An Efficient Knapsack-Based Approach for Calculating the Worst-Case Demand of AVR Tasks

Transcription

1 An Efficient Knapsack-Based Approach for Calculating the Worst-Case Demand of AVR Tasks Sandeep Kumar Bijinemula *, Aaron Willcock, Thidapat Chantem *, Nathan Fisher * Department of Electrical and Computer Engineering, Virginia Tech, USA Department of Computer Science, Wayne State University, USA This research was supported in part by the US National Science Foundation (CNS Grant Nos & ) and a Thomas C. Rumble Graduate Fellowship from Wayne State University.

2 Holiday Thievery Imagine your favorite store. You have 100 seconds to take anything for free Where would you start? What would you take? 2

3 Holiday Thievery Goal: Maximize value over 100 seconds Increase value Decrease time < < 3

4 Value of Items Holiday Thievery Function Engine Control Module (ECM) Items per Second 4

5 Execution Times of the Modes Engine Control: Adaptive Variable-Rate (AVR) Tasks c 1 Engine Control Module (ECM) c 2 c 3 c 4 c 5 Image Provided by: Zephyris. File:4StrokeEngine_Ortho_3D_Small.gif c M ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max c execution time ω angular speed Speeds Biondi et al. [ICCPS 15] 5

6 Execution Times of the Modes Engine Control: Adaptive Variable-Rate (AVR) Tasks c 1 c 2 c 3 Jobs released at Top Dead Center (TDC) Jobs due in one revolution (one full rotation) α + c 4 Higher Lower rate WCET of job per release job c 5 c M Less Higher time rate between of job job releases ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds α + ω max Biondi et al. [ICCPS 15] 6

7 Execution Times of the Modes Engine Control: Adaptive Variable-Rate (AVR) Tasks c 1 c 2 c 3 Jobs released at Top Dead Center (TDC) Jobs due in one revolution (one full rotation) α c 4 Higher rate WCET of job per release job c 5 c M More Higher time between rate of job job releases ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max α 7 Biondi et al. [ICCPS 15] 7

8 Execution Times of the Modes Engine Control: Adaptive Variable-Rate (AVR) Tasks c 1 c 2 c 3 c 4 Higher rate WCET of job per release job Jobs released at Top Dead Center (TDC) Jobs due in one revolution (one full rotation) But how do we calculate worst-case demand? α c 5 c M More Higher time between rate of job job releases ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max α 8 Biondi et al. [ICCPS 15]

9 Execution Times of the Modes Demand of AVR Tasks c 1 c 2 Which point gives the worstcase demand? c 3 c 4 c 5 Calculation of worst-case demand is not straightforward. c M ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max Speeds Biondi et al. [ICCPS 15] 9

10 Execution Times of the Modes Motivation and Problem Statement c 1 Motivation:? c 2 c 3 Existing calculations of exact worstcase demand for AVR tasks are compute intensive. c 4 Problem Statement: c 5 c M Calculate the exact worst-case demand of an AVR task over a time interval δ. ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max 10

11 Approach & Contributions 1. Worst-case AVR task demand Bounded precedence constraint knapsack problem (BPCKP). 2. Symmetric acceleration + Engine kinematics Reduced search space. α α + 3. Results significantly outperform the state-of-the-art in computation time.? 11

12 Outline 1. Approach and Contributions 2. Background & Related Work 3. Precedence Constraints & Search Space Reduction 4. Worst-Case AVR Demand as Knapsack Problem 5. Evaluation & Results 6. Conclusion & Future Work 12

13 Background & Related Work 13

14 Execution Times of the Modes AVR Tasks Speed or WCET? c 1 c 2 c 3 + Allows WCET to scale inversely with speed? How can we calculate worst-case demand? c 4 c 5 c M Right Boundary (RB) RB Speeds (ω rb ) Modes / Steps ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max Speeds Biondi et al. [ICCPS 15] 14

15 Execution Times of the Modes Digraph Real-Time (DRT) Approach c 1 c 2 + Exact worst-case demand for a select set of speeds? How can we compute worst-case demand faster? c 3 c 4 c 5 c M ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds Mohaqeqi et al. [ECRTS 17] ω max 15

16 Related Works Model Adaptive-Variable Rate / Rhythmic Biondi et al. [ICCPS 15], Kim et al. [ICCPS 12] Constant Acceleration between Jobs Variable Acceleration between Jobs Schedulability Guo and Baruah [ICCPS 15] Exact Worst-case Demand Biondi et al. [ECRTS 14] Exact Worst-case Demand via DRT Mohaqeqi et al. [ECRTS 17] ω peak α α α + α + α Exact Worst-case Demand via Knapsack (This work) α ω 1 ω 1 ω 0 ω 0 16

17 Related Works Model Adaptive-Variable Rate / Rhythmic Biondi et al. [ICCPS 15], Kim et al. [ICCPS 12] Constant Acceleration between Jobs Variable Acceleration between Jobs Schedulability Guo and Baruah [ICCPS 15] Exact Worst-case Demand Biondi et al. [ECRTS 14] Exact Worst-case Demand via DRT Mohaqeqi et al. [ECRTS 17] Exact Worst-case Demand via Knapsack (This work) α α + α = maximum deceleration α + = maximum acceleration 17

18 Outline 1. Approach and Contributions 2. Background & Related Work 3. Precedence Constraints & Search Space Reduction 4. Worst-Case AVR Demand as Knapsack Problem 5. Evaluation & Results 6. Conclusion & Future Work 18

19 Precedence Constraints & Search Space Reduction 19

20 Execution Times of the Modes Acceleration Limits (on AVR) c 1 α α + Cannot reach ω 5 in a single rotation. c 2 c 3 α α + c 4 c 5 c M ω min c execution time ω angular speed ω 0 ω 1 ω 2 ω 3 ω 4 ω 5 ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max Reachable Speeds 20

21 Search Space Reduction: Dominant Sequences A dominant sequence is a sequence of job releases which maximizes demand over a time interval δ. 21

22 Execution Times of the Modes Dominant Sequence Visualization c 1 c 2 c 3 Max. Accel. (α + ) Max. Decel. (α ) ω peak Var. Accel. ω 1 ω 0 c 4 c 5 c M A dominant sequence is a sequence of job releases which maximizes demand over a time interval δ. ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max 22

23 Execution Times of the Modes Dominant Sequence Visualized c 1 c 2 c 3 Max. Accel. (α + ) Max. Decel. (α ) ω peak Var. Accel. ω 1 ω 0 c 4 c 5 c M A dominant sequence is a sequence of job releases which maximizes demand over a time interval δ. ω min c execution time ω angular speed ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max 23

24 Dominant Sequence Max. Accel. (α + ) Max. Decel. (α ) ω peak Var. Accel. ω 1 ω 0 > > We only search for dominant sequences! 24

25 Properties of a Dominant Sequence c 1 c 2 c 3 c 4 c 5 Lemma 1 : Dominant sequences do not contain any decreasing subsequences. c 1 c 2 c 3 c 4 c 5 Max. Accel. (α + ) Max. Decel. (α ) ω peak Var. Accel. ω 1 ω 0 c M c M ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max Decreasing sequences Speeds can be transformed into non-decreasing Speeds sequences with equal or greater demand! 25

26 Properties of a Dominant Sequence c 1 c 2 c 3 c 4 c 5 Lemma 1 : Dominant sequences do not contain any decreasing subsequences. c 1 c 2 Lemma 1: Only non-decreasing c 3 sequences allowed. c 4 c 5 Max. Accel. (α + ) Max. Decel. (α ) ω peak Var. Accel. ω 1 ω 0 c M c M ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Decreasing sequences Speeds can be transformed into non-decreasing Speeds sequences with equal or greater demand! ω max 26

27 Properties of a Dominant Sequence Lemma 2 : The first job of a dominant sequence begins at a right boundary. n jobs in the middle of a step n jobs at the right boundary speed c m 1 c m ω rbx Same execution time, shorter time 27

28 Properties of a Dominant Sequence Lemma 2 : The first job of a dominant sequence begins at a right boundary. n jobs in the middle of a step n jobs at the right boundary speed c m 1 Lemma 2: Start at a right boundary speed (RB). c m ω rbx Same execution time, shorter time 28

29 Properties of a Dominant Sequence Lemma 3 : Dominant sequences have fixed choices for next speeds. c m 1 Decision Speed c m 1 Option 1 Max. Accel. Use Max. Accel. c m ω rbx When α + travels past right boundary, variable accel. may be used. c m ω rbx Option 2 Single Var. Accel. n RBs Multiple and Max. Accel. c m 1 c m 1 c m c m ω rbx ω rbx 29

30 Properties of a Dominant Sequence Lemma 3 : Dominant sequences have fixed choices for next speeds. c m 1 Use Max. Accel. c m Decision Speed c m 1 Lemma 3: Use maximum acceleration. Additionally, at decision speeds, use variable acceleration c m to reach RB. ω rbx ω rbx Can release multiple jobs at RB. When α + travels past right boundary, variable accel. may be used. Option 1 Max. Accel. c m 1 Option 2 Single Var. Accel. c m 1 n RBs Multiple and Max. Accel. c m ω rbx c m ω rbx 30

31 Execution Times of the Modes Dominant Sequence Visualization c 1 c 2 c 3 c 4 c 5 c M ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 ω max Speeds 31

32 Dominant Sequence Visualization Max. Accel. (α + ) Max. Decel. (α ) c m 2 ω peak Var. Accel. ω 1 ω 0 c m 1 c m Lemma 1: Only non-decreasing sequences allowed. ω rbx Lemma 2: Start at a right boundary speed (RB). ω rbx+1 Lemma 3: Use maximum acceleration. Additionally, at decision speeds, use variable acceleration to reach RB. Can release multiple jobs at RB. 32

33 A Visual Comparison c m 2 c m 1 c m ω rbx ω rbx+1 ω rbx ω rbx+1 Sequence in our approach Sequences in previous work Search space in our approach is significantly reduced. 33

34 Worst-Case Demand as Knapsack 34

35 Jobs as Items, Time as Knapsack ID Profit Weight Initial Speed, Final Speed WCET Min. Interarrival (time) Max 100 kg Knapsack = Duration of time 35

36 Execution Times of the Modes Transformation Visualization c 1 c 2 c 3 c 4 c 5 0 ω j, ω k t c M ω min ω rb1 ω rb2 ω rb3 ω rb4 ω rb5 Speeds ω max 0 Precedence Constraints should be followed too t 36

37 Bounded Precedence Constraint Knapsack Problem BPCKP Right boundary speed c(ω j ) T(ω j, ω j ) T(ω j, ω k ) c(ω j ) Execution time T - Job inter-arrival time c(ω j ) c(ω k ) T(ω j, ω j ) T(ω k, ω l ) c(ω j ) c(ω l ) Decision Speed Middle of a step T(ω j, ω k ) T(ω l, ω m ) c(ω k ) T(ω m, ω n ) c(ω m ) T(ω m, ω o ) c(ω n ) c(ω o ) T(ω n, ω n ) c(ω n ) 37

38 BPCKP c(ω j ) c(ω j ) Execution time T - Job inter-arrival time T(ω j, ω j ) T(ω j, ω k ) c(ω j ) c(ω k ) Multiple jobs at RBs Restricted by the length of the time-interval T(ω j, ω j ) T(ω k, ω l ) c(ω j ) c(ω l ) T(ω j, ω k ) T(ω l, ω m ) c(ω k ) T(ω m, ω n ) c(ω m ) T(ω m, ω o ) Solve the resultant BPCKP with Dynamic Programming T(ω n, ω n ) c(ω n ) c(ω n ) c(ω o ) 38

39 Time complexity Number of Unique Items in Knapsack: m number of modes O(m ω 2 2 max ω min ) 2α max Search complexity Pseudo polynomial Same as any bounded precedence constraint knapsack problem 39

40 Evaluation and Results 40

41 AVR Task Set: Table Comparison Task Set in Previous Work i th mode ω rbm ω rbi c(ω rbm ) ω rbi is reachable from ω rbi k in integer rotations Decreases the number of decision speeds A More Challenging Task Set ω rbi is not reachable from ω r k in integer rotations 6500 i th mode ω rbm ω rbi c(ω rbm )

42 AVR Task Set: Visual Comparison Previous Work Task Set c m 2 # Decision Speeds less than # Modes c m 1 Fewer sequences in the search space c m ω rbx ω rbx+1 More Challenging Task Set c m 2 # Decision Speeds equal to # Modes c m 1 More sequences in the search space c m ω rbx ω rbx+1 42

43 Algorithm Runtime Comparison System: Python 3.6.5, 3.4 GHz quad-core processor, 8 GB RAM Experiment: Calculate worst-case demand for 100 intervals from [0,1s] Previous Task Set More Challenging Task Set DRT Alg. 3 min. 31 sec. 17 min. 2 sec. Our Alg sec sec. Performance Improvement

44 Random Taskset Alg. Runtimes 44

45 Public Artifact 1. Recreate paper results RTSS18KAVR 2. Customize task sets 3. Graphical comparison 4. Python w/ Markdown documentation 45

46 Future Work 1. Relax the symmetric acceleration assumptions α α + 2. Additional problem dimensions 3. AVR tasks with different phases θ 4. Approximation algorithms ~ 46

47 Conclusion 1. Worst-case AVR task demand Bounded precedence constraint knapsack problem (BPCKP). 2. Symmetric acceleration + Engine kinematics Reduced search space. 3. Results significantly outperform the state-of-the-art in computation time. Questions? α α + sandeepb@vt.edu aaron.willcock@wayne.edu tchantem@vt.edu fishern@wayne.edu? 47