Optimal Reliability Allocation

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1 Optmal Relablty Allocato Wley Ecyclopeda of Operatos Research ad Maagemet Scece Yashwat K. Malaya Computer Scece Dept. Colorado State Uversty, Fort Colls CO Phoe: , FAX Y. K. Malaya, "Optmal Relablty Allocato," Wley Ecyclopeda of Operatos research ad Maagemet Scece, Joh Wley & Sos, Ja. 4, 20. Abstract--- The overall relablty of a complex system, desged as a assembly of subsystems, depeds o the subsystem relablty metrcs. The cost of each subsystem s a fucto of ts relablty. Ths artcle cosders the problem of allocatg the relablty values to mmze the total cost whle achevg the relablty target. The relablty allocato problem s applcable to mechacal ad electrcal systems as well as computer hardware ad software. The approach volves expressg the system relablty terms of the subsystem relablty values ad specfyg the cost fucto for each subsystem, whch allows settg up a optmzato problem. Software relablty allocato s examed as a detaled example ad some prelmary apportomet approaches are preseted. Whle may cases, t s possble to use exact optmzato methods, a complex case may requre use of teratve approaches. I. INTRODUCTION A large system s mplemeted by usg a set of tercoected subsystems. Whle the archtecture of the overall system s ofte fxed because of the fuctoal requremets, mplemetato choces are geerally avalable for dvdual subsystems. A desger eeds to acheve the target relablty whle mmzg the total cost, or alteratvely maxmze the relablty whle usg oly the avalable budget. Itutvely, some of the lowest relablty compoets may eed specal atteto to rase the overall relablty level. Such optmzato problems arse whle desgg a complex software or a hardware system. Such problems also arse mechacal or electrcal systems. A umber of studes sce 960 have examed some of the computatoal aspects of such problems []. All the subsystems are essetal a o-redudat system. However, a dvdual subsystem ca ofte be made more relable by usg a more costly mplemetato. Ths addtoal cost may represet wder colums a buldg or more thorough testg of software. I redudat mplemetatos, hgher relablty ca sometmes be acheved by usg several copes of a subsystem, such that the system corporates a parallel or a k-out-of- cofgurato. The problem formulato s cosdered the ext secto, followed by approaches used for settg up a optmzato problem. As a detaled example, software relablty allocato s examed detal wth a umercal llustrato. The last secto cosders relablty allocato geeral complex systems.

2 II. Allocato as a Optmzato Problem We cosder a system that has bee desged at a hgher level as a assembly of approprately coected subsystems. The fuctoalty of each subsystem s ofte uque; however, there ca be several choces for may of the subsystems that provde the same fuctoalty but dfferetly relablty levels. Here we cosder the problem formulato for a commo ad wdely applcable case. Let there be subsystems SS, =,..,, each wth relablty R ad cost C. Let the cost C be a fucto of the relablty gve by f ( R ), ths s geerally referred to as the cost fucto. Let C s ad R s represet the total system cost ad the overall relablty ad let R be the specfed target relablty. If all the subsystems are essetal to the system ad f ther falures are statstcally depedet, the system ca be modeled as a seres system. The cost mmzato problem ca be stated as: Mmze Cs C Subject to f ( R ) (). e. R R Rs R sce R S R (2) Note that equato () assumes that the cost of tercoectg the subsystems s eglgble. A alteratve problem would be to maxmze R s whle keepg C s less tha or equal to the allocated cost budget. Depedg o the type of the system, the th subsystem SS may have several mplemetato choces wth dfferet relablty values: A. The subsystem ca be made more relable by extedg a cotuous attrbute (for example dameter of a colum buldg or tme spet for software testg). B. Dfferet veders may offer ther ow mplemetatos of SS at dfferet costs. C. It may be possble to use multple copes of SS (for example double wheels of a truck) to acheve hgher relablty. Ofte the umber of copes s costraed betwee a mmum (ofte oe) ad a maxmum umber because of mplemetato ssues. I the frst case, both cost ad the relablty ca be vared cotuously, whereas the other two cases, the choces are dscrete. I the frst case, we ca defe a cotuous cost fucto. I the secod case too, the market forces may mpose a cost fucto. I the thrd case, the subsystem may be modeled as a parallel or k-out-of- system for relablty evaluato, provded the falures are statstcally depedet. A umber of publcatos o relablty allocato cosder oly the thrd case, where the optmzato problem becomes a teger optmzato problem. It becomes a 0- optmzato problem whe choces are dscrete ad a compoet from a gve lst of caddates s ether used or ot used [2]. 2

3 III. The Cost fucto ad problem set-up It s reasoable to assume that the cost fucto f would satsfy these three codtos [3]: ) f s a postve fucto 2) f s o-decreasg, thus hgher relablty wll come at a hgher cost. 3) f creases at a hgher rate for hgher values of R The thrd codto leads to the fact that t ca be very expesve to acheve the relablty value of. I fact, for software, t has bee show that t s feasble to acheve ultra-hgh relablty software [4]. Note that the cost fucto geeral wll also be a fucto of the relatve complexty of the subsystem. I some cases, the cost fucto ca be derved from basc cosderatos, as we wll do below for software relablty. I other cases, t may be derved emprcally by complg data for dfferet choces. The cost fucto s ofte stated terms of the relablty, for example, the cost fucto proposed by Mettas [3] s gve terms of the maxmum ad mmum achevable relablty values. I some cases, there are choces that ca be made that ca ehace relablty wth lttle or o addtoal cost, sometmes by usg ewer techology. For example, some dscpled software developmet processes have bee foud to yeld more relable software wth the same effort. We assume that such choces have already bee made, ad optmzato s sought by explotg the avalable choces of attrbutes. The cost fucto ca also be gve terms of the falure rate as llustrated below for software relablty allocato. A trasformato of equato (2) ca be obtaed by logarthms of both sdes of the equato [5, 6, 7]. l( R ) l( R ) (3) The trasformato equato (3) ca reduce the problem to a lear optmzato problem. I some cases, the term l(r ) also has a well-defed physcal sgfcace ad s termed the falure rate. Whe the falure rate of a subsystem SS s costat, ts relablty s gve by a expoetal relatoshp R (t) = exp(-λ t), the system falure rate s gve by the summato of the subsystem falure rates ad hece equato (3) ca be restated as (4) The falure rate s tself a frequetly used relablty metrc. I some cases such as software relablty egeerg, t s the falure rate that s ofte specfed [8, 9]. The cost fucto of a subsystem ca also be gve terms of ts falure rate. If the cost C s gve by the fucto f (λ ), equato () ca be restated as 3

4 Mmze C S C f ( ) (5) For example, let us cosder software relablty. Whe a software s tested, defects t are foud ad removed by debuggg. A program tested more thoroughly wll have fewer bugs ad hece hgher relablty. Several models relatg software relablty growth wth the tme spet testg, have bee proposed ad valdated. These are termed software relablty growth models (SRGMs). For the popular expoetal software relablty growth model [8, 9, 0], the falure rate as a fucto of testg tme d s gve by ( d) 0 exp( d) where λ 0 ad β are the model parameters. It should be oted that λ 0 depeds o tal defect desty ad β depeds versely o program sze []. If we assume that the varable cost s domated by the testg tme, the cost s gve by the followg fucto, whch satsfes the three codtos metoed above. IV. Relablty Allocato for Basc Seres ad Parallel Systems 0 d( ) l (6) The relablty allocato problem for two basc relablty structures seres ad parallel ca be solved by learzg the costrats [, 2, 7]. I a seres system, the costrat s gve equato 2 above, whch ca be learzed by rewrtg t as gve equato (3) above, whch may the be solved relatvely easly. A example s gve below for software relablty allocato. I a parallel system, fuctoally detcal subsystems are cofgured such that correct operato of at least oe of them assures a correctly fuctog system. It s assumed that ay overhead mplemetg such a system s eglgble. I real systems, the overhead volved wll result a lower level of relablty. I a umber of studes, the problem assumes that the relablty of a subsystem ca be creased by usg fuctoally detcal compoets parallel [2, 7]. For parallel systems, the costrat s gve by R ( ) (7) R The costrat ca be learzed by usg logarthms of the complemets of relablty. Thus, equato (7) ca be rewrtte as l( R ) l( R ) (8) 4

5 Elegbede et al. have recetly show [7] that f the cost fucto satsfes the three propertes gve above, the cost s optmal f the all the parallel compoets have the same cost. For software, computer hardware ad mechacal systems, the umber of dscrete parallel compoet s lkely to be very small. V. Relablty Allocato Software Systems Let us exame the problem of software relablty allocato [6], whch serves as a good llustrato. Typcally, a software cossts of sequetally executed compoets, oly oe of them s uder executo at a tme. Each compoet ca be depedetly tested ad debugged to reduce the falure rate below a target value. I some cases, the relablty of a compoet ca be further creased by replcato. For replcato to be effectve, each replcated verso must be developed depedetly such that the falures are relatvely depedet. The mpact of replcato ca be evaluated by assumg statstcal depedece. However, t has bee show that sometmes the statstcal correlato ca be sgfcat, requrg aalyss that s more complex. I ths example, we cosder a o-redudat mplemetato of software, dvded to sequetal compoets [6], whch s a commo case. Let us assume that a compoet s uder executo for a fracto x of the tme where x = [5]. Software relablty s ofte descrbed terms of the falure rates. The the relablty allocato problem ca be wrtte as Mmze 0 C l (9) Subject to (0) x Soluto: let us solve the problem posed by equatos (9) ad (0) by usg the Lagrage multpler approach by fdg the mmum of F (,..., ) C (... ) where s the Lagrage multpler. The ecessary codtos for the mmum to exst are () the partal dervatves of the fucto F are equal to zero, () > 0 ad () x λ +x 2λ 2+ x λ = λ [6]. Equatg the partal dervatves to zero ad usg the thrd codto, the solutos for the optmal falure rates are foud as followg x 2 2x2 x x x () The optmal testg tmes of the dvdual modules, d, = to, are gve by equato (2) below. 5

6 0x d l ad d 0 x l x (2) Note that d s postve f λ λ 0. The testg tme for a module must be o-egatve. If ay of the testg tmes obtaed usg (3) are egatve, the optmzato problem may have be solved teratvely [6] or usg specal approaches. Sometmes a reused subsystem has a sgfcatly hgher relablty because of past testg, whch may result λ λ 0. Sce the reused subsystem s already relable eough, t would make sese to sped ay testg effort o the other subsystems. I software relablty egeerg, the assumptos volved formulato of the expoetal model mply that the parameter β s versely proportoal to the software sze [8, ], measured terms of the les of code. I the examples below, we assume that value of x s proportoal to the subsystem code sze. The values of λ ad λ 0 do ot deped o sze but deped o the tal defect destes [] at the begg of testg. Thus f the expoetal model deed holds, the equato () states that the optmal values of the post-test falure rates λ, λ are equal. I addto, f the tal defect destes are also all equal for all the compoets, the the optmal test tmes for each module s proportoal to ts sze. The parameter values eeded for (2) ad (3) may be determed by some tal testg or usg emprcal relatoshps. Example : A software system uses fve fuctoal compoets B-B5. Ths example has bee costructed assumg szes, 2, 3, 0 ad 20 KLOC (thousad les of code) respectvely, ad the tal defect destes of 0, 0, 0, 5 ad 20 defects per KLOC respectvely. Let us assume that measured parameter values are gve the top three rows, whch are the puts to the optmzato problem. The soluto obtaed usg equatos () ad (2) are gve the two bottom rows. The test cost has bee mmzed so that the overall falure rate s less tha or equal to 0.04 per ut tme. Here the tme uts ca be perso hours of testg tme, or hours of CPU tme used for testg. Note that the optmal values of λ for the fve compoets are equal, eve though they started wth dfferet tal values. Ths mples a substatal part of the test effort s allocated to largest compoets. The total cost terms of testg s 4984 ut tme. Compoet B B 2 B 3 B 4 B 5 β λ x Optmal λ Optmal d If the total test tme were equally dstrbuted for all fve compoets, t would have resulted sgfcatly hgher falure rate of per ut tme. 6

7 VI. Relablty Apportomet rules The dscusso the secto above suggests some prelmary rules may be used for obtag tal relablty apportomets before a more thorough optmzato. Some apportomet rules have bee suggested the lterature [5]. Equal relablty apportomet: Oe ca test a set of software compoets, such that at the ed they all dvdually have the falure rate equal to the target falure rate for the system. Newly developed modules eed to be tested utl ther relablty equals the relablty of the exstg modules. Complexty based apportomet: The software sze tself s a complexty metrc. Thus, the avalable test tme ca be apportoed proporto to the software sze. Impact based apportomet: A compoet that s executed more frequetly, or s more crtcal terms of falures, should be assged more resources. Note that the aalyss above, x ca be chose to reflect the product of frequecy ad crtcalty. VII. Relablty Allocato for Complex Systems Some systems ca have complex relablty allocato choces ad may requre a teratve approach [, 2, 7, 2]. Such a approach s also eeded f the objectve fucto s mult-objectve ad cludes both total cost ad the system relablty []. A teratve approach volves the followg steps [2]: ) Desg the system usg fuctoal subsystems. 2) Perform a tal apportomet of cost or relablty attrbutes based o sutable apportomet rules or prelmary computato. 3) Predct system relablty. 4) Determe f reallocato s feasble ad wll ehace the objectve fucto. If so, perform reallocato. 5) Repeat utl optmalty s acheved. 6) See f ths meets the objectves. If ot, cosder returg to step ad revsg the desg at a hgher level. 7) Falze the desg wth recommeded relablty allocato ad the cost projectos. Several optmzato methods ca be used for steps 2-5 above. These ca be classfed to three approaches []. ) Exact methods: Whe the problem s ot large, exact methods ca be desrable. I geeral, the problem ca be a o-lear optmzato problem. I a few cases, the problem ca be trasformed to a lear problem, as show the example above. 2) Heurstcs based methods: Several heurstcs for relablty allocato have bee developed. May of them are based o detfyg the varable to whch the soluto s most sestve ad varyg ts value. 3) Metaheurstc algorthms: These algorthms are based o artfcal reasog. The best kow of them are geetc algorthms, smulated aealg ad tabu-search. These algorthms ca be useful whe the search space s large ad approxmate results are sought. 7

8 Relablty allocato systems wth replcated subsystems ca ecouter correlated falures ad thus would eed a more careful modelg. I software, such correlato ca be sgfcat [3]. The factors that affect techology depedat relablty attrbutes such as defect desty software [4] have bee studed by researchers ad are stll beg researched. Several software tools, both geeral purpose [5] ad specal purpose [6], have bee developed that ca smplfy settg up ad solvg the optmal allocato problem. Relablty allocato problem ca also be formulated to address other relablty attrbutes lke avalablty or mataablty. REFERENCES [] Kuo W, Prasad VR, A Aotated Overvew of System-Relablty Optmzato, IEEE Trasactos o Relablty, Jue 2000, 49: [2] Majety SRV, Dawade M, Rajgopal J, Optmal Relablty Allocato wth Dscrete Cost-Relablty Data for Compoets. Operatos Research, Nov-Dec 999, 47: [3] Mettas A, Relablty allocato ad optmzato for complex systems. Proceedgs Aual Relablty ad Mataablty Symposum, Los Ageles, CA, Jauary 2000, [4] Butler RW, Fell GB, The feasblty of quatfyg the relablty of lfe-crtcal real-tme software, IEEE Tras. Software Egeerg, 993, 9:3 2. [5] Lakey PB, Neufelder AM, System ad Software Relablty Assurace Notebook; Rome Laboratory, 996, Rome NY, [6] Lyu, MR, Ragaraja S, va Moorsel, APA, Optmal allocato of test resources for software relablty growth modelg software developmet. IEEE Trasactos o Relablty, Ju 2002, 5: [7] Elegbede AOC, Chegb C, Adjallah KH, Yalaou F, Relablty allocato through cost mmzato, IEEE Trasactos o Relablty, March 2003, 52:06-. [8] Musa JD, Iao A, Okumoto K, Software Relablty, Measuremet, Predcto, Applcato, McGraw-Hll, 897. [9] Lyu, MR, Ed., Hadbook of Software Relablty Egeerg, McGraw-Hll, 995. [0] Goel AL, Okumoto K, Tme-Depedet Error Detecto Rate Model for Software ad Other Performace Measures, IEEE Tras. o Relablty, August 979, 28: [] Malaya YK, Deto J, What Do the Software Relablty Growth Model Parameters Represet? It. Symp. o Software Relablty Egeerg, 997, [2] NASA Documet, Orgazatoal Istructo: Relablty Allocato, QDR-008 Rev. E, Oct, [3] Da YS, Xe M ad Poh KL, Modelg ad aalyss of correlated software falures of multple types, IEEE Trasactos o Relablty, vol. 54, pp , [4] Neufelder A, The Facts About Predctg Software Defects ad Relablty, The RAC Joural, Aprl 8, 2002, -4, [5] RelaSoft, Relablty Importace ad Optmzed Relablty Allocato (Aalytcal)

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