New Schedulability Tests for RealTime 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:
MIGRATION
PRIORITY
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
Global queue
(ordered by relative deadline)
t5
t4
t3
t2
t1
CPU1
t1
CPU2
t2
CPU3
t3
The first m tasks are scheduled upon the m CPUs
Multiprocessor DM
Global queue
(ordered by relative deadline)
t5
t54
t43
t2
t1
CPU1
t1
CPU2
t2
CPU3
t43
When a task finishes its execution, the next one in
the queue is scheduled on the available CPU
Multiprocessor DM
Global queue
(ordered by relative deadline)
t5
t45
t34
t2
t1
CPU1
t1
CPU2
t2
CPU3
t34
t3
When a higher priority task arrives, it preempts the
task with highest deadline among the executing tasks
Multiprocessor DM
Global queue
(ordered by relative deadline)
t5
t54
t43
t32
t21
Task t4 “migrated” from
CPU3 to CPU1
CPU1
t41
CPU2
t2
CPU3
t34
When another task ends its execution, the
preempted task can resume its execution
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

Efficient implementations

Good performances on average

Schedulability tests (sufficient conditions):

Andersson, Baruah, Jonsson (2002)  ABJ test

Baker (2003)  BAK test
Dhall’s effect
Example: m processors, n=m+1 tasks, Di = Ti
t1 ,…, tm = (1,T-1)
tm+1 = (T,T)
DEADLINE
MISS
T
RM can fail at very low utilizations
The ABJ test


For implicit deadline systems (Di = Ti) using RM
Linear complexity
A 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:
bi = f(ti ,tk)
lk = 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?
DEADLINE
MISS
CPU3
CPU2
CPU1
rk
tk
I3,k I1,k
I2,k I5,k
I4,k I3,k
tk
I3,k
I6,k
I5,k
I2,k
I7,k I8,k
Ik > (Dk-Ck)
tk
rk+Dk
Ik = Total interference suffered by task tk
Ii,k = Interference of task ti on task tk
The BCL test
DEADLINE
MISS
CPU3
CPU2
CPU1
rk
tk
I3,k I1,k
I2,k I5,k
I4,k I3,k
tk
I3,k
I6,k
I5,k
I2,k
I7,k I8,k
SIi,k > m(Dk-Ck)
tk
rk+Dk
for all i,k: Ii,k ≤ Ik
IDEA: It is sufficient to consider at most the portion Dk-Ck
of each term Ii,k in the sum
The BCL test for DM
A task set is schedulable with DM on m processors
if and only if, for every task tk :


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
A task set is schedulable with DM on m processors
if, for every task tk :
bi = f(ti ,Dk)
Complexity is O(n2)
lk = Ck /Dk
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 processors
U=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 implicit deadlines is schedulable with
RM-US[1/3] on m ≥ 2 identical processors if:
A task set with constrained deadlines is schedulable
with DM-DS[1/3] on m ≥ 2 identical processors if:
Existing schedulability bounds for
SMPs
M=number of processors
U=worst-case total utilization
[Carpenter, Funk, Holman, Srinivasan, Anderson, Baruah]
Conclusions

Extended BAK test for DM:


Improved ABJ test:




BCL test that better behaves with heavy tasks
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
THE END
Marko: [email protected]
Michele: [email protected]
Peppe: [email protected]