A Task Scheduling Algorithm Based on PSO for Grid Computing

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1 Internatonal Journal of Computatonal Intellence esearch. ISSN Vol.4, No.1 (28, pp esearch Inda Publcatons A Tas Scheduln Alorthm Based on PSO for Grd Computn Le Zhan 1, Yuehu Chen 2, unyuan Sun 1, Shan Jn 1 Bo Yan 1 1 Networ Center, Unversty of Jnan Jwe oad 16, Jnan, 2522, P.. Chna {zhanle,, sunry, jnshan, yanbo}@ujn.edu.cn 2 School of Informaton Scence Enneern, Unversty of Jnan Jwe oad 16, Jnan, 2522, P.. Chna yhchen@ujn.edu.cn Abstract: Grd computn s a hh performance computn envronment to solve larer scale computatonal dems. Grd computn contans resource manaement, tas scheduln, securty problems, nformaton manaement so on. Tas scheduln s a fundamental ssue n achevn hh performance n rd computn systems. However, t s a b challene for effcent scheduln alorthm desn mplementaton. In ths paper, a heurstc approach based on partcle swarm optmzaton alorthm s adopted to solvn tas scheduln problem n rd envronment. Each partcle s represented a possble soluton, the poston vector s transformed from the contnuous varable to the dscrete varable. Ths approach ams to enerate an optmal schedule so as to et the mnmum completon tme whle completn the tass. The results of smulated eperments show that the partcle swarm optmzaton alorthm s able to et the better schedule than enetc alorthm. Keywords: partcle swarm optmzaton, tas scheduln, rd computn 1. Introducton Wth the development of the networ technoloy, rd computn used to solve larer scale comple problems becomes a focus technoloy. Tas scheduln s a challenn problem n rd computn envronment [1. If lare numbers of tass are computed on the eoraphcally dstrbuted resources, a reasonable scheduln alorthm must be adopted n order to et the mnmum completon tme. So tas scheduln whch s one of NP-Complete problems becomes a focus by many of scholars n rd computn area. Heurstc optmzaton alorthm s wdely used to solve a varety of NP-complete problems. Abraham et al Braun et al [2 presented three basc heurstcs mpled by Nature for Grd scheduln, namely Genetc Alorthm (GA [3-7, Smulated Annealn (SA [8-9 Tabu Search (TS [1, heurstcs derved by a combnaton of there three alorthms. GA SA are powerful stochastc optmzaton methods, whch are nspred form the nature. GA s smulated the evolutonary natural selecton process. The better soluton of eneraton s evaluated accordn to the ftness value the cdates wth better ftness values are used to create further solutons throuh crossover mutaton processes. Smulated annealn s based on the process of annealn about the sold matter n physcs. Both methods are vald have been appled n varous felds due to ther stron converence propertes. Partcle Swarm Optmzaton (PSO [11 s one of the latest evolutonary optmzaton technques nspred by nature. It s smulated the process of a swarm of brds preyn. It has the better ablty of lobal searchn has been successfully appled to many areas such as neural networ trann, control system analyss desn, structural optmzaton so on. It also has fewer alorthm parameters than both enetc alorthm smulated alorthm. Furthermore, PSO alorthm wors well on most lobal optmal problems. In ths paper, PSO alorthm s employed to solve the scheduln problem n a rd envronment. Throuh a seral of smulated eperments, our results show that PSO alorthm s effectve for tas scheduln n computatonal rd. Ths paper s oranzed as follows. In Secton 2, rd scheduln ssues are manly dscussed. Partcle swarm optmzaton alorthm s ntroduced n secton 3. In Secton 4, PSO alorthm for tas scheduln problem n rd envronment s ven. Eperment settns results are

2 38 Le Zhan et al dscussed n Secton 5 some conclusons are ven n Secton Grd scheduln ssues A computatonal rd s a hardware software nfrastructure that provdes dependable, consstent, pervasve, nepensve access to hh-end computatonal capabltes [1. A resource n the computatonal rd s somethn that s requred to carry out an operaton, for eample, a processor used for data processn. The resource manaement of computatonal rd s responsble for resource dscovery allocaton of a tas to a partcular resource. Usually t s easy to et the nformaton about the ablty to process data of the avalable resource. In ths paper, we wll dscuss the problem that n tass wor on m computn resources wth an objectve of mnmzn the completon tme utlzn the resources effectvely. If the number of tass s less than the number of resources n rd envronment, the tass can be allocated on the resources accordn to the frst-come-frst-serve rule. If the number of tas s more than the number of resources, the allocaton of tass s to be made by some scheduln schemes. Consdern the number of tass s more than the computn resources n ths paper, so one tas can not be assned to dfferent resource, mplyn that the tas s not allowed to be mrated between resources[17. Usually t s able to et the nformaton about the avalable resources va resource manaement n the rd envronment. T To formulate the problem, defne ={1,2,3, n} as n j ndependent tass permutaton j={1,2,3, m} as m P, j computn resources. Suppose that the processn tme for tas computn on j resource s nown. The completon C( tme represents the total cost tme of completon. The objectve s to fnd an permutaton matr = ( j j, wth, =1 f resource performs tas j f, otherwse, C, j m ( = Subject to m = 1 j= 1 =, whch mnmzes the total costs n P, j *, j = 1, j T, j j {,1 },, j T ( The mnmal C (1 (2, (3 represents the lenth of schedule whole tass worn on avalable resources. The scheduln constrants (2 uarantee that each tas s assned to eactly one resource. We wll dscuss that a new optmal schedule s able to fnd the mnmal completon tme. 3. Partcle swarm optmzaton alorthm The partcle swarm optmzaton [12 whch s one of the latest evolutonary optmzaton technques was ntroduced n 1995 by Kennedy Eberhart. PSO alorthm s an adaptve method that can be used to solve optmzaton problem. Conductn search uses a populaton of partcles. Each partcle corresponds to ndvdual n evolutonary alorthm. A floc or swarm of partcles s romly enerated ntally, each partcle s poston representn a possble soluton pont n the problem space. Each partcle has an updatn poston vector X updatn velocty vector V by movn throuh the problem space. Kennedy Eberhart proposed the formula of updatn poston vector X : v + 1 = 4 V And the formula of updatn velocty vector : v+ 1 = w v + c1r1 (p + c2 (p 5 Where c1 c2 are postve constant r1 are unformly dstrbuted rom number n [,1. The velocty V vector s rane of [ V,V ma ma [13. At each teraton step, a functon F s calculated by X poston vector evaluatn each partcle s qualty. The vector P represents ever the best poston of each partcle P represents the best poston obtaned so far n the populaton. Chann velocty ths way enables the partcle to search around ts ndvdual best poston P, updatn lobal best poston P, untl searn for the lobal best poston n the lmted teraton. 4. PSO alorthm for scheduln problem n computatonal rd In rd envronment, scheduln whch s one of a varety of NP-complete problems s a very complcated ssue. The am of ths problem s how to mprove the effcency of resource how to mnmze the completon tme at the same tme. PSO can be mplemented to solve varous functon optmzaton problems, or some problems whch can be transformed to functon optmzaton problems. For solvn the tas scheduln problem by usn PSO alorthm, we use the small poston value (SPV rule [14 whch borrowed from the rom ey representaton to

3 A Tas Scheduln Alorthm Based on PSO for Grd Computn 39 solve the tas scheduln problem. The SPV rule can convert the contnuous poston values to dscrete permutaton n PSO alorthm. A populaton of partcles s romly enerated ntally. X Each partcle denoted as (=1, 2, 3,, n wth ts poston, velocty, ftness value represents a potental soluton about resource scheduln. The poston of each partcle s represented by n number of dmensons as = [ 1, 2, n where s the poston value of partcle wth respect to the n dmenson, the velocty s v [,,, represented by = v1 v2 vn v where s the velocty of partcle wth respect to the n dmenson. Based on SPV rules, the contnuous poston convert to a S permutaton of sequences, whch s a sequence of tass X S mpled by the partcle. s represented by s = [ s1, s2, sn s, where s the sequence of tas of partcle n the processn order wth respect to the n r [,,, dmenson. Defne = r1 rn as the n dmenson r tas processn on the resources. The ftness value evaluated by ftness functon represents the partcle s qualty based on the current sequence. The personal best value denoted as P represents the best searchn poston of the partcle so far. For each partcle n the populaton, P can be determned updated at each teraton step. In rd f ( resource scheduln wth the ftness functon X where s the correspondn sequence of partcle, the personal best P of the partcle s obtaned such that f ( f ( 1 where s the correspondn sequence of personal best P.For each partcle, the p [,,, personal best s denoted as = p1 p2 pn where p s the poston value of the partcle best wth respect to the n dmenson. The best partcle n the whole swarm s assned to the lobal best denoted as G. The G can be f ( f ( obtaned such that, for = 1, 2, 3,, m, where s the correspondn sequence of partcle best P. In eneral, we defne the ftness functon of the lobal best f ( best = f ( best as, the lobal best s defned as = [, 2, n s the poston value of the 1 where lobal best wth respect to the n dmenson. Each partcle updates ts velocty vector based on the eperences of the personal best the lobal best n order to update the poston of each partcle wth the velocty currently updated n search space. Each partcle eeps trac of ts own best poston the swarm eeps trac of the lobal best poston. In addton, a local search may be appled to a certan roup of partcles n the swarm to enhance the eplotaton of searchn space. If the mnmum computn tme whch s able to complete the whole tass s found or fnsh wth mamum number of teraton, the process wll be termnated. The pseudo code of PSO alorthm for tas scheduln n rd computn system s ven as follows, BEGIN { Intalze parameters Intalze populaton romly Intalze each partcle poston vector velocty vector Fnd a permutaton accordn to each partcle s poston Evaluate each partcle fnd the personal best the lobal best Do { Update each partcle s velocty poston Fnd a permutaton accordn to the updated each partcle s poston Evaluate each partcle update the personal best the lobal best Apply the local search } Whle (!Stop crteron } END 4.1. Soluton representaton For tas scheduln alorthm n rd envronment, one of the most mportant ssues s how to represent a soluton. The soluton representaton tes up wth the PSO alorthm performance. We defne one partcle as a possble soluton n the populaton. And dmenson n correspondn to n tass, each dmenson represents a tas. The poston vector of each partcle maes transformaton about the contnuous poston. We use the smallest poston value, namely, the SPV rule s used frst to fnd a permutaton correspondn to X the contnuous poston. For the n tass m resource problem, each partcle represents a reasonable scheduln [,,, scheme. The poston vector = 1 2 n has a contnuous set of values. Based on the SPV rule, the contnuous poston vector can be transformed a dspersed s [,,, value permutaton = s1 s2 sn. Then the operaton r [,,, vector = r1 rn s defned by the follown formula:

4 4 Le Zhan et al = S mod m 6 Table 1 llustrates the soluton representaton of partcle X of PSO alorthm for 9 tass 3 processors. Based on s SPV rules, s formed to a permutaton. Usn the formulas (6, r11 = s11 mod 3= 2 where refers to the sequence number of computn processor. We defne that the start number sequence s zero. Dmenson Table 1 : Soluton epresentaton s Intal Populaton The ntalzed populaton of partcles s constructed romly for PSO alorthm. The ntalzed contnuous poston values contnuous veloctes are enerated by the follow formula [14 = ( mn + ma mn * r (7 Where mn = -.4, ma = 4. r s a unform rom number between 1. v = v ( v v mn + ma mn * r (8 Where v v mn = -.4, ma = 4. r s a unform rom number between 1. We thn the populaton sze s the number of dmensons. Snce the objectve s to mnmze the completon tme, the ftness functon value s the completon tme value whch s decoded from the operaton repetton vector for partcle. r f 4.3. The flow of PSO alorthm for tas scheduln ssues Intal populaton romly of PSO alorthm s the frst step of ths alorthm. The formulas (4 (5 are used to construct the ntal contnuous poston values velocty value of each partcle. The complete flow of the PSO alorthm for the tas scheduln of rd can be summarzed as follows, Step1: Intalzaton. Set the contents about ths alorthm: ma c 1 =c 2 =2w ter=1 Defne the number of actve resource the lst of tass. The dmenson of PSO alorthm s the number of tass. Intalze poston vector velocty vector of each partcle romly = [1, 2, n v = [v1, v2, vn Apply the SPV rule to fnd the permutaton s = [s1,s2, sn Apply the formula (6 to fne the operaton vector r = [r1,, rn Evaluate each partcle I n the swarm usn the objectve functon fnd the best ftness value amon the whole f ( swarm. Set the lobal best value Step 2: Update teraton varable. ter = ter + 1 Step 3: Update nerta weht. ter ter-1 ω = ω * β Step 4: Update velocty. G = f ( Table 2 : Local Search Appled to Permutaton before eparn Dmenson s r s r Table 3 : Local Search Appled to Permutaton after eparn Dmenson s r s r.

5 A Tas Scheduln Alorthm Based on PSO for Grd Computn 41 Table 4 : Parameter settns of PSO GA alorthm Alorthm Parameter descrpton Parameter Value PSO GA Sze of Swarm 3 Self-reconton coeffcent c1 2 Socal coeffcent c2 2 Weht w.9.4 Ma Velocty 1 Sze of populaton 3 Probablty of crossover.8 Probablty of mutaton.3 Scale for mutatons.1 Table 5 : An eample of the best result based on PSO alorthm for (3, 1 esource Tas T1 T2 T3 T4 T5 T6 T7 T8 T9 T Apply the formula (5 update velocty of each partcle Step 5: Update poston. Apply the formula (4 update poston of each partcle Step 6: Fnd permutaton. Apply the SPV rule to fnd the s [,,, permutaton = s1 s2 sn Step 7: Fne operaton vector Apply the formula (6 to fne the operaton r [,,, vector = r1 rn Step 8: Update personal best. Each partcle s evaluated by usn the operaton vector to f ( ( see f the personal best wll mprove. If best f, f ( ( then best = f Step 9: Update lobal best. f ( best f ( best f ( best = f ( best If, then Step 1: Stoppn crteron If the number of teraton eceeds the mamum number of teraton, then stop otherwse o to step Nehborhood of PSO In the PSO alorthm, local search s appled to the permutaton drectly. However, t volates the SPV rule needs a repar alorthm. The nehborhood structures [14 s showed n Table 2 where r 2 = 1 r 4 = 2 are nterchaned. As shown n Table 2, applyn a local search to the permutaton volates the SPV rule because the permutaton s a result of the partcle s poston values. After completn the local search, a partcle should be repared n order to the SPV rule s not volated. Table 3 shows a new permutaton acheved by chann the poston values accordn to the SPV rule after reparn. The values of postons permutaton are nterchaned n terms of ther dmensons. When s 1 = 1 s = 3 3 are nterchaned, ther correspondn 1 =.99 = 3.72 are nterchaned respectvely for dmensons =1 =3 to eep the partcle consstent wth the SPV rule. The advantae of ths approach s due to the fact that the repar alorthm s only needed after evaluatn all the nehbors n a permutaton. The performance of the local search alorthm depends on the choce of the nehborhood structure. In ths paper, the varable nehborhood search (VNS method presented by Mladenovc Hansen [18 s adopted. Usually the two nehborhood structures are employed. S1: Interchane two tass between a b dmensons, (a b S2: emove the tas at the a dmenson nsert t n the b dmenson, (a b Where a b are the rom nteer numbers between 1 the number of tass. Two swaps two nterchanes are used to dversfy the lobal best soluton before applyn the local search. The modfcaton s mportant to drect the search towards the lobal optma snce the lobal best soluton remans the same after some teratons, probably at a local mnmum. In addton, neural moves are allowed n the VNS local search n order to restart the search from a dfferent permutaton wth the same functon value. 5. Epermental settns results In our eperments we conducted a seral of eperments to test ths alorthm on a smulated rd envronment. We compared the performance of PSO alorthm wth enetc alorthm (GA [15-16 that have many smlartes. Genetc alorthm s an evolutonary natural selecton process. The cdate soluton of each eneraton s evaluated accordn to the hh ftness value s used to create further solutons va crossover mutaton procedures. The epermental parameter settns of PSO GA alorthms are descrbed n Table 3. We consdered a fnte number of processors n our small scale rd envronment assumed that the processn speeds of each processor the cost tme of each tas are nown. Each eperment was repeated 1 tmes wth dfferent rom seeds. We recorded the completon tme values of the best solutons throuhout the optmzaton teratons a mnmum cost tme of all tass completed.

42 Le Zhan et al In order to analyze the performance of tas scheduln alorthm, frstly we had an eperment wth a small tas scheduln problem. There s 3 resources s 1 tass.

6 42 Le Zhan et al In order to analyze the performance of tas scheduln alorthm, frstly we had an eperment wth a small tas scheduln problem. There s 3 resources s 1 tass. The speeds of the 3 resources are 4, 3, 2 the cost tme of each tas s 19, 23, 24, 2, 2, 25, 16, 21, 24, 15. The results of GA alorthm runnn 1 tmes were {26, 25.4, 25.8, 25.8, 25, 25, 25.8, 26, 25.4, 25}, wth an averae value of The results of PSO alorthm were {25, 25, 25.4, 25.4, 25, 25, 25, 25, 25, 25.2}, wth an averae value of PSO alorthm provded the best result 7 tmes, whle GA alorthm provded the best result 3 tmes. Table 2 shows an eample of the best tas scheduln results based on PSO alorthm about (3, 1, whch 1 means the tas s assned to the respectve resource n rd envronment. We test the tas scheduln problem from 5 processors, 1 tass to 2 processor 2 tass. In PSO alorthm, the parameters were set that the number of partcle s 3, the self-reconton coeffcent c 1 socal coeffcent c 2 are 2, the weht s lnear decreased from.9 to.4. For GA, the sze of the populaton s 3. Fure 1 shows the completon tme of PSO GA about 5 processors 1 tass. It dsplays that PSO usually had better averae completon tme values than GA. Fure 2 shows three types of test data of runnn dfferent numbers of tass. The curves of T1, T2 T3 denote the results about runnn 5 processors, 1 processors 2 processors. Table 6 shows the best result of GA PSO alorthm about s types of test data. It shows PSO usually spent the shorter tme to complete the scheduln than GA alorthm. It s to be noted that PSO usually spent the shorter tme to accomplsh the varous tas scheduln tass had the better result compared wth GA alorthm. Fure 1 : Performance of PSO GA scheduln alorthm about 5 processors 1 tass. Averae eecuton tme Number of tass T1 T2 Fure 2 : The performance of curves of dfferent numbers of processors runnn dfferent numbers of tass T3 Table 6 : Performance of GA PSO alorthm Problem GA Alorthm PSO Alorthm Completon tme Tme Completon tme Tme 5* * * * * * Conclusons In ths paper, scheduln alorthm based on PSO s proposed for tas scheduln problem on computatonal rds. Each partcle represents a feasble soluton. The poston vector s transformed from the contnuous values to the dscrete values based on SPV rules, accordnly, a permutaton formed. Our approach s to enerate an optmal schedule so as to complete the tass n a mnmum tme as well as utlzn the resources n an effcent way. We evaluate the performance of our proposed approach compared t wth enetc alorthm under the same condton. From the smulated eperment, the result of PSO alorthm s better than GA. Smulaton results demonstrate that PSO alorthm can et better effect for a lare scale optmzaton problem. Tas scheduln alorthm based on PSO alorthm can be appled n the computatonal rd envronment. 7. eferences [1 Foster C. Kesselman (edtors, The Grd: Blueprnt for a Future Computn Infrastructure, Moran Kaufman Publshers, USA, 1999.

7 A Tas Scheduln Alorthm Based on PSO for Grd Computn 43 [2 Abraham,. Buyya B. Nath, Nature's Heurstcs for Scheduln Jobs on Computatonal Grds, The 8th IEEE Internatonal Conference on Advanced Computn Communcatons (ADCOM 2, pp ,Cochn, Inda, December 2,. [3 Y. Gao, H.Q on J.Z. Huan, Adaptve rd job scheduln wth enetc alorthms, Future Generaton Computer Systems, pp Elsever,21(25. [4 M. Aarwal,.D. Kent A. Nom, Genetc Alorthm Based Scheduler for Computatonal Grds, n Proc. of the 19 th Annual Internatonal Symposum on Hh Performance Computn Systems Applcaton (HPCS 5,,pp Guelph, Ontaro Canada, May 25. [5 S. Son, Y. Kwo, K. Hwan, Securty-Drven Heurstcs A Fast Genetc Alorthm for Trusted Grd Job Scheduln, n Proc. of 19 th IEEE Internatonal Parallel Dstrbuted Processn Symposum (IPDPS 5, pp.65-74, Denver, Colorado USA, Aprl 25. [6 Lee Wan, Howard Jay Seel, Vwan P. oychowdhury, Anthony A. Macejews, Tas Matchn Scheduln n Heteroeneous Computn Envronments Usn a Genetc- Alorthm-Based Approach, Journal of Parallel Dstrbuted Computn 47, pp.8-22(1997, Foster I. Kesselman C. [7 V. D. Martno M. Mllott, Sub optmal scheduln n a rd usn enetc alorthms, Parallel Computn, pp , Elsever,3(24. [8 J.E. Orosz S.H. Jacobson, Analyss of statc smulated annealn alorthm, Journal of Optmzaton theory Applcatons, pp ,Sprner,115(1(22. [9 E. Tr, Y. Collette P.Sarry, A theoretcal study on the behavor of smulated annealn leadn to a new cooln schedule, pp.77-92, European Journal of Operatonal esearch, Elsever, 166(25. [1. Braun, H. Seel, N. Bec, L. Bolon, M. Maheswaran, A. euther, J. obertson, M. Theys, B. Yao, D. Hensen. Freund, A Comparson of Eleven Statc Heurstcs for Mappn a Class of Independent Tass onto Heteroeneous Dstrbuted Computn Systems, pp , J. of Parallel Dstrbuted Computn, vol.61, No. 6,21. [11 Kennedy J. Eberhart. Swarm Interllnece, Moran Kaufmann, 21. [12 J. Kennedy. C. Eberhard, Partcle swarm optmzaton, Proc. of IEEE Int l Conf. on Neural Networs, pp , Pscataway, NJ, USA,, [13 J.F. Schute A.A. Groenwold, A study of lobal optmzaton usn partcle swarms, Journal of Global Optmzaton, pp.93-18, Kluwer Academc Publsher,31(25. [14 M. Fath Tasetren, Yun-Cha Lan, Mehmet Sevl, Gunes Gencylmaz, Partcle Swarm Optmzaton Dfferental Evoluton for Snle Machne Total Wehted Tardness Problem, Internatonal Journal of Producton esearch, pp , vol. 44, no. 22, 26. [15 M. Weczored,. Prodan T. Fahrner, Scheduln of Scentfc Worflows n the ASKALON Grd Envronment, n ACM SIGMOD ecord, pp.56-62,vol.34,no.3, September 25. [16 S. Km J.B. Wessman, A Genetc Alorthm Based Approach for Scheduln Decomposable Data Grd Applcatons, n Proc. of the 24 Internatonal Conference on Parallel Processn (ICPP 4, pp , Montreal, Quebec Canada, Auust 24. [17 Ln Jan Nn Wu Hu Zhon,Scheduln n Grd Computn Envronment Based on Genetc Alorthm, Journal of Computer esearch Development, pp ,vol. 4,No.12,Dec 24. [18 Mladenovc, N. P. Hansen. (1997. Varable Nehborhood Search. Computers Operatons esearch, 24, [19 Acnowledment [2 Ths research was partally supported by the Natural Scence Foundaton of Chna under rant number , Unversty of Jnan under rant number Y617 Y529X.

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