Spotlight talk A Brief History of Speedup Factors for Uniprocessor - PowerPoint PPT Presentation

Spotlight talk A Brief History of Speedup Factors for Uniprocessor EDF and Fixed Priority Scheduling Robert I. Davis Real-Time Systems Research Group, University of York, UK

Scope Single processor system  Execution time of all tasks scales linearly with processor clock  speed Sporadic task model  Static set of n tasks  i with priorities 1.. n  Bounded worst-case execution time C i  Independent sporadic arrivals: minimum inter-arrival time T i  Relative deadline D i  Independent execution (no resource sharing)  Three classes of task set: Implicit- ( D i =T i ), Constrained- ( D i ≤ T i ),  Arbitrary-deadline

Resource augmentation metric Speedup factor – two perspectives  Speedup factor for algorithm A versus algorithm B  # 1 Speedup factor is the minimum factor by which it is  necessary to increase the processor speed so that any task set that was schedulable under algorithm B becomes schedulable under algorithm A # 2 Speedup factor is the maximum factor by which the  execution times of a set of tasks, that are only just schedulable under algorithm A can be increased and the task set remain just schedulable under algorithm B

Background: Scheduling algorithms & optimality Pre-emptive  EDF-P is an optimal uniprocessor scheduling algorithm for  arbitrary-deadline sporadic tasks EDF-P dominates FP-P, EDF-NP, and FP-NP Non-pre-emptive  No work-conserving non-preemptive algorithm is optimal  Inserted idle time is necessary for optimality  EDF-NP is optimal in a weak sense that it can schedule any  task set for which a feasible work-conserving non-preemptive schedule exists EDF-NP dominates FP-NP

Background: Scheduling algorithm optimality 1400 Random Optimal Priorities Priorities Fixed Priority Scheduling 1200  1000  Priority assignment important 800 Frequency 600 400 Optimal priority assignment (FP-P)  200  Implicit-deadlines – Rate-Monotonic 0 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Breakdown Utilisation  Constrained-deadlines – Deadline Monotonic  Arbitrary-deadlines – Audsley’s Optimal Priority Assignment algorithm Optimal priority assignment (FP-NP)   All 3 cases – Audsley’s algorithm

Landscape of scheduling algorithms and speedup factors I nterested in comparing EDF and Fixed Priority (FP) scheduling in the preemptive and non-preemptive cases EDF-P FP-P ( optimal ) FP-NP EDF-NP

Previous results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP Early results: Liu & Layland 1973, [1], [2], [3] from 2009/10 As of Jan 2015: Taskset FP-P v. EDF-P FP-NP v. EDF-NP Constraints Speedup factor Speedup factor [Priority ordering] Lower bound Upper bound Lower bound Upper bound 1/ln(2) 1/ Ω 2 Implicit-deadline ≈ 1.44269 ≈ 1.76322 [RM] [OPA] 1/ Ω 1/ Ω 2 Constrained-deadline ≈ 1.76322 ≈ 1.76322 [DM] [OPA] 1/ Ω 2 1/ Ω 2 Arbitrary-deadline ≈ 1.76322 ≈ 1.76322 [OPA] [OPA] Open Problems

Recent results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP ECRTS 2015 [6] Taskset FP-P v. EDF-P FP-NP v. EDF-NP Constraints Speedup factor Speedup factor [Priority ordering] Lower bound Upper bound Lower bound Upper bound 1/ln(2) 1/ Ω Implicit-deadline ≈ 1.44269 ≈ 1.76322 [RM] [OPA] 1/ Ω 1/ Ω Constrained-deadline ≈ 1.76322 ≈ 1.76322 [DM] [OPA] 1/ Ω 2 1/ Ω 2 Arbitrary-deadline ≈ 1.76322 ≈ 1.76322 [OPA] [OPA]

Recent results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP Real-Time Systems Sept 2015 [7] Taskset FP-P v. EDF-P FP-NP v. EDF-NP Constraints Speedup factor Speedup factor [Priority ordering] Lower bound Upper bound Lower bound Upper bound 1/ln(2) 1/ Ω Implicit-deadline ≈ 1.44269 ≈ 1.76322 [RM] [OPA] 1/ Ω 1/ Ω Constrained-deadline ≈ 1.76322 ≈ 1.76322 [DM] [OPA] 2 2 Arbitrary-deadline [OPA] [OPA]

Non-preemptive scheduling Non-preemptive scheduling suffers from the long task  problem  C D  If task set is not schedulable max min  Without accounting for this, speedup factor is arbitrarily large Express speedup factor in a way that is parametric  in C max / D min  Simplest form that gives a finite speedup factor

Recent results: Speedup factors for non-preemptive scheduling RTSS Dec 2015 [9] (also results from [4]) Taskset FP-NP v. EDF-P FP-NP v. EDF-NP v. Constraints FP-P EDF-P Sub-optimality [Priority ordering] Speedup Sub- Lower bound Upper bound factor optimality Implicit-deadline Open Problem C [RM] [OPA] C C  max   1 max max 2 1 D D D Constrained-deadline min min min [DM] [OPA] C Arbitrary-deadline C C  max   1 max max 2 2 D [OPA] [OPA] D D min min min

Recent results: Speedup factors for FP-P v. FP-NP RTSOPS July 2015 [5] Taskset FP-P v. FP-NP Constraints Speedup factor [Priority ordering] Lower bound Upper bound 1.34 1/ln(2) Implicit-deadline ≈ 1.44269 (expt) [RM] [OPA] 1/ Ω Constrained-deadline ≈ 1.76322 [DM] [OPA] 2 Arbitrary-deadline 2 [OPA] [OPA] Open Problem

and Finally… … currently under review [10] (also results from [6] and [8]) All of the results below (upper bounds) still hold for FP-P / FP-NP with DM priority assignment and simple linear schedulability tests Taskset FP-P v. EDF-P FP-NP v. EDF-NP Constraints Speedup factor Speedup factor [Priority ordering] Lower bound Upper bound Lower bound Upper bound 1/ln(2) 1/ Ω Implicit-deadline ≈ 1.44269 ≈ 1.76322 [RM] [RM] 1/ Ω 1/ Ω Constrained-deadline ≈ 1.76322 ≈ 1.76322 [DM] [DM] 2 2 Arbitrary-deadline [DM] [DM]

References [10] G. von der Bruggen, J.-J. Chen, and W.-H. Huang, “Exact Speedup Factors for Linear-Time Schedulability Tests for Fixed-Priority Preemptive and Non-preemptive Scheduling” Under review. [9] R.I. Davis, A. Thekkilakattil, O. Gettings, R. Dobrin, S.Punnekkat, "Quantifying the Exact Sub-Optimality of Non- Preemptive Scheduling”. In Real-Time Systems Symposium (RTSS ) , Dec 2015. [8] J.-J. Chen, W.-H. Huang, and C. Liu. k2U: A general framework from k-point effective schedulability analysis to utilization-based tests. In Real-Time Systems Symposium (RTSS) , Dec 2015. [7] R.I. Davis, A. Burns, S. Baruah, T. Rothvoss, L. George, O. Gettings "Exact comparison of fixed priority and EDF scheduling based on speedup factors for both pre-emptive and non-pre-emptive paradigms”. Real-Time Systems , Vol 51, Issue 5, Pages 566-601, Sept 2015. [6] G. von der Bruggen, J.-J. Chen, and W.-H. Huang. Schedulability and optimization analysis for non-preemptive static priority scheduling based on task utilization and blocking factors. In Euromicro Conference on Real-Time Systems (ECRTS) , pages 90–101, July 2015. [5] R. I. Davis , O. Gettings, A. Thekkilakattil, R. Dobrin, S. Punnekkat, "What is the Exact Speedup Factor for Fixed Priority Pre-emptive versus Fixed Priority Non-pre-emptive Scheduling?”. In Real-Time Scheduling Open Problems Seminar (RTSOPS), , pages 23-24, July 2015. [4] Fathi Abugchem, Michael Short, and Donglai Xu. A note on the suboptimality of non-preemptive real-time scheduling. Embedded Systems Letters, IEEE, PP(99):1–1, April 2015 [3] R. I. Davis, L. George, P. Courbin “Quantifying the Sub-optimality of Uniprocessor Fixed Priority Non-Pre-emptive Scheduling”. In Real-Time and Network Systems (RTNS'10) , pages 1-10, Nov 2010. [2] R.I. Davis, T. Rothvoß, S.K. Baruah, A. Burns “Quantifying the Sub-optimality of Uniprocessor Fixed Priority Pre- emptive Scheduling for Sporadic Tasksets with Arbitrary Deadlines”. In Real-Time and Network Systems (RTNS'09) , pages 23-31, Oct 2009. [1] R.I. Davis, T. Rothvoß, S.K. Baruah, A. Burns "Exact Quantification of the Sub-optimality of Uniprocessor Fixed Priority Pre-emptive Scheduling”. Real-Time Systems , Vol 43, No 3, pages 211-258, Nov 2009.