Multiple-Resource Periodic Scheduling Problem: how much fairness is necessary?

1
Multiple-Resource Periodic Scheduling Problem
how much fairness is necessary?

• Dakai Zhu, Daniel Mossé and Rami Melhem

The work is supported by DARPA through the PARTS
project.
2
Preface

• Power-aware computing for parallel systems
• Frame based task sets
• Shared Memory Systems
• Traditional AND-model applications TPDS03
• AND/OR-model applications to appear in TPDS
• Distributed Systems IPDPS03
• Periodic task sets
• Global scheduler P-fair Baruah96

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Problem

• Scheduling
• n periodic tasks, each task Ti with (ci , pi)
• ci , pi are integers (i.e.,multiples of system
  time unit)
• Weight wi ci / pi System U ?wi
• m resources (m ? U)
• Constraints at any time
• One resource ? one task (resources are not
  shareable)
• One task occupies one resource (tasks are not
  parallel)
• Optimal
• Full utilization (U m)
• Scheduling overhead
• Number of Context switches

4
Previous Work

• EDF/RM dynamic/static priority
• Optimal for single resource
• The famous task set Dhall78
• (n 1) tasks with (2?, 1) 1 task with (1, 1 ?)
• U (2?(n-1) 1/(1 ?) )/n ? 0 with big n
• Harmonic tasks periods are multiples of each
  other
• Mancini94 and Khemka97

5
Previous Work (cont.)

• PF proportional fairness Baruah96
• Proportional progress for tasks at ANY time
• At time t either ?wi t? or ?wi t?
• Absolute allocation error less than one time
  unit
• Optimal full utilization, U m
• Variations of P-fair algorithms
• PD Baruah95
• PD2 Anderson01
• ER-fair Anderson00

Scheduling decision at EVERY time unit !
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An Example

• T1(1, 3), T2(2, 5), T3(2, 5), T4(2, 3),
  T5(1, 5)
• mU?wi 2, LCM 15

• PF schedule Baruah96
• 15 scheduling points
• 27 context switches

Deadline miss only happens at period boundaries !
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B-Fair A new approach

• Idea
• Schedule ONLY at period boundaries
• Allocate resources to tasks for the time units
  within two consecutive boundaries
• Maintain fairness at boundary time bk (B-Fair)
• Ti get either ?wi bk ? or ?wi bk ?
• Two-step allocation
• Mandatory units to keep fairness
• Optional units for future urgent tasks

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The Example

• T1(1, 3), T2(2, 5), T3(2, 5), T4(2, 3),
  T5(1, 5)

At boundary time 0 allocate 0 ? 3)
T1 T2 T3 T4 T5
R1
R2
0
1
2
3
4
5
6
7
8
10
9
11
12
13
14
15
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BF Algorithm

• Define a set of boundaries b0, , bL
• bk lt bk1 b0 0, bL LCM
• ?k, ?Ti, bk is a multiple of pi

• At bk, we allocate bk , bk1)
• Mandatory units
• Optional units

• mik max 0, ?RWik wi(bk1 bk)?
• RWik Remaining work for Ti before bk
• RWik wi bk allocated units for Ti
• m(bk1 bk) gt ? mik

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BF Algorithm (cont.)

• Eligible tasks
• Pending workPWikRWik wi(bk1 bk) - mik gt 0
• Not fully allocated mik lt bk1 bk
• Dynamic priority urgency of future
• Character string
• Time Factor (TF)

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BF Algorithm (cont.)

• For eligible tasks Ti and Tj
• Compare ?(Ti, k) and ?(Ti, k)
• Character by character gt 0 gt
• Bigger string has higher priority
• If tie and last character is 0 arbitrary
• If tie and last character is
• Compare time factor (TF)
• smaller time factor has higher priority
• If tie again arbitrary, e.g., smaller index
• High priority tasks get one optional unit each

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The Example (cont.)

• T1(1, 3), T2(2, 5), T3(2, 5), T4(2, 3),
  T5(1, 5)

T1 T2 T3 T4 T5
Compared BF vs. PF Scheduling Points 7
vs. 15 Context Switches 24 vs. 27
R1
R2
0
1
2
3
4
5
6
7
8
10
9
11
12
13
14
15
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Complexity of BF

• For each scheduling point O(npmax)
• Mandatory units allocation O(n)
• Optional units O(npmax)
• Priority compare O(pmax)
• Linear selection from n tasks O(n) Blum73

• Improved BF O(n)
• Improve priority comparison O(1)
• If ?k(Ti) ?k(Tj) , consider two
  counter-task with weight of 1-wi and 1-wj,
  respectively.
• ?k(CTi) ?k(CTj) - compare their time
  factors (TF)
• Bigger TF gets higher priority
• Efficient P-fair PD has O(minn, m log n )
• Baruah95

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Number of Scheduling Points

• Up to 100 tasks
• Min period 10
• Max period
• 20 ? 100
• LCM ? 232

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Cycles per Scheduling Point

• Up to 100 tasks
• Min period 10
• Max period 100
• ci is randomly chosen from 1 to pi i.e., m ?
  n/2
• Conservative implementation

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Total Time to Generate Schedules

• Up to 100 tasks
• Min period 10
• Max period 100
• ci is randomly chosen from 1 to pi i.e., m ? n/2

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Conclusion

• Notion of boundary-fairness
• BF is optimal full system utilization
• Fewer scheduling points than P-fair algorithms
• Complexity per scheduling point is O(n)
• Comparable to PD O (min n, m log n )
• Less total time to generate schedules
• Compared with PD less than 100 tasks

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Multiple-Resource Periodic Scheduling Problem
how much fairness is necessary?

• Dakai Zhu, Daniel Mossé and Rami Melhem

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Analysis of B-Fair

• Ahead, behind, punctual task sets
• RWik lt 0, gt 0, 0 respectively
• Pre-ahead, pre-behind task set
• PWik lt 0, gt 0 respectively

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Analysis of B-Fair (cont.)

• No tasks miss the deadline if m ? ? (wi)
• When allocating bk , bk1) m ? (wi)
• No over requirement
• ? mik ? m(bk1 bk)
• Enough eligible tasks
• At least m(bk1 bk) ? mik
• Otherwise, non-empty ahead set and behind set for
  each previous boundary, which contradicts at b0

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Performance Evaluation (cont.)

• Generally, prime number points are not scheduling
  point for B-Fair
• Prime number theory big number x
• x / ln(x) prime numbers smaller than x
• LCM gt 1/lnLCM
• 103 ? gt14.5
• 106 ? gt7.24
• 109 ? gt 5