Compound System Reliability Test Design Based on Bayesian Approach and Monte Carlo Method Anyang Zhang1,a and Qi Liu2,b

Transcription

1 Iteratoal Idustral Iormatcs ad Computer Egeerg Coerece (IIICEC 5) Compoud System Relablty est Desg Based o Bayesa Approach ad Mote Carlo Method Ayag Zhag,a ad Q Lu,b, College o Iormatoal Systems ad Maagemet, Natoal versty o Deese echology, Chagsha, P R Cha a super_879@63com Keywords: Relablty test desg; Iormato uso; CSRD; Bayesa approach; Mote Carlo method Abstract Aalyzg the curret lack o relablty test desg ad cosderg the producer s rsk ad cosumer s rsk, the relablty test desg model or compoud system (CSRD) based o ormato uso was costructed Aalyzg the requremets ad costrat codtos o relablty test desg, the ormato uso techology betwee bomal dstrbuted subsystems ad ther system was preseted by meas o Bayesa approach ad Mote Carlo method Accordg to posteror rsk crtera, the ormulas o producer s rsk ad cosumer s rsk were deduced he soluto o computg the optmum test pla was gve by Matlab sotware Fally, a example o mssle lauchg stallato was gve to demostrate the valdty o the model Itroducto I the eld o relablty test, the evaluato o the system relablty o product wth hgh relablty ad log le s oe o the mportat problems, especally or the complex system such as arcrats, mssles ad satelltes hereore, due to hgh cost ad log test tme, t s mproper to perorm relablty test by the tradtoal test method [-3] As the Bayesa theory has the advatage o reduce actual test umbers, t has bee wdely studed the eld o relablty test recet years ad gradually recogzed by the teratoal academc commuty Belkacem et al [4] used the Bayesa etwork approach to ormulate the relablty model o systems wth ucerta structures Saku [5] obtaed Bayes estmators or the ukow parameter o a Iverse Raylegh dstrbuto uder symmetrc ad asymmetrc lear expoetal loss uctos usg a o-ormatve pror Ca et al [6] developed a redudat sotware system or subsea blowout preveters by Bayesa etworks However, may crcumstace, the applcato o these methods may ot gve satsactory result he exstg methods have at least the ollowg problem: Whe ecouterg complex compoud system, we caot obta the pror dstrbuto o system rom the subsystems by aalytc method, so the posteror dstrbuto o system caot be deduced by Bayesa theory he subsystems ad system s test cost should be totally take to cosderato, but the curret researches pay ewer attetos o the total cost o relablty test Accordg to the above shortcomgs, ths paper presets a subsystem-to-system ormato uso method or compoud system by meas o Bayesa approach ad Mote Carlo method Ad CSRD (the relablty test desg model or compoud system) s bult to gure out the mmum cost o the compoud system whe cosderg the costrats At last, oe umercal example s used to demostrate the valdty o the model CSRD Descrpto he Basc Assumptos I order to costruct the model precsely, ve assumptos are preset as ollowg ) he assumpto o model 5 he authors - Publshed by Atlats Press 49

2 Supposg that a compoud system s made up o m subsystems, the relablty parameter o each subsystem s R (,,, m) ad R m deotes the compoud system s I ths paper, the varables whose subscrpt s m descrbe the characterstc o the compoud system I all the alures are depedet, the the compoud system structure ucto s R m gr (, R,, Rm) () ) he assumpto o pror ormato Pror ormato s people s realzato o ukow parameters beore perormg relablty tests I ths paper, all the subsystems ad system s test results are success/alure data, ad ca be aalyzed by bomal dstrbuto hereore, supposg all the subsystems ad system are bomal system, the pror dstrbuto o R (,,, ) s beta dstrbuto B( α, β ), where α ad β are hyper-parameters π ( R )(,,, m) deotes the pror dstrbuto oth () subsystem ad π ( R m ) deotes the compoud system s Aother pror dstrbuto π () ( R m ) o the compoud system s derved rom the subsystems by ormato uso techology 3) he assumpto o relablty parameter I the relablty demostrato test, the cosumer requres a desred value R ad a lowest accepted value R o the system s relablty, whch s equvalet to test the ollowg statstcal hypothess H : R R H: R < R () 4) he assumpto o test cost Durg system relablty demostrato test, we suppose that the testg cost s a lear ucto o the actual test umber o the subsystems ad system, e I α β α β (3) C C C C (, ) C (, ) C otal relablty test cost I C th subsystem s (system s) tal relablty test cost, or,,, he tal relablty test cost cludes all the cost whch s ecessary to start the relablty test, such as cost o test aclty costructo, sta trag ee, test prepare ee, etc C th subsystem s (system s) actual ut relablty test cost, or,,, he ut test cost cludes all the cost whch s ecessary to operate the actual relablty test, such as stas salary, subsystem s (system s) cost prce, aclty charge or use, etc th subsystem s (system s) actual relablty test umber, or,,, m System s relablty alure test umber α (, ) Producer s rsk o system β (, ) Cosumer s rsk o system C α System s relablty test cost whch producer s rsk leads to, whe rejectg H C β System s relablty test cost whch cosumer s rsk leads to, whe rejectg H 5) he assumpto o rsks I ths artcle, the posteror rsk crtera s used to calculate the produce s rsk ad the cosumer s rsk [7] Accordg to the posteror rsk crtera, the deto o the two types o rsks s Posteror Producer ' s Rsk P( H est s Faled) (4) Posteror Cosumer ' s Rsk P( H est s Passed) Accordg to equato (4), α (, ) ad β (, ) ca be deduced, e P( > m Rm ) π( Rm αm, βm ) α(, ) (5) P( > R ) π( R α, β ) 493

3 P( R ) π( R α, β ) P( R ) π( R α, β ) β (, ) (6) ( α, β ) he two parameters o system posteror dstrbuto ater ormato uso Costructo o CSRD Based o the Secto ad the mmzato o total test cost, CSRD ca be bult as I α β M( C) M C C C α(, ) C β(, ) (7) St I C >, C >,,,, α β C >, C > L (8) N N,,,, α(, ) α β(, ) β N he maxmum test umbers that ca be provded by the producer ad cosumer L N he mmum test umbers that ca be accepted by the producer ad cosumer α he sgcace level o producer s rsk β he sgcace level o cosumer s rsk Iormato Fuso ad Soluto o CSRD I ths artcle, we ocus o two ways o ormato uso he rst s how the ormato o subsystems s used to the system s, ad the secod s how to combe the system pror ormato π () ( R m ) rom subsystems wth the system heret pror ormato π () ( R m ) ) Soluto o the rst way Accordg to Bayesa theory ad the assumpto o pror ormato, the posteror dstrbuto o R(,,, m) ca be calculated, e π( R α, β ) R ( R) π( R (, )) π( R α, β ) (9) π( R, ) ( ) α β R R So the posteror dstrbuto o R s B( α, β ) o complex compoud system, t s dcult to gure out the aalytcal soluto o ts pror ormato π () ( R m ) hereore, the paper uses Mote Carlo aalyss to obta the estmato o system pror ormato, ad the the dstrbuto π () ( R m ) ca be gve by the goodess o t he steps o Mote Carlo (SMC) are as ollow Step : Set the value o smulato tmes N, ad set k ; Step : By Matlab, we ca calculate the relablty smulato value o each subsystem respectvely accordg to π( R α, β ), e ( k) ( k) ( k),,, m () Step 3: Accordg to the equato(), the relablty smulato value o system s ( k) ( k) ( k) ( k) g (,,, ) () m m 494

4 Step 4: Judge: k < N, the k k ad move to Step ( k ) () Step 5: o { m }( k,,, N), π ( R m ) ca be gured out by ttg o dstrbuto ad parameter estmato ) Soluto o the secod way Weghtg uso algorthm [8] s used to solve the uso o system pror dstrbuto Supposg that the source o pror dstrbuto ad the accuracy o pror dstrbuto are ully aalyzed, we ca obta the weght valuesw, W, orw W hereore, the system pror dstrbuto s () () π( R ) Wπ ( R ) Wπ ( R ) () So, the posteror dstrbuto o R m s π ( R ) LR ( (, )) π ( R (, )) π ( R ) L( R (, )) (3) LR ( (, )) he lkelhood ucto o R m Accordg to equato () ad (3), the π () () ( Rm ( m, m )) W ( Rm ) W ( Rm ) L( Rm ( m, m )) m(( m, m ) π ) π π (4) m((, ) π ) he margal desty o R m, π π m((, ) ) ( R ) L( R (, )) o h th pror dstrbuto, ts posteror dstrbuto ca be calculated, e π ( R ) LR ( (, )) π ( R (, )) ( h, ) (5) π ( R ) L( R (, )) Accordg to equato (4), dee ((, ) π ) π ( ) ( (, )) Wm ((, ) π ) (, ) (, ) Wm h ((, ) π ) h m R L R (6) λ (7) Accordg to equato (3), (5), (6) ad (7), the posteror dstrbuto o R m s h h π( R (, )) λ (, ) π ( R (, )) (8) We ca see that the system posteror dstrbuto s the weghtg sum o the posteror dstrbuto rom deret pror ormato source o solve the CSRD, t requres calculatg the equato (5) ad (6) rst We eed to kow actual subsystem s (system s) test results (, )(,,, ), but test desg stage, the actual test umbers ad alure test umbers (,,, ) are ukow, ad eed to be solved It s dcult to d the soluto o,,, m or all possble,,, m So, practce, the alure test umbers ca be xed, ad or gve,,, m, to d the proper actual test umbers,,, m Nowadays, the subsystems ad system are usually very relable ad ther relablty are usually very hgh, especally or the systems whch has passed the relablty growth test ad evromet stress test So, the alure test umbers ca be xed at or other umber whch s much less tha the actual test umber he equato (5) ad (6) ca be deduced urther as ollow, whe m 495

5 α(,) R β α ( R m ) R m β α ( R m ) R m (9) R β α ( R m ) Rm β (,) () β α ( R m ) R Because the calculato cludes the tegral operato o α (, ) ad β (, ), t s dcult to solve the CSRD (equato (7) ad (8)) by aalytcal method So ths paper, the umercal method s suggested to solve the CSRD, such as usg Matlab he steps are as ollows Step : Accordg to test evromet, set the values o I, α β C,,, L, C C C N N, α, β, R, R,, or,,, Set the system pror dstrbuto π () ( R m ) ad the weghtg valuesw, W Step : Accordg to the pror ormato o subsystems, such as hstorc test results, expert s ormato, corm the subsystems pror dstrbutos π ( R)(,,, m) Step 3: Accordg to equato (9), gure out the subsystems posteror dstrbutos π ( R (, )) sg SMC method ( Chapter 3), we ca obta oe o system s pror dstrbuto π () ( R m ) Step 4: Accordg to equato (6), (7) ad (8), we ca obta the system s posteror dstrbutoπ ( R (, )) L Step 5: Set the costra codto o actual test umbers, deote the lower lmt as N, upper L lmt as N, or,,, o test pla, N, N N, or,,, Dete a data set m (,,, ), where deote the actual test umbers o subsystem (system), or,,, Lst all the possble value o (,,, m ), deote the result as a set S, the L L L L L L S {( N, N,, N ),( N, N,, N ),,( N, N,, N )} Deote the k th elemet L set S as Sk ( ), or k,,, ( N N ) Step 6: Set k, let the tal test costsc L Step 7: Judge: k ( N N ), the move to Step 8, else move to Step Step 8: o Sk ( ), accordg to the equato (5) ad (6), calculateα (, ) ad β (, ) Step 9: Iα(, ) αad, the move to Step, else let k k, move to Step 7; Step : Accordg to the equato (3), calculate the test costc, ad judge IC IC C, let k k, move to Step 7 < C, Sk ( ) wll be the curret optmum actual test combato, deote the optmum test pla as (( O, ),,( O, ),,( O, )), where O O O S( k) ((, ),,(, ),,(, )) LetC C, k k, move to Step 7 Step : Ed calculato Numercal Example Aalyss A mssle lauchg stallato cossts o oe target chael (subsystem ), two mssle chaels (subsystem ad subsystem 3), ad our lauchg stallatos (subsystems 4-7) he mssle 496

6 lauchg stallato s a compoud system as show Fgure, ad every subsystem relablty test results are bomal dstrbuted ad depedet Laucher 4 Mssle chael Laucher 5 arget chael Laucher 6 Mssle chael 3 Laucher 7 Fgure he relablty block dagram o the mssle lauchg stallato able Parameter value assgmet Parameter Value I C (5,,,,,,,) C (3,5,5,5,5,5,5,) L N (,,,,,,,) N (8,,,7,6,7,8,3) α β ( C, C ) (4,4) 8 8 ( α, β ) (,) ( R, R ) (98,9) (,,,,,,,) Accordg to the steps o algorthm, able lsts parts o the possble test pla he mmum o k s 8 L, ad the maxmum s ( N N )6838 able Parts o the calculato results k α β C

7 It s obvous that S(96877) s the lowest cost pla Compared wth other test plas, such as S(968386) ad S ( ), these test plas caot satsy ether the rsk costras or the mmum cost costra hereore, most mportatly, whe desgg a relablty test pla, the desger eeds to cosder the test cost ad related costras, ad d a optmum test pla Cocluso I ths research, a Bayesa relablty test desg to the compoud system, whch cossts o bomal subsystems, s coducted o acheve ths purpose, ths paper presets the CSRD methodology or vercato o system relablty by meas o ormato uso ad Mote Carlo method, ad also the example o the mssle lauchg system to demostrate the proposed model Comparg wth curret methods, the advatages o the proposed methods ca be summarzed: () () the proposed model takes two types o pror ormato o the system ( π ( R m ) ad π () ( R m ) ) to cosderato ad completes two types o ormato uso ( Secto ) o the basc o Bayesa theory ad Mote Carlo method hereore, the posteror dstrbuto o the compoud system s gured out at last; () the proposed model ocuses o the mmum cost o tegral relablty test, so the test desgers ca d out a optmum test pla accordg to the totally cost Reerece [] G S, "Optmal est Desg or Relablty Demostrato," Operatos Research, vol, pp 36-4, 974 [] A I K, "he Desg o Optmum Compoet est Plas the Demostrato o a Seres System Relablty," Computatoal Statstcs & Data Aalyss, vol 4, pp 8-9, Oct 99 [3] F O ad e al, "he desg o optmum compoet test plas or system relablty," Computatoal Statstcs & Data Aalyss, vol 5, Jul 6 [4] A Z ad e al, "Bayesa relablty models o Webull systems: State o the art," Appl Math Comput Sc, vol, pp 585-6, [5] S Dey, "Bayesa Estmato o the Parameter ad Relablty Fucto o a Iverse Raylegh Dstrbuto," Malaysa Joural o Mathematcal Sceces, vol 6, pp 3-4, [6] B Ca, Y Lu, Q Fa, ad Y Zhag, "Applcato o Bayesa Networks to Relablt Evaluato o Sotware System or Subsea Blowout Preveters," Iteratoal Joural o Cotrol ad Automato, vol 6, pp 47-59, 3 [7] M S Hamada, G W Alyso, ad R C Shae, Byesa Relablty New York: Sprger, 8 [8] V P Savchuk ad H F Martz, "Bayes Relablty Estmato sg Multple Sources o Pror Iormato: Bomal Samplg," IEEE rasactos o Relablty, vol 43, pp 38-44,