Darbha-Agrawal-Homogeneous-Multiprocessor-Scheduling.pdf

2 starting from the exit node the latest allowable

This preview shows page 4 - 5 out of 9 pages.

2) Starting from the exit node, the latest allowable start and completion times are computed by using (7)-(9) and are shown in Table 1. 3) The array queue for this DAG is: queue = [10, 8, 5, 9, 2, 7, 3, 6, 4, 1]. Starting from the first node in queue , i.e., node 10, the first cluster is generated. The search process visits through the favorite predecessors of each task. The first processor is allocated tasks 1, 4, 6, 9, and 10. The next search is started from the first un- assigned task in queue , which is task 8. The favorite predecessor of task 8 is task 6, which has already been assigned to processor 1, and the value of last (8) ³ lact (6) is 2, which is greater than or equal to c 6,8 . Thus, task 6 is not critical for task 8, and it is not necessary to dupli- cate task 6 on the same processor as task 8. Conse- quently, task 5 is chosen to be allocated onto the same processor as task 8. The rest of the allocations are gen- erated by following this process until all tasks have been assigned to a processor. The processor allocation and the schedule times obtained by following this algorithm are shown in Fig. 5. For this example DAG, it can be observed that the schedule time generated is equal to the level of the entry node of the DAG. Thus, the TDS algorithm generates an optimal schedule for the example DAG. 3 R ESULTS In this section, the performance of TDS algorithm is re- ported. There are two cases here. In the first case, the DAG satisfies the condition stated in Section 2. If that condition is satisfied, then earliest completion time is guaranteed and is proven below. The algorithm has also been applied to some practical DAGs. In this case, the condition is not necessarily satisfied and the optimal solution may or may not be obtained. 3.1 Condition Is Satisfied A DAG consists of fork and join nodes. The fork nodes can be transformed to achieve the earliest possible schedule time as shown in Fig. 6. The problem arises when schedul- ing the join nodes, because only one predecessor can be assigned to the same processor as the join node. In this sub- section, it is proven that, for join nodes which satisfy the condition, the schedule time obtained by scheduling the join node on the same processor as its favorite predecessor, is optimal. The rest of the predecessors of the join node are each assigned to a separate processor. T HEOREM 3.1. Given a join node satisfying the condition stated in Section 2 , the TDS algorithm gives minimum possible schedule time. P ROOF . Fig. 7 shows an example join node. According to the condition, tasks m and n have the highest and the sec- ond-highest values of { ect ( j ) + c j , i j ° pred ( i )}. It is as- sumed that task m is assigned to processor m and task n is assigned to processor n . Since task m , has the highest value of ect ( m ) + c m , i , task i is also assigned to processor m , and est ( i ) = max { ect ( m ), ect ( n ) + c n , i }.
Image of page 4

Subscribe to view the full document.

Image of page 5

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern