Robert Davis Real-Time Systems Research Group, University of York

1. Global Fixed Priority Pre-emptive Scheduling:What is the arrival pattern that leads to the worst-case response time? Robert Davis Real-Time Systems Research Group, University of York

2. Question scope • Homogeneous Multiprocessor Real-Time Systems • Global scheduling • Single global run-queue • Pre-emption and migration • Fixed priority scheduling • Tasks have unique priorities • All jobs of a task have the same fixed priority • What pattern of job arrivals leads to the worst-case response time for a particular task?

3. Basic task model • Task model • Static set of n tasks tk with priorities 1..n (1 is the highest) • Bounded worst-case execution time Ck • Minimum inter-arrival time or period Tk • Relative deadline Dk • implicit-deadline Dk = Tk • constrained-deadline DkTk • arbitrary-deadline • Independent • Each task gives rise to a potentially infinite sequence of jobs • Worst-case response time Rk is the longest time from arrival to completion of any job of task tk for any valid arrival pattern

4. System model • Multiprocessor system • m identical processors • Global fixed priority pre-emptive scheduling • At any given time, the m highest priority ready jobs execute • Migration is permitted, but a job can only execute on one processor at a time

5. Task models • Concrete periodic tasks with synchronous initial release • First job of every task arrives at time 0 • Subsequent jobs arrive strictly periodically Tk after the arrival of the previous job of the same task • Non-Concrete periodic tasks • Arrival times of the first job of each task is unknown • Subsequent jobs arrive strictly periodically Tk after the arrival of the previous job of the same task • Sporadic tasks • Arrival times of all jobs are unknown a priori • Subsequent jobs may arrive at any time once a minimum inter-arrival time Tk has elapsed since the arrival of the previous job of the same task

6. What is known? • Global FPPS is a completion time predictable algorithm • [Ha and Liu 1994] showed that later completion times (or longer response times) cannot be obtained by reducing the execution time of any job • Worst-case response time for concrete periodic tasks with synchronous /asynchronous initial release • [Cucu and Goossens 2006, 2007] showed that worst-case response times can be determined by simulating the schedule over the LCM of task periods (or longer in the arbitrary deadline / asynchronous case) and using worst-case execution times • However the LCM can be very large…

7. 1 2 3 4 4 4 What is known? • Synchronous arrival sequence does not necessarily result in the longest response time (unlike in the Uniprocessor case) [Lauzac et al. 1998] • Lower priority tasks have no effect on when higher priority tasks execute • So we can think in terms of when a lower priority task can and cannot run 1 3 P1 2 4 P2

8. Recent work • Sporadic and non-concrete periodic tasksets (Generalises a more specific result in [Guan et al. 2009]) • Theorem: A worst-case response time for task tk occurs when arrival of a job of task tk at time t is coincident with all m processors becoming busy with higher priority tasks (i.e. in the time interval [t-ε,t)not all m processors were busy) • Proof: (By contradiction) Assume that the worst-case response time occurs for a job of task tk which arrives at time x, not compliant with the theorem, and this response time is strictly greater that that for any such arrival at a compliant time t.

9. Recent work • Case 1: At time x, not all m processors are busy with higher priority tasks. We can move the arrival time forward to the next compliant time t without decreasing the response time • Case 2: At time x, all m processors are busy with higher priority tasks and have been busy since the last compliant time t. We can move the arrival time back to last compliant time t without decreasing the response time Case 1: Response time Case 2: Response time

10. Recent work • This theorem tells us something useful about the pattern of arrivals that leads to the worst-case response time • Specific case used to improve sufficient schedulability tests [Guan et al. 2009] • State-of-the-art sufficient tests are still pessimistic with respect to • The amount of higher priority task execution falling within an interval • The time for which that amount of higher priority task execution occupies all m processors

11. Research Question • What is the arrival pattern of higher priority tasks that leads to the worst-case response time? 1 3 1 1 4 P1 2 5 2 P2 4 6 3 P2 7 Response time

12. Research Question • Fully determining this arrival pattern provides an exact schedulability test • No exact test is currently known for non-concrete periodic tasksets, except in theory for simulating all distinct combinations of arrival patterns over the LCM, which is intractable for any reasonable sized examples • No exact test is currently known for sporadic tasksets • Any properties of the arrival pattern that we can derive provide an opportunity to improve upon existing sufficient schedulability tests for global FPPS