1. Efficient Online Schedulability Tests for Real-Time Systems
Tei-Wei Kuo, Senior Member, IEEE, Li-Pin Chang,
Yu-Hua Liu, and Kwei-Jay Lin, Senior Member, IEEE

2. Schedulability Test
Liu & Layland first introduced the concept of achievable utilization factor U
Online v.s. offline
Kuo & Mok, Han & Tyan,

3. Utilization factor Proposed by Liu & Layland Under the process model
Preemptive processes
Periodically request
Parameters : (Ci, Pi)
U = Σ (Ci/Pi)
Sufficient condition for schedulablily test

4. Schedulability Test Based on the Liu and Layland Model
Some definition
Division Graph
Offspring set
Root Reduced set & RSR

5. Division Graph

7. Offspring set

8. Root

9. Reduced Set

10. Example {τi | i = 1~5 } = {3, 5, 15, 20, 60} Root :60
Sevreal offspring of τ5 , e.g. {3,5}, {3,5,60}
For {τ1,τ2, τ3} , τ (which period is τ3 and utilization factor is equal to the sum of utilization factor of τ1,τ2, τ3) is an RSR of {τ1,τ2, τ3} .
{τ1,τ2, τ3} is a reduced set of τ

11. Schedulability Theorems

12. Schedulability Theorems

15. Schedulablility Test Algorithm

16. example
T3 is schedulable according to utilization bound, but T4, T5 is not
guaranteed schedulable because U4 > U(4), U5 > U(5), where Ui
Is the utilization factor of Ti and U(i) is the achievable utilization
of i processes
Actually, T4 and T5 is schedulable ! Why and How ?

17. Schedulability test for multiframe model
Multiframe ? τ: ((C0, C1, … CN), pi)
Peak execution time
Peak utilization factor

18. AM Multiframe model
AM means Accumulative monotonic

19. AM multiframe model Critical instant : same as LL model
Root & offspring : same as LL model
RSR :