Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs Uso e Dagramas e Esao UML para la Moelacó el Desempeño e Programas Paralelos Jorge Orega Aroa Deparameo e Maemácas, Facula e Cecas, UNAM loa@fcecas.uam.mx Arcle receve o Jue 28, 27; accepe o Ocober 24,27 Absrac There are may possbles o esg a parallel program orer o oba he bes performace possble. The seleco of a program srucure, as a orgasao of processes, mpacs o he performace o be acheve, a epes o he problem o be solve. Now, orer o selec a program srucure as he bes erms of performace, he sofware esger requres performace moellg echques o evaluae ffere aleraves. If he srucure of he parallel program ca be moelle as a se of eracg processes, escrbe erms of UML Sae Dagrams, hs paper preses a performace moellg o esmae he average execuo me of a parallel program. Performace moellg s acheve by calculag he average execuo me of a parallel program, escrbe as a se of processes whch ru wh eermscally a expoeally srbue execuo mes. Keywors:.Performace moellg, parallel program, UML Sae Dagram Resume Hay muchas posblaes para señar u programa paralelo a f e obeer el meor esempeño posble. La seleccó e ua esrucura el programa, así como ua orgazacó e procesos, mpaca sobre el esempeño a lograrse, y epee el problema a resolver. Ahora be, para seleccoar ua esrucura el programa como la meor e érmos e esempeño, el señaor e sofware requere e éccas e moelacó para evaluar ferees opcoes. S la esrucura e u programa paralelo puee moelarse como u couo e procesos eracvos, escros e érmos e Dagramas e Esao e UML, ese arículo presea ua moelacó para esmar el empo e eecucó promeo e u programa paralelo, escro como u couo e procesos que corre e empos e eecucó co srbucoes eermísca y expoecal. Palabras clave: Moelacó e esempeño, programa paralelo, Dagrama e Esao e UML Irouco Durg he las few years, parallel compug has bee propose as a poeal soluo for he creasgly complex problems several research a evelopme areas le quaum chemsry, flu mechacs, weaher forecasg, a ohers. Desgg a programmg parallel programs requres a exraorary effor of he sofware esger, who has o balace bewee he complexy of he parallel mplemeao a he performace expecaos. A he al sage of parallel sofware evelopme, he sofware esger cous oly wh he formao of he problem o solve, he avalable parallel harware plaform, a he programmg laguage o use. Base solely o hs formao a o he sofware esger experece, a parallel program s commoly esge a mplemee. Bu, as parallel programmg represes a hgh cos erms of evelopme effor a me, woul be a avaage o cou wh quafable formao before furher seps are ae urg esg a mplemeao. Hece, he sofware esger coul be able o selec a program srucure or aoher, regarg he parallelsm coae he problem a ha. I geeral, a sofware esger oes o ow avace whch of he varous parallel srucures, escrbe as a se of eracg processes, woul have he esre execuo me o a gve parallel plaform. Thus, he sofware esger faces wo aleraves:. The sofware esger ca mpleme he varous parallel srucures. The parallel harware plaform s avalable, so he mplemeaos are possble. Neverheless, hs approach requres a lo of effor a me o es every possble soluo, a herefore, es o be very expesve erms of boh, me a effor. Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
2 Jorge Orega Aroa 2. Isea, he sofware esger ca moel he varous parallel srucures, a ry o f he bes oe by evaluag he moels, usg performace smulao moels. Ths paper preses a approach, base o he seco alerave, o obag a average rume of a parallel program. The basc assumpo s ha he whole parallel program cosss of processes whose saes are combe o oba a overall sae of he parallel program. The rumes of he saes are moelle by a raom varable a s srbuo fuco. Moreover, he moel s base o he epeecy bewee he saes of he processes. Ths s escrbe usg UML Sae Dagrams Booch, e al., 998; Fowler & Sco, 997). Fgure shows he UML Sae Dagram of a smple parallel program cossg of wo processes, A a B. Process A has wo saes a a a 2, whereas process B has wo saes b a b 2. The agram caes ha processes A a B execue smulaeously saes a a b. Process A ca ge o sae a 2 oly afer fshg sae a. However, process B ca oly ge o sae b 2 whe boh processes A a B have respecvely fshe saes a a b. Process A a a 2 Process B b b 2 Fg.. UML Sae Dagram of a smple parallel program The aalyss of hs of agrams es o be very complex whe creasg he umber of parallel processes a her saes. However, f coul be fou a equvale sae agram whch cosers he saes of he parallel program as a sgle ey base o he varous possble sae combaos of s processes, a also, coul be measure he rume srbuos of all processes urg such saes, he woul be possble o compue he srbuo of he overall parallel program rume. Moreover, orer o oba more realsc moels, s ecessary o moel he behavour wh a sae usg srbuo fucos ha approxmae o measure emprcal srbuo fucos. The obecve of hs paper s o prese a aalyss meho whch ca be apple o compue he srbuo of he overall parallel program rume base o a equvale sae agram of he parallel program a measure aa abou he rume srbuos of he parallel processes. Seco 2 preses some relae wor he areas of Relably Egeerg, Performace Egeerg, a Parallel Programmg. Seco 3 explas how o compue he average rume of a program whch cosss of processes wh expoeally srbue rume varables Kleroc, 975). Seco 4 preses he aalyss meho ha allows o approxmae he overall parallel program rume by moellg he processes rumes usg expoeally a eermscally srbue raom varables. Fally, Seco 5 preses he execuo of smulao moels ha solve he Hea Equao problem, as a case suy o valae he meho. 2 Relae Wor Several oher smlar approaches have bee evelope for moellg he performace a relably of sofware sysems, wheher hese mae use of reuco of sae agrams for Relably Egeerg Bllgo & Alla, 992), mae use of UML agrams for Performace Egeerg Pooley & Kg, 999), or are use for basc parallel programmg Lu e al., 998). Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs 2 Bllgo a Alla 992) mae use of space sae agrams a ewor moellg echques for evaluag he relably of a sysem, a moel a/or a compoe. Mosly, hese agrams are use o represe falure-repar processes. The essece s o erve a se of equaos suable for seres-parallel sysems: a) seres sysem, all compoes mus operae for sysem success, a b) for a parallel sysem, oe compoe ee o wor for sysem success. So, hese equaos are use o euce he probably of he sysem o be ow sae or up sae, by reucg he ffere probables of resg each of he sysem saes, ervg he approxmae sae probables for each moel a seres-parallel sysem. I a smlar way, here we mae use of UML sae agrams as space sae agrams for epcg he parallel compoes saes. However, he way whch he sae probables are reuce s ffere: saes are o reuce usg equaos of he seres or parallel probably of resece of each compoe each sae, bu he sae of he sysem s globally cosere by coserg he preceece of saes, a hece, he sysem sae s moelle as a sgle ey, obag a equvale UML agram whch aes o coserao oly preceece of saes o moel he sysem s performace. Pooley a Kg 999) prese a horough revso of UML, a s poeal o be use Performace Egeerg. They prove a bref bu escrpo of each UML agram as a poeal moellg ool for performace. Neverheless, hey oly shallowly escrbe how o explo use case agrams, mplemeao agrams, sequece agrams, collaborao agrams, acvy agrams, a sae agrams. They compleme hese UML agrams wh queug moels orer o erve performace moels. Uforuaely, hey o o eeper o furher escrbg ay of he UML agrams for moellg performace. I he parcular case of UML sae agrams, hey ulmaely meo ha hs approach requres a lo of wor, provg o furher formao abou. I he prese paper, we exclusvely focus o UML sae agrams o reuce he saes of a parallel sysem, obag a equvale sae agram, whch s acually use for performace moellg. I fac, he reame gve here goes beyo smply coserg he sae of he parallel compoes, eeper o a aalyss of he saes wh he agram a her reuco o a sgle equvale UML agram. The equvale saes of hs UML agram are moelle by eermscally a expoeally srbue varables. Lu e al. 998) perhaps prove he closes approach o he oe presee here. They also mae use of space sae agrams a seres-parallel reuco, as well as expoeally srbue varables, for ervg performace moels for smple for-o parallel programs execug o a mulprocessor evrome. Neverheless, hey o o ae o coserao aoher or more realsc parallel program srucure. For-o programs are commo, bu hey e o eglec he commucao bewee parallel compoes. Hece, hese programs o o cover oher ffere ypes of parallel sysems, such as Commucag Sequeal Elemes Orega-Aroa, 2), whch hghly epes o he commucao bewee parallel compoes. I hs paper, UML sae agrams are erve recly by he preceece relaos of he parallel program srucure. Thus, such sae agrams reflec he behavour of he parallel program epeg o boh, compuao wh compoes) a commucao whch affecs he preceece of compuaos). So, epeg o he orgazao of parallel compoes, ffere sae agrams are obae. These agrams are reuce regarg he preceece of saes o a smpler moel of compuao, whch s acually use for performace moellg by ag o coserao varaos of he me cosume by he saes of he parallel program as a whole. The followg secos expla how execuo mes are moelle usg eermscally a expoeally srbue varables. 3 Usg Expoeally Dsrbue Varables for Moellg Execuo Tmes I orer o aalyse a sae agram wh oly expoeally srbue rumes, ca be use he sae space meho Thomasa & Bay, 986). I hs meho, every sae s of he sae space s characerse by he se of processes P s) { P, L, P } of he parallel program whch execue smulaeously sae s. The rume of process P s a expoeally srbue raom varable wh parameer. The esy a srbuo fucos of he rume are respecvely f ) a F ). The whole sysem chages from sae s o sae s' f process P s Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
22 Jorge Orega Aroa he frs o chage sae. The ew se of processes P s' ) rug he sae s' resuls from P s) \{ P} oe wh he se of sae of processes S s P ) whch ca sar f P s he frs o chage sae s Fgure 2): P s' ) P s) \ { P }) Ss P ) s S s P ), P 2,..., P s P, P 2,..., P pp s)) pp 2 s)) s 2 P, S s P 2 ),..., P pp s)) s P, P 2,..., S s P ) Fg. 2. UML Sae Dagram of a sage of he equvale sae agram as par of he sae space Coserg he example Fgure, he UML Sae Dagram for he equvale sae agram s show Fgure 3. a a 2, b 2 a 2 a, b a 2, b b b 2 Fg.3. Equvale UML Sae Dagram of he sae space for he example Fgure Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs 23 I orer o calculae he average rume of he whole parallel program, s requre o efe he average servce me sae s a he probably of P beg he frs process whch chages sae s. Therefore, he average rume of he whole parallel program E[S] ) ca be recursvely calculae usg he followg expresso: where: E [ S] P P S ) p P S)) E[ P S)] E[ S' ]) E[ P s)] s he average servce me of process P sae s uer he coo ha process o chage sae; p P s)) s he probably ha P s he frs process o chage sae s ; a S s he frs sae of he whole parallel program he sae space. P s he frs As he expoeal srbuo has he memoryless propery Kleroc, 975), he behavour of he parallel program sae s s epee of s hsory. Usg hs propery, s obae for p s)) ha: P a for E[ P s)] ha: p P s)) p Λ f ) E[ P s)] p < F )) Λ f ) f ) < F )) F )) I s oceable ha E[ P s)] s equal o he frs mome of he srbuo of he mmum m,..., ), a s epee of he process whch chage sae frs. Oly he brachg probables p P s)) epes o. Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
24 Jorge Orega Aroa 4 Usg Deermscally a Expoeally Dsrbue Varables for Moellg Execuo Tmes Moellg he rume of a process a parcular sae usg oly a smple expoeally srbuo s o a very realsc approach for he process real behavour. Hece, he use of oher arbrary srbuo fucos are propose here as par of he moels, a hus, coul be possble o aalyse he moels by he phase meho. Moreover, he use of phase ype srbuos, le he Erlag srbuo Kleroc, 975), may smplfy he aalyss. I geeral, here are parallel programs wh processes whose saes have rume srbuos wh a small varace. Here, s propose he use of Erlag- srbuos E Kleroc, 975), whch requres a hgh umber of phases ) for s compuao. Neverheless, moels ha use Erlag- srbuos e o become racable because of he so-calle sae space exploso Thomasa & Bay, 986). To avo hs problem, he umber of phases mus be reuce a fe. Ths ca be oe by approxmag he E -srbuo wh s frs mome E a varace V ) by a sae wh a eermsc phase wh parameer a a expoeal phase wh parameer Kleroc, 975). Therefore, he umber of phases of oe oe s reuce from o wo Fgure 4). Moel of a sae wh E srbue rume 2 Approxmao E Approxmao by a sae wh eermscally srbue rume a a sae wh expoeally srbue rume Fg. 4. Approxmao of a sae wh E srbue rume wh wo saes wh eermscally srbue rume a a sae wh expoeally srbue rume E. Approxmag he rume of he sae of a process by a eermsc a a expoeal phase mples ha he moelle rume always has he mmum of me. Ths s a beer moel of he real behavour ha a smple expoeal srbuo fuco wh a posve probably for all posve rumes. Noce ha for V a Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs 25 E V, he frs wo momes of he -srbuo a of he approxmae srbuo are he same. E Moreover, a beer approxmao ca be acheve by approxmag wh a eermsc srbuo a a - srbuo wh parameers 2 V a E 2V. The seco expoeal phase causes a slower crease of he srbuo fuco a me. Thus, as he expoeal phase creases, e o be close a closer o he orgal E -srbuo. Now, s requre o approxmaely aalyse a sae agram cossg of saes whch moel her rume by eermscally a/or expoeally srbue raom varables. Neverheless, oce ha he memoryless propery of he expoeally srbue raom varables s los, sce whe roucg eermscally srbue varables, he sae space meho oly allows a approxmao o he exac value. Therefore, because he eermscally srbue varables o o accomplsh he memoryless propery, he behavour a parcular sae epes o all prevous saes, a sce here are eermscally srbue phases rug s, hey have bee rug from he sar. Coserg hese epeeces, he moellg complexy creases o a po alreay uracable for small examples. Hece, s ow requre o approxmaely oba he me whch he parallel program remas sae s', coserg he use of he approxmao above. Le P s) { P, P2, L, P, P, L, P } be he se of processes whch smulaeously execue sae s of he parallel program. Le us coser ha he phases of a sae of he process P ) have eermscally srbue servce mes wh parameer. Now, m, 2, L, ) s he mmum of he eermsc phases a Pm he process have he shores eermsc rume sae s. Therefore, m. For < he servce me s a expoeally srbue raom varable wh parameer. The rug ass s' are obae as P s' ) { P', P' 2, L, P', P, LP } \ { P } U Ss P ) wh parameer ' E[ P s)] for a for >. I s oceable ha, for, he raom varable ' s approxmae by a eermscally srbue varable wh parameer ' mae use of he Drac fuco. To oba he brachg probably a he expece remag me ay sae, le us δ ) Kleroc, 975): E 2, δ ), f f δ ) Usg hs fuco a he prevous efos, hree cases ca be sgushe:. m. Ths meas coserg all eermsc phases excep he shores, a hece: p s)) P Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
26 Jorge Orega Aroa 2.. Ths meas coserg he shores eermsc phase, a hus: m s P E e e F f s P < Λ )] [ ) )) ) p )) p δ 3.. Ths meas coserg he expoeally srbue phases, so: < Λ m m e F F f s P )) )) ) p )) p Λ < < m e e s P E p )] [ Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
5 Case Suy The Hea Equao Problem Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs 27 The Hea Equao problem s o calculae he hea ffuso hrough a subsrae, usg a parallel program Orega- Aroa, 2). Le us coser he smples case, whch he Hea Equao s use o moel he hea srbuo o a oe-mesoal boy, a h subsrae, such as a wre. Dffere ervals expose a ffere emperaure, eermg a parcular srbuo a ffere mes. The hea ffuso s obae as aa represeg he way whch he emperaure of each erval vares hrough me, eg o crease or ecrease epeg o he exchage of hea wh oher ervals. A smple meho evelope for ervg a umercal soluo o he Hea Equao s he meho of fe ffereces Ges e al., 994; Orega-Aroa, 2). Coser he scree form for he oe-mesoal hea equao: Δ A, ) A, ) A, ) 2A, ) A, )) 2 Δx where represes me seps a caes wre subervals. The umercal soluo s ow compue smply by calculag he value for each erval a a gve me frame, coserg he emperaure from boh s prevous a s ex ervals Orega-Aroa, 2). Fgure 5 shows a escrpo of he Maager-Worers paer Orega-Aroa, 24) a he Commucag Sequeal Elemes paer Orega-Aroa, 2) as wo Archecural Paers for Parallel Programmg Orega- Aroa & Robers, 998) whose rumes are compare whe solvg a parcular problem. These wo archecural paers are use o oba wo ffere soluos for he Hea Equao problem, represee as he equao above, o a cluser of 6 compuers Ges e al., 994). Noce ha he UML Sae Dagram for each archecural paer represes he aa epeeces ha such a paer escrbes. For example, he aa epeeces of he Commucag Sequeal Elemes paer cosra ha a process o sage mus wa ul s ow preecessor a he preecessor of s lef a s rgh eghbour have chage sae o sage. O he oher ha, he Maager-Worers paer proposes ha every process o sage mus fsh compug before chagg sae o level. Noce ha for he Maager-Worers paer, he saes mare wh S are sychrosao saes, whch are cosere o cause o elay hs meas, hey are eermscally srbue wh parameer ). Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
28 Jorge Orega Aroa Maager-Worers paer S S S S S Sage Sage 2 Sage 3 Sage N Commucag Sequeal Elemes paer Fg. 5. UML Sae Dagrams for wo Archecural Paers for Parallel Programmg I orer o compare he approxmae rumes wh smulao resuls, e smulaos have bee performe for each moel a for boh archecural paers, coserg he varaos o a) he umber of processes, b) he umber of phases ) for he Erlag- srbuos E ), a c) varaos of he parameers a for he eermsc phase a he expoeal phase, respecvely. These varaos represe ffere worloas for he parallel sysem. Table shows oly four of hese varaos, whch are cosere releva for he prese aalyss, sce hey accomplsh he -es crera for comparg he wo ses of values Wess, 999; Mogomery, 99). The errors bewee approxmae a smulao resuls le bewee % a.5% of he greaer resul hese smulaos. Noce ha he prese meho shoul be heorecally exac f he smulao moel cosss of oly eermscally or oly expoeally srbue rumes. Some commes abou he smulaos a her resuls:. The comparsos bewee approxmae resuls, exac values, a smulao resuls are obae for he Commucag Sequeal Elemes paer Fgure 5. I s suppose ha all processes have ecally srbue rumes, as worloa. The accuracy of he approxmao s ese by worloa srbuos of ype Erlag. I orer o be able o compare resuls, a have bee chose o ge he frs mome cosa, for ffere varaces. Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
Usg UML Sae Dagrams for Moellg he Performace of Parallel Programs 29 Table. Comparso of exac, approxmae, a smulao resuls Worloa Exac De- Smulao Saes Rume approxmao resul E.25) 256 39.84 39.84 39.64±.27 E.5) 2 52 33.75 33.77 33.35±.42 E.) Too hgh compuaoal 28.27 28.26±.37 4 coss for calculag 25.43 25.67±.53 E 82.) The e-approxmao colum preses he compue rume for approxmae moels. The srbue rumes are approxmae by a moel wh a eermscally srbue rume a E ) a expoeally srbue phase. A exac resuls s obae oly whe he process rumes are moelle by E ) or E 2 ) srbue raom varables. I he case of E 4 ) a E 8 ) srbue rume, he compua o coss of a exac Marova aalyss are oo hgh erms of he umber of saes he sae space. The approxmao resuls a smulao resuls are compare o oba formao abou he qualy of he approxmao meho. The rumes he smulao moels are also approxmae by a moel wh a eermscally srbue rume a a expoeally srbue phase. The approxmae rumes le he.99 cofece erval of he smulao resuls Table. 2. The comparsos bewee approxmae rumes of a moel srucure wh he Commucag Sequeal Elemes paer a he approxmae rumes of a moel srucure wh he Maager-Worers paer are obae for varous worloas, represee by he parameers mome a varace, whe solvg he Hea Equao. Table 2. Comparso of CSE a MW rumes. Worloa Rume CSE Rume MW secos) secos) Mome 5 4.66±.43 46.63±.38 Varace25 Mome 5 3.27±.26 32.46±.22 Varace5 Mome 5 27.42±.22 29.5±.8 Varace2.5 Mome 5 2.±.7 2.97±.2 Varace I s oceable from Table 2 ha MW moel preses always a hgher execuo rume ha he CSE moel. The fferece of he oal expece rume creases wh he creasg varaces of he process rumes. I he case of cosa process rumes, he wo moels have very smlar expece rumes. Ths s, for he shores rume, he varace of he process rumes s. 6 Cocluso Ths paper preses a meho o approxmaely compue he rume of a parallel program. The meho allows o evaluae moels ha are srucurally more complex erms of processes a her saes. Usg eermscally a expoeally srbue rumes, more realsc moels of he real behavour ca be obae ha usg oly phase ype srbuos. Thus, he meho s compose of wo approxmaos: Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546
2 Jorge Orega Aroa. Rumes are approxmae by a eermscally srbue a a expoeally srbue rume varable. 2. The overall rume s obae by a approxmae evaluao of he moel. The expermes for solvg he Hea Equao wh CSE a MW moels have show ha he approxmao resuls ffer less ha.5 perce from exac Marova resuls. 7 Refereces. Bllo, R., a Alla, R.N. 992). Relably Evaluao of Egeerg Sysems: Coceps a Techques. Sprger. 2. Booch, G., Rumbaugh, J., a Jacobso I. 998). The Ufe Moelg Laguage User Gue. Aso- Wesley. 3. Fowler, M., a Sco, K. 997). UML Dslle. Aso-Wesley Logma Ic., Reag MA. 4. Ges, A., Beguel, A., Dogarra, J., Jag, W., Mache, R., a Sueram, V. 994). PVM: Parallel Vrual Mache. A User s Gue a Tuoral for Newore Parallel Compug. The MIT Press. 5. Kleroc, L. 975). Queueg Sysems. Volume : Theory. Joh Wley a Sos. 6. Lu, J.C.S., Muz, R.R., a Towsley, D. 998). Compuer performace bous of for-oprograms uer a mulprocessor evrome. IEEE Trasacos o Parallel a Dsrbue Sysems, Vol. 9 parallel No. 3. 7. Mogomery, D.C. 99). Desg a Aalyss of Expermes. Joh Wley & Sos, Ic. 8. Orega-Aroa, J., a Robers, G. 998). Archecural Paers for Parallel Programmg. Proceegs of he 3 r Europea Coferece o Paer Laguages of Programmg a Compug EuroPLoP 98). 9. Orega-Aroa, J. 2). The Commucag Sequeal Elemes paer. A Archecural Paer for Doma Parallelsm. Proceegs of he 7 h Coferece o Paer Laguages of Programmg PLoP 2).. Orega-Aroa, J. 24). The Maager-Worers paer. A Acvy Parallelsm Archecural Paer for Parallel Programmg. Proceegs of he 9 h Europea Coferece o Paer Laguages of Programmg a Compug EuroPLoP 24).. Pooley, R., a Kg, P. 999) The Ufe Moelg Laguage a Performace Egeerg. IEE Proceegs Sofware 462). 2. Thomasa, A. a Bay, P. 986). Aalyc Queueg Newor Moels for Parallel Processg of Tas Sysems. IEEE Trasacos o Compuers, December 986. 3. Wess, B. 999). Iroucory Sascs. Aso-Wesley. Jorge L. Orega Aroa s a full-me lecurer of he Deparme of Mahemacs, Faculy of Sceces, UNAM. He obae a BSc. Elecroc Egeerg, as well as a MSc Compuer Scece, a UNAM, a a PhD from he Uversy College Loo UCL), U.K. Hs research eress clue Sofware Archecure a Desg, Sofware Paers, a Parallel Processg. Compuacó y Ssemas Vol. No. 3, 28, pp 99-2 ISSN 45-5546