1. Progress Report 07/30

2. Virtual Core Scheduling Problem For every time period, the hypervisor scheduler is given a set of virtual cores with their operating frequency. Generate a scheduling plan, such that the power consumption is minimized, and the performance is guaranteed. 2

3. Scheduling Plan A set of a i,j which indicates the amount of time executing virtual core j on physical core i in a time interval. The execution order of each virtual core on a physical core. 3

4. Current Solution Two phases: ◦ Use integer programming to find a feasible set of a i,j. ◦ Decide the execution order on each physical core.  A virtual core cannot appear in two or more physical core on the same time.

5. Example vCPU 0 (50,40,0, 0) t=100 t=0 vCPU 3 (10,10,20, 20) vCPU 1 (20,20,20, 20) vCPU 4 (10,10,10, 10) vCPU 2 (10,10,20, 20) vCPU 5 (0, 0,10, 10)

6. Open-shop Problem “Open-shop scheduling problem (OSSP) is a scheduling problem in which a given set of jobs must each be processed for given amounts of time at each of a given set of workstations, in an arbitrary order, and the goal is to determine the time at which each job is to be processed at each workstation.” [1] [1] Open-shop scheduling: http://en.wikipedia.org/wiki/Open_shop_scheduling

7. Open-shop Problem(Cont.) O2||C max can be solved in polynomial time. ON||C max (N > 2) ◦ May be solved in polynomial time when all nonzero processing times are equal. ◦ Otherwise, NP-hard problem. O|pmtn| C max ◦ With preemption

8. Open-shop Model Define: ◦ Load of machine i ML i = ◦ Load of job j JL j = ◦ Lower bound on C max LB = max{max m i=1 ML i, max n j=1 JL j } ◦ tight: job j(machine i) with JL j (ML i )= LB ◦ slack: not tight, JL j (ML i )< LB

9. Open-shop Model(Cont.) “Decrementing Set” D ◦ For each tight job and machine, exactly one operation. ◦ For each job and machine with slack, at most one operation. ◦ An operation is a job-machine pair.  Job running on the machine.

10. Algorithm O|pmtn|C max REPEAT ◦ Calculate a decrementing set D ◦ Calculate a maximum value △ with  △ ≦ min (i,j) ∈ D p ij  △ ≦ LB – ML i if i has slack and no operation in D  △ ≦ LB – JL j if j has slack and no operation in D ◦ schedule the operations in D for △ time units in parallel ◦ Update p, LB, ML, JL UNTIL all operations have been scheduled

11. Example

12. Summary The second part of our problem can be formulated into Open-shop problem. ◦ Since virtual cores can preempt each other, we can apply algorithm O|pmtn| C max to find a feasible scheduling in polynomial time.