Adaptive Divisible Load Scheduling in Computational Grids Luis de la Torre 1 Héctor de la Torre 2 1

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1 Adaptve Dvsbe Load Schedung n Computatona Grds Lus de a Torre Héctor de a Torre 2 Scence and Technoogy Schoo, Unversdad Metropotana, San Juan, PR 2 Schoo of Engneerng, Turabo Unversty, Gurabo, PR Ana G. Mendez Unversty System. de@suagm.edu, hectorfabo50@hotma.com Abstract Severa probems n scence and engneerng admt computatona soutons that are mpementabe over Grd computng patforms. One probem, freuenty faced by mpementers s how to dvde and dstrbute the workoad nto chunks among the Grd workers, the so-caed oad schedung probem. Most commony researches have studed ths phenomena departng from a statcs approach. Ths assumpton s not fuy functona n Grd envronment where the resources are non-dedcated. Ths research proposed a methodoogy to ntegrate a statstc heterogeneous patform schedung wth a dynamc resources predcton to dstrbute the workoad dependng on future avaabe resources. The SCOW agorthm s ntegrated to a tendency-based method, a mechansm to predct CPU utzaton. These mpementatons can retro ament the statstc schedung agorthm to produce accuracy estmaton to the Grd resources changes. Kewords: Schedung, grd computng, dvsbe oad, dvsbe task, makespan, Throughput. Introducton Dvsbe workoad conssts of workoads that can be parttoned nto arbtrary tasks or chunks. These chunks n many cases are a core task that s repeated a number of tmes over dfferent data. In snge-program mutpedata (SPMD) stye, these tasks are mpemented as nested seuences of do-oops around the core task. Usuay, a master process scheduer the chunks across a partcpatng workers (round of data nstaments), so the executon tme of the entre oad (makespan) s mnmum. In ths research, s assumed that the dstrbuton process use the network connecton n a seuenta fashon [3]. Each data nstament s foowed by a receve and a compute operaton, both performed by the recevng worker. The p workers can compute and receve the next tasks concurrenty. In SPMD mpementatons, rounds are controed by an externa do-oop, whch mposes the perodc character of the job's executon. The man parameters n a SPMD mpementaton are thus, the number of rounds, denoted beow by m, the number of workers nvoved n the concurrent computatons, denoted beow by and the chuck szes x to the worker, p. Two approaches domnate among the methodooges deveoped for schedung of master-workers oad. These methodooges are: steady state schedung (SSS)[6, 7], and dvsbe oad theory (DLT) [2, 3, 4.]. Both methodooges assume dedcated resources. These assumptons make the agorthm poor n rea tme envronment such as Grd computng patforms wth non-dedcated workers. SCOW s a perodc user-eve scheduer that tunes some seected parameters n a snge-program mutpedata mpementaton of a master-worker parae souton. SCOW mnmzes the job make-span under ether maxma producton per perod, or perfect worker utzaton. Ths paper presents the theoretca foundatons of SCOW to maxma producton per perod mprovng wth the mxed tendency based strategy for predctng the CPU utzaton of workers. UMR [2] s a DLT mut-round agorthm for schedung dvsbe oads on parae computng systems. For homogeneous systems, the method uses unform rounds meanng that n each round, each worker receves the same amount of work. There are two versons for the UMR dea. In the orgna, caed UMR, the unform amount of work s ncreased wth each round. In a revsed verson, caed UMR2, the unform amount s ncreased or decreased dependng on a parameter. If > the amount s ncreased, and decreased f <. The UMR scheduer mantans perfect worker utzaton throughout the executon, but f < there s no perfect worker utzaton for URM2. Both methods mode the system wth a set of affne euatons expressng executon tmes n terms of oad. These euatons are smar n sprt to the one used for SCOW. The mode gves the amount of work for the frst round and, through a recursve formua, the ncrements or decrements per round. Authors n [3] mprove the UMR by predct the workers CPU utzaton wth a mxed tendency based strategy.

2 Ths paper s organzed as foows: Secton 2 dscusses the mode. Secton 3 descrbes the Theoretca Foundatons of SCOW. Secton 4 shows the Makespan mnmzaton probem to deveop a mutround dvsbe oad schedung agorthm for affne cost modes. In secton 5, the oca tasks CPU utzaton n a CPU predcton strategy s ncorporated. SCOWS was evauated wth extensve smuaton n Secton 6 and the executons s dscuss ater n on secton 7. Fnay, Secton 8 concudes the paper and dscusses future drectons. 2. Mode As stated n, s assumed a tota number X of core tasks. These core tasks can be aggomerated to produce dfferent chunks szes (porton of job). These jobs are ndependent n the sense that nether orderng between them, nor synchronzaton among them s necessary. 2.. Notaton As ustrated n Fgure, the STARAFFINE network conssts of p + processor, P= {P 0,P,P 2,..., Pp}. The master processor s denoted P 0 whe the p workers are abeed as P, p. There are p communcaton nks from the master P 0 to each one of the workers P. Let x be the number of unts of core tasks sent to worker P. (x) measures the tme unts that takes for a oad x to be moved from the master to the th worker n affne mappng mode. Each worker performs two operatons, as we. These operatons are message recepton and the actua executon of the job, referred as computaton. The worker spends w (x) tme unts n executng x core tasks, w (x) s supposed to be an affne mappng. w 2 P 0 P P 2 P P p Fgure. Heterogeneous Star Graph 2.2. Archtectura Mode As mentoned before, wthn a round, the master performs a seuence of data sends operatons. Each data retreva and send s foowed by the data transmssons () over the network. Receve and compute (w) operatons are performed by the workers upon the arrva of the data package. As a resut, two w 2 w w p p major concurrent tme segments are dstngushed wthn a round: L, tme spent by a network nk n transmttng a round of data chunks; W maxmum tme spent by a workers n competng the recepton of the data and executon of the correspondng data chunk. Ths research assumes the fu overap, snge-port mode. In ths mode, the master uses the network connecton n a seuenta fashon and the workers can perform the computaton concurrenty wth data recepton. One of the assumptons n ths research s that the workers are non-dedcated processors. In Grd envronmenta the CPU power s dstrbuted between oca task and the Grd users Affne mappng Ths subsecton s a bref dscusson of the affne maps n whch the mathematca mode s based. The executon tmes to each operatons of data communcaton, and tasks executon vary as an affne mappng on the number of aggomerated core tasks x. Ths s, (x) = x + L () w (x) = w x + W (2) for p, where L s the nta cost of estabshng a connecton between the master P 0 and worker, s the send tme assocate to the data of a snge core task; W s the overhead (startup tme) of the computaton n processor and w corresponds to the executon tme of a snge core task. 3. Theoretca Foundatons of SCOW SCOW s desgned as a perodc user-eve scheduer for aocatng aggomerated core tasks on parae heterogeneous computng systems. Ths means that the mathematca framework behnd SCOW s desgned to return optma constant vaues of the three parameter descrbe above m,, and x to a master-worker SPMD mpementaton; under some specfc constrants. Indeed, SCOW mnmzes the make-span of the job under ether maxma producton per round or perfect system utzaton []. In ths research, the maxma producton per perod SCOW abty s seected, refers to a dstrbuton of aggomerated core tasks across the workers that maxmzes the tota number of tasks competed n a round. 3.. Maxma perodc producton In ths secton a bref descrpton of scheduer theory s presented. The maxma producton probem (MP) s a probem that mposes a restrcton n the perod to fnd the best approxmaton to the maxmum number of task performed.

3 3.2. Probem Suppose p (3) m x X where m s a number of round of data nstaments. The (MP) probem s stated as foows: Gven a tme perod T, fnd a subset of + workers such as Maxmze Subject to x (4) 3.3. Souton Let MAXTASK(T) be the optma souton of the prevous probem. The next theorem provdes a cose form souton for the MP probem n homogeneous patform. Theorem. Let T be a rea nonnegatve numbers, w =w and = for a and Then L ( x ) T (5) W max{ w ( x ) / } T (6) x 0, (7) y w ( T) (9) T / ( y) (0) T T ( y) () MAXTASK(T) = y max{0, ( T )} (2) The prevous theorem has a possbe extenson to the heterogeneous probems. At ths moment the best souton s an approxmaton theorem. Theorem 2. Let T be a rea nonnegatve number, p be a postve nteger. The method:. Sort the workers by ncreasng communcaton tmes. Renumber them so that 2 p. 2. Let y =w - (T) for p and the argest ndex so that ( ). y T If <p, et T =T- ( ) y ; otherwse et T =0. Return the vaues to construct the foowng neuates: L. MAXT ASK( T) y max{0, ( T )}.If the perod T s arge enough MAX T ASK( T) y max{0, ( T )} L Fgure 2. Gantt Char nterpretaton to the proposes souton The theorem 2 gves an approxmate to the optma souton to the MP probem. In secton 4 ths approxmaton s used as a restrcton to formuate a makespan mnmzaton probem. Fgure 2 s a graphca representaton to the souton gven by the theorem 2. Ths souton foows the prncpe of bandwdth-centrc [3, 4] because the prortes do not depend on the workers computaton capabtes, ony on ther communcaton capabtes. Esewhere, the number of seect processor s drecty affected by the computaton capacty. In Grd computng the computatona capacty depends on the oca CPU utzaton. Ths tendency s estmated usng predcton strategy descrbed n secton Last round modfcaton A common condton to get an optma schedue s that a processor fnshes the work at the same tme. Modfcatons of the ast round are used to mpose the condton that a processors end operatng at the same tme[]. Ths ast round modfcaton ntroduces a constant makespan reducton of /2T. 4. Makespan mnmzaton The makespan mnmzaton probem constraned to maxma producton (MMP-MP) souton approxmaton and the workers order descrbed n theorem 2 s formuate as foow: Mnmze (T) = (M+/2)T + (Y ) subject to X v y y (3) T w ( y ), (4) T ( y ) (5) T w ( y ) (6) y 0, (7) p (8)

4 The probem s soved by Lagrange mutpers [9]. Ths souton s stated n the next theorem. Theorem 3: Let X be a nonnegatve rea number and a postve nteger (+ p). Then the souton to the MMP-MP probem wthout restrcton 6 wth + processor s, where and b() w X b ( ) ( ) ( ) T a 2 w a a ( ) (9) j j j wj (20) w j b( ) W j j j Lj j wj j j wj W (9) the theorem 3 permt reformuates the MMP-MP, n the probem to fndng the mnma vaue of (T) wth rangng over the subset the that satsfyng w ( y ) T (20) 5. CPU predcton strategy In Grd computng typcay the resources are nondedcated, that s, the avaabty of the fu processng speed s no guaranteed. Let S, the fu processor speed. The oca task executon generates a CPU utzaton, f the Utzaton can be predcted them the ActuaSpeed can be computed as foows: ActuaSpeed = S * (00%-Utzaton) (2) Ths ActuaSpeed s used as retro-amentaton nformaton to the statc scheduer. To predct the CPU oad and utzaton s used a tme seres predcton approach [0, ] that has been probed the effectve emprcay. The dea of ths predcton strategy s based on the assumpton that f the current vaue ncreases, the next vaue w aso ncrease, and f the current vaue decreases, the next vaue w aso decrease. Formay, we can wrte: If (U T- < U T ) IncrementVaueAdaptaton() P T+ = U T + IncrementVaue Ese f(u T- > U T ) DecrementFactorAdaptaton() P T+ = U T IncrementVaue Where, U T : the measured utzaton at measurement T, P T+ : the predcted utzaton for measurement U T+, H: the number of hstorca data ponts used n the predcton. Increment vaue and decrement factor can be cacuated as: Procedure: INNCREMENTVALUEADAPTATION() Mean = (/n) ReaIncVaue =U T U T- ; NormaInc = IncrementVaue + (ReaIncVaue IncrementVaue) AdaptDegree; f (U T < Mean) IncrementVaue = NormaInc; Ese PastGreater = (number of data ponts>ut)/h TurnngPontInc=IncrementVaue PastGreater IncrementVaue=Mn(NormaInc, TurnngPontInc) AdaptDegree can range from 0 to and expresses the adaptaton degree of the varaton. The best vaues for nput parameters such as AdaptDegree and DecrementFactor are determned emprcay. 6. Experment In tabe the group of numerca vaues seected to perform the smuaton s presented. The vaues are used to predct the correspondng SCOW-MP and UMR verson deveop n [3]. These predctons are made usng the mathematca euatons underyng them and a random varabe to CPU smuaton s aso generated usng gamma functon Tabe : Smuaton Parameter Parameter Number of workers Vaues p=0,20,30,40 Aggomerated tasks X = 000 Computatona rate randonvarave S =.5+randonvarave w = /S s aso generated usng gamma functon wth fxed arrva tme anda=.5 and beta = Transfer rate B =.p to.p + ; =/b Computatona atency Communcaton atency clat = 0:03; W = clat; nlat = 0:03; L = nlat 7. Resut It s worth remarkng that UMR methods and the optma number of rounds, and perform no dscretzaton on the amount of work per round. Thus, n order to make the comparsons possbe, SCOW dscretzatons are made on the number of rounds and not on the amount of aggomerated tasks per round. The randomy chosen vaues are shown n Tabe.

5 The numerca predcton of the performances of UMR and SCOW are shown n Tabe 2. Tabe 2: Comparson Between SCOW and UMR SCOW-MP UMR UMR2 Normazed Make-span Normazed Workers CPU Utzaton Tabe 2 shows the comparson between SCOW-MP, UMR and UMR2, averaged over smar (n the number of workers) experment. A the scheduer was mproved by a ast round modfcaton n order keep consstent the comparson. The frst row shows the rato of make-span acheved for the 3 scheduers. The second row shows the smar rato for the system utzaton, but at ths tme the rato s nverted, because the maxma vaues s the best. The man observaton s that SCOW-MP outperforms UMR and UMR2 on average. The SCOW-MP s the best agorthm n Make-span and system utzaton. 8. Concusons Many researches n maxma throughput n nea mode can be found n the terature. These resuts are deveoped to the probem formuated for fxed szes tasks. In ths research the probem s exported to affne mode and n contrast the goa s maxmze the producton, that s, the tota number of tasks processed. The contrbuton of ths research ncudes an optma souton n the homogeneous case and approxmate souton n the heterogeneous case, ntegrate wth a CPU predcton strategy to perform a scheduer reabe n a Grd envronment. The makespan and system utzaton of two agorthms s aso compared. The resuts show that the proposed SCOW-MP agorthms outperform the compettors. Future work ncudes the deveopment of a strategy to predct the network utzaton; due to SCOW s a bandwdthcentrc agorthm. 9. Acknowedgments Ths work was made possbe by fundng from the Carbbean Computer Center of Exceence (CCCE) under NSF Award number CNS Thanks are due to Dr. Juan F Arrata and Dr. Ova Prmera-Pedrozo from the Unversdad Metropotana- Cupey for ther support. References. L. de a Torre, Schedung dvsbe tasks under producton or utzaton constrants, PhD dss., Unversty of Puerto Rco, Mayaguez, Puerto Rco Y. Yang, K. van der Raadt, H. Casanove, Mutround Agorthms for Schedung Dvsbe Loads, IEEE Transactons on Parae and Dstrbuted Systems, Vo. 6, No., C. Banno, O. Beaumont, L. Carter, J. Ferrante, A. Legrand and Y. Robert, Schedung Strateges for Master-save Taskng on Heterogeneous Processor Patforms, IEEE Transactons on Parae and Dstrbuted Systems, Vo. 5, No. 4, pp , M. Drozdowsk and P. Wonewcz Optmum Dvsbe Load Schedung on Heterogeneous Stars wth Lmted Memory, European Journa of Operaton Research, Vo. 72, No. 2, N. Jones and P. Pevzner An Introducton to Bonformatcs Agorthms, MIT Press, V. Bharadwaj, D. Ghose, V. Man, and T.G. Robertazz. Schedung Dvsbe Loads n Parae and Dstrbuted Systems. IEEE Computer Socety Press, O. Beaumont, H. Casanova, A. Legrand, Y. Robert, Y. Yang: Schedung Dvsbe Loads on Star and Tree Networks: Resuts and Open Probems. IEEE Trans. on Parae and Dstrbuted Systems, vo. 6, no. 3, 2005, D. Bertsekas. Constraned Optmzaton and Lagrange Mutper Methods. Athena Scentfc, Bemont, Mass., D. Bertsekas, edtor. Constraned Optmzaton and Lagrange Mutper Methods. Athena Scent_c, Bemont, Mass., L. Yang, J.M. Schopf, and I. Foster, Conservatve Schedung: Usng Predcted Varance to Improve Schedung Decson n Dynamc Envronments, SuperComputng 2003, Phoenx, Arzona USA November L.Yang, I. Foster, and J.M. Schopf, Homeostatc and Tendency-Based CPU Load Predctons, Internatona Parae and Dstrbuted Processng Symposum (IPDPS'03), Nce,France, Apr Yang Yang and Henr Casanova, UMR: A Mut- Round Agorthm for Schedung Dvsbe Workoads; Proceedng of the Internatona Parae and Dstrbuted Processng Symposum (IPDPS 03), Nce, France, Apr Sad Enaffar and Nguyen The Loc. "Enabng Dynamc Schedung n Computatona Grds by Predctng CPU Utzaton"; WSEAS Transactons on Communcatons Issue 2, Voume 4, pages December 2005.