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214 23rd International Conference on Computers and Industrial EngineeringT h e A u t o r e g r e s s i v e I n t e g r a t e d M o v i n g A v e r a g e m o d e l s ( A R I M A ) , p i o n e e r e d b y B o x -

J e n k i n s ( B o x e t a l . , 1994) , ea r l be a su i tab le a l t e rna t ive in mode l ing the fa i lu re pa t te rn . Byi te ra t ive ly ad jus t ing the we igh ts in th i s t ime se r ie s mode l , au toco r re la t ion be tween the fa i lu reda ta can be exp lo red and be t te r e s t ima te s can be ob ta ined . In th i s pape r , we examine thea p p l i c a ti o n o f ti m e s e r ie s m o d e l i n g i n r ep a i r a b le s y s t e m a n a l y s is . A c o m p a r a t i v e s t u d yb e t w e e n D u a n e a n d A R I M A m o d e l s f o r f o r e c a s t i n g f a i l u r e s i s d i s c u s s e d , w i t h e m p h a s i s o nt h e i r p r e d i c t iv e p e r f o r m a n c e . A n e x a m p l e o n a n a l y s i n g t h e f a i lu r e d a t a o f a m e c h a n i c a ls y s t e m i s p r e se n t e d .

A P P R O A C H E S T O F A I L U R E D A T A A N A LY S IST h e D u a n e M o d e lT h e D u a n e m o d e l , a l s o k n o w n a s t h e p o w e r l a w m o d e l , b e l o n g s t o t h e f a m i l y o fn o n h o m o g e n e o u s P o i s s o n p r o c e s s ( N H P P ) . T h i s i s t h e m o s t c o m m o n l y u s e d s t o c h a s t i cp r o c e s s f o r m o d e l i n g r e l i a b il i t y g r o w t h . T h e p r i m a r y r e a s o n o f it s p o p u l a r i ty o r i g i n a t e s f r o mthe ea r ly s tud ie s r epo r ted by J . T . Duane (Duane , 1964) . He pos tu la ted tha t the p lo t o fc u m u l a t i v e n u m b e r o f f a i lu r e s N ( t ) ve rsus the cum ula t ive ope ra t ing hou rs t on a log - log sca lecan be f i t t ed by a s t r a igh t l ine i f the mo de l i s va l id . The g rad ien t b g ives the ind ica t ion o f theg r o w t h r a te . T h i s m o d e l , b e i n g e m p i r i c a l, p r o v i d e s a r e a s o n a b l y s i m p l e m e t h o d f o r r e l ia b i l i t yg row th fo recas t ing o f r epa i rab le sy s tem s . M ore s ig n i f ican t ly , it s s tr eng th l i e s in it s g raph ica la p p r o a c h a n d e a s y i n t e r p r e t a t i o n ( X i e a n d Z h a o , 1 9 9 6 ) . T h e m e a n v a l u e f u n c t i o n h a s t h el inea r fo rm

I n m ( t ) = I n a + b I n t , w h e r e a _> O , b _> O .S u c h g r a p h i c a l t e c h n i q u e p r o v i d e s a s i m p l e w a y f o r m o d e l v a l i d a t i o n a n d p a r a m e t e re s t im a t i o n . V a l u e s o f b < 1 i n d i c a te s t h a t t h e s y s t e m i s e x h i b i ti n g r e l ia b i l i ty g r o w t h t r e n d ,wh ereas b > 1 show s tha t the sys tem i s de te r io ra t ing .T h e A R I M A M o d elT h e B o x - J e n k i n s A u t o r e g r e s s i v e I n t e g r a t e d M o v i n g A v e r a g e ( A R I M A ) m e t h o d o l o g y i s w e l le s tab l i shed in the s ta t i s t i ca l li t e ra tu re . How ever , i t s u se in the re l i ab i l i ty f ie ld i s r a the r l imi ted .O n l y i n r e c e n t y e a r s d i d w e s e e m o r e w o r k b e e n r e p o r te d , i n p a r t i cu l a r o n u s i n g t i m e s e r i e sm o d e l s f o r s o f t w a r e r e l i a b i l it y g ro w t h ( s e e f o r e x a m p l e S i n g p u r w a l l a , 1 9 80 , C h a t t er j e e , e t a l . ,1997) . An i t e ra t ive th ree - s tage p rocess , i . e . th rough mode l iden t i f i ca t ion , pa rame te re s t i m a t i o n a n d d i a g n o s t i c c h e c k i s r e q u ir e d t o d e t e r m i n e t h e a d e q u a c y o f t h e p r o p o s e d m o d e l .A n e x c e l l e n t r e f e r e n c e c a n b e f o u n d i n B o w e r m a n a n d O ' C o n n e l l ( 1 9 9 3 ).L e t { X t . k , k = O , 1 , 2 , 3 . . . . . . . . . . . p } b e t h e s ta t iona ry t ime se r ie s unde r s tudy . In r e l i ab i l i tya n a l y s i s o f r e p a i ra b l e s y s t e m s , X t r e p re s e n ts t h e t i m e b e t w e e n f a i l u r es o r th e n u m b e r o ff a i lu r e s p e r t i m e i n te r v a l. I t c a n b e e x p r e ss e d a s A R I M A ( p , q ), w h i c h i s a l i n e a r c o m b i n a t i o no f pas t va lues o f Xt and e r ro r s 6 t

x , = 7 " 0 + 4 x ,.i + 4 2 x , . 2 + . . . . . . + 4 p x t . p - O i 6 , - i - 0 2 6, . - . . . . . . - O q 6 t . q + e ,

w h e r e p a n d q a r e t h e o r d e r o f t h e a u t o r e g re s s i v e m o d e l a n d m o v i n g a v e r a g e m o d e lrespe ct ive ly , {41. 42........ 4p, Or, 02........ O q } a r e th e regres s ion we igh ts to be e s t ima ted , ~ '0r e p r e s en t s t h e t r e n d c o m p o n e n t , a n d {6t, st . t , ct .2 . . .. . .. . 6 t .q} a re the random e r ro r s .

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2 3 r d I n t e r n a t i o n a l C o n f e r e n c e o n C o m p u t e r s a n d I n d u s t r i a l En g i n e e r i n g 2 1 5The ARIMA model ing i s essen t ia l ly an exp lora to ry da ta -or ien ted approach tha t has thef lex ib i l i ty o f f i t t ing an appropr ia te mode l which i s adap ted f rom the s t ruc tu re o f the da tai t se l f . Wi th the a id o f au tocor re la t ion func t ion and par t i a l au tocor re la t ion func t ion , thes tochas t ic na tu re o f the t ime se r ies can be approx imate ly modeled ; f rom which in format ionsuch as t rend , random var ia t ions , pe r iod ic component , cyc l ic pa t te rns and se r ia l cor re la t ioncan be d i scovered . As a resu l t, fo recas ts o f the fu tu re va lues o f the se r ies, wi th so me degreeof accurac y , can be read i ly ob ta ined .In repa i rab le sys tem s ana lys i s , i t i s r easonab le to assum e tha t the cu r ren t in te r fa i lu re t ime Xti s re la ted to the m os t recen t Xt .l . o r ev en ex tend in to the pas t va lues Xt.2...... X t.p. Th is isbecause the nex t in te r fa i lu re t ime i s dependen t on the cur ren t l eve l o f repa i r done and to acer ta in ex ten t , the p rev ious con t inued eng ineer ing e f fo r t s. T here fore , the bas ic assum pt ion o findependence o f the res idua ls i s v io la ted due to cor re la t ion and the use o f the t rad i t iona lNHPP model i s ques t ionab le . The t ime se r ies t echn ique imposes no res t r i c t ive assumpt ionson the fa i lu re p rocess and has the capab i l i ty o f exp lo i t ing th i s in te rdepen dency and ide n t i fythe t rend component , i . e . r e l i ab i l i ty g rowth , and thus p roduces super io r fo recas t s . Bes ides ,the re i s no need to a rb i t ra r i ly assume a mode l p r io r to da ta ana lys i s . In fac t , the mode l i si t e ra t ive ly se lec ted based on the under ly ing da ta s t ruc tu re .

A N IL L U S T R A T I V E E X A M P L EIn genera l , two c lasses o f fa i lu re in format ion a re encounte red ; a se r ies o f spec i f ic t imes a tw h i c h f a il u r es o c c u r ( u n g r o u p e d d a t a ) a n d c o u n t s o f t h e n u m b e r o f f a il u re s r e c o r d e d i n s o m eordered t ime in te rva l , e .g . hours , days (g rouped da ta ) . As the l a t t e r i s more f requen t lyenco unte red in the f i e ld wh ere the ac tua l fa i lu re t ime i s no t ava i lab le , we presen t an ex am pleo f a g r o u p e d f a il u re d a t a o f a m e c h a n i c a l sy s te m . C o m p a r i s o n is m a d e t o b o t h A R I M A a n dDuane models . The da ta se t and the f i t t ed models a re dep ic ted in Tab le 1 ; a l l da ta a re readf rom le f t to r igh t. A sum ma ry o f the p red ic t ive e r ro rs , based on the m ean abso lu te dev ia t ion( M A D ) , f r o m b o t h m o d e l s i s g i v e n i n T a b l e 2.

Tab le 1 . Fa i lu re da ta o f a mech an ica l sys tem.N um be r of fa i lures per interval : 3 , 2 , 9 , 17, 16, 14, 12, 26, 21, 15A R I M A m o d e l: X t = 2 X t . l - X t . 2 + 6 t - 0 . 9 9 6 t . iD u a n e m o d e l : m ( t ) = 0 . 8 1 t 1 8 , R 2 = 0 . 9 7 8

Table 2 . P red ic t ive per fo rm ance com par i son .Fa i lu re Actua l fa i lu res Pred ic ted Abso lu te e r ro r Abso lu te e r ro ri n te r v a l p e r i n te r v a l ( A R I M A ) ( A R I M A ) ( D u a n e )

11 15 14.1 0.9 18.912 18 13.1 4.9 29 .613 15 12.2 2.8 45 .214 7 11.3 4.3 70.815 3 10.4 7.4 102.3

"M A D " 4 .1 5 3 .4

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216 23rd International Conference on Computers and Industrial EngineeringFrom the resul t s in Tables 1 and 2 , s ince the under ly ing s tructure of da ta se t 1 has chan ged,Duan e m ode l i s n o t capab le o f de t ec t ing th i s change and henc e r e su lt ed i n poor fo recast s. T heassumpt ion of extending the pas t fa i lure pa t tern in to the fu ture i s no longer va l id . On theother hand, A R IM A m ode l provides a be t ter f it to the da ta w i th re la tive ly low absolute errors .

In our inves t iga t ion , a l though we have examined only a se t of grouped fa i lure da ta , s imi lars tudies condu cted on u ngro uped fa i lure da ta ; i .e . in the fo rm of in ter fa ilure tim e o f fa i lureeven t s , have show n tha t the p red i ct i ve pe r fo rmance f rom A RIM A m ode l s i s a s goo d a s t ha tf rom D uane m ode l , i f no t be tt er . Even w hen D uane mo de l f it s t he da t a r easonab ly we l l , thet ime ser ies m od el i s s t il l com parable . In part icular, for da ta se ts where the assu mp t ion ofmonotonica l ly increas ing or decreas ing fa i lure in tens i ty as in Duane model i s v io la ted , there sul ts o f t he AR IM A m ode l is even more appea ling . Usua l ly , a pa r s imon ious AR IM A mo de lof lo we r orders wo uld suf f ice in providing a reasonab le f i t to the fa i lure data.

C O N C L U D IN G R E M A R K SIn th is s tudy, the t ime ser ies approach based on A RIM A m odels i s inves t iga ted for repai rablesys tem fa i lure analys is and forecas t ing . Sat i s fac tory predic t ive per formance i s achieved ascom pared to t he Du ane m ode l . W i th t he adven t o f compute r techno logy today , a l t houghARIM A mode l ing i s ma themat i ca l ly soph i s t i ca t ed in t heo ry , t he i t e r a t i ve mode l bu i ld ingproces s and hen ce accu rat e fo recast s can be a ided and m ade s imp le r by the ease o f m anyuser - f r iendly s ta t is t ica l sof tware pack ages such as SA S, Statgraphics, Stat ist ica, . . . .etc. T h e s ecommercia l ly avai lable tools have cer ta in ly br idged the gap for re l iabi l i ty prac t i t ioners incharac ter iz ing the fa i lure process . As evident f rom the resul t s , the ARIMA model ingapproach can be a prom is ing a l te rnat ive for repai rable sys tem analys is .

REFERENCES

Asche r , H.an d Feingo ld , H. (1984) . Repairable Systems Reliability, M a r c e l D e k k e r , N e wYork .Bowerman, B.L. and O'Connel l , R. T. (1993) . Forecas t ing and Time Ser ies: An Appl iedApproach , D uxbury P ress , B e lmon t , CA.Box, G. E. P. , Jenkins, G. M. and Reinsel , G. C. (1994). Time Series Analysis, Forecastingand Control, Prent ice Hal l , En glew ood Cl i ffs , N.J .Chat ter jee , S ., M isra, R. B . and A lam, S . S . (1997) . Pred ic t ion of sof tware re l iabi l i ty us ing anautoregres s ive process , International Jo urna l o f System Science, 28(2) , 211-216.Duane, J . T . (1964) . Le arning curve approach to re l iabi l ity moni tor ing , 1EE E Transact ions onAerospace , 2 , 563-566.Singpurw al la , N. D. (1980). Es t ima t ing re liabi l ity growth (or de ter iora t ion) us ing t im e ser iesanalys is , Naval Research Logist ics Quarterly, 27, 1-14.Xie, M. and Z hao, M . (1996) . R el iabi li ty grow th p lo t - an underut i l ized tool in re l iabi li tyanalys is , M icroelectronics & Reliability, 36(6), 797-805.