Optimal Reliability Allocation

Transcription

1 Optmal Relablty Allocato Yashwat K. Malaya Departmet of Computer Scece Colorado State Uversty

2 Relablty Allocato Problem Allocato the relablty values to subsystems to mmze the total cost whle achevg the relablty target. Wdely applcable Software systems Electrcal systems Mechacal systems Implemetato choces Dscrete Cotuous

3 Relablty Allocato Software A software system cossts of may fuctoal modules Some reused, probably wth lower defect destes Some are ew, wth hgher defect destes Some are voked more ofte To crease relablty Addtoal testg Replcated usg -verso programmg? What s the best strategy?

4 Optmal Relablty Allocato System composed of subsystems: Subsystem cost a fucto of relablty System relablty depeds o subsystems Falure rate as a relablty measure Commos systems: seres ad parallel Software system relablty Fractoal executo tme Lagrage multpler: closed form optmal soluto Parameter depedece: sze, defect desty Apportomet & geeral approach

5 Problem Formulato System S has subsystems Ss, =,... Each subsystem SS has a specfc fuctoalty Several choces wth same fuctoalty, but dfferetly relablty levels. C f ( R ) Mmze system cost C s C f ( R ) Subject to

6 Cost mmzato problem thus system R For a seres Subject to S ST s ST R R R R R s R f C C ) ( Mmze

7 Subsystem mplemetato choces Subsystem ca be made more relable by extedg a cotuous attrbute dameter of a colum buldg tme spet for software testg. Dfferet veders mplemetatos of SS at dfferet costs. Multple copes of SS to acheve hgher relablty. double wheels of a truck Number of copes s costraed betwee oe ad a practcal umber because of mplemetato ssues.

8 The Cost fucto Cost fucto f should satsfy these three codtos: f s a postve fucto f s o-decreasg, thus hgher relablty wll come at a hgher cost. f creases at a hgher rate for hgher values of R Relablty vs Cost Steep cost crease Mettas A, Relablty allocato ad optmzato for complex systems. Pro A Relablty ad Mataablty Symp, Jauary 2000, 26-22

9 I terms of falure rate Takg log of both sdes, ad sce R (t) = e -λt Statg cost as a fucto of falure rate R ST R ) l( ) l( S f C C ) ( ST

10 I terms of falure rate: SRGM expoetal software relablty growth model ( d) 0 exp( d) λ 0 depeds o tal defect desty β depeds versely o program sze Restatg t as Cost fucto d( l 0 ) Assumes costat developmet cost, thus eglected

11 Seres ad Parallel Systems: learlzato Costrat Learzato smplfes the calculatos. Seres system l( R ST ) l( R ) Parallel system: log of urelabltes R ST ( R ) Elegbede: If cost fucto satsfes 3 propertes gve above, the cost s optmal f all parallel compoets have the same cost. l( R ST ) (l( R )

12 Relablty Allocato for Software Systems a block s uder executo for a fracto x of the tme where x = Relablty allocato problem Mmze C 0 l

13 Soluto usg Lagrage multpler solutos for the optmal falure rates optmal values of test tmes d ad d, ST x x x x x ST x d 0 l 0 l x x d

14 Observatos: Software relablty allocato A reused subsystem have a hgher relablty because of past testg causg λ λ 0 ad hece egatve d. Soluto: apply allocato problem oly to modules wth postve d. If x s proportoal to the subsystem code sze, the optmal values of the post-test falure rates λ, λ are equal.

15 Ex: Optmal: Software wth 5 blocks λ ST 0.04 Block B B 2 B 3 B 4 B 5 Sze KSLOC I Defect desty β λ x Optmal λ Optmal d Optmal whe all modules have the same falure rate!

16 Ex: Equal testg λ ST 0.04 Block B B 2 B 3 B 4 B 5 Sze KSLOC I Defect desty β λ x λ Equal d If Total test tme s equally dstrbuted for all 5 blocks, system wll have sgfcatly hgher falure rate of per ut tme

17 Ex: Testg oly B5 λ ST 0.04 Block B B 2 B 3 B 4 B 5 Sze KSLOC I Defect desty β λ x λ Equal d If Total test tme s allowed oly for block B5, system wll have hgher falure rate of per ut tme

18 Illustrato usg excel See Excel sheet relallocatoexamples.xls Try chagg etres. 8

19 Commo Apportomet rules Equal relablty apportomet: At ed they all dvdually have falure rate equal to target falure rate for the system Complexty based apportomet test tme apportoed proporto to the software sze Impact based apportomet: A compoet executed more frequetly, or more crtcal, should be assged more resources

20 Relablty Allocato for Complex Systems A teratve approach Desg the system usg fuctoal subsystems. Perform a tal apportomet of cost or relablty attrbutes based o sutable apportomet rules or prelmary computato. Predct system relablty. Is reallocato feasble ad wll ehace the objectve fucto. If so, perform reallocato. Repeat utl optmalty s acheved. Does ths meets objectves? If ot, retur to step ad revsg the desg at a hgher level..

21 Coclusos Relablty allocato: cosder how cost vares wth relablty. Software testg: cost log(/falure rate) sze Relablty allocato systems wth replcated subsystems ca ecouter correlated falures ad thus would eed a more careful modelg. 2