On Advantages of Scheduling using Genetic Fuzzy Systems

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1 On Advantages of Schedulng usng Genetc Fuzzy Systems Carsten Franke, Joachm Leppng, and Uwe Schwegelshohn Computer Engneerng Insttute, Dortmund Unversty, Dortmund, Germany (emal: {carsten.franke, joachm.leppng, Abstract. In ths paper, we present a methodology for automatcally generatng onlne schedulng strateges for a complex schedulng objectve wth the help of real lfe workload data. The schedulng problem ncludes ndependent parallel jobs and multple dentcal machnes. The objectve s defned by the machne provder and consders dfferent prortes of user groups. In order to allow a wde range of objectve functons, we use a rule based schedulng strategy. There, a rule system classfes all possble schedulng states and assgns an approprate schedulng strategy based on the actual state. The rule bases are developed wth the help of a Genetc Fuzzy System that uses workload data obtaned from real system nstallatons. We evaluate our new schedulng strateges agan on real workload data n comparson to a probablty based schedulng strategy and the EASY standard schedulng algorthm. To ths end, we select an exemplary objectve functon that prortzes some user groups over others. 1 Introducton In ths paper, we address the development of a methodology to automatcally generate schedulng strateges for Massvely Parallel Processng (MPP) systems that consder the provders preferences. The schedulng problem conssts of n ndependent non-clarvoyant jobs that are submtted by dfferent users over tme. The schedulng strategy s responsble to assgn the avalable processors of the MPP system to those jobs. However, the machne provders n real scenaros have dfferent relatonshps to the varous users or user groups. Those dfferent relatonshps lead to dfferent prortzatons of the users and ther correspondng jobs. Consequently, the schedulng strategy needs to ncorporate those prortes durng the schedulng process. Many nstallatons use parttons [14] or quotas [31] to mplement ths knd of prortzatons of dfferent user groups. However, those attempts result n a low system utlzaton n most of the cases [14]. Hence, we present the development of a rule based schedulng system that s able to generate schedules wth a hgher qualty n terms of the provder preferences whle not decreasng system utlzaton. To our knowledge, there s no smlar work that s able to ncorporate the user group prortzatons n a smlar way. Born Carsten Ernemann.

2 The development of schedulng strateges for MPP systems s based on workload traces orgnatng from real nstallatons, see for example Hene et al. [16]. Such workload data nclude all hdden job dependences, patterns and feedback mechansms. For MPP systems several workloads are avalable, see the standard workload archve mantaned by Fetelson [13], that are for nstance descrbed by Chapn [5]. Although those data are rather old they suffce for our purpose. So far, workload models are rarely used to develop schedulng algorthms as they are not able to descrbe workload traces wth an acceptable accuracy, see Song et al. [28] and the gven references there. The onlne job schedulng on MPPs s usually non-clarvoyant as the processng tme p j of job j s not avalable at ts release date r j. However, users are often requred to provde estmates p j of the processng tme that are manly used to determne faulty jobs whose processng tme exceeds the estmate. Further, parallel jobs on MPPs are typcally not moldable or malleable, that s, they need concurrent and exclusve access to m j m machnes durng the whole executon phase. The value m j s provded at the release date r j by the user. Fnally, the completon tme of job j n a schedule S s denoted by C j (S). As preempton s not allowed n many MPPs, each job starts ts executon at tme (C j (S) p j ). Unfortunately, the avalable workload data do not provde any user group nformaton nor defne any complex schedulng objectve. To address the user group problem, we are usng the work of Song et al. [29], who have shown that users can be reasonably well parttoned nto 5 groups for all avalable MPP workload traces. Those groups are dfferentated wth respect to job characterstcs and frequency of job submssons. Wthn ths work, we wll also use 5 dfferent user groups. However, we wll use the user s resource consumpton as the dfferentaton crteron. The bnary functon (j) s used to state whether job j belongs to user group ( (j) = 1) or not ( (j) = 0). We present a methodology to automatcally generate a rule based schedulng system that s able to produce good qualty schedules wth respect to a gven complex provder objectve. Note that our methodology s not restrcted to a specfc user group selecton. The ndvdual preferences of the machne provders are expressed usng a complex objectve functon that s generated by combnng well known smple basc objectves. Even f dfferent provders use the same objectve functons for the varous groups, the transformaton of a generc mult-objectve scenaro nto a specfc schedulng problem wth a sngle objectve depends on the actual prortes assgned to the user groups and s lkely to be ndvdual. Hence, we focus on the development of a sutable methodology and do not generate a sngle schedulng strategy. Wthout loss of generalty, we exemplarly select a complex objectve functon to demonstrate the feasblty of our approach. Here, we present a rule based schedulng that s able to adapt to varous scenaros. So far, the use of rule based systems n schedulng envronments s rare. Nevertheless, frst attempts [10, 4] have shown the feasblty of such an approach. However, those schedulng systems are all based on sngle smple objectve evaluaton functons that are not optmzed. 2

3 The proposed schedulng process s dvded nto two steps. In the frst step, the queue of watng jobs s reordered accordng to a sortng crteron. Then an algorthm uses ths order to schedule the jobs onto the avalable machnes n the second step. Based on the present schedulng state, the rules determne the sortng crteron and the schedulng algorthm. In order to guarantee general applcablty, the system classfes all possble schedulng states. Ths classfcaton consders the schedulng decsons n the past, the actual schedule, and the current watng queue. Note that we have chosen some classfcaton features exemplarly. Other possble features can be used as well for ths task. Our feature selecton only serves the purpose to llustrate our methodology. As already stated n many other publcatons, see for example Ernemann et al. [6, 7], a local schedulng decson nfluences the allocaton of future jobs. Hence, the effect of a sngle decson cannot be determned ndvdually. Therefore, the whole rule base s only evaluated after the complete schedulng of all jobs belongng to a workload trace. Ths has a sgnfcant nfluence on the learnng method to generate ths rule base as ths type of evaluaton prevents the applcaton of a supervsed learnng algorthm, see Hoffmann [17]. Instead, the reward of a decson s delayed and determned by a crtc. Furthermore, the generaton of an approprate stuaton classfcaton s not known n advance and must be generated mplctly whle constructng the rule based schedulng system. The varous desgn concepts for Fuzzy logc controllers often use Evolutonary Algorthms to adjust the membershp functon as well as to defne the output behavor of ndvdual rules, see, for example, Hoffmann [17]. Especally Genetc Fuzzy Systems have been proven to deal wth such classfcaton and automatc rule base generaton problems n a sutable way. All those Genetc Fuzzy Systems ether encode sngle rules (Mchgan approach, Bonarn [3]) or complete rule bases (Pttsburgh approach, Smth [27]). Wthn ths work, the determnaton of a Genetc Fuzzy System s realzed usng the Pttsburgh approach. In ths case, each ndvdual represents a whole rule base. Durng the evoluton, the ndvdual rules are adjusted n order to better ft to the gven stuatons. Furthermore, we wll present a Coevolutonary approach that uses two rule bases, one for the determnaton of the sortng crteron and one for the schedulng algorthm that s appled. Both rule bases evolve ndependently wth the only excepton that durng the qualty assgnment one ndvdual from each rule base must be selected. We use an Evoluton Strateges for the optmzaton of the rule based schedulng system. Ths s n contrast to the majorty of Genetc Fuzzy Systems, see Hoffmann [17]. As our membershp functons nclude real valued parameters, Evoluton Strateges are superor to Genetc Algorthms n ths case, see Bäck and Schwefel [1]. To fnally show the results of our approach, we use a lnear prorty functon whch favors user group 1 over user group 2 over all other user groups. The choce of another prorty functon may lead to dfferent results but does not affect the feasblty of our methodology. Due to the lack of a schedulng strategy support- 3

4 ng prorty functons, no prorty functons are avalable n practce. Therefore, we had to defne one. For the evaluaton of the derved schedulng strategy we present the dstance of ths schedule from the Pareto front of all feasble schedules for ths workload, as generated by Ernemann et al. [8]. Although the generaton of an approxmate Pareto front s not subject of ths paper, two restrctons must be noted: 1. For real workloads, we are only able to generate approxmate Pareto fronts. Therefore, schedules of ths front are not guaranteed to be lower bounds. 2. The schedules are generated off-lne. On-lne methods may not be able to acheve as good results due to the on-lne constrants. On purpose, we selected a crteron where user groups wth a hgh computng demand are preferred over user groups wth a low demand. Then classcal schedulng algorthms wll typcally generate acceptable results. Ths s not true for a prortzaton of a user group wth a low resource demand. Moreover, we also show the results of the best conventonal strategy that does not support prortes. The remander of ths paper s organzed as follows. In Secton 2, we ntroduce the underlyng schedulng system, Evolutonary Algorthms, and Genetc Fuzzy Systems n more detal. The schedulng objectves and features are presented n Secton 3. Then the model of our approach s descrbed n Secton 4. Ths s followed by a detaled analyss of the system behavor and an evaluaton of the results. The paper ends wth a bref concluson. 2 Background Ths secton ntroduces the man schedulng algorthms and ther applcaton wthn our rule based schedulng system. Furthermore, the concept of Evoluton Strateges s presented. Those strateges are used to optmze the rule based schedulng system. 2.1 Schedulng Concepts As already mentoned, schedulng strateges of hgh performance parallel computers need to pay more attenton to certan users or user groups n order to acheve a hgher degree of satsfacton for them. Prorty or membershp nformaton are not avalable n the workloads. Hence, we use the resource consumpton as a groupng crteron such that user group 1 represents all users wth a hgher resource consumpton whereas all users n group 5 have a very low resource demand. Detals of the user group defntons are provded by Ernemann et al. [8]. As already ntroduced, a state of a schedulng system manly conssts of the current schedule, that descrbes the actual allocaton of processor nodes to certan jobs, the schedulng results acheved so far, and the queue of watng jobs. Ths watng queue s typcally ordered. 4

5 In most cases, a statc orderng lke sortng by submsson tme or sortng by estmated runtme s appled. In some other cases, the watng queue s dynamcally reordered dependng on the system state by usng a more complex sortng crteron that may for nstance consder lmts of the watng tme. The varous schedulng algorthms manly dffer n the way they select the next job from the sorted watng queue to nsert t nto the exstng schedule, that s, they obey dfferent restrctons when choosng the next job. Ths results n dfferent algorthmc complextes and correspondngly dfferent executon tmes for the schedulng algorthms. In the followng, we present four selected schedulng algorthms n ncreasng order of algorthmc complexty. Note that the frst three algorthms use a statcally sorted watng queue whle the last algorthm dynamcally reorders ths queue. Frst Come Frst Serve (FCFS) starts the frst job of the watng queue whenever enough dle resources are avalable. Thus, ths algorthm has a constant complexty as the scheduler always only tests whether the frst job can be started mmedately f a job n the schedule has completed ts executon or a new job has rsen to the top of the watng queue. Lst Schedulng as ntroduced by Graham [15] s not appled n ths work. However, t serves as the basc template for the two backfllng varants. By applyng Lst Schedulng, the scheduler tres to fnd the frst job wthn the queue of watng jobs, that can be started on the currently dle resources. Agan, the algorthm uses the sorted queue. The complexty s hgher than n the case of FCFS as n the worst case, the whole queue s tested each tme the schedulng procedure s ntated. EASY Backfllng (EASY) s smlar to the orgnal Lst Schedulng. However, f the frst job wthn the watng queue cannot be started mmedately the algorthm estmates the completon tme of ths job. To ths end, a runtme estmaton provded by the user s needed. Then EASY tres to fnd an allocaton for the followng jobs of the watng queue on the currently dle resources whle ensurng that the frst job s not further delayed. Ths algorthm requres more tme than Lst Schedulng, as the scheduler needs to estmate the processng of the frst job n case that t cannot be started drectly. Conservatve Backfllng (CONS) extends the concept of EASY. Here, the scheduler tres to fnd the next job wthn the watng queue, that can be started mmedately whle ensurng that no prevous job wthn the queue s further delayed. Ths results n a much hgher complexty of the schedulng algorthm as n the worst case, the completon tme of all jobs wthn the watng queue except of the last job must be estmated each tme the schedulng process s ntated. Greedy Schedulng (Greedy) uses a dynamcally sorted watng queue contrary to the already ntroduced schedulng algorthms. To ths end, the algorthm defnes a complex sortng crteron. Each tme, the Greedy schedulng process n started, the queue s sorted accordng to ths crteron. Then, a smple 5

6 FCFS s appled. The complexty of ths algorthm s potentally hgh as the executon of the sortng functon for each job wthn the watng queue may be computatonally expensve. Furthermore, the necessary sortng of all jobs must be taken nto account. Greedy has the advantage to specfy user or user group dependent preferences wthn the complex sortng crteron. In our case, the complex sortng functon wthn the Greedy algorthm tres to schedule jobs of the user groups 1 and 2 earler unless jobs from other user groups are already watng for a very long tme. Ths sortng crteron s modeled accordng to our schedulng objectve. For more detals on the used sortng crteron, see Ernemann et al. [8]. 2.2 Evoluton Strateges To ntegrate those schedulng algorthms nto an approprate rule base system, we use Evoluton Strateges, see Beyer and Schwefel [2], whch are a subclass of Evolutonary Algorthms. Those algorthms are stochastc search methods that mmc the behavor of natural bologcal evoluton. They operate on a populaton of µ ndvduals and apply genetc operators lke selecton, mutaton and recombnaton to breed λ offsprng ndvduals from those µ parent ndvduals. Wthn ths paper, we do not provde a deeper nsght nto Evoluton Strateges. Furthermore, for all detals about specfc genetc operators, we smply refer to references n the remander of ths paper. 2.3 Fuzzy Systems Wthn ths work, we am to generate rule based schedulng systems. To ths end, several approaches can be used. On the one hand, a statc approach of defnng strct boundares for certan features and assgnng a correspondng combnaton of sortng crtera and schedulng algorthm s possble. On the other hand, one may apply the more flexble approach of generatng a Genetc Fuzzy System. In our case, nether precse knowledge about the assgnment of certan schedulng strateges to certan stuatons nor tranng data are avalable. Furthermore, ndvdual schedulng decsons cannot be evaluated drectly, but only after all jobs have been assgned to resources, see Secton 1. Hence, the award for the assgnment of schedulng strateges to stuatons s gven by a crtc only at the end of schedulng a whole workload trace. Furthermore, the generaton of an approprate stuaton classfcaton s not known n advance and has to be generated mplctly durng the generaton of the rule based schedulng system. 3 Schedulng Objectves and Features Wthn ths secton, we wll ntroduce several smple schedulng objectves, whch have been used to construct more complex evaluaton functons for the whole schedulng procedure. However, our methodology s not lmted to the presented objectve and can be extended to any other crtera. 6

7 Furthermore, we apply several features to classfy possble schedulng stuatons wthn our rule based schedulng system. The concept of ths work can be extended to other features as well. Note that objectves evaluate the whole schedulng process at the end of a smulaton whle features only descrbe the current state of the system. As mentoned n Secton 1, the complex objectve functon of a machne provder n our case s based on ndvdual propertes of users or user groups. Therefore, both the objectve and the feature set refer to those propertes and to the overall performance of the whole system. Frst, we ntroduce some defntons and notatons. (p j m j ) as the Resource Consumpton of a sngle job j, τ the set of all n jobs wthn our schedulng system, ξ(t) the set of already fnshed jobs at tme t, π(t) the set of runnng jobs at tme t, and ν(t) the set of watng jobs at tme t. 3.1 Schedulng Objectves Durng the development of schedulng systems, an evaluaton functon s needed n order to descrbe the acheved qualty. We generate our evaluaton functon by combnng smple, commonly used schedulng objectves. Wthn ths work, we exemplarly use 7 of those smple objectves. Overall Utlzaton (U): U = ( p j m j j τ ) (1) m max j(s)} mn j(s) p j } j τ j τ Average Weghted Response Tme (AWRT) over all jobs of all users: p j m j (C j (S) r j ) j τ AWRT = (2) p j m j j τ AWRT objectve for user groups 1 to 5: p j m j (C j (S) r j ) (j) j τ AWRT = p j m j (j) j τ, {1, 2,...5} (3) As we have the AWRT for the 5 user groups, the AWRT for all users, and the utlzaton U the complex objectve functon n our system can be defned by usng those 7 smple objectves. 7

8 3.2 Feature Defntons Next, we present 7 features that are used for classfcaton of system states wthn our rule base schedulng system. In order to defne our frst feature, the Average Weghted Slowdown, we need to ntroduce the Slowdown (SD j ) for a sngle job j wthn schedule S: SD j = C j(s) r j p j (4) SD j wll reach ts mnmum value of 1 f job j does not wat before t starts executon. Then the release date s dentcal wth the job s start tme. Normally, the range of ths feature can be lmted to the nterval of [1,100] as values greater than 10 occur very rarely n practce. The feature Average Weghted Slowdown (SD) for all already processed jobs j ξ(t) uses the same weghtng as defned for the AWRT. SD = j ξ(t) p j m j (C j (S) r j ) j ξ(t) p 2 j m j Ths measure ndcates the average delay of jobs between ther release and start tme for the past. Further, ths feature represents the schedulng decsons n the past as only already fnshed jobs are used to calculate ths feature. Here, we have not lmted the wndow for SD. In practcal cases, a lmtaton to, for nstance, the last month may be approprate. The Momentary Utlzaton (U m ) of the whole parallel computer at tme t: U m = m j j π(t) The Proportonal Resource Consumpton of the Watng Queue for User Group (PRCWQ ): PRCWQ = j ν(t) m p j m j (j) j ν(t) (5) (6) p j m j (7) Note that the real processng tme p j s unknown for the jobs n the watng queue. Therefore, we use the estmated processng tme p j nstead. PRCWQ represents the relatve part of the estmated resources consumpton of user group to all jobs wthn the watng queue. Remember, we are usng 5 user groups wthn our system. Hence, those fve feature values represent the expected future of the system. Usng these features, the schedulng system s enabled to react on a changed demand of the varous user groups. 8

9 4 Rule Based Schedulng Systems As stated n Secton 1, local schedulng decsons nfluence the allocaton of future jobs. Hence, the effect of a sngle decson cannot be determned ndvdually. Therefore, the whole rule base s only evaluated after the complete schedulng of all jobs belongng to a workload trace. Ths has a sgnfcant nfluence on the learnng method to generate ths rule base as the evaluaton prevents the applcaton of a supervsed learnng algorthm. Instead, the reward of a decson s delayed and determned by a crtc. Furthermore, the generaton of an approprate schedulng stuaton classfcaton s not known n advance and has to be generated mplctly durng the generaton of the rule base schedulng system. For a rule based schedulng approach, every possble schedulng state must be assgned to a correspondng stuaton class that s descrbed usng the already ntroduced features. A complete rule base RB conssts of a set of rules R. Each rule R contans a condtonal and a consequence part. The condtonal part descrbes the condtons for the actvaton of the rule usng the defned features. The consequence part represents the correspondng schedulng strategy recommendaton. In order to specfy all schedulng states n an approprate fashon each rule defnes certan parttons of the feature space wthn the condtonal part. The rule base system must contan at least one actvated rule for each possble system state. As already mentoned, the schedulng strategy specfes 1. a sortng crteron for the watng queue ν(t) and 2. a schedulng algorthm that uses the order of ν(t) to schedule one or more jobs. We use the term strategy to descrbe the whole schedulng process that conssts of both steps. An algorthm only descrbes the procedure of the second step that uses the already sorted watng queue. Frst, the chosen sortng crteron s used to determne the sequence of jobs wthn the watng queue. Second, the selected schedulng algorthm s used to fnd a processor allocaton for at least one job of the sorted watng queue. We have chosen four dfferent sortng crtera. Those sortng crtera are only examples that are used to demonstrate our rule based schedulng approach. Other sortng crtera are possble and could easly be ncorporate nto the system. Our four sortng crtera are: Increasng Number of Requested Processors: Preference of jobs wth lttle parallelsm and therefore hgher utlzaton. Ths sortng provdes the potental gan of beng able to nsert many jobs nto the current schedule as jobs wth a smaller amount of requested processors are often easer to schedule. Increasng Estmated Run Tme: Preference of short jobs and therefore hgher job throughput. Decreasng Watng Tme: Preference of long watng jobs. Ths sortng crteron provdes a hgher farness as the jobs are processed accordng to ther submsson. Jobs wth a hgher watng tme are selected frst. 9

10 Decreasng User Group Prorty: Preference of jobs from users wth a hgher resource demand. The sortng by user groups provdes a hgher rankng for all jobs of users wth a hgher overall resource demand accordng to ther user group assgnment. Ths crteron reflects our objectve functon. The selected schedulng algorthm s one of the four methods presented n Secton 2.1. Note that Greedy s already a complete schedulng strategy whle the other schedulng algorthms of Secton 2 must be supplement wth a sortng crteron of ν(t). Agan, the set of schedulng algorthm can be extended for other rule base systems. The general concept of the rule based schedulng approach s depcted n Fgure 1. As 4 dfferent sortng crtera wth 3 possble schedulng algorthms and the combned Greedy strategy are avalable, we have to chose one of 13 strateges for each possble system state. However, t s not practcable to test all possble assgnments n all possble states. For example, lets assume a very coarse dvson of each feature nto only 2 parttons. Then 13 possble strateges and 7 features result n smulatons f all combnatons n all possble stuaton states are tested. Addtonal problems occur durng the generaton of a rule based schedulng system as the number and reasonable parttons of features, that are requred to descrbe the stuaton classes n an approprate way, are generally unknown n advance. Hence, we ntroduce three possble approaches to derve a rule based schedulng system usng only a lmted number of smulatons. 4.1 Probablty Drven Rule Base Development A rgd rule base system uses N F features wth a fxed number of ntervals for each feature ω. That s, each feature ω has (N part,ω 1) statc bounds, that dvde the possble value range of ω nto N part,ω parttons. The statc bounds are specfed before the assgnment of sortng crtera and schedulng algorthms to the varous stuaton classes are extracted. The concept of such fxed parttons s shown n Fgure 2. Generally, a larger number of parttons N part,ω of a feature ω potentally leads to a more accurate rule set whle more stuaton classes must be optmzed. Overall, ths results n N r stuaton classes that must be provded to cover all possble system states wth N r = N F ω=1 N part,ω. The descrbed rgd rule based system actvates only a sngle rule n any system state. Hence, the output recommendaton of ths sngle actvated rule s the output of the whole schedulng system. In ths work, we assume one dvson of the ntervals of SD and PRCWQ 1 to PRCWQ 5 respectvely. Ths leads to two parttons n each case. Further, we use two dvsons for the U m feature, resultng n three parttons. Overall, ths 10

11 Controller Schedulng Algorthm Rule Selecton Sortng Crteron Rule Base R 1 R 2 R 3 R 4 R 5 R 6 R 7 R 8 R 9 R Nr Current Schedule Classfcaton Past Schedulng Decsons Current Watng-Queue Rule Applcaton Tme Order of Executon Sortng Crteron Machnes m Schedulng Algorthm Schedulng System Fg.1. General Concept of the Rule Based Schedulng Approach. Feature B R 1 R 2 Feature A a R 3 R 4 b Fg.2. Example parttonng of the feature space and the resultng set of rules R 1... R 4. produces (N r = = 192) dfferent stuaton classes that are needed to buld a complete rule base. 11

12 Furthermore, we have evaluated several dfferent dvson values for the stuaton class features. The parttons whch acheved the best results are used for the rgd rule base system development,see Table 1. Feature Intervals SD [1-2], ]2-100] U m[%] [0-75], ]75-85], ]85-100] PRCWQ 1[%] [0-20], ]20-100] PRCWQ 2[%] [0-20], ]20-100] PRCWQ 3[%] [0-25], ]25-100] PRCWQ 4[%] [0-25], ]25-100] PRCWQ 5[%] [0-25], ]25-100] Table 1. Feature Parttons for the Rgd Rule Based Schedulng Systems. Such a rgd rule based schedulng system has the advantage of a smple mplementaton and easy nterpretaton. Future schedulng development may beneft from knowledge ganed through ths knd of nterpretaton. The selected schedulng algorthms and sortng crtera for a certan schedulng stuaton can drectly be extracted from the correspondng rules wthout further computaton. Rule bases are generated by assgnng potental schedulng strateges to rules n a random fashon such that each schedulng strategy s assgned to each rule the same number of tmes. Hence, not all rule bases are generated n a completely random way. Remember that the condtonal part s rgd and does not vary. Thus, a sngle rule descrbes a sngle schedulng stuaton class completely. Then we use those rule bases to produce schedules for the gven workload data and evaluate those schedules wth the help of the complex schedulng objectve. Thus, each schedule results n a scalar objectve value. The assgnment of a specal schedulng strategy to a rule s evaluated by addng the scalar objectve values of all schedules that were generated usng ths assgnment. Fnally, we buld the resultng rule base by assgnng the schedulng strateges wth the smallest sum of the objectve values to the ndvdual rule as we assume a mnmzaton of the objectve functon. Ths approach s able to reduce the number of requred smulatons sgnfcantly as we only approxmate the optmal assgnments. In general, the performance can be ncreased by generatng more rule bases. However, the trade-off between a better performance and more requred smulatons should be kept n mnd. A parameter p descrbes how often a schedulng strategy s assgned to a sngle rule. Ths parameter p nfluences the number of requred smulatons that s gven by the product of the number of possble schedulng strateges (N Ω ) and the parameter p. In our smulatons p = 50 turned out to be a good compromse between the requred number of smulatons and the schedulng qualty. Ths results n our case n (13 50 = 650) smulatons whch s sgnfcantly less than the requred number of smulaton for all possble assgnments. 12

13 Unfortunately, the fxed dvson of the whole feature space has a crtcal nfluence on the performance of the schedulng system. At the moment, no mechansm s avalable that automatcally adjusts the defned parttons. Usng ths approach, we avod the excessve amount of smulatons that must be performed n order to generate the rule base. Further, our approach pays attenton to the cooperaton aspect of the rules wthn the fnal rule base as the evaluaton of the assgnment of a specal schedulng strategy to the consequence part of a rule s based on several smulatons wth varyng strategy assgnments for all other rules. 4.2 Schedulng Strateges Based on Genetc Fuzzy Systems The prevously presented schedulng system has several drawbacks regardng the generaton of an approprate rule based schedulng system. Manly, the statc number of feature parttons and the statc pre-defned bounds for these parttons are not flexble enough and may lead to bad schedulng results. Furthermore, the whole feature space needs to be dvded and approprate schedulng strateges assgned to each ndvdual partton. Hence, the number of rules cannot be vared. Consequently, we need a method that automatcally adjusts the partton of the feature space and assgns approprate schedulng strateges to the resultng regons n parallel. Genetc Fuzzy Systems, see Hoffmann [17], provde the capabltes to solve those problems. As already mentoned wthn the ntroducton, our Genetc Fuzzy System uses the Pttsburgh approach to encode a whole rule base n a sngle ndvdual. Further, we parameterze the resultng system wth Evoluton Strateges. Before the dfferent rule base encodng schemes are explaned n detal, we ntroduce the encodng of ndvdual rules and detal the computaton of the fnal Fuzzy controller output. Codng of Fuzzy Rules Our Genetc Fuzzy Systems are based on the tradtonal Takag-Sugeno-Kang (TSK) model [30] for Fuzzy systems. The used codng schemes and learnng technques are adapted and slghtly modfed from the work of Juang et al. [21] and Jn et al. [20]. For a sngle rule R, every feature ω of all N F features s modeled from a Gaussan Membershp Functon, (µ (ω), σ (ω) )-GMF g (ω) (x) = σ (ω) 1 (x µ (ω) exp 2π 2 2σ (ω) ) 2. In Fgure 3 a sample (5,0.75)-GMF s depcted. A feature s then represented by a par of real values µ (ω) and σ (ω). The µ (ω) value s the center of the feature GMF that s covered by the rule R. Therefore, ths value defnes a doman n the feature space where the nfluence of the rule 13

14 g (x) 0.3 σ µ x Fg.3. Gaussan Membershp Functon wth µ (ω) = 5 and σ (ω) = s very hgh. Note that, when usng a so defned GMF as feature descrpton, the condton g (ω) (z)dz = 1 {1,...N r } ω {1,...N F } always holds. In other words, for ncreasng σ (ω) values, the peak value of the GMF decreases because the ntegral remans constant. Usng ths property of a GMF, we are able to reduce the nfluence of a rule for a certan feature completely by settng σ (ω) to a very hgh value. Theoretcally for σ (ω), a rule has no nfluence for ths feature anymore. Wth ths approach, t s also possble to establsh a knd of default value that s used f no other peaks are defned n a feature doman. Based on ths feature descrpton, a sngle rule can be descrbed by { } R = g (1) (x), g (2) (x),... g (NF) (x), Ω (R ). The consequence part Ω of every rule R, {1,...N r }, ncludes a weghted recommendaton for all N Ω possble outputs. Therefore, the consequence part of rule R s descrbed by a vector Ω(R ) = ( w 1 w 2...w NΩ ). We restrct the possble weght values to elements of the set { 5, 1, 0, 1, 5}. The value 5 represents a partcularly unfavorable connecton whle 5 s partcularly favorable one. The other possble weghts can be nterpreted accordngly. We use a non-lnear weght scalng n order to force dstnct recommendatons. When consderng the superposton of those weghts smlar weghts may lead to almost ndstngushable recommendatons. Furthermore, we also nclude 0 as possble weght to express that a rule behaves completely neutral wth respect 14

15 to the recommendaton of a schedulng strategy for a gven stuaton. Ths may also reduce the number of overall rules. The man advantage of usng several GMFs for descrbng a sngle rule s the automatc coverage of the possble feature space. In contrast to the rgd approach, even one rule gves a schedulng strategy for all possble system states. Hence, t s the focus of ths approach to fnd a meanngful set of N r rules that generates a good rule base system RB. Thus, RB = {R 1, R 2,... R Nr } s a complete rule base consstng of N r rules. Computaton of the Controller Decson For a gven system state, we compute the superposton of the weghted output consequence parts of all rules. The system state s represented by the actual feature vector x = ( x 1 x 2... x ω... x NF ) T of N F feature values. Then we compute the degree of membershp φ (x ω ) = g (ω) (x ω ) of the ω-th feature of rule R for all N r rules and all N F features. The multplcatve superposton of all these values as AND -operaton leads to an overall degree of membershp φ (x) = N F ω=1 N F g (ω) (x ω ) = ω=1 σ (ω) 1 exp 2π { (x ω µ (ω) ) 2 2σ (ω)2 for rule R. For all N r rules together, the correspondng values φ (x) are collected n a membershp vector } φ(x) = ( φ 1 (x) φ 2 (x)... φ Nr (x) ). Next, we construct a matrx C e N F N r of the weghted consequences Ω(R ), {1,...N r } of all rules by usng the weghted consequence vectors for all ndvdual rules R. Ths yelds C e N F N r = [ Ω(R 1 ) Ω(R 2 )... Ω(R Nr ) ]. Now, we can compute the weght vector Ψ by multplyng the membershp vector φ(x) by the transposed matrx C e T : Ψ = φ(x) C e T = ( Ψ 1 Ψ 2... Ψ NΩ ). 15

16 The vector Ψ contans the superpostoned weght values for all N Ω possble schedulng strategy recommendatons, that s, Ψ contans 13 elements. Fnally, we choose the schedulng strategy wth the hghest overall value as the output of the rule base system, that s arg max 1 h N Ω {Ψ h }. As already mentoned, wthn the Pttsburgh approach, each ndvdual represents a complete rule base. We construct such a complete rule base RB wth a fxed number of rules N r. A sngle rule conssts of (2 N F ) elements per rule wthn the condtonal part. Furthermore, we nclude the vector Ω(R ) for the consequence part, whch conssts of N Ω = 13 elements. Thus, a rule based schedulng system wth constant number of rules can also be modeled usng the followng encodng. As such, R 1 R 2... R Nr {}}{{}}{ o k = { µ (1) 1 σ (1) 1,..., µ (NF) 1 σ (NF) 1, Ω(R 1 ), µ (1) 2 σ (1) 2 }{{}}{{},...,µ(nf) N r σ (NF) N r, Ω(R Nr )} GMF Ω 1... Ω NΩ s the codng scheme of the object parameter vector o k of ndvdual a k whch s a complete rule base. Hence, the number of elements u wthn the object parameter vector o k of the ndvdual a k can be computed by u = N r (2 N F + N Ω ). We have chosen a non-sotropc mutaton, see Bäck and Schwefel [2], as ths allows the ndvdual adaptaton of the mutaton for the dfferent dmensons. Therefore, each object parameter of the ndvduals conssts of a correspondng strategy parameter that specfes ts mutaton strength. Further, we apply a standard Evoluton Strategy wth µ = 3 parent and λ = 21 offsprng ndvduals. The rato of 1/7 s suggested by Schwefel [26]. Further, we do not use any recombnaton. Wthn our Evoluton Strategy, we used 40 generatons wth a randomly ntalzed frst generaton. Our (3+21)-Evoluton Strategy leads to 3+(40 21) = 843 evaluatons for the development of a sngle rule base. We use a constant number of rules N r = 10 for each rule base. Ths results n a constant number of object and strategy parameters wthn each ndvdual. Hence, u = N r (2 N F + N Ω ) = 10 ( ) = 270 parameters must be determned. Thus, the two exogenous learnng rates for the non-sotropc mutaton are defned as: see Bäck and Schwefel [2]. τ 0 = 1 2 u = 0.043, and τ 1 = 1 2 u = 0.174, 16

17 Coevolutonary Genetc Fuzzy System Development As presented n Secton 4, the rule based schedulng system needs to determne for each schedulng state a correspondng sortng crteron and a schedulng algorthm. In the prevously ntroduced rule based schedulng systems, a whole schedulng strategy, consstng of both, a sortng crteron and a schedulng algorthm, was assgned to the dfferent schedulng states. However, ths combned assgnment s not necessary. Moreover, the assgnment of the same sortng crteron to two schedulng states wthn the features space does not always lead to the assgnment of the same schedulng algorthm. Ths motvates the usage of a Coevolutonary Algorthm as the assgnment problem can easly be decomposed nto two subproblems. Concept of Cooperatve Coevolutonary Algorthms Coevolutonary Algorthms potentally lead to better solutons compared wth standard Evolutonary Algorthms, f the problem can be decomposed nto two subproblems, see for example Jansen et. al [19]. Furthermore, Potter and De Jong [25] have proven that Coeveolutonary Algorthms acheve better results wth fewer generatons compared wth standard Evolutonary optmzaton technques. In ths work, we apply the commonly called Cooperatve Coevolutonary Algorthm (CCA), see Pareds [24]. Ths model uses two dstnct speces. Both speces are genetcally solated. Hence, the genetc operatons are only appled to ndvduals of the same speces. The two dfferent speces are evolved n two dfferent populatons n parallel by usng standard Evoluton Strateges. However, durng the ftness evaluaton, two ndvduals of each speces must cooperate. In general, ths concept allows a larger number of speces. Frst, two speces wth µ ndvduals each are randomly generated. Then, the ndvduals of both speces are evaluated by randomly combnng two ndvduals, one from each speces. Note that other selecton schemes are also possble and dscussed n the lterature, see for example Panat et al. [22]. However, those methods need more evaluatons and addtonal smulatons n our case. In order to avod ths effort, we use our smple heurstc. After evaluaton, the genetc operators produce λ offsprngs for each speces separately. Then, the resultng offsprng ndvduals are agan evaluated by a randomly chosen cooperaton. Fnally, normal evolutonary selecton determnes the next parent generaton. Rule Based Schedulng Development by applyng Coevolutonary Algorthms As already mentoned, our schedulng problem can be decomposed nto two separate subproblems. Ths concept s shown n Fgure 4. Contrary to the general rule based schedulng, see Fgure 1, we use two separate rule bases wthn the same feature space. One determnes the sortng crteron dependng on the system state and the other calculates the schedulng algorthm. However, the parttonng of ths feature space dffers between the two speces. To ths end, the dfferent values are determned separately for the two speces. The resultng schedulng system s expected to react on certan system states very accurately. GMF-µ (ω) and GMF-σ (ω) 17

18 Controller Rule Selecton Schedulng Algorthm Rule Base Schedulng R 1 R 2 R 3 R Nr Sortng Crteron Rule Base Sortng R 1 R 2 R 3 R Nr Current Schedule Classfcaton Past Schedulng Decsons Current Watng-Queue Rule Applcaton Tme Order of Executon Sortng Crteron Machnes m Schedulng Algorthm Schedulng System Fg. 4. General Concept of the Rule Based Schedulng Approach wth Dedcated Rule Bases for Schedulng and Sortng. Such a coevolutonary approach yelds several potental advantages for the resultng schedulng system and for the extracton process of approprate rule bases. Frst, each of the two separate rule bases has fewer output recommendatons. In detal, for the sortng crteron as well as for the schedulng algorthm, we have only N Ω = 4 possble output recommendatons nstead of 13 as n the combned scenaro. Ths reduces the length of the ndvduals wthn the populatons and enables a better and faster adaptaton. However, note that the sortng crteron s redundant f the Greedy schedulng algorthm s selected snce Greedy ncludes ts own sortng. Second, as the feature space partton can be optmzed for both speces separately, fewer rules mght be requred for each speces. The Evoluton Strateges for both populatons are dentcal. We apply the Pttsburgh approach wth the same genetc operators and no recombnaton for both populatons. In detal, we use a constant number of rules N r = 10 and a (3+21)-Evoluton Strategy for both populatons. The optmzaton s lmted to 18

19 40 generatons. Consequently, each ndvdual wthn the populatons conssts of u = N r (2 N F + N Ω ) = 10 ( ) = 180 object parameters. Hence, we adapt the learnng rates for the non-sotropc mutaton to τ 0 = 1 1 = 0.053, and τ 1 = = 0.193, 2 u 2 u see Bäck and Schwefel [2]. 5 Evaluaton For the evaluaton, we execute varous dscrete event smulatons wth real parallel computer workload traces. To ths end, sx well known workloads are selected. They were recorded at the Cornell Theory Center (CTC) [18], the Royal Insttute of Technology (KTH) [23] n Sweden, the Los Alamos Natonal Lab (LANL) [11] and the San Dego Supercomputer Center (SDSC 00/ SDSC 95/ SDSC 96) [12, 32]. Each of these workloads provdes nformaton about the job requests for the computatonal resources. In order to make those workloads comparable they are scaled to a standard machne confguraton wth 1024 processors as descrbed by Ernemann et al. [9]. The characterstcs of the used workloads are presented n Table 2. Identfer CTC KTH LANL SDSC 00 SDSC 95 SDSC 96 Machne SP2 SP2 CM-5 SP2 SP2 SP2 Perod 06/26/96-09/23/96-04/10/94-04/28/98-12/29/94-12/27/95-05/31/97 08/29/97 09/24/96 04/30/00 12/30/95 12/31/96 Processors (m) Jobs (n) Table 2. Scaled Workload Traces from Standard Workload Archve [13] usng the Scalng Procedure by Ernemann et al. [9]. As no real lfe objectve functons are avalable from the workload traces, we exemplarly use the objectve functon (f obj = 10 AWRT 1 +4 AWRT 2 ). Clearly, ths objectve prortzes user groups 1 and 2, wth user group 1 havng a hgher prorty than user group 2. As already mentoned, we present our acheved results relatve to the Pareto front (PF) of all feasble schedules for the smulated workloads. Noteworthy, the Pareto front was generated off-lne and t cannot be taken for granted that ths front can be reached by our proposed onlne schedulng systems at all. Therefore, we refer to ths front as a reference for the best achevable soluton. Note that our Pareto front s only an approxmaton as t s derved by heurstcs. Although we do not know the real Pareto front, the hgh densty of our approxmaton ndcates that the qualty of the approxmaton s very good, see Fgure 7. 19

20 In Table 3, the absolute results are presented. We show the AWRT values for the user groups, the objectve and the overall Utlzaton. It s obvous that all Approach AWRT 1 AWRT 2 AWRT 3 AWRT 4 AWRT 5 U f obj PF EASY PITTS CCA PROB Table 3. AWRT, f obj (n Seconds), and U (n %) of the Pareto Front (PF), EASY Schedulng, the Pttsburgh Approach (PITTS), the Cooperatve Coevolutonary Algorthm (CCA), and the Probablty Procedure (PROB) for the CTC Workload. proposed concepts acheve better results than the EASY standard algorthm. We restrcted the comparson to EASY as ths s n most cases the best schedulng algorthm for the examned workloads wth respect to the AWRT objectve. Note that U remans constant and s not affected by the rule based schedulng concept although t s not explctly ncluded n the objectve. As such we are able to prortze dfferent user groups wthout any reducton of the system utlzaton. Further, the results show that we are very close to the off-lne generated Pareto front, see Fgure 7. In Fgure 5 we presents the results for all sx examned workloads. The very smple and rgd probablty drven procedure s already able to mprove the objectve sgnfcantly. Apart from the KTH workload the rule system mproves the objectve value by 10 % on average compared to EASY schedulng. However, the two Genetc Fuzzy Systems outperform ths procedure. It s noteworthy that the on-lne rule based schedulng systems produce schedules almost as good as those acheved n the off-lne case. Despte the approxmatve character of the Pareto front, one can reasonably say that the results are qute close to the fronts of all workloads. Workload AWRT 1 AWRT 2 AWRT 3 AWRT 4 AWRT 5 U f obj CTC KTH LANL SDSC SDSC SDSC Table 4. AWRT and Utlzaton Improvements Acheved wth the Genetc Fuzzy System n Comparson to the EASY Schedulng Algorthm (n Percent). The results lsted n Table 4 demonstrate that the objectve mprovements really result n a shorter AWRT for the desred user groups. As we have already shown that the results are close to the Pareto front we now compare the 20

21 Objectve Improvements n % PROB PITTS CCA CTC KTH LANL SDSC00 SDSC95 SDSC96 Fg. 5. Objectve Improvements of all 3 Approaches n Comparson to EASY Schedulng. Genetc Fuzzy System, created regardng to the Pttsburgh approach, wth the EASY standard algorthm. The mprovements of the AWRT n the gray shaded columns show that t s possble to shorten AWRT 1 and AWRT 2 sgnfcantely compared to EASY for most workloads. Apart from the SDSC 00 workload ths prortzaton s realzed wthout deteroraton of the utlzaton. In Fgure 6, we exemplarly show the acheved AWRT mprovements for all 3 approaches for the CTC workload. We can realze the desred group prortzaton wth all proposed approaches. Note that we lmted ths chart at the y-axs as the AWRT values for user group 5 are extremely large. As the utlzaton remans constant these user groups have to pay the prce for the short AWRT of the favored user groups. Ths s acceptable as we dd not take these user groups nto account for our objectve formulaton. Fnally, we show n Fgure 7 the AWRT values of the two user groups to prortze. Ths chart also depcts the Pareto front of all feasble schedules. Remember that we have 7 smple objectves. Each pont wthn ths chart represents a feasble schedule that s not domnated by any other generated feasble soluton wthn the 7-dmensonal objectve space. As we show only a projecton of the actual 7-dmensonal Pareto front approxmaton, the elements cover an area n ths 2-dmensonal chart. As the EASY standard algorthm does not favor any user groups, the acheved AWRT values are located n the md of the projected front area. Wth the proba- 21

22 20 10 AWRT Improvements n % AWRT 1 AWRT 2 PITTS COEA PROB AWRT 3 AWRT 4 AWRT 5-40 Fg.6. AWRT Improvements of all 3 Approaches n Comparson to EASY Schedulng and the CTC Workload. blty drven procedure, t s already possble to move the AWRT values towards the actual front. Obvously, ths approach s capable to mprove AWRT 2 sgnfcantly but t does only slghtly mprove AWRT 1. However, the two proposed Genetc Fuzzy Systems almost reach the front n our example. Thereby, the CCA leads to a lttle bt better results than the classc Pttsburgh approach. 5.1 Estmaton of Computatonal Effort to Establsh the Rule Based Schedulng System Our chosen objectve s just an example and the proposed methods can be used wth any other objectve as well. However, we restrcted our analyss to these example as ths already requred a hgh computatonal effort. For the probablty drven procedure, we smulated (50 13 = 650) rule systems per workload. Of course ths value s scalable by choosng a smaller number of guaranteed partcpatons, but values smaller 50 dd not yeld good results. Nevertheless, ths procedure establshed the rule bases wth a comparatvely small number of smulatons. The Genetc Fuzzy Systems are realzed by an (3+21)-Evoluton Strategy. In order to obtan good results, we smulated 40 evolutonary generatons. Therefore, ( = 843) objectve evaluatons per workload were necessary. 22

23 AWRT 2 n Seconds PF EASY CCA PITTS PROB AWRT 1 n Seconds Fg.7. AWRT 1 versus AWRT 2 of all 3 Approaches, the EASY Standard Algorthm, and the Pareto Front of all Feasble Schedules for the CTC Workload. Further, a sngle smulaton of a complete workload takes about 4 hours computng tme on average. For the Genetc Fuzzy System ths resulted n 4 months computng tme per workload and objectve assumng only one avalable machne for the schedulng strategy generaton. Obvously, we are only able to present our results here because we used a compute cluster wth 120 processors. Wth ths nstallaton, we can smulate all objectve evaluatons n parallel as they are completely ndependent from each other. Therefore, the smulaton of one objectve and one workload takes approxmately one week. Furthermore, the parallel computaton of the sx workloads can also be realzed. Despte the hghly parallel executon of our smulatons t stll took more than 4 months to obtan the results presented n ths paper. Nevertheless, the presented effort estmates are only related to the generaton of the rule bases. But remember that the executon of our schedulng algorthm n the runtme envronment s not slower than the executon of a conventonal schedulng algorthm. 6 Concluson In ths paper, we have presented a novel approach to automatcally generatng onlne schedulng systems for a complex provder defned objectve. The schedulng systems are based on rules that nclude standard schedulng algorthms. We 23

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