Global Fixed Priority Pre-emptive Scheduling:
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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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Robert Davis Real-Time Systems Research Group, University of York

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Robert davis real time systems research group university of york

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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P1

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P2


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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    1

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    P1

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    2

    P2

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    P2

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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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