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

1. Compositional & Parallel Real Time Systems CoPaRTS Fixed-Priority Schedulabiltiy of Arbitrary-Deadline Sporadic Tasks upon Periodic Resources FarhanaDewan Nathan Fisher Wayne State University RTCSA, August 22nd, 2012

2. Compositional & Parallel Real Time Systems Outline CoPaRTS • 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

3. Compositional & Parallel Real Time Systems Setting: Compositional RTS CoPaRTS Component C • Workload W • Component-level Scheduling Algorithm A • Real-time Interface I Global Scheduler C A I C1 C2 C3 W … A1 A2 A3 I3 I2 I1 𝝉1 𝝉2 𝝉n W1 W2 W3 … … … 𝝉1 𝝉2 𝝉n 𝝉1 𝝉2 𝝉n 𝝉1 𝝉2 𝝉n

4. Compositional & Parallel Real Time Systems Setting [Interface]: Periodic Resource Model CoPaRTS (Explicit-Deadline) Periodic Resource Model • Periodic resource, Ω=(Π, Θ, ∆) [Easwaran et al., RTSS07] • Θ units of processing capacity in deadline ∆ of every Πperiod • Assume Θ≤Π Interface Bandwidth • Fraction of system’s resource supply required by a component • Interference of a component on other components • For periodic resource: Θ/Π t

5. Compositional & Parallel Real Time Systems Setting[Workload]: Sporadic Task System CoPaRTS • Each component is a sporadic task system, τ= {τ1, τ2 …, τn} • Example: τ1=(2,3,5) Sporadic Tasks • Characterized by the tupleτi=(ei , di , pi ) • Worst case execution requirement, ei • Relative deadline, di • Minimum interarrivalserperation, or Period, pi • τi,jis the j-th job of τi, with arrival time ai,j and abs. deadline di,j Arbitrary task deadlines di≤ pi or di> pi (2) (2) (2) (2) (2) (2) 0 5 10 15 20 25 t

6. Compositional & Parallel Real Time Systems Setting[Component-Level Scheduler]: Fixed-Priority CoPaRTS • 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

7. Compositional & Parallel Real Time Systems Problem: MIB-RT CoPaRTS Minimization 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 C A I … W τ1 τn Problem: Find interface ΘandΠ that minimize interface bandwidthΘ/Πwhile ensuring τ isFixed-Priority Schedulable on Ω.

8. Compositional & Parallel Real Time Systems Problem:Sub-Problems CoPaRTS • To solve MIB-RT, we need to address two sub-problems: • Capacity Determination • Period Selection

9. Compositional & Parallel Real Time Systems Sub-Problem: Capacity Determination CoPaRTS 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)).

10. Compositional & Parallel Real Time Systems Capacity Determination [Background]: Request-Bound Function CoPaRTS Request Bound Function RBF(τ1,t) • RBF(τi,t): Maximum cumulative execution requests of all jobs of τiarriving within the interval of t Example:τ1=(e1, d1,p1) For sporadic task τi: RBF(τi,t) = ⌈ t/pi ⌉.ei t

11. Compositional & Parallel Real Time Systems 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(τ1,t) W3(t) t RBF(τ2,t) RBF(τ3,t) t t t Cumulative Request Bound Function, Wi (t)

12. Compositional & Parallel Real Time Systems 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) Testing set points

13. Compositional & Parallel Real Time Systems Capacity Determination [Background]: Supply-Bound Function CoPaRTS 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. sbf usbf “no-supply period” t

14. Compositional & Parallel Real Time Systems Capacity Determination [Prior Results] CoPaRTS • 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 Tasks No prior result in compositional setting!

15. Compositional & Parallel Real Time Systems Capacity Determination [Exact Schedulability Test] CoPaRTS • Response-time based approach [using uniprocessorschedulability 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, determinebusy period • For each job in the busy period, check whether crbf is less than supply at each testing set point Exact test is potentially exponential

16. Compositional & Parallel Real Time Systems Capacity Determination [Solution] CoPaRTS Goal • Address the computational inefficiency of the exact schedulability test Solution • Develop a polynomial-time parametric sufficient scheme • based on testing set points

17. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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

18. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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

19. Compositional & Parallel Real Time Systems Solution [Step 1] CoPaRTS • Consider τ contains 3 tasks: • τ1(1, 5, 2) • τ2(1, 10, 4) • τ3(1, 15, 8) • k=3 Testing set points reduced to polynomial Reduce Testing Set Points • Approximate RBF and hence cumulative RBF • Given parameter k, for each task, approximate RBF after k-1 steps

20. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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

21. Compositional & Parallel Real Time Systems Solution [Step 2] CoPaRTS • Determine intersection of Wi,j with sbf Schedulability of first active job • For each testing set point ta determine whether the first active job τi,jwith deadline before tameets its deadline

22. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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

23. Compositional & Parallel Real Time Systems Solution [Step 3] CoPaRTS sbf usbf Number of Active Jobs that finished execution in [ta-1, ta] 3 jobs will finish execution within [ta-1, ta] • Intersection of parallel line segments with usbf have same horizontal distance ta-1 ta t

24. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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

25. Compositional & Parallel Real Time Systems Solution [Step 4] CoPaRTS usbf ϕi ϕi 1. 2. Sufficient Test Maximum horizontal distance between usbf and sbf intersections • Ensure that all active jobs from step 3 meet their deadline ɸi,j di,j t

26. Compositional & Parallel Real Time Systems Capacity Determination [Solution]: Sufficient Schedulability Test CoPaRTS 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 Complexity: O(kn2logkn)

27. Compositional & Parallel Real Time Systems Simulation [Parameters] CoPaRTS Parameters • Compared 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. Compositional & Parallel Real Time Systems Simulation [Results]:Comparison with Exact, Approximate CoPaRTS Approximate test with linear approximation of ``no-supply period’’ performs worse than the sufficient!

29. Compositional & Parallel Real Time Systems Simulation [Results]:Comparison with Exact, Approximate CoPaRTS • Sufficient test performs better than both exact and approximate • Iterative exact test takes higher time for higher utilization

30. Compositional & Parallel Real Time Systems Simulation [Results]:Comparison with Exact, Approximate CoPaRTS • Relative error of the approx-imate test is twice as that of the sufficient test

31. Compositional & Parallel Real Time Systems Conclusion CoPaRTS • 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

32. Compositional & Parallel Real Time Systems Thank You! CoPaRTS Questions? farhanad@wayne.edu

33. Compositional & Parallel Real Time Systems References CoPaRTS • [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.