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 Ci



Independent sporadic arrivals: minimum inter-arrival time Ti



Relative deadline Di



Independent execution (no resource sharing)



Three classes of task set: Implicit- (Di=Ti), Constrained- (Di≤Ti), 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



Fixed Priority Scheduling

 Priority assignment important



Optimal priority assignment (FP-P)

 Implicit-deadlines – Rate-Monotonic  Constrained-deadlines – Deadline Monotonic  Arbitrary-deadlines – Audsley’s Optimal

Priority Assignment algorithm



Optimal priority assignment (FP-NP)

 All 3 cases – Audsley’s algorithm

Optimal Priorities Random Priorities 200 400 600 800 1000 1200 1400 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 Frequency Breakdown Utilisation

Landscape of scheduling algorithms and speedup factors

I nterested in comparing EDF and Fixed Priority (FP) scheduling in the preemptive and non-preemptive cases

FP-P

EDF-P (optimal)

EDF-NP FP-NP

Previous results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP

Taskset Constraints [Priority ordering] FP-P v. EDF-P Speedup factor

Lower bound Upper bound

Implicit-deadline [RM] [OPA]

1/ln(2) ≈ 1.44269

Constrained-deadline [DM] [OPA]

1/Ω ≈ 1.76322

Arbitrary-deadline [OPA] [OPA]

1/Ω ≈ 1.76322 2

FP-NP v. EDF-NP Speedup factor

Lower bound Upper bound

1/Ω ≈ 1.76322 2 1/Ω ≈ 1.76322 2 1/Ω ≈ 1.76322 2

Early results: Liu & Layland 1973, [1], [2], [3] from 2009/10 As of Jan 2015:

Open Problems

Recent results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP

Taskset Constraints [Priority ordering] FP-P v. EDF-P Speedup factor

Lower bound Upper bound

Implicit-deadline [RM] [OPA]

1/ln(2) ≈ 1.44269

Constrained-deadline [DM] [OPA]

1/Ω ≈ 1.76322

Arbitrary-deadline [OPA] [OPA]

1/Ω ≈ 1.76322 2

FP-NP v. EDF-NP Speedup factor

Lower bound Upper bound

1/Ω ≈ 1.76322 1/Ω ≈ 1.76322 1/Ω ≈ 1.76322 2

ECRTS 2015 [6]

Recent results: Speedup factors for FP-P v. EDF-P and FP-NP v. EDF-NP

Taskset Constraints [Priority ordering] FP-P v. EDF-P Speedup factor

Lower bound Upper bound

Implicit-deadline [RM] [OPA]

1/ln(2) ≈ 1.44269

Constrained-deadline [DM] [OPA]

1/Ω ≈ 1.76322

Arbitrary-deadline [OPA] [OPA]

2

FP-NP v. EDF-NP Speedup factor

Lower bound Upper bound

1/Ω ≈ 1.76322 1/Ω ≈ 1.76322 2

Real-Time Systems Sept 2015 [7]

Non-preemptive scheduling



Non-preemptive scheduling suffers from the long task problem

 If task set is not schedulable  Without accounting for this, speedup factor is arbitrarily

large



Express speedup factor in a way that is parametric in

 Simplest form that gives a finite speedup factor

min max

D C 

min max / D

C

FP-NP v. FP-P Speedup factor

Recent results: Speedup factors for non-preemptive scheduling

Taskset Constraints [Priority ordering] FP-NP v. EDF-P Sub-optimality

Lower bound Upper bound

Implicit-deadline [RM] [OPA] Constrained-deadline [DM] [OPA] Arbitrary-deadline [OPA] [OPA]

min max

1 D C 

min max

2 D C 

min max

2 D C 

min max

1 D C 

min max

2 D C 

EDF-NP v. EDF-P Sub-

• ptimality

min max

1 D C 

Open Problem

RTSS Dec 2015 [9] (also results from [4])

Recent results: Speedup factors for FP-P v. FP-NP

Taskset Constraints [Priority ordering] FP-P v. FP-NP Speedup factor

Lower bound Upper bound

Implicit-deadline [RM] [OPA]

1.34 (expt) 1/ln(2) ≈ 1.44269

Constrained-deadline [DM] [OPA]

1/Ω ≈ 1.76322

Arbitrary-deadline [OPA] [OPA] 2

2

Open Problem

RTSOPS July 2015 [5]

and Finally…

Taskset Constraints [Priority ordering] FP-P v. EDF-P Speedup factor

Lower bound Upper bound

Implicit-deadline [RM] [RM]

1/ln(2) ≈ 1.44269

Constrained-deadline [DM] [DM]

1/Ω ≈ 1.76322

Arbitrary-deadline [DM] [DM]

2

FP-NP v. EDF-NP Speedup factor

Lower bound Upper bound

1/Ω ≈ 1.76322 1/Ω ≈ 1.76322 2

… 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

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.