Soft Real-Time Semi-Partitioned Scheduling with Restricted

Migrations on Uniform Heterogeneous Multiprocessors

Kecheng YangJames H. Anderson

Dept. of Computer ScienceUNC-Chapel Hill

Motivation

•Soft real-time (SRT) semi-partitioned scheduling algorithms with restricted migrations have been previously proposed for homogeneous multiprocessors.

•Heterogeneous multiprocessors are emerging. E.g., the big.LITTLE technology from ARM.

image source: http://www.arm.com/Big.Little_New.gif

Contributions

•EDF-sh ( semi-partitioned scheduling for heterogeneous multiprocessors) is the first SRT scheduling algorithm with restricted migrations for heterogeneous multiprocessors.

•Develop a schedulability condition.• Improve the task assignment strategy.•Derive tardiness bounds.• Evaluate the schedulability and tardiness bounds.

Multiprocessor Platforms Categories• Identical: All processors of speed 1.0.

•Uniform Heterogeneous: Processor p has speed sp.

•Unrelated Heterogeneous: Task i executes with speed sp,i on processor p.

We assume there are m processors indexed by speed, i.e., s1 ≥ s2 ≥ … ≥ sm.

This paper

Real-Time Tasks

•Task model: sporadic tasks with implicit deadlines.• Task i is specified by (Ci, Ti), where • Ci is defined by its worst-cast execution time (WCET) on a

unit-speed processor (i.e., if it entirely executes on processors p, its WCET would be Ci/sp), • and Ti is it period. • Its utilization is defined as ui = Ci/Ti, which can be greater

than 1 if there is a processor with a speed greater than 1.•Also, we assume there are n tasks indexed by

utilization, i.e., u1 ≥ u2 ≥ … ≥ un.

Ci

Ci

0 Ci

timeC i2

if Sp=2

if Sp=1

Conditions

Feasibility Schedulability

∑𝑖=1

𝑛

𝑢𝑖≤∑𝑖=1

𝑚

𝑠𝑚 ∑𝑖=1

𝑛

𝑢𝑖≤∑𝑖=1

𝑚

𝑠𝑚

∑𝑖=1

𝑘

𝑢𝑘≤∑𝑖=1

𝑘

𝑠𝑘 ∑𝑢𝑖>𝑠 𝑘

𝑢𝑖≤ ∑𝑠𝑝>𝑠 𝑘

𝑠𝑝

for k = 1, 2, …, m for k = 1, 2, …, m

Processor p is “fast enough” for task i if sp≥ui.

a necessary condition for only executing tasks on fast enough processors without overutilizing them.

s1 = 3, s2 = 1;u1 = 2, u2 = 2.

Semi-Partitioned Scheduling• In semi-partitioned scheduling,

most tasks are partitioned, or fixed, onto processors, but those tasks that cannot be feasibly partitioned are allowed to migrate.

•Our algorithm EDF-sh is such a semi-partitioned scheduling algorithm.

EDF-sh

•EDF-sh has two phases, an assignment phase and an execution phase.

•Assignment phase:• Consider tasks from heaviest to lightest.• If the considered task can be fixed then fix it (fixed task).• Otherwise allocate shares on processors from fastest to

slowest to match its utilization (migrating task).

τ4=(4,3)

100%

Exampleτ1=(3,1)

τ2=(11,6)τ3=(5,3)

τ5=(1,2

)

τ7=(1,3)τ4=(4,3) τ7=(1,3

)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

Both tasks and processors are indexed decreasingly w.r.t utilization or speed.

Fix a task when possible

Migrating task are assigned to processors from fastest to slowest

τ4=(4,3)τ7=(1,3)τi=(Ci,Ti): τ1=(3,1), τ2=(11,6), τ3=(5,3), τ4=(4,3), τ5=(1,2), τ6=(2,6), τ7=(1,3).

τ1=(3,1)τ2=(11,6) τ3=(5,3)

τ5=(1,2

)

τ4=(4,3)τ7=(1,3)τ4=(4,3) τ7=

(1,3)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

100%

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

ExampleKey properties for the assignment:1. On any single processor, there are at most two migrating tasks.2. A task will not assign a share to a not-fast-enough processor.

Processor p is “fast enough” for task i if sp≥ui.

Restricted Migrations

•Fixed tasks do not migrate.

•Migrating tasks do migrate, but the migrations can only happen on job boundaries.•We assign jobs to processors.

I.e., a migrating task can migrate among processors;however, a single job can only execute on the processor it was assigned to.• Given the shares of a migrating task on each processor, we

have a mechanism to systematically assign the jobs of this task to those processors, maintaining the long-term workload on each processor to match its corresponding share.

If a migrating task has been assigned one half to processor 1 and one half to processor 2, then we just need to assign all the odd-indexed jobs to processor 1

and all the even-indexed jobs to processor 2.

τ1=(3,1)τ2=(11,6) τ3=(5,3)

τ5=(1,2

)

τ4=(4,3)τ7=(1,3)τ4=(4,3) τ7=

(1,3)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

100%

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

Example the one-half-one-half case we discussed in the prior slide

For τ4, every 8 jobs, 6 → Processor 1, 1 → Processor 2,1 → Processor 3.

EDF-sh

•Execution phase:•Migrating tasks are statically prioritized over fixed tasks.

• Fixed tasks are prioritized against each other on by EDF.

• On a migrating task's last processor (the highest-indexed processor that has non-zero share of this migrating task), its priority is lower than the other migrating task, but still higher than fixed ones.

τ1=(3,1)τ2=(11,6) τ3=(5,3)

τ5=(1,2

)

τ4=(4,3)τ7=(1,3)τ4=(4,3) τ7=

(1,3)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

100%

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

Example High PriorityThe last processor of τ4.

Priority: τ7 >τ4 >fixed ones.

Low Priority

Tardiness Bounds

•The tardiness bounds are computed inductively. The order is as follows.•Migrating task(s) that do not share their last processor

with other migrating tasks. (base case)•Migrating task(s), if on their last processor, the bound for

the other migrating task has already been computed. (inductively)• Fixed task(s).

τ1=(3,1)τ2=(11,6) τ3=(5,3)

τ5=(1,2

)

τ4=(4,3)τ7=(1,3)τ4=(4,3) τ7=

(1,3)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

100%

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

ExampleFirst, compute a bound for τ7, since it does not share its last

processor.

Finally, we compute the bounds for fixed tasks by the already computed bounds for τ4 and

τ7.

τ4 shares its last processor with τ7, so we need τ7’s bound to compute τ4

’s.

Proof

•Key idea: introduce a hypothetical situation.• Tardiness in the hypothetical situation upper bounds

that in the real situation.• Derive a upper bound on the tardiness for the

hypothetical situation.

Real Situation ≤ Hypothetical Situation ≤ Bounds

τ1=(3,1)τ2=(11,6) τ3=(5,3)

τ5=(1,2

)

τ7=(1,3) τ7=(1,3

)

12

16

16

533

s1 = 4Processor Capacity

Processor Speed

100%

0%

s2 = 2 s3 = 2 s4 = 1

τ4=(4,3) 16

1

116

τ6=(2,6

)

16

13

Example

τ4=(4,3)43

τ4=(4,3)τ4=(4,3)

τ4=(4,3)

faster processorhighest priority

slower processormay not be the highest priority

faster processorhighest priority

slower processormay not be the highest priority

Schedulability

total speed =36

Tardiness Bounds

total speed =36

Multiples of periods

Future Works

•Eliminate the utilization constraints to develop an optimal scheduling algorithm for uniform heterogeneous multiprocessors.

• Implement the schedulers on real heterogeneous hardware.

•Consider CPU frequency scaling, where an identical multiprocessor may function as a heterogeneous multiprocessor.

Thank you!Questions?

Schedulability

all the platforms in evaluation were set of total speed

=36

Four different platform settings.π1={6,6,6,6,3,3,3,3}π2={8,8,4,4,4,4,3,3}π3={8,7,6,5,4,3,2,1}

π4={15,3,3,3,3,3,3,3}

the percentage of randomly generated

feasible task sets that satisfied our

proposed schedulability

conditions

Tardiness Bounds

all the platforms in evaluation were set of total speed

=36

The ratio of the tardiness

bounds and corresponding tasks’ periods

Two different platform settings.π1={6,6,6,6,3,3,3,3}π2={8,8,4,4,4,4,3,3}