A Survey of Hard Real-Time Scheduling for Multiprocessor...

A Survey ofA Survey of

Hard Real-Time Scheduling for Multiprocessor Systems

Xinyun Jiang and Robert Dick

Power consumption and multicoreCritical instant

Global schedulingSchedulability tests

Limitations and future directions

Outline

1. Power consumption and multicore

2. Critical instant

3. Global scheduling

4. Schedulability tests

5. Limitations and future directions

2 Jiang and Dick EECS 598-13




Motivation

Paper focuses on homogeneous multiprocessor systems.

Question

Why use multicore processors instead of just making unicore processorsfaster?





Sources of embedded systems research problems and ideas

Changes in applications.

Changes in implementation technologies.





Process scaling implications

Change Power implicationFeature size reduction, density increases Power increases

More devices used Power increasesFrequency increases Power increasesVoltage decreases Power reduces

Net effect is to increase power density.

Can’t increase frequency because power at air-cooling limit.





Device power density history

Year of announcement

1950 1960 1970 1980 1990 2000 2010

Pow

er d

ensi

ty (

Wat

ts/c

m2 )

0

2

4

6

8

10

12

14

Bipolar

CMOS

VacuumIBM 360

IBM 370 IBM 3033

IBM ES9000

Fujitsu VP2000

IBM 3090S

NTT

Fujitsu M-780

IBM 3090

CDC Cyber 205IBM 4381

IBM 3081Fujitsu M380

IBM RY5

IBM GP

IBM RY6

Apache

Pulsar

Merced

IBM RY7

IBM RY4

Pentium II(DSIP)

T-Rex

Squadrons

Pentium 4

Mckinley

Prescott

Jayhawk(dual)

IBM Z9





Multicore to the rescue

Problems.

Can’t increase frequency.

Need more speed.

Global communication delay in big uniprocessors is hard.

Designing big, fast uniprocessors is hard.

Multicore solution.

Stamp down a bunch of cores.

They can process many instructions per cycle.

Clock at a moderate rate.

Good power density.

Good theoretical throughput.





Multicore problem

If you want speed, need to stop assuming one thing happens at a time.

Software components (threads) must have explicit synchronization.

Harder to reason about, develop, and debug.

Harder to reason about resource use and contention in real-time systems.

HW engineers, “I’m sure the software folks will figure something out.”


ClassificationofMultiprocessorSystems

• Heterogeneous• Differentprocessors• Rateofexecutionbedifferentondifferentprocessors

• Homogeneous(paperfocus)• Identicalprocessors• Rateofexecutionbethesameacrossallprocessors

• Uniform• Rateofexecutiondependsonspeedoftheprocessor

DemandBoundFunctionBaruah etal.,[1990]

• Maximumamountoftaskexecutionthatcanbereleasedandhastocompleteintimeinterval[0, 𝑡)

• ℎ 𝑡 = ∑ max(0, @ABCDC

+ 1)GHIJ 𝐶H

• 𝑘 + 1:largestnon-negativeintegersuchthat𝑘 ⋅ 𝑇H + 𝐷H ≤ 𝑡

• Notschedulableifℎ 𝑡 > 𝑡

ProcessorLoad

• 𝑙𝑜𝑎𝑑 𝜏 = max∀@X(@)@

• Task-setFeasibility->𝑙𝑜𝑎𝑑 𝜏 ≤ 𝑚

• Considerthetasksystembelow

• ℎ 𝑡 = 2• 𝑙𝑜𝑎𝑑 = 2• Under-estimatethedemandof𝜏J in 0,1• Effectivecombineddemand=3

LimitsinthepreviousworkBaker,Cirinei [2006]

Maxmin DemandBaker,Cirinei [2006]

Maximumofminimumamountoftimethatataskmustexecutewithinaspecifictimeintervalinordertomeetallitsdeadlines.

𝐶H = 2,𝐷H= 3,𝑇H= 7

• ℎ∗ 𝑡 = ℎ 𝑡 + ∑ max(0, 𝒕 − 𝒋𝒕 ⋅ 𝑻𝒊 − 𝑫𝒊 + 𝑪𝒊)GHIJ

• 𝑗@ = max 0, @ABCDC

+ 1

• Throwforward

Optimality ExistorNot?Hong,Leung[1988]

ProblemDefinition

• Online schedulerwithtaskshave>1distinctdeadline• Online schedulerwithtaskshaveacommondeadline• Offline scheduler

HardnessofSolving

• No optimalonlineschedulerexists

• Optimalonlineschedulerexists

• Optimalofflineschedulerexists𝑶(𝒏𝟑)




Outline


2. Critical instant








Rate mononotic scheduling (RMS)

Single processor.

Independent tasks.

Differing arrival periods.

Schedule in order of increasing periods.

No fixed-priority schedule will do better than RMS.

Guaranteed valid for loading ≤ ln 2 = 0.69.

For loading > ln 2 and < 1, correctness unknown.

Usually works up to a loading of 0.88.





Rate monotonic scheduling

1973, Liu and Layland derived optimal scheduling algorithm(s) for thisproblem.

Schedule the job with the smallest period (period = deadline) first.

Analyzed worst-case behavior on any task set of size n.

Found utilization bound: U(n) = n · (21/n − 1).

0.828 at n = 2.

As n→∞, U(n)→ log 2 = 0.693.

Result: For any problem instance, if a valid schedule is possible, theprocessor need never spend more than 31% of its time idle.





Optimality and utilization for limited case

Simply periodic: All task periods are integer multiples of all lesser taskperiods.

In this case, RMS/DMS optimal with utilization 1.

However, this case rare in practice.

Remains feasible, with decreased utilization bound, for in-phase tasks witharbitrary periods.





Rate monotonic scheduling

Constrained problem definition.

Over-allocation often results.

However, in practice utilization of 85%–90% common.

Lose guarantee.

If phases known, can prove by generating instance.





Critical instants

Definition

A job’s critical instant a time at which all possible concurrent higher-priorityjobs are also simultaneously released.

Useful because it implies latest finish time





Definitions

Period: T .

Execution time: C .

Process: i .

Utilization: U =∑m

i=1Ci

Ti.

Assume Task 1 is higher priority than Task 2, and thus T1 < T2.





Critical instants

C1

T1

C1

C2

C1

T1 T1

Case 1 Case 2

T2





Case 1 I

All instances of higher-priority tasks released before end of lower-priority taskperiod complete before end of lower-priority task period.

C1 ≤ T2 − T1

⌊T2

T1

⌋.

I.e., the execution time of Task 1 is less than or equal to the period ofTask 2 minus the total time spent within the periods of instances of Task 1finishing within Task 2’s period.

Now, let’s determine the maximum execution time of Task 2 as a function ofall other variables.

Number of T1 released =⌈T2

T1

⌉so C2,max = T2 − C1

⌈T2

T1

⌉.





Case 1 II

I.e., the maximum execution time of Task 2 is the period of Task 2 minus thetotal execution time of instances of Task 1 released within Task 2’s period.





Case 1 III

In this case,

U = U1 + U2

=C1

T1+

C2,max

T2

=C1

T1+

T2 − C1

⌈T2

T1

⌉T2

=C1

T1+ 1−

C1

⌈T2

T1

⌉T2

= 1 + C1

(1

T1− 1

T2

⌈T2

T1

⌉)





Case 1 IV

Is 1T1− 1

T2

⌈T2

T1

⌉≤ 0?

If T2 = T1 + ε, this is

1

T1− 1

T1 + ε

⌈T1 + ε

T1

⌉=

1

T1− 2

T1 + εwhich is less than or equal to zero.

Thus, U is monotonically nonincreasing in C1.





Case 2 I

Instances of higher-priority tasks released before end of lower-priority taskperiod complete after end of lower-priority task period.

C1 ≥ T2 − T1

⌊T2

T1

⌋.

C2,max = T1

⌊T2

T1

⌋− C1

⌊T2

T1

⌋.

U1 = C1/T1.

U2 = C2/T2 = T1

T2

⌊T2

T1

⌋− C1

T2

⌊T2

T1

⌋.

U = U1 + U2 = T1

T2

⌊T2

T1

⌋+ C1

(1T1− 1

T2

⌊T2

T1

⌋).





Minimal U I

C1 = T2 − T1

⌊T2

T1

⌋.

U = 1− T1

T2

(⌈T2

T1

⌉− T2

T1

)(T2

T1−⌊T2

T1

⌋).

Let I =⌊T2

T1

⌋and

f = T2

T1.

Then, U = 1− f (1−f )I+f .

To maximize U, minimize I , which can be no smaller than 1.

U = 1− f (1−f )1+f .





Minimal U II

Differentiate to find mimima, at f =√

2− 1.

Thus, Umin = 2(√

2− 1)≈ 0.83.

Is this the minimal U? Are we done?





Proof sketch for RMS utilization bound

Consider case in which no period exceeds twice the shortest period.

Find a pathological case: in phase

Utilization of 1 for some duration.

Any decrease in period/deadline of longest-period task will causedeadline violations.

Any increase in execution time will cause deadline violations.






See if there is a way to increase utilization while meeting all deadlines.

Increase execution time of high-priority task.

e′i = pi+1 − pi + ε = ei + ε.

Must compensate by decreasing another execution time.

This always results in decreased utilization.

e′k = ek − ε.

U ′ − U =e′ipi

+e′kpk− ei

pi− ek

pk= ε

pi− ε

pk.

Note that pi < pk → U ′ > U.






Same true if execution time of high-priority task reduced.

e′′i = pi+1 − pi − ε.

In this case, must increase other e or leave idle for 2 · ε.

e′′k = ek + 2ε.

U ′′ − U = 2εpk− ε

pi.

Again, pk < 2→ U ′′ > U.

Sum over execution time/period ratios.






Get utilization as a function of adjacent task ratios.

Substitute execution times into∑n

k=1ekpk

.

Find minimum.

Extend to cases in which pn > 2 · pk .





Notes on RMS

DMS better than or equal RMS when deadline 6= period.

Why not use slack-based?

What happens if resources are under-allocated and a deadline is missed?





Multiprocessor breaks critical instant

P1

P2

1

2

3

4

1

3

2 4P1

P2


UtilizationBound

• Performancemetric:Worst-caseutilizationbound

• Theminimumutilizationofanyimplicit-deadlinetaskset thatisonlyjustschedulableaccordingtoalgorithmA

• Sufficient,notnecessaryschedulability test

𝑈jkD =𝑚 + 12

• Periodictaskset,implicitdeadline

• Suppose• 𝑚 + 1 tasks,𝑚 processors,1 executiontime,period2• lmJ

nlutilizationperprocessor

• lmJn

overallutilization• Ifexecutiontimebecomes1 + 𝜖 ->unschedulable

UtilizationBound Andersson etal[2001]

PartitionedScheduling

Advantages• Onlyeffectoneprocessorunderworstcase• Nopenaltyintermsofmigrationcost• Separaterun-queueperprocessor• Uniprocessortechniquescanbeapplied

Disadvantages• Taskallocationproblem------ NPhard




Outline


2. Critical instant








Global scheduling

Task migration.

Fewer preemptions.

Tasks anywhere can use spare time.

Dhall and Liu ’78.

Terrible worst-case EDF utilization bound for periodic tasks with implicitdeadlines.

Numerous algorithms proposed with utilization bounds slightly tighter than(m + 1)/2.





Pfair

Proportionate fairness.

Each task makes progress proportionate to its utilization.

Theoretical utilization bound of m.

However, that ignores

(frequent) migration and

per-quanta and migration decision algorithm.





EDZL

Lee ’94 Earliest Deadline until Zero Laxity

Act like EDF until a task will miss its deadline.

Set those to highest priority.

Can schedule any possible set of ready tasks, but not necessarily futuretasks.

Chao ’08

Proved that utilization bound is at most m(1− 1/e).


HybridScheduling–Semipartitioned ApproachAndersson,Tovar[2006]

• 𝑈pqr = 𝑆𝐸𝑃 = vwwmJ

, 𝑘 < 𝑚1, 𝑘 = 𝑚

• Utilizationboundof66%

• Atmost4preemptionsperjob




Outline


2. Critical instant








Schedulability tests

Instead of considering worst-case task set, use pseudo-random or benchmarkbased performance.

Drawback

Naıve pseudo-random task sets don’t look like real-world task sets.

Advantage

Closer to real-world average case performance than worst-case.

More appropriate for making decisions about soft real-time systems. . .

. . . unless application task sets are constrained.





Outline


2. Critical instant









Representations don’t match complex real-world task sets.

Edmonds and Pruhs ’09

Tasks have phases, which have degrees of parallelization.

Large gaps between theoretical bounds and algorithm results for sporadictask models.






Can also sometimes model complex task structures with minor changes toexisting models, such as grouping.

E.g., pre-/post-computation and streaming.

b) pre− and post−computation

K

J J1/3

J3/3

J2/3 K1/3

K2/3

K3/3

J1/3

J2/3

J3/3

K1/3

K2/3

K3/3

a) conventional

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3 kb

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c) streaming

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