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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. Question scope. Homogeneous Multiprocessor Real-Time Systems Global scheduling Single global run-queue

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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

question scope
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?
basic task model
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
system model
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
task models
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
what is known
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…
what is known1

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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

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recent work
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.
recent work1
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

recent work2
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
research question
Research Question
  • What is the arrival pattern of higher priority tasks that leads to the worst-case response time?

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Response time

research question1
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
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