Two-type Heterogeneous Multiprocessor Scheduling: Is there a Phase Transition?

1. Two-type Heterogeneous Multiprocessor Scheduling: Is there a Phase Transition? Gurulingesh Raravi, Björn Andersson and Konstantinos Bletsas CISTER-ISEP Research Centre Polytechnic Institute of Porto 1

2. Introduction • System Model • Computing Platform • Two-type Heterogeneous platform • A platform with two unrelated processor types • Task set • implicit-deadline sporadic tasks • Assumptions • Independent Tasks • No Migrations • No job parallelism 2

3. Introduction • Phase Transition • Transition of a system from one state to another upon changing some system parameters 3

4. Introduction • Phase Transition • Transition of a system from one state to another upon changing some system parameters • Phase Transition in Real-Time Scheduling • Transition of a system from “almost surely schedulable” state to “almost surely not schedulable” state upon changing the task set characteristics 4

5. Introduction • Phase Transition • Transition of a system from one state to another upon changing some system parameters • Phase Transition in Real-Time Scheduling • Transition of a system from “almost surely schedulable” state to “almost surely not schedulable” state upon changing the task set characteristics • Uni-processor Rate Monotonic Scheduling: • U*RM: Utilization threshold • U(τ) ≤ U*RM then τ is almost surely schedulable • U(τ) > U*RM then τ is almost surely not schedulable • Identical multiprocessor scheduling 5

6. The Problem • Does there exist a phase transition behavior for the two-type heterogeneous multiprocessor scheduling problem? • Is there a threshold for a parameter (or combination of parameters) which classifies the task set from “almost surely schedulable” state to “almost surely not schedulable” state 6

7. Some Insights • Simulations and Observations: • Simulation setup 7

8. Some Insights at most 15 tasks and 4 processors (2 of each type) • Simulations and Observations: • Simulation setup Generate a random problem instance 8

9. Some Insights at most 15 tasks and 4 processors (2 of each type) • Simulations and Observations: • Simulation setup Generate a random problem instance Using ILP formulation Is there a feasible assignment? Z<=1 9

10. Some Insights at most 15 tasks and 4 processors (2 of each type) • Simulations and Observations: • Simulation setup Generate a random problem instance Using ILP formulation Is there a feasible assignment? Z<=1 NO 10

11. Some Insights at most 15 tasks and 4 processors (2 of each type) • Simulations and Observations: • Simulation setup Generate a random problem instance Using ILP formulation Is there a feasible assignment? Z<=1 NO Using Exhaustive Enumeration: ratio = Nsucc/Nvalid YES Compute the “success ratio” 11

12. Some Insights at most 15 tasks and 4 processors (2 of each type) • Simulations and Observations: • Simulation setup Generate a random problem instance Using ILP formulation Is there a feasible assignment? Z<=1 NO Using Exhaustive Enumeration: ratio = Nsucc/Nvalid YES Compute the “success ratio” • Repeat till 10000 feasible task sets are found 12

13. Some Insights • Simulations and Observations: • Observations • Plotted for 10000 feasible task sets 13

14. Some Insights • Simulations and Observations: • Observations (for 10000 feasible task sets) • Observations: • No sharp threshold • Fluctuations/peaks in the range 0 ≤ Z ≤ 0.4 is probably due to imbalanced task generation 14

15. The Question • Observations (for 10000 feasible task sets) • Questions: • Is there a phase transition? • Yes: What parameters should we observe? • No: What is its implication? • considering such a behavior has been observed for: • uni-processor (RM) and identical multiprocessor scheduling • Any insights will be useful  15

16. Few Questions • Observations (for 10000 feasible task sets) • Questions: • Is there a phase transition? • Yes: What parameters should we observe? • No: What is its implication? • Since such a behavior has been observed for: • uni-processor and identical multiprocessor scheduling • Any insights will be useful  Thank You ! 16