new schedulability tests for real- time task sets scheduled by deadline monotonic on multiprocessors...
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New Schedulability Tests for Real-Time task sets scheduled by Deadline Monotonic on Multiprocessors
Marko Bertogna, Michele Cirinei, Giuseppe Lipari
Scuola S.Anna, Pisa, Italy
Overview
Real-time multiprocessing Deadline-Monotonic (DM) for
multiprocessors Existing schedulability tests for RM/DM An improved test for DM Existing schedulability bounds Improving the bound for fixed priority
global scheduling
Real-time scheduling for multiprocessor platforms Platform: identical, uniform or
heterogeneous Migration and priorities:
MIGRATIONPRIORITY
Full At job boundaries
Not allowed (partitioning)
Static RM-global, DM-global, …
RM-global, DM-global, …
RM-FFDU, DM-WFIU, …
Job-level dynamic
EDF-global, fpEDF, …
EDF, fpEDF, … EDF-FFDU, EDF-WFIU…
Unrestricted dynamic
pfair algorithms, LLF, …
not useful not useful
Multiprocessor DM
CPU1
CPU2
CPU3
Global queue(ordered by relative deadline)
The first m tasks are scheduled upon the m CPUs
Multiprocessor DM
CPU1
CPU2
CPU3
Global queue(ordered by relative deadline)
When a task finishes its execution, the next one in the queue is scheduled on the available CPU
Multiprocessor DM
CPU1
CPU2
CPU3
Global queue(ordered by relative deadline)
When a higher priority task arrives, it preempts the task with highest deadline among the executing tasks
Multiprocessor DM
CPU1
CPU2
CPU3
Global queue(ordered by relative deadline)
When another task ends its execution, the preempted task can resume its execution
Task “migrated” from
CPU3 to CPU1
Why fixed priority global scheduling?
Advantages: Load balancing Number of preemptions Simple implementation Easy rescheduling Reclaiming
Disadvantages: Cache affinity: HW mitigates migration cost Utilization bound lower than pfair algorithms
RM for uniprocessor systems
Optimality among fixed priority systems
Bounded number of preemptions
Efficient implementations
Easy sufficient schedulability test:
RM uniprocessor: necessary and sufficient test
Response Time Analysis: Repeat:
Until: Pseudopolynomial complexity
RM on multiprocessors
Low utilization bound (Dhall’s effect) Bounded number of preemptions/migrations
Good performances on average Schedulability tests (sufficient conditions):
Andersson, Baruah, Jonsson (2002) ABJ test
Baker (2003) BAK test
Efficient implementations
T
Dhall’s effect
Example: m processors, n=m+1 tasks, Di = Ti
1 ,…, m = (1,T-1) m+1 = (T,T)
RM can fail at very low utilizations
DEADLINEMISS
The ABJ test
For implicit deadline systems (Di = Ti) using RM
Linear complexityA task set is schedulable with RM on a platform with m identical processors if:
1.
2.
Total utilization
The BAK test
For constrained deadline systems (Di
Ti) Quadratic complexity
A task set is schedulable with EDF on a platform with m identical processors if:
i = f(i ,k) k = Ck /Dk
Toward a better schedulability test for RM/DM
Improve BAK when heavy tasks are considered
Extend the ABJ test: for arbitrary task utilizations for constrained deadline systems
Can BAK be improved?
Ik > (Dk-Ck)
k
k
k
DEADLINEMISS
CPU1CPU2CPU3
rkrk+Dk
Ik = Total interference suffered by task k
I2,k
I1,kI2,k
I3,kI4,k
I5,k
I6,k
I8,k
I5,k
I3,k
I7,k
I3,k
Ii,k = Interference of task i on task k
The BCL test
Ii,k > m(Dk-Ck)
k
k
k
DEADLINEMISS
CPU1CPU2CPU3
rkrk+Dk
I2,k
I1,kI2,k
I3,kI4,k
I5,k
I6,k
I8,k
I5,k
I3,k
I7,k
I3,k
IDEA: It is sufficient to consider at most the portion Dk-Ck of each term Ii,k in the sum
for all i,k: Ii,k ≤ Ik
The BCL test for DM
A task set is schedulable with DM on m processors if and only if, for every task k :
Computing each Ii,k requires exponential time
To reduce the complexity: bound the interference with the load give an upper bound on the load
Derive a sufficient condition to be checked for every task
The BCL test for DM
i = f(i ,Dk) k = Ck /Dk
Complexity is O(n2)
A task set is schedulable with DM on m processors if, for every task k :
Can ABJ be improved?
New analysis for constrained deadline systems and priorities according to DM
Improvement over ABJ: Preperiod deadline systems Arbitrary individual task utilization Higher global utilization Introduce to a better schedulability bound
for the fixed priority global scheduling class of algorithms
Density and utilization based test for RM/DM
A task set with constrained deadlines is schedulable with DM on m ≥ 2 identical processors if:
A task set with implicit deadlines is schedulable with RM on m ≥ 2 identical processors if:
Improvement over existing bounds
Bound more general than ABJ: taking we have
as ABJ.
Corrected (and extended) Baker’s bound [RTSS’03]
Existing schedulability bounds for SMPs
M=number of processorsU=worst-case total utilization
[Carpenter, Funk, Holman, Srinivasan, Anderson, Baruah]
Hybrid algorithms
Treat differently heavy and light tasks Allow to overcome Dhall’s effect
ALGORITHM RM-US[Uth]- if (Ui>Uth) task has maximum priority- else task has priority according to RM
ALGORITHM DM-DS[λth]- if (λi>λth) task has maximum priority- else task has priority according to DM
RM-US[1/3] and DM-DS[1/3]
A task set with constrained deadlines is schedulable with DM-DS[1/3] on m ≥ 2 identical processors if:
A task set with implicit deadlines is schedulable with RM-US[1/3] on m ≥ 2 identical processors if:
Existing schedulability bounds for SMPs
M=number of processorsU=worst-case total utilization
[Carpenter, Funk, Holman, Srinivasan, Anderson, Baruah]
Conclusions
Extended BAK test for DM: BCL test that better behaves with heavy tasks
Improved ABJ test: generalized to constrained deadline systems extended to arbitrary task utilizations/densities increased the schedulability bound for RM/DM
Proposed hybrid algorithms (RM-US, DM-DS): improved the schedulability bound of the fixed
priority global scheduling class of algorithms
