Evaluating Rliability Systms by Using Wibull & Nw Wibull Extnsion Distributions Mushtak A.K. Shikr مشتاق عبذ الغني شخير Univrsity of Babylon, Collg of Education (Ibn Hayan), Dpt. of Mathmatics Abstract : In this papr w valuat th rliability systms by using (Wibull & Nw Wibull Extnsion) distributions, and by using both of thm w dtrmin th Rliability Function and th associatd functions such as cumulativ distribution function, hazard function tc., thn w discuss th rlationship btwn thm and how ffctivnss this to calculat th rliability function and associatd functions. الخالصة : في هزا البحث تم تييمي ةيمأل امة بمأل خاامتخذاي تويبم تببمع تتويبم تببمع البوام السذبمذ تخاامتخذاي همزبم التمويب يم تم الحصول على دالأل الثيأل تالذتال رات ال القأل كذالأل التسبي االحتباليأل تدالأل الخطر... الخ.ة ةاقشنا ال القأل خيم هزبم التويب يم تكيفيأل تأةير رلك على إبساد دالأل الثيأل تالذتال البرتبطأل خها. 1.Introduction Many rsarchrs work to comput th rliability systms by using Wibull distributions as David (2010), Tang (2004) and Xi (2003). Whn manufacturrs claim that thir products ar vry rliabl thy ssntially man that th products can function as rquird for a long priod of tim, whn usd as spcifid. In ordr to assss and improv th rliability of an itm w nd to b abl to masur it. Thus a mor formal dfinition is rquird. Th Wibull distribution (Wibull, 1951), namd aftr th Swdish Profssor Waloddi Wibull, is prhaps th most frquntly usd lif tim distribution for liftim data analysis mainly bcaus of not only its flxibility of analyzing divrs typs of phnomna, but also its simpl and straightforward mathmatical forms compard with othr distributions. Th Wibull distribution is gnralization from xponntial distribution. This distribution is appropriat for a systm or complx componnt mad up of svral parts. Nw Wibull xtnsion distribution is subsquntly introducd in this papr. This modl is rgardd as an xtnsion of Wibull distribution which has bathtub shapd failur rat function. It also contains an analysis of th proprtis of th modl. 2. Liftim Following a Wibull Distribution Whn th liftims hav a Wibull distribution thn it's p.d.f. is ] Quk S-T., Ang A. H-S.,1986[ : 1 t f ( t xp (1) Whr and ar paramtrs. Th scal paramtr,, rflcts th siz of th units in which th random variabl, t, is masurd. Th shap paramtr,, causs th shap of th distribution to vary. By changing th valu of w can gnrat widly varying st of curvs to modl ral liftim failur distributions. Th ffcts of diffrnt scal and shap paramtrs on th Wibull distribution ar shown in fig.(1,2):
Fig.(1):Effct of diffrnt shap paramtr, Spcial cas: If = 1 thn th pdf collapsing is : Fig.(2):Effct of diffrnt scal paramtr, f t ( whr 1 1 which is an xponntial distribution with rat (or man ) 2.1 Associatd functions By dpnding quation (1), w can find th following quations : (i) Cumulativ Distribution Function is givn by : t F( 1 (2) Whn = 1 F(, w gt : (ii) Rliability Function R ( (iii) hazard function 1 t f ( h( h( t 1 i. h( t t t (3) (4) 1 (iv) Man ti Btwn failur MTBF is MTBF 1 (5) whr dnots th gamma function. Tabls of gamma functions ar availabl to assist in calculating th man tim btwn failurs.
If x > 2, (x)= (x-1) (x-1) (v) cumulativ hazard function is t t H( ln ln (6) 2.2 Tim-dpndant hazard function For a Wibull liftim distribution, th hazard function is givn by 1 h ( t This will giv various forms for th hazard function dpnding on th valu of, for mor dtails s ] Tang Yong, 2004 [, Exampl (1):- Th liftim of a componnt (in thousands of hours) has a Wibull distribution with α = 0.5 and β = 2, Find th following : (i) th probability that th componnt will fail bfor 1000 hours of th opration. (ii) MTBF. Solution : t t (i) Hr 0.5 R ( = Thn 1)= -4 = 0.01 Rquird probability = 1-0.01 0.99 1 (ii) MTBF 1 = 0.5 (1+1/2) =0.5.(0.886227) = 0.443 (thousands of hours ) i.. 443 hours. 2 3. A Nw Wibull Extnsion Distribution Th rliability function of nw Wibull xtnsion is givn by ] Castrn J.V., 2001 [, ]Paul Barringr, P.E., 2000 [ t xp 1 xp (7) for any λ, α,β > 0, t 0. Th nw Wibull xtnsion is drivd from Chn s modl (for mor dtails s ]Paul Barringr, P.E.,2000 [ ), this xtnsion modl has th Wibull distribution as a spcial and asymptotic cas, and hnc it can b considrd as a Wibull xtnsion. Th corrsponding failur rat function of Wibull xtnsion modl has th following form: 1 t t h( xp (8) Th shaps of th failur rat function, which can b of bathtub shap, will b dmonstratd. For th nw Wibull xtnsion distribution, th cumulativ distribution function is givn by:
t F( 1 1 xp 1 xp (9) and th pdf is givn by: 1 t t t f ( xp 1 xp (10) 3.1 Charactristic of failur rat function To study th shap of th failur rat function, W firstly tak th drivativ of th failur rat function and w gt ] Lai C. D., Min Xi and Murthy D. N. P.,2003 [,] Tang Yong,2004 [ 2 t t t h ( xp.. 1 (11) Th shap of th failur rat function dpnds only on th shap paramtr β. Hnc, th following two cass will b considrd. Cas 1: β 1 i). In this cas, for any t > 0, h ( > 0, thrfor h( is a monotonically incrasing function. ii). h(0) = 0 if β > 1 and h(0) = λ, if β = 1. iii). h( + as t +. Cas 2: 0 < β <1 i). Lt h (b3 ) = 0, thn w hav th quation : b 1 (12) and by solving this quation, a chang point of th failur rat can b obtaind as 1 1 b (13) W can s that whn 0 < β <1, b3 xist and it is finit. Whn t < b3, h ( < 0, th failur rat function is monotonically dcrasing; whn t > b3, h ( > 0, th failur rat function is monotonically incrasing. Hnc, th failur rat function has a bathtub shap proprty. ii). h( + for t 0 and t + ; iii). Th chang point b3 incrass as th shap paramtr β dcrass from 1 to 0. Figur (3) shows th plots of th failur rat function for Wibull xtnsion modl at svral diffrnt paramtr combinations. From Figur (3), w can obsrv th failur rat function has an incrasing function whn β 1, and h( is a bathtub shapd function whn 0 < β <1. Figur (3): Plots of th failur rat function with λ = 2, α = 100 and β changing from 0.4 to 1.2
3.2 Man Tim Btwn Failur Th xpctd tim to failur of th Wibull xtnsion distribution, or th man tim btwn failur (MTBF) can b xprssd as ] Castrn J. V., 2001 [,]Constantin Tarcola, Adrian Paris, Cristian Andrscu, 2008 [ : MTBF tf ( dt 0 0 dt (14) t xp 1 xp dt 0 Exampl(2): W can solv prvious xampl by using Wibull xtnsion distribution with t = 1 and w will gt from q.(7) 1) xp 0.5 154.5 12 26.7 2.510 xp 4. Rlationship Btwn Nw Wibull Extnsion Distribution and Wibull Distribution Th nw modl is rlatd to Wibull distribution in an intrsting way ] C. D. Lai, Min Xi and Murthy D. N. P., 2003 [,]Constantin Tarcola, Adrian Paris, Cristian Andrscu, 2008 [, ] Polpo A., Coqu M. A., Prira CAB., 2009 [ Wibull distribution can b sn as an asymptotic cas of th nw distribution. Whn α is too larg thn : t t t 1 xp 1 1 0t (15) Thrfor, th rliability function can b approximatd by t 1 xp 1 xp t (16) which is a standard two-paramtr Wibull distribution with a shap paramtr of β, and a scal paramtr of α β 1 /λ. That is, in th limiting cas whn α approachs infinity whil α β 1 /λ rmains constant, th nw distribution rmains a standard two paramtr Wibull distribution. In this limiting cas, th Wibull xtnsion modl is capabl of handling both incrasing and dcrasing failur rats, which ar in fact, spcial cass of bathtub curv. A furthr spcial cas is, whn β = 1, α is larg nough, w hav t R t ( ) xp 1 xp t Hnc, th modl rducs to th xponntial distribution with paramtr λ. It is wll-known that th xponntial distribution has a constant failur rat, which is again, a vry spcial cas of bathtub curv ] Tang Yong, 2004 [. 5. Conclusions On of th good proprtis of Wibull distribution is that it can hav diffrnt monotonic typs of hazard rat shaps so that it can b applid to diffrnt kinds of products. From Equation (4), it is clar that th shap of hazard rat function dpnds solly on th shap paramtr. Thr ar som rlationships btwn Wibull distribution and othr distributions. For xampl whn β =1, it is rducd to xponntial distribution. Whn β =2, it has th form of Rayligh distribution. Whn β >3.6, Wibull distribution is vry similar to normal distribution and Whn th scal paramtr α bcoms vry larg or approachs infinity Wibull distribution can b sn as an asymptotic cas of th nw Extnsion Wibull distribution.
Th Rliability Systm by Using Wibull distribution is largs than th nw Extnsion Wibull distribution as in xampls (1)& (2) rspctivly. Rfrncs 1. Castrn J. V., 2001 "Powr systm rliability assssmnt using th Wibull-Markov modl", A Thsis, Chalmrs Univrsity Of Tchnology, Götborg, Swdn. 2. Constantin Tarcola, Adrian Paris and Cristian Andrscu, 2008, "Comparison of Rliability Modls", PSG procding and Dynamical Systms(DGDS-2008) and th V-th, Colloq. of mathmatics in Enginring and numrical physics (MENP-5)-math,sctions, August 29-sptmbr 2,, magnolia, Romania Pp.150-155. www.mathm.pub.ro/proc/bsgp-16/p16-ta.pdg 3. Lai C. D., Xi Min and Murthy D. N. P., 2003, "A Modifid Wibull Distribution", IEEE Transactions On Rliability, Vol. 52, No. 1, (33-37). www.muir_massy.ac.nz/bitstram/ 4. Paul Barringr, P.E., 2000, "Rliability Enginring Principls", Barringr & Associats, Inc., USA. www.barringr.com/pdf/rep-brochur.pdf 5. Polpo A., Coqu M. A. and Prira CAB., 2009,"Wibull Componnts: sris & Paralll Systms",50th Annivrsary clbration, th Florida stat univrsity,. www.fsu.du/50th/prira.pdf 6. Quk S-T. and Ang A. H-S., 1986 "Structural Systm Rliability by Th Mthod of Stabl Configuration ", National Scinc Foundation, Univrsity of Illinois, ISSN: 0069-4274, NO. 529, Washington. 7. Tang Yong, 2004," Extndd Wibull Distributions In Rliability Enginring", A thsis, National Univrsity Of Singapor. 8. Trindad, David C., 2010," Applid Rliability Tchniqus For Rliability Analysis With Applid Rliability Tools", A thsis, Santa Clara Univrsity. 9. Wibull W., 1951, "A statistical distribution function of wid applicability, J.Appl. Mch-Trans, ASME 18 (3):239-297.