1. How Many Boundaries Are Required to Ensure Optimality in Multiprocessor Scheduling? Geoffrey Nelissen Shelby Funk Dakai Zhu Joёl Goossens

2. Optimal Multiprocessor Scheduling Boundary-based schedulers have two distinct phases – During the boundary phase execution times are determined – During the execution phase scheduling is done using these execution times All known optimal scheduling algorithms are boundary-based – E.g., Pfair, BF, LLREF, DP-Fair, EKG

3. Example 6 boundaries 16 preemptions 12 migrations 1 boundary 8 preemptions 8 migrations

4. Why Use Boundary-Based Scheduling? In general, optimal scheduling algorithms do not exist for multiprocessor platforms – When all deadlines are equal, optimal algorithms do exists E.g., McNaughton’s wrap-around algorithm Using boundaries, all sub-jobs to share deadlines Allocated execution “reasonably close” to the fluid schedule – While execution may diverge from the fluid schedule between boundaries, the divergence is bounded at all boundaries

5. The Problem With Boundary-Based Schedules All boundary-based schedules break jobs into pieces This introduces scheduling overhead due to preemptions and migrations Removing boundaries can reduce preemptions and migrations Question: How many boundaries can we remove and still maintain optimality?

6. Conjecture Assume τ is a periodic task set that is feasible on m processors and contains at most 2m-1 tasks. One task is statically allocated to each processor. Remaining tasks are permitted to migrate. There exists an optimal algorithm that can schedule τ using boundaries only at the migratory task deadlines.

7. Example

8. Questions Is the conjecture true? – If so, what is the optimal algorithm? Does the conjecture continue to hold if there are more than 2m-1 tasks? Can we reduce the number of boundaries using an efficient scheduling algorithm, or is this a lower bound for optimal scheduling algorithms? Can this idea be extended to sporadic task sets?