Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic Tasks upon Periodic Resources

Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic

Tasks upon Periodic ResourcesFarhana DewanNathan Fisher

Wayne State University

RTCSA, August 22nd, 2012

CoPaRTS

2

Outline Setting:

Compositional Real-Time Systems Sporadic Tasks with Arbitrary Deadline Fixed Priority Scheudling

Problem: Interface Selection for Minimization of Interface Bandwidth (MIB-RT) Capacity Determination

Solution: Sufficient Schedulability Test Algorithm Simulation results

CoPaRTS

3

Setting: Compositional RTS

…

A

𝝉n𝝉2 𝝉1I

C

W

Global Scheduler

…

A1

𝝉1

I1

C1

W1𝝉2

𝝉n

…

A2

I2

C2

W2𝝉

1𝝉2

𝝉n

…

A3

I3

C3

W3𝝉

1𝝉2

𝝉n

Component C Workload W Component-level Scheduling Algorithm A Real-time Interface I

CoPaRTS

4

CoPaRTS

Setting [Interface]: Periodic Resource Model

(Explicit-Deadline) Periodic Resource Model

Periodic resource, Ω=(Π, Θ, ∆) [Easwaran et al., RTSS07] Θ units of processing capacity in deadline ∆ of every Π period Assume Θ ≤ Π

2 3 4t

Interface Bandwidth Fraction of system’s resource supply required by a component Interference of a component on other components For periodic resource: Θ/Π

5

Setting [Workload]: Sporadic Task System Each component is a sporadic task system, τ= {τ1, τ2 …, τn}

Example: τ1 =(2,3,5)

Sporadic TasksCharacterized by the tuple τi=(ei , di , pi ) Worst case execution requirement, ei Relative deadline, di Minimum interarrival serperation, or Period, pi τi,j is the j-th job of τi, with arrival time ai,j and abs.

deadline di,j

0 5 10 15 20 25

(2) (2) (2) (2) (2) (2)

t

Arbitrary task deadlines

di ≤ pi or di > pi

CoPaRTS

6

Setting [Component-Level Scheduler]: Fixed-Priority

Each task is associated with a pre-assigned priority (indexed by priority)

All jobs generated from a task inherit its priority

Within every allocation to component C, schedule active job with the highest priority

CoPaRTS

7

Problem: MIB-RTMinimization of Interface Bandwidth

(MIB-RT)Given: Component C=(W, A) Find: Interface I such that

Workload W is A-schedulable upon component C with respect to interface I

Interface bandwidth is minimized

…

A

τnτ1

IC

W

Problem: Find interface Θ and Π that minimize interface bandwidth Θ/Π while ensuring τ is Fixed-Priority Schedulable on Ω.

CoPaRTS

8

Problem:Sub-Problems

To solve MIB-RT, we need to address two sub-problems:1. Capacity Determination2. Period Selection

CoPaRTS

9

Sub-Problem: Capacity Determination

Capacity Determination Algorithm (A)Given:

• Sporadic Task System: τ • Fixed Period: Π

Find: Minimum capacity Θ(A, Π ,τ) such that τ is FP-schedulable upon resource Ω =(Π , Θ(A, Π ,τ)).

CoPaRTS

10

Capacity Determination [Background]: Request-Bound Function

Request Bound Function RBF(τi,t): Maximum cumulative execution requests of all jobs of τi arriving within the interval of t

For sporadic task τi: RBF(τi,t) = ⌈ t/pi ⌉.ei

Example: τ1 =(e1, d1, p1)

1p 12p 14p

1e12e13e14e15e16e

16p

RBF(τ 1

,t)

t13p 15p

CoPaRTS

11

Capacity Determination [Background]: Cumulative Request-Bound Function

CoPaRTS

Consider τ contains 3 tasks: τ1(1, 5, 2) τ2(1, 10, 4) τ3(1, 15, 8)

RBF(τ 2

,t)

2p 23p22p 24p

2e

22e

23e

24e

t 3p 33p32p

3e

32e

33e

34e

RBF(τ 3

,t)

t

1p 12p 13p

1e12e13e14e15e16e

15p

RBF(τ 1

,t)

t14p

W3(t

)

tCumulative Request Bound Function, Wi (t)

12

Capacity Determination [Background]: Cumulative Request-Bound Function

CoPaRTS

Testing set

points


13

Capacity Determination [Background]: Supply-Bound Function

Supply Bound Function sbf(Ω=(Π, Θ, Δ), t): Minimum execution supply a component may receive over any interval of length t executed upon EDP resource Ω.

CoPaRTS

usbf

2

3

2

22 23

2 23

24 t

sbf

“no-supply period”

14

Capacity Determination [Prior Results]

Constrained-Deadline Tasks Exact schedulability test [Easwaran et al.,

RTSS’07] Sufficient schedulability test [Shin and Lee, ACM

TECS’08] Approximate schedulability test [Dewan and

Fisher, RTAS’10]

Arbitrary-Deadline TasksNo prior result in compositional

setting! CoPaRTS

15

Capacity Determination [Exact Schedulability Test] Response-time based approach [using uniprocessor

schedulability test] Model ``no-supply’’ period of resource Ω as a special

highest priority task Apply schedulability test to modified task system

Exact test [Lehoczky, RTSS’90] Approximate test [Fisher and Baruah, ECRTS’05]

Search capacity in the range [0, Π] Testing set based approach [shown in the paper]

For each task, determine busy period For each job in the busy period, check whether crbf is

less than supply at each testing set pointCoPaRT

S

Exact test is potentially exponential

16

Capacity Determination [Solution]

Goal Address the computational inefficiency of the exact schedulability test

Solution Develop a polynomial-time parametric sufficient schemebased on testing set points

CoPaRTS

17

Capacity Determination [Solution]: Sufficient Schedulability Test

For each task in priority order: Step 1: Reduce the number of testing set

points Step 2: Determine schedulability of first

active job between testing set points Step 3: Determine number of active jobs

between testing set points Step 4: Perform sufficient test

CoPaRTS

18






CoPaRTS

19

Solution [Step 1]

CoPaRTS

Reduce Testing Set Points Approximate RBF and hence cumulative RBF Given parameter k, for each task, approximate RBF after k-1 steps

Testing set points reduced to polynomial


k=3

20






CoPaRTS

21

Solution [Step 2]

Determine intersection of Wi,j with sbf

CoPaRTS

Schedulability of first active job For each testing set point ta determine whether the first active job τi,j with deadline before ta meets its deadline

jijit

aji

tji

t

jit

aji

tji

tji

t

dlltD

tDDl

a

aa

a

aaa

,,

,,

,1

,,,

1))1(

)(,max 11

22






CoPaRTS

23

2

3

t

sbf usbf

ta-1 ta

Solution [Step 3]

CoPaRTS

Number of Active Jobs that finished execution in [ta-1, ta] Intersection of parallel line segments with usbf have same horizontal distance

3 jobs will finish execution within [ta-1, ta]

24






CoPaRTS

25

Solution [Step 4]Sufficient Test Ensure that all active jobs from step 3 meet their deadline

CoPaRTS

usbf

t

Maximum horizontal distance

between usbf and sbf

intersections

1. jit

jijia

d ,,, 1

2. iip

di,jɸi,j

ϕi ϕi

26






CoPaRTS

Complexity: O(kn2 logkn)

27

Simulation [Parameters]

CoPaRTS

ParametersCompared response time based exact test, approximate test and our sufficient test

System utilization, U(τ) = [0.1-0.9], UUnifast [Bini and Buttazzo, ECRTS04] to generate task utilizations

For each utilization randomly generate task system parameters

Workload size, n = 10 System utilization Task period, pi = [5-30] Task deadline, di = [5-100] EDP period, Π =10; EDP deadline, Δ = Π Approximation parameter, k=[1-20] Each point in the plot is the average of 1000

simulation runs

28

Simulation [Results]:Comparison with Exact, Approximate

CoPaRTS

Approximate test with linear approximation of ``no-supply period’’ performs worse than the sufficient!

29


CoPaRTS

Sufficient test performs better than both exact and approximate Iterative exact test takes higher time for higher utilization

30


CoPaRTS

Relative error of the approx-imate test is twice as that of the sufficient test

31

Conclusion Addressed: MIB-RT for a larger class of tasks

Fixed-priority-scheduled sporadic tasks with arbitrary deadline

Developed: Polynomial-time sufficient test Verified: Simulation showed better

performance than straightforward approximate test

Future Work: Tighter results for this setting Multiprocessor compositional frameworks

CoPaRTS

32

Thank You!

[email protected]

CoPaRTS

33

References [Lehoczky, RTSS’90] J. P. Lehoczky. Fixed priority scheduling of periodic tasks with

arbitrary deadlines. In Proceedings of the IEEE Real-Time Systems Symposium, pages 201-209, December 1990.

[Fisher and Baruah, ECRTS‘05] N. Fisher and S. Baruah. A fully polynomial-time approximation scheme for feasibility analysis in static-priority systems with arbitrary relative deadlines. In Proceedings of the EuroMicro Conference on Real-Time Systems, Spain, July 2005.

[Easwaran et al., RTSS’07] A. Easwaran, M. Anand, and I. Lee. Compositional analysis framework using EDP resource models. In Proceedings of the IEEE Real-Time Systems Symposium, Tucson, Arizona, December 2007.

[Dewan and Fisher, RTAS’10] F. Dewan and N. Fisher. Approximate bandwidth allocation for fixed-priority-scheduled periodic resources. In Proceedings of the IEEE Real-Time Technology and Application Symposium, Stockholm, Sweden 2010.

[Shin and Lee, ACM TECS’08] I. Shin and I. Lee. Compositional real-time scheduling framework with periodic resource model. ACM Transactions on Embedded Computing Systems, 7(3), April 2008.

[Okwudire et al. ETFA’10] C. Okwudire, M. van den Heuvel, R. Bril, and J. Lukkien. Exploiting harmonic periods to improve linearly approximated response-time upper bounds. In IEEE Conference on Emerging Technologies and Factory Automation, September 2010.

CoPaRTS

Date post:	23-Feb-2016
Category:	Documents
Upload:	stasia
View:	18 times
Download:	0 times

Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic Tasks upon Periodic Resources

Documents