Load Prediction Using Hybrid Model for Computational Grid

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1 Load Predicion Using Hybrid Model for Compuaional Grid Yongwei Wu, Yulai Yuan, Guangwen Yang 3, Weimin Zheng 4 Deparmen of Compuer Science and Technology, Tsinghua Universiy, Beijing 00084, China, 3, 4 {wuyw, ygw, zwm-dcs}@singhua.edu.cn yuan-yl05@mails.singhua.edu.cn Absrac Due o he dynamic naure of grid environmens, schedule algorihms always need assisance of a long-ime-ahead load predicion o make decisions on how o use grid resources efficienly. In his paper, we presen and evaluae a new hybrid model, which predics he n-sep-ahead load saus by using inerval values. This model inegraes auoregressive (AR) model wih confidence inerval esimaions o forecas he fuure load of a sysem. Meanwhile, wo filering echnologies from signal processing field are also inroduced ino his model o eliminae daa noise and enhance predicion accuracy. The resuls of experimens conduced on a real grid environmen demonsrae ha his new model is more capable of predicing n-sep-ahead load in a compuaional grid han previous works. The proposed hybrid model performs well on predicion advance ime for up o 50 minues, wih significan less predicion errors han convenional AR model. I also achieves an inerval lengh accepable for ask scheduler. I. INTRODUCTION Grid compuing [] is he high-performance and inerne-based infrasrucure ha aggregaes geographically disribued and diverse resources o deliver compuaional power o users in a ransparen way, supporing large-scale, resource-inensive and disribued applicaions. Task scheduling is an imporan facor on improving resource usage efficiency in such a dynamic disribued environmen. Obviously, sysem load predicion can be used o forecas ask run ime [8] and guide scheduling sraegies, hus o achieve high performance and more efficien resource usage [] [3] [4]. In mos previous works, load predicion models, such as mean-based mehods, median-based mehods, auoregressive (AR) models, polynomial fiing, ec., use poin value o represen fuure load saus of worksaions, clusers and grids. Papers [5] [6] [0] describe such poin value predicion models and heir performances. Some oher works, like [], implemen inerval value predicion on srucural model o forecas sysem performance in a disribued producion environmen. Though poin value predicion sraegy has been widely adaped by recen sysem load predicion models, his kind of sraegy do have some drawbacks for highly disribued environmens. Because grid asks usually ake long run ime, ask scheduler in a compuaional grid needs a comparaively long-ime-ahead predicion or an n-sep-ahead predicion wih large sep inervals. Poin value can hardly cover load variabiliy in such a long ime frame (i.e., sep inerval), and herefore hese convenional poin value predicion models do no perform well on n-sep-ahead load predicion for large n. The inerval value predicion model proposed in [] addresses hese wo problems of poin value predicion and provides valuable addiional informaion ha can be used by a ask scheduler. Bu i does no achieve a reasonable predicion inerval lengh, which is imporan for he ask scheduler. Furhermore, mos of he previous predicion works also ignore some problems in he hisory load daa, which significanly affecs he predicion accuracy. One problem is he load measuremen error, which inherenly exiss in all measuremen mehods and can be amplified by a predicion model; anoher problem is he load daa noise inroduced by he load flucuaion in a compuaional grid. The endency of hisory daa is used by some endency-rack models, like AR model or polynomial fiing, o forecas fuure saus. Bu he fuure load endency migh be disored or concealed in he noisy daa, and herefore misleads he endency rack models, which will consequenly impair predicion accuracy. Our conribuion in his paper is ha we propose a new hybrid model, which can forecas far more n-sep-ahead load saus han previous models and mehods, wih comparaively less predicion errors and accepable predicion inerval lengh for ask scheduler and oher uiliies. To deal wih he dynamic naure of compuaional grids and he needs of ask schedulers, we inegrae AR model wih confidence inerval esimae in our hybrid model, where AR model is used as a basic n-sep-ahead poin value predicion mehod, and he confidence inerval of fuure load saus is esimaed based on boh hisorical and prediced poin values. In order o enhance predicion accuracy, Kalman Filer [] is used o minimize he measuremen errors of hisory load daa, and Savizky-Golay filer funcions [7] are used as he endency revealing ool for daa noise eliminaing. The resuls of our experimens on a real compuaional grid environmen demonsrae ha his hybrid model is of an excellen capabiliy on n-sep-ahead load predicion, and i also achieves much smaller predicion mean square error han convenional AR model. Furhermore, n-sep-ahead load predicion is achieved for large n in our hybrid model as well. The corresponding predicion advance ime can be up o 50 minues, wih low predicion mean square error (0.04 on average) and accepable confidence inerval lengh (less han 0%) for he ask scheduler. The res of his paper is organized as follows. Secion II inroduces he relaed work. Secion III analyses he causes of /07/$ IEEE 35 8 h Grid Compuing Conference

2 measuremen errors and describes an error minimize ool Kalman filer. Secion IV presens our hybrid model on load predicion, and also inroduces a signal smoohing algorihm, Savizky-Golay filer, which is used o eliminae daa noise and reveal he endency of he daa more clearly. Secion V describes he experimen resuls of our hybrid model on a real compuaional grid. Finally, we conclude our work and describe our fuure work in secion VI. II. RELATED WORK Previous works [7] [0] indicae ha load has such properies as self-similariy and epochal behaviour, and is srongly correlaed over ime, which implies ha sysem load is consisenly predicable from he pas behaviour. Therefore, correcly correlaing he hisory daa wih he fuure values is he kernel o make accurae predicions. Several load predicion sraegies in he disribued environmen have been proposed in he pas. [4] exends he predicion by using seasonal variaion and Markov model-based mea-predicor in addiion o seasonal variaion for -sep-ahead predicion. [5] proposes a muli-resource predicion model ha uses boh auocorrelaion and cross correlaion o achieve higher predicion accuracy. The Nework Weaher Service (NWS) provides a dynamically monioring and forecasing mehod o implemen -sep-ahead predicion on nework and compuaional resources [5]. NWS uses various predicion mehods, such as mean-based mehods, median-based mehods, and AR mehods o forecas he fuure sysem saus a he same ime. NWS racks he accuracy of all predicors, and selecs he one exhibiing he lowes cumulaive error measure a any given momen o generae a forecas. In his way, NWS auomaically idenifies he bes forecasing echnique for any given resource. In Dinda s paper [6], he predicion power of several linear models, including AR, MA, ARMA, ARIMA, and ARFIMA are evaluaed in deail. Their resuls show ha simple, pracical models such as AR are sufficien for load predicion and AR(6) models or beer are recommended for CPU load predicion. In [6], several homeosaic and endency-rack -sep-ahead predicion sraegies are presened and evaluaed. Homeosaic mehods assume ha he load of a sysem always remains seady in a given ime frame, while endency-rack sraegies are based on he assumpion ha he endency of he load variabiliy exiss in he load daa, and can be properly revealed by he endency-rack models. The predicion of fuure value is adjused according o he magniudes of he las load measuremen and he las predicion error, and he evaluaions of his echnique show ha i ouperforms NWS for CPU load predicion. The predicion sraegies menioned above in his secion are all using poin value o represen he fuure load saus. Convenional poin value predicion models are ofen inaccurae since hey can only represen one poin in a range of possible behaviours. In [], sochasic inerval values are inroduced in he area of sysem load predicion, which can address his problem. Auhors of he paper define sochasic values using inervals, and hen hey define he inerval arihmeic formulas and implemen inerval value on he poin value predicion model. The inerval value predicion model in [] performs more effecively han poin value predicion models in producion environmens. III. MEASUREMENT ERROR AND ERROR MINIMIZATION Measuremen error is unavoidable in every measuremen mehod. On one hand, he monioring sensors bring some degrees of disurbers ino arge sysem; on he oher hand, some measuremen mehods gain performance value by esimaion, for example, compuing CPU availabiliy from he average load of he sysem, or using he value of a ime poin in a period o represen he average performance of ha period. Because of hese, measuremen would ineviably be inaccurae. Even worse, a predicion model would amplify hese measuremen inaccuracies in he predicion resul. In his secion, we ignore he errors caused by he monioring sensors, considering ha hese errors are relaively small compared wih hose caused by he measuremen mehods. In his secion, he measuremen error repored by he Compuer Science and Engineering Deparmen a UCSD and measuremen error observed by us in Bioinformaics Applicaion Grid (BioGrid) of ChinaGrid [0] are discussed firs, followed by he inroducion of he soluion o he problem, which use he famous Kalman filer o minimize he error observed in he BioGrid. A. Measuremen mehods and errors observed by UCSD In [0], hree kinds of measuremen mehods are given and he resuls gahered by monioring a collecion of worksaions and compue servers in he Compuer Science and Engineering Deparmen a UCSD are repored. To deermine he accuracy of hese hree mehods, hey compared he measuremens of hese hree mehods wih he readings generaed by an independen en-second, CPU bound process which hey refer o as he es process. The mean absolue measuremen errors of hese hree mehods observed for differen hoss a UCSD vary from as small as 3.% o as big as 4.3%, which would ineviably deeriorae he predicion accuracy. B. Measuremen mehods and errors observed in he BioGrid of ChinaGrid The BioGrid of ChinaGrid (see secion V for deails) uses a simple mehod o generae he average performance value of every fixed ime frame. The monior cenre gahers various performance daa from he nodes of he Grid every 60 seconds, and uses he poin value observed a he beginning of he monior inerval o represen he average performance of his ime frame. Because he asks running on he BioGrid are ime-consuming and he CPU load of each node in he BioGrid is sable in a moderae period, we assume ha he load daa deeced by he node sensors a a cerain ime poin can be used o represen he average load of he ime frame o which his ime poin belongs o. We also use he es process mehod in [0] o deermine he measuremen accuracy. We se a es process o repor 36

3 CPU load every 0 seconds, whereas he original BioGrid sensors repor he monioring informaion every 60 seconds. Fig. deails he measuremen errors observed from a node of he BioGrid. Fig. CPU load measuremen errors observed on one node (in Tsinghua Universiy) of he BioGrid. Mean absolue Measuremen Error is , and Mean Square Measuremen Error is C. Using Kalman filer o minimize measuremen errors In 960, R.E. Kalman proposed a recursive soluion o he discree-daa linear filering problem in his famous paper []. Kalman filer has been exensively researched and applied, paricularly in he area of auonomous or assised navigaion [9]. I provides an efficien recursive mehod o esimae he sae of a process, and minimizes mean square error. Kalman filer uses feedback conrol o esimae a process: firsly he filer esimaes he process sae a cerain ime and secondly obains feedback from he measuremens. Thus we can divide he equaions of Kalman filer ino wo groups: ime updae equaions and measuremen updae equaions. The specific equaions for ime and measuremen updaes are presened in [] and [9]. Our work applies Kalman filer o minimize measuremen errors, and herefore enhance predicion accuracy, which is evaluaed in secion V. Kalman filer has wo iniial parameers, process noise covariance Q and measuremen noise covariance R. In pracice, hese wo parameers migh vary wih each ime sep or measuremen, bu we assume hey are consans here. Presuming a very small process variance, we le Q = e 5, and fix he measuremen variance R = (0.06) = Because his is he rue average measuremen error variance we observed from he pervious load daa in he BioGrid of ChinaGrid, we would expec he bes performance in erms of balancing responsiveness and esimae variance. IV. PREDICTION Task schedule and load balance sraegies can benefi a lo from accurae load predicion. Because he load variabiliy and resource consuming siuaion on a compuaional grid are srongly correlaed over ime and have he properies such as self-similariy, period sabiliy, long ime load predicion should be feasible. In his secion, we inroduce our hybrid model on load predicion for compuaional grid, which inegraes AR model wih confidence inerval esimae. We use AR model as he basic linear model o predic an n-sep-ahead load poin value, and hen hisory load daa and predicion load daa are used o esimae he possible load inerval of an n-sep-ahead fuure ime frame. Even wih Kalman filer applied o eliminae he measuremen errors, he noise sill exiss in he load daa, and he endency of load variabiliy is concealed in he flucuaing daa noise. We need a smoohing algorihm o expose he feaures of load daa, and provide a reasonable saring approach for parameric fiing. If he curve of load daa is smoohed, he endency of fuure sae would be more obvious and he predicion would be more accurae. A. Smooh hisory load daa using Savizky-Golay filer Savizky-Golay smoohing filer [7] is ypically used o "smooh ou" a noisy signal wih a large frequency span. In our work, we use Savizky-Golay smoohing filer o smooh load daa in several seps of our predicion model. Savizky-Golay filer can be regarded as a generalized moving average. The filer coefficiens are derived by implemening an unweighed linear leas square fi using a polynomial of a given degree. A higher degree polynomial makes i possible o achieve a high level of smoohing wihou aenuaing daa feaures. For frequency daa, his filer is effecive a preserving he high-frequency componens of he signal. In our work, we se he parameers of Savizky-Golay filer wih span=0, pd (polynomial degree) = 4. B. Predic fuure sae using AR models and inerval value In [5], he performance predicion minimum mean square error and mean percenage error values for each nework seing of differen predicion models are summarized. Alhough bes predicor of each performance characerisic is, in general, no obvious and varies from resource o resource, wih a series evaluaion in [6], AR models are recommended o predic compuing resource relaed meric, such as CPU load. ) Using AR models o predic a fuure poin value The main idea behind using a linear ime series model in load predicion is o rea he sequence of periodic samples of performance, <Z >, as a realizaion of a sochasic process ha can be modelled by linear saionary models, such as AR, MA, ARMA, ARIMA, ARFIMA, ec. [6]. The coefficiens of he model can be esimaed by observing pas daa sequence. If he acion of he model can cover mos of he daa sequence variabiliy, is coefficiens could be used o esimae fuure daa sequence values wih low mean squared error. AR model is a model used o find an esimaion of a daa based on previous inpus of he daa. The general form of a ph-order AR(p) model is as follows: p X = ϕ X + ε i= i i (4.) 37

4 AR model consiss of wo pars: an error or noise parε, and an auoregressive summaion ϕ i X. The summaion i represens a fac ha curren inpu value depends only on he previous inpu values. The variable p is he order of AR model. The higher he order of AR model, he more accurae a represenaion will be. As he order of he model approaches infiniy, we ge almos an exac represenaion of he inpu daa; however, he compuaional expense on calculaing he AR(p) coefficiens increases wih he p order. Therefore, we should balance he predicion accuracy and he compuaional expense. In [6], he auhors colleced a large number of Hz benchmark load races, which capure all he dynamics of load signal, and subjeced hem o a deailed saisical analysis, drew a conclusion ha AR(6) models or beer are recommended for hos load predicion. The problem in AR(p) analysis is o derive he "bes" values for φi, given a series X -i. The majoriy of mehods assume ha he series X -i is linear and saionary. By convenion he series X -i is assumed o be zero mean, if no his is simply anoher erm ε in he summaion of he equaion (4.). Even for relaively large values of p, wih he Burg algorihm [3] ha we used o compue he AR(p) coefficiensϕ i, his can be done almos insananeously. Then he -sep-ahead value X can be easily compued. However, in highly disribued compuaional grid environmens, -sep-ahead load forecasing can no saisfy he demands of ask scheduler. The furher he load predicion of a compuaional grid can reach, he more appropriae he ask scheduler can make scheduling sraegies and balance he load. So wha he scheduler in hese large disribued environmen needs is he n-sep-ahead predicion (n =,3,4 or even 30,40,50..), which can forecas he load variabiliy far more ahead before i really happens. AR(p) model aemps o predic an oupu X of a sysem based on he previous inpus (X -, X -, X -3...) and he coefficiens ( ϕ, ϕ,... ϕ p ). Before we use AR(p) model o implemen he n-sep-ahead predicion, he following wo imporan assumpions have o be addressed: The coefficiens of AR(p) model represen he variabiliy of he hisory load daa X -, X -,, and hey are also suiable o represen he endency of he load in a fuure period, and he variabiliy of fuure unknown daa X, X +, X +n-, and X +n-. The predicion of he X +n- is based on he daa X +n-, X +n-3,, X, X -, X -,,and only daa X -, X -, is he rue measured daa; X, X +,, X +n- mus be compued sep by sep before predicing he X +n-, and hen X, X +,, X +n- can be assumed as he rue daa when we predic he X +n-. Based on hese wo assumpions, he n-sep-ahead performance poin value predicion is generaed using he following algorihm in Table I. p i= TABLE I N-STEP-AHEAD AR(P) MODEL PREDICTION ALGORITHM ϕ ϕ,... ϕ, Compue he AR coefficiens p based on he rue monioring daa X -, X -, ; for i = o n use he AR coefficiensϕ, ϕ,... ϕ p and X -+i-, X -+i-,, o compue he X -+I ; end ) Confidence inerval value predicion AR(p) model predics he fuure load saus by using poin value. In pracice, poin value is ofen an ideal esimae, or a value ha is accurae only for a given ime frame []. I ofen represens he average load in a given ime frame, and can no reflec he variabiliy of a sysem in his ime frame. In he highly disribued sysems like grids, poin values can no convey enough dynamic load informaion for he ask scheduler o make scheduling sraegies. Anoher ype of value wih he form of X±Y, or X~X, is referred as inerval value. The load of a compuaional grid varies over ime, and he inerval value can provide an esimae of his variabiliy in a given ime frame. Fig. CPU load variabiliy of THU node cluser of he BioGrid Unforunaely, i is difficul o cover all he daa of a given ime frame in an inerval value. Someimes he load can be exremely abnormal and flucuan in a wide range, as shown in Fig.. If we wan o use an inerval value o represen all he load variabiliy in a ime frame, i would lenghen he inerval and be meaningless for grid ask scheduler. In [], many large-sample-size real phenomena generae disribuions which are close o normal disribuion. Therefore we can use confidence inerval o esimae his large sample size normal disribuion of load, and for he small sample size daa, -disribuion is more suiable. This approach can eliminae some abnormal behaviour of load variabiliy, and dramaically shoren he inerval lengh in hese siuaions. Confidence inervals represen a means of providing a range of values in which he flucuaion value can be expeced o lie. This esimaed range is calculaed from a given se of sample daa. Here, we se he sample size o be CIwindow+N, where CIwindow (Confidence inerval window) is he basic sample windows size, and N is he addiional windows size, which 38

5 includes he n-sep-ahead predicion values generaed by AR(p) model. We se he confidence level o be 95% in our experimens. The confidence inerval value here is described as a pair of wo poin values (X lower (), X upper ()), where X lower () denoes he lower limi of he load predicion of ime frame, while X upper () means he upper limi of he load predicion of ime frame. In pracice, suddenly abnormal load flucuaion in a compuaional grid does no happen frequenly and ofen lass ransiorily, which makes his phenomenon hard o be prediced. In his paper, we assume ha load is sable in mos of he ime, and herefore he load confidence inerval predicion values should also display as wo smoohness curves which reflec he sabiliy of he load. So afer we compue he n-sep-ahead load confidence inerval, we use Savizky-Golay filer inroduced in formula 4. o smooh he inerval daa, he lower limi daa and he upper limi daa, o do some correcions on he original load confidence inerval predicion values. TABLE II LOAD PREDICTION PROCESS OF HYBRID MODEL. Using Kalman filer o minimize he measuremen errors of he hisory daa X measure (), generae he more accuracy daa X filer ().. Using Savizky-Golay filer o smooh he daa X filer () wih he former daa X filer (-), X filer (-),., generae he daa X smooh (). ϕ ϕ,... ϕ, 3. Compue he AR(p) coefficiens p based on he X smooh (), X smooh (-) 4. For he given n-sep-ahead parameer N, using he X smooh (), X smooh (-) and AR(p) coefficiens ϕ, ϕ,... ϕ p o compue he predic fuure value of X predic (+), X predic (+) X predic (+n). 5. Compue he confidence inerval (X lower (+n), X upper (+n)) wih sample size CIwindow+N, a confidence level 95%, using he daa X filer (-CIwindow+),X filer (-), X filer (-), X predic (), X predic (+) X predic (+n). 6.Using Savizky-Golay filer o smooh he predicion inerval value (X lower (+n), X upper (+n)) wih he former daa (X lower (+n-), X upper (+n-)), (X lower (+n-), X upper (+n-)),, generae he smoohed predicion inerval value(x smoohed_lower (+n), X smoohed_upper (+n)) 3) The whole predicion process The whole process of our hybrid model on n-sep-ahead load predicion is described in Table II. V. EVALUATION We evaluaed our hybrid model in an real grid environmen, BioGrid. This compuaional grid provides bioinformaics supercompuing services for bioinformaics researchers hrough Web inerface ransparenly. I is par of he China Educaion and Research Grid Projec ChinaGrid, an imporan projec funded by Chinese Minisry of Educaion. ChinaGrid aims o inegrae heerogeneous mass resources disribued in he China Educaion and Research Nework (CERNET) ino a public plaform for research and educaion in China. The BioGrid has 7 nodes disribued in 6 disan ciies of China (Beijing, Shanghai, Wuhan, Ji nan, Lanzhou, Xi an), and each of hem is a cluser wih processor number from 64 o 56. Wihou he loss of generaliy, we use a se of 7 series of CPU load daa of he BioGrid from he ChinaGrid Super Vision (CGSV [9]) o evaluae our hybrid model, he oal sample size is These CPU load daa is gahered from he fron-end machines of 7 nodes in he BioGrid every 60 seconds, and he CPU load of each node(cluser) is denoed as 0% o 00%, using poin value o represen he average CPU load of every ime frame. All he following evaluaion resuls are generaed from his se of CPU load daa. A. The opimal parameers of he AR(p) model There is no doub ha more accurae poin value predicion in our work will lead o a nicer confidence inerval value forecasing. In he conclusion of [6], AR(6) model or beer are appropriae for he hos load predicion. We measured he accuracy of he AR n-sep-ahead predicion wih differen n, where he resul of his measuremen reveals he capabiliy of AR(p) model in he furher n-sep-ahead predicion. Define he mean square predicion error as (5.) MSEpo in_ value_ predicion( ) = ( errorpo in_ value_ predicion( i) ) (5.) + i= 0 TABLE III MEAN SQUARE ERROR OF AR(P) N-STEP-AHEAD POINT VALUE PREDICTION, CPU LOAD DATA FROM THE BIOGRID, TOTAL 00% P sep ahead 0 sep ahead 30 sep ahead 50 sep ahead From Table III we noice ha AR(p) model performs well a p 6 and sep ahead beween o 0, where he mean square error could be smaller han The resuls sugges ha AR(p) model can no predic much furher fuure CPU load of he node in a compuaional grid. Considering he predicion accuracy and compuaional expense, we use p=3 in our evaluaions below. B. The accuracy of he confidence inerval value predicion using hybrid model Unlike he poin value predicion using AR(p) model which esimaes a possible poin value of a fuure ime poin, confidence inerval value predicion gives a view of possible values in a fuure ime frame. If he rue CPU load measured by he monioring sysem in a fuure ime frame falls in he prediced confidence inerval value, we define he predicion accuracy of his siuaion as 00%, in oher words, he 39

6 predicion error is 0. The mean square error of he confidence inerval value predicion is defined as (5.), and Fig. 3 shows he evaluaion resuls. MSEin erval _ value _ predicion( ) = ( errorin erval _ value _ predicion( i) ) (5.) + i= 0 The errorin erval _ value _ predicion ( i) here is slighly differen from he error in he poin value predicion. We compue he errorin erval _ value _ predicion( i) using he definiion of (5.3). if X lower (i) <= X rue (i) <= X upper (i) errorin erval _ value _ predicion ( i) = 0 ; Else errorin erval_ value_ predicion( i) (5.3) = Min{ abs( X ( i) X ( i)), abs( X ( i) X ( i))} rue lower rue upper confidence inerval lengh of he predicion, which is ofen ignored in previous inerval predicion models. If he confidence inerval is oo wide, i may be meaningless for he ask scheduler. For example, he range of CPU load is described as 0% o 00%, bu he predicion load confidence inerval lengh of a fuure ime is as wide as 60%. This predicion is so fain ha he ask scheduler can hardly arrange he ask dispaching based on i. On he conrary, if his inerval lengh of predicion can be limied as small as 5% or 0%, i is much easier for he scheduler o recognize he fuure saus of he sysem CPU load. (a) (a) (b) (b) Fig. 3 Mean square error of hybrid model predicion C. The predicion confidence inerval lengh using hybrid model As shown in Fig. 3, our hybrid model is of an ousanding forecasing capabiliy. I illusraes ha we should use as small as possible he CIwindow o achieve small predicion mean square error. Bu here is anoher problem abou he Fig. 4 Mean predicing confidence inerval lengh Fig. 4 demonsraes he influence of he parameers, CIwindow and N, on he predicion inerval lengh. Noiceably, he mean confidence inerval lengh is almos below 0% when he CIwindow is bigger han 5. As a compromise beween he predicion accuracy and inerval lengh, we se he CIwindow as 0 when using our model o forecas he CPU load of he BioGrid. D. Comparison beween hybrid model and AR model We compared he predicion accuracy of our hybrid model and AR model on a se of 7 series of CPU load daa colleced from he BioGrid. The predicion mean square error of our 40

7 Fig. 5 Predicion errors compare on 7 series of CPU load daa colleced from he BioGrid hybrid model (using equaion (5.)) and AR model (using equaion (5.)) on his se of load daa are shown in Fig. 5. The experimenal resuls show ha he hybrid model ouperforms AR model on he CPU load predicion. The predicion mean square error of hybrid model is 77.07% less on average han ha of AR model. We also evaluaed he predicion mean square error of he inerval median predicion by using he hybrid model (using equaion (5.)). This variaion from our hybrid model also shows beer predicion accuracy ha he predicion mean square error is 5.3% less on average han ha of AR(3) model. In hese evaluaions, we se he parameer of he hybrid model as P=3, CIwindow =0. E. The power of Kalman filer in minimizing he measuremen errors In secion III, we analyse he causes of he measuremen errors. Measuremen error is unavoidable and may be amplified by predicion models, hus impairing he predicion accuracy. In our work, we use Kalman filer o minimize hese measuremen errors. We compare he mean square error of our hybrid model using Kalman filer wih he one wihou using Kalman filer. The resuls are shown in Fig. 6, which demonsraes ha Kalman filer works very well on minimizing he measuremen errors and he predicion accuracy is improved consequenly. BioGrid. Sample ime is from :7 o :08, oal 30 sample poin, wih sample inerval of 60 seconds, in Tsinghua Universiy node. Fig. 7(b), (c), (d) are he 0, 30 and 50 sep-ahead CPU load confidence inerval predicion by hybrid model, using he same parameer p=3 and CIwindow=0. Table IV summarizes he mean square error of hese predicions a differen n-sep-ahead and he mean inerval lengh. (a) Real CPU load (b) 0-sep-ahead predicion (c) 30-sep-ahead predicion Fig. 6 Mean square error amplificaion wihou using Kalman filer, where Z-label = MSE no _ kalman _ filer MSEu sin g _ kalman _ filer F. A sample of CPU load predicion using hybrid model Fig. 7 is a sample of he CPU load predicion using our hybrid model. Fig. 7(a) is he CPU load measuremen daa of he (d) 50-sep-ahead predicion Fig. 7 CPU load confidence inerval predicion using hybrid model 4

8 TABLE IV CONFIDENCE INTERVAL PREDICTION ERRORS AND INTERVAL LENGTH STATISTICS OF FIGURE 7 Evaluaion merics 0-sep -ahead 30-sep -ahead 50-sep -ahead Confidence inerval predicion MSE Mean inerval lengh of predicion (confidence 7.34% ±0.% 8.05% ±0.9% 6.7% ±0.8% level 95%) Fig. 7 and Table IV show ha our hybrid model works excellenly in CPU load predicion, which is always he boleneck for a compuaional grid. Using he parameer p=3 and CIwindow=0, he inerval lengh is limied o below 0% and he mean square error is smaller han 0.05 even for 50-sep-ahead (50 minues ahead) CPU load. VI. CONCLUSION AND FUTURE WORK To predic he load of a compuaional grid, we have developed a hybrid model which inegraes AR model wih he confidence inerval esimae. Whereas he poin value predicion is always he ideal esimae of he load in he fuure ime poin, he confidence inerval predicion adoped in his hybrid model can reflec he load variabiliy in a fuure ime frame and convey more informaion o he ask scheduler of a compuaional grid. In order o enhance he predicion accuracy, we use Kalman filer o minimize he load measuremen errors, and Savizky-Golay filer o smooh he hisory daa. The evaluaion resuls demonsrae ha hese noise eliminaing ools perform very well and lead o a significan improvemen on predicion accuracy. Considering he rade-off beween he predicion accuracy and he confidence inerval lengh, we use parameers p=3 and CIwindow = 0 o predic he CPU load of he BioGrid. The predicion advance ime can be even 50-sep-ahead long, wih significan less predicion mean square error han he convenional AR model and has accepable inerval lengh for he schedule algorihm. The opimal parameer values may be slighly differen according o he differen compuaional applicaions and programs running on a compuaional grid, and we will evaluae he predicion performance variabiliy of our hybrid model under differen applicaions and parameers in he fuure. Some machine learning mechanism and parameer auo-adapaion funcion could also be added in our model o fi in differen condiions. ACKNOWLEDGEMENT This Work is suppored by Naural Science Foundaion of China (904006, 9040, , 90606, ), Naional Key Basic Research Projec of China (004CB38000, 003CB37007), Naional High Technology Developmen Program of China (006AA0A08, 006AA0A, 006AA0A0), and IST programme of he European Commission (DG Informaion Sociey and Media, projec n 03444). The auhors acknowledge Ms. Jing Zhu of UCSD and Dr. Ying Zhao of Tsinghua Universiy for heir revision o his paper. REFERENCES [] I. Foser and C. 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Available: hp://chinagrid.hus.edu.cn or hp:// 4