OPTIMAL REDUNDANCY ALLOCATION FOR INFORMATION MANAGEMENT SYSTEMS

Relablty ad Qualty Cotol Pactce ad Expeece OPTIMAL REDUNDANCY ALLOCATION FOR INFORMATION MANAGEMENT SYSTEMS Ceza VASILESCU PhD, Assocate Pofesso Natoal Defese Uvesty, Buchaest, Roaa E-al: caesav@ca.o Abstact: Relablty allocato eques defg elablty objectves fo dvdual subsystes ode to eet the ultate goal of elablty. Idvdual elablty objectves set fo softwae developet ust lead to a adequate ato of te-legth, level of dffculty ad sks, as well as decease developet pocess total cost. Thus, edudacy esues eetg the elablty equest by toducg a suffcet quatty of spae equpet. But the sae te, ths soluto leads to a cease weght, sze ad cost. The a of ths pape s the vestgato of elablty allocato to specfc sets of softwae applcatos () ude the ccustaces of zg developet ad pleetato costs by usg the Roe Reseach Laboatoy ethodology ad by coplyg wth the codtos of costs zato tggeed by the toducto of edudaces [GHITA 00]. The pape aalyses the ways whch the softwae elablty allocato gadual ethodology ca be exteded. It also aalyses the ssue of optal syste desg tes of elablty allocato by usg stuets of atheatcal pogag ad appoaches the vaato of elablty ad syste cost by takg to accout the edudacy toduced the syste. Ths pape s also gog to povde a exaple of calculus whch uses a epesetatve softwae syste ad llustates the ethodology of optal allocato of specfc sets of softwae applcatos elablty. Key wods: Relablty allocato; Optal edudacy; Icease of softwae applcatos elablty; Applcato softwae tools Itoducto Relablty allocato eques defg elablty objectves fo dvdual subsystes ode to eet the ultate goal of elablty. Idvdual elablty objectves set fo softwae developet ust lead to a adequate ato of te-legth, level of dffculty ad sks, as well as decease developet pocess total cost. Thus, edudacy esues eetg the elablty equest by toducg a suffcet quatty of spae equpet. But the sae te, ths soluto leads to a cease

Relablty ad Qualty Cotol Pactce ad Expeece weght, sze ad cost. I ths espect, softwae elablty allocato gadual ethodology [ROME 97] ca be exteded to clude the appoach used [GHITA 96]. The latte aalyses the ssue of optal syste desg tes of elablty allocato by usg stuets of atheatcal pogag ad appoaches the vaato of elablty ad syste cost by takg to accout the edudacy toduced the syste. Cosequetly, the a of ths pape s the vestgato of elablty allocato to specfc sets of softwae applcatos () ude the ccustaces of zg developet ad pleetato costs by usg the Roe Reseach Laboatoy ethodology ad by coplyg wth the codtos of costs zato tggeed by the toducto of edudaces [GHITA 00]. Befoe poceedg ay futhe soe theoetcal clafcatos ae eeded. Fstly, elablty allocato as vewed by [ROME 97] efes to allottg elablty specfcatos at syste level to softwae odule level (be thee a o-edudat cofguato). Relablty allocato as vewed by [GHITA 00] efes to the optal allocato of edudacy ode to each the elablty level set though elablty specfcatos. I coclusos, the copleetaty of the two appoaches s woth etog. Secodly, wth the cotext of foato aageet systes, the te edudacy efes both to the exstece of seveal specfc sets of softwae applcatos () developed ad desged depedetly ad whch have the sae fuctos, ad to testg ad upgadg these sets. All ths cosdeed, ths pape s also gog to povde a exaple of calculus whch uses a epesetatve softwae syste ad whch llustates the dea of the possblty of egg the two ethodologes. Moeove, the cocluso that s to be daw s that the odelg of the elablty cease by techologcal eas (.e. by testg ad upgadg the softwae) ad by edudacy s a ecessty. The Icease of Softwae Applcatos Relablty though Redudacy The hypothess udelyg the aalyss of the softwae elablty cease of the foato aageet systes s that these systes ae pat of those systes that ae fault-toleat. I ths espect, edudacy (vewed as the use wth a syste of oe eleets tha ecessay fo ts fuctog ode to have the syste u flawlessly eve the pesece of beakdows/falues [SERB 96]) s the basc eleet that assues the elablty of these systes. Othe eleets ay coce hadwae o softwae subsystes ad ca be taced at ay level, statg fo dvdual copoets up to the whole syste (hadwae ad/ softwae). Wth egad to elablty, the foato aageet systes softwae has a heachcal fuctoal patto, begg wth the Msso Specfc Tools Set (MSTS), Softwae Applcatos sets () ad the softwae odules wth the, all of whch cludg edudat copoets ad echass to eestablsh the fuctog. The basc ethods fo the fault toleace theoy fo the hadwae feld ca be adapted ad appled to the softwae of the foato aageet systes. Thus, ode to assue ts toleace to falues, ecodg logcal fuctos by usg edudat codes, eo ecogto ad eo eoval by sceeg faults wth the help of ultple (edudat) softwae odules stalled dffeet syste equpets o fuctoal ecofguato of

Relablty ad Qualty Cotol Pactce ad Expeece the syste by actvatg a spae softwae eleet that s to eplace the faled eleet ca be used. These ethods udele the suggestos ade by the foato aageet systes desges to use the followg basc fos of edudat softwae achtectues (fos that assue a cease elablty egadless of the heachcal fuctoal level - MSTS,, softwae odule): - The tple odula edudacy. It cludes thee detcal fuctoal odules that cay out sla tasks. The esults ae subject to the pocess kow as votg that scees a possble eoeous fuctog of oe of the odules. - Duplcato wth copaso. It s based o two fuctoal odules that assue cayg out sla tasks. If due to the paallel fuctog esults (outputs) dffe, dagoss pocedues ae caed out to detfy the faulty odule. - Dyac edudacy. It cotas seveal odules wth sla fuctos. Howeve, oly a pat of the fuctos ae opeatoal, wheeas the othe s o stad-by. Whe a falue s detfed the oes o stad-by becoe opeatoal ad take ove the tasks of the faulty oes. All these thee basc fos of edudat softwae achtectues ae to be foud the pleetato of the specfc sets of softwae applcatos (). I ode to evaluate the latte s elablty pefoace ths pape stats fo the hypothess that sceeg faults s stataeous ad that the faults of the dvdual copes of s ae depedet. Moeove, I a to eploy elablty logcal odels covetoally epeseted a way sla to those specfc to the evaluato of the elablty fuctos fo edudat hadwae stuctues. The followg exaples dsplay the evaluato of elablty pefoace usg as bblogaphy the evaluato of elablty fuctos of edudat stuctues [SERB 96]. Exaple The tple odula edudacy ade up of detcal s The tple odula edudacy s ade up of thee detcal s whee R ( t) s the elablty fucto ad a vote whee R V ( t) s ts elablty fucto. The elablty fucto of the tple odula edudacy ca be odeled by statg fo the logcal elablty odel (fg. ) R (t) R (t) R V (t) V R (t) Fgue. The elablty logcal odel fo the tple odula edudacy

Relablty ad Qualty Cotol Pactce ad Expeece Fo a good fuctog of the softwae syste, at least s ad the V vote ust fucto coectly. The fuctog pobablty of the edudat syste ude dscusso s gve by the geeal foula [SERB 96] fo k-out-of- systes: whch s () t C R() t ( R() t ) R RV k R R V ( ) () t R () t R ( t) Exaple The tple odula edudacy ade up of o- detcal s The thee s pefo the sae fuctos but they ae dffeet tes of desg ad pleetato. The elablty logcal odel s sla to the oe fg., wth the obsevato that the s have dffeet elabltes whch ae gve the otato R () t. I ode to calculate the elablty fucto the ethod of exhaustve eueato of syste states s used. I table the pobabltes of coect fuctog of the syste ad the pobabltes assocated to these evets ae peseted. Table. The pobabltes of good fuctog of the syste wth o-detcal s Seq. The evets assug the good fuctog The pobablty of the evet. R ( t) R ( t) R ( t). R ( t) R ( t) ( R ( t) ). 4. R ( t) R ( t) ( ) ( ) R () t ( R t ) R ( t) R ( t) The good fuctog of the syste s assued by jog all fou evets. They ae copatble wth oe aothe ad thus the pobablty of the good fuctog of the tple odula edudacy s: R() t RV () t [ R () t R ( t) R ( t) + R ( t) R ( t) ( R ( t) ) + + R () t ( R () t ) R () t + ( R () t ) R () t R ()] t RV () t [ R () t R () t + R () t R () t + R () t R () t - R () t R () t R ()] t I the two exaples, the odelg of the elablty fucto of the edudaces does ot take to accout the staces of eo-copesato. Cosequetly, 4

Relablty ad Qualty Cotol Pactce ad Expeece the pobablty of the evet to have falues, < falues, < falues ad the thee s to fucto: P R λt λt λt () t ( ) R () ( ) () ( ) t R t!!! whee ( λ ) s the falue ate of a. Thee s a ube of P peutatos fo the tple (,, ) wth a vew to detfyg the eos of the thee s:, P, sau 6, > > Each tple s assocated wth a P,, codtoed pobablty defed as: The syste fuctos coectly f t cotas, o eos, whee ca be set to ay value, < ad <. Cosequetly, the elablty fucto of the s calculated accodg to the elato: R () () ( ) P () ( ) λt λt () ( λt t P R t R t R t ) R R 0 0 0 () t 0 0 0,, P P,, () t P P R () t + R () t + 000 0,0,0 0 P P,, ( λt)! ( λt) + +!!! + +!!! P 00 P!,0,0 ( λt) By ackowledgg that fo softwae systes thee s a expoetal epatto fo λt u te, fo whch R () t e, t esults: o R () t λt +! ( λt) 0! ( λt) R () t! R () t By eplacg, t esults: () R () t + R () t ( R () t ) + R ( t) R t! + 0 P! P,, ( λt) + +!!! 5

Relablty ad Qualty Cotol Pactce ad Expeece Exaple Dyac edudacy Be thee a dyac edudacy ade up of two s, a basc (fuctoal) oe - ad a spae oe -. The spae ca be fuctog o o stad-by ad ca be detcal (o ot) wth the fuctoal oe. The followg otatos ae to be used: R () t R () t R ( t) - the elablty fucto of the basc ; - the elablty fucto of the spae fuctog ; - the elablty fucto of the spae stadby. The logcal odel of the dyac edudacy s peseted fg.. R (t) R C R (t) Fgue. The logcal elablty odel of the dyac edudacy The dyac edudacy ca successfully fucto o log-te f the followg evets take place:. (basc ) fuctos well fo the (0, t) te duato; fo the pobablty of ths evet we gve the otato P R ( t) ;. fals at te oetτ, wheeτ < t ; (spae ) s pope fuctog codto ad t woks well fo the te teval ( τ, t). The pobablty of falue wth the fte sall te teval ( τ, τ + dτ ) s f ( τ ) dτ, ad the pobablty of at the τ oet ad of the fuctog fo the τ oet utl the t oet, wth fuctog codto at the τ oet s: f ( τ ) R R ( τ ) R ( t τ ) dτ If 0 < τ < t, the pobablty of the P coposed evet s: t P f ( τ ) R ( τ ) R ( t τ ) dτ R 0 The two evets ae copatble. Thus, the pobablty of a good fuctog of the dyac edudacy s: t R() t R () t + f ( τ ) R ( τ ) R ( t τ ) dτ R 0 6

Relablty ad Qualty Cotol Pactce ad Expeece The Optal Allocato of Applcato Softwae Redudaces The poble of elablty allocato ssues dug the stage of povsoal elablty evaluato. The pape [GHITA 00] offes solutos fo the optal allocato of elablty fo geeal stuatos by tacklg the topc of objects ade up copoet equpets ad puts fowad a way of choosg the type of edudacy that best eets the elablty equeet. I what follows I would lke to deal wth the ssue of adaptg the ethodology of elablty allocato to the elablty of specfc sets of softwae applcatos ad to peset a ethodology- adequate calculato poga that would eable solvg case studes. The fst thg ude cosdeato s the poble of avalablty allocato (adapted afte [SERB 96]) f the IT syste s desged as a seal coecto of (paallel) edudaces of subsystes. Usually, syste desg stats by toducg a u ube of fuctoally ecessay equpets ts stuctue. The esultg stuctue s, fo the elablty pot of vew, a seal oe. Sce seal stuctues have the lowest elablty, they ay ot eet elablty equeets ad, cosequetly, the desge s to cease syste elablty statg fo edudacy the ube of eleets. By gvg the otato of D ( ) to the avalablty of equpet ube, equpet whch has " " edudat (sae type of) equpets ad the otato of ( l,,..., ) to edudacy at poduct level, whee s the ube of equpets, t esults that D() s expessed as: D D Avalablty calculato D ( ) depeds o the type of edudacy pactced (edudacy though the desg of paallel systes, out of, o by usg spae equpet). I the fst two alteatves, edudat equpets wok ude the sae codtos as basc equpet does. O the oe had, that assues a techcally ease soluto. Howeve, the ssug elablty s less good copaed to the last alteatve. Fo ths alteatve of paallel edudacy D + ( ) d fo 0, t esults D d d λ λ + μ whee D s the avalablty of a equpet of type. Though edudacy, the elablty equeets ca be et by toducg eough spae equpet. Noetheless, weght, sze ad cost cease. If we gve the otato of C () to the cost of edudat equpets wth the syste, the latte s calculated as follows: C ( ) ( ) c whee c s the cost of a equpet of type. 7

Relablty ad Qualty Cotol Pactce ad Expeece Fo the cost elato t esults that the fucto ceases ootoously as agast ay copoet. Of all solutos, the oe that eets the elablty codto at the lowest cost ust be chose. I cocluso, the poble of the optal desg of the syste s foulated as follows: of all edudat solutos, oe ust fd the soluto that zes the cost C(), be thee a estcto, whch D D() s calculated accodace wth the elatos above. The elablty equests fo ca be expessed as follows: P P o whee K D K D P s the pobablty of good fuctog; K D s the avalablty coeffcet; P ad K D ae the u values of elablty dcatos. The elablty equeet ca be thus et by [GHITA 96] [GHITA 00]: a) ceasg syste s copoets elablty; b) ceasg (povg) syste s epaablty; c) usg soe elablty edudaces. The thd alteatve s gog to be dscussed oe detals the followg paagaphs. Usually softwae desg stats fo the basc pcple of a u ad fuctoally ecessay ube of odules wth the syste. Relablty aalyss pots out that the latte s a seal stuctue of low elablty. Relablty cease dug the desg stage s doe by havg a edudacy toduced as fa as the ube of odules s coceed. If P ( ) s the otato fo the pobablty of good fuctog of a that has edudat odules of the sae type ad the edudacy at the level of the, whee epesets the whole set of s s gve the otato (,,..., ) ube of s, t esults that P ( ) s expessed though the elato: P ( ) P ( ). Calculatg pobabltes P ( ) depeds o the type of edudacy eployed: edudacy by desgg systes of paallel softwae odules; edudacy by desgg systes of out of softwae odules; the use of spae softwae. The last alteatve has the advatage of assug a elablty cease supeo to the othe two fo whch the systes of edudat softwae odules wok the sae ae as the basc oes. 8

Relablty ad Qualty Cotol Pactce ad Expeece edudacy s: whee P The pobablty of a k k ( ) C P ( P ) k + good fuctog fo the fst two alteatves of + k P - pobablty of good fuctog of a odel of type ; P λt e (a expoetal epatto law follows); wth λ - falue testy of the odule type ad t - sso duato. If the syste s of a paallel type ad f > the syste s of a -out- of- type. As fo the edudacy though spae softwae odules, each odule togethe wth the edudat odules fos a kt that fals whe the + odules fal. follows: whee P I ths case, the pobablty to have a exact ube of k falues s calculated as k λ t ( k) P( ν k) ( λ t) e, λ - falue testy of odules; ν - ube of type faled odules. But P ( ) P( ) P ν, thus esultg the elato: k λ ( ) ( t) ( e t λ )/ k! k 0 The pevous foula s vald f we ae to accept the hypothess accodg to whch falue ad odule eplaceet s stataeously doe though a spae odule ad that pobablty equals. By havg the calculus foulas the good fuctog of aalyzed t esults that they ae ootoously ceasg fuctos. I cocluso, egadless of the P level, thee s the possblty of eachg the desed elablty level by cludg eough edudat softwae odules. l P ( ) s l P ( ) Howeve, oe obsevato ust be ade ths espect: by toducg ay ube of edudat softwae odules wth the stuctue of a, ts coplexty ad cost autoatcally cease. I all softwae systes, total cost educto s a effcecy cteo uaously accepted. Cosequetly, a optal equlbu betwee the desed elablty fo a, the ube of edudat softwae odules ad the cost of ths actvty eeds to be eached. I the geeal cocept of cost we clude the desg/ developet costs, softwae ateace/ explotato costs ad dowte costs. By gvg the cost of toducg wth the edudat odules the otato C ( ), ts value ca be estated as follows: 9

Relablty ad Qualty Cotol Pactce ad Expeece whee C ( ) c, c s the cost of a odule of type. Fo the cost elato t esults that the fucto ceases ootoously as copaed wth ay copoet. Fgue pesets a qualtatve gaph of elablty ad cost vaato as copaed wth the edudacy eployed. Each dot o the gaph coespods to a vecto, ad by ceasg the ube of copoets the dots ove up ad to the ght. The optal desg of the s doe by selectg the dots that go ove P (edudat dots that eet the elablty equeet) of the ost cost-effectve alteatve. Thus, t esults that the poble of fdg a optal desg soluto petas to atheatcal pogag (optu wth estctos) ad that t dsplays ceta patculates [GHITA 00]: t s a poble of o-lea pogag - P ( ) s ot a lea fucto as copaed wth aguet ; t s a poble of whole ubes pogag- aguets ae whole ubes. Cosequetly, the poble of elablty allocato s a poble of whole ubes o- lea pogag that s to be woked out by usg specfc ethods. Fgue. Vaato of elablty ad cost accodace wth the edudacy I the gaph dsplayed fgue thee s a le of dots o the uppe sde called doat vectos ad they ae optal solutos as copaed wth the othe dots. Thus the optal soluto s the fst vecto fo the le of doat vectos that go ove level P. It esults that detfyg the optal soluto s a atte of usg the appopate ethods by whch soe doat vectos ae obtaed. A vecto s called doat f the followg codtos ae et: ' ' P > P C > C. ( ) ( ) ( ) ( ). P( ) P( ' ) C( ) C( ' ) The detfcato of the doat vectos s doe by usg the fuctos: P ( k + ) ( k) l c P( k ) ϕ 0

Relablty ad Qualty Cotol Pactce ad Expeece whch evaluate the pobablty cease pe ut of cost fo a copoet wth k edudaces. All pocedues ae wokable f the fuctos ϕ ( k) ae covex ad fo the pevously etoed stuctues (- out- of- systes o spae oes) ϕ ( k) ae covex. The pocedue below supples a le of doat vectos ( k) ( ( k), ( k),..., ( k) ), k,,, N whch. () (0, 0,, 0). (k+) s ecuetly deduced as follows ( k + ) ( k) + daca I ( k + ) ( k) daca I whee I s the dex ube that axzes fucto ϕ ( k) ; (f thee ae oe dces I, ode to obta axu possble oe of the s selected as dex I);. Algoth stall esults fo: N k : P k P { ( ( )) } I cocluso, the pocedue leads to a le of doat vectos deduced oe fo the othe by addg oe ut fo each aguet that eaches the geatest cease pobablty pe ut of cost. The cha begs wth the detcal ull vecto ad eds wth the fst vecto that eets the elablty codto (C). Ths pocedue supples a cha of doat vectos that does ot ecessaly clude all possble doat vectos betwee the detcal vald vecto ad vecto ( k ). Cosequetly, t does ot always supply a optal soluto, but a quas-optal oe. The pocedue has the advatage of copletely takg algoth fo ad of beg easy to pleet o a copute. Its a dsadvatage esdes the fact that t stats fo vecto (0, 0,..., 0) ad thus a ube of steps ust be take towads fdg the fst doat vecto that eets the elablty ad cost equests. A oe dect ethod (wth fewe steps) towads obtag a cha of doat vectos s the oe ecoeded [BARLOW 9] ad whch volves usg oe of the followg pocedues. Pocedue It s sla to the pocedue pevously descbed ad t helps detee the whole cha of doat vectos by statg fo vecto (0, 0,..., 0) ad successvely toducg edudaces accodace wth the cease cteo. Pocedue It s a opeatoal alteatve that helps detee oe doat vecto that coespods to the posed level of pobablty P. It s based o the ( ),..., patculaty that lets pobablty P be, thee s a costat value so that all the copoets of the doat vecto eet the codto: k k P. { : ϕ ( ) < δ ( )} ϕ k ae postve ad ootoously deceasg ad ( P ) > 0 Sce fuctos ( ) δ, the pevous elato always assues fdg copoets. The advatage of ths pocedue cossts dectly supplyg vecto that coespods to the posed

Relablty ad Qualty Cotol Pactce ad Expeece pobablty level P. The dsadvatage les ot offeg ay clue as to the ae of choosg the costat value δ ( P ), whch s doe though successve tals. Pocedue It cossts jog pevous pocedues by usg the advatages. If a level of pobablty doat vecto ( ) δ usg pocedue. P s posed, a estate value fo the costat ( P ) δ ad ts coespodg ae establshed though successve tals, so that P ( t, ( )) < P δ by Oce ( δ ) s establshed, by usg pocedue the cha of doat vectos s establshed ts tu utl the doat vecto obtaed. P, P s that eets codto ( t ) Case Study: The Methodology of Optal Allocato of Relablty I ths sub-chapte we gve a exaple that llustates the ethodology of optal allocato of elablty [VASILESCU 05] by usg the optzed ethod of doat vectos calculato that was explaed detal the pevous paagaphs (pocedues -). () () () (k) (k) (k).... () () ().. Modulul () Modulul () Modulul () Fgue 4. Specfc set of softwae applcatos () wth edudaces at the softwae odules level I ode to set the bass of ths calculus, hee ae the tal data of the poble. We aalyze a IT syste, whch a coad ad cotol actvty s suppoted though a specfc set of softwae applcatos () cosstg of thee softwae odules (fgue 4). Table depcts the falue testes ad the specfc costs. Table. Specfc set of softwae applcatos - tal data I λ (hou - ) c (u.c.) 0,0008 00 0,0005 00 0,000 50

Relablty ad Qualty Cotol Pactce ad Expeece Fo axu geealty we pefeed expessg the c values ut costs (u.c.). Fo a effectve aalyss of ths case study ad ode to obta elevat esults we aalyzed the fuctog of the thee s fo the te legth of t000 hous. The elablty of the softwae applcatos set s ceased by toducg a edudacy wth ts odules. The type of edudacy chose s the oe based o toducg soe spae softwae odules. The elablty equeet s P 0,95. Poble: The optal desg of the by choosg the edudacy alteatve that eets the elablty equeet ad that volves the lowest cost fo the edudat odules. The soluto to ths poble s gve by pocedue, fo P 0,95. I ode to establsh a oetatve value fo the costat ( P ) δ we take P 0,75 < P 0,95 as a pobablty ad assue that all ts copoets, P, P P ae equal. Cosequetly, P P 0,75 0, 9. The teeday esults ae povded table 4. As we otce fo the table, P ( 7) 0, 8867 s the closest value to P 0,9 ad cosequetly/ t esults that, ax k : P( k) < P 7. Table 4. Establshg the oetatve value of the costat ( ) k P (k) R (k) 0 0,008 0,008 0,095 0,0477 0,0948 0,45 0,57 0,94 4 0,80 0,476 5 0,747 0,650 6 0,98 0,7908 7 0,0959 0,8867 8 0,0575 0,944 If { k : ϕ ( k) < δ ( P )} δ ϕ, the oetatve value s: ϕ ( 7) P l c P ( 8) ( 7) δ - teeday esults P 0,004 By usg pocedue we obta the copoets of the doat vecto that have to eet the codto: { k : ϕ ( k) < δ ( P )},

Relablty ad Qualty Cotol Pactce ad Expeece As a esult, the fstϕ that eets the pevous codto s ϕ () 8 0, 00060, aely the fst doat vecto ( 7,8,6) ϕ that eets the pevous codto s ϕ ( 6) 0, 00079 wth ts coespodg pobablty P 0,6 < 0, 95.. It esults the Moeove, by eployg the ule of cease fo pocedue the esult s the cha of doat vectos peseted table 5 ad the coespodg values P ( ) ad C ( ). Table 5. Cha of doat vectos P ( ) C ( ) 7 8 6 0,695 500 7 9 6 0,7906 5600 7 0 6 0,88945 5900 7 6 0,9460 600 8 6 0,97565 6400 We otce that the fst doat vecto that eets the codto P ( ) 0, 95 at the lowest cost s ( 8,,6), wheeas ts coespodg pobablty s ( ) 0, 975 C u.c. the cost of ( ) 6400 P at We also obseve that by usg pocedue we eeded 5 steps to obta the esult, wheeas fo the pocedue we would have eeded exta steps. I ode to pleet foulas ad do the calculatos we used Mcosoft Excel due to the possblty t offes to toduce tal data a apd ae, ad also because of the elegat ad explct layout fo the esults povded by ts speadsheets. Excel speadsheet cells explaato: D4:D6 - testy of falues the odules that wee gve the otatos, ad ; E4:E6 - odules specfc cost; H - sso duato hous; B9:B - ube of edudat odules toduced; D9:D (H9:H, L9:L) - values of odules elablty; fo exaple, cell D0 cotas the foula D9+POWER($F$4$D$4$H$;B0)/FACT(B0)EXP(-($F$4$D$4$H$)); E9:E (I9:I, M9:M) - values of the elablty cease fo odules pe ut cost; fo exaple, cell E0 cotas the foula (/$E$4)LN((D/D0)); C5:E5 (C9:E9) - chas of doat vectos; fo exaple, cell C6 cotas the foula IF(MAX(E7;I7;M7)E7;C5+;C5); F5:F9 - values of oveall elablty; fo exaple, cell F5 cotas the foula D6H6L6; G5:G9 - values of oveall costs; fo exaple, cell G5 cotas the foula C5$E$4+D5$E$5+E5$E$6. 4

Relablty ad Qualty Cotol Pactce ad Expeece Fgue 6. Optal allocato of the elablty by usg the ethod of calculatg the doat vectos- esults 5

Relablty ad Qualty Cotol Pactce ad Expeece I cocluso, ode to eet the posed elablty equeet a specfc set of softwae applcatos tools s eeded. The latte cludes: eght odules type oe, eleve odules type two ad sx odules type thee. The cost of the s 6400 ut cost. Refeeces. BARLOW, R., PROSCHAN, F. Matheatcal Theoy of Relablty, Joh Wley, New Yok, 998. GHITA, A., IONESCU, V. Metode de calcul î fabltate, Edtua Acadee Tehce Mltae, Buchaest, 996. GHITA, A., IONESCU, V., BICA, M. Metode de calcul î eteabltate, Edtua Acadee Tehce Mltae, Buchaest, 000 4. SERB, A. Sstee de calcul toleate la defecta Edtua Acadee Tehce Mltae, Buchaest, 996 5. VASILESCU, C. Alocaea opta a fabltat setulo specfce de aplcat softwae d ssteele C4ISR, The 0 th Iteatoal Scetfc Cofeece, Acadea Fotelo Teeste, Sbu, Novebe 4-6, 005 6. Syste ad Softwae Relablty Assuace Notebook, poduced fo Roe Laboatoy, New Yok, 997 7. TR-9-5 - Softwae Relablty Measueet ad Test Itegato Techques, poduced fo Roe Laboatoy, New Yok, 99 Ceza Vaslescu has gaduated the Faculty of Electocs ad Ifoato Scece wth the Mltay Techcal Acadey - Buchaest 997. He holds a PhD dploa Copute Scece fo 006. He has gaduated 00 the Advaced Maageet Poga ogazed by Natoal Defese Uvesty of Washgto D.C., USA. Also, he has eceved the US Depatet of Defese Chef Ifoato Offce (CIO) cetfcato fo the Ifoato Resouces Maageet College of Washgto D.C. Cuetly he s the head of the IT&C Offce wth the Regoal Depatet of Defese Resouces Maageet Studes - Basov ad assocate pofesso at the Natoal Defese Uvesty - Buchaest. He s the autho of oe tha 0 joual atcles ad scetfc pesetatos at cofeeces the felds of hadwae/softwae elablty, coad ad cotol systes ad foato esouces aageet. He has coodated as poga aage the actvty of establshg Roaa of a teatoal educatoal poga the feld of foato esouces aageet, collaboato wth uvestes fo USA. Besde hs eseach actvty, he has coodated Ta the Taes ad Educate the Educatos actvtes wth teatoal patcpato. Ma publshed books: - Ifoato Maageet, Mltay Techcal Acadey Publshg House, Buchaest, 006. - Ifoato Techology fo Maageet, Regoal Cete of Defese Resouces Maageet Publshg House, Basov, 00. Codfcatos of efeeces: [BARLOW 98] BARLOW, R., PROSCHAN, F. Matheatcal Theoy of Relablty, Joh Wley, New Yok, 998 [GHITA 96] GHITA, A., IONESCU, V. Metode de calcul î fabltate, Edtua Acadee Tehce Mltae, Buchaest, 996 [GHITA 00] GHITA, A., IONESCU, V., BICA, M. Metode de calcul î eteabltate, Edtua Acadee Tehce Mltae, Buchaest, 000 [ROME 9] TR-9-5 - Softwae Relablty Measueet ad Test Itegato Techques, poduced fo Roe Laboatoy, New Yok, 99 [ROME 97] Syste ad Softwae Relablty Assuace Notebook, poduced fo Roe Laboatoy, New Yok, 997 [SERB 96] SERB, A. Sstee de calcul toleate la defecta Edtua Acadee Tehce Mltae, Buchaest, 996 [VASILESCU 05] VASILESCU, C. Alocaea opta a fabltat setulo specfce de aplcat softwae d ssteele C4ISR, The 0 th Iteatoal Scetfc Cofeece, Acadea Fotelo Teeste, Sbu, Novebe 4-6, 005 6