Relablty Allocato Yashwat K. Malaya omputer Scece Dept. olorado State Uversty Fort olls O 80523 USA malaya@cs.colostate.edu Phoe: 970-49-703, 970-49-2466 Abstract---A system s geerally desged as a assembly of subsystems, each wth ts ow relablty attrbutes. The overall system relablty s a fucto of the subsystem relablty metrcs. The cost of the system s the sum of the costs for all the subsystems. Ths artcle exames possble approaches to allocate the relablty values such that the total cost s mmzed. The problem s very geeral ad s applcable for mechacal systems, electrcal systems, computer hardware ad software. The problem 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. The artcle also exames some prelmary apportomet approaches that have bee proposed. I some cases, t s possble to use exact optmzato methods. I geeral, a complex case wll requre use of teratve approaches. I. INTRODUTION May systems are mplemeted by usg a set of tercoected subsystems. Whle the archtecture of the overall system ca ofte be fxed, dvdual subsystems may be mplemeted dfferetly. A desger eeds to ether acheve the target relablty whle mmzg the total cost, or 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 a optmzato problem may arse whle desgg a complex software or a computer system. Such problems also arse mechacal or electrcal systems. A umber of studes sce 960 have examed such problems [kuo00]. I a o-redudat system, all the subsystems are essetal, however ofte a dvdual subsystem ca 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 subsystems, such that form a parallel or k-out-of- cofgurato. The ext secto cosder the problem formulato, followed by approaches used for settg up a optmzato problem. As a example, software relablty allocato s examed detal wth two umercal llustratos. The last secto cosders relablty allocato complex systems. II. Problem Formulato We assume that a system has bee desged at a hgher level as a assembly of approprated coected subsystems. I geeral the fuctoalty of each subsystems ca be uque, however there ca be several choces for may of the subsystems provdg 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. Let the cost be a fucto of the relablty gve by f ( R. Let s ad R s represet the total system cost ad the overall relablty ad 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 Subject to R s Rs R f ( 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 s less tha the allocated cost budget. The th subsystem SS ca may have several mplemetato choces wth dfferet relablty values: A. By extedg a cotuous attrbute (for example dameter of a colum buldg or tme spet for software testg the subsystem ca be made more relable. B. Dfferet veders may offer ther ow mplemetatos of SS at dfferet costs.. 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. Note that the frst case, both cost ad the relablty ca be vared cotuously, where as the other two cases, the choces are dscreet. 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 dscreet ad a compoet from a gve lst of caddates s ether used or ot used [majety99]. III. Approaches for Problem Set-up It s reasoable to assume that the cost fucto f would satsfy these three codtos [mettas00]: f s a postve defte fucto 2 f s o-decreasg 3 f creases at a hgher rate for hgher values of R The thrd codto suggests that the t ca be very expesve to acheve the relablty value of. I fact for software, t has be show that uder some assumptos, t s feasble to acheve ultra-hgh relablty software [butler93]. 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 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 [mettas00] s gve by ( R, f, R R exp ( f, m,,max R R R,max,m R 2
Where R,m ad R,max are the mmum ad maxmum values of R ad f s parameter ragg betwee 0 ad that represets the relatve dffculty of creasg a compoet s relablty. The cost fucto ca also be gve terms of the falure rate as llustrated below for software relablty allocato. A useful trasformato of equato (2 ca be obtaed by logarthms of the R values equato [lakey96, lyu02, elegbede03]. l( (3 R l( R The trasformato equato (3 ca sometmes reduce the problem to a lear optmzato problem. The term l(r ca also have a well defed physcal sgfcace some cases as the falure rate. Whe the falure rate of a subsystem SS s costat, ts relablty s gve by a expoetal relatoshp R (t exp( t, the system falure rate s gve by the summato of the subsystem falure rates ad hece equato (3 ca be restated as λ λ (4 The falure rate tself a major relablty attrbute. I some cases such as software relablty egeerg, t s the falure rate that s ofte specfed [musa87, lyu97]. The cost fucto of a subsystem ca also be gve terms of ts falure rate. If the cost s gve by the fucto f (, equato ( ca be restated as Mmze S f (λ (5 For example, whe software relablty growth s gve by the popular expoetal relablty growth model (SRGM [goel79, musa87, lyu97], the falure rate as a fucto of testg tme d s gve by λ ( d λ0 exp( d Where 0 ad are the SRGM parameters. If we assume that the cost s domated by the testg tme, the cost s gve by the followg fucto whch satsfes the three codtos metoed above. d λ l λ 0 λ IV. Relablty Allocato Approaches for Basc Seral ad Parallel Systems ( (6 The relablty allocato problem for two basc relablty structures seres ad parallel ca be solved by learzg the costrats [kuo00, majety99, elegbede03]. I a seres system, the costrat s gve equato 2 above, whch ca be learzed by rewrtg t as R l( R l( (7 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, 3
the problem assumes that the relablty of a subsystem ca be creased by usg a fuctoally detcal compoets parallel [majety99, elegbede03]. For parallel systems, the costrat s gve by R ( (8 R The costrat ca be learzed by usg logarthms of the complemets of relablty. Thus equato (8 ca be rewrtte as R (l( R l( (9 Elegbede et al. have recetly show [elegbede99] 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 for Software Systems As a detaled example, we exame the problem of software relablty allocato [lyu02]. Typcally a software cossts of a sequetally executed blocks, such that oly oe s uder executo at a tme. Each block ca be depedetly tested ad debugged to reduce the falure rate below a target value. I some cases, the relablty of a block 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 the statstcal correlato teds to be sgfcat requrg more complex aalyss. Here ths example we cosder the commo case, a o-redudat mplemetato of software, dvded to sequetal blocks [lyu02]. Let us assume that a block s uder executo for a fracto x of the tme where Σx [lakey96]. The the relablty allocato problem ca be wrtte as Mmze λ0 l λ (0 Subject to λ λ ( x Soluto: let us solve the problem posed by equatos 0 ad by usg the Legrage multpler approach by fdg the mmum of F λ,... λ + θ ( λ +... + λ λ ( where θ s the Legrage multpler. The ecessary codtos for the mmum to exst are ( the partal dervatves of the fucto F are equal to zero, ( θ > 0 ad ( x +x 2 2 + x [lyu02]. Equatg the partal dervatves to zero ad usg the thrd codto, the solutos for the optmal falure rates are foud as followg λ λ x λ2 λ λ λ 2x2 x x x (2 4
The testg tmes of the dvdual modules are gve by equato 0. The optmal values of d ad d, are gve by λ0 x d l λ ad d λ 0 x l λ x Note that d s postve f 0. The testg tme for a block must be o-egatve. If ay of the testg tmes (3 are egatve, the optmzato problem must be solved teratvely [lyu02]. I software relablty egeerg, the assumptos volved formulato of the expoetal model mply that the parameter s versely proportoal to the software sze [musa87, malaya97], whe measures terms of the les of code. The value of x ca be reasoably assumed to be proportoal to the code sze. The values of ad 0 do ot deped o sze but deped o the tal defect destes [malaya97]. Thus f the expoetal model deed holds, the equato (2 state that the optmal values of the post-test falure rates, are equal. Also f the tal defect destes are also all equal for all the blocks, the the optmal test tmes for each module s proportoal to ts sze. Example : A software system uses fve fuctoal blocks B-B5. We costruct ths example assumg szes, 2,3,0 ad 20 KLO (thousad les of code respectvely, ad the tal defect destes of 20,20, 20, 25 ad 30 defects per KLO 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 (2 ad (3 are gve the two bottom rows. Let us ow mmze the test cost such the overall falure rate s less tha or equal to 0.06 per ut tme. (3 Block B B 2 B 3 B 4 B 5 7 0-3 3.5 0-3 2.333 0-3 7 0-4 3.5 0-4 0 0.4 0.4 0.4 0.75 0.2 x 0.028 0.056 0.083 0.278 0.556 Optmal 0.06 0.06 0.06 0.06 0.06 Optmal d 2.043 242.085 363.28.529 0 3 3.579 0 3 Note that the optmal values of for the fve modules are equal, eve though they start wth dfferet tal values. Ths requres a substatal part of the test effort allocated to largest blocks. The total cost s 5.835 0 3. Example 2: Ths s detcal to the prevous example wth oe dfferece, the umbers the secod row for 0 are obtaed assumg all the fve module have the same defect desty value of 20/KLO. Block B B 2 B 3 B 4 B 5 7 0-3 3.5 0-3 2.333 0-3 7 0-4 3.5 0-4 0 0.4 0.4 0.4 0.4 0.4 x 0.028 0.056 0.083 0.278 0.556 Optmal 0.06 0.06 0.06 0.06 0.06 Optmal d 2.043 242.085 363.28.2 0 3 2.42 0 3 We ote that for ths example the optmal testg tme s exactly proportoal to the software sze, block B 5 s tested for 20 tmes the tme used for B. The fal falure rates are detcal for the fve blocks. 5
The above dscusso suggest some prelmary rules may be used for obtag tal apportomets. Some apportomet rules have bee suggested the lterature [lakey96]. Equal relablty apportomet: For example appled to sequetal software blocks, such that they each have the falure rate equal to the target falure rates. omplexty based apportomet: For example, 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 block that s executed more frequetly, or s more crtcal terms of falures, should be assged more resources. VI. Relablty Allocato for omplex Systems I practce, may cases ca be complex ad may requre a teratve approach [asa04, kuo00, majety99, elegbede03]. Such a approach s also eeded f the objectve fucto s mult-objectve ad cludes both total cost ad the system relablty [kuo00]. These steps are based o [asa04]: 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. 7 Falze the desg wth recommeded relablty allocato ad the cost projectos. The optmzato methods used steps 2-5 above ca be classfed to three approaches [kuo00]. 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 cremetg 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. Some geeral purpose [relasoft03] ad specal purpose [lyu03] software tools 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 [gurov95] or mataablty. REFERENES [kuo00] Kuo W, Prasad VR, A Aotated Overvew of System-Relablty Optmzato, IEEE Trasactos o Relablty, Jue 2000, 49: 76-87. [majety99] Majety SRV, Dawade M, Rajgopal J, Optmal Relablty Allocato wth Dscrete ost-relablty Data for ompoets. Operatos Research, Nov-Dec 999, 47: 899-906. 6
[mettas00] Mettas A, Relablty allocato ad optmzato for complex systems. Proceedgs Aual Relablty ad Mataablty Symposum, Los Ageles, A, Jauary 2000, 26-22 [butler93] Butler RW, Fell GB, The feasblty of quatfyg the relablty of lfe-crtcal real-tme software, IEEE Tras. Software Egeerg, 993, 9:3 2. [lakey96] Lakey PB, Neufelder AM, System ad Software Relablty Assurace Notebook; Rome Laboratory, 996, Rome NY, 6.-6.24 [lyu02] 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: 83-92 [elegbede03] Elegbede AO, hegb, Adjallah KH, Yalaou F, Relablty allocato through cost mmzato, IEEE Trasactos o Relablty, March 2003, 52:06-. [musa87] Musa JD, Ia A, Okumoto K, Software Relablty, Measuremet, Predcto, Applcato, McGraw- Hll, 897. [lyu97] Lyu, MR, Ed., Hadbook of Software Relablty Egeerg, McGraw-Hll, 995. [malaya97] Malaya YK, Deto J, What Do the Software Relablty Growth Model Parameters Represet? It. Symp. o Software Relablty Egeerg, 997, 24-35. [goel79] Goel AL, Okumoto K, Tme-Depedet Error Detecto Rate Model for Software ad Other Performace Measures, IEEE Tras. o Relablty, August 979, 28: 206-2. [asa04] NASA Documet, Relablty Allocato, Oct, 2004, foa.msfc.asa.gov/docs/nas8-0079/s&maoi/r- Relablty/QD-R-008.pdf [gurov95] Gurov SV, Utk LV, Shubsky IB, Optmal relablty allocato of redudat uts ad repar facltes by arbtrary falure ad repar dstrbutos. Mcroelectrocs Relablty, Volume 35, Number 2, December 995, 45-460 [relasoft03] RelaSoft, Relablty Importace ad Optmzed Relablty Allocato (Aalytcal http://www.webull.com/systemrelweb/blocksmtheory.htm, 2003. [asa04] NASA Documet, Relablty Allocato, Oct, 2004, foa.msfc.asa.gov/docs/nas8-0079/s&maoi/r- Relablty/QD-R-008.pdf 7