HASSO-PLATTNER-INSTITUT für Softwaresystemtechnik GmbH an der Universität Potsdam Multiprocessor Scheduling “Global Multiprocessor Scheduling of Aperiodic Tasks using Time- Independent Priorities” Alexander Küchler

Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 2 Agenda  Definitions  Previous work  Problems, assumptions and goals  Results  ‘Calculating’ U threshold  Strategy for using the bound  References

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 3 Definitions I  Global/dynamic scheduling = a process can be migrate from one processor to another at run- time (assumption: no penalty for migration)  Aperiodic tasks = subgroup of periodic tasks, where no statements for arrival can be made  Time-independent priorities = assigned priority does not depend of its absolute arrival time  Preemptive scheduling = a task can be interrupted and continued later

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 4 Definitions II  Multiprocessor with m processors and a set of n independent tasks  Each task T n has an arrival time A n a worst-case execution time C n a deadline D n, a synthetic utilization U n =C n /D n a priority P n  Total utilization U is the sum of all U n of tasks that have arrived but not yet reached their deadline

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 5 Definitions III  V(t) is a task set with A i ≤t<A i +D i  Synthetic utilization is

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 6 Previous work I  (1) shows categorization of time-independent and non-time-independent scheduling policies for aperiodic tasks  (2) shows optimal utilization bound of for liquid tasks (C i  0 and C i /D i  0) on multiprocessor machines  (3) shows avoidance of Dhall’s effect for periodic tasks by dividing tasks into two categories (U i U threshold (m) = heavy; U i U threshold (m) = light)

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 7 Previous work I – Dhall’s effect

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 8 Previous work II  (4) shows similar problem for periodic tasks with a result of ln(1+y)=(1-y)/(1+y) which is approx. 37,482% for m  

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 9 Problems, assumptions and goals  Problems: Dhall’s effect shows that deadline monotonic priority assignment may lead to missed deadlines even if U is very low  Assumptions: Preemptive scheduling is possible Time-independent priorities for each task (depending on C and D, not A!)  Goals: All tasks in the task set meet their deadlines

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 10 Results I  Optimal utilization is a function of the number of processors m: with  U threshold has two meanings: threshold for synthetic utilization in the admission control of incoming tasks: accept or reject threshold for categorization of light and heavy tasks

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 11 Results II  U threshold is an always valid bound (but not always optimal for all multiprocessors)  Generalize previous single-processor results [1] to multiprocessors  Extend previous multiprocessor results by considering not only liquid tasks [2] which are a special case of the generalized formulas

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 12 ‘Calculating’ U threshold I  The idea behind the calculation is that one: compute US(t) for a worst-case critically schedulable task pattern this depends on the number of processors and is a function of x=U n =C n /D n  The calculation based upon one lemma and two theorem’s …

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 13 ‘Calculating’ U threshold II Lemma 1: A task T x and a task pattern where each T i that has higher priority than T x has A i ≥A x can be modified to a task pattern with same US(t) and block interference on T x

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 14 ‘Calculating’ U threshold II (Lemma 1)

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 15 ‘Calculating’ U threshold II (Theorem 1) Theorem 1: There is a worst-case critically schedulable task pattern, such that no task T i in this task pattern has an arrival time A i <A n : that means that we can disregard all tasks with an early arrival, because they do not influence the block interference

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 16 ‘Calculating’ U threshold II (Theorem 1)

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 17 ‘Calculating’ U threshold II (Theorem 1)

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 18 ‘Calculating’ U threshold II (Theorem 1)

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 19 ‘Calculating’ U threshold II (Theorem 2) Theorem 2: Create m copies of each task  disregarding T n, the load on each processor is identical  reduce of the problem to a single-processor case

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 20 ‘Calculating’ U threshold III Important: by splitting each task into m new tasks block interference from high priority tasks occurs [1] shows the worst-case how to handle the problem An quadratic equation will be solved to get an optimal utilization bound for the single-processor case from [1] Regard for the problem of heavy tasks: m heavy tasks with large D may lead to missed deadlines  priority assignment strategy to avoid Dhall’s effect:  1. consider high utilization  2. consider short deadline

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 21 Strategy for using the bound I  Accept incoming tasks when current An accepted task T i is light if  Use m: accept incoming tasks when current An accepted task T i is light if

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 22 Strategy for using the bound II  Keep track of among current tasks: accept incoming tasks if for liquid tasks: accept for [2]  In admission test: for very hard tasks with increase US(t) with only in the interval A i ≤t≤A i +D i, so you reduce synthetic utilization and can accept more tasks

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 23 References (1)“Schedulability Analysis and Utilization Bounds for Highly Scalable Real- Time Services”; Abdelzaher/Lu (2)“The Aperiodic Multiprocessor Utilization Bounds for Liquid Tasks”; Abdelzaher et.al (3)“Static-priority scheduling on multiprocessors”; Andersson/Baruah/Jonsson (4)“Analyzing Fixed-Priority Multiprocessor Scheduling”; Lundberg

Alexander Küchler Multiprocessor Scheduling Prof. Dr. Lars Lundberg, Prof. Dr. Andreas Polze 24 Questions? Thank you very much for your attention. Any questions left? Ask me!