1. 1 Uwe Schwiegelshohn, 2 Andrei Tchernykh, 1 Ramin Yahyapour 1 Technische Universität Dortmund, Germany uwe.schwiegelshohn@udo.edu, ramin.yahyapour@udo.edu 2 CICESE Research Center, Ensenada, Baja California, Mexico chernykh@cicese.mx CIRM-Marseille-Luminy, May 12 - 16, 2008 Online Scheduling in Grids

2. Computational Grid 2CICESE Parallel Computing Laboratory (by Christophe Jacquet)

3. Grid Model 3CICESE Parallel Computing Laboratory An encompassing and precise representation of a Grid is usually too complex to address various problems occurring in Grids. Application of a suitable model that considers important properties of a Grid. Important propertiesUnconsidered properties Heterogeneity of Grid (Machines with different numbers of identical processors) Heterogeneity of processors Fixed job parallelism (rigid jobs)Variable parallelism Non clairvoyant schedulingEstimation of processing times Online operation Advance reservation, multi-site allocation Utilization (makespan)User priorities

4. Grid Model 4CICESE Parallel Computing Laboratory Each job J is described by a triple : release date, size (degree of parallelism), execution time on processors. Job must be executed on processors on one machine without interruption (space sharing mode). GP m | size j | C max P m | r j, size i | C max is referred to as PS while the scheduling on a set of parallel machines GP m | r j, size i | C max is referred to as MPS. The Grid contains m machines. Machine M i has size m i if it comprises m i processors. All processors in the Grid are identical.

5. List Scheduling Processors Time Non clairvoyant scheduling CICESE Parallel Computing Laboratory5

6. Processors Time C max (LIST)=17 C max *=9 CICESE Parallel Computing Laboratory6 List Scheduling

7. C max (LIST)/C max * ≤ 2-1 / m –All jobs are sequential and have release date 0. Graham 1966 –Jobs have release date 0 and may be parallel. Garey, Graham 1975 –Jobs are parallel and submitted over time (online scheduling) Naroska, Schwiegelshohn 2002 Does the same bound hold for Grids as well? CICESE Parallel Computing Laboratory7 List Scheduling on Parallel Processors

8. CICESE Parallel Computing Laboratory8 Applicability to Grids There is no polynomial time algorithm that always produces schedules S with C max (S)/C max ∗ < 2 for GP m | size i | C max and all input data unless P = NP.

9. Applicability to Grids 2 machines with m processors each All jobs have processing time 1 and different degrees of parallelism –Total requirement of all jobs: 2m processors Consider an arbitrary algorithm A. machine 2machine 1 C max (A)=1  C max *=1: optimal solution C max (A)=2 and C max *=2: optimal solution C max (A)=2 and C max *=1: optimal solution machine 2machine 1 CICESE Parallel Computing Laboratory9

10. How do we know whether C max *=2 applies? –Partition: NP-hard –There is no algorithm A with polynomial time complexity guaranteeing C max (A)/C max * < 2. Scheduling in Grids is inherently more difficult than scheduling on a single parallel processor. CICESE Parallel Computing Laboratory10 Applicability to Grids

11. Time Machines with different numbers of processors C max (LIST)=4 CICESE Parallel Computing Laboratory11 List Scheduling in the Grid

12. Time Machines with different numbers of processors C max *=2 CICESE Parallel Computing Laboratory12 List Scheduling in the Grid

13. C max (LIST)/C max * = (k+1)/2 Analysis of the problem –Jobs with little parallelism occupy large machines which are not available for highly parallel jobs. –In case of few highly parallel jobs it is inefficient to prevent jobs with little parallelism from using these large machines. Simple approach –Increased priority for highly parallel jobs –Arranging jobs in descending order of their parallelism Fairness is neglected. CICESE Parallel Computing Laboratory13 Problems of List Scheduling

14. Sorting in Order of Parallelism Processors Time Predominantly execution of sequential jobs Few available processors for parallel jobs CICESE Parallel Computing Laboratory14

15. Does Ordering the Jobs Help? We are interested in an algorithm that does not use a single list of jobs. –Some machines are blocked from executing some jobs under certain circumstances. CICESE Parallel Computing Laboratory15

16. Online Job Stealing Scheduling in Grids

17. Does Ordering the Jobs Help? We assume a machine indexing such that m i−1 ≤ m i holds Three sets of jobs are considered o Set Ai contains all jobs that cannot execute on the previous (next smaller) machine and require more than 50% of the processors of machine Mi. o Set Bi contains all jobs that cannot execute on the previous machine but require at most 50% of the processors of machine Mi. o Set Hi contains all jobs that require more 50% of the processors of machine Mi but can also be executed on the previous machine. CICESE Parallel Computing Laboratory17

18. Grid Scheduling Algorithm 2. A job is assigned to the first machine that can execute it. Group A: >= half of the processors on this machine are required. Group B: < half of the processors on this machine are required. 1. The machines are arranged in ascending order of processor numbers. CICESE Parallel Computing Laboratory18

19. 3. Any machine applies a priority order when selecting jobs for execution: Jobs of its group A Jobs of its group B Jobs that are enabled for execution on its previous machine. CICESE Parallel Computing Laboratory19 Grid Scheduling Algorithm

20. Theoretical evaluation –C max (LIST)/C max * < 3 in the offline case –C max (LIST)/C max * < 5 in the online case U.Schwiegelshohn, A.Tchernykh, R.Yahyapour Online Scheduling in Grids. IEEE, IPDPS’08, 2008 CICESE Parallel Computing Laboratory20 Performance of the Algorithm

21. Conclusion Common list scheduling does not work well in Grids. Jobs should receive priority on the machines that provide the right amount of parallelism. Jobs with less parallelism are only executed on these machines if better suited jobs are not available. The presented algorithm has a constant worst case bound and relatively small gap. CICESE Parallel Computing Laboratory21

22. Adaptive Admissible Allocation

23. Two Level Grid Model 23CICESE Parallel Computing Laboratory Grid Workload Broker Allocation Local queue Local scheduler node We regard MPS as two stage (two layer) scheduling MPS = MPS_Allocation + PS.

24. Allocation 24CICESE Parallel Computing Laboratory For each job: first be the minimum i such that node is able to execute a job. last is the maximum i set of nodes first, first+1,..., last is a set M-available. … m1m1 m2m2 m3m3 m4m4 m5m5m first(J j ) = 2last(J j ) = m M-available

25. 25CICESE Parallel Computing Laboratory … m1m1 m2m2 m3m3 m4m4 m5m5m first(J j ) = 2 last(J j ) = m M-available M-admis last(J j ) = 5 If last is the minimum r such that Allocation

26. 26CICESE Parallel Computing Laboratory 1 f 0 f l 0 l m a*m(f,m) (1-a)*m(f,m) a*m(f 0,m)(1-a)*m(f 0,m)

27. 27CICESE Parallel Computing Laboratory For a set of machines with identical processors, and for a set of rigid jobs with admissible range the competitive factor of Min_LB-a + Best_PS is for for

28. 28CICESE Parallel Computing Laboratory Competitive factor

29. 29CICESE Parallel Computing Laboratory Competitive factor

30. 30CICESE Parallel Computing Laboratory Competitive factor A.Tchernykh, U.Schwiegelshohn, R.Yahyapour, N.Kuzurin. Online Hierarchical Job Scheduling in Grids. IEEE, CoreGrid’08, EuroPar, 2008

31. Thank you