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

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Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic

Tasks upon Periodic ResourcesFarhana DewanNathan Fisher

Wayne State University

RTCSA, August 22nd, 2012

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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

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Setting: Compositional RTS

𝝉n𝝉2 𝝉1I

Global Scheduler

W1𝝉2

W2𝝉

1𝝉2

W3𝝉

1𝝉2

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

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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

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)

Arbitrary task deadlines

di ≤ pi or di > pi

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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

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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

τnτ1

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

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Problem:Sub-Problems

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

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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)).

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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

RBF(τ 1

t13p 15p

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Capacity Determination [Background]: Cumulative Request-Bound Function

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

RBF(τ 2

2p 23p22p 24p

t 3p 33p32p

RBF(τ 3

1p 12p 13p

1e12e13e14e15e16e

RBF(τ 1

tCumulative Request Bound Function, Wi (t)

Capacity Determination [Background]: Cumulative Request-Bound Function

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Testing set

points

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.

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“no-supply period”

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

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

Exact test is potentially exponential

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

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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

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Solution [Step 1]

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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

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Solution [Step 2]

Determine intersection of Wi,j with sbf

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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

)(,max 11

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sbf usbf

ta-1 ta

Solution [Step 3]

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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]

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

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Maximum horizontal distance

between usbf and sbf

intersections

1. jit

d ,,, 1

2. iip

di,jɸi,j

ϕi ϕi

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Complexity: O(kn2 logkn)

Simulation [Parameters]

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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

Simulation [Results]:Comparison with Exact, Approximate

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

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

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

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

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Thank You!

Questions?farhanad@wayne.edu

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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.

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

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