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Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

Alex F. Mills Department of Statistics and Operations Research James H. Anderson Department of Computer Science. Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling. RTAS 2010. Multicore Global Scheduling. Sporadic task model with implicit deadlines.

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Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

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  1. Alex F. Mills Department of Statistics and Operations Research James H. Anderson Department of Computer Science Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling RTAS 2010

  2. Multicore Global Scheduling • Sporadic task model with implicit deadlines. • Soft real-time system. • Bounded deadline tardiness: • jobs complete within x of deadline • tasks receive correct processor share • Previous work on bounded deadline tardiness: • no utilization loss when provisioning on worst-case utilization • Devi and Anderson (2005): GEDF • Leontyev and Anderson (2007): GFIFO, EDZL, LLF, and others. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  3. An Odd Couple • Bounded tardiness: • Not a conservative model of correctness. • Tasks may miss deadlines arbitrarily often. • Nonetheless, acceptable for soft real-time. • Provisioning on worst-case execution time: • Conservative model of execution. • Insufficient tool support. • Highly pessimistic on multicore. • Result: wasted processor capacity. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  4. Our Contribution Observed maximum is a biased estimator of population maximum. Observed mean is an unbiased estimator of population mean. WCET > 164ms Average ET ≈ 17ms • Goal: introduce a stochastic execution model • provision tasks based on average utilization. • non-conservative. • flexible. • Example: a video decoding task [need to get citation from Hennadiy’s paper] • Period: 25ms • Maximum observed ET: 164ms • Average observed ET: 17ms • Unschedulable under worst-case provisioning (U>1). • Schedulable under average-case provisioning. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  5. Our Contribution Execution of jobs follows a probability distribution. Tardiness of jobs follows a probability distribution. We derive upper bounds on mean and quantiles of the tardiness distribution. Result: a generalization of tardiness-bound results. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  6. Worst-Case Provisioning:Proof Technique 1 2 3 4 PS GEDF 1 2 3 4 • Ideal system (processor sharing): • tasks allocated a processor share based on worst-case utilization • jobs always finish exactly on time • Track performance of global scheduler against ideal system. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  7. Worst-Case Provisioning:Proof Technique 1 2 3 4 PS GEDF 1 2 3 4 number of processors WCET • Measure difference in allocation to t in PS and GEDF (lag) at arbitrary deadline di,j. Exists x so that lag never exceeds mx+eiat a deadline. • Inductively show: at any di,j, allocation to ti in PS and GEDF differs by no more than x+ei. • No tardiness in PS, so tardiness in GEDF isbounded by x+ei. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  8. Worst-Case Provisioning:Proof Technique 1 2 3 4 PS GEDF 1 2 3 4 • Red time point is: • Deadline of t1,2. • Completion time of t1,2. • Completion of all work with equal or higher priority than t1,2. • All analysis can be done at job deadlines. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  9. Average Case Provisioning:Proof Technique PS PS 1 2 3 4 1 2 3 4 • Provision processor share on average execution time. • Some jobs will miss deadlines in PS. • Deadline of t1,2. • Completion time of t1,2. • Completion of all work with equal or higher priority than t1,2. A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

  10. Average-Case Provisioning:Proof Technique • Deadline of t1,2. • Completion time of t1,2. • Completion of all work with equal or higher priority than t1,2. PS 1 2 3 4 A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling • Deterministic analysis: • Exists x so that lag at any green time point is bounded by a function of x, and corresponding purple time point. • Inductively, show that GEDF completion time bounded by x plus purple time point. • Need: relationship with redtime point.

  11. Average-Case Provisioning:Proof Technique • Deadline of t1,2. • Completion time of t1,2. • Completion of all work with equal or higher priority than t1,2. PS 1 2 3 4 A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling • Processor share based on average utilization. • Worst case: jobs of t1 always execute longer than average. • Time between redand purple(tardiness in PS) could be arbitrarily large. • Law of large numbers: probability of this is zero. • Model as a stochastic system.

  12. Analysis of Ideal System v3 1/p3 v2 1/p2 v1 1/p1 PS vi≥ ui 1 2 3 4 A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling • Model PS as a system of n parallel G/G/1 queues • General arrival time distribution. • General service time distribution. • We need execution times of successive jobs to be independent. • Waiting time in queue corresponds to tardiness in PS.

  13. Analysis of Ideal System v3 1/p3 v2 1/p2 v1 1/p1 PS vi≥ ui 1 2 3 4 A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling Known bound for mean waiting time in G/G/1. Depends only on means and variances of execution time. Can also derive a bound on quantiles/percentiles of the tardiness distribution.

  14. Example • Worst case: • every task over-utilized • total worst case utilization is > 26 • Average case: • no tasks over-utilized • requires < 4 processors • Our work shows: global earliest-deadline first can schedule this system on 4 processors • expected tardiness of every job of every task is bounded (e.g., for task 1 expected tardiness < 100 time units) • percentiles of tardiness distribution are also bounded A. F. Mills & J. H. Anderson: Stochastic Analysis of Multiprocessor Global Soft Real-Time Scheduling

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