A Novel Discrete Differential Evolution Algoritnm for Task Scheduling in Heterogeneous Computing Systems

Transcription

1 Proceedngs of he 2009 IEEE Inernaonal Conference on Sysems, Man, and Cybernecs San Anono, TX, USA - Ocober 2009 A Novel Dscree Dfferenal Evoluon Algornm for Task Schedulng n Heerogeneous Compung Sysems Qnma Kang * School of Elecroncs and Informaon Engneerng Tongj Unversy Shangha , Chna Hong He School of Informaon Engneerng Shandong Unversy a Weha Weha , Chna Absrac Task schedulng s one of he core seps o effecvely explo he capables of dsrbued heerogeneous compung sysems. In hs paper, a novel dscree dfferenal evoluon (DDE) algorhm s presened o address he ask schedulng problem. The encodng schemes and he adapaon of classcal dfferenal evoluon algorhm for dealng wh dscree varables are dscussed as well as he echnque needed o handle boundary consrans. The performance of he proposed DDE algorhm s showed by comparng wh a genec algorhm, whch s a well-known populaon-based probablsc heursc, on a large number of randomly generaed nsances. Expermenal resuls ndcae ha he proposed DDE algorhm has generaed beer resuls han GA n erms of boh soluon qualy and compuaonal me. Keywords Dscree dfferenal evoluon algorhm; Heerogeneous compung; Task schedulng; Genec algorhm I. INTRODUCTION Heerogeneous compung sysems (HCS) are composed of a heerogeneous sue of machnes wh vared compuaonal capables nerconneced by hgh-speed neworks, and have emerged as a powerful plaform o execue compuaonally nensve applcaons ha have dverse compuaonal requremens [1]. One of he major challenges for harnessng he compung power of HCS o acheve hgh performance for asks s he schedulng problem,.e., we need address ha each ask should be mapped o s sued machne archecure. The mappng mehods can be grouped no wo caegores: on-lne mode and bach-mode [2]. In he on-lne mode, a ask s mapped ono a machne as soon as arrves a he scheduler. In he bach mode, asks are no mapped ono he machnes as hey arrve; nsead hey are colleced no a se ha s examned for mappng a prescheduled mes called mappng evens. The ndependen se of asks ha s consdered for mappng a he mappng evens s called a mea-ask [1, 3]. In hs sudy, we consder he mea-ask schedulng problem, whch s o fnd a ask assgnmen scheme ha mnmzes he schedule lengh of a mea-ask. * The auhor s also afflaed o School of Informaon Engneerng, Shandong Unversy a Weha, Weha , Chna. The mea-ask schedulng problem s known o be NPcomplee [2]. Therefore, only small-szed nsances of he schedulng problem can be solved opmally whn a reasonable compuaonal me usng exac algorhms. Heursc algorhms, on he oher hand, have generally accepable compuaon me and memory requremens o oban a near-opmal or opmal soluon. For hs reason, mos research focused on developng heursc algorhms. Some of he well-known fas consrucve heurscs, whch have been repored n he leraure, nclude Mn-mn, Max-mn, Sufferage [3], Qos Guded Mn-mn [4], Segmened Mn-mn [5], ec. In recen years, a growng body of leraure suggess he use of mea-heursc search mehods for combnaoral opmzaon problems (COPs). Several mea-heursc algorhms ha have been denfed as havng grea poenal o address praccal opmzaon problems nclude Genec Algorhms (GA) [6, 7], Smulaed Annealng (SA), Tabu Search (TS) and Parcle Swarm Opmzaon (PSO), ec. Consequenly, over he pas few years, several researchers have demonsraed he applcably of hese mehods o ask assgnmen problem [8-11]. A good overvew of he ask assgnmen algorhms can be found n [12]. Due o he nracable naure of he machng and schedulng problem and s mporance n HCS, new effcen echnques are always desrable o oban he bes possble soluon whn an accepable compuaonal me. Dfferenal evoluon (DE) s a populaon-based meaheursc algorhm recenly proposed by Sorn and Prce [13] for global opmzaon over connuous spaces. In he muaon process of a DE algorhm, he weghed dfference beween wo randomly seleced populaon members s added o a hrd member o generae a muaed soluon. Then, a crossover operaor follows o combne he muaed soluon wh a arge soluon so as o generae a ral soluon. Thereafer, a selecon operaor s appled o compare he fness funcon value of boh compeng soluons, namely, arge and ral soluons o deermne who can survve for he nex generaon. Snce s nvenon, DE has been appled wh hgh success o many numercal opmzaon problems. Moreover, dfferen varans of he basc DE model have been proposed o mprove s effcency over connues problems. Despe he smplcy and he hgh effcency of DE, s applcaon on he soluon of COPs wh dscree decson /09/$ IEEE 5151

2 varables s sll lmed. The major obsacle of successfully applyng a DE algorhm o solve COPs n he leraure s due o s connuous naure. To remedy hs drawback, hs research nroduces a novel dscree dfferenal evoluon (DDE) algorhm for solvng mea-ask schedulng problem o effecvely explo he capables of heerogeneous compung sysems. The remanng paper s organzed as follows. Secon 2 gves a bref nroducon o DE. Secon 3 descrbes he formulaon of mea-ask schedulng problem. Secon 4 nroduces he dscree dfferenal evoluon algorhm. Secon 5 repors he comparave performance and convergence analyss. Fnally, secon 6 summarzes he concludng remarks. II. Bref Inroducon o DE As wh all evoluonary opmzaon algorhms, DE sars wh a populaon of D-dmensonal search-varable vecors called ndvduals, where D corresponds o he number of problem s parameers, and each vecor represens poenal soluon for he opmzaon problem a hand. I fnds he global opma by ulzng he dsance and drecon nformaon accordng o he dfferenaons among populaon. Currenly, here are several varans of DE, whch vary based on he base vecor o be perurbed, he number and selecon of he dfference vecors and he ype of crossover operaors. The common forma of DE s DE/x/y/z, where x represens a base vecor o be perurbed, y s he number of dfference vecors used for perurbaon of x, and z sands for he ype of crossover beng used (bn: bnomal; exp: exponenal). Le denoe he number of eraons, he -h vecor of he populaon a he curren generaon can be represened by X = ( X, X,..., X ). Le PS denoe he number of,1,2, D ndvduals n he nal populaon, namely populaon sze, and he populaon sze s usually kep consan hroughou eraons. For a mnmzaon problem, he basc procedure of DE, whch s denoed as DE/rand/1/bn [13], can be gven below: Sep 1: Inalze populaon. For each search-varable, here may be a ceran range whn whch value of he parameer should le for beer search resuls. A he very begnnng of a DE run or a = 0, problem parameers or ndependen varables are nalzed somewhere n her feasble numercal range. Therefore, f he j-h parameer of he gven problem has s lower and upper bound L as x j and x U j respecvely, hen we may nalze he j-h componen of he -h populaon members as 0 L U L X = x + rand(0,1) ( x x ) (1), j j j j where rand(0,1) represens a unformly dsrbued random value ha ranges from 0 o 1. Sep 2: Muaon phase. In he muaon phase, for each arge vecor X, {1, 2,, PS}, a muan vecor V s obaned by V = X + F ( X X ) (2) r1 r2 r3 where r 1, r 2, r 3 {1, 2,, PS} are muually dsnc random ndces and are also dfferen from he curren arge ndex. The vecor X s known as he base vecor o be perurbed, and r1 F > 0 s a scalng parameer. Sep 3: Crossover phase. For each arge X, a ral vecory s formed as:, (0,1) V j f rand < CR or j = k Y, j= (3) X, j else where k s a randomly chosen neger n he se {1, 2,..., D}; he subscrp j represens he j-h componen of respecve vecors; rand(0,1)(0, 1), drawn randomly for each j; CR(0, 1)s crossover probably ha affecs he convergence rae and robusness of he search process. Sep 4: Selecon phase. The selecon scheme of DE also dffers from he oher evoluonary algorhms. In he selecon phase, he funcon 1 value of he ral vecor, f ( Y + ), s compared o f ( X ), o deermne whch one of he arge vecor and he ral vecor wll survve n he nex generaon. The process may be oulned as: Y f f ( Y ) < f ( X ) + 1 X = (4) X else where f s he funcon o be mnmzed. So f he new ral vecor yelds a beer value of he fness funcon, replaces s arge n he nex generaon; oherwse he arge vecor s reaned n he populaon. Hence he populaon eher ges beer or remans consan bu never deeroraes. Recombnaon (muaon and crossover) and selecon connues unl a soppng creron s sasfed, and hen he bes ndvdual of he populaon found so far s repored as he fnal soluon. The scheme descrbed above s only one varan of he basc DE algorhm. There are some oher DE varans, manly dfferng n he way hey creae he muan vecor (Equaon (2)). The creaon of he -h muan vecor n anoher wellknown varan called DE/bes/1/bn [13] s gven by he followng relaon: V = X + F ( X X ) (5) where X bes bes r1 r2 s he bes vecor of he curren populaon. III. META-TASK SCHEDULING PROBLEM A mea-ask s defned as a collecon of ndependen asks ha s consdered for mappng a he schedulng evens, and makespan he compleon me for he enre mea-ask [12]. A ypcal example of mea-ask schedulng s he mappng of an arbrary se of ndependen asks from dfferen users wang o execue on a heerogeneous sue of machnes. Each ask n a mea-ask may have assocaed properes, such as a deadlne or a prory. I s assumed ha each machne execues a sngle ask a a me, n he order n whch he asks arrved, and he 5152

3 sze of he mea-ask (number of asks o execue), N, and he number of machnes n he HCS, M, are sac and known a pror. Therefore, ask schedulng problem can be defned as N asks are assgned on M heerogeneous resources wh he objecve of mnmzng he compleon me and ulzng he resources effecvely. To formulae he problem, defne P = {p 1, p 2,, p M }) as he se of M heerogeneous machnes and T = { 1, 2,, N } as he se of N ndependen asks o be assgned o he machnes. Because of he heerogeneous naure of he machnes and dssmlar naure of he asks, he expeced execuon mes of a ask on dfferen machnes are dfferen. Every ask has an expeced me o compue (ETC) on a specfc machne. We assume ha he ETC of each ask on each machne s known based on user-suppled nformaon, ask proflng and analycal benchmarkng. The assumpon of such ETC nformaon s a common pracce n schedulng research (e.g. [2-5]). So we can oban an N M ETC marx. ETC (, j) s he esmaed execuon me for ask on machne j. The oal execuon me used by machne p j s he sum of he expeced execuon mes of hose asks ha are mapped o he machne. The makespan of a schedule s he maxmum of he oal execuon mes of hose machnes. To calculae he makespan of a schedule, we frs nroduce a vecor of machne compleon me whch sze s M [14]. Formally, for a machne p m and a schedule S, he compleon me of p m s defned as follows: compleon[ m] = ETC( j, m) (6) S[ j] = m, j T where S[j]=m denoes he ask j s assgned o machne m under he schedule S. We can use he compleon me of machnes o compue he makespan as follows: makespan = max{ compleon[ m] p P} (7) m The objecve of hs work s o fnd a schedule scheme wh he mnmum makespan. IV. THE PROPOSED DDE ALGORITHM A. Soluon Represenaon and Inal Swarm Generaon One of he key ssues n desgnng a successful DE algorhm s he soluon represenaon,.e. fndng a suable mappng beween problem soluon and DE ndvdual. The naural codng for ask schedulng problems s he neger vecors as used n GAs [12]. In hs paper, each ndvdual S correspondng o a feasble soluon of he underlyng problem s represened as a vecor of sze N n whch s j h poson (an neger value) ndcaes he machne o whch ask j s assgned: S[j] = m, m{1, 2,..., M}. The DE s a populaon-based searchng paradgm ha explores he soluon space by recombnng a number of ndvduals, called a populaon. An nal populaon s randomly generaed for erave mprovemen accordng o equaon (8), whch s presened for he j-h dmenson of he - h parcle. 0 X = INT (1 + rand(0,1) M ) 1 PS,1 j N (8), j where INT s a funcon for converng a real-value o an neger value by runcaon and rand(0,1) s a unform random number n he range (0,1). B. Fness Value In DE algorhm, all ndvduals a each eraon sep are evaluaed accordng o a measure of soluon qualy, called fness. For he problem a hand, we use he makespan of an ndvdual as s fness value. See secon 2. C. The Proposed DDE Algorhm In s canoncal form, he DE algorhm s only capable of handlng connuous varables. In order o apply he DE algorhm o neger opmzaon problems, s crucal o desgn a suable encodng scheme ha maps he floang-pon vecors o neger vecor soluons. Several approaches have been used o deal wh dscree varable opmzaon. Mos of hem round off he varable o he neares avalable value before evaluang each ral vecor,.e., neger values are used o evaluae he objecve funcon, even hough DE self may sll works nernally wh connuous floang-pon values [15]. Anoher wdely used represenaon echnque ha deals wh floang-pon vecors for dscree opmzaon problems s he random-keys encodng. Accordng o hs echnque, each soluon s encoded as a vecor of N floang-pon numbers. The componens of he vecor are sored and her order n he vecor deermnes he fnal soluon o he problem. However, hs encodng scheme has been demonsraed o be poor sued whn he DE algorhm o address dscree opmzaon problems [16]. Moreover, he echnques used o deal wh sequencng opmzaon problem canno be appled drecly o ask schedulng n HCS generally. I s mporan o noe ha DE s self-adapve. A he begnnng of he evoluon process, he muaon operaor of DE favors exploraon snce paren ndvduals are far away o each oher. As he evoluonary process proceeds o he fnal sage, he muaon operaor favors exploaon n ha he populaon converges o a small regon. As a resul, he adapve search sep enables he evoluon algorhm o perform global search wh a large search sep a he begnnng and refne he populaon wh a small search sep a he end. Therefore, exendng DE o opmzaon of neger varables s raher easy. Only he scalng parameer s se o 1. Based on hs dea, we propose a dscree varan of DE denoed as DE//1/bn. In he proposed algorhm, a muan vecor V s obaned by V = X + X X (9) r1 r2 where r 1, r 2 {1, 2,, PS} are muually dsnc random ndces and are also dfferen from he curren arge ndex. The ral vecory s formed as:, (0,1) V j f rand < CR Y, j= (10) X bes, j else Noe ha we use he -h vecor as base vecor for he -h arge ndvdual, and hen he muan vecor s combned wh he bes ndvdual of he curren populaon for yeldng he ral vecor a each eraon sep. 5153

4 D. Boundary Consrans I s mporan o noce ha he recombnaon operaon of DE s able o exend he search ousde of he nalzed range of he search space, whch leads o llegal soluons. For he problem under nvesgaon, s essenal o ensure ha parameer values le nsde her allowed ranges afer recombnaon. A smple way o guaranee hs s o replace parameer values ha volae boundary consrans wh random values generaed whn he feasble range: INT (1 + rand (0,1) M / 2) f V, j< 1 V, j= INT( M /2 + rand(0,1) ( M /2 + 1)) (11) f V, j> M E. The DDE Algorhm The deals of he proposed DDE algorhm are presened n Fg. 1. The algorhm sars wh an nal populaon of PS ndvduals. Each ndvdual vecor corresponds o a canddae soluon under nvesgaon. Then, all of he ndvduals are eravely mproved unl one of soppng crera s sasfed. When he algorhm s ermnaed, he ncumben global bes ndvdual and he correspondng fness value are oupu as he opmal ask schedulng scheme and he mnmum makespan. DDE { Inalze parameers. Generae PS ndvduals a random usng equaon (8). Evaluae he ndvduals and deermne he bes ndvdual. Repea { Muaon sep: Generae muan ndvduals usng equaon (9); Tes boundary consrans for he muan ndvduals. If volaon occurs, he value s dragged o he feasble range usng equaon (11). Crossover sep: Generae ral ndvduals usng equaon (10). Selecon sep: Deermne he ndvduals of he nex generaon usng equaon (4); Deermne he bes ndvdual of he populaon. } unl one of he soppng crera s sasfed; Reurn he bes ndvdual and s fness value. } Fgure 1. Pseude Code of DDE Algorhm. V. EXPERIMENT RESULTS To evaluae he effcency and effecveness of he proposed algorhm, nensve expermens have been conduced. A large smulaon daase s creaed whch feaures dfferen mappng suaons. The proposed algorhm s compared wh he genec algorhm (GA) proposed by Braun e al. [12] because boh of hem are populaon-based evoluonary algorhms. Boh he algorhms for mea-ask schedulng are coded n MATLAB6.5 and expermens are execued on a Penum Dual 1.6GHz processor wh 1GB man memory runnng under Wndows XP envronmen. The GA s mplemened wh roulee wheel selecon, onepon crossover operaor, a muaon operaor and an elsm mechansm for he comparson purpose. The parameers of GA are se as follows: crossover probably = 0.8, muaon probably = 0.1, populaon sze = 100 and her GA fnalzes when eher 1000 eraons have been execued, or, when he chromosome ele have no vared durng 150 eraons. The crossover rae CR n DDE s deermned expermenally by ncremenng from 0.1 o 0.9 n 0.1 seps. In he smulaon rals esed, he performance peak s deeced wh CR equal o 0.4. The sze of he nal populaon s se o 100 and we use he same soppng crera as ha used n GA for he far comparson purpose. The smulaon model presened n [12] s also employed here for our comparson sudy. In hs model, characerscs of he ETC marces are vared n an aemp o smulae varous possble heerogeneous compung envronmens. Ths knd of varaon s mplemened by seng he value scope of random numbers used o produce ETC marces. Frsly, an N 1 baselne column vecor B s generaed, where each elemen B(), 1N, s a unform random number beween 1 and b. Then, he rows of he ETC marx are consruced. Each elemen ETC(, j), 1N, 1 jm, of ETC marx s creaed by akng baselne value, B(), and mulplyng by a unform random number x(, j), whose value s beween 1 and r. Therefore, any gven value n he ETC marx s whn he range [1, br]. To denoe varous knds of mappng suaons, he marces are classfed no 12 dfferen ypes accordng o hree mercs: ask heerogeney, machne heerogeney and conssency. The ask heerogeney descrbes he amoun of varance among he execuon mes of he asks n he mea-ask for a gven machne, wo possble values are defned: hgh and low. Machne heerogeney descrbes he possble varaon of he runnng me of a parcular ask across all he machnes, and agan has wo values: hgh and low. Each of he dfferen ask and machne heerogenees s modeled by usng dfferen b and r values: hgh ask heerogeney s represened by seng b = 3000 and low ask heerogeney s modeled usng b = 100. Hgh machne heerogeney s represened by seng r = 1000, and low machne heerogeney s modeled usng r = 10. An ETC marx s consdered conssen when, f a machne m execues ask k faser han machne m j, hen m execues all he asks faser han m j. Inconssency means ha a machne s faser for some asks and slower for some ohers. An ETC marx s consdered sem-conssen f conans a conssen sub-marx. All nsances conss of 256 asks and 16 machnes and are labeled u_x_yyzz whose meanng s he followng: u means unform dsrbuon (used n generang he marces). x means he ype of nconssency (c conssen, nconssen and s means sem-conssen). 5154

5 TABLE I. COMPARATIVE RESULTS FOR DDE AND GA Insrance GA DDE Improvmen NRT Mavg(s) Msd Tavg(s) Mavg(s) Msd Tavg(s) over GA u hh u hlo u loh u lolo 8.833* * * * * * * * % 20.00% 29.05% 24.13% u_c_hh u_c_hlo u_c_loh u_c_lolo 8.289* * * * * * * * % 10.04% 22.99% 12.76% 1.05 u_s_hh u_s_hlo u_s_loh u_s_lolo 9.392* * * * * * * * % 11.99% 20.12% 15.14% 1.03 Average 19.96% 1.03 yy ndcaes he heerogeney of he asks (h hgh, and lo low). zz ndcaes he heerogeney of he resources (h hgh, and lo low). Ths se of nsances has been acually consdered as he mos dffcul one for he schedulng problem n heerogeneous envronmens and s he man reference n he leraure. The sze of he nal populaon s se o 100 and we use he same soppng crera as ha used n GA for he far comparson purpose. As performance mercs, we consder he qualy of he soluon (makespan) and he amoun of CPU me used for he benchmarks. The percenage mprovemen s compued as: makespan makespan 1 DDE GA 100% (13) and he normalzed runnng me (NRT) s compued by dvdng he oal GA algorhm runnng me wh DDE algorhm runnng me for he benchmark under consderaon. Moreover, boh GA and DDE are sochasc-based algorhms and each ndependen run of he same algorhm on a parcular esng nsance may yeld a dfferen resul, we hus calculae he average makespan and he average compuaon me over 20 ndependen runs of each algorhm for every problem nsance. And he sandard devaon of makespans s also compued for each nsance. All expermenal resuls are summarzed n able 1, where Mavg, Msd and Tavg, respecvely, denoe he average makespan, he sandard devaon of makespans and he average compuaon me. I can be observed ha he proposed DDE algorhm ouperforms he GA n erms of boh soluon qualy and compuaonal me for all he nsances. Improvemen of DDE over GA s beween 10.04% and 29.05% wh an average of 19.96%. Moreover, GA on he average runs 1.03 mes slower han DDE echnque. The DDE s also superor n erms of sandard devaon. These resuls ndcae ha he proposed DDE algorhm s a vable alernave for solvng he ask schedulng problem. These resuls are no compared wh he radonal mehods snce earler sudes of Braun e al. [12] show ha GA based algorhm for mea-ask assgnmen problems ouperform oher radonal approaches such as SA and TS. Also, our prelmnary ess ndcae ha PSO based algorhm [11] proposed for homogeneous sysems s poorly sued for medum o large-szed problems of mea-ask schedulng problems n HCS. To analyze he convergence behavor of he proposed algorhms, we have recorded he varaon of he makespans of he wo algorhms a each eraon sep. Fg. 2 dsplays a ypcal run of wo algorhms for solvng a problem nsance wh he number of asks beng equal o 256 and ha of machnes 16. From he evoluonal processes of he algorhms, we fnd ha DDE delvers beer performance han GA n erms of soluon qualy. The graph also ndcaes GA evolves slower han DDE afer approxmaely 300 eraon seps and he decremen of makespan almos sagnaes. Fgure 2. Convergence properes of GA and DDE. VI. CONCLUSIONS In dsrbued heerogeneous compung sysems, qualfed ask schedulng among processors s an mporan sep for effcen ulzaon of resources. In hs paper, a new heursc algorhm called DDE algorhm s proposed for he ask schedulng problem n heerogeneous compung envronmens. DE s one of he recen evoluonary opmzaon mehods. I 5155

6 has been wdely used n a wde range of applcaons. Unlke he sandard DE, he DDE algorhm employs an neger-valued soluon vecor and work on he dscree doman. The performance of DDE algorhm s evaluaed n comparson wh a well-known GA algorhm for a number of randomly generaed mappng problem nsances. The resuls showed ha he DDE algorhm soluon qualy s beer han ha of GA n all cases. Moreover, he DDE algorhm runs faser as compared wh GA. These resuls ndcae ha he proposed DDE algorhm s an aracve alernave for solvng he ask schedulng problem. A naural exenson o hs work would be hybrdzng DDE wh a local search heursc and a wdespread search heursc o promoe a search for a global opmum. Moreover, DDE applcaon o oher combnaoral opmzaon problems needs furher nvesgaon. REFERENCES [1] H. J. Segel and S. Al, Technques for mappng asks o machnes n heerogeneous compung sysems. J. Sys. Archec., vol. 46, pp , [2] P. Luo, K. V. L and Zh. Zh. Sh, A revs of fas greedy heurscs for mappng a class of ndependen asks ono heerogeneous compung sysems. J. Parallel Dsrb. Compu., vol. 67, pp , [3] M. Maheswaran, S. Al, e al. Dynamc mappng of a class of ndependen asks ono heerogeneous compung sysems. In 8h IEEE Heerogeneous Compung Workshop (HCW '99), San Juan, Puero Rco, vol , [4] X. S. He, X. H. Sun and G. Laszewsk, QoS guded mn-mn heursc for grd ask schedulng. J. Compu Sc. Technol., vol. 18, pp , [5] M. Y. Wu, W. Shu and H. Zhang, Segmened mn-mn: A sac mappng algorhm for mea-asks on heerogeneous compung sysems. In Proccedng of he 9h Heerogeneous Compung Workshop (HCW'00), Cancun, Mexco, pp , [6] D. E. Goldberg, Genec algorhms n search, opmzaon, and machne learnng. Addson-Wesley Longman Publshng Co., Inc. Boson, MA, USA, January [7] J. H. Holland, Adapaon n naural and arfcal sysems: An nroducory analyss wh applcaons o bology, conrol, and arfcal nellgence. The Unversy of Mchgan Press, Ann Arbor, [8] R. Subraa, Y. A. Zomaya and B. Landfeld, Arfcal lfe echnques for load balancng n compuaonal grds. J. Compu. Sys. Sc., vol. 73, pp , [9] G. Aya and Y. Hamam, Task allocaon for maxmzng relably of dsrbued sysems: a smulaed annealng approach. J. Parallel Dsrb. Compu., vol. 66, pp , [10] W. H. Chen and C. S. Ln, A hybrd heursc o solve a ask allocaon problem. Compu. Oper. Res., vol. 27, pp , [11] A. Salman, I. Ahmad and S. Al-Madan, Parcle swarm opmzaon for ask assgnmen problem. Mcroprocess. Mcrosy., vol. 26, pp , [12] T. D. Braun, H. J. Segel, e al. A comparson of eleven sac heurscs for mappng a class of ndependen asks ono heerogeneous dsrbued compung sysem. J. Parallel Dsrb. Compu., vol. 6, pp , [13] R. Sorn and K. Prce, Dfferenal evoluon a smple and effcen heursc for global opmzaon over connuous spaces. J. Global Opm., vol. 11, pp , [14] J. Carreero and F. Xhafa, Usng genec algorhms for schedulng jobs n large scale grd applcaons. J. of Technologcal and Economc Developmen A Research Journal of Vlnus Gedmnas Techncal Unversy, vol. 12, pp.11 17, [15] G. Onwubolu and Davendra D. Schedulng flow shops usng dfferenal evoluon algorhm. Eur. J. Oper. Res., vol. 171, pp , [16] C. N. Andreas and L. O. Sors, Dfferenal evoluon for sequencng and schedulng opmzaon. J Heurscs, vol. 12, pp ,