William Hawkins and Tarek Abdelzaher Presented By: Farhana Dewan

1. Towards Feasibility Region Calculus: An End-to-end Schedulability Analysis of Real-Time Multistage Execution William Hawkins and TarekAbdelzaher Presented By: FarhanaDewan

2. Introduction • System Model • Generalized Stage Delay Theorem • Proof of the Theorem • Usage of Feasibility Region • Simulation Results • Conclusion Outline CSC 8260

3. Aperiodicdistributed system- system is large complex, workload irregular • Less attention than periodic counter part • This paper presents • Analytic framework for computing end-to-end feasibility • Fixed-priority scheduling • Based on generalized stage delay theorem • Maximum fraction of end-to-end deadline a task can spend at a resource as a function of utilization of that resource • Sum of such fractions are less than 1 • Feasibility region is considered as a volume in multi-dimensional space where each dimension is utilization of one resource • Extends uni-dimensional schedulable region to multi-dimensional representation for distributed systems • Generalizes concurrent infinitesimal tasks to arbitrary set of finite tasks Introduction CSC 8260

4. Distributed real-time systems • Performance sensitive server farms • Radar data processing back ends • Sensor networks • Different classes of traffic traverse several stages of distribute processing • Task must exit the system within specified per-class end 2 end latency constraints • Utilization bound of resource for centralized system • U ≤ Ubound • For distributed systems resource stage i has utilization Ui • f(U1 …,Un) ≤ Cbound • Cbound systems capacity to meet deadlines Introduction CSC 8260

5. Goal: • Simple schedulability analysis technique for distributed rtsto satisfy e2e timing constraints • Conditions are sufficient • Fast dynamic admission control • Acyclic resource system • No feedback cycle in overall task flow graph • Synthetic utilization • Non-acyclic resource system • Instantaneous utilization Introduction CSC 8260

6. Distributed system, task Ti arrive, require execution to N (subset of) resources • Aijarrival time of Ti at stage j, 1≤j ≤N • Ai arrival time of task to the system, Ai1 • Di e2e deadline for Ti • Cijcomputation time of Ti at stage j • Set of current tasks V(t)={Ti|Ai≤t<Ai+Di} • Instantaneous utilization Uj • Synthetic utilization Uj System Model CSC 8260

7. Urgency inversion factor αj for stage j • Less urgent task assigned greater priority • αj= min (Dlo /Dhi ) over all tasks executing at stage j such that priority(Thi)>priority(Tlo) • Blocking factor βij • Maximum amount of time task i can be blocked at stage j due to lower priority task holding critical resource • Maximum normalized blocking factor • γj = max (βij/Di) Definitions CSC 8260

8. End to end schedulabiltiy condition: ΣjFj ≤1 Generalized Stage Delay Theorem CSC 8260

9. Stage j processing n concurrent tasks • Instantaneous utilization at stage j for task Tm • To obtain lower bound, ignore lower priority tasks Proof CSC 8260

10. Consider task Tn at stage j • Worst case delay at stage j, Qnj • B is the end of last processor gap • tf time at which Tn departs stage j • L = An –B offset of arrival of Tn on j • For worst case arrival scenario, L=0 • Max amount of time critical task is preempted by tasks with absolute deadline prior to tf, Proof (cont.) CSC 8260

11. Busy period • Rearranging and substituting, we obtain instantaneous utilization Uj Proof (cont.) CSC 8260

12. Worst case arrival sequence • T= A1j – Anj, Aij – Anj = T + Σh=1 Chj • Qnj = T + Σi=1 Cij Proof (cont.) CSC 8260

13. Proof (cont.) CSC 8260

14. Di is bounded by Dn/αj and Σ is minimized when Cij = C for i=1 to n-1 Proof (cont.) CSC 8260

15. Delay: • To obtain fraction of deadline, divide by deadline • F to be worst case bound, last term must be maximized Proof (cont.) CSC 8260

16. Generalized Stage Delay Theorem Corollary CSC 8260

17. Stages of computation: resources can be CPU, communication links, disks • Scheduling policy at each resource • αj and γj must be pre computed • Admission controller: based on generalized stage delay theorem or corollary • Feasibility region calculus: to build admission controller • Each task arriving the system, utilization is added to Uj of each stage to be traversed by the task • Check fractional delay, if greater than 1, don’t admit, reverse the utilization modification • AC checks that the system operates in feasibility region • Complexity: linear in terms of number of stages, fractional stage delay in constant time Usage of Feasibility Region CSC 8260

18. Simulator: distributed real-time system with arbitrary tasks • Admission Controller: for either of the cases • For each arriving task, its utilization is tentatively added to every stage j it will traverse during computation. • The generalized stage delay theorem, or its corollary if applicable, is used to check whether ΣjFj≤ 1 over the stages to be traversed. • If so, the task is admitted. If not, the task is rejected and its utilization is removed from further consideration. • Task granularity: ratio of total computation time and deadline • Load: sum of computation time of all tasks divided by simulation time Simulation Results CSC 8260

19. Resources has increasing ids, a task leaving stage x never requires resource from stage i, 0≤i≤x • Pipeline, 1 to 5 stages, each task must be executed by each stage from 1 to 5 in order • Deadlines are drawn from uniform distribution • Task granularity is 1/100 • Load is varied from 60% to 200% • Corollary of generalized stage delay theorem is used • No task misses deadline • Each point in the plot is average of 100 simulation runs • Utilization is high for all offered loads, independent of no of stages, AC is not pessimistic Acyclic Task System CSC 8260

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22. Task may receive computation from same resource more than once (Ex- resource is database) • Each task in the experiment traverses more than 1 stage in the system • Task granularity 1/100, computation time approximately equal at each stages • Load is varied between 60% and 200% • Utilization of system with 1 stage is higher than that of 2,3 or 4 stages • Lower priority task suffer from delay, whether delay is from higher priority task in same stage or other • Stage delay corollary can be used as heuristic in admission controller to improve utilization Non-Acyclic Task System CSC 8260

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25. Stage delay theorem • System with very large number of concurrent task • Calculates feasibility region based on utilization in the stages • Generalized Stage delay theorem • Calculates feasibility region based on utilization and concurrent tasks in the stages • Two stage pipeline distributed rts • 6 task classes, arrival time and deadline from uniform distribution • For moderate number of tasks and very small granularity gsdt performs better Comparing Stage Delay Theorem with Generalized Stage Delay Theorem CSC 8260

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27. Presented: Analytical framework for computing the end-to-end feasibility regions of distributed aperiodic task systems under independent fixed-priority scheduling • Extended: the previous derivations of uni-dimensional schedulability regions for single processors • Generalized: the results for infinite number of concurrent liquid tasks to arbitrary sets of finite tasks • Applicable to more realistic acyclic and non-acyclic workloads Conclusion CSC 8260

28. The results can be extended to: • Other categories of scheduling policies such as EDF • Systems that accept some percentage of deadline misses (soft real-time systems), relaxed schedulability conditions can be derived • System where tasks need multiple resources simultaneously Future Work CSC 8260