Performance Analysis of Work-Conserving Schedulers for Minimizing Total Flow-Time with Phase Precedence

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1 Perforance Analysis of Work-Conserving Schedulers for Miniizing Total low-tie with Phase Precedence Yousi Zheng, Prasun Sinha, Ness B Shroff, Departent of Electrical and Coputer Engineering, Departent of Coputer Science and Engineering The Ohio State University, Colubus, OH 3 Abstract We consider the proble of iniizing the total flow-tie of ultiple jobs in a pool of ultiple hoogeneous achines, where the jobs arrive over tie and have to be served with phase precedence This is a coon occurrence in job scheduling for the increasingly popular data center oriented systes, where jobs need to be processed through Map and Reduce procedures before leaving the syste or this proble, one can construct an arrival pattern such that no scheduler can achieve a constant copetitive ratio However, what we find is that by using a slightly weaker perforance etric, which we call the efficiency ratio, we can provide bounds on the perforance We say that a scheduler achieves an efficiency ratio of γ when the flow-tie incurred by that scheduler divided by the iniu flow-tie achieved over all possible schedulers is less than or equal to γ alost surely, when the tie slots or job arrivals go to infinity Under soe weak assuptions, we show a surprising property that all workconserving schedulers for the flow-tie proble with phase precedence have a constant efficiency ratio in both preeptive and non-preeptive scenarios We provide nuerical results to support our analysis I INTRODUCTION In any widely used systes, the workload of arriving jobs can be divided into several phases Soe phases need to finish before other phases can be started One coon exaple of such a syste is the MapReduce fraework, which is designed to process assive aounts of data in a cluster of achines [] It is widely used for applications such as search indexing, distributed searching, web statistics generation, and data ining MapReduce has two eleental phases: Map and Reduce A critical consideration for the design of the scheduler is the precedence between the Map and Reduce tasks or each job, the Map tasks need to be finished before starting any of its Reduce tasks [], [] To iniize the total flow-tie, it is well known that the Shortest-Reaining-Processing-Tie scheduler is optial in one achine case [3] or in ultiple achines with work preserving alleable tasks[] The related scenarios are studied: scheduling alleable tasks [], scheduling chainstructured tasks [], and scheduling with release tie constraint [7] However, they focus on the coplexity analysis, and they cannot be directly applied to the ulti-phase setting or the MapReduce fraework, soe scheduling solutions have also been proposed [8], [9], [], [], [], but analytical We gratefully acknowledge Dr Zizhan Zheng for his helpful suggestions The work has been partially supported through a grant fro the Ary Research Office MURI award W9N bounds on perforance have been derived only in soe of these works [], [], [] However, rather than focusing directly on the flow-tie, for deriving perforance bounds, [], [] have considered a slightly different proble of iniizing total copletion tie and [] has assued speedup of the achines In this paper, we directly analyze the perforance of total delay flow-tie in the syste In an attept to iniize this, we introduce a new etric to analyze the perforance of schedulers called efficiency ratio Based on this new etric, we analyze and design schedulers that can provide perforance guarantees The contributions of this paper are as follows: To directly analyze the total delay in the syste, we propose a new etric to easure the perforance of schedulers, which we call the efficiency ratio Section II or bounded workload of Phase, we then show a surprising property that for the flow-tie proble any work-conserving scheduler has a constant efficiency ratio in the preeptive as well as in the non-preeptive scenario precise definitions provided in Section II Sections III and IV We relax the assuption of Phase fro bounded workload to light-tailed distributed workload, and get the sae properties in both preeptive and nonpreeptive scenarios Section V or the typical work-conserving schedulers, we show that the convergence of efficiency ratios through siulations Section VI II SYSTEM MODEL AND EICIENCY RATIO A Syste Model Consider a pool of N achines There are n jobs arriving into the syste, and each achine can only process one job at a tie Tie is slotted and each achine can run one unit of workload in each tie slot We assue that the distribution of job arrivals in each tie slot is iid, and the arrival rate is λ Each job i contains ultiple tasks, which can be classified into two phases: Phase and Phase Its tasks of Phase need to be finished before starting any of its tasks of Phase or each job i, Phase brings M i units of workload and Phase brings R i units of workload Each

2 task of Phase has unit of workload, however, each task of Phase can have ultiple units of workload Assue that M i } are iid with expectation M, and R i } are iid with expectation R We assue that the traffic intensity ρ<, ie, λ< N Assue the oent generating function of MR workload of arriving jobs in a tie slot has finite value in soe neighborhood of In tie slot t for job i, i,t and r i,t achines are scheduled for the tasks of Phase and, respectively We assue that job i contains K i tasks of Phase, and the workload of the Phase task k of job i is R k i Thus, for any job i, K i k R k i R i In tie slot t for job i, r k i,t achines are scheduled for the task k of Phase As each task of Phase ay consist of ultiple units of workload, it can be processed in either preeptive or non-preeptive fashion based on the type of scheduler Definition : A scheduler is called preeptive if the tasks of Phase belonging to the sae job can run in parallel on ultiple achines, can be interrupted by any other task, and can be rescheduled to different achines in different tie slots A scheduler is called non-preeptive if each task of Phase can only be scheduled on one achine and, once started, it ust keep running without any interruption The flow-tie i of job i is equal to f r i a i, where f r i is the finish tie of the reduce tasks and a i is the arrival tie of job i The objective of the scheduler is to deterine the assignent of jobs in each tie slot, such that the cost of delaying the jobs or equivalently the flow-tie is iniized or the preeptive scenario, the proble definition is as follows: in i,t,r i,t st n i f r i a i n i,t r i,t N, r i,t, i,t, t, i f i ta i i,t M i, f i r tf i r i,t R i, i,, n} In the non-preeptive scenario, the tasks of Phase cannot be interrupted by other jobs Once a task of Phase begins execution on a achine, it has to keep running on that achine without interruption until all its workload is finished Also, the optiization proble in this scenario is siilar to Eq, with additional constraints representing the non-preeptive nature, as shown below: or exaple, in the MapReduce fraework as the Map tasks are independent and have sall workload [9], such an assuption is valid in i,t,r k i,t st n i i f r i a i n K i i,t r k i,t N, t, f i k ta i i,t M i, i,t, i,, n}, f i r tf i r k i,t r k i,t R k i, i,, n}, k,, K i }, or, r k i,t t if < s r k i,s <Rk i The proof of the following theore is in the technical report [] Theore : The scheduling proble both preeptive and non-preeptive is NP-coplete in the strong sense B The copetitive ratio is often used as a easure of perforance in a wide variety of scheduling probles or our proble, the scheduling algorith S has a copetitive ratio of c, if for any total tie T, any nuber of arrivals n in the tie T, any arrival tie a i of each job i, any workload M i and R i of the Phase and tasks with respect to each arrival job i, the total flow-tie S T,n,a i,m i,r i ; i n} of scheduling algorith S satisfies the following: S T,n,a i,m i,r i ; i n} T,n,a i,m i,r i ; i n} c, where T,n,a i,m i,r i ; i n} in S T,n,a i,m i,r i ; i n}, S is the iniu flow-tie of an optial off-line algorith or our proble, we can construct an arrival pattern such that no scheduler can achieve a constant copetitive ratio c The quick exaple is given in technical report [] We now introduce a slightly weaker notion of perforance, called the efficiency ratio Definition : We say that the scheduling algorith S has an efficiency ratio γ, if the total flow-tie S T,n,a i,m i,r i ; i n} of scheduling algorith S satisfies the following: S T,n,a i,m i,r i ; i n} T γ, wp T,n,a i,m i,r i ; i n} Later, we will show that for the quick exaple, a constant efficiency ratio γ can still exist eg, the non-preeptive scenario with light-tailed distributed Reduce workload in Section IV

3 or each job i, R i R ax And in the last slot of the previous interval, the axiu nuber of finished jobs of Phase is N Thus, R j, N R ax Now, let s consider the case of k> ig A exaple schedule of a work-conserving scheduler III WORK-CONSERVING SCHEDULER IN THE PREEMPTIVE SCENARIO In this section, we analyze the perforance of workconserving scheduler in the preeptive scenario In this section, we study the case in which the workload of Phase of each job is bounded by a constant, ie, there exists a constant R ax, st, R i R ax, i Consider the total scheduled nuber of achines over all the tie slots If all the N achines are scheduled in a tie slot, we call this tie slot a coplete tie slot; otherwise, we call this tie slot a incoplete tie slot We define the j th interval to be the interval between the j st incoplete tie slot and the j th incoplete tie slot We define the first interval as the interval fro the first tie slot to the first incoplete tie slot Thus, the last tie slot of each interval is the only incoplete tie slot in the interval Let K j be the length of the j th interval, as shown in ig irstly, we show that different oents of the length K j of interval j is bounded by different constants, as shown in Lea Lea : If the scheduler is work-conserving, then for any given nuber H, there exists a constant B H, such that E[Kj H] <B H, j Proof: Since K j is a positive rando variable, then E[Kj H ] if H Thus, we only focus on the case that H> In the j th interval, consider the t th tie slot in this interval Assue that the total workload including Map and Reduce tasks of the arrivals in the t th tie slot of this interval is W j,t Also assue that the unavailable workload of Phase at the end of the t th tie slot in this interval is R j,t, ie, the workload of Phase whose corresponding tasks of Phase are finished in the t th tie slot in this interval is R j,t The reaining workload of the Reduce function left by the previous interval is R j, Then, the distribution of K j will be as follows: P K j k P P P k W j,t R j, R j,k k N t k W j,t N R ax k N t k W j,t t k N N R ax k By the assuptions, the total both Phase and arriving workload W j,t in each tie slot is iid, and the distribution does not depend on the interval index j or the index of tie slot t in this interval Then, for any interval j, we know that P K j k P k W s s k N N R ax k, where W s } is a sequence of iid rando variables with the sae distribution of W j,t Since the traffic intensity ρ<, E[W s ]λm R <N or any >, there exists a k, such that for any k>k, Then for any k>k,wehave N R ax k P K j k P k W s s k N, We take <N E[W s ] and corresponding k or any k>k N Rax, by Craer-Chernoff Theore [3], we can achieve that P k W s s k N e kln, 3 P K j k P W j, R j, R j, N, k W j,t R j, R j,k k N, t k W j,t R j, R j,k <kn t where the rate function la is shown as below: la sup θa loge[e θw s ] θ Since the oent generating function of workload in a tie slot has finite value in soe neighborhood around,

4 for any <<N λm R, ln > Thus, E[Kj H ] k H P K j k k k k H P K j k k k H k H e kln kk k H P K j k N R H ax k H e kln Since ln >, k H e kln is bounded Thus, given H, E[Kj H] is bounded by a constant B H, for any j, where B H is given as below N R H ax B H in,n λmr k H e kln Now, we directly obtain the expression for first and second oents of K j Corollary : E[K j ] is bounded by a constant B, E[Kj ] is bounded by a constant B, for any j The expressions of B and B are shown as below, where the rate function la is defined in Eq N R ax B in,n λmr e ln, e ln e ln N R ax B in,n λmr eln 3e ln e ln e ln 3 Proof: By substituting H and H in Eq, we directly achieve the corollary We add soe duy workload of Phase in the beginning tie slot of each interval, such that the reaining workload of the previous interval is equal to N R ax The added workload are only scheduled when no real job can be scheduled Then, in the new artificial syste, the total flow-tie of the real jobs doesn t change However, the total nuber of intervals ay decrease, because the added workload ay erge soe adjacent intervals Corollary : In the artificial syste, the length of new intervals K j,,, } also satisfies the sae results of Lea and Corollary Proof: The proof is siilar to the proofs of Lea and Corollary Reark : In the artificial syste constructed in Corollary, the length of each interval has no effect on the length of other intervals, and it is only dependent on the job arrivals and the workload of each job in this interval Based on our assuptions, the job arrivals in each tie slot are iid Also, the workload of jobs are also iid Thus, the rando variables K j, j,, } are iid Theore : or any work-conserving scheduling policy, the efficiency ratio is not greater than B B wp, when n goes to ax, p λ, ρ λ}ax, N ρ } infinity B and B can be achieved by Corollary, and p is the probability that no job arrives in a tie slot Proof: Consider the artificial syste constructed in Corollary Since we have ore duy workload of Phase in the artificial syste, the total flow-tie of the artificial syste is not less than the total flow-tie of the workconserving scheduling policy Note that, in each interval, all the job arrivals in this interval ust finish Phase in this interval, and all their workload of Phase will finish before the end of the next interval In other words, for all the arrivals in the j th interval, Phase is finished in K j tie slots, and Phase is finished in K j K j tie slots, as shown in ig This is also true for the artificial syste with duy workload of Phase, ie, all the arrivals in the j th interval of the artificial syste can be finished in K j K j tie slots Let A j be the nuber of arrivals in the j th interval The total flow-tie j of the work-conserving scheduler for all the arrivals in the j th interval is not greater than A j K j K j Siilarly, the total flow-tie j for all the arrivals in the j th interval of the artificial syste is not greater than à j K j K j, where Ãj is the nuber of arrivals in the j th interval of the artificial syste Let be the total flow-tie of any work-conserving scheduler, and be the iniized total flow-tie Assue that there are a total of intervals for the n arrivals Correspondingly, there are a total of intervals in the artificial syste Also, based on the construction of the artificial syste, and e j T K j is equal to the finishing tie of all the n jobs or any scheduler, the total flow-tie is at least ˆT, which is the nuber of tie slots in which the nuber of scheduled achines is at least Thus, the total flow-tie j of the optial scheduler should be at least ˆT or any scheduling policy, each job needs at least tie slot to schedule its tasks of Phase and tie slot to schedule its tasks of Phase Thus, the lower bound of the each job s flow-tie is, assuing that the Map and Reduce workload of any job is at least unit each Thus, for the optial scheduling ethod, its total flow-tie j should not be less than A j Since EM i R i >, ρ<, then and

5 e e e e ˆT à j Kj e e ˆT/ e e à j K j K j à j K j Since K j, j,, } are iid distributed rando variables, then Ãj K j, j,, } are also iid Based on Strong Law of Large Nuber SLLN, we can achieve that e e à j Kj ] wp E [Ãj Kj or Ãj K j,,, }, they are identically distributed, but not independent However, all the odd ter are independent, and all the even ter are independent Based on SLLN, we can achieve that e e à j K j wp E[Ãj K j ] 7 Based on Eqs and 7, we can achieve that e e à j Kj e à j K j wp E[Ãj K j ]E[Ãj K j ] E[E[à j K j K j ]] E[à j ]E[ K j ] E[ K j E[Ãj K j ]] E[E[Ãj K j ]]E[ K j ] λe[ K j ]E[ Kj ] or these n jobs, there are incoplete tie slots before the tie T or each incoplete tie slot, there are at ost N achines are assigned Then, N K j N s Arrivals before T } W s Thus, W s K j s Arrivals before T } N 8 T Since the workload W s of each job is iid, by SLLN, we have N λm R P K j N ρ T K j wp Thus, P K j wp, ie, } ax, N ρ wp or any work-conserving scheduler, ˆT is not less than the total nuber of tie slots with at least arrival, we have ˆT p T w p At the sae tie, for each tie slot in which the nuber of scheduled achines is greater than, there are at ost N achines are scheduled Then, we can get N ˆT n M i R i Then, i T ρ Thus, ˆT λmr N T wp λe[ K j ]E[ Kj ] } wp, N ρ axρ, p } ax λb B } wp axρ, p } ax, N ρ Siilarly, e e / j à j K j K j A j j e e e 9 à j K j K j A j With Strong Law of Large Nuber SLLN, we can get A j A j / K j K j λ } K j w p λ ax, N ρ Thus, λb B λ ax, N ρ } wp Based on the Eqs 9 and, we can achieve that the efficiency ratio of any work-conserving scheduling policy is B B ax, p λ, λ}ax, ρ N ρ } IV WORK-CONSERVING SCHEDULER IN THE NON-PREEMPTIVE SCENARIO The definitions of B, B, p,,,, j, j, K j, K j, A j, à j, W j,t, and R j,t in this section are sae as Section III Lea : Lea, Corollary, Corollary in Section III are also valid in the non-preeptive scenario Proof: Different fro preeptive scenario in Section III, the constitution of R j,t are different, because it contains two parts irst, it contains the unavailable workload of Phase which are just released by Phase which are finished in the last tie slot of each interval Second, it also

6 ig The arrivals in the j th interval finish Phase functions in K j tie slots and Phase in K j K j R ax tie slots contains the workload of Phase which cannot be processed in this tie slot, even if there are idle achines, because the tasks of Phase are non-preeptive However, we can also achieve that the reaining workload of Phase fro previous interval satisfies R j, N R ax, because the total nuber of reaining unfinished tasks of Phase and new available tasks of Phase in this tie slot is less than N All the following steps are sae Theore 3: In the non-preeptive scenario, any workconserving scheduling policy has an efficiency ratio of B B BR ax, p λ, λ}ax, ρ N ρ } Proof: In the non-preeptive scenario, in each interval, all the workload of Phase of job arrivals in this interval is also finished in this interval, and all their tasks of Phase will be started before the end of the next interval In other words, for the job i in the j th interval, the tasks of Phase are finished in K j tie slots, and the tasks of Phase are finished in K j K j R i tie slots, as shown in ig And this is also true for the artificial syste with duy workload of Phase, ie, all the arrivals in the j th interval of the artificial syste can be finished in K j K j R i So, i in the j th interval is less than A j K j K j R i Siilarly, i in the j th interval of the artificial syste is less than Ãj K j K j R i Using the sae technique as in the proof of Theore we can show that, à j K j K j n R i i K j S j à j K n j K j K j S j λ[b B B R ] } wp, ax ρ, p } ax, N ρ n n R i i and λ[b B B R ] } wp λ ax, N ρ Thus, based on the Eqs and, we obtain that any work-conserving scheduler has the efficiency ratio of B B BR ax, p λ, λ}ax, ρ N ρ } V LIGHT-TAILED DISTRIBUTED WORKLOAD O PHASE In this part, we assue the workload R of Phase are light-tailed distributed, ie, r, such that P R r α exp βr, r r, where α, β > are two constants We use the sae definitions of W j,t, R j,t and K j as Section III Lea 3: If the scheduling algorith is work-conserving, then for any given nuber H, there exists a constant B H, such that E[K H j ] <B H, j Proof: Siilarly to Eqs and 3, we can achieve P K j k P k W j,t R j, k N t Let R be the axial reaining workload of jobs in the previous interval, then k P K j k P W j,t N R k N [ P k t W j,t N R k N R r t P R r P k ] W j,t t k N Thus, we can achieve E[Kj H ] k H P K j k k k H k [P P R r k W j,t t k [k H P k N r k N k W j,t t k P R r 3 N r ] } P R r k N N r ]} k The next steps are siilar to the proof of Lea, we skip the details and directly show the following result or any <<N λm R, we know that ln > Thus, we can achieve

7 E[Kj H ] [ P R N r r H k H e kln ]} k H e kln P R r N r } H k H e kln k H e kln rr r P R r P R r k H e kln N r } H αe βr N r } H N r } H N r } H Thus, given H, E[K H j ] is bounded by a constant B H, for any j, where B H is given as below B H in,n λmr k H e kln r rr αe βr H N r H N r Corollary 3: E[K j ] is bounded by a constant B, E[Kj ] is bounded by a constant B, for any j The expressions of B and B are shown as below, where the rate function la is defined in Eq B B in,n λmr in,n λmr r N r r α αn e β csch β, e ln α e ln e ln r N r r α N r r r αn 8 sinh βcsch β α e β αn csch β eln 3e ln e ln e ln 3 Proof: By substituting H and H in Eq, we directly achieve the result by siilar steps of Corollary Reark : ollowing the sae proof steps, we can get the sae results as Theore and Theore 3 The difference is that we need use B and B instead of B and B Also, following the sae steps, we can directly extend the results to ore than phases VI SIMULATION RESULTS A Siulation Setting We evaluate the convergence of efficiency ratio for both preeptive and non-preeptive scenarios We consider a data center with N achines, and choose Poisson process with arrival rate λ jobs per tie slot as the job arrival process We choose unifor distribution and exponential distribution as exaples of bounded workload and light-tailed distributed workload, respectively or short, we use Expμ to represent an exponential distribution with ean μ, and use U[a, b] to represent a unifor distribution on a, a,, b,b} We choose the total tie slots to be T 8, and the nuber of tasks in each job is up to We choose typical schedulers to evaluate the convergence of efficiency ratio: The scheduler: Jobs with saller unfinished workload are always scheduled first Since the exact scheduler is not always feasible in the proble with phase precedence, the scheduler here only schedules the available part of the workload The scheduler: It is the default scheduler in any systes, like Hadoop All the jobs are scheduled in their order of arrival The scheduler: It is also a widely used scheduler The assignent of achines are scheduled to all the waiting jobs in a fair anner However, if soe jobs need fewer achines than others in each tie slot, then the reaining achines are scheduled to the other jobs, to avoid resource wastage and to keep the scheduler work-conserving The scheduler: Jobs with larger unfinished workload are always scheduled first Roughly speaking, the perforance of this scheduler represents in a sense how poorly even soe work-conserving schedulers can perfor B Convergence of In the siulations, the efficiency ratio of a scheduler is obtained by the total flow-tie of the scheduler over the lower bound of the total flow-tie in T tie slots Thus, the real efficiency ratio should be saller than the efficiency ratio given in the siulations irst, we evaluate the exponentially distributed workload We choose the workload distribution of Phase for each job as Exp and the workload distribution of Phase for each job as Exp The convergence of efficiency ratios of different schedulers are shown in ig 3 or different workload, we choose workload distribution of Phase as Exp3 and the workload distribution of Phase as

8 8 8 Total Tie T a Preeptive Scenario 8 8 Total Tie T b Non-Preeptive Scenario ig 3 Convergence of s Exponential Distribution, Large Reduce 8 Total Tie T a Preeptive Scenario 8 Total Tie T b Non-Preeptive Scenario ig Convergence of s Exponential Distribution, Sall Reduce Exp The efficiency ratios of schedulers are shown in ig Then, we evaluate the uniforly distributed workload We choose the workload distribution of Phase for each job as U[, 9] and the workload distribution of Phase for each job as U[, 7] The convergence of efficiency ratios of different schedulers are shown in ig To evaluate for a saller workload of Phase, we choose workload distribution of Phase as U[, ] and the workload distribution of Reduce as U[, ] The convergence of efficiency ratios of different schedulers are shown in ig ro igs 3-, we observe that the efficiency ratios of these typical work-conserving schedulers are sall constants 3 ig Reduce 8 Total Tie T 3 ig Reduce a Preeptive Scenario 8 8 Total Tie T b Non-Preeptive Scenario Convergence of s Unifor Distribution, Large 8 Total Tie T a Preeptive Scenario 3 8 Total Tie T b Non-Preeptive Scenario Convergence of s Unifor Distribution, Sall VII CONCLUSION In this paper, we study the proble of iniizing the total flow-tie of a sequence of jobs with phase precedence, where the jobs arrive over tie and need to be processed through Phase and Phase before leaving the syste Since no on-line algorith can achieve a constant copetitive ratio in soe scenarios, we define a weaker etric of perforance called the efficiency ratio and propose a corresponding technique to analyze on-line schedulers Under assuption of bounded workload of Phase, we then show a surprising property that for the flow-tie proble any work-conserving scheduler has a constant efficiency ratio in both preeptive and non-preeptive scenarios urtherore, we relax the assuption to light-tailed distributed workload of Phase The siulation results show that the efficiency ratios of typical work-conserving schedulers converge when tie goes to infinity, for both preeptive and non-preeptive scenarios, and for both bounded and light-tailed distributed workload of Phase We believe that the efficiency ratio etric and the proof technique presented here can be used for odeling and analyzing a wide range of online solutions REERENCES [] J Dean and S Gheawat, Mapreduce: Siplified data processing on large clusters, in Proc of Sixth Syposiu on Operating Syste Design and Ipleentation, OSDI, Deceber, pp 37 [] B Moseley, A Dasgupta, R Kuar, and T Sarlos, On scheduling in ap-reduce and flow-shops, in Proc of the 3rd ACM syposiu on Parallelis in algoriths and architectures, SPAA, June, pp [3] K R Baker and D Trietsch, Principles of Sequencing and Scheduling Hoboken, NJ, USA: John Wiley & Sons, 9 [] edited by Joseph Y-T Leung, Handbook of Scheduling: Algoriths, Models, and Perforance Analysis Boca Raton, L, USA: Chapan and Hall/CRC, [] M Caraia and M Drozdowski, Scheduling alleable tasks for ean flow tie criterion, Institute of Coputing Science, Poznan University of Technology, Tech Rep, [Online] Available: [] J Du, J Y-T Leung, and G H Young, Scheduling chain-structured tasks to iniizing akespan and ean flow tie, Inforation and Coputation, vol 9, no, pp 9 3, June 99 [7], Miniizing ean flow tie with release tie constraint, Theoretical Coputer Science, vol 7, no 3, pp 37 3, October 99 [8] M Zaharia, D Borthakur, J S Sara, K Eleleegy, S Shenker, and I Stoica, Job sechduling for ulti-user apreduce clusters, University of Califonia, Berkley, Tech Rep, April 9 [9] J Tan, X Meng, and L Zhang, Perforance analysis of coupling scheduler for apreduce/hadoop, in Proc of IEEE Infoco, March [] H Chang, M Kodiala, R R Kopella, T V Lakshan, M Lee, and S Mukherjee, Scheduling in apreduce-like systes for fast copletion tie, in Proc of IEEE Infoco, March, pp [] Chen, M Kodiala, and T Lakshan, Joint scheduling of processing and shuffle phases in apreduce systes, in Proc of IEEE Infoco, March [] Y Zheng, P Sinha, and N Shroff, Perforance Analysis of Work-Conserving Scheduler in Miniizing lowtie Proble with Two-Stages Precedence, Ohio State University, Tech Rep, June, zhengy/tr3pdf [Online] Available: zhengy/tr3pdf [3] A Shwartz and A Weiss, Large Deviations for Perforance Analysis: Queues, Counication and Coputing New York, NY, USA: Chapan and Hall, 99