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Space-Efficient Scheduling of Stochastically Generated Tasks Tom Brzdil 1 Javier Esparza 2 Stefan Kiefer 3 Michael Luttenberger 2 1 Masaryk University, Brno (Czech Republic) 2 TU Mnchen (Germany) 3 University of Oxford (UK) ICALP ,

  1. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY X Y Y Y Y ֒ − → X Y ֒ − → ε X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  2. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY X Y Y Y Y ֒ − → X Y ֒ − → ε X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  3. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY X Y Y Y Y ֒ − → X Y ֒ − → ε X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  4. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY X Y Y Y Y ֒ − → X Y ֒ − → ε X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  5. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY X Y Y Y Y ֒ − → X Y ֒ − → ε X Y Y X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 4 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  6. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY Each finite tree has its X Y Y Y Probability (does not depend on scheduler) Y ֒ − → X Time (does not depend on scheduler) Y ֒ − → ε X Y Y Space (depends on scheduler) X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε Time: 10 (number of nodes) Space: 4 4 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  7. Another Derivation X X ֒ − → XY X Y X ֒ − → ε Y ֒ − → YY Each finite tree has its X Y Y Y Probability (does not depend on scheduler) Y ֒ − → X Time (does not depend on scheduler) Y ֒ − → ε X Y Y Space (depends on scheduler) Raises questions like: What is the expected time and the expected space? X ⇒ XY ⇒ XYY ⇒ XYYY XYYY ⇒ XYY ⇒ XY ⇒ X ⇒ XY ⇒ XX ⇒ X ⇒ ε How are these random variables distributed? Time: 10 (number of nodes) Space: 4 4 Time has been studied before. We focus on space. Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  8. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  9. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  10. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  11. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  12. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  13. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  14. Related Work: Branching Processes X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ Branching Processes have been extensively studied. They are models for biological or physical systems. But they assume an unbounded number of “processors”. Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  15. Termination Probability X 0 . 2 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 2 0 . 3 0 . 3 Y − → YY X Y Y Y ֒ 0 . 1 0 . 8 0 . 1 0 . 6 0 . 3 Y − → X ֒ 0 . 6 X Y Y Y − → ε ֒ 0 . 8 0 . 6 0 . 6 Each tree has its probability. The sum of these probabilities is the “termination probability”. Is it always 1? Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  16. Termination Probability and the function f A task system induces a vector f ( x ) . � x � f x ( x , y ) � � For our example: x = and f ( x ) = with y f y ( x , y ) 0 . 2 � X − → XY ֒ f x ( x , y ) = 0 . 2 xy + 0 . 8 0 . 8 X − → ε ֒ 0 . 3  − → YY Y ֒    f y ( x , y ) = 0 . 3 y 2 + 0 . 1 x + 0 . 6 0 . 1 − → X Y ֒  0 . 6  Y − → ε  ֒ Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  17. Termination Probability and the function f A task system induces a vector f ( x ) . � x � f x ( x , y ) � � For our example: x = and f ( x ) = with y f y ( x , y ) 0 . 2 � X − → XY ֒ f x ( x , y ) = 0 . 2 xy + 0 . 8 0 . 8 X − → ε ֒ 0 . 3  − → YY Y ֒    f y ( x , y ) = 0 . 3 y 2 + 0 . 1 x + 0 . 6 0 . 1 − → X Y ֒  0 . 6  Y − → ε  ֒ Proposition (well-known, see [Harris]) The termination probability is the (first component of the) least fixed point of f . Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  18. The function f � termination probability = 1 The subcritical case: expected time = finite 1 . 2 0 . 4 X − → XX ֒ 1 0 . 6 f ( x ) = 0 . 4 x 2 + 0 . 6 X − → ε ֒ 0 . 8 0 . 6 0 . 4 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  19. The function f � termination probability < 1 The supercritical case: expected time = ∞ 1 . 2 0 . 7 X − → XX ֒ 1 0 . 3 X − → ε ֒ f ( x ) = 0 . 7 x 2 + 0 . 3 0 . 8 0 . 6 0 . 4 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  20. The function f � termination probability = 1 The critical case: expected time = ∞ 1 . 2 0 . 5 X − → XX ֒ 1 0 . 5 X − → ε ֒ f ( x ) = 0 . 5 x 2 + 0 . 5 0 . 8 0 . 6 0 . 4 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  21. The function f � termination probability = 1 The critical case: expected time = ∞ 1 . 2 0 . 5 X − → XX ֒ 1 0 . 5 X − → ε ֒ f ( x ) = 0 . 5 x 2 + 0 . 5 0 . 8 0 . 6 Assumption 0 . 4 We assume termination probability = 1 in the following, i.e., subcritical or critical. 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  22. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1 S op ( t 0 ) + 1 , S op ( t 1 ) � � max , S op ( t ) = S op ( t 1 ) + 1 , S op ( t 0 ) � � max Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  23. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1   � S op ( t 0 ) + 1 , S op ( t 1 ) � max ,     S op ( t ) = S op ( t 1 ) + 1 , S op ( t 0 ) � � max     Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  24. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1   � S op ( t 0 ) + 1 , S op ( t 1 ) � max ,     S op ( t ) = S op ( t 1 ) + 1 , S op ( t 0 ) � � max     Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  25. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1   � S op ( t 0 ) + 1 , S op ( t 1 ) � max ,     S op ( t ) = S op ( t 1 ) + 1 , S op ( t 0 ) � � max     Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  26. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1   � S op ( t 0 ) + 1 , S op ( t 1 ) � max ,     S op ( t ) = S op ( t 1 ) + 1 , S op ( t 0 ) � � max     Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  27. Optimal Scheduling Given a tree t with two children t 0 , t 1 . What is the optimal scheduling? t t 0 t 1   � S op ( t 0 ) + 1 , S op ( t 1 ) � max ,     S op ( t ) = min S op ( t 1 ) + 1 , S op ( t 0 ) � � max     Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  28. Optimal Scheduling Given a tree t with just one child t 0 . What is the optimal scheduling? t t 0 S op ( t ) = Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  29. Optimal Scheduling Given a tree t with just one child t 0 . What is the optimal scheduling? t t 0 S op ( t ) = S op ( t 0 ) Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  30. Optimal Scheduling So we can determine S op for any tree t :  � � max { S op ( t 0 ) + 1 , S op ( t 1 ) } ,  min if t has two children t 0 , t 1   max { S op ( t 1 ) + 1 , S op ( t 0 ) }   S op ( t ) = S op ( t 0 ) if t has one child t 0     1 if t has no children.  Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  31. Optimal Scheduling So we can determine S op for any tree t :  � � max { S op ( t 0 ) + 1 , S op ( t 1 ) } ,  min if t has two children t 0 , t 1   max { S op ( t 1 ) + 1 , S op ( t 0 ) }   S op ( t ) = S op ( t 0 ) if t has one child t 0     1 if t has no children.  What is the distribution of S op , if trees are randomly generated? Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  32. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 1 f ( x ) = 0 . 4 x 2 + 0 . 6 0 . 8 0 . 6 0 . 4 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  33. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 1 0 . 8 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  34. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 1 0 . 8 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 ν ( 0 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  35. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 1 0 . 8 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 ν ( 0 ) ν ( 1 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  36. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 1 0 . 8 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 ν ( 2 ) ν ( 0 ) ν ( 1 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  37. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 Proposition (Etessami,Yannakakis, 2005) 1 Newton’s method converges to the least solution. 0 . 8 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 ν ( 2 ) ν ( 0 ) ν ( 1 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  38. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 Proposition (Etessami,Yannakakis, 2005) 1 Newton’s method converges to the least solution. 0 . 8 The least solution is = 1. Why is he talking about Newton’s method?? 0 . 6 g ( x ) = f ( x ) − x 0 . 4 0 . 2 ν ( 2 ) ν ( 0 ) ν ( 1 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  39. Newton’s Method Let g ( x ) := f ( x ) − x and apply Newton’s method to g ( x ) = 0: 1 . 2 Proposition (Etessami,Yannakakis, 2005) 1 Newton’s method converges to the least solution. 0 . 8 The least solution is = 1. Why is he talking about Newton’s method?? 0 . 6 g ( x ) = f ( x ) − x 0 . 4 Theorem � � S op ≤ k = ν ( k ) Pr 0 . 2 ν ( 2 ) ν ( 0 ) ν ( 1 ) 0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  40. Tail Bounds for the Optimal Scheduler Theorem � S op ≤ k � = ν ( k ) Pr It follows: � � S op ≥ k = 1 − ν ( k − 1 ) Pr � � The ν ( k ) converge to 1, so Pr S op ≥ k goes to 0. But how fast?? Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  41. Tail Bounds for the Optimal Scheduler Theorem � S op ≤ k � = ν ( k ) Pr It follows: � � S op ≥ k = 1 − ν ( k − 1 ) Pr � � The ν ( k ) converge to 1, so Pr S op ≥ k goes to 0. But how fast?? Corollary (follows from KLE’07, EKL ’08) � � S op ≥ k ∈ O ( d k ) general task systems: Pr ( d < 1 ) � � S op ≥ k ∈ O ( d 2 k ) subcritical task systems: Pr ( d < 1 ) Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  42. Online Scheduling X 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  43. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  44. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ X Y 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  45. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ X Y 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  46. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ X X Y 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  47. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ X X Y 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  48. Online Scheduling X 0 . 2 X − → XY ֒ X Y 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ X X Y 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  49. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X ֒ 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  50. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY X ֒ 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  51. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ Y X 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  52. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X Y ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  53. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X Y ֒ Y X 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  54. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X Y Y ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  55. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X Y ֒ Y 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  56. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X Y ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  57. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ X 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  58. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  59. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ X 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  60. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε X ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  61. Online Scheduling The urn model is more appropriate: 0 . 2 X − → XY ֒ 0 . 8 X − → ε ֒ 0 . 3 Y − → YY ֒ 0 . 1 Y − → X ֒ X 0 . 6 Y − → ε ֒ X ⇒ XY ⇒ XYY ⇒ XY ⇒ XX ⇒ X ⇒ ε Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  62. Weights: An Auxiliary Notion Let v > 1 be a vector with v ≥ f ( v ) . Choose h > 1 and for all types X a weight w X with h w X = v X for all types X . X Y w X w Y Denote by W the maximum weight of a derivation. For instance: X ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε yields X X Y Y Y Y Y Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  63. Weights: An Auxiliary Notion Let v > 1 be a vector with v ≥ f ( v ) . Choose h > 1 and for all types X a weight w X with h w X = v X for all types X . X Y w X w Y Denote by W the maximum weight of a derivation. For instance: X ⇒ XY ⇒ Y ⇒ YY ⇒ Y ⇒ ε yields X X Y W = 2 · w Y Y Y Y Y Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  64. An Upper Bound for All Online Schedulers (Recall: v > 1 with v ≥ f ( v ) and h w X = v X .) One can show by a martingale argument: ≤ v X 0 � � Pr W ≥ k h k Note: Whenever S ≥ k then W ≥ k · w min . So we obtain: v X 0 v X 0 � � � � ≤ Pr W ≥ k · w min ≤ h k · w min = Pr S ≥ k v mink Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  65. An Upper Bound for All Online Schedulers (Recall: v > 1 with v ≥ f ( v ) and h w X = v X .) One can show by a martingale argument: ≤ v X 0 � � Pr W ≥ k h k Note: Whenever S ≥ k then W ≥ k · w min . So we obtain: v X 0 v X 0 � � � � ≤ Pr W ≥ k · w min ≤ h k · w min = Pr S ≥ k v mink Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  66. An Upper Bound for All Online Schedulers (Recall: v > 1 with v ≥ f ( v ) and h w X = v X .) One can show by a martingale argument: ≤ v X 0 � � Pr W ≥ k h k Note: Whenever S ≥ k then W ≥ k · w min . So we obtain: v X 0 v X 0 � � � � ≤ Pr W ≥ k · w min ≤ h k · w min = Pr S ≥ k v mink Theorem Let v > 1 with v ≥ f ( v ) . Then v X 0 � � S σ ≥ k Pr ≤ v mink for all online schedulers σ and all k ∈ N . Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  67. An Upper Bound for All Online Schedulers (Recall: v > 1 with v ≥ f ( v ) and h w X = v X .) One can show by a martingale argument: ≤ v X 0 � � Pr W ≥ k h k Note: Whenever S ≥ k then W ≥ k · w min . So we obtain: v X 0 v X 0 � � � � ≤ Pr W ≥ k · w min ≤ h k · w min = Pr S ≥ k v mink Theorem Let v > 1 with v ≥ f ( v ) . Let u > 1 with u ≤ f ( u ) . Then v X 0 c � � S σ ≥ k u max k ≤ Pr ≤ v mink for all online schedulers σ and all k ∈ N . Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  68. An Upper Bound for All Online Schedulers Example Consider the task system from the beginning: f x ( x , y ) = 0 . 2 xy + 0 . 8 f y ( x , y ) = 0 . 3 y 2 + 0 . 1 x + 0 . 6 � 1 � 1 . 4 � � One can show: f has two fixed points: and . 1 2 . 2 c ≤ 1 . 4 � � S σ ≥ k So: 2 . 2 k ≤ Pr holds for all σ . 1 . 4 k Theorem Let v > 1 with v ≥ f ( v ) . Let u > 1 with u ≤ f ( u ) . Then v X 0 c � � S σ ≥ k u max k ≤ Pr ≤ v mink for all online schedulers σ and all k ∈ N . Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  69. An Upper Bound for All Online Schedulers Example Consider the task system from the beginning: f x ( x , y ) = 0 . 2 xy + 0 . 8 f y ( x , y ) = 0 . 3 y 2 + 0 . 1 x + 0 . 6 � 1 � 1 . 4 � � One can show: f has two fixed points: and . 1 2 . 2 c ≤ 1 . 4 � � S σ ≥ k So: 2 . 2 k ≤ Pr holds for all σ . 1 . 4 k Theorem Let v > 1 with v ≥ f ( v ) . Let u > 1 with u ≤ f ( u ) . Then v X 0 c � � S σ ≥ k u max k ≤ Pr ≤ v mink for all online schedulers σ and all k ∈ N . Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  70. A Light-First Scheduler For Our Example For the upper bound we used: W ≥ k · w min . Whenever S ≥ k then � � 1 . 4 We have v = , so w min = w X . 2 . 2 We say, X is the lightest type. 0 . 2 X − → XY ֒ Light-First Scheduler: 0 . 8 X − → ε ֒ Process the lightest type (here: X ) whenever it is in the pool. 0 . 3 Y − → YY ֒ In our example, the light-first scheduler guarantees: 0 . 1 Y − → X ֒ at any time at most one X -task in the pool. 0 . 6 Hence, with the light-first scheduler: Y − → ε ֒ W ≥ 1 · w X + ( k − 1 ) · w Y . Whenever S ≥ k then Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

  71. A Light-First Scheduler For Our Example For the upper bound we used: W ≥ k · w X . Whenever S ≥ k then � � 1 . 4 We have v = , so w min = w X . 2 . 2 We say, X is the lightest type. 0 . 2 X − → XY ֒ Light-First Scheduler: 0 . 8 X − → ε ֒ Process the lightest type (here: X ) whenever it is in the pool. 0 . 3 Y − → YY ֒ In our example, the light-first scheduler guarantees: 0 . 1 Y − → X ֒ at any time at most one X -task in the pool. 0 . 6 Hence, with the light-first scheduler: Y − → ε ֒ W ≥ 1 · w X + ( k − 1 ) · w Y . Whenever S ≥ k then Tomáš Brázdil, Javier Esparza, Stefan Kiefer, Michael Luttenberger Space-Efficient Scheduling of Stochastically Generated Tasks

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