumr nd Kdyn[] del with profit nlysis of two unit non identicl system with degrdtion nd replcement while ureri et l[5] studied cost enefit nlysis of computer system with priority to softwre replcement over hrdwre repir... Ojectives Mny uthors studied redundnt system exposes to different filure types. However, such systems re suject to different types of repirs like s d s old, imperfect repir, worse repir, worst repir mening tht in course of time their condition flls from higher to lower, nd susequently fil. However some repirs (perfect) cn restore the system operting condition to s good s new The question rise here is whether such perfect repirs cn ffect some mesures of system effectiveness like men time to system filure (MTF) nd system vilility. The purpose of the present pper is to discuss reliility nd vilility evlution of -out-of- system under perfect repir condition. The system is filed if two of its units filed. The system is nlyzed using Kolmogorov s equtions method. Grphs re plotted to highlight importnt results. Nottions Constnt type i filure rte i repir rte F Ri Filed unit under type i repir witing for type i repir i =,, i Constnt type i F Wi Filed unit Unit in tndy O Unit in opertion Assumptions. The system consist of components/units. Initilly two units re in operle condition of full cpcity. The system is filed when the numer of working component goes down elow. Filure nd repir time follow exponentil distriution 5. Repir is s good s new(erfect repir).. Method tte of the ystem Fig. chemtic digrm of the system ( O, O, O ), ( FR, O, O ), ( O, FR, O ), ( FR, O, F W), R ( O, F, F ) 6 R W ( O, O, F ), 5( FR, FW, O ),.. Men Ti me to ystem Filure Anlysis From Fig. ove, define t i () to e the proility tht the system t time t, ( t ) is in stte i. Let t () e the proility row vector t time t, the initil condition for this pper re 6 Figure. chemtic digrm of the system = ( ) ( ) ( ) ( ) ( ) 5( ) 6( ) = [,,,,,, ] we otin the following differentil equtions: () [,,,,,, ] d =- ( + + ) d =- ( + + ) t () + t () + t () + t 5() d =- ( + + ) () t d =- ( + ) t () + t () + t () d =- ( + ) + + () t d5 =- ( + ) t 5() + t () + t () d5 =- ( + ) t 5() + t () + t () d6 =- 6 + () t () the system of differentil equtions cn e written in mtrix s form s = A Where 5

3 Irhim Yusuf et l.: Evlution of Reliility nd Avilility Chrcteristics of -out of - 6 tndy ystem under erfect Repir Condition Ø- ( + + ) ø ( ) -( + + ) A = -( + ) -( + ) -( + ) - º It is difficult to evlute the trnsient solutions hence following[,,7] we delete the rows nd columns of soring stte of mtrix A nd tke the trnspose to produce new mtrix, sy Q. The expected time to rech n soring stte is otined from Øø MTF = ()( -Q - ) º Where Ø- ( + + ) ø -( + + ) Q = - ( + + ) º -( + ) The stedy stte men time to system filure is: N MTF = D N = ( + + )( + + )( + ) + ( + + )( + ) + ( + + )( + ) + ( + + )( + + ) D = Avilility Anlysis For the nlysis of vilility cse of Fig. using the sme initil conditions for this prolem s º 6 = ( ) ( ) ( ) ( ) ( ) ( ) ( ) = [,,,,,, ] () [,,,,,, ] 5 6 The differentil equtions cn e expressed s Ø ø Ø- ( + + ) øøø ( ) -( + + ) = -( + ) -( + ) ( ) º - º 6 5 The stedy-stte vilility is given y ()

4 7 Americn Journl of Mthemtics nd ttistics, (5): -9 A( ) = ( ) + ( ) + ( ) + ( ) () In the stedy stte, the derivtives of the stte proilities ecome zero so tht A = () which in mtrix form Ø- ( + + ) øøø Øø ( ) -( + + ) -( + ) = -( + ) -( + ) 5 - º º 6 º Using the following normlizing condition ( ) ( ) ( ) ( ) ( ) ( ) + ( ) = (5) 5 6 We sustitute (5) in ny of the redundnt rows in () to give Ø- ( + + ) øø ( ) ø Øø -( + + ) ( ) - ( + + ) ( ) -( + ) ( ) = -( + ) ( ) -( + ) 5 ( ) 6 ( ) º º º We solve for the system of equtions in the mtrix ove to otin the stedy-stte proilit ies The stedy-stte vilility is given y.. Busy eriod Anlysis ( ), ( ), ( ), ( ), ( ) A( ) = Using the sme initil condition s for the reliility cse: º 6 = ( ) ( ) ( ) ( ) ( ) ( ) ( ) = [,,,,,, ] () [,,,,,, ] 5 6 The differentil equtions cn e expressed s Ø ø Ø- ( + + ) øøø ( ) -( + + ) = ( ) - + -( + ) -( + ) º - º In the stedy stte, the derivtives of the stte proilities ecome zero nd this will enle us to compute stedy stte usy :

5 Irhim Yusuf et l.: Evlution of Reliility nd Avilility Chrcteristics of -out of - 8 tndy ystem under erfect Repir Condition nd B( ) = - ( ) (6) A =, which in mtrix form Ø- ( + + ) øøø Øø ( ) -( + + ) -( + ) = -( + ) -( + ) 5 - º º 6 º We solve for ( ) Using the following normlizing condition ( ) ( ) ( ) ( ) ( ) ( ) + ( ) = (7) 5 6 We sustitute (7) in ny of the redundnt rows in () to give Ø- ( + + ) øø ( ) ø Øø -( + + ) ( ) - ( + + ) ( ) -( + ) ( ) = -( + ) ( ) -( + ) 5 ( ) 6 ( ) º º º The stedy stte usy period B ( ) is therefore B ( ) = rofit nlysis Following[,]the expected profit per unit time incurred to the system in the stedy-stte is given y: rofit =totl revenue from system using - totl cost due to repir F = RA( )- CB( ) (8) Where F: is the profit incurred to the system R: is the revenue per unit up time of the system C : is the cost per unit time which the system is under repir. Results Fig. : shows reltion etween type I repir rte nd MTF of the system Fig. : shows reltion etween type I repir rte nd vilility of the system Fig. : shows the reltion etween type I repir rte nd profit function of the system Tle : shows the reltion etween type I repir rte nd the profit of the system.. Discussion nd Conclusions.. Discussion For the study of system ehvior, we plot grphs in Fig. to for MTF, system vilility nd profit function with respect to while other prmeters re kept fixed s =.5, =., =.5, =.6, =., R =, C =, From Fig. it is cler tht MTF increses with increse in the vlue of wh ich reflects the effect of perfect repir on MTF. From Fig. the system vilility increses with increse in the vlue of. Thus, the more the repir is perfect, the more the system is ville for opertion. In Fig. it cn e oserved tht with increse in the vlue of the profit lso increse. Thus, the more the system is in perfect condition the more it is effective. me rgument cn lso e oserve in tle where increse in the vlue of led to the increse in profit function.

6 9 Americn Journl of Mthemtics nd ttistics, (5): -9 T le. Relt ion et ween t ype I repir rt e nd the profit rofit Avlility Figure. effect of type I repir rte on system vilility rofit Figure. effect of type I repir rte on profit function.. Conclusions Effect of on Avlility Effect of on rofit In this study, we developed the explicit expressions for MTF, system vilility, usy period nd profit function for -out-of- system sujected to perfect repir. Grphs were plotted to show the effect of perfect repir on importnt mesures of system effectiveness. imultions results show tht perfect repir is importnt prticulrly in incresing men time to system filure, vilility, nd profit function nd system performnce s whole. REFERENCE [] El-id, K.M.,. ( 8).Cost nlysis of system with preventive mintennce y using Kolmogorov s forwrd equtions method. Ame. J. of App. ci. 5(), 5- [] Hggg, M.Y., (9). Cost nlysis of system involving common cuse filures nd preventive mintennce.j. Mths. And tt. 5(), 5- [] Hggg, M.Y., (9). Cost nlysis of k-out-of-n repirle system with dependent filure nd stnd y support using kolmogorov s forwrd equtions method. J. Mths. And tt. 5(), -7 [] El-id, K.M., nd M.. El-hereny. ( 5).rofit nlysis of two unit cold stndy system with preventive mintennce nd rndom chnge in units. J. Mths nd tt., () 7-77 [5] El-id, K.M., nd M.. El-hereny. (5).Evlution of reliility nd vilility chrcteristics of two different systems y using liner first order differentil equtions. J. Mths nd tt., () 9- [6] Gupt., R. nd Mittl., M. (6). tochstic nlysis of compound redundnt system involving humn filure. J. Mths nd tt. (), pp 7-. [7] Wng, K., Hseih, C., nd Liou, C. H. (6). Cost enefit nlysis of series systems with cold stndy components nd repirle service sttion. Qulity technology nd quntittive mngement. Vol. (), pp [8] Bhtti, J., Chitkr, A., nd Bhrdwj, N., (). rofit nlysis of two unit cold stndy system with two types of filures under inspection policy nd discreet distriution. Interntionl Journl of science nd engineering reserch, Vol. (), pp -7. [9] Kumr, Jitender., Kdyn, M.. nd Mlik,.C., (). Cost enefit nlysis of two unit prllel system suject to degrdtion fter repir. Applied mthemticl sciences, (56), pp [] Gupt, R., Goel, C.K. nd Tomer, A. (). Anlysis of two unit stndy system with correlted filure nd repir nd rndom ppernce nd disppernce of repirmn. [] Mlik,.C., Bhrwj, R.K. nd Grewl, A.. (). roilistic nlysis of system of two non identicl prllel units with priority to repir to repir suject to inspection. Journl of reliility nd sttisticl studies, vol. (), pp - [] Mhmoud, M.A.W. nd Moshref, M.E. (). On two unit cold stndy system considering hrdwre, humn error filures nd preventive mintennce, Mthemtics nd Computer modeling, 5(5-6), pp [] Yusuf, I. nd Bl,.I. (). tochstic modeling of two unit prllel system under two types of filures. Interntionl Journl of Ltest trends in Mthemtics, Vol. (), pp -5 [] Kumr, J. nd Kdyn, M.. (). rofit nlysis of system of non identicl units with degrdtion nd replcement. Interntionl journl of computer ppliction, vol. (), pp 9-5 [5] ureri, J.K., Mlik,.C. nd Annd, J. (). Cost enefit nlysis of computer system with priority to softwre replcement over hrdwre repir. Applied Mthemticl ciences, vol. 6 (75), pp 7-7.