Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic Tasks upon Periodic Resources Farhana Dewan Nathan Fisher Wayne State University RTCSA, August 22 nd , 2012 CoPaRTS
Feb 23, 2016
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
…
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
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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
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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
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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
…
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.
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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
16p
RBF(τ 1
,t)
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
,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)
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Capacity Determination [Background]: Cumulative Request-Bound Function
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Testing set
points
Consider τ contains 3 tasks: τ1(1, 5, 2) τ2(1, 10, 4) τ3(1, 15, 8)
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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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usbf
2
3
2
22 23
2 23
24 t
sbf
“no-supply period”
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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
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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
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Exact test is potentially exponential
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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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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
Consider τ contains 3 tasks: τ1(1, 5, 2) τ2(1, 10, 4) τ3(1, 15, 8)
k=3
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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 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
jijit
aji
tji
t
jit
aji
tji
tji
t
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a
aa
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1))1(
)(,max 11
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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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2
3
t
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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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 4]Sufficient Test Ensure that all active jobs from step 3 meet their deadline
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usbf
t
Maximum horizontal distance
between usbf and sbf
intersections
1. jit
jijia
d ,,, 1
2. iip
di,jɸi,j
ϕi ϕi
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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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Complexity: O(kn2 logkn)
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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
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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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Simulation [Results]:Comparison with Exact, Approximate
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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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Simulation [Results]:Comparison with Exact, Approximate
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Relative error of the approx-imate test is twice as that of the sufficient test
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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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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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