A comparative study of scheduling algorithms for the multiple deadline-constrained workflows in heterogeneous computing systems with time windows

Transcription

1 Proceda Computer Scence Volume 29, 2014, Pages ICCS th Internatonal Conference on Computatonal Scence A comparatve study of schedulng algorthms for the multple deadlne-constraned workflows n heterogeneous computng systems wth tme wndows 1 1 ITMO Unversty, St Petersburg, Russa kbochenna@gmalcom Abstract Schedulng tasks wth precedence constrants on a set of resources wth dfferent performances s a well-known NP-complete problem, and a number of effectve heurstcs has been proposed to solve t If the start tme and the deadlne for each specfc workflow are known (for example, f a workflow starts executon accordng to perodc data comng from the sensors, and ts executon should be completed before data acquston), the problem of multple deadlne-constraned workflows schedulng arses Takng nto account that resource provders can gve only restrcted access to ther computatonal capabltes, we consder the case when resources are partally avalable for workflow executon To address the problem descrbed above, we study the schedulng of deadlne-constraned scentfc wo rkflo ws n non-dedcated heterogeneous envronment In ths paper, we ntroduce three schedulng algorthms for mappng the tasks of multple workflows wth dfferent deadlnes on the statc set of resources wth prevously known free tme wndows Smulaton experments show that schedulng strateges based on a proposed staged scheme gve better results than merge-based approach consderng all workflows at once Keywords: deadlne-constraned workflows, workflow schedulng, tme wndows 1 Introducton Over the past decade, the problem of schedulng nterrelated tasks n heterogeneous dstrbuted systems (such as grds and clouds) has ganed partcular attenton due to the ncreased use of scentfc workflows n a varety of subject areas n condtons of contnuous growth of avalable computng power Today there are a lot of statc and dynamc schedulng approaches desgned for dfferent combnatons of workflows and resources propertes (eg, parallelsm level, graph shape, data nterchange ntensty, arrval patterns of tasks, homogenety of processors performance and communcaton speed, on-demand resources accessblty) Goals of schedulng process are Selecton and peer-revew under responsblty of the Scentfc Programme Commttee of ICCS 2014 c The Authors Publshed by Elsever BV do:101016/jprocs

2 determned based on both the selected nfrastructure archtectural concepts [1] (possbltes gven by resource provders) and customers requrements (QoS lmtatons) In general, modern workflow management systems operate under condtons of hgh unpredctablty both of workload and resource accessblty, whch leads to the necessty to use dynamc schedulng algorthms for vrtualzed envronments However, for problems n whch we have good estmates of tasks executon tmes, exactly know the nformaton about resource set compound, avalable tmes of resources and moments when workflows start, t s advsable to use statc schedulng methods whch can provde better schedules than dynamc ones due to takng nto account a workflow structure The shnng example of such a problem s a regular operatng process of an urgent computng system when perodcally ncomng peces of data should be processed usng workflow mechansm wthout volatng user defned deadlne [2] (whch can concde wth the moment of recevng next chunk of data) For nstance, ths scheme s commonly used n urban flood decson support systems [3], [4] Typcal modelng workflow conssts of 5-25 tasks and can nclude stages of meteorologcal data processng, smulaton and predcton of sea level and wave parameters, and varous decson support scenaros New nstance of regular operatng process s started every several hours to obtan up-to-date forecast based on a recent data Non-volatng the deadlne of ths process guarantees a relevance of the results whch s crucal for an effectve flood preventon In normal operaton mode the regular processes often use a specfc statc set of resources whch can be non-dedcated to workflows executon At the same tme, nformaton about avalable CPU tme wndows can be provded n advance, e f one reserves computatonal capactes for scentfc or educatonal purposes Also, perodcal nature of regular calculatng process provdes an opportunty to gather a comprehensve statstcs about executon tmes of sngle tasks In case of several processes (each wth ts own data ncomng frequency and ts own maxmum makespan) the schedulng problem s to fnd a mappng of tasks to processors maxmzng the effcency of resource utlzaton whle meetng all workflows deadlnes (and maybe some other users constrants) The contrbutons of ths paper are as follows: 1) formulaton of the schedulng problem (shortly descrbed above); 2) development of three algorthms for schedulng of multp le deadlne-constraned workflows n heterogeneous platforms wth tme wndows; 3) smulaton experments on the sets of synthetc workflows to compare the effectveness of these algorthms accordng to the proposed metrcs 2 Related works Over the past decade, there have been proposed a lot of algorthms n the area of sngle deadlneconstraned workflow schedulng Most of these algorthms are desgned to solve the problem of mnmzng the workflow executon cost/makespan whle respectng ther servce level agreements (SLAs) to ensure the QoS complance, and moreover they are focused on dynamc resource allocaton usng vrtual machnes Bossche et al[5] and Genez et al [6] descrbed the formulaton of ths problem n terms of nteger lnear programmng and proposed a set of heurstcs to solve t Smlar problem statement can be found n [7], where authors ntroduced an SCB (Server Count Bound) heurstc to fnd the mnmum resource count whch s requred to execute workflow wthout volatng the deadlne Yu et al [8] studed sngle workflow schedulng wth tme and cost constrants and proposed a genetc algorthm based on heurstc for mnmzng executon tme whle meetng user s budget constrants Zheng and Sakellarou [9] proposed a heurstc BHEFT as an extenson of HEFT algorth m for market -orented envronments wth non-dedcated resources In the paper [10] authors ntroduced a novel sngle workflow schedulng algorthm PEFT whch outperforms well-known lstbased heurstcs n terms of makespan, effcency and frequency of best results Many researchers use the dea of assgnng sub-deadlnes to ndvdual tasks (f we meet all the sub-deadlnes, we wll automatcally meet the whole workflow deadlne) Yu et al [11] and Yuan et 510

3 al [12] suggested to group the tasks accordng to ther level (e, path length from the nput task) and to fnd a schedule meetng group deadlnes (Deadlne Top Level, DTL and Deadlne Bottom Level, DBL algorthms) Schedulng method proposed n [13] s also based on tasks groupng Wang et al [14] developed a heurstc algorthm Deadlne Early Tree (DET), where tasks of crtcal path are scheduled usng dynamc programmng Determnaton of the certan tasks deadlnes s usually produced usng one of two approaches: 1) provdng the earlest startng tme and latest fnsh tme usng the network plannng methods, 2) settng the deadlnes proportonal to the computatonal complexty of tasks Thus, the Partal Crtcal Path (PCP) algorthm [15] n tally fnds the deadlnes for the crtcal path tasks, and then uses a recursve procedure to set other tasks deadlnes Later two modfcatons, IC-PCP and IC-PCPD2, has been developed n [16] Only few studes have consdered multple workflows schedulng problem As showed n [17], there are two man approaches to schedule a set of workflows: 1) mergng the tasks of all workflows nto a sngle composte workflow and applyng any sngle workflow schedulng algorthm (eg, SOT (Serve on Tme algorthm)[18]); 2) orderng sngle workflows accordng to some metrc and then schedulng t consequently (eg, FPFT/FPCT (Farness Polcy based on Fnshng/Concurrent Tme) algorthms [19]) Mao et al [20] proposed an algorthm to mnmze the multple workflows executon cost whle meetng the deadlnes, ntended for scalable vrtualzed computatonal envronment In [21] there s proposed an approach to cost- and deadlne-constraned schedulng of ensembles of nter-related workflows n resources-on-demand cloud envronment For scentfc workflow schedulng on partally avalable resources wth prelmnary reserved tme wndows, Luo et al [22] proposed three algorthms for sngle data-parallel workflo ws (exact branchand-cut based and two heurstcs, MHEFT-RSV and MHEFT-RSV-BD) To the author s best knowledge, develop ment of mu ltple deadlne-constraned workflows schedulng algorthm on heterogeneous platforms wth predefned tme wndows s stll an open research problem, and no conclusve results have been reported n the lterature on ths topc 3 Multple deadlne-constraned workflows schedulng wth tme wndows: problem formulaton Scentfc workflows typcally use two types of parallelsm whch can be exploted to get a hgher beneft from the large computng power [23]: task parallelsm (when dfferent tasks of a partcular workflow can be executed concurrently) and data parallelsm (when each task can be executed on more than one processor at the tme) Task-parallel workflows are usually represented as DAG (Drected Acyclc Graph) whle mxed (task and data parallel) applcatons use PTG (Parallel Task Graph) model In ths paper we consder DAG representaton of workflows wth non-preemptve tasks, so one task can be executed only on one processor wthout suspendng and mgratng to other processor Thus, each workflow n the set WFSet can be represented as a graph WF V, E, 1, NWF where N WF number of workflows n WFSet, V 1,, Np ndexes of tasks of -th workflo w, E e, j V V a set of Nt number of tasks of -th workflow, edges representng lnks between workflow tasks WFSet s a set of tasks Task, 1, N, j 1, Nt Each task can be executed on some set of resource types RT jk, k 1, Nrtj ( Nrt j s number of resource types for j -th task of -th workflow) Also, estmates of executon tmes t jk are known for each possble combnaton of task and resource type Ths estmates can be obtaned by prelmnary proflng, handlng data from prevous runs or usng parametrc performance models of partcular tasks j j WF 511

4 [24] If performances of partcular resources (n FLOPS) and FLOP count for each task are specfed, we can use ths nformaton to calculate approxmate estmates Gven a computer platform that conssts of N R resources related to dfferent N T resource types (where NR, 1, NT number of resources of -th resource type) Dfferent resources are assgned to one resource type, f they have the same techncal characterstcs (processors/cores count, RAM sze) and there s a hgh-speed network connecton between them In ths work we assume that gven workflows are not communcaton-ntensve, so we can neglect a data transfer tme Let T be the length of plannng perod All workflows have the known start tmes tbegn, 0 tbegn T and deadlnes deadlne, tbegn deadlne T where 1, NWF The smplest way to measure the effcency of resources utlzaton s to fnd a rato of tme occuped by scheduled tasks to ntally avalable t me In ths case, tme ntervals wth the same length have the same value of objectve functon regardless of these placements We consder more sophstcated case when effectveness of resource usage s characterzed by utlty functon g whch reflects preferred loadng tme ( 0;1 t normalzed tme) If we assume that the maxmum resource utlzaton T level s equal to 1, and all resources have the same utlty functon, then g d, (31) N R 1 Nproc where Nproc number of processors for -th resource type Total effcency of resource utlzaton for resource type R n the perod s W e Int j jk R g d jk 1 ; 2 Nproc, 1, 2, (32) j1 k1 b Intj Int 1, j 2 number of ntervals occuped by scheduled tasks for j -th processor of -th resource type n the perod 1 ; 2, b e jk, jk begnnng and end of the k -th nterval In ths work we use a smple decreasng lnear functon as utlty functon: 2 g 1 (33) where R 1 Nproc So, for the same resource type, the earler s the start tme of a partcular task, the greater s the correspondng resource utlzaton The goal of schedulng s to maxmze the overall resource utlzaton level for a gven perod 0 ;T wthout volatng deadlnes of workflows ( tend s a fnshng tme for -th workflo w n the obtaned schedule): Nr,0,1 max W W R, (34) 1 Wf, 1, N WF deadlne tend (35) 512

5 4 Consdered schedulng algorthms As we mentoned before, there are two man approaches to multple workflows schedulng (the one s based on mergng all tasks nto a bg workflow, and the other takes nto account afflaton of tasks to workflows) Our hypothess s that n the case of deadlne-constraned workflows the second approach should gve better results snce t becomes possble to consder the partcular deadlnes To check t, we have developed algorthms mplementng merge-based and workflow-based approaches Note that they both should use some addtonal algorthm for a sngle workflow schedulng 41 Sngle deadlne-constraned workflow schedulng algorthm for heterogeneous resources wth tme wndows where P j j j Snce volaton of the deadlne s undesrable, the purpose of an algorthm s to maxmze the tme between actual workflow completon and ts deadlne (on the other hand, ncreasng ths margn allows to ncrease the possblty of meetng the deadlne f some tasks fnsh ther executon later than planned) More formally, the schedulng goal for a gven Wf, 1, NWF s to maxmze deadlne max tend P tend s the fnshng tme of j -th task There are a lot of wellknown and effcent algorthms for mnmzng the makespan of workflow executon on a set of heterogeneous resources (such as HEFT and PEFT[10]) In ths work we propose lst-based heurstc SmpleSched whch s smlar to those algorthms due to basc steps (prortzng and assgnng tasks to workers) The man dfference s that we use the value of deadlne whle buldng a queue of tasks It s crucal to consder the deadlnes of partcular workflows n the algorthm because our merge-based approach s based on schedulng tasks from dfferent workflows at once, and prorty of the task should reflect accumulated executon tme (or computatonal cost) as well as requred fnshng tme of the workflow Then, SmpleSched heurstc conssts of the followng steps: 1) Determnaton of a tentatve latest fnshng tme (sub-deadlne) for each task; 2) Creaton of a prorty queue of tasks consderng sub-deadlnes and tasks precedence constrants; 3) Mappng each task from queue to the resource whch provdes the earlest fnshng tme for ts executon If estmaton tmes t jk are gven, evaluaton of latest fnshng tmes can be done by network analyss methods (note that f the mnmum earlest fnshng tme s greater than the gven deadlne, workflow cannot be fnshed n tme) Consder the case when we have nformaton about resources performance and FLOP count for each task Let LFT be the latest fnshng tme of -th task of a gven workflow, calc computatonal cost of -th task (n FLOP), amount maxmum accumulated computatonal cost from ntal task to -th task (ncludng ts own cost), n and out sets of nput/output ndexes of tasks for -th task Then defnng the sub-deadlnes for all tasks can be descrbed as follows ( used s a set of task ndexes wth already calculated values ofamount, lnked s a lst of tasks for whch at least one of ts nput tasks s n used set, lnked used ) The descrpton of the algorthm for calculatng the subdeadlnes for all tasks of a sngle workflow s gven n Table 1 1 used : n, lnked j : used, j out 2 used amount calc 3 Whle lnked repeat steps current lnked 0 513

6 5 lnked lnked \ current 6 If ncurrent used, amount max amount j calc, used used current, jncurrent out current, lnked lnked lnked Else lnked lnked current 7 Fnd such that j used amount amount j 8 LFT deadlne 9 used, LFT amount amount deadlne Table 1 Algorthm for calculatng the sub-deadlnes of workflow tasks by ther computatonal costs When the latest fnshng tmes are calculated, tasks are placed n prorty queue n ascendng order of ts sub-deadlnes (t s also necessary to take nto account that package should not be scheduled before any of ts nput packages) After constructon the queue packages are mapped consequently EST Task max tend pred be the earlest possble start tme of -th task of the j - Let j Task j th workflow as a maxmum fnshng tme of ts already scheduled predecessors, and EST Task, R EST Task the earlest start tme of ths task on the k -th resource type j k j consderng actual busy tme wndows The task assgns to the resource Rk whch provdes earler fnshng tme for t ( Rk : EFT Taskj, Rk mnest Taskj, Rk tjk ) k In fact, the merge-based algorthm s SmpleSched heurstc (descrbed above) appled to workflow constructed from all avalable packages Snce packages belongng to dfferent workflows are not lnked, one can calculate sub-deadlnes ndependently for each partcular workflow, and then merge all packages nto queue So, merge-based algorthm mplctly takes nto account workflow deadlnes by prmarly schedulng tasks wth earler sub-deadlnes, but n a stuaton when several tasks of dfferent workflows have smlar LFTs and compete for resources (due to ther lack) t provdes no guarantees that resources wll be gven to packages whch workflow has earler deadlne To llustrate prevously mentoned statement about necessty of takng nto account the deadlnes whle prortzng the tasks of multple workflows at once, we consder the example lsted below Fgure 1 Test example (2 workflows) Notaton: a/b (c), a - tme of executon on R1, b - tme of executon on R2, с - average executon tme 514

7 Assume that we have two workflows as shown at Fgure 1, and a set of 3 resources (1 resource of type 1 (R1), and 2 resources of type 2 (R2)) Let s compare the results of HEFT (wthout takng nto account the communcaton costs) and SmpleSched algorthms Table 1 shows the ranks of tasks accordng to HEFT algorthm and LFTs of tasks accordng to SmpleSched The ranks for HEFT are calculated usng the formula: rank Task w max rank Task j, (41) jsucctask a b where w an average executon tme of -th task 2 Task ndex Rank (HEFT) LFT (SmpleSched) Workflow Workflow Table 1 Values of prortzaton crtera for HEFT and SmpleSched Due to Table 1, the order of assgnng the tasks to processors wll be: ) for HEFT: 1, 3, 5, 2, 6, 4, 7; ) for SmpleSched 1, 5, 2, 6, 3, 7, 4 The results of assgnng the tasks to processors wth earlest fnshng tme for each task are summarzed on Fgure 2 Fgure 2 SmpleSched vs HEFT for the sets of deadlne-constraned workflows For the test example HEFT provdes the schedule where Task6 and Task7 volate the deadlne, and SmpleSched gves the schedule wthout volatng the deadlne At the same tme, makespan for workflow 1 ncreases n case of SmpleSched only by 1 tme unt whle workflow 2 reduces ts makespan by 4 tme unts n comparson wth HEFT If we suppose that overall number of workflows s n, and each workflow contans m tasks, the computatonal complexty of SmpleSched wll be Om n 515

8 42 Stage-based approach to a multple deadlne-constraned workflow schedulng wth tme wndows The man dea of the second approach s to consder workflow deadlnes as much as possble If one s gong to schedule multple workflows consequently, one should know the order n whch workflows wll take access to resources selecton (n other words, t s necessary to prortze workflows) The most obvous soluton of the problem s to assgn prortes accordng to ts deadlnes Ths approach does not seem to be the best for several reasons ncludng possble presence of multple workflows wth the same or smlar deadlnes whch have consderably dfferent computatonal complextes and/or startng tmes Suppose that we are gven prortzaton crtera, e a certan metrc applcable to workflow schedule whch extreme value s used to choose a workflow for resource allocaton The man concept of stage-based approach s that schedulng procedure conssts of several steps (by number of workflows), and each step nvolves a choce of one workflow n accordance wth prortzaton crtera, schedulng t and fxng busy tme wndows on selected resources In ordered scheme, a partcular workflow can be scheduled wth any algorthm for mnmzng makespan of a sngle workflow (e, HEFT) There s only one restrcton that selected algorthm should assgn the tasks to the processors wth consderaton of reserved tme wndows Let Scheduled be the set of workflows numbers that were already scheduled, Unschedule d the set of workflows numbers that requre schedulng, Intervals current set of resources busy ntervals, Schedule Intervals schedule for -th workflow taken by sngle workflow schedulng algorthm for current tme wndows, Prorty Schedule value of prortzaton crtera of Schedule Assume that workflow selecton s carred out accordng to mnmum prortzaton crtera value Then an algorthm mplementng stage-based approach (t s denoted as StagedScheme) can be descrbed as follows 1 Scheduled, Unschedule d, Intervals Intervals nt, where Intervals nt s n tal busy tme wndows 2 For all Wf, 1, NWF Unscheduled Unscheduled 3 For all Unscheduled Schedule Intervals get the schedule 4Fnd : j PrortySchedule Prorty Schedule j 5 Scheduled Scheduled, Unscheduled Unscheduled \ 6 Intervals Intervals Schedule 7 Repeat steps 3-6 whle Unschedule d where, j Unscheduled Table 2 The StagedScheme algorthm for a multple deadlne-constraned workflows schedulng wth tme wndows O m, If we suppose that computatonal complexty of schedulng one workflow of m packages s and we should schedule n workflows of m packages, the computatonal complexty of StagedScheme s Om n Om n Om1 Om Om n 1 n 1 2 ~ Om n 1 n 2 516

9 5 Smulaton technque To study the effectveness of proposed schedulng algorthms we have generated synthetc test data whch parameters and values are summarzed n Table 3 The goal of the smulaton was to compare the merge-based and ordered schemes under the crcumstances of wde varance of number of workflows and tasks on the statc set of resources Ths approach can show us the man tendences n behavor of studed algorthms by fxng the parameters of resources and varyng the amount of computatons and the partton of these computatons nto workflows and tasks Workflows test examples were obtaned wth DA GGEN workflow generator wh ch allows us to generate random graphs of tasks wth dfferent shapes of graphs The most mportant parameters affectng the shape of the graph are wdth (maxmum number of tasks that can be executed concurrently), densty (number of dependences between tasks of consecutve graph levels), regularty (regularty of the dstrbuton of tasks between the dfferent levels) and jump (maxmum number of levels spanned by task communcatons) Obtaned test examples were supplemented wth the values of package computatonal cost whch were chosen so that on a partcular set of resources one package has executon tme n the range of sec To receve test examples of resources we mplemented a resource generator whch provdes sets of non-dedcated resources wth heterogeneous processors performances We ve chosen so hgh level of heterogenety of resources (ten tmes dfference as a maxmum) because modern systems often nclude not only CPUs but GPUs as computng devces, and nequalty of performance parameters of these devces can be sgnfcant Parameter name Possble values Workflows parameters Workflows count 25; 50; 75; ; 400 Task computatonal cost (GFLOP) Number of tasks per workflow 5; 20; 50 Wdth of the DAG 01; 02; 08 DAG densty 02; 08 DAG regularty 02; 08 DAG jump 1, 2, 4 Resources parameters Processor performance (GFLOPS) 5-50 Part of non-dedcated tme per processor 025; 05; 075 Non-dedcated ntervals count per processor 0-3 Number of resources types 1; 2; 4; 8 Number of resources per resource type 1; 2; 4; 8; 16 Number of processor per resource 1; 2; 4 Table 3 Parameters of Synthetc Test Data Estmaton of comparatve effcency of algorthms was made by the calculaton of several metrcs: 1 Average reserved tme NWF reserve _ tme 1 Rtme NWF, (51) 0, deadlne tend where reserve _ tme deadlne tend, deadlne tend 2 Volated deadlnes per number of workflows n the set 3 Loadng effcency 517

10 free W Intervals busy Eff, (52) W Intervals where Intervals busy tme wndows used by workflows, Intervals free ntal free tme wndows 4 The executon tme of algorthm Each set of experments was carred out for a specfc number of tasks per workflow (for example, 20) and dfferent number of workflows (from 25 to 400) Each workflow set ncluded approxmately the same number of workflows for dfferent DAG wdths Deadlnes of all workflows were equal to the end of plannng perod All experments were performed on the same set of resources (8 resource types, 16 resources per type, 4 processors per resource, part of non-dedcated tme per processor 025) to get the most precse estmates of executon tmes of the algorthms Each experment was carred out twce for two dfferent plannng perods Frst length of the perod (denoted as wde makespan) has been selected so that all workflows n maxmum set ( e 400 workflows) could fnsh executon wthout volatng the deadlnes To lmt a wde makespan, we suppose that loadng effcency should be at least 08 for the maxmum workflows set Experments wth wde makespan were used to estmate average reserved tmes and loadng effcences Secon d plannng length of the perod (denoted as tght makespan) was taken as 75% from wde makespan It has been used to estmate the part of volated deadlnes per workflow count 6 Smulaton results We denote the studed algorthms as follows: 1) Merge-based an algorthm mplementng mergebased approach; 2) MaxReserved an algorthm mplementng the staged scheme and usng maxmum reserved tme as a prortzaton crtera; 3) MnEff an algorthm mplementng the staged scheme and usng mnmum effcency as a prortzaton crtera All three algorthms use SmpleSched for a sngle workflow schedulng Snce a sngle set of experments was carred out for a partcular number of tasks per workflow, we use ths value to denote the set (for example, MaxReserved 20) In Fgure 3 we show a rato of the average reserved tme (whch was calculated usng (51)) to the makespan Note that n all cases MnEff provdes better reserved tmes than two other algorthms, especally for the set of 5-tasks workflows where ts maxmum advantage s up to 13% Merge-based algorthm shows the lowest values and faster decrease of reserved tme than stage-based algorthms n all sets of experments Fgure 3 Rato of the average reserved tme to makespan 518

11 Fgure 2 shows the effcences taken for a wde makespan Staged-based algorthms demonstrate near effcences for all experments (except MnEff 5) Maxmum advantages for MaxReserved algorthm n comparson wth Merge-based s respectvely 6,79% for 5-tasks sets, 636% for 20- tasks sets and 1491% for 50- tasks sets Note that on both fgures descrbed above maxmum dfferences between algorthms appear for experments wth number of workflows more than 200 It can be explaned by the fact that the ncrease of number of workflows (wth the same resource set) leads to the ncrease of competton for resources Fgure 4 Effcences for wde makespan Fgure 5 shows the rato of volated deadlnes to workflows count for a 50- tasks test set for a tght makespan Merge-based algorthm demonstrates absolutely the worst results wth maxmum dsadvantage equal to 32,8% and average dsadvantages equal to 19,28% for MaxReserved and 15,2% for MnEff Effcency estmates have shown that loadng for 275 workflows (and hgher) was more than 95% for staged-based algorthms It explans a sharp rse of the metrc begnnng at that pont (there were no more avalable tme slots) We should note that loadng effcency of Merge-Based was n average 10% less than of both MaxReserved and MnEff (despte the fact that t had more unscheduled tasks) Fgure 5 Rato of volated deadlnes to number of workflows for a tght makespan 519

12 Fnally, Fgure 4 shows the comparson of algorthms runtmes Our experments confrmed the estmates of computatonal complexty gven n Secton 4 Maxmum executon tmes are: for tasks workflows: Merge-Based 0272 sec; average stage-based 16,33 sec; for tasks workflows: Merge-Based 1338 sec; average stage-based 55,75 sec; for tasks workflows: Merge-Based 5148 sec; average stage-based 223,56 sec Fgure 6 Executon tmes of algorthms 7 Conclusons and future work In ths paper we ntroduced an algorthm for a sngle deadlne-constraned workflow schedulng for heterogeneous platforms wth tme wndows and a staged scheme for a multple deadlne-constraned workflow schedulng We run a seres of experments based on synthetc test examples n order to analyze and compare three proposed schedulng strateges n accordance wth four metrcs Smulaton results ndcate that the staged-based approach shows better results than merge-based for all schedule qualty crtera (average reserved tme, effcency and meetng the deadlnes) However, when computng envronment s relatvely free, we can use a merged-based algorthm due to ts lnear computatonal complexty and close to stage-based results Future work n ths area mght nclude the mprovement of proposed schedulng formulaton and algorthms consderng the account costs of data transfer between the resource types and presence of data-parallel tasks We am to work out a fast reschedulng algorthm whch should be appled n case of delay and tasks falures, as well as to study the nfluence of the chosen prortzaton crtera on the qualty of obtaned schedules Also, the authors plan to embed the developed algorthms n a schedulng system of CLAVIRE [25] e-scence nfrastructure platform Ths work was fnancally supported by Government of Russan Federaton, Grant 074-U01 References [1] S V Kovalchuk, P A Smrnov, S V Maryn, T N Tchurov, and V A Karbovsky, Deadlnedrven Resource Management wthn Urgent Computng Cybernfrastructure, Proceda Comput Sc, vol 18, pp , Jan

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